2-fourier-wavelets-hmm-svm.pdf

Im jk ω0t

e

3

kω0t Re

There was a celebrated Fourier at the Academy of Science, whom posterity has forgotten; and in some garret an obscure Fourier, whom the future will recall. Victor Hugo Les Misèrables

Fourier Analysis and Synthesis 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8

Introduction Fourier Series Fourier Transform FFT Discrete Fourier Transform Short Time Fourier Transform Fast Fourier Transform 2-D Discrete Fourier Transform Discrete Cosine Transfrom

J

3.9 Applications Spectrogram Based Digital Filters Digital BASS for Audio Radar Signal Processing Pitch Modification Image Compression Watermarking Bibliography, Exercises

ean Baptiste Joseph Fourier (1768-1830) studied the mathematical theory of heat conduction in his major work, The Analytic Theory of Heat, (Théorie analytique de la chaleur). He established the partial differential equation governing heat diffusion and solved it using an infinite series of trigonometric functions. The description of a signal in terms of elementary trigonometric functions had a profound effect on the way signals are analysed. The Fourier method is the most extensively applied signal-processing tool. This is because the transform output lends itself to easy interpretation and manipulation, and leads to the concept of frequency analysis. Furthermore even biological systems such as the human auditory system perform some form of frequency analysis of the input signals. This chapter begins with an introduction to the complex Fourier series and the Fourier transform, and then considers the discrete Fourier transform, the Fast Fourier transform, the 2-D Fourier transform and the discrete cosine transform. Important engineering issues such as the trade-off between the time and frequency resolutions, problems with finite data length, windowing and spectral leakage are considered. The applications of the Fourier transform include filtering. telecommunication, music processing, pitch modification, signal coding and signal synthesis feature extraction for pattern identification as in speech recognition, image processing, spectral analysis in astrophysics, radar signal processing.

2

Chap.3 Fourier Analysis and Synthesis

3.1 Introduction The objective of signal transformation is to express a signal as a combination of a set of basic “building block” signals, known as the basis functions. The transform output should lend itself to convenient analysis, interpretation and manipulation. A useful consequence of transforms, such as the Fourier and the Laplace, is that differential analysis on the time domain signal become simple algebraic operations on the transformed signal. In the Fourier transform the basic building block signals are sinusoidal signals with different periods giving rise to the concept of frequency. In Fourier analysis a signal is decomposed into its constituent sinusoids, i.e. frequencies, the amplitudes of various frequencies form the socalled frequency spectrum of the signal. In an inverse Fourier transform operation the signal can be synthesised by adding up its constituent frequencies. It turns out that many signals that we encounter in daily life such as speech, car engine noise, bird songs, music etc. have a periodic or quasi-periodic structure, and that the cochlea in the human hearing system performs a kind of harmonic analysis of the input audio signals. Therefore the concept of frequency is not a purely mathematical abstraction in that biological and physical systems have also evolved to make use of the frequency analysis concept. The power of the Fourier transform in signal analysis and pattern recognition is its ability to reveals spectral structures that may be used to characterise a signal. This is illustrated in Fig. 3.1 for the two extreme cases of a sine wave and a purely random signal. For a periodic signal the power is concentrated in extremely narrow bands of frequencies indicating the existence of structure and the predictable character of the signal. In the case of a pure sine wave as shown in Fig. 3.1.a the signal power is concentrated in one frequency. For a purely random signal as shown in Fig 3.1.b the signal power is spread equally in the frequency domain indicating the lack of structure in the signal. P XX (f)

x(t)

t (a)

P XX (f)

x(t)

t (b)

Figure 3.1 The concentration or spread of power in frequency indicates the correlated or random character of a signal: (a) a predictable signal, (b) a random signal.

Sec. 3.2 Fourier Series

3

Notation: In this chapter the symbols t and m denote continuous and discrete time variables, and f and k denote continuous and discrete frequency variables respectively. The variable ω=2πf denotes the angular frequency in units of rad/s and is used interchangably (within a scaling of factor of 2π) with the frequency variable f in units of Hz.

3.2 Fourier Series: Representation of Periodic Signals The following three sinusoidal functions form the basis functions for the Fourier analysis x1 (t ) = cosω 0 t (3.1) x 2 (t ) = sinω 0 t (3.2) (3.3) x3 (t ) = cosω 0 t + j sinω 0 t = e jω0t

Fig. 3.2.a shows the cosine and the sine components of the complex exponential (cisoidal) signal of Eq. (3.3), and Fig. 3.2.b shows a vector representation of the complex exponential in a complex plane with real (Re) and imaginary (Im) dimensions. The Fourier basis functions are periodic with an angular frequency of ω0 rad/s and a period of T0=2π/ω0=1/F0 seconds, where F0 is the frequency in Hz. The following properties make the sinusoids the ideal choice as the elementary building block basis functions for signal analysis and synthesis:

(i) Orthogonality; two sinusoidal functions of different frequencies have the following orthogonal property : Im sin(kωot)

jk ω0t

e

cos(kωot)

kω0t Re

t T0

(a)

(b)

Figure 3.2 - Fourier basis functions: (a) real and imaginary parts of a complex sinusoid, (b) vector representation of a complex exponential.

4

Chap.3 Fourier Analysis and Synthesis

∞

∞

∞

1 1 ∫−∞sin(ω1t ) sin(ω 2 t ) dt = − 2 −∫∞cos(ω1 + ω 2 )t dt + 2 −∫∞cos(ω1 − ω 2 )t dt = 0 (3.4) For harmonically related sinusoids the integration can be taken over one period. Similar equations can be derived for the product of cosines, or sine and cosine, of different frequencies. Orthogonality implies that the sinusoidal basis functions are independent and can be processed independently. For example in a graphic equaliser we can change the relative amplitudes of one set of frequencies, such as the bass, without affecting other frequencies, and in subband coding different frequency bands are coded independently and allocated different number of bits. (ii) Sinusoidal functions are infinitely differentiable. This is important, as most signal analysis, synthesis and manipulation methods require the signals to be differentiable. (iii) Sine and cosine signals of the same frequency have only a phase difference of π/2 or equivalently a relative time delay of a quarter of one period i.e. T0/4. Associated with the complex exponential function e jω 0 t is a set of harmonically related complex exponential of the form

[1, e

± jω 0 t

,e ± j2ω 0t ,e ± j3ω 0t , K

]

(3.5)

The set of exponential signals in Eq. (3.5) are periodic with a fundamental frequency ω0=2π/T0=2πF0 where T0 is the period and F0 is the fundamental frequency. These signals form the set of basis functions for the Fourier analysis. Any linear combination of these signals of the form ∞

∑c e k

jkω 0t

(3.6)

k = −∞

is also periodic with a period of T0. Conversely any periodic signal x(t) can be synthesised from a linear combination of harmonically related exponentials. The Fourier series representation of a periodic signal are given by the following synthesis and analysis equations:

x(t ) =

∞

∑c e k

k = −∞

jkω 0t

k = L − 1,0,1,L

Synthesis equation (3.7)

Sec. 3.2 Fourier Series

1 ck = T0

T0 / 2

5

∫ x(t )e

− jkω 0t

dt

k = L − 1,0,1,L

Analysis equation

(3.8)

−T0 / 2

The complex-valued coefficient ck conveys the amplitude (a measure of the strength) and the phase of the frequency content of the signal at kω0 Hz. Note from the analysis Eq. (3.8), that the coefficient ck may be interpreted as a measure of − jkω t the correlation of the signal x(t) and the complex exponential e 0 . The set of complex coefficients … c−1, c0, c1, … are known as the signal spectrum. Eq. (3.7) is referred to as the synthesis equation, and can be used as a frequency synthesizer (as in music synthesizers) to generate a signal as a weighted combination of its elementary frequencies. The representation of a signal in the form of Eq. (3.7) as the sum of its constituent harmonics is also referred to as the complex Fourier series representation. Note from Eqs. (3.7) and (3.8) that the complex exponentials that form a periodic signal occur only at discrete frequencies which are integer multiples, i.e. harmonics, of the fundamental frequency ω0. Therefore the spectrum of a periodic signal, with a period of T0, is discrete in frequency with discrete spectral lines spaced at integer multiples of ω0=2π/T0. Example 3.1 Given the Fourier synthesis Eq. (3.7), obtain the frequency analysis Eq. (3.8). Solution: Multiply both sides of Eq. (3.7) by e − jmω0t and integrate over one period to obtain T0 / 2

∫ x(t )e

−T0 / 2

− jmω 0t

dt =

T0 / 2

∞

∑ c ∫e k

k = −∞

jkω 0t

e

− jmω 0t

dt =

∑ c ∫e k

k = −∞

−T0 / 2

T0 / 2

∞

j( k − m )ω 0t

dt

(3.9)

−T0 / 2

From the orthogonality principle the integral in the r.h.s of Eq. (3.9) is zero unless k=m in which case the integral is equal to T0. Hence cm =

1 T0

T0 / 2

∫ x(t )e

−T0 / 2

− jmω 0t

dt

(3.10)

6

Chap.3 Fourier Analysis and Synthesis | ck |

x(t)

t

-1

1

k

-103

103

f H

T0=1 ms

(a)

(b) Figure 3.3 - A sinewave and its magnitude spectrum.

Example 3.2 Find the frequency spectrum of a 1 kHz sinewave shown in Fig 3.3.a x(t ) = sin( 2000πt ) − ∞
x(t ) =

∞

∑c e k

jk 2000πt

k = −∞

= L + c −1e − j2000πt + c0 + c1e j2000πt + L

now the sine wave can be expressed as 1 1 x(t ) = sin(2000πt ) = e j2000πt − e − j2000πt 2j 2j

(3.12)

(3.13)

Equating the coefficients of Eqs. (3.12) and (3.13) yields c1 =

1 1 , c −1 = − and c k ≠ ±1 = 0 2j 2j

( 3.14)

Fig 3.3.b shows the magnitude spectrum of the sinewave, where the spectral lines c1 and c−1 correspond to the 1 kHz and −1 kHz frequencies respectively. Solution B: Substituting sin( 2000πt ) =

1 j2000πt 1 − j2000πt in the Fourier e − e 2j 2j

analysis Eq. (3.8) yields 1 ck = T0 =

T0 / 2

 1 j 2000πt 1 − j2000πt  − jk 2000πt  e  e dt − e 2j 2j  −T0 / 2 

1 2 jT0

∫

T0 / 2

∫

e j (1− k ) 2000πt dt −

−T0 / 2

1 2 jT0

T0 / 2

∫e

−T0 / 2

− j(1+ k ) 2000πt

dt

(3.15)

Sec. 3.2 Fourier Series

7

Since sine and cosine functions are positive-valued over one half a period and negative-valued over the other half, it follows that Eq. (3.15) is zero unless k=1 or k=−1. 1 1 c1 = and c −1 = − and c k ≠ ±1 = 0 (3.16) 2j 2j Example 3.3 Find the frequency spectrum of a periodic train of pulses with amplitude of 1.0, a period of 1.o kHz and a pulse 'on' duration of 0.3 milliseconds. Solution: The pulse period T0=1/F0=0.001 s, and the angular frequency ω0=2πF0=2000π rad/s. Substituting the pulse signal in the Fourier analysis Eq. (3.8) gives

1 ck = T0

T0 / 2

∫

x(t ) e − jkω 0t dt =

−T0 / 2

e − jk 2000πt = − j2πk

t = 0.00015

t = −0.00015

0.00015

1 e − jk 2000πt dt 0.001 −0.00015

∫

(3.17)

e j0.3πk − e − j0.3πk sin(0.3πk ) = = j2πk πk

For k=0 as c0=sin(0)/0 is undefined, differentiate the numerator and denominator of Eq. (3.17) w.r.t. to the variable k (strictly this can only be done for a continuous variable k i.e. when the period T0 tends to infinity) to obtain c0 =

0.3π cos(0.3π 0)

π

(3.18

= 0.3

x(t)

t c(k) 0 .3 0 .2 5 0 .2 0 .1 5 0 .1 0 .0 5 0 -0 .0 5

1

2

k

-0 .1

Figure 3.4 - A rectangular pulse train and its discrete frequency ‘line’ spectrum.

8

Chap.3 Fourier Analysis and Synthesis

Example 3.4 For the example 3.3 write the formula for synthesising the signal up to the Nth harmonic, and plot a few examples for the increasing number of harmonics. Solution: The equation for the synthesis of a signal upto the Nth harmonic content is given by x(t ) =

N

∑c

ke

jkω 0 t

= c0 +

k =− N

= c0 +

N

∑c e k

jkω 0 t

+

k =1

N

∑ [Re(c

N

∑c

−k e

− jkω 0 t

k =1

k

) + j Im(c k )][cos(kω 0 t ) + j sin( kω 0 t )]

k

) − j Im(c k )][cos(kω 0 t ) − j sin(kω 0 t )]

k =1

+

N

∑ [Re(c k =1

= c0 +

N

∑ [2 Re(c

k

(3.19)

) cos(kω 0 t ) − 2 Im(c k ) sin( kω 0 t )]

k =1

The following MatLab code generates a synthesised pulse train composed of N harmonics. In this example there are 5 cycles in an array of 1000 samples. Fig. 3.5 shows the waveform for the number of harmonics equal to; 1,3,6, and 100. NHarmonics=100; Ncycles=5; Nsamples=1000; y(1:Nsamples)=0.3; j=1:Nsamples; for k=1:NHarmonics x(j)=(2*sin(0.3*pi*k)/(pi*k))*cos(k*2*pi*Ncycles*j/Nsamples); y=y+x; plot(y); pause; end

x(t)

x(t)

(a)

(b) t

t

x(t)

x(t)

(c)

(d)

t

t

Figure 3.5 Illustration of the Fourier synthesis of a periodic pulse train, and the Gibbs phenomenon, with the increasing number of harmonics in the Fourier synthesis: (a) N=1, (b) N=3, (c) N=6, and (d) N=100.

Sec. 3.3 Fourier Transform

9

3.2.1 Fourier Synthesis of Discontinuous Signals: Gibbs Phenomenon

The sinusoidal basis functions of the Fourier transform are smooth and infinitely differentiable. In the vicinity of a discontinuity the Fourier synthesis of a signal exhibits ripples as shown in the Fig 3.5. The peak amplitude of the ripples does not decrease as the number of harmonics used in the signal synthesis increases. This behaviour is known as the Gibbs phenomenon. For a discontinuity of unity height, the partial sum of the harmonics exhibits a maximum value of 1.09 (that is an overshoot of 9%) irrespective of the number of harmonics used in the Fourier series. As the number of harmonics used in the signal synthesis increases, the ripples become compressed toward the discontinuity but the peak amplitude of the ripples remains constant.

3.3 Fourier Transform: Representation of Aperiodic Signals The Fourier series representation of periodic signals consist of harmonically related spectral lines spaced at the integer multiples of the fundamental frequency. The Fourier representation of aperiodic signals can be developed by regarding an aperiodic signal as a special case of a periodic signal with an infinite period. If the period of a signal is infinite, then the signal does not repeat itself and is aperiodic. Now consider the discrete spectra of a periodic signal with a period of T0, as

c(k)

x(t) Ton

(a)

Toff

t

T0=Ton+Toff

1 T0

k X(f)

x(t)

Toff = ∞ (b)

t

f

Figure 3.6 – (a) A periodic pulse train and its line spectrum, (b) a single pulse from the periodic train in (a) with an imagined ‘off’ duration of infinity; its spectrum is the envelope of the spectrum of the periodic signal in (a).

10

Chap.3 Fourier Analysis and Synthesis

shown in Fig. 3.6.a. As the period T0 is increased, the fundamental frequency F0=1/T0 decreases, and successive spectral lines become more closely spaced. In the limit as the period tends to infinity (i.e. as the signal becomes aperiodic) the discrete spectral lines merge and form a continuous spectrum. Therefore the Fourier equations for an aperiodic signal, (known as the Fourier transform), must reflect the fact that the frequency spectrum of an aperiodic signal is continuous. Hence to obtain the Fourier transform relation the discretefrequency variables and operations in the Fourier series Eqs. (3.7) and (3.8) should be replaced by their continuous-frequency counterparts. That is the discrete summation sign

Σ should be replaced by the continuous summation integral

∫

,

the discrete harmonics of the fundamental frequency kF0 should be replaced by the continuous frequency variable f, and the discrete frequency spectrum ck must be replaced by a continuous frequency spectrum say X ( f ) . The Fourier synthesis and analysis equations for aperiodic signals, the so-called Fourier transform pair, are given by ∞

x(t ) =

∫ X ( f )e

j2πft

df

(3.20)

X ( f ) = x(t )e − j2πft dt

(3.21)

−∞ ∞

∫

−∞

Note from Eq. (3.21), that X ( f ) may be interpreted as a measure of the − j2πft

. correlation of the signal x(t) and the complex sinusoid e The condition for existence and computability of the Fourier transform integral of a signal x(t) is that the signal must have finite energy ∞

∫

2

x(t ) dt < ∞

(3.22)

−∞

Example 3.5 Derivation of inverse Fourier transform Given the Fourier transform Eq. (3.21) derive the Fourier synthesis Eq. (3.20). Solution: Consider the Fourier analysis Eq. (3.8) for a periodic signal and Eq. (3.21) for its non-periodic version (consisting of one period only). Comparing these equations reproduced below

ck = we have

1 T0

T0 / 2

∫

x(t )e − j2πkF0 t dt

− T0 / 2

∞

∫

X ( f ) = x(t )e − j2πft dt −∞

Sec. 3.3 Fourier Transform

11

ck =

1 X (kF0 ) T0

(3.23)

as T0 → ∞

where F0=1/T0. Using Eq. (3.23) the Fourier synthesis Eq. (3.7) for a periodic signal can be rewritten as x(t ) =

∞

∑ X ( k / T )e 0

j2πkF0 t

∆F

(3.24)

k = −∞

where ∆F=1/T0=F0 is the frequency spacing between successive spectral lines of the spectrum of a periodic signal as shown in Fig. 3.6. Now as the period T0 tends to infinity, ∆F=1/T0 tends to zero, then the discrete frequency variables kF0 and ∆F should be replaced by a continuous frequency variable f, and the discrete summation sign by the continuous integral sign. Thus Eq. (3.24) becomes ∞

x(t ) ⇒ T0 →∞

∫ X ( f )e

j2πft

(3.25)

df

−∞

Example 3.6 The spectrum of an Impulse Function

Consider the unit-area pulse p(t) shown in Fig 3.7.a. As the pulse width ∆ tends to zero the pulse tends to an impulse. The impulse function shown in Fig 3.7.b is defined as a pulse with an infinitesimal time width as 1 / ∆ t ≤ ∆ / 2  0 t > ∆ / 2

δ (t ) = limit p (t ) =  ∆→0

p(t) 1/∆

∆

(a)

(3.26)

δ(t) As ∆

∆(f)

0

t

t

(b)

Figure 3.7 (a) A unit-area pulse, (b) the pulse becomes an impulse as spectrum of the impulse function.

f

(c) ∆ → 0 , (c) the

12

Chap.3 Fourier Analysis and Synthesis

It is easy to see that the integral of the impulse function is given by ∞

1

∫ δ (t )dt = ∆ × ∆ =1

(3.27)

−∞

The important sampling property of the impulse function is defined as ∞

∫ x(t )δ (t − T ) dt = x(T )

(3.28)

−∞

The Fourier transform of the impulse function is obtained as ∞

∫

∆ ( f ) = δ (t )e − j 2πft dt = e 0 = 1

(3.29)

−∞

The impulse function is used as a test function to obtain the impulse response of a system. This is because as shown in Fig 3.7.c an impulse is a spectrally rich signal containing all frequencies in equal amounts. Example 3.7 Find the spectrum of the 10 kHz periodic pulse train shown in Fig. 3.8.a with an amplitude of 1 mV and expressed as

x(t ) =

∞

∑

m = −∞

A× limit p(t − mT0 ) × ∆ = ∆ →0

∞

∑ A δ (t − mT )

(3.30)

0

m = −∞

where p(t) is a unit-area pulse with width ∆ as shown in Fig. 3.7.a, and the function δ(t-mT0) is now assumed to have a unit amplitude representing the area under the impulse. Solution: T0=1/F0=0.1 ms and A=1 mV. For the time interval −T0/2
x(t )

... T0

... t

1/T0

(a) (b) Figure 3.8 A periodic impulse train x(t) and its Fourier series spectrum ck.

k

Sec. 3.3 Fourier Transform

1 ck = T0

T0 / 2

∫ x(t )e

− jkω 0t

−T0 / 2

13

1 dt = T0

T0 / 2

∫ Aδ (t )e

− jkω 0t

dt =

−T0 / 2

A = 10 V T0

(3.31)

As shown in Fig. 3.8.b the spectrum of a periodic impulse train in time is a periodic impulse train in frequency Example 3.8 The Spectrum of a Rectangular Function: Sinc Function

The rectangular pulse is particularly important in digital signal analysis and digital communication. For example, the spectrum of a rectangular pulse can be used to calculate the bandwidth required by pulse radar systems or communication systems that transmit pulses. The Fourier transform of a rectangular pulse Fig. 3.9.a of duration T seconds is obtained as ∞

T /2

−∞

−T / 2

R ( f ) = ∫ r (t ) e − j2πft dt =

∫ 1.e

− j2πft

dt

e j2πfT / 2 − e − j2πfT / 2 sin(πfT ) = =T = T sinc(πfT ) j2πf πfT

(3.32)

Fig. 3.9.b shows the spectrum of the rectangular pulse. Note that most of the pulse energy is concentrated in the main lobe within a bandwidth of 2/T. However there are pulse energy in the side lobes that may interfere with other electronic devices operating at the side lobe frequencies. Matlab code for drawing the spectrum of a rectangular pulse. % Pulsewidth T, Number of frequency samples N, Frequency resolution for plot df. T=.001;df=10; N=1000; for i=1:N if (i~=N/2)x(i)=T*sin(2*pi*df*T*(i-N/2))/(2*pi*df*T*(i-N/2));end end x(N/2)=T; plot(x); R(f) r(t) T 1

T

(a)

t

-2 T

-1 T

1 T

2 T

3 T

(b) Figure 3.9 A rectangular pulse and its spectrum.

f

14

Chap.3 Fourier Analysis and Synthesis

e–σtIm[e–j2πft]

e–σtIm[e–j2πmf]

e–σtIm[e–j2πmf]

m

σ>0

m

m

σ =0 Figure 3.10 – The Laplace basis functions.

σ <0

3.3.1 The Relation Between the Laplace and the Fourier Transforms

The Laplace transform of x(t) is given by the integral ∞

X ( s ) = ∫ x(t )e − st dt

(3.33)

0−

where the complex variable s=σ+jω, and the lower limit of t=0− allows the possibility that the signal x(t) may include an impulse. The inverse Laplace transform is defined by σ 1 + j∞

x(t ) =

∫ X (s)e σ

st

ds

(3.34)

1 − j∞

where σ1 is selected so that X(s) is analytic (no singularities) for s>σ1. The basis functions for the Laplace transform are damped or growing sinusoids of the form e − st = e −σt e − jωt as shown in Fig. 3.10. These are particularly suitable for transient signal analysis. The Fourier basis functions are steady complex exponential, e − jωt , of time-invariant amplitudes and phase, suitable for steady state or time-invariant signal analysis. The Laplace transform is a one-sided transform with the lower limit of integration at t = 0 − , whereas the Fourier transform Eq. (3.21) is a two-sided transform with the lower limit of integration at t = −∞ . However for a one-sided signal, which is zero-valued for t < 0 − , the limits of integration for the Laplace and the Fourier transforms are identical. In that case if the variable s in the Laplace transform is replaced with the frequency variable jω then the Laplace integral becomes the Fourier integral. Hence for a one-sided signal, the Fourier transform is a special case of the Laplace transform corresponding to s=jω and σ=0. The relation between the Fourier and the Laplace transforms are discussed further in Chapter 4. 3.3.2 Properties of the Fourier Transform

Sec. 3.3 Fourier Transform

15

There are a number of Fourier Transform properties that provide further insight into the transform and are useful in reducing the complexity of the solutions of Fourier transforms and inverse transforms. These are: Linearity The Fourier transform is a linear operation, this mean the principle of superposition applies. Hence if: z (t ) = ax(t )+by (t ) (3.35) then Z ( f ) = aX ( f )+bY ( f ) (3.36) Symmetry This property states that if the time domain signal x(t) is real (as is often the case in practice) then X ( f ) = X * (− f ) (3.37)

Where the superscript asterisk * denotes the complex conjugate operation. From Eq. (3.37) it follows that Re{ X ( f ) } is an even function of f and Im{ X ( f ) } is an odd function of f. Similarly the magnitude of X ( f ) is an even function and the phase angle is an odd function. Time Shifting and Frequency Modulation (FM) Let X ( f ) = F [ x(t ) ] be the Fourier transform of x(t). If the time domain signal x(t) is delayed by an amount T0, the effect on its spectrum X ( f ) is a phase shift of

e − j2πT0 f as

F [x(t − T0 )] = e − j2πfT

0

X(f )

(3.38)

Conversely if X(f) is shifted by an amount F0, the effect is

F −1 [X ( f

− F0 )] =e j2πF0t x(t )

(3.39)

Note that the modulation Eq. (3.39) states that multiplying a signal x(t) by e j 2πF0t translates the spectrum of x(t) onto the frequency F0, this is the frequency modulation (FM) principle. Differentiation and Integration Let x(t) be a continuous time signal with Fourier transform X ( f ) . ∞

x(t ) =

∫ X ( f )e

−∞

j2πft

df

(3.40)

16

Chap.3 Fourier Analysis and Synthesis

Then by differentiating both sides of the Fourier transform Eq. (3.40) we obtain ∞

d x(t ) = j2πf X ( f ) e j2πft df 14243 dt −∞

∫

(3.41)

FourierTransform of d x ( t ) / dt

That is multiplication of X ( f ) by the factor j2πf in the frequency domain is equivalent to differentiation of x(t) in time. Similarly division of X ( f ) by j2πf is equivalent to integration of the function of time x(t) t

F

1

∫ x(τ )dτ ←→ j2πf X ( f )+πX (0)δ ( f )

(3.42)

−∞

Where the impulse term on the right-hand side reflects the dc or average value that can result from the integration. Time and Frequency Scaling If x(t) and X(f) are Fourier transform pairs then

1 f F x(αt ) ←→ X  α α 

(3.43)

For example try to say something very slowly, then α >1, your voice spectrum will be compressed and you may sound like a slowed down tape or disc, you can do the reverse and the spectrum would be expanded and your voice shifts to higher frequencies, This property is further illustrated in section 3.9.4. Convolution The convolution integral of two signals x(t) and h(t) is defined as ∞

∫

y (t ) = x(τ ) h(t − τ )dτ

(3.44)

−∞

The convolution integral is also written as y (t ) = x(t ) * h(t )

(3.45)

where asterisk * denotes the convolution operation. The convolution integral is used to obtain the time-domain response of linear systems to arbitrary inputs as will be discussed in later sections.

Sec. 3.3 Fourier Transform

17

The Convolution Property of the Fourier Transform. It can be shown that convolution of two signals in the time domain corresponds to multiplication of the signals in the frequency domain, and conversely multiplication in the time domain corresponds to convolution in the frequency domain. To derive the convolutional property of the Fourier transform take the Fourier transform of the convolution of the signals x(t) and h(t) as ∞ ∞ ∞   x(τ )h(t − τ )dτ  e − j2πft dt = x(τ ) e − j2πfτ dτ h(t − τ ) e − j2πf (t −τ ) dt   −∞  −∞ −∞ −∞  = X ( f )H ( f ) ∞

∫ ∫

∫

∫

(3.46)

Duality Comparing the Fourier transform and the inverse Fourier transform relations we observe a symmetric relation between them. In fact the main difference between Eqs. (3.20) and (3.21) is a negative sign in the exponent of e − j2πft in Eq (3.21). This symmetry leads to a property of Fourier transform known as the duality principle and stated as F x(t ) ←→ X ( f ) (3.47) F X (t ) ←→ x( f )

As illustrated in Fig 3.11. the Fourier transform of a rectangular function of time r(t) has the form of a sinc pulse function of frequency sinc( f ) . From the duality X (f) x(t)

T

-2 T

t

T

-1 T

1 T

2 T

3 T

f

x(f)

X (t)

T

t T1

-1 T1

1 T1

f

Figure 3.11 Illustration of the principle of duality.

principle the Fourier transform of a sinc function of time sinc(t) is a rectangular function of frequency R(f).

18

Chap.3 Fourier Analysis and Synthesis

Parseval's Theorem: Energy Relationship in Time and Frequency

Parseval’s relation states that the energy of a signal can be computed by integrating the squared magnitude of the signal either over the time domain or over the frequency domain. If x(t) and X(f) are a Fourier transform pair, then ∞

Energy =

∫

∞

∫

| x(t ) | 2 dt = | X ( f ) | 2 df

−∞

(3.48)

−∞

This expression referred to as Parseval's relation follows from a direct application of the Fourier transform. Example 3.9 The Spectrum of a Finite Duration Signal

Find and sketch the frequency spectrum of the following finite duration signal x(t ) = sin( 2πF0 t ) − NT0 / 2 ≤ t ≤ NT0 / 2

(3.49)

where T0=1/F0 is the period and ω0= 2π/T0. Solution: Substitute for x(t) and its non-zero valued limits in the Fourier transform Eq. (3.21) NT0 / 2

X(f ) =

∫ sin(2πF t )e 0

− j 2πft

dt

− NT0 / 2

(3.50)

substituting sin(2πF0 t ) = (e j2πF0t − e − j2πF0t ) / 2 j in (3.50) gives

NT0 / 2

∫

X(f ) =

− NT0

e j2πF0t − e − j2πF0t − j2πft e dt 2 j /2

NT0 / 2

=

∫

− NT0

NT0 / 2

e − j2π ( f − F0 )t dt − 2j /2 − NT

Evaluating the integrals yields

∫

0

e − j2π ( f + F0 )t dt 2j /2

(3.51)

Sec. 3.3 Fourier Transform

X(f ) = =

19

e− jπ ( f − F0 ) NT0 − e jπ ( f − F0 ) NT0 e− jπ ( f + F0 ) NT0 − e jπ ( f + F0 ) NT0 − 4π ( f − F0 ) 4π ( f + F0 )

j −j sin (π ( f − F0 ) NT0 ) + sin (π ( f + F0 ) NT0 ) 2π( f − F0 ) 2π( f + F0 ) (3.52)

Matlab code for drawing the spectrum of a finite duration sinwave. % Sinewave Period T0, Number of Cycles in the window N T0=.001; F0=1/T0; N=2000; for Nc=1:4; for f=1:999 if ((f-f0)~=0) x(Nc,f)=Nc*T0*sin(pi*(f-F0)*T0*Nc)/(pi*(f-F0)*T0*Nc); end end x(Nc,1000)=Nc*T0; end plot(x); X(f) Nc=4

Nc=2 Nc=1

F0

f Nc=3

Figure 3.12 The spectrum of a finite duration sine wave with the increasing length of observation. Nc is the number of cycles in the observation window. Note that the width of the main lobe depends inversely on the duration of the signal.

The signal spectrum can be expressed as the sum of two shifted sinc functions.

X ( f ) = − jNT0 sinc(π ( f − F0 ) NT0 ) + j NT0 sinc(π ( f + F0 ) NT0 )

(3.53)

Note that energy of a finite duration sinewave is spread in frequency between the main lobe and the side lobes of the sinc function. Fig. 3.12 demonstrates that as the window length increases the energy becomes more concentrated in the main lobe and in the limit for an infinite duration window the spectrum tends to an impulse positioned at the frequency F0.

20

Chap.3 Fourier Analysis and Synthesis

Example 3.10 Calculation of the bandwidth for transmission of data at a rate of rb bits per second. In its simplest form binary data can be represented as a sequence of amplitude modulated pulses as N −1

x(t ) = ∑ A(m) p (t − mTb )

()

m =0

where A(m) may be +1 or a –1 and

1 p (t ) =  0

t ≤ Tb / 2

()

t >0

The Fourier transform of x(t) is given by N −1

N −1

m=0

m=0

X ( f ) = ∑ A(m) P( f )e − j 2πmTb f = P( f ) ∑ A(m)e − j 2πmTb f

()

The power spectrum of this signal is obtained as ∞  ∞  E [X ( f ) X * ( f )] = E  ∑ A(m) P( f )e − j 2πmTb f ∑ A(n) P * ( f )e j 2πmTb f  n = −∞  m = −∞ 

= P( f )

2

N −1 N −1

∑ ∑ E[ A(m) A(n)]e

()

− j 2π ( m − n )Tb f

m =0 n =0

Now assuming that the data is uncorrelated we have

1 E[ A(m) A(n)] =  0

m=n m≠n

()

Substituting Equations () in () we have

E [ X ( f ) X * ( f )] = N P ( f )

2

()

From Equation () the bandwidth required from a sequence of pulses is basically the same as the bandwidth of a single pulse. From Figure (3.11) the main lobe width is 2rb.

Sec. 3.3 Fourier Transform

21

3.3.3 Fourier Transform of a Sampled Signal

v

A sampled signal x(m) can be modelled as

x ( m) =

∞

∑ x(t )δ (t − mT ) s

m = −∞

(3.54)

where m is the discrete time variable and Ts is the sampling period. The Fourier transform of x(m), a sampled version of a continuous signal x(t), can be obtained from Eq. (3.21) as ∞

X ( f )=

∞

∑ x(t )δ (t − m)e − j 2πft dt =

∫

−∞ m = −∞

=

∞

∑ x(mT )e

m = −∞

s

∞

∞

∑ ∫ x(t )δ (t − mT )e

m = −∞ −∞

s

− j 2πmf / Fs

− j 2πft

dt (3.55)

For convenience of notation and without loss of generality it is often assumed that sampling frequency Fs=1/Ts=1, hence

Xs( f ) =

∞

∑ x(m)e

− j 2πmf

(3.56)

m = −∞

The inverse Fourier transform of a sampled signal is defined as 1/ 2

x ( m) =

∫X

s

( f ) e j2πfm df

(3.57)

−1 / 2

Note that x(m) and X s ( f ) are equivalent in that they contain the same information in different domains. In particular, as expressed by the Parseval's theorem, the energy of the signal may be computed either in the time or in the frequency domain a

Signal Energy =

∞

1/ 2

m = −∞

−1 / 2

∑ x 2 (m) = ∫ | X s ( f ) |2 df

(3.58)

Example 3.10 Show that the spectrum of a sampled signal is periodic with a period equal to the sampling frequency Fs.

22

Chap.3 Fourier Analysis and Synthesis

Solution: substitute f+kFs for the frequency variable f in Eq. (3.56)

X ( f + kFs ) =

∞

∑ x(mT )e s

− j2πm

( f + kFs ) Fs

∞

=

∑ x(mT )e

− j2πm

s

m = −∞

m = −∞

f Fs

− j2πm

kFs Fs

e1 424 3= X(f ) =1

(3.59)

Fig. 3.13.a shows the spectrum of a band-limited continuous-time signal. As shown in Fig. 3.13.b after the signal is sampled its spectrum becomes periodic.

Xs(f)

X(f)

...

... 0

-2Fs

f

-Fs

(a)

0

Fs

2Fs

f

(b)

Figure 3.13 The spectrum of : (a) a continuous signal, and (b) its sampled version.

x(0) x(1) x(2)

x(N–2) x(N – 1)

Discrete Fourier Transform . . .

N–1

X(k) =

∑ m=0

2 π kn x(m) e N –j

. . .

X(0) X(1) X(2)

X(N – 2) X(N– 1)

Figure 3.14 Illustration of the DFT as a parallel-input parallel-output signal processor.

Sec. 3.5 Short-Time Fourier Transform

23

3.4 Discrete Fourier Transform (DFT) When a non-periodic signal is sampled, its Fourier transform becomes a periodic but continuous function of frequency, as shown in Eq. (3.59). The discrete Fourier transform (DFT) is derived from sampling the Fourier transform of a discrete-time signal. For a finite duration discrete-time signal x(m) of length N samples, the discrete Fourier transform (DFT) is defined as N uniformly spaced spectral samples N −1

X ( k ) = ∑ x ( m) e

−j

2π mk N

k = 0, . . ., N−1

(3.60)

m=0

Comparing Eqs. (3.60) and (3.56) we see that the DFT consists of N equi-spaced samples taken from one period (2π) of the continuous spectrum of the discrete time signal x(m). The inverse discrete Fourier transform (IDFT) is given by 2π

x ( m) =

j mk 1 N −1 X (k ) e N ∑ N k =0

m= 0, . . ., N−1

(3.61)

A periodic signal has a discrete spectrum. Conversely any discrete frequency spectrum belongs to a periodic signal. Hence the implicit assumption in the DFT theory, is that the input signal x(m) is periodic with a period equal to the observation window length of N samples. Example: Derivation of inverse discrete Fourier transform Obtain the inverse DFT from the DFT equation. The discrete Fourier transform (DFT) is given

by N −1

X ( k ) = ∑ x ( m) e

−j

2π km N

k = 0, . . ., N−1

m =0

Multiply both sides of the DFT equation by e-j2πkn/N and take the summation as

24

Chap.3 Fourier Analysis and Synthesis N −1

∑ X (k ) e

+j

2π kn N

k =0

=

N −1 N −1

∑∑ x(m) e

−j

2π km N

e

+j

2π kn N

k =0 m=0

N −1

N −1

−j

2π

k ( m−n)

= ∑ x ( m) ∑ e N k =0 m=0 14 4244 3 N if m = n 0 otherwise

Using the orthogonality principle,the inverse DFT equation can be derived as 2π −j km 1 N −1 x ( m) = ∑ X ( k ) e N N k =0

3.4.1 Time and Frequency Resolutions: The Uncertainty Principle

Signals such as speech, music or image are composed of nonstationary  i.e. time-varying and/or space varying  events. For example speech is composed of a string of short-duration sounds called phonemes, and an image is composed of various objects. When using the DFT it is desirable to have a high enough time and space resolution in order to obtain the spectral characteristics of each individual elementary event or object in the input signal. However there is a fundamental trade-off between the length, i.e. the time or space resolution, of the input signal and the frequency resolution of the output spectrum. The DFT takes as the input a window of N uniformly spaced time domain samples [x(0), x(1), …, x(N−1)] of duration ∆T=N.Ts, and outputs N spectral samples [X(0), X(1), …, X(N−1)] spaced uniformly between zero Hz and the sampling frequency Fs=1/Ts Hz. Hence the frequency resolution of the DFT spectrum ∆f, i.e. the space between successive frequency samples, is given by

∆f =

F 1 1 = = s ∆T NTs N

(3.62)

Note that the frequency resolution ∆f and the time resolution ∆T are inversely proportional in that they can not both be simultanously increased, in fact ∆T∆f=1. This is known as the uncertainty principle. Example 3.11 A DFT is used in a DSP system for the analysis of an analog signal with a frequency content of up to 10 kHz. Calculate: (i) the minimum sampling rate Fs required, and (ii) the number of samples required for the DFT to achieve a frequency resolution of 10 Hz at the minimum sampling rate.

Sec. 3.5 Short-Time Fourier Transform

25

Solution: (i) Sampling rate > 2×10 kHz, say 22 kHz, and

∆f =

(ii)

Fs N

10 =

22000 N

N ≥ 2200 .

Example 3.12 Write a MATLAB program to explore the spectral resolution of a signal consisting of two sinewaves, closely spaced in frequency, with the varying length of the observation window. Solution: In the following program the two sinwaves have frequencies of 100 Hz and 110 Hz, and the sampling rate is 1 kHz. We experiment with two time windows of length N1=1024 with a theoretical frequency resolution of ∆f=1000/1024=0.98 Hz, and N2=64 with a theoretical frequency resolution ∆f=1000/64=15.7 Hz.

Amplitude

2

1

0

- 1

- 2

2 0 0

0

6 0 0

4 0 0

1 0 0 0

8 0 0

T im e

(a) 450 Amplitude

Amplitude

80 60

300

40

150 20

0

10

20

30

40

50

60

70

0

100

200

300

(b)

400

500

600

Feq k

Feq k

(c)

Figure 3.15 Illustration of time and frequency resolutions: (a) sum of two sinewaves with 10 Hz difference in their frequencies, (b) the spectrum of a segment of 64 samples from demonstrating insufficient frequency resolution to separate the sinewaves, (c) the spectrum of a segment of 1024 samples has sufficient resolution to show the two sinewaves.

Fs=1000; F1=100; F2=110; N=1:1024; N1=1024; N2=64; x1=sin(2*pi*F1*N/Fs); x2=sin(2*pi*F2*N/Fs); y=x1+x2;

26

Chap.3 Fourier Analysis and Synthesis

Y1=abs(fft(y(1:N1)));Y2=abs(fft(y(1:N2))); figure(1); plot(y); figure(2); plot(Y1(1:N1/2)); figure(3); plot(Y2(1:N2/2));

3.4.2 The Effect of Finite Length Data on DFT (Windowing)

In a practical situation we have either with a short length signal, or with a long signal of which the DFT can only handle one segment at a time. Having a short segment of N samples of a signal or taking a slice of N samples from a signal is equivalent to multiplying the signal by a unit-amplitude rectangular pulse window of N samples. Therefore an N-sample segment of a signal x(m) is equivalent to

x w (m) = w(m) x(m)

(3.63)

where w(m) is a rectangular pulse of N samples duration given is

1 0≤m ≤ N − 1 w(m)= 0 otherwise

(3.64)

Multiplying two signals in time is equivalent to the convolution of their frequency spectra. Thus the spectrum of a short segment of a signal is convolved with the spectrum of a rectangular pulse as X w (k ) = W (k ) * X (k ) (3.65) The result of this convolution is some spreading of the signal energy in the frequency domain as illustrated in the next example. Example 3.13 Find the DFT of a rectangular window given by

1 0 ≤ m ≤ N − 1 w(m)= 0 otherwise

(3.66)

Solution: Taking the DFT of w(m), and using the convergence formula for the partial sum of a geomertic series, described in the appendix A, we have

W (k ) =

N −1

∑

m =0

w(m)e

−j

2π mk N

=

1 − e − j2πk 1− e

−j

2π k N

=e

−j

( N −1) πk N

sin(πk ) sin(πk / N )

(3.67)

Note that for the integer values of k, w(k) is zero except for k=0. Example 3.14 Find the spectrum of an N-sample segment of a complex sinewave with a fundamental frequency F0=1/T0.

Sec. 3.5 Short-Time Fourier Transform

27

Solution: Taking the DFT of x(m) = e − j 2πF0 m we have

X (k ) =

N −1

∑e

− j2πF0 m

m =0

e

−j

2π mk N

=

N −1 − j2π ( F − k ) m 0 N e

∑

m =0

−j 1 − e − j2π ( NF0 − k ) = =e − j2π ( NF0 − k ) / N 1− e

( N −1) π ( NF0 − k ) N

sin(π ( NF0 − k )) sin(π ( NF0 − k ) / N )

(3.68)

Note that for integer values of k, X(k) is zero at all samples but one k=0. 3.4.3 End-Point Effects in DFT; Spectral Energy Leakage and Windowing

In DFT the input signal is assumed to be periodic, with a period equal to the length of the observation window of N samples. Now, for a sinusoidal input signal if there is an integer number of cycles within the observation window, as in Fig. 3.16.a, then the assumed periodic waveform is the same as an infinite length pure sinusoid. But if the observation window contains a non-integer number of cycles of a sinusoid then the assumed periodic waveform will not be a pure sine wave and will have end-point discontinuities. The spectrum of the signal then differs from the spectrum of sinewave as illustrated in Fig. 3.16.b. The overall effects of finite length window and end-point discontinuities are:

time

time

2 cycles

F0

(a)

2.25 cycles

Frequency

F0

Frequency

(b)

Figure 3.16 The DFT spectrum of exp(j2πfm) : (a) an integer number of cycles with in the N-sample analysis window, (b) a non-integer number of cycles in the window.

1. The spectral energy which could have been concentrated at a single point, or in a narrow band of frequencies is spread over a larger band of frequencies.

28

Chap.3 Fourier Analysis and Synthesis

2. A smaller amplitude signal, located in frequency near a larger amplitude signal, may be obscured by one of the larger signal’s side-lobes. That is sidelobes of a large amplitude signal may interfere with the main-lobe of a nearby small amplitude signal. The end-point problems may be alleviated using a window that gently drops to zero. One such window is a raised cosine window of the form 2πm  0≤m≤ N − 1 α − (1 − α ) cos w(m) =  N  0 otherwise

(3.69)

For α=0.5 we have the Hanning window also known as the raised cosine window 2πm N For α=0.54 we have the Hamming window wHan (m) = 0.5−0.5cos

wHam (m) = 0.54 − 0.46 cos

2πm N

0≤m≤ N − 1

(3.70)

0 ≤ m ≤ N −1

(3.71)

0 -5 -1 0 -1 5 -2 0 -2 5

0

T im e

6 4

-3 0

0

F re q u en c y

F

s

/2

0 -1 0 -2 0 -3 0 -4 0 -5 0 -6 0

0

0

6 4

F

s

/2

0 -1 0 -2 0 -3 0 -4 0 -5 0

0

6 4

-6 0 -7 0 -8 0

0

F

s

/2

Figure 3.17 (a) Rectangular window frequency response, (b) Hamming window frequency response, (c) Hanning window frequency response.

3.4.3 Spectral Smoothing

Sec. 3.5 Short-Time Fourier Transform

29

The spectrum of a short length signal can be interpolated to obtain a smoother spectrum. Interpolation of the frequency spectrum X(k) is achieved by zeropadding of the time domain signal x(m). Consider a signal of length N samples [x(0), . . ., x(N−1)]. Increase the signal length from N to 2N samples by padding N zeros to obtain the padded sequence [x(0), . . ., x(N−1), 0, . . ., 0]. The DFT of the padded signal is given by 2 N −1

X (k ) =

∑ x ( m) e

−j

N −1

−j

2π mk 2N

m=0

=

∑ x ( m) e

π N

k = 0, . . ., 2N−1

(3.72)

mk

m =0

The spectrum of the zero-padded signal, Eq. (3.72), is composed of 2N spectral samples; N of which, [X(0), X(2), X(4), X(6), . . . X(2N−2)] are the same as those that would be obtained from a DFT of the original N samples, and the other N samples [X(1), X(3), X(5), X(6), . . . X(2N−1)] are interpolated spectral lines that result from zero-padding. Note that zero padding does not increase the spectral resolution, it merely has an interpolating or smoothing effect in the frequency domain, as illustrated in Fig 3.18.

r(m)

|R(f)| without zero padding Padded zero’s Time

|R(f)|

Frequency |R(f)| with 20 padded zeros

with 10 padded zeros

Frequency

Frequency

Figure 3.18 Illustration of the interpolating effect, in the frequency domain, of zero padding a signal in the time domain.

30

Chap.3 Fourier Analysis and Synthesis

time

Figure 3.19 - Segmentation of speech using Hamming window for STFT.

3.5 Short-Time Fourier Transform In Fourier transform it is assumed that the signal is stationary, meaning the signal statistics, such as the mean, the power, and the power spectrum, are time-invariant. Most real life signals such as speech, music and image signals are nonstationary in that their amplitude, power, spectral composition and other features changes continuously with time. To apply Fourier transform to nonstationary signals the signal is divided into appropriately short-time windows, such that within each window the signal can be assumed to be time-invariant. The Fourier transform applied to the short signal segment within each window is known as the short-time Fourier Transform. Fig. 3.19 illustrates the segmentation of a speech signal into a sequence of overlapping, hamming windowed, short segments. The choice of window length is a compromise between the time resolution and the frequency resolution. For Audio signals a time window of about 25 ms, corresponding to a frequency resolution of 40 Hz is normally adopted.

walker.qxp 3/24/98 2:16 PM Page 658

Fourier Analysis and Wavelet Analysis James S. Walker

I

n this article we will compare the classical methods of Fourier analysis with the newer methods of wavelet analysis. Given a signal, say a sound or an image, Fourier analysis easily calculates the frequencies and the amplitudes of those frequencies which make up the signal. This provides a broad overview of the characteristics of the signal, which is important for theoretical considerations. However, although Fourier inversion is possible under certain circumstances, Fourier methods are not always a good tool to recapture the signal, particularly if it is highly nonsmooth: too much Fourier information is needed to reconstruct the signal locally. In these cases, wavelet analysis is often very effective because it provides a simple approach for dealing with local aspects of a signal. Wavelet analysis also provides us with new methods for removing noise from signals that complement the classical methods of Fourier analysis. These two methodologies are major elements in a powerful set of tools for theoretical and applied analysis. This article contains many graphs of discrete signals. These graphs were created by the computer program FAWAV, A Fourier–Wavelet Analyzer, being developed by the author.

James S. Walker is professor of mathematics at the University of Wisconsin-Eau Claire. His e-mail address is [email protected]. The author would like to thank Hugo Rossi, Steven Krantz, his colleague Marc Goulet, and two anonymous reviewers for their helpful comments during the writing of this article.

658

NOTICES

OF THE

Frequency Information, Denoising As an example of the importance of frequency information, we will examine how Fourier analysis can be used for removing noise from signals. Consider a signal f (x) defined over the unit interval (where here x stands for time). The periodP1 Fourier series exi2π nx , with pansion n∈Z cn e R 1 of f is defined by cn = 0 f (x)e−i2π nx dx. Each Fourier coefficient, cn , is an amplitude associated with the frequency n of the exponential ei2π nx. Although each of these exponentials has a precise frequency, they all suffer from a complete absence of time localization in that their magnitudes, |ei2π nx | , equal 1 for all time x . To see the importance of frequency information, let us examine a problem in noise removal. In Figure 1(a)[top] we show the graph of the signal

(1)

f (x) = 2

(5 cos 2π νx) [ e−640π (x−1/8) 2

2

+ e−640π (x−3/8) + e−640π (x−4/8) 2

2

+ e−640π (x−6/8) + e−640π (x−7/8) ] where the frequency, ν , of the cosine factor is 280 . Such a signal might be used by a modem for transmitting the bit sequence 1 0 1 1 0 1 1 . The Fourier coefficients for this signal are shown in Figure 1(b)[top]. The highest magnitude coefficients are concentrated around the frequencies ±280. Suppose that when this signal is received, it is severely distorted by added noise; see Figure 1(a)[middle]. Using Fourier analysis, we can remove most of this noise. Computing the noisy signal’s Fourier coefficients, we obtain the graph shown in Figure 1(b)[middle]. The original signal’s largest magnitude Fourier coefficients are clusAMS

VOLUME 44, NUMBER 6

walker.qxp 3/24/98 2:16 PM Page 659

Figure 1. (a)[top] Signal. (b)[top] Fourier coefficients of signal. (a)[middle] Signal after adding noise. (b)[middle] Fourier coefficients of noisy signal and filter function. (b)[bottom] Fourier coefficients after multiplication by filter function. (a)[bottom] Recovered signal. tered around the frequency positions ±280. The Fourier coefficients of the added noise are localized around the origin, and they decrease in magnitude until they are essentially zero near the frequencies ±280 . Thus, the original signal’s coefficients and the noise’s coefficients are well separated. To remove the noise from the signal, we multiply the noisy signal’s coefficients by a filter function, which is 1 where the signal’s coefficients are concentrated and 0 where the noise’s coefficients are concentrated. We then recover essentially all of the signal’s coefficients; see Figure 1(b)[bottom]. Performing a Fourier series partial sum with these recovered coefficients, we obtain the denoised signal, which is shown in Figure 1(a)[bottom]. Clearly, the bit sequence 1 0 1 1 0 1 1 can now be determined from the denoised signal, and the denoised signal is a close match of the original signal. In the section “Signal Denoising” we shall look at another example of this method and also discuss how wavelets can be used for noise removal.

Signal Compression As the example above shows, Fourier analysis is very effective in problems dealing with frequency location. However, it is often very ineffective at representing functions. In particular, there are severe problems with trying to analyze transient signals using classical Fourier methods. For example, in Figure 2(a)[top] we show a discrete signal obtained from M = 1024 values {fj = F(j/M)}M−1 of the function F(x) = j=0 5

for n = − 12 M + 1, . . . , 0, . . . , 12 M . The discrete Fourier coefficients ˆ fn can be calculated by a fast Fourier transform (FFT) algorithm and are the discrete analog of the Fourier coefficients cn fn is just a for F , when fj = F(j/M) . Moreover, ˆ Riemann sum approximation of the integral that defines cn .1 The magnitudes of the discrete Fourier coefficients for this transient damp down to zero very slowly (their graph is a very wide bell-shaped curve with maximum at the origin). Consequently, to represent the transient well, one must retain most if not all of these Fourier coefficients. In Figure 2(a)[bottom] we show the results obtained from trying to compress the transient a discrete partial sum P104 by computing ˆ i2π nj using only one-fifth of the n=−104 fn e Fourier coefficients.2 Clearly, even a moderate compression ratio of 5:1 is not effective. Wavelets, however, are often very effective at representing transients. This is because they are designed to capture information over a large range of scales. A wavelet series expansion of a function f is defined by X n/2 βn ψ (2n x − k) k2 n,k∈Z

with

βn k =

Z∞ −∞

f (x) 2n/2 ψ (2n x − k) dx .

The function ψ (x) is called the wavelet, and the coefficients βn k are called the wavelet coefficients. The function 2n/2 ψ (2n x − k) is the

2

e−10 π (x−.6) . For this example, we compute the fn defined by discrete Fourier series coefficients ˆ ˆ fn = M JUNE/JULY 1997

P −1 M−1 j=0

fj

e−i2π nj/M

1For further discussion of discrete Fourier coefficients,

see [2] or [11].

P512

ˆ i2π nj , which uses all of the disn=−511 fn e crete Fourier coefficients, equals fj. 2The sum

NOTICES

OF THE

AMS

659

walker.qxp 3/24/98 2:17 PM Page 660

Figure 2. (a)[top] Signal. (a)[bottom] Fourier series using 205 coefficients, 5:1 compression. (b)[top] Wavelet coefficients for signal. (b)[bottom] Wavelet series using only largest 4% in magnitude of wavelet coefficients, 25:1 compression. wavelet shrunk by a factor of 2n if n is positive (magnified by a factor of 2−n if n is negative) and shifted by k2−n units. The factor 2n/2 in the expression 2n/2 ψ (2n x − k) preserves the L2-norm. Since the wavelet series depends on two parameters of scale and translation, it can often be very effective in analyzing signals. These parameters make it possible to analyze a signal’s behavior at a dense set of time locations and with respect to a vast range of scales, thus providing the ability to zoom in on the transient behavior of the signal. For example, let us examine the earlier transient using a discretized version of a wavelet series. We shall use a Daubechies order 4 wavelet (Daub4 for short; see the section “Daubechies Wavelets”). In Figure 2(b)[top] we show all of the 1024 wavelet coefficients of this transient and observe that most of these coefficients are close to 0 in magnitude. Consequently, by retaining only the largest magnitude coefficients for use in a wavelet series, we obtain significant compression. In Figure 2(b)[bottom] we show the reconstruction of the transient using only the top 4% in magnitude of the wavelet coefficients, a 25:1 compression ratio. Notice how accurately the transient is represented. In fact, the maximum error at all computed points is less than 9.95 × 10−14 . There is an important application here to the field of signal transmission. By transmitting only these 4% of the wavelet coefficients, the information in the signal can be transmitted 25 times faster than if we transmitted all of the original signal. This provides a considerable boost in efficiency of transmission. We shall look at more examples of compression in the section “Compression of Signals”, but first we shall describe how wavelet analysis works. 660

NOTICES

OF THE

AMS

The Haar Wavelet In order to understand how wavelet analysis works, it is best to begin with the simplest wavelet, the Haar wavelet. Let 1A (x) denote the indicator function of the set A , defined by 1A (x) = 1 if x ∈ A and 1A (x) = 0 if x ∈ / A . The ψ is defined by Haar wavelet ψ (x) = 1[0, 1 ) (x) − 1[ 1 ,1) (x) . It is 0 outside of 2

2

[0, 1), so it is well localized in time, and it satisfies Z∞ Z∞ ψ (x) dx = 0, |ψ (x)|2 dx = 1. −∞

−∞

The Haar wavelet ψ (x) is closely related to the function φ(x) defined by φ(x) = 1[0,1) (x) . This function φ(x) is called the Haar scaling function. Clearly, the Haar wavelet and scaling function satisfy the identities

ψ (x) = φ(2x) − φ(2x − 1),

(2)

φ(x) = φ(2x) + φ(2x − 1),

and the scaling function satisfies

Z∞ −∞

φ(x) dx = 1,

Z∞ −∞

|φ(x)|2 dx = 1.

The Haar wavelet ψ (x) generates the system of functions {2n/2 ψ (2n x − k)} . It is possible to show directly that {2n/2 ψ (2n x − k)} is an orthonormal basis for L2 (R), but it is more illuminating to put the discussion on an axiomatic level. This axiomatic approach leads to the Daubechies wavelets and many other wavelets as well. We begin by defining the subspaces {Vn }n∈Z of L2 (R) in the following way: VOLUME 44, NUMBER 6

walker.qxp 3/24/98 2:17 PM Page 661

n Vn = step functions in L2 (R), constant o k k+1 , ), k ∈ Z . 2n 2n This set of subspaces {Vn }n∈Z satisfies the following five axioms [6]: on the intervals [

Axioms for a Multi-Resolution Analysis (MRA) Scaling: f (x) ∈ Vn if and only if f (2x) ∈ Vn+1. Inclusion: Vn ⊂ Vn+1, for each n.   S Density: closure Vn = L2 (R) . n∈Z T Vn = {0}. Maximality: n∈Z

Basis: ∃φ(x) such that {φ(x − k)}k∈Z is an orthonormal basis for V0 . To satisfy the basis axiom, we shall use the Haar scaling function φ defined above. Then, by combining the scaling axiom with the basis axiom, we find that {2n/2 φ(2n x − k)}k∈Z is an orthonormal basis for Vn . But the totality of all these orthonormal bases, consisting of the set {2n/2 φ(2n x − k)}k,n∈Z, is not an orthonormal basis for L2 (R) because the spaces Vn are not mutually orthogonal. To remedy this difficulty, we need what are called wavelet subspaces. Define the wavelet subspace Wn to be the orthogonal complement of Vn in Vn+1 . That is, Wn satisfies the equation Vn+1 = Vn ⊕ Wn where ⊕ denotes the sum of mutually orthogonal subspaces. From the density axiom and repeated application ofLthe last equation, we obtain L2 (R) = V0 ⊕ ∞ V0 in a n=0 Wn . Decomposing L similar way, we obtain L2 (R) = n∈Z Wn . Thus, L2 (R) is an orthogonal sum of the wavelet subspaces Wn. Using (2) and the MRA axioms, it is easy to prove the following lemma.

Note that these wavelets have period 1 . Fur˜ for and thermore, ψ n < 0, n,k ≡ 0 ˜ ˜ ψn,k+2n = ψn,k for all k ∈ Z and n ≥ 0. On the in˜ terval [0, 1), the periodic Haar wavelets ψ n,k sat˜ n/2 n isfy ψn,k (x) = 2 ψ (2 x − k) for n ≥ 0 and k = 0, 1, . . . , 2n − 1. So we have the following theorem as a consequence of Theorem 1.

˜ Theorem 2. The functions 1 and ψ n,k for n ≥ 0 n and k = 0, 1, . . . , 2 − 1 are an orthonormal basis for L2 [0, 1) . Remark. In the section “Daubechies Wavelets” we will make use of periodized scaling func˜ n,k, defined by tions, φ

Fast Haar Transform The relation between the Haar scaling function φ and wavelet ψ leads to a beautiful set of relations between their coefficients as bases. Let {αkn } and {βn k } be defined by Z∞ αkn = f (x) 2n/2 φ(2n x − k) dx, −∞ Z∞ (5) βn = f (x) 2n/2 ψ (2n x − k) dx. k −∞

Substituting 2n x in place of x in the identities in (2), we obtain n+1 1 2n/2 φ(2n x) = √ [2 2 φ(2n+1 x)] 2 n+1 1 + √ [2 2 φ(2n+1 x − 1)] 2

n+1 1 2n/2 ψ (2n x) = √ [2 2 φ(2n+1 x)] 2 n+1 1 − √ [2 2 φ(2n+1 x − 1)]. 2

It then follows that

{2n/2 ψ (2n x − k)}k,n∈Z

(3)

˜ (x) = ψ n,k

X j∈Z

JUNE/JULY 1997


2n/2 ψ 2n (x + j) − k .


2n/2 φ 2n (x + j) − k

for n ≥ 0 and k = 0, 1, . . . , 2n − 1.

It follows from the scaling axiom that {2n/2 ψ (2n x − k)}k∈Z is an orthonormal basis for Wn. Therefore, since L2 (R) is the orthogonal sum of all the wavelet subspaces Wn, we have obtained the following result.

are an orthonormal basis for L2 (R). This orthonormal basis is the Haar basis for L2 (R). There is also a Haar basis for L2 [0, 1). To ˜ obtain it, we first define periodic wavelets ψ n,k by

X j∈Z

Lemma 1. The functions {ψ (x − k)}k∈Z are an orthonormal basis for the subspace W0 .

Theorem 1. The functions

˜ n,k (x) = φ

(4)

(6)

1 n+1 + αkn = √ α2k 2 1 n+1 √ α2k βn − k = 2

1 n+1 √ α2k+1 , 2 1 n+1 √ α2k+1 . 2

This result shows that the nth level coefficients st αkn and βn k are obtained from the (n + 1) level n+1 coefficients αk through multiplication by the following orthogonal matrix: NOTICES

OF THE

AMS

661

walker.qxp 3/24/98 2:17 PM Page 662

  O=

(7)

√1 2

√1 2

√1 2

−1 √ 2



cient α00 and a single wavelet coefficient β00 , and at this last step the permutation P2 is unnecessary. The complete transformation, denoted by H , satisfies

 .

Successively applying this orthogonal matrix O , we obtain {αkn } and {βn k } starting from some highest level coefficients {αkN } for some large N . Because of the density axiom, N can be chosen enough to approximate f by P large N N/2 φ(2N x − k) in L2-norm as closely k∈Z αk 2 as desired. Let us now discretize these results. Suppose that we are working with data {fj }M−1 j=0 associM−1 ated with the time values {xj = j/M}j=0 on the unit interval. If we shrink the Haar scaling function φ(x) = 1[0,1) (x) enough, it covers only the first point, x0 = 0 . Consequently, by choosing a large enough N , we may assume that our scaling coefficients {αkN } satisfy αkN = fk , for k = 0, 1, . . . , M − 1 . Assuming that M is a power of 2 , say M = 2R , it follows that N = R . The next step involves expressing the coefficient relations in (6) in a matrix form. Let A ⊕ B stand for the orthogonal  sum  of the matrices A A0 and B, that is, A ⊕ B = 0 B . Now let HM denote the M × M orthogonal matrix defined by HM = O ⊕ O ⊕ · · · ⊕ O , where the orthogonal matrix sums are applied M/2 times and O is the matrix defined in (7). Then, by applying the coefficient relations in (6) and using the fact that fk = αkR, we obtain

HM [f0 , f1 , . . . , fM−1 ]T h = α0R−1 , βR−1 , α1R−1 , βR−1 , 0 1 T , βR−1 . . . . , αR−1 1 1 2

M−1

2

2

M−1

...

, αR−1 } 1 M−1 2

2

Discrete Haar Series The fast Haar transform can be used for computing partial sums of the discretized version of the following Haar wavelet series in L2 [0, 1) :

M−1

4

} . These scaling coefficients {α0R−2 , . . . , αR−2 1 M−1

operations continue until we can no longer divide the number of components by 2 . At the R = log2 M step, we obtain a single scale coeffiNOTICES

α00 +

∞ 2X −1 X

n/2 βn ψ (2n x − k). k2

Let us assume, as in the previous section, that we have a discrete signal {fj }M−1 j=0 associated with the time values {xj = j/M}M−1 j=0 on the unit interval. Substituting these time values into (8) and restricting the upper limit of n, we obtain

OF THE

α00 CM

+

n R−1 −1 X 2X

n/2 βn ψ (2n xj − k). k CM 2

n=0 k=0

R−2 R−2 wavelet coefficients {β0 , . . . , β 1 M−1 } and

662

Therefore, the inverse operation is also an O( M ) operation.

fj =

to obtain the next level

4

· · · (H2T ⊕ IM−2 ).

M−1

If we go to the next lower level, the transformations just described are repeated, only now the matrices used are HM/2 and PM/2 , and they operate only on the M/2 -length vector

{α0R−1 ,

T T T T H −1 = HM PM (HM/2 ⊕ IM/2 ) (PM/2 ⊕ IM/2 )

n=0 k=0

 T R−1 R−1 = α0R−1 , . . . , αR−1 , β , . . . , β . 1 1 0 2

where IN is the N × N identity matrix. These matrix multiplications can be performed rapidly on a computer. Multiplying by HM requires only O(M) operations, since HM consists mostly of zeroes. Similarly, the permutation PM requires O(M) operations. Therefore, the whole transformation requires O(M) + O(M/2) + · · · + O(2) = O(M) operations. The transformation H is called a fast Haar transform. It should be noted that FFTs, which have revolutionized scientific practice during the last thirty years, are O(M log M) algorithms. Since each Hk is an orthogonal matrix, and so is every permutation Pk , it follows that H is invertible. Its inverse is

n

To sort the coefficients properly into two groups, we apply an M × M permutation matrix PM as follows:  T R−1 R−1 PM α0R−1 , βR−1 , . . . , α , β 1 1 0 M−1

· · · (PM/2 ⊕ IM/2 ) (HM/2 ⊕ IM/2 ) PM HM

(8)

M−1

2

H = (H2 ⊕ IM−2 )

AMS

The right side of this equation is just the transformation H −1 H applied to {fj } . The first part of this transformation, H {fj }, produces the 0 0 1 1 R−1 , and the coefficients α0 , β0 , β0 , β1 , . . . , β 1 2

M−1

second part, the application of H −1 , reproduces the original data {fj } . The constant CM is a scale factor which ensures that the constant vector CM and the vectors {CM 2n/2 ψ (2n xj − k)} are unit vectors in RM , using the p standard inner product. Consequently, CM = 1/M . There are many ways of forming partial sums of discretized Haar series. The simplest ones conVOLUME 44, NUMBER 6

walker.qxp 3/24/98 2:17 PM Page 663

Figure 3. (a)[top] Haar series partial sum, 229 terms. (b)[top] Fourier series partial sum, 229 terms. (a)[bottom] Haar series partial sum, 92 terms. (b)[bottom] Daub4 series partial sum, 22 terms. sist of multiplying the data by H , then setting some of the resulting coefficients equal to 0 , and then multiplying by H −1 . A widely used method involves specifying a threshold. All coefficients whose magnitudes lie below this threshold are set equal to 0 . This method is frequently used for noise removal, where coefficients whose magnitudes are significant only because of the added noise will often lie below a well-chosen threshold. We shall give an example of this in the section “Signal Denoising”. A second method keeps only the largest magnitude coefficients, while setting the rest equal to 0 . This method is convenient for making comparisons when it is known in advance how many terms are needed. We used it in the compression example in the section “Signal Compression”. A third method, which we shall call the energy method, involves specifying a fraction of the signal’s energy, where the energy is the square root of the sum of the squares of the coefficients.3 We then retain the least number of the largest magnitude coefficients whose energy exceeds this fraction of the signal’s energy and set all other coefficients equal to 0 . The energy method is useful for theoretical purposes: it is clearly helpful to be able to specify in advance what fraction of the signal’s energy is contained in a partial sum. We shall use the energy method frequently below. Let us look at an example. Suppose our signal is {fj = F(j/8192)}8191 j=0 where F(x) = x1[0,.5) (x) + (x − 1)1[.5,1) (x) . In Figure 3(a)[top] we show a Haar series partial sum, 3The Haar transform is orthogonal, so it makes sense

to specify energy in this way.

JUNE/JULY 1997

created by the energy method, which contains 99.5% of the energy of this signal. This partial sum, which used 229 coefficients out of a possible 8192, provides an acceptable visual representation of the signal. In fact, the sum of the squares of the errors is 2.0 × 10−3 . By comparison a 229 coefficient Fourier series partial sum suffers from serious drawbacks (see Figure 3(b)[top]). The sum of the squares of the errors is 4.5 × 10−1 , and there is severe oscillation and a Gibbs’ effect near x = 0.5 . Although these latter two defects could be ameliorated using other summation methods ([11], Ch. 4), there would still be a significant deviation from the original signal (especially near x = 0.5 ). This example illustrates how wavelet analysis homes right in on regions of high variability of signals and that Fourier methods try to smooth them out. The size of a function’s Fourier coefficients is related to the frequency content of the function, which is measured by integration of the function against completely unlocalized basis functions. For a function having a discontinuity, or some type of transient behavior, this produces Fourier coefficients that decrease in magnitude at a very slow rate. Consequently, a large number of Fourier coefficients are needed to accurately represent such signals.4 Wavelet series, however, use compactly supported basis functions which, at increasing levels of resolution, have rapidly decreasing supports and can zoom in on transient behavior. The transient behaviors contribute to the magnitude of only a small portion of the wavelet coefficients. 4In recent years, though, significant improvements

have been achieved using local cosine bases [3, 7, 9, 5, 1].

NOTICES

OF THE

AMS

663

walker.qxp 3/24/98 2:17 PM Page 664

Figure 4. (a) Magnitudes of highest level coefficients for a function F: [top] Haar coefficients, [middle] Daub4 coefficients; [bottom] Graph of F. (b) Sums of squares of all coefficients: [top] Haar coefficients, [middle] Daub4 coefficients; [bottom] Graph of F. Vertical scales for highest level coefficients and sums of squares are logarithmic. Consequently, a small number of wavelet coefficients are needed to accurately represent such signals. The Haar system performs well when the signal is constant over long stretches. This is because the Haar wavelet is supported on R ∞[0, 1) and satisfies a 0th order moment condition, −∞ ψ (x) dx = 0 . Therefore, if the signal {fj } is constant over an interval a ≤ xj < b such that [k2−n , (k + 1)2−n ) ⊂ [a, b) , then n the wavelet coefficient βk equals 0 . For example, suppose {fj = F(j/8192)}8191 j=0 where

F(x) = (8x − 1)1[.125,.25) (x) + 1[.25,.75) (x) + (7 − 8x)1[.75,.875) (x). In Figure 3(a)[bottom] we show a Haar series partial sum which contains 99.5% of the energy of this signal and uses only 92 coefficients out of a possible 8192. The fact that the signal is constant over three large subintervals of [0, 1) accounts for the excellent compression in this example. In order to obtain wavelet bases that provide considerably more compression, we need a compactly supported wavelet ψ (x) which has more moments equal to zero. That is, we want

(9)

Z∞ −∞

xj ψ (x) dx = 0, for j = 0, 1, . . . , L − 1

for an integer L ≥ 2 . We say that such a wavelet has its first L moments equal to zero. For example, a Daub4 wavelet has its first 2 moments equal to zero. Using a Daub4 wavelet series for the signal above, it is possible to capture 99.5% of the energy using only 22 coefficients! See Figure 3(b)[bottom]. This improvement in compression is due to the fact that the 664

NOTICES

OF THE

AMS

Daub4 R ∞wavelet is supported R ∞ on [0, 3) and satisfies −∞ ψ (x) dx = 0 and −∞ xψ (x) dx = 0. Consequently, if the signal is constant or linear over [a, b) which contains an interval [k2−n , 3(k + 1)2−n ) , then the wavelet coefficient βn k will equal 0. In Figure 4(a) we show graphs of the magnitudes of the highest level Haar coefficients and Daub4 coefficients. Each magni−12 tude |β12 k | is plotted at the x -coordinate k2 12 for k = 0, 1, . . . , 2 − 1 . These graphs show that the highest level Haar coefficients are near 0 over the constant parts of F , while the highest level Daub4 coefficients are near 0 over the constant and linear parts of F . In Figure 4(b) we show graphs of the sums of the squares of all the coefficients, which show that almost all the Daub4 coefficients are near 0 over the constant and linear parts of F , while the Haar coefficients are near 0 only over the constant parts of F . Furthermore, the largest magnitude Daub4 coefficients are concentrated around the locations of the points of nondifferentiability of F . This kind of local analysis illustrates one of the powerful features of wavelet analysis. Looking again at Figure 3(b)[bottom], we see that the most serious defects of the Daub4 compressed signal are near the points where F is nondifferentiable. If, however, we consider the interval [0.4, 0.6] where F is constant, the Daub4 compressed signal values differ from the values of F by no more than 1.2 × 10−15 at all of the 1641 discrete values of x in this interval. In contrast, a Fourier series partial sum using 23 coefficients differs by more than 10−3 at 1441 of these 1641 values of x . The Fourier series partial sum exhibits oscillations of amplitude 6.5 × 10−3 around the value 1 over this subinVOLUME 44, NUMBER 6

walker.qxp 3/24/98 2:18 PM Page 665

Figure 5. (a)[top] Signal. (b)[top] 37-term Daub4 approximation. (a)[bottom] 257-term Fourier cosine series approximation. (b)[bottom] Highest level Daub4 wavelet coefficients of signal. terval. Because the Fourier coefficients for F are only O (n−2 ) , using just the first 23 coefficients produces an oscillatory approximation to F over all of [0, 1), including the subinterval [0.4, 0.6]. The highest magnitude wavelet coefficients, however, are concentrated at the corner points for F , and their terms affect only a small portion of the partial sum (since their basis functions are compactly supported). Consequently, the wavelet series provides an extremely close approximation of F over the subinterval [0.4, 0.6]. A major defect of the Haar wavelet is its discontinuity. For one thing, it is unsatisfying to use discontinuous functions to approximate continuous ones. Even with discrete signals there can be undesirable jumps in Haar series partial sum values (see Figure 3(a)[bottom]). Therefore, we want to have a wavelet that is continuous. In the next section we will describe the Daubechies wavelets, which have their first L ≥ 2 moments equal to zero and are continuous.

Daubechies Wavelets It is possible to generalize the construction of the Haar wavelet so as to obtain a continuous scaling function φ(x) and a continuous wavelet ψ (x) . Moreover, Daubechies has shown how to make them compactly supported. We will briefly sketch the main ideas; more details can be found in [4, 5, 8, 10]. Generalizing from the case of the Haar wavelets, we require that φ(x) and ψ (x) satisfy

Z∞ (10)

−∞

Z∞

φ(x) dx = 1, |φ(x)|2 dx = 1, −∞ Z∞ |ψ (x)|2 dx = 1. −∞

JUNE/JULY 1997

The MRA axioms tell us that φ(x) must generate a subspace V0 and that V0 ⊂ V1 . Therefore,

(11)

X

φ(x) =

p ck 2 φ(2x − k)

k∈Z

for some constants {ck } . A wavelet ψ (x) , for which {ψ (x − k)} spans the wavelet subspace W0 , can be defined by5

(12)

ψ (x) =

X

p (−1)k c1−k 2 φ(2x − k).

k∈Z

Equations (11) and (12) generalize the equations in (2) for the Haar case. The orthogonality of φ and ψ leads to the following equation

X

(13)

(−1)k c1−k ck = 0.

k∈Z

This equation, and the second equation in (14) below, imply the orthogonality of the matrices, WN, used in the fast wavelet transform which we shall discuss later in this section. Combining (11) with the first two integrals in (10), it follows that

(14)

p ck = 2,

X k∈Z

X

|ck |2 = 1.

k∈Z

Similarly, assuming that L = 2 , the equations in (9) combined with (12) imply

(15)

X

(−1)k ck = 0,

k∈Z

X

k(−1)k ck = 0.

k∈Z

5 A simple proof, based on the MRA axioms, that

{ψ (x − k)} spans W0 can be found in [10].

NOTICES

OF THE

AMS

665

walker.qxp 3/24/98 2:18 PM Page 666

And, for L > 2 , equations (9) and (12) yield additional equations similar to the ones in (15). There is a finite set of coefficients that solves the equations in (14) and (15), namely, √ √ 1+ 3 3+ 3 √ , c1 = √ , c0 = 4 2 4 2 √ √ (16) 3− 3 1− 3 √ √ c2 = , c3 = 4 2 4 2

R1 ˜ n,k (x) dx . And, we periodi˜ kn by α ˜ kn = 0 f (x)φ α cally extend f with period 1 , also denoting this periodic extension by f . Then, for n ≥ 0 and k = 0, 1, . . . , 2n − 1 , we have Z1 X
n ˜k = f (x)2n/2 φ 2n (x + j) − k dx α 0

(20)

with all other ck = 0 . Using these values of ck , the following iterative solution of (11)

(17)

φ0 (x) = 1[0,1) (x), p X φn (x) = ck 2 φn−1 (2x − k), k∈Z

for n ≥ 1, converges to a continuous function φ(x) supported on [0, 3]. It then follows from (12) that the wavelet ψ (x) is also continuous and compactly supported on [0, 3]. This wavelet ψ we have been referring to as the Daub4 wavelet. The set of coefficients {ck } in (16) is the smallest set of coefficients that produce a continuous compactly supported scaling function. Other sets of coefficients, related to higher values of L , are given in [4] and [12]. Once the scaling function φ(x) and the wavelet function ψ (x) have been found, then we proceed as we did above in the Haar case. We define the coefficients {αkn } and {βn k } by the equations in (5), where now φ and ψ are the Daubechies scaling function and wavelet, respectively. The scaling identity (11) and the wavelet definition (12) yield the following coefficient relations: X n+1 αkn = cm αm+2k ,

(18)

m∈Z

βn k

=

X

n+1 (−1)m c1−m αm+2k .

m∈Z

In order to perform calculations in L2 [0, 1) , we ˜ define the periodized wavelet ψ n,k and the pe˜ riodized scaling function φn,k by equations (3) and (4), only now using the Daubechies wavelet ψ and scaling function φ in place of the Haar wavelet and scaling function. Theorem 2 remains valid using these periodic wavelets, but the proof is more involved (see section 4.5 of [5] or section 3.11 of [8]). Therefore, for each f ∈ L2 [0, 1) we can write n

(19)

f (x) =

˜ 00 α

+

∞ 2X −1 X

˜ ˜n ψ β k n,k (x),

n=0 k=0

R1

˜n = ˜ 00 = 0 f (x) dx β α where and k R1 ˜ 0 f (x)ψn,k (x) dx . We also define the coefficients 666

NOTICES

OF THE

AMS

=

j∈Z

X Z j+1

f (x − j)2n/2 φ(2n x − k) dx

j∈Z j

= αkn . ˜ n = βn and Similar arguments show that β k k n n n n ˜ n =β ˜ for n ≥ 0 and ˜ k+2n = α ˜ k and β α k+2 k k = 0, 1, . . . , 2n − 1 . After periodizing (11) and (12), it follows that X n+1 ˜ kn = ˜ m+2k α cm α , (21)

m∈Z

˜n = β k

X

n+1 ˜ m+2k (−1)m c1−m α .

m∈Z

Remark. In the section “The Haar Wavelet” we saw, for the Haar wavelet ψ , that ˜ (x) = 2n/2 ψ (2n x − k) ψ n≥0 for and n,k k = 0, 1, . . . , 2n − 1 . Almost exactly the same result holds for the Daubechies wavelets. For instance, if ψ is the Daub4 wavelet, then ψ is supported on [0, 3]. It follows, for n ≥ 2 , that on the ˜ −n ]. On unit interval ψ n,0 is supported on [0, 3 · 2 the unit interval, we then have ˜ (x) = 2n/2 ψ (2n x − k) for k = 0, 1, . . . , 2n − 3 . ψ n,k Hence, for n ≥ 2 , the periodized Daub4 wavelets ˜ 2 ψ n,k are identical in L [0, 1) with the wavelet n/2 n functions 2 ψ (2 x − k) , except when k = 2n − 2 and 2n − 1 . Similar results hold for all the Daubechies wavelets. We can discretize the series in (19) by analogy with the Haar series. The coefficient relations in (21) yield a fast wavelet transform, W , an orthogonal matrix defined by

W = (W2 ⊕ IM−2 ) · · · (PM/2 ⊕ IM/2 ) (WM/2 ⊕ IM/2 ) PM WM where each matrix WN is an N × N orthogonal matrix (as follows from (13) and the second equation in (14)). The matrix WN is used to produce αkN−1 } and the (N − 1)st level coefficients {˜ N−1 th ˜ {βk } from the N level scaling coefficients {˜ αkN } as follows:

h iT ˜ 2NN −1 ˜ 0N , . . . , α WN α h iT ˜ N−1 , . . . , α ˜ N−1 ˜ 0N−1 , β ˜ 2N−1 = α . N−1 −1 , β2N−1 −1 0 VOLUME 44, NUMBER 6

walker.qxp 3/24/98 2:18 PM Page 667

If we use the coefficients c0 , c1 , c2 , c3 defined in (16), then for N > 2 , WN has the following structure :

 c +c c +c  2 0 3 1 For N = 2, W2 = c1 +c3 −c0 −c2 . For other Daubechies wavelets, there are other finite coefficient sequences {ck } , and the matrices WN are defined similarly. The permutation matrix PN is the same one that we defined for the Haar transform and is used to sort the (N − 1)st level coefficients so that WN−1 can be applied to the αkN−1 }. scaling coefficients {˜ As initial data for the wavelet transform we can, as we did for the Haar transform, use discrete data of the form {fj }M−1 j=0 . The equations in (15) then provide a discrete analog of the zero moment conditions in (9) [for L = 2]; hence the wavelet coefficients will be 0 where the data is linear. In the last section, we saw how this can produce effective compression of signals when just the 0th and 1st moments of ψ are 0 . It is often the case that the initial data are values of a measured signal, i.e. {fk } = {F(xk )} , for xk = k2−n , where F is a signal obtained from a measurement process. As shown in the previous section, we can interpret the behavior of the discrete case based on properties of the function F . A measured signal R ∞F is often described by a convolution: F(x) = −∞ g(t)µ(x − t) dt, where g is the signal being measured and µ is called the instrument function. Such convolutions generally have greater regularity than a typical function in L2 (R). For instance, if g ∈ L1 (R) and is supported on a finite interval and µ = 1[−r ,r ] for some positive r , then F is continuous and supported on a finite interval. By a linear change of variables, we may then assume that F is supported on [0, 1]. The data {F(xk )} then provide approximations for the highest level scale coefαkR } . If we assume that F has period 1 , ficients {˜ then ˜ kR = 2−R/2 α

Z∞ −∞

  F(x) 2R φ 2R (x − xk ) dx

Z∞ −∞

F(x) 2R φ(2R (x − xk )) dx ≈ F (xk ) .

This approximation will hold for all period 1 continuous functions F and will be more accurate the larger the value of 2R = M . The higher the order of a Daubechies wavelet, the more of its moments are zero. A Daubechies wavelet of order 2L is defined by 2L nonzero coefficients {ck } , has its first L moments equal to zero, and is supported on the interval [0, 2L − 1] . Generally speaking, the more moments that are zero, the more wavelet coefficients that are nearly vanishing for smooth functions F . This follows from considering Taylor expansions. Suppose F(x) has an L term Taylor expansion about the point xk = k2−n . That is,

F(x) =

L−1 X j=0

+

1 (j) F (xk ) (x − xk )j j!

1 (L) F (tx ) (x − xk )L L!

where tx lies between x and xk . Suppose also that ψ is supported on [−a, a] and that ψ has its first L moments equal to 0 and that |F (L) (x)| is bounded by a constant B on [(k − a)2−n , (k + a)2−n ] . It then follows that

B ˜n | ≤ p |β k L + 1/2 L!

(22)



a 2n

L+1/2 .

This inequality shows why ψ (x) having zero moments produces a large number of small wavelet coefficients. If F has some smoothness on an interval ˜ n corresponding to (c, d), then wavelet coefficients β k n the basis functions ψ (2 (x − xk )) whose supports are contained in (c, d) will approach 0 rapidly as n increases to ∞ . In addition to the Daubechies wavelets, there is another class of compactly supported wavelets called coiflets. These wavelets are also constructed using the method outlined above. A coiflet of order 3L is defined by 3L nonzero coefficients {ck } and has its first L moments equal to zero and is supported on the interval [−L, 2L − 1] . A coiflet of order 3L is distinguished from a Daubechies wavelet of order 2L in that, in addition to ψ having its first L moments equal to zero, the scaling function φ for the coiflet L − 1 moments vanishing. In particular, also R ∞ has j φ(x) dx = 0, for j = 1, . . . , L − 1. For a coiflet of x −∞ order 3L , supported on [−a, a] , an argument similar to the one that proves (22) shows that

B |˜ αkR − CM F(xk )| ≤ p L + 1/2L!



a 2R

L+1/2 .

≈ CM F (xk ) .

√ Here we have replaced 1/ 2R by the scale factor CM and used JUNE/JULY 1997

This inequality provides a stronger theoretical justification for using the data {CM F(xk )} in place of NOTICES

OF THE

AMS

667

walker.qxp 3/24/98 2:18 PM Page 668

Figure 6. (a)[top] Signal. (b)[top] Signal’s coiflet30 transform, 7th level coefficients lie above the dotted line. (b)[middle] 7th level coefficients. (a)[bottom] Signal with added noise. (b)[bottom] Noisy signal’s coiflet30 transform. The horizontal lines are thresholds equal to ±0.15. the highest level scaling coefficients, beyond the argument we gave above for Daubechies wavelets. The construction of coiflets was first carried out by Daubechies and named after Coifman (who first suggested them).

Compression of Signals One of the most important applications of wavelet analysis is to the compression of signals. As an example, let us use a Daub4 series to compress the signal {fj = F(j/1024)}1023 j=0 where F(x) = − log |x − 0.2|. See Figure 5(a)[top]. For this signal, a partial sum containing 99% of the energy required only 37 coefficients (see Figure 5(b)[top]). It certainly provides a visually acceptable approximation of {fj } . In particular, the sharp maximum in the signal near x = 0.2 seems to be reproduced quite well. The compression ratio is 1024: 37 ≈ 27: 1 , which is an excellent result considering that we also have 99% accuracy. In addition, wavelet analysis has identified the singularities of F . Notice in Figure 5(b)[bottom] the peak in the wavelet coefficients is near x = 0.2 , where F has a singularity, and the largest wavelet coefficient is near x = 1 , where the periodic extension of F has a jump discontinuity. Turning to Fourier series, since the even periodic extension is continuous, we used a discrete Fourier cosine series to compress this signal. In Figure 5(a)[bottom] we show a 257-term discrete Fourier cosine series partial sum for {fj } . Even using seven times as many coefficients as the wavelet series, the cosine series cannot reproduce the sharp peak in the signal. Better results could be obtained in this case by either seg668

NOTICES

OF THE

AMS

menting the interval and performing a cosine expansion on each segment, or by using a smoother version of the same idea involving local cosine bases [3, 9, 12, 1, 5]. One way to quantify the accuracy of these approximations is to use relative R.M.S. differences. Given two sets of data {fj }M−1 j=0 and {gj }M−1 j=0 , their relative R.M.S. difference, relative to {fj } , is defined by

v v uM−1 uM−1 uX .u u uX 2 D(f , g) = t |fj − gj | t |fj |2 . j=0

j=0

For the example above, if we denote the wavelet approximation by f w , then D(f , f w ) = 9.8 × 10−3 . For the Fourier cosine series apf c , we have proximation, call it c −2 D(f , f ) = 2.7 × 10 . A rule of thumb for a visually acceptable approximation is to have a relative R.M.S. difference of less than 10−2. The approximations in this example are consistent with this rule of thumb. We can also do more localized analysis with R.M.S. differences. For example, over the subinterval [.075, .325] centered on the singularity of F , we find that D(f , f w ) = 9.7 × 10−3 and D(f , f c ) = 3.2 × 10−2. These numbers confirm our visual impression that the wavelet series does a better job reproducing the sharp peak in the signal. Or, using the subinterval [.25, .75], we get D(f , f w ) = 1.0 × 10−2 and D(f , f c ) = 3.3 × 10−3 , confirming our impression that both series do an adequate job approximating {fj } over this subinterval. VOLUME 44, NUMBER 6

walker.qxp 3/24/98 2:18 PM Page 669

Figure 7. (a)[top] Denoised signal using wavelet analysis. (a)[bottom] Denoised signal using Fourier analysis. (b)[top] Fourier coefficients of noisy signal and filter function. (b)[middle] Moduli-squared of Fourier coefficients of original signal. (b)[bottom] Moduli-squared of Fourier coefficients of wavelet denoised signal. Although in the examples we have discussed so far Fourier analysis did not compress the signals very well, we do not wish to create the impression that this will always be true. In fact, if a signal is composed of relatively few sinusoids, then Fourier analysis will provide very good compression. For example, consider the signal {fj = f (j/1024)}1023 j=0 where f (x) is defined in (1) with ν = 280 . The Fourier coefficients for f are graphed in Figure 1(b)[top]. They tend rapidly to 0 away from the frequencies ±280 ; hence the signal is composed of relatively few sinusoids. By computing a Fourier series partial sum that uses only the 122 Fourier coefficients whose frequencies are within ±30 of ±280 , we obtained a signal g that was visually indistinguishable from the original signal. In fact, D(f , g) = 5.1 × 10−3 . However, by compressing {fj } with the largest 122 Daub4 wavelet coefficients, we obtained D(f , f w ) = 2.7 × 10−1 and the compressed signal f w was only a crude approximation of the original signal. The reason that compactly supported wavelets perform poorly in this case is that the large number of rapid oscillations in the signal produce a correspondingly large number of high magnitude wavelet coefficients at the highest levels. Consequently, a significant fraction of all the wavelet coefficients are of high magnitude, so it is not possible to significantly compress the signal using compactly supported wavelets. This example illustrates that wavelet analysis is not a panacea for the problem of signal compression. In fact, much work has been done in creating large collections of wavelet bases and Fourier bases and choosing for each signal a basis which best compresses it [12, 9, 3, 5]. JUNE/JULY 1997

Signal Denoising Wavelet analysis can also be used for removing noise from signals. As an example, we show in Figure 6(a)[top] a discrete signal {f (j/1024)}1023 j=0 where f (x) is defined by formula (1) with ν = 80. Each term of the form 2

(5 cos 2π νx)e−640π (x−k/8)

we shall refer to as a blip. Notice that each blip is concentrated around x = k/8 , since 2 e−640π (x−k/8) rapidly decreases to 0 away from x = k/8 . This signal can be interpreted as representing the bit sequence 1 0 1 1 0 1 1 . In Figure 6(a)[bottom] we show this signal after it has been corrupted by adding noise. In Figure 6(b)[top] we show the coiflet30 wavelet coefficients for the original signal. The rationale for using wavelets to remove the noise is that the original signal’s wavelet coefficients are closely correlated with the points near x = k/8 where the blips are concentrated. To demonstrate this, we show in Figure 6(b)[middle] a graph of the 7th ˜7 } level wavelet coefficients {β k 27 −1 −7 corresponding to the points {k2 }k=0 on the unit interval. Comparing this to Figure 6(a)[top], we can see that the positions of this level’s largest magnitude wavelet coefficients are closely correlated with the positions of the blips. Similar graphs could also be drawn for other levels, but the 7th level coefficients have the largest magnitude. In Figure 6(b)[bottom] we show the coiflet30 transform of the noisy signal. In spite of the noise, the 7th level coefficients clearly stand out, although in a distorted form. By introducing a threshold, in this case 0.15 , we can retain these 7th level coefficients and remove NOTICES

OF THE

AMS

669

walker.qxp 3/24/98 2:18 PM Page 670

most of the noise. In Figure 7(a)[top] we show the reconstructed signal obtained by computing a partial sum using only those coefficients whose magnitudes do not fall below 0.15 . This reconstruction is not a flawless reproduction of the original signal, but nevertheless the amount of noise has been greatly reduced, and the bit sequence 1 0 1 1 0 1 1 can be determined. In Figure 7(a)[bottom] we show the denoised signal obtained by filtering the Fourier coefficients of the noisy signal (see Figure 7(b)[top]) using the method of denoising described in the section “Frequency Information, Denoising”. In contrast to the wavelet denoising, the Fourier denoising has retained a significant amount of noise in the spaces between the blips. The source of this retained noise is that most of the original noise’s Fourier coefficients are of uniform magnitude distributed across all frequencies. Consequently, the filter preserves noise coefficients corresponding to frequencies that were not present in the original signal. These coefficients generate sinusoids that oscillate across the entire interval [0, 1]. The noise’s wavelet coefficients also have almost uniform magnitude, but the thresholding process eliminates them all, except the ones modifying the 7th level coefficients of the original signal. Since these coefficients’ wavelet basis functions are compactly supported, this causes distortions in the recovered signal that are limited to neighborhoods of the positions of the 7th level coefficients. Consequently, there is still noise distorting the blips, but very little noise in between them. It is also interesting to observe that the wavelet reconstructed signal and the original signal have similar frequency content. In Figure 7(b)[middle] and Figure 7(b)[bottom], we have graphed the moduli-squared of the Fourier coefficients of the original signal and of the wavelet denoised signal, respectively. These graphs show that the frequencies of the wavelet reconstruction are, like the frequencies of the original signal, concentrated around ±80 with the highest magnitude frequencies located precisely at ±80 . This shows that the coiflet30 wavelet has the ability to extract frequency information. Much work has been done in refining this ability, including the development of another class of bases called wavelet packets [12, 9, 3, 5].

References [1] P. Auscher, G. Weiss, and M. V. Wickerhauser, Local sine and cosine basis of Coifman and Meyer and the construction of smooth wavelets, Wavelets: A Tutorial in Theory and Applications (C. K. Chui, ed.), Academic Press, 1992. [2] W. Briggs and V. E. Henson, The DFT. An owner’s manual for the Discrete Fourier Transform, SIAM, 1995. [3] R. Coifman and M. V. Wickerhauser, Wavelets and adapted waveform analysis, Wavelets: Mathematics and Applications (J. Benedetto and M. Frazier, eds.), CRC Press, 1994. [4] I. Daubechies, Ten lectures on wavelets, SIAM, 1992. [5] E. Hernandez and G. Weiss, A first course on wavelets, CRC Press, 1996. [6] S. Mallat, Multiresolution approximation and wavelet orthonormal bases of L2 (R), Trans. Amer. Math. Soc. 315 (1989), 69–87. [7] H. Malvar, Signal processing with lapped transforms, Artech House, 1992. [8] Y. Meyer, Wavelets and operators, Cambridge Univ. Press, 1992. [9] ———, Wavelets: Algorithms and applications, SIAM, 1993. [10] R. Strichartz, How to make wavelets, MAA Monthly 100, no. 6 (June–July 1993). [11] J. Walker, Fast Fourier transforms, second edition, CRC Press, 1996. [12] M. V. Wickerhauser, Adapted wavelet analysis from theory to software, IEEE and A. K. Peters, 1994.

Conclusion In this paper we have tried to show how the two methodologies of Fourier analysis and wavelet analysis are used for various kinds of work. Of course, we have only scratched the surface of both fields. Much more information can be found in the references and their bibliographies. 670

NOTICES

OF THE

AMS

VOLUME 44, NUMBER 6

1. Wavelet-based Image Coding: An Overview

23

This spectral density is shown in Figure 10. From the shape of this density we see that in order to obtain segments in which the spectrum is flat, we need to partition the spectrum finely at low frequencies, but only coarsely at high frequencies. The subbands we obtain by this procedure will be approximately vectors of white noise with variances proportional to the power spectrum over their frequency range. We can use an procedure similar to that described for the KLT for coding the output. As we will see below, this particular partition of the spectrum is closely related to the wavelet transform.

4 Wavelets: A Different Perspective 4.1

Multiresolution Analyses

The discussion so far has been motivated by probabilistic considerations. We have been assuming our images can be reasonably well-approximated by Gaussian random vectors with a particular covariance structure. The use of the wavelet transform in image coding is motivated by a rather different perspective, that of approximation theory. We assume that our images are locally smooth functions and can be well-modeled as piecewise polynomials. Wavelets provide an efficient means for approximating such functions with a small number of basis elements. This new perspective provides some valuable insights into the coding process and has motivated some significant advances. We motivate the use of the wavelet transform in image coding using the notion of a multiresolution analysis. Suppose we want to approximate a continuous-valued square-integrable function f(x) using a discrete set of values. For example, f(x) might be the brightness of a one-dimensional image. A natural set of values to use to approximate f(x) is a set of regularlyspaced, weighted local averages of f(x) such as might be obtained from the sensors in a digital camera. A simple approximation of f(x) based on local averages is a step function approximation. Let φ(x) be the box function given by φ(x) = 1 for x ∈ [0, 1) and 0 elsewhere. A step function approximation to f(x) has the form Af(x) =



fn φ(x − n),

(1.22)

n

where fn is the height of the step in [n, n + 1). A natural value for the heights fn is simply the average value of f(x) in the interval [n, n + 1).  n+1 This gives fn = n f(x)dx. We can generalize this approximation procedure to building blocks other than the box function. Our more generalized approximation will have the

24

Geoffrey M. Davis, Aria Nosratinia

form Af(x) =



˜ − n), f(x)φ(x − n). φ(x

(1.23)

n

˜ Here φ(x) is a weight function and φ(x) is an interpolating function cho˜ − n) = δ[n]. The restriction on φ(x) ensures that sen so that φ(x), φ(x our approximation will be exact when f(x) is a linear combination of the ˜ functions φ(x − n). The functions φ(x) and φ(x) are normalized so that  2 2 ˜ |φ(x)| dx = |φ(x)| dx = 1. We will further assume that f(x) is periodic with an integer period so that we only need a finite number of coefficients to specify the approximation Af(x). We can vary the resolution of our approximations by dilating and conj ˜ tracting the functions φ(x) and φ(x). Let φj (x) = 2 2 φ(2j x) and φ˜j (x) = j ˜ j x). We form the approximation Aj f(x) by projecting f(x) onto the 2 2 φ(2 span of the functions {φj (x − 2−j k)}k∈Z , computing  Aj f(x) = f(x), φ˜j (x − 2−j k)φj (x − 2−j k). (1.24) k

Let Vj be the space spanned by the functions {φj (x−2−j k)}. Our resolution j approximation Aj f is simply a projection (not necessarily an orthogonal one) of f(x) onto the span of the functions φj (x − 2−j k). For our box function example, the approximation Aj f(x) corresponds to an orthogonal projection of f(x) onto the space of step functions with step width 2−j . Figure 11 shows the difference between the coarse approximation A0 f(x) on the left and the higher resolution approximation A1 f(x) on the right. Dilating scaling functions give us a way to construct approximations to a given function at various resolutions. An important observation is that if a given function is sufficiently smooth, the differences between approximations at successive resolutions will be small. Constructing our function φ(x) so that approximations at scale j are special cases of approximations at scale j + 1 will make the analysis of differences of functions at successive resolutions much easier. The function φ(x) from our box function example has this property, since step functions with width 2−j are special cases of step functions with step width 2−j−1 . For such φ(x)’s the spaces of approximations at successive scales will be nested, i.e. we have Vj ⊂ Vj+1 . The observation that the differences Aj+1 f −Aj f will be small for smooth functions is the motivation for the Laplacian pyramid [26], a way of transforming an image into a set of small coefficients. The 1-D analog of the procedure is as follows: we start with an initial discrete representation of a function, the N coefficients of Aj f. We first split this function into the sum Aj f(x) = Aj−1 f(x) + [Aj f(x) − Aj−1 f(x)]. (1.25) Because of the nesting property of the spaces Vj , the difference Aj f(x) − Aj−1 f(x) can be represented exactly as a sum of N translates of the func-

1. Wavelet-based Image Coding: An Overview 1

1

0.9

0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1 0 0

25

0.1

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

FIGURE 11. A continuous function f (x) (plotted as a dotted line) and box function approximations (solid lines) at two resolutions. On the left is the coarse approximation A0 f (x) and on the right is the higher resolution approximation A1 f (x).

tion φj (x). The key point is that the coefficients of these φj translates will be small provided that f(x) is sufficiently smooth, and hence easy to code. Moreover, the dimension of the space Vj−1 is only half that of the space Vj , so we need only N2 coefficients to represent Aj−1 f. (In our box-function example, the function Aj−1 f is a step function with steps twice as wide as Aj f, so we need only half as many coefficients to specify Aj−1 f.) We have partitioned Aj f into N difference coefficients that are easy to code and N 2 coarse-scale coefficients. We can repeat this process on the coarse-scale coefficients, obtaining N2 easy-to-code difference coefficients and N4 coarser scale coefficients, and so on. The end result is 2N − 1 difference coefficients and a single coarse-scale coefficient. Burt and Adelson [26] have employed a two-dimensional version of the above procedure with some success for an image coding scheme. The main problem with this procedure is that the Laplacian pyramid representation has more coefficients to code than the original image. In 1-D we have twice as many coefficients to code, and in 2-D we have 43 as many.

4.2

Wavelets

We can improve on the Laplacian pyramid idea by finding a more efficient representation of the difference Dj−1 f = Aj f − Aj−1 f. The idea is that to decompose a space of fine-scale approximations Vj into a direct sum of two subspaces, a space Vj−1 of coarser-scale approximations and its complement, Wj−1 . This space Wj−1 is a space of differences between coarse and fine-scale approximations. In particular, Aj f − Aj−1 f ∈ W j−1 for any f. Elements of the space can be thought of as the additional details that must be supplied to generate a finer-scale approximation from a coarse one. Consider our box-function example. If we limit our attention to functions on the unit interval, then the space Vj is a space of dimension 2j . We can

26

Geoffrey M. Davis, Aria Nosratinia 0.5 0.4 0.3 0.2 0.1 0 −0.1 −0.2 −0.3 −0.4 −0.5 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

FIGURE 12. D f (x), the difference between the coarse approximation A0 f (x) and the finer scale approximation A1 f (x) from figure 11. φ (x) 1

ψ (x) 1

1

1

-1

-1

FIGURE 13. The Haar scaling function and wavelet.

decompose Vj into the space Vj−1 , the space of resolution 2−j+1 approximations, and Wj−1 , the space of details. Because Vj−1 is of dimension 2j−1 , Wj−1 must also have dimension 2j−1 for the combined space Vj to have dimension 2j . This observation about the dimension of Wj provides us with a means to circumvent the Laplacian pyramid’s problems with expansion. Recall that in the Laplacian pyramid we represent the difference Dj−1 f as a sum of N fine-scale basis functions φj (x). This is more information than we need, however, because the space of functions Dj−1 f is spanned by just j N −j ˜j 2 basis functions. Let ck be the expansion coefficient φ (x − 2 k), f(x) in the resolution j approximation to f(x). For our step functions, each coefficient cj−1 is the average of the coefficients cj2k and cj2k+1 from the k resolution j approximation. In order to reconstruct Aj f from Aj−1 f, we only need the N2 differences cj2k+1 − cj2k . Unlike the Laplacian pyramid, there is no expansion in the number of coefficients needed if we store these differences together with the coefficients for Aj−1 f. The differences cj2k+1 − cj2k in our box function example correspond (up to a normalizing constant) to coefficients of a basis expansion of the space of details Wj−1 . Mallat has shown that in general the basis for Wj consists of translates and dilates of a single prototype function ψ(x), called a wavelet [27]. The basis for Wj is of the form ψj (x − 2−j k) where j ψj (x) = 2 2 ψ(x).

1. Wavelet-based Image Coding: An Overview

27

Figure 13 shows the scaling function (a box function) for our box function example together with the corresponding wavelet, the Haar wavelet. Figure 12 shows the function D0 f(x), the difference between the approximations A1 f(x) and A1 f(x) from Figure 11. Note that each of the intervals separated by the dotted lines contains a translated multiple of ψ(x). The dynamic range of the differences D0 f(x) in Figure 12 is much smaller than that of A1 f(x). As a result, it is easier to code the expansion coefficients of D0 f(x) than to code those of the higher resolution approximation A1 f(x). The splitting A1 f(x) into the sum A0 f(x) + D0 f(x) performs a packing much like that done by the Karhunen-Lo`eve transform. For smooth functions f(x) the result of the splitting of A1 f(x) into a sum of a coarser approximation and details is that most of the variation is contained in A0 f, and D0 f is near zero. By repeating this splitting procedure, partitioning A0 f(x) into A−1 f(x) + D−1 f(x), we obtain the wavelet transform. The result is that an initial function approximation Aj f(x) is decomposed into the telescoping sum Aj f(x) = Dj−1 f(x) + Dj−2 f(x) + . . . + Dj−n f(x) + Aj−n f(x). (1.26) The coefficients of the differences Dj−k f(x) are easier to code than the expansion coefficients of the original approximation Aj f(x), and there is no expansion of coefficients as in the Laplacian pyramid.

4.3

Recurrence Relations

For the repeated splitting procedure above to be practical, we will need an efficient algorithm for obtaining the coefficients of the expansions Dj−k f from the original expansion coefficients for Aj f. A key property of our scaling functions makes this possible. One consequence of our partitioning of the space of resolution j approximations, Vj , into a space of resolution j − 1 approximations Vj−1 and resolution j − 1 details Wj−1 is that the scaling functions φ(x) possess self-similarity properties. Because Vj−1 ⊂ Vj , we can express the function φj−1 (x) as a linear combination of the functions φj (x − n). In particular we have  φ(x) = hk φ(2x − k). (1.27) k

Similarly, we have ˜ φ(x) =



˜hk φ(2x ˜ − k)

k

ψ(x)

=



gk φ(2x − k)

k

˜ ψ(x) =

 k

˜ g˜k φ(2x − k).

(1.28)

28

Geoffrey M. Davis, Aria Nosratinia

These recurrence relations provide the link between wavelet transforms and subband transforms. Combining ( 4.3) and ( 4.3) with ( 4.1), we obtain a simple means for splitting the N expansion coefficients for Aj f into the N2 expansion coefficients for the coarser-scale approximation Aj−1 f and the N2 coefficients for the details Dj−1 f. Both the coarser-scale approximation coefficients and the detail coefficients are obtained by convolving the coefficients of Aj f with a filter and downsampling by a factor of 2. For the coarser-scale approximation, the filter is a low-pass filter with taps given by ˜ −k . For the details, the filter is a high-pass filter with taps g˜−k . A related h derivation shows that we can invert the split by upsampling the coarserscale approximation coefficients and the detail coefficients by a factor of 2, convolving them with synthesis filters with taps hk and gk , respectively, and adding them together. We begin the forward transform with a signal representation in which we have very fine temporal localization of information but no frequency localization of information. Our filtering procedure splits our signal into lowpass and high-pass components and downsamples each. We obtain twice the frequency resolution at the expense of half of our temporal resolution. On each successive step we split the lowest frequency signal component in to a low pass and high pass component, each time gaining better frequency resolution at the expense of temporal resolution. Figure 14 shows the partition of the time-frequency plane that results from this iterative splitting procedure. As we discussed in Section 3.5, such a decomposition, with its wide subbands in the high frequencies and narrow subbands at low frequencies leads to effective data compression for a common image model, a Gaussian random process with an exponentially decaying autocorrelation function. The recurrence relations give rise to a fast algorithm for splitting a finescale function approximation into a coarser approximation and a detail function. If we start with an N coefficient expansion Aj f, the first split requires kN operations, where k depends on the lengths of the filters we use. The approximation AJ−1 has N2 coefficients, so the second split requires Initial time/ frequency localization

Time/frequency localization after first split

Time/frequency localization after second split

first high pass

first high pass

frequency

split split

second high pass

first low pass second low pass time

FIGURE 14. Partition of the time-frequency plane created by the wavelet transform.

1. Wavelet-based Image Coding: An Overview

29

k N2 operations. Each successive split requires half as much work, so the overall transform requires O(N ) work.

4.4

Wavelet Transforms vs. Subband Decompositions

The wavelet transform is a special case of a subband transform, as the derivation of the fast wavelet transform reveals. What, then, does the wavelet transform contribute to image coding? As we discuss below, the chief contribution of the wavelet transform is one of perspective. The mathematical machinery used to develop the wavelet transform is quite different than that used for developing subband coders. Wavelets involve the analysis of continuous functions whereas analysis of subband decompositions is more focused on discrete time signals. The theory of wavelets has a strong spatial component whereas subbands are more focused in the frequency domain. The subband and wavelet perspectives represent two extreme points in the analysis of this iterated filtering and downsampling process. The filters used in subband decompositions are typically designed to optimize the frequency domain behavior of a single filtering and subsampling. Because wavelet transforms involve iterated filtering and downsampling, the analysis of a single iteration is not quite what we want. The wavelet basis functions can be obtained by iterating the filtering and downsampling procedure an infinite number of times. Although in applications we iterate the filtering and downsampling procedure only a small number of times, examination of the properties of the basis functions provides considerable insight into the effects of iterated filtering. A subtle but important point is that when we use the wavelet machinery, we are implicitly assuming that the values we transform are actually fine-scale scaling function coefficients rather than samples of some function. Unlike the subband framework, the wavelet framework explicitly specifies an underlying continuous-valued function from which our initial coefficients are derived. The use of continuous-valued functions allows the use of powerful analytical tools, and it leads to a number of insights that can be used to guide the filter design process. Within the continuous-valued framework we can characterize the types of functions that can be represented exactly with a limited number of wavelet coefficients. We can also address issues such as the smoothness of the basis functions. Examination of these issues has led to important new design criteria for both wavelet filters and subband decompositions. A second important feature of the wavelet machinery is that it involves both spatial as well as frequency considerations. The analysis of subband decompositions is typically more focused on the frequency domain. Coefficients in the wavelet transform correspond to features in the underlying function in specific, well-defined locations. As we will see below, this explicit use of spatial information has proven quite valuable in motivating

30

Geoffrey M. Davis, Aria Nosratinia

some of the most effective wavelet coders.

4.5

Wavelet Properties

There is an extensive literature on wavelets and their properties. See [28], [23], or [29] for an introduction. Properties of particular interest for image compression are the the accuracy of approximation , the smoothness, and the support of these bases. The functions φ(x) and ψ(x) are the building blocks from which we construct our compressed images. When compressing natural images, which tend contain locally smooth regions, it is important that these building blocks be reasonably smooth. If the wavelets possess discontinuities or strong singularities, coefficient quantization errors will cause these discontinuities and singularities to appear in decoded images. Such artifacts are highly visually objectionable, particularly in smooth regions of images. Procedures for estimating the smoothness of wavelet bases can be found in [30] and [31]. Rioul [32] has found that under certain conditions that the smoothness of scaling functions is a more important criterion than a standard frequency selectivity criterion used in subband coding. Accuracy of approximation is a second important design criterion that has arisen from wavelet framework. A remarkable fact about wavelets is that it is possible to construct smooth, compactly supported bases that can exactly reproduce any polynomial up to a given degree. If a continuousvalued function f(x) is locally equal to a polynomial, we can reproduce that portion of f(x) exactly with just a few wavelet coefficients. The degree of the polynomials that can be reproduced exactly is determined by the ˜ number of vanishing moments of the dual wavelet  ψ(x). The dual wavelet ˜ ˜ ψ(x) has N vanishing moments provided that xk ψ(x)dx = 0 for k = 2 ˜ 0, . . . , N . Compactly supported bases for L for which ψ(x) has N vanishing moments can locally reproduce polynomials of degree N − 1. The number of vanishing moments also determines the rate of convergence of the approximations Aj f to the original function f as the resolution goes to infinity. It has been shown that f −Aj f ≤ C2−jN f (N) where N ˜ is the number of vanishing moments of ψ(x) and f (N) is the N th derivative of f [33, 34, 35]. The size of the support of the wavelet basis is another important design criterion. Suppose that the function f(x) we are transforming is equal to polynomial of degree N − 1 in some region. If ψ˜ has has N vanishing moments, then any basis function for which the corresponding dual function lies entirely in the region in which f is polynomial will have a zero coefficient. The smaller the support of ψ˜ is, the more zero coefficients we will obtain. More importantly, edges produce large wavelet coefficients. The larger ψ˜ is, the more likely it is to overlap an edge. Hence it is important that our wavelets have reasonably small support.

1. Wavelet-based Image Coding: An Overview

31

There is a tradeoff between wavelet support and the regularity and accuracy of approximation. Wavelets with short support have strong constraints on their regularity and accuracy of approximation, but as the support is increased they can be made to have arbitrary degrees of smoothness and numbers of vanishing moments. This limitation on support is equivalent to keeping the analysis filters short. Limiting filter length is also an important consideration in the subband coding literature, because long filters lead to ringing artifacts around edges.

5 A Basic Wavelet Image Coder State-of-the-art wavelet coders are all derived from the transform coder paradigm. There are three basic components that underly current wavelet coders: a decorrelating transform, a quantization procedure, and an entropy coding procedure. Considerable current research is being performed on all three of these components. Before we discuss state-of-the-art coders in the next sections, we will describe a basic wavelet transform coder and discuss optimized versions of each of the components.7

5.1

Choice of Wavelet Basis

Deciding on the optimal wavelet basis to use for image coding is a difficult problem. A number of design criteria, including smoothness, accuracy of approximation, size of support, and filter frequency selectivity are known to be important. However, the best combination of these features is not known. The simplest form of wavelet basis for images is a separable basis formed from translations and dilations of products of one dimensional wavelets. Using separable transforms reduces the problem of designing efficient wavelets to a one-dimensional problem, and almost all current coders employ separable transforms. Recent work of Sweldens and Kovaˇcevi´c [36] simplifies considerably the design of non-separable bases, and such bases may prove more efficient than separable transforms. The prototype basis functions for separable transforms are φ(x)φ(y), φ(x)ψ(y), ψ(x)φ(y), and ψ(x)ψ(y). Each step of the transform for such bases involves two frequency splits instead of one. Suppose we have an N × N image. First each of the N rows in the image is split into a lowpass half and a high pass half. The result is an N × N2 sub-image and an N × N2 high-pass sub-image. Next each column of the sub-images is split into a low-pass and a high-pass half. The result is a four-way partition 7 C++ source code for a coder that implements these components is available from the web site http://www.cs.dartmouth.edu/∼gdavis/wavelet/wavelet.html.

32

Geoffrey M. Davis, Aria Nosratinia

of the image into horizontal low-pass/vertical low-pass, horizontal highpass/vertical low-pass, horizontal low-pass/vertical high-pass, and horizontal high-pass/vertical high-pass sub-images. The low-pass/low-pass subimage is subdivided in the same manner in the next step as is illustrated in Figure 17. Unser [35] shows that spline wavelets are attractive for coding applications based on approximation theoretic considerations. Experiments by Rioul [32] for orthogonal bases indicate that smoothness is an important consideration for compression. Experiments by Antonini et al [37] find that both vanishing moments and smoothness are important, and for the filters tested they found that smoothness appeared to be slightly more important than the number of vanishing moments. Nonetheless, Vetterli and Herley [38] state that “the importance of regularity for signal processing applications is still an open question.” The bases most commonly used in practice have between one and two continuous derivatives. Additional smoothness does not appear to yield significant improvements in coding results. Villasenor et al [39] have systematically examined all minimum order biorthogonal filter banks with lengths ≤ 36. In addition to the criteria already mentioned, [39] also examines measures of oscillatory behavior and of the sensitivity of the coarse-scale approximations Aj f(x) to translations of the function f(x). The best filter found in these experiments was a 7/9tap spline variant with less dissimilar lengths from [37], and this filter is one of the most commonly used in wavelet coders. There is one caveat with regard to the results of the filter evaluation in [39]. Villasenor et al compare peak signal to noise ratios generated by a simple transform coding scheme. The bit allocation scheme they use works well for orthogonal bases, but it can be improved upon considerably in the biorthogonal case. This inefficient bit allocation causes some promising biorthogonal filter sets to be overlooked. For biorthogonal transforms, the squared error in the transform domain is not the same as the squared error in the original image. As a result, the problem of minimizing image error is considerably more difficult than in the orthogonal case. We can reduce image-domain errors by performing bit allocation using a weighted transform-domain error measure that we discuss in section 5.5. A number of other filters yield performance comparable to that of the 7/9 filter of [37] provided that we do bit allocation with a weighted error measure. One such basis is the Deslauriers-Dubuc interpolating wavelet of order 4 [40, 41], which has the advantage of having filter taps that are dyadic rationals. Both the spline wavelet of [37] and the order 4 Deslauriers-Dubuc wavelet have 4 vanishing moments in both ψ(x) ˜ andψ(x), and the basis functions have just under 2 continuous derivatives in the L2 sense. One new very promising set of filters has been developed by Balasingham and Ramstad [42]. Their design procedure combines classical filter design

1. Wavelet-based Image Coding: An Overview

33

Dead zone x x=0

FIGURE 15. Dead-zone quantizer, with larger encoder partition around x = 0 (dead zone) and uniform quantization elsewhere.

techniques with ideas from wavelet constructions and yields filters that perform significantly better than the popular 7/9 filter set from [37].

5.2

Boundaries

Careful handling of image boundaries when performing the wavelet transform is essential for effective compression algorithms. Naive techniques for artificially extending images beyond given boundaries such as periodization or zero-padding lead to significant coding inefficiencies. For symmetrical wavelets an effective strategy for handling boundaries is to extend the image via reflection. Such an extension preserves continuity at the boundaries and usually leads to much smaller wavelet coefficients than if discontinuities were present at the boundaries. Brislawn [43] describes in detail procedures for non-expansive symmetric extensions of boundaries. An alternative approach is to modify the filter near the boundary. Boundary filters [44, 45] can be constructed that preserve filter orthogonality at boundaries. The lifting scheme [46] provides a related method for handling filtering near the boundaries.

5.3

Quantization

Most current wavelet coders employ scalar quantization for coding. There are two basic strategies for performing the scalar quantization stage. If we knew the distribution of coefficients for each subband in advance, the optimal strategy would be to use entropy-constrained Lloyd-Max quantizers for each subband. In general we do not have such knowledge, but we can provide a parametric description of coefficient distributions by sending side information. Coefficients in the high pass subbands of a wavelet transform are known a priori to be distributed as generalized Gaussians [27] centered around zero. A much simpler quantizer that is commonly employed in practice is a uniform quantizer with a dead zone. The quantization bins, as shown in Figure 15, are of the form [n∆, (n + 1)∆) for n ∈ Z except for the central bin [−∆, ∆). Each bin is decoded to the value at its center in the simplest case, or to the centroid of the bin. In the case of asymptotically high rates, uniform quantization is optimal [47]. Although in practical regimes these dead-zone quantizers are suboptimal, they work almost as well as Lloyd-

34

Geoffrey M. Davis, Aria Nosratinia

Max coders when we decode to the bin centroids [48]. Moreover, dead-zone quantizers have the advantage that of being very low complexity and robust to changes in the distribution of coefficients in source. An additional advantage of these dead-zone quantizers is that they can be nested to produce an embedded bitstream following a procedure in [49].

5.4

Entropy Coding

Arithmetic coding provides a near-optimal entropy coding for the quantized coefficient values. The coder requires an estimate of the distribution of quantized coefficients. This estimate can be approximately specified by providing parameters for a generalized Gaussian or a Laplacian density. Alternatively the probabilities can be estimated online. Online adaptive estimation has the advantage of allowing coders to exploit local changes in image statistics. Efficient adaptive estimation procedures are discussed in [50] and [51]. Because images are not jointly Gaussian random processes, the transform coefficients, although decorrelated, still contain considerable structure. The entropy coder can take advantage of some of this structure by conditioning the encodings on previously encoded values. A coder of [49] obtains modest performance improvements using such a technique.

5.5

Bit Allocation

The final question we need to address is that of how finely to quantize each subband. As we discussed in Section 3.2, the general idea is to determine the number of bits bj to devote to coding subband j so that the total distortion j Dj (bj ) is minimized subject to the constraint that j bj ≤ b. Here Dj (b) is the amount of distortion incurred in coding subband j with b bits. When the functions Dj (b) are known in closed form we can solve the problem using the Kuhn-Tucker conditions. One common practice is to approximate the functions Dj (b) with the rate-distortion function for a Gaussian random variable. However, this approximation is not very accurate at low bit rates. Better results may be obtained by measuring Dj (b) for a range of values of b and then solving the constrained minimization problem using integer programming techniques. An algorithm of Shoham and Gersho [52] solves precisely this problem. For biorthogonal wavelets we have the additional problem that squared error in the transform domain is not equal to squared error in the inverted image. Moulin [53] has formulated a multiscale relaxation algorithm which provides an approximate solution to the allocation problem for this case. Moulin’s algorithm yields substantially better results than the naive approach of minimizing squared error in the transform domain. A simpler approach is to approximate the squared error in the image by weighting the squared errors in each subband. The weight wj for subband

1. Wavelet-based Image Coding: An Overview

35

j is obtained as follows: we set a single coefficient in subband j to 1 and set all other wavelet coefficients to zero. We then invert this transform. The weight wj is equal to the sum of the squares of the values in the resulting inverse transform. We allocate bits by minimizing the weighted sum w D (b ) rather than the sum j j j j j Dj (bj ). Further details may be found in Naveen and Woods [54]. This weighting procedure results in substantial coding improvements when using wavelets that are not very close to being orthogonal, such as the Deslauriers-Dubuc wavelets popularized by the lifting scheme [46]. The 7/9 tap filter set of [37], on the other hand, has weights that are all nearly 1, so this weighting provides little benefit.

5.6

Perceptually Weighted Error Measures

Our goal in lossy image coding is to minimize visual discrepancies between the original and compressed images. Measuring visual discrepancy is a difficult task. There has been a great deal of research on this problem, but because of the great complexity of the human visual system, no simple, accurate, and mathematically tractable measure has been found. Our discussion up to this point has focused on minimizing squared error distortion in compressed images primarily because this error metric is mathematically convenient. The measure suffers from a number of deficits, however. For example, consider two images that are the same everywhere except in a small region. Even if the difference in this small region is large and highly visible, the mean squared error for the whole image will be small because the discrepancy is confined to a small region. Similarly, errors that are localized in straight lines, such as the blocking artifacts produced by the discrete cosine transform, are much more visually objectionable than squared error considerations alone indicate. There is evidence that the human visual system makes use of a multiresolution image representation; see [55] for an overview. The eye is much more sensitive to errors in low frequencies than in high. As a result, we can improve the correspondence between our squared error metric and perceived error by weighting the errors in different subbands according to the eye’s contrast sensitivity in a corresponding frequency range. Weights for the commonly used 7/9-tap filter set of [37] have been computed by Watson et al in [56].

6 Extending the Transform Coder Paradigm The basic wavelet coder discussed in Section 5 is based on the basic transform coding paradigm, namely decorrelation and compaction of energy into a few coefficients. The mathematical framework used in deriving the wavelet transform motivates compression algorithms that go beyond the traditional

36

Geoffrey M. Davis, Aria Nosratinia

mechanisms used in transform coding. These important extensions are at the heart of modern wavelet coding algorithms of Sections 7 and 9. We take a moment here to discuss these extensions. Conventional transform coding relies on energy compaction in an ordered set of transform coefficients, and quantizes those coefficients with a priority according to their order. This paradigm, while quite powerful, is based on several assumptions about images that are not always completely accurate. In particular, the Gaussian assumption breaks down for the joint distributions across image discontinuities. Mallat and Falzon [57] give the following example of how the Gaussian, high-rate analysis breaks down at low rates for non-Gaussian processes. Let Y [n] be a random N -vector defined by   X if n = P Y [n] = X if n = P + 1(modN ) (1.29)  0 otherwise Here P is a random integer uniformly distributed between 0 and N − 1 and X is a random variable that equals 1 or -1 each with probability 12 . X and P are independent. The vector Y has zero mean and a covariance matrix with entries  2  N for n = m 1 E{Y [n]Y [m]} = for |n − m| ∈ {1, N − 1} (1.30)  N 0 otherwise The covariance matrix is circulant, so the KLT for this process is the simply the Fourier transform. The Fourier transform of Y is a very inefficient reprek sentation for coding Y . The energy at frequency k will be |1+e2πi N |2 which means that the energy of Y is spread out over the entire low-frequency half of the Fourier basis with some spill-over into the high-frequency half. The KLT has “packed” the energy of the two non-zero coefficients of Y into roughly N2 coefficients. It is obvious that Y was much more compact in its original form, and could be coded better without transformation: Only two coefficients in Y are non-zero, and we need only specify the values of these coefficients and their positions. As suggested by the example above, the essence of the extensions to traditional transform coding is the idea of selection operators. Instead of quantizing the transform coefficients in a pre-determined order of priority, the wavelet framework lends itself to improvements, through judicious choice of which elements to code. This is made possible primarily because wavelet basis elements are spatially as well as spectrally compact. In parts of the image where the energy is spatially but not spectrally compact (like the example above) one can use selection operators to choose subsets of the wavelet coefficients that represent that signal efficiently. A most notable example is the Zerotree coder and its variants (Section 7).

1. Wavelet-based Image Coding: An Overview

37

More formally, the extension consists of dropping the constraint of linear image approximations, as the selection operator is nonlinear. The work of DeVore et al. [58] and of Mallat and Falzon [57] suggests that at low rates, the problem of image coding can be more effectively addressed as a problem in obtaining a non-linear image approximation. This idea leads to some important differences in coder implementation compared to the linear framework. For linear approximations, Theorems 3.1 and 3.3 in Section 3.1 suggest that at low rates we should approximate our images using a fixed subset of the Karhunen-Lo`eve basis vectors. We set a fixed set of transform coefficients to zero, namely the coefficients corresponding to the smallest eigenvalues of the covariance matrix. The non-linear approximation idea, on the other hand, is to approximate images using a subset of basis functions that are selected adaptively based on the given image. Information describing the particular set of basis functions used for the approximation, called a significance map, is sent as side information. In Section 7 we describe zerotrees, a very important data structure used to efficiently encode significance maps. Our example suggests that a second important assumption to relax is that our images come from a single jointly Gaussian source. We can obtain better energy packing by optimizing our transform to the particular image at hand rather than to the global ensemble of images. The KLT provides efficient variance packing for vectors drawn from a single Gaussian source. However, if we have a mixture of sources the KLT is considerably less efficient. Frequency-adaptive and space/frequency-adaptive coders decompose images over a large library of different bases and choose an energy-packing transform that is adapted to the image itself. We describe these adaptive coders in Section 8. Trellis coded quantization represents a more drastic departure from the transform coder framework. While TCQ coders operate in the transform domain, they effectively do not use scalar quantization. Trellis coded quantization captures not only correlation gain and fractional bitrates, but also the packing gain of VQ. In both performance and complexity, TCQ is essentially VQ in disguise. The selection operator that characterizes the extension to the transform coder paradigm generates information that needs to be conveyed to the decoder as “side information”. This side information can be in the form of zerotrees, or more generally energy classes. Backward mixture estimation represents a different approach: it assumes that the side information is largely redundant and can be estimated from the causal data. By cutting down on the transmitted side information, these algorithms achieve a remarkable degree of performance and efficiency. For reference, Table 1.1 provides a comparison of the peak signal to

38

Geoffrey M. Davis, Aria Nosratinia

TABLE 1.1. Peak signal to noise ratios in decibels for coders discussed in the paper. Higher values indicate better performance. Lena (bits/pixel) Barbara (bits/pixel) Type of Coder 1.0 0.5 0.25 1.0 0.5 0.25 JPEG [59] 37.9 34.9 31.6 33.1 28.3 25.2 Optimized JPEG [60] 39.6 35.9 32.3 35.9 30.6 26.7 Baseline Wavelet [61] 39.4 36.2 33.2 34.6 29.5 26.6 Zerotree (Shapiro) [62] 39.6 36.3 33.2 35.1 30.5 26.8 Zerotree (Said & Pearlman) [63] 40.5 37.2 34.1 36.9 31.7 27.8 Zerotree (R/D optimized) [64] 40.5 37.4 34.3 37.0 31.3 27.2 Frequency-adaptive [65] 39.3 36.4 33.4 36.4 31.8 28.2 Space-frequency adaptive [66] 40.1 36.9 33.8 37.0 32.3 28.7 Frequency-adaptive + zerotrees [67] 40.6 37.4 34.4 37.7 33.1 29.3 TCQ subband [68] 41.1 37.7 34.3 – – – Bkwd. mixture estimation (EQ) [69] 40.9 37.7 34.6 – – –

noise ratios for the coders we discuss in the paper.8 The test images are the 512×512 Lena image and the 512×512 Barbara image. Figure 16 shows the Barbara image as compressed by JPEG, a baseline wavelet transform coder, and the zerotree coder of Said and Pearlman [63]. The Barbara image is particularly difficult to code, and we have compressed the image at a low rate to emphasize coder errors. The blocking artifacts produced by the discrete cosine transform are highly visible in the image on the top right. The difference between the two wavelet coded images is more subtle but quite visible at close range. Because of the more efficient coefficient encoding (to be discussed below), the zerotree-coded image has much sharper edges and better preserves the striped texture than does the baseline transform coder.

7 Zerotree Coding The rate-distortion analysis of the previous sections showed that optimal bitrate allocation is achieved when the signal is divided into subbands such that each subband contains a “white” signal. It was also shown that for typical signals of interest, this leads to narrower bands in the low frequencies and wider bands in the high frequencies. Hence, wavelet transforms have very good energy compaction properties. This energy compaction leads to efficient utilization of scalar quantizers. However, a cursory examination of the transform in Figure 17 shows that a significant amount of structure is present, particularly in the fine scale coefficients. Wherever there is structure, there is room for compression, and 8 More

current numbers may be found on the web at http://www.icsl.ucla.edu/∼ipl/psnr results.html

1. Wavelet-based Image Coding: An Overview

39

FIGURE 16. Results of different compression schemes for the 512 × 512 Barbara test image at 0.25 bits per pixel. Top left: original image. Top right: baseline JPEG, PSNR = 24.4 dB. Bottom left: baseline wavelet transform coder [61], PSNR = 26.6 dB. Bottom right: Said and Pearlman zerotree coder, PSNR = 27.6 dB.

40

Geoffrey M. Davis, Aria Nosratinia

advanced wavelet compression algorithms all address this structure in the higher frequency subbands. One of the most prevalent approaches to this problem is based on exploiting the relationships of the wavelet coefficients across bands. A direct visual inspection indicates that large areas in the high frequency bands have little or no energy, and the small areas that have significant energy are similar in shape and location, across different bands. These high-energy areas stem from poor energy compaction close to the edges of the original image. Flat and slowly varying regions in the original image are well-described by the low-frequency basis elements of the wavelet transform (hence leading to high energy compaction). At the edge locations, however, low-frequency basis elements cannot describe the signal adequately, and some of the energy leaks into high-frequency coefficients. This happens similarly at all scales, thus the high-energy high-frequency coefficients representing the edges in the image have the same shape. Our a priori knowledge that images of interest are formed mainly from flat areas, textures, and edges, allows us to take advantage of the resulting cross-band structure. Zerotree coders combine the idea of cross-band correlation with the notion of coding zeros jointly (which we saw previously in the case of JPEG), to generate very powerful compression algorithms. The first instance of the implementation of zerotrees is due to Lewis and Knowles [70]. In their algorithm the image is represented by a treestructured data construct (Figure 18). This data structure is implied by a dyadic discrete wavelet transform (Figure 19) in two dimensions. The root node of the tree represents the scaling function coefficient in the lowest frequency band, which is the parent of three nodes. Nodes inside the tree correspond to wavelet coefficients at a scale determined by their height in the tree. Each of these coefficients has four children, which correspond to the wavelets at the next finer scale having the same location in space. These four coefficients represent the four phases of the higher resolution basis elements at that location. At the bottom of the data structure lie the leaf nodes, which have no children. Note that there exist three such quadtrees for each coefficient in the low frequency band. Each of these three trees corresponds to one of three filtering orderings: there is one tree consisting entirely of coefficients arising from horizontal high-pass, vertical low-pass operation (HL); one for horizontal low-pass, vertical high-pass (LH), and one for high-pass in both directions (HH). The zerotree quantization model used by Lewis and Knowles was arrived at by observing that often when a wavelet coefficient is small, its children on the wavelet tree are also small. This phenomenon happens because significant coefficients arise from edges and texture, which are local. It is not difficult to see that this is a form of conditioning. Lewis and Knowles took this conditioning to the limit, and assumed that insignificant parent nodes always imply insignificant child nodes. A tree or subtree that contains (or

1. Wavelet-based Image Coding: An Overview

FIGURE 17. Wavelet transform of the image “Lena.”

LL3

LH3

HL3

HH3

LH2 LH1 HH2 HL2

HH1 HL1 Typical Wavelet tree

FIGURE 18. Space-frequency structure of wavelet transform

41

42

Geoffrey M. Davis, Aria Nosratinia H

2

H

2

0

H

0

Input Signal

H

0

1

1

H

1

2

2

H

Low frequency Component

2

Detail Components

2

FIGURE 19. Filter bank implementing a discrete wavelet transform

is assumed to contain) only insignificant coefficients is known as a zerotree. Lewis and Knowles used the following algorithm for the quantization of wavelet coefficients: Quantize each node according to an optimal scalar quantizer for the Laplacian density. If the node value is insignificant according to a pre-specified threshold, ignore all its children. These ignored coefficients will be decoded as zeros at the decoder. Otherwise, go to each of its four children and repeat the process. If the node was a leaf node and did not have a child, go to the next root node and repeat the process. Aside from the nice energy compaction properties of the wavelet transform, the Lewis and Knowles coder achieves its compression ratios by joint coding of zeros. For efficient run-length coding, one needs to first find a conducive data structure, e.g. the zig-zag scan in JPEG. Perhaps the most significant contribution of this work was to realize that wavelet domain data provide an excellent context for run-length coding: not only are large run lengths of zeros generated, but also there is no need to transmit the length of zero runs, because they are assumed to automatically terminate at the leaf nodes of the tree. Much the same as in JPEG, this is a form of joint vector/scalar quantization. Each individual (significant) coefficient is quantized separately, but the symbols corresponding to small coefficients in fact are representing a vector consisting of that element and the zero run that follows it to the bottom of the tree. While this compression algorithm generates subjectively acceptable images, its rate-distortion performance falls short of baseline JPEG, which at the time was often used for comparison purposes. The lack of sophistication in the entropy coding of quantized coefficients somewhat disadvantages this coder, but the main reason for its mediocre performance is the way it generates and recognizes zerotrees. As we have noted, whenever a coefficient is small, it is likely that its descendents are also insignificant. However, the Lewis and Knowles algorithm assumes that small parents always have small descendents, and therefore suffers large distortions when this does not hold because it zeros out large coefficients. The advantage of this method is that the detection of zerotrees is automatic: zerotrees are determined by measuring the magnitude of known coefficients. No side information is required to specify the locations of zerotrees, but this simplicity is obtained at the cost of reduced performance. More detailed analysis of this tradeoff gave

1. Wavelet-based Image Coding: An Overview

43

rise to the next generation of zerotree coders.

7.1

The Shapiro and Said-Pearlman Coders

The Lewis and Knowles algorithm, while capturing the basic ideas inherent in many of the later coders, was incomplete. It had all the intuition that lies at the heart of more advanced zerotree coders, but did not efficiently specify significance maps, which is crucial to the performance of wavelet coders. A significance map is a binary function whose value determines whether each coefficient is significant or not. If not significant, a coefficient is assumed to quantize to zero. Hence a decoder that knows the significance map needs no further information about that coefficient. Otherwise, the coefficient is quantized to a non-zero value. The method of Lewis and Knowles does not generate a significance map from the actual data, but uses one implicitly, based on a priori assumptions on the structure of the data. On the infrequent occasions when this assumption does not hold, a high price is paid in terms of distortion. The methods to be discussed below make use of the fact that, by using a small number of bits to correct mistakes in our assumptions about the occurrences of zerotrees, we can reduce the coded image distortion considerably. The first algorithm of this family is due to Shapiro [71] and is known as the embedded zerotree wavelet (EZW) algorithm. Shapiro’s coder was based on transmitting both the non-zero data and a significance map. The bits needed to specify a significance map can easily dominate the coder output, especially at lower bitrates. However, there is a great deal of redundancy in a general significance map for visual data, and the bitrates for its representation can be kept in check by conditioning the map values at each node of the tree on the corresponding value at the parent node. Whenever an insignificant parent node is observed, it is highly likely that the descendents are also insignificant. Therefore, most of the time, a “zerotree” significance map symbol is generated. But because p, the probability of this event, is close to 1, its information content, −p log p, is very small. So most of the time, a very small amount of information is transmitted, and this keeps the average bitrate needed for the significance map relatively small. Once in a while, one or more of the children of an insignificant node will be significant. In that case, a symbol for “isolated zero” is transmitted. The likelihood of this event is lower, and thus the bitrate for conveying this information is higher. But it is essential to pay this price to avoid losing significant information down the tree and therefore generating large distortions. In summary, the Shapiro algorithm uses three symbols for significance maps: zerotree, isolated zero, or significant value. But using this structure, and by conditionally entropy coding these symbols, the coder achieves very

Geoffrey M. Davis, Aria Nosratinia

0

0

0

0

0

0

0

x

x

1

1

x

x

x

x

x

x

x

x

x

x

0

0

0

1

0

0

0

0

0

0

0

0

0

0

0

0

....

x

1

1

1

1

0

0

1

0

0

0

0

0

....

x

x

x

x

x

1

1

1

1

1

1

0

0

....

......

x

0

....

0

......

0

......

0

......

0

......

1

......

1

......

LSB

1

......

MSB

......

44

Wavelet Coefficients in scan order

FIGURE 20. Bit plane profile for raster scan ordered wavelet coefficients.

good rate-distortion performance. In addition, Shapiro’s coder also generates an embedded code. Coders that generate embedded codes are said to have the progressive transmission or successive refinement property. Successive refinement consists of first approximating the image with a few bits of data, and then improving the approximation as more and more information is supplied. An embedded code has the property that for two given rates R1 > R2 , the rate-R2 code is a prefix to the rate-R1 code. Such codes are of great practical interest for the following reasons: • The encoder can easily achieve a precise bitrate by continuing to output bits when it reaches the desired rate. • The decoder can cease decoding at any given point, generating an image that is the best representation possible with the decoded number of bits. This is of practical interest for broadcast applications where multiple decoders with varying computational, display, and bandwidth capabilities attempt to receive the same bitstream. With an embedded code, each receiver can decode the passing bitstream according to its particular needs and capabilities. • Embedded codes are also very useful for indexing and browsing, where only a rough approximation is sufficient for deciding whether the image needs to be decoded or received in full. The process of screening images can be speeded up considerably by using embedded codes: after decoding only a small portion of the code, one knows if the target image is present. If not, decoding is aborted and the next image is requested, making it possible to screen a large number of images quickly. Once the desired image is located, the complete image is decoded. Shapiro’s method generates an embedded code by using a bit-slice approach (see Figure 20). First, the wavelet coefficients of the image are

1. Wavelet-based Image Coding: An Overview

45

indexed into a one-dimensional array, according to their order of importance. This order places lower frequency bands before higher frequency bands since they have more energy, and coefficients within each band appear in a raster scan order. The bit-slice code is generated by scanning this one-dimensional array, comparing each coefficient with a threshold T . This initial scan provides the decoder with sufficient information to recover the most significant bit slice. In the next pass, our information about each coefficient is refined to a resolution of T /2, and the pass generates another bit slice of information. This process is repeated until there are no more slices to code. Figure 20 shows that the upper bit slices contain a great many zeros because there are many coefficients below the threshold. The role of zerotree coding is to avoid transmitting all these zeros. Once a zerotree symbol is transmitted, we know that all the descendent coefficients are zero, so no information is transmitted for them. In effect, zerotrees are a clever form of run-length coding, where the coefficients are ordered in a way to generate longer run lengths (more efficient) as well as making the runs self-terminating, so the length of the runs need not be transmitted. The zerotree symbols (with high probability and small code length) can be transmitted again and again for a given coefficient, until it rises above the sinking threshold, at which point it will be tagged as a significant coefficient. After this point, no more zerotree information will be transmitted for this coefficient. To achieve embeddedness, Shapiro uses a clever method of encoding the sign of the wavelet coefficients with the significance information. There are also further details of the priority of wavelet coefficients, the bit-slice coding, and adaptive arithmetic coding of quantized values (entropy coding), which we will not pursue further in this review. The interested reader is referred to [71] for more details. Said and Pearlman [72] have produced an enhanced implementation of the zerotree algorithm, known as Set Partitioning in Hierarchical Trees (SPHIT). Their method is based on the same premises as the Shapiro algorithm, but with more attention to detail. The public domain version of this coder is very fast, and improves the performance of EZW by 0.3-0.6 dB. This gain is mostly due to the fact that the original zerotree algorithms allow special symbols only for single zerotrees, while in reality, there are other sets of zeros that appear with sufficient frequency to warrant special symbols of their own. In particular, the Said-Pearlman coder provides symbols for combinations of parallel zerotrees. Davis and Chawla [73] have shown that both the Shapiro and the Said and Pearlman coders are members of a large family of tree-structured significance mapping schemes. They provide a theoretical framework that explains in more detail the performance of these coders and describe an algorithm for selecting a member of this family of significance maps that is optimized for a given image or class of images.

46

Geoffrey M. Davis, Aria Nosratinia

7.2

Zerotrees and Rate-Distortion Optimization

In the previous coders, zerotrees were used only when they were detected in the actual data. But consider for the moment the following hypothetical example: assume that in an image, there is a wide area of little activity, so that in the corresponding location of the wavelet coefficients there exists a large group of insignificant values. Ordinarily, this would warrant the use of a big zerotree and a low expenditure of bitrate over that area. Suppose, however, that there is a one-pixel discontinuity in the middle of the area, such that at the bottom of the would-be zerotree, there is one significant coefficient. The algorithms described so far would prohibit the use of a zerotree for the entire area. Inaccurate representation of a single pixel will change the average distortion in the image only by a small amount. In our example we can gain significant coding efficiency by ignoring the single significant pixel so that we can use a large zerotree. We need a way to determine the circumstances under which we should ignore significant coefficients in this manner. The specification of a zerotree for a group of wavelet coefficient is a form of quantization. Generally, the values of the pixels we code with zerotrees are non-zero, but in using a zerotree we specify that they be decoded as zeros. Non-zerotree wavelet coefficients (significant values) are also quantized, using scalar quantizers. If we saves bitrate by specifying larger zerotrees, as in the hypothetical example above, the rate that was saved can be assigned to the scalar quantizers of the remaining coefficients, thus quantizing them more accurately. Therefore, we have a choice in allocating the bitrate among two types of quantization. The question is, if we are given a unit of rate to use in coding, where should it be invested so that the corresponding reduction in distortion is maximized? This question, in the context of zerotree wavelet coding, was addressed by Xiong et al. [74], using well-known bit allocation techniques [1]. The central result for optimal bit allocation states that, in the optimal state, the slope of the operational rate-distortion curves of all quantizers are equal. This result is intuitive and easy to understand. The slope of the operational rate-distortion function for each quantizer tells us how many units of distortion we add/eliminate for each unit of rate we eliminate/add. If one of the quantizers has a smaller R-D slope, meaning that it is giving us less distortion reduction for our bits spent, we can take bits away from this quantizer (i.e. we can reduce its step size) and give them to the other, more efficient quantizers. We continue to do so until all quantizers have an equal slope. Obviously, specification of zerotrees affects the quantization levels of nonzero coefficients because total available rate is limited. Conversely, specifying quantization levels will affect the choice of zerotrees because it affects the incremental distortion between zerotree quantization and scalar quantization. Therefore, an iterative algorithm is needed for rate-distortion opti-

1. Wavelet-based Image Coding: An Overview

47

mization. In phase one, the uniform scalar quantizers are fixed, and optimal zerotrees are chosen. In phase two, zerotrees are fixed and the quantization level of uniform scalar quantizers is optimized. This algorithm is guaranteed to converge to a local optimum [74]. There are further details of this algorithm involving prediction and description of zerotrees, which we leave out of the current discussion. The advantage of this method is mainly in performance, compared to both EZW and SPHIT (the latter only slightly). The main disadvantages of this method are its complexity, and perhaps more importantly, that it does not generate an embedded bitstream.

8 Frequency, space-frequency adaptive coders 8.1

Wavelet Packets

The wavelet transform does a good job of decorrelating image pixels in practice, especially when images have power spectra that decay approximately uniformly and exponentially. However, for images with non-exponential rates of spectral decay and for images which have concentrated peaks in the spectra away from DC, we can do considerably better. Our analysis of Section 3.5 suggests that the optimal subband decomposition for an image is one for which the spectrum in each subband is approximately flat. The octave-band decomposition produced by the wavelet transform produces nearly flat spectra for exponentially decaying spectra. The Barbara test image shown in Figure 16 contains a narrow-band component at high frequencies that comes from the tablecloth and the striped clothing. Fingerprint images contain similar narrow-band high frequency components. The best basis algorithm, developed by Coifman and Wickerhauser [75], provides an efficient way to find a fast, wavelet-like transform that provides a good approximation to the Karhunen-Lo`eve transform for a given image. As with the wavelet transform, we start by assuming that a given signal corresponds to a sum of fine-scale scaling functions. The transform performs a change of basis, but the new basis functions are not wavelets but rather wavelet packets [76]. Like wavelets, wavelet packets are formed from translated and dilated linear combinations of scaling functions. However, the recurrence relations they satisfy are different, and the functions form an overcomplete set. Consider a signal of length 2N . The wavelet basis for such a signal consists of a scaling function and 2N − 1 translates and dilates of the wavelet ψ(x). Wavelet packets are formed from translates and dilates of 2N different prototype functions, and there are N 2N different possible functions that can be used to form a basis. Wavelet packets are formed from recurrence relations similar to those for

48

Geoffrey M. Davis, Aria Nosratinia

wavelets and generalize the theoretical framework of wavelets. The simplest wavelet packet π0 (x) is the scaling function φ(x). New wavelet packets πj (x) for j > 0 are formed by the recurrence relations  π2j (x) = hk πj (2x − k) (1.31) k

π2j+1 (x) =



gk πj (2x − k).

(1.32)

k

where the hk and gk are the same as those in the recurrence equations ( 4.3) and ( 4.3). The idea of wavelet packets is most easily seen in the frequency domain. Recall from Figure 14 that each step of the wavelet transform splits the current low frequency subband into two subbands of equal width, one high-pass and one low-pass. With wavelet packets there is a new degree of freedom in the transform. Again there are N stages to the transform for a signal of length 2N , but at each stage we have the option of splitting the low-pass subband, the high-pass subband, both, or neither. The high and low pass filters used in each case are the same filters used in the wavelet transform. In fact, the wavelet transform is the special case of a wavelet packet transform in which we always split the low-pass subband. With this increased flexibility we can generate 2N possible different transforms in 1D. The possible transforms give rise to all possible dyadic partitions of the frequency axis. The increased flexibility does not lead to a large increase in complexity; the worst-case complexity for a wavelet packet transform is O(N log N ).

8.2

Frequency Adaptive Coders

The best basis algorithm is a fast algorithm for minimizing an additive cost function over the set of all wavelet packet bases. Our analysis of transform coding for Gaussian random processes suggests that we select the basis that maximizes the transform coding gain. The approximation theoretic arguments of Mallat and Falzon [57] suggest that at low bit rates the basis that maximizes the number of coefficients below a given threshold is the best choice. The best basis paradigm can accommodate both of these choices. See [77] for an excellent introduction to wavelet packets and the best basis algorithm. Ramchandran and Vetterli [65] describe an algorithm for finding the best wavelet packet basis for coding a given image using rate-distortion criteria. An important application of this wavelet-packet transform optimization is the FBI Wavelet/Scalar Quantization Standard for fingerprint compression. The standard uses a wavelet packet decomposition for the transform stage of the encoder [78]. The transform used is fixed for all fingerprints, however, so the FBI coder is a first-generation linear coder.

1. Wavelet-based Image Coding: An Overview

49

The benefits of customizing the transform on a per-image basis depend considerably on the image. For the Lena test image the improvement in peak signal to noise ratio is modest, ranging from 0.1 dB at 1 bit per pixel to 0.25 dB at 0.25 bits per pixel. This is because the octave band partitions of the spectrum of the Lena image are nearly flat. The Barbara image (see Figure 16), on the other hand, has a narrow-band peak in the spectrum at high frequencies. Consequently, the PSNR increases by roughly 2 dB over the same range of bitrates [65]. Further impressive gains are obtained by combining the adaptive transform with a zerotree structure [67].

8.3

Space-Frequency Adaptive Coders

The best basis algorithm is not limited only to adaptive segmentation of the frequency domain. Related algorithms permit joint time and frequency segmentations. The simplest of these algorithms performs adapted frequency segmentations over regions of the image selected through a quadtree decomposition procedure [79, 80]. More complicated algorithms provide combinations of spatially varying frequency decompositions and frequency varying spatial decompositions [66]. These jointly adaptive algorithms work particularly well for highly nonstationary images. The primary disadvantage of these spatially adaptive schemes are that the pre-computation requirements are much greater than for the frequency adaptive coders, and the search is also much larger. A second disadvantage is that both spatial and frequency adaptivity are limited to dyadic partitions. A limitation of this sort is necessary for keeping the complexity manageable, but dyadic partitions are not in general the best ones.

9 Utilizing Intra-band Dependencies The development of the EZW coder motivated a flurry of activity in the area of zerotree wavelet algorithms. The inherent simplicity of the zerotree data structure, its computational advantages, as well as the potential for generating an embedded bitstream were all very attractive to the coding community. Zerotree algorithms were developed for a variety of applications, and many modifications and enhancements to the algorithm were devised, as described in Section 7. With all the excitement incited by the discovery of EZW, it is easy to automatically assume that zerotree structures, or more generally interband dependencies, should be the focal point of efficient subband image compression algorithms. However, some of the best performing subband image coders known today are not based on zerotrees. In this section, we explore two methods that utilize intra-band dependencies. One of them uses the concept of Trellis Coded Quantization (TCQ). The other uses both

50

Geoffrey M. Davis, Aria Nosratinia D0 D0

D1

D2

D3

D0

D1

D2

D3

D2

D0

D2

S

0 D1

D3

D1

D3

S

1

FIGURE 21. TCQ sets and supersets

inter- and intra-band information, and is based on a recursive estimation of the variance of the wavelet coefficients. Both of them yield excellent coding results.

9.1

Trellis coded quantization

Trellis Coded Quantization (TCQ) [81] is a fast and effective method of quantizing random variables. Trellis coding exploits correlations between variables. More interestingly, it can use non-rectangular quantizer cells that give it quantization efficiencies not attainable by scalar quantizers. The central ideas of TCQ grew out of the ground-breaking work of Ungerboeck [82] in trellis coded modulation. In this section we describe the operational principles of TCQ, mostly through examples. We will briefly touch upon variations and improvements on the original idea, especially at the low bitrates applicable in image coding. In Section 9.2, we review the use of TCQ in multiresolution image compression algorithms. The basic idea behind TCQ is the following: Assume that we want to quantize a stationary, memoryless uniform source at the rate of R bits per sample. Performing quantization directly on this uniform source would require an optimum scalar quantizer with 2N reproduction levels (symbols). The idea behind TCQ is to first quantize the source more finely, with 2R+k symbols. Of course this would exceed the allocated rate, so we cannot have a free choice of symbols at all times. In our example we take k = 1. The scalar codebook of 2R+1 symbols is partitioned into subsets of 2R−1 symbols each, generating four sets. In our example R = 2; see Figure 21. The subsets are designed such that each of them represents reproduction points of a coarser, rate-(R − 1) quantizer. Thefour subsets are designated D0 , D1 , D2 , and D3 . Also, define S0 =  D0 D2 and S1 = D1 D3 , where S0 and S1 are known as supersets. Obviously, the rate constraint prohibits the specification of an arbitrary symbol out of 2R+1 symbols. However, it is possible to exactly specify, with R bits, one element out of either S0 or S1 . At each sample, assuming we know which one of the supersets to use, one bit can be used to determine the active subset, and R − 1 bits to specify a codeword from the subset. The choice of superset is determined by the state of a finite state machine, described by a suitable trellis. An example of such a trellis, with eight

1. Wavelet-based Image Coding: An Overview

51

D0 D2 D1 D3 D0 D2 D1 D3 D2 D0 D3 D1 D2 D0 D3 D1

FIGURE 22. An 8-state TCQ trellis with subset labeling. The bits that specify the sets within the superset also dictate the path through the trellis.

states, is given in Figure 22. The subsets {D0 , D1 , D2 , D3 } are also used to label the branches of the trellis, so the same bit that specifies the subset (at a given state) also determines the next state of the trellis. Encoding is achieved by spending one bit per sample on specifying the path through the trellis, while the remaining R − 1 bits specify a codeword out of the active subset. It may seem that we are back to a non-optimal rate-R quantizer (either S0 or S1 ). So why all this effort? The answer is that we have more codewords than a rate-R quantizer, because there is some freedom of choosing from symbols of either S0 or S1 . Of course this choice is not completely free: the decision made at each sample is linked to decisions made at past and future sample points, through the permissible paths of the trellis. But it is this additional flexibility that leads to the improved performance. Availability of both S0 and S1 means that the reproduction levels of the quantizer are, in effect, allowed to “slide around” and fit themselves to the data, subject to the permissible paths on the trellis. Before we continue with further developments of TCQ and subband coding, we should note that in terms of both efficiency and computational requirements, TCQ is much more similar to VQ than to scalar quantization. Since our entire discussion of transform coding has been motivated by an attempt to avoid VQ, what is the motivation for using TCQ in subband coding, instead of standard VQ? The answer lies in the recursive structure of trellis coding and the existence of a simple dynamic programming method, known as the Viterbi algorithm [83], for finding the TCQ codewords. Although it is true that block quantizers, such as VQ, are asymptotically as efficient as TCQ, the process of approaching the limit is far from trivial for VQ. For a given realization of a random process, the code vectors generated by the VQ of size N − 1 have no clear relationship to those with vector dimension N . In contrast, the trellis encoding algorithm increases

52

Geoffrey M. Davis, Aria Nosratinia

the dimensionality of the problem automatically by increasing the length of the trellis. The standard version of TCQ is not particularly suitable for image coding, because its performance degrades quickly at low rates. This is due partially to the fact that one bit per sample is used to encode the trellis alone, while interesting rates for image coding are mostly below one bit per sample. Entropy constrained TCQ (ECTCQ) improves the performance of TCQ at low rates. In particular, a version of ECTCQ due to Marcellin [84] addresses two key issues: reducing the rate used to represent the trellis (the so-called “state entropy”), and ensuring that zero can be used as an output codeword with high probability. The codebooks are designed using the algorithm and encoding rule from [85].

9.2

TCQ subband coders

In a remarkable coincidence, at the 1994 International Conference in Image Processing, three research groups [86, 87, 88] presented similar but independently developed image coding algorithms. The main ingredients of the three methods are subband decomposition, classification and optimal rate allocation to different subsets of subband data, and entropy-constrained TCQ. These works have been brought together in [68]. We briefly discuss the main aspects of these algorithms. Consider a subband decomposition of an image, and assume that the subbands are well represented by a non-stationary random process X, whose samples Xi are taken from distributions with variances σi2 . One can compute an “average variance” over the entire random process and perform conventional optimal quantization. But better performance is possible by sending overhead information about the variance of each sample, and quantizing it optimally according to its own p.d.f. This basic idea was first proposed by Chen and Smith [89] for adaptive quantization of DCT coefficients. In their paper, Chen and Smith proposed to divide all DCT coefficients into four groups according to their “activity level”, i.e. variance, and code each coefficient with an optimal quantizer designed for its group. The question of how to partition coefficients into groups was not addressed, however, and [89] arbitrarily chose to form groups with equal population.9 However, one can show that equally populated groups are not a always 9 If for a moment, we disregard the overhead information, the problem of partitioning the coefficients bears a strong resemblance to the problem of best linear transform. Both operations, namely the linear transform and partitioning, conserve energy. The goal in both is to minimize overall distortion through optimal allocation of a finite rate. Not surprisingly, the solution techniques are similar (Lagrange multipliers), and they both generate sets with maximum separation between low and high energies (maximum arithmetic to geometric mean ratio).

1. Wavelet-based Image Coding: An Overview

53

a good choice. Suppose that we want to classify the samples into J groups, and that all samples assigned to a given class i ∈ {1, ..., J} are grouped into a source Xi . Let the total number of samples assigned to Xi be Ni , and the total number of samples in all groups be N . Define pi = Ni /N to be the probability of a sample belonging to the source Xi . Encoding the source Xi at rate Ri results in a mean squared error distortion of the form [90] Di (Ri ) = 52i σi2 2−2Ri

(1.33)

where 5i is a constant depending on the shape of the pdf. The rate allocation problem can now be solved using a Lagrange multiplier approach, much in the same way as was shown for optimal linear transforms, resulting in the following optimal rates: Ri =

52 σ 2 R 1 + log2 J i i 2 2 pj J 2 j=1 (5j σj )

(1.34)

where R is the total rate and Ri are the rates assigned to each group. Classification gain is defined as the ratio of the quantization error of the original signal X, divided by that of the optimally bit-allocated classified version. 52 σ 2 Gc =  J (1.35) 2 2 pj j=1 (5j σj ) One aims to maximize this gain over {pi }. It is not unexpected that the optimization process can often yield non-uniform {pi }, resulting in unequal population of the classification groups. It is noteworthy that nonuniform populations not only have better classification gain in general, but also lower overhead: Compared to a uniform {pi }, any other distribution has smaller entropy, which implies smaller side information to specify the classes. The classification gain is defined for Xi taken from one subband. A generalization of this result in [68] combines it with the conventional coding gain of the subbands. Another refinement takes into account the side information required for classification. The coding algorithm then optimizes the resulting expression to determine the classifications. ECTCQ is then used for final coding. Practical implementation of this algorithm requires attention to a great many details, for which the interested reader is referred to [68]. For example, the classification maps determine energy levels of the signal, which are related to the location of the edges in the image, and are thus related in different subbands. A variety of methods can be used to reduce the overhead information (in fact, the coder to be discussed in the next section makes the management of side information the focus of its efforts) Other issues include alternative measures for classification, and the usage of arithmetic coded TCQ. The coding results of the ECTCQ based subband coding are some of

54

Geoffrey M. Davis, Aria Nosratinia

the best currently available in the literature, although the computational complexity of these algorithms is also considerably greater than the other methods presented in this paper.

9.3

Mixture Modeling and Estimation

A common thread in successful subband and wavelet image coders is modeling of image subbands as random variables drawn from a mixture of distributions. For each sample, one needs to detect which p.d.f. of the mixture it is drawn from, and then quantize it according to that pdf. Since the decoder needs to know which element of the mixture was used for encoding, many algorithms send side information to the decoder. This side information becomes significant, especially at low bitrates, so that efficient management of it is pivotal to the success of the image coder. All subband and wavelet coding algorithms discussed so far use this idea in one way or another. They only differ in the constraints they put on side information so that it can be coded efficiently. For example, zerotrees are a clever way of indicating side information. The data is assumed from a mixture of very low energy (zero set) and high energy random variables, and the zero sets are assumed to have a tree structure. The TCQ subband coders discussed in the last section also use the same idea. Different classes represent different energies in the subbands, and are transmitted as overhead. In [68], several methods are discussed to compress the side information, again based on geometrical constraints on the constituent elements of the mixture (energy classes). A completely different approach to the problem of handling information overhead is explored in [69, 91]. These two works were developed simultaneously but independently. The version developed in [69] is named Estimation Quantization (EQ) by the authors, and is the one that we present in the following. The title of [91] suggests a focus on entropy coding, but in fact the underlying ideas of the two are remarkably similar. We will refer to the the aggregate class as backward mixture-estimation encoding (BMEE). BMEE models the wavelet subband coefficients as non-stationary generalized Gaussian, whose non-stationarity is manifested by a slowly varying variance (energy) in each band. Because the energy varies slowly, it can be predicted from causal neighboring coefficients. Therefore, unlike previous methods, BMEE does not send the bulk of mixture information as overhead, but attempts to recover it at the decoder from already transmitted data, hence the designation “backward”. BMEE assumes that the causal neighborhood of a subband coefficient (including parents in a subband tree) has the same energy (variance) as the coefficient itself. The estimate of energy is found by applying a maximum likelihood method to a training set formed by the causal neighborhood. Similar to other recursive algorithms that involve quantization, BMEE has to contend with the problem of stability and drift. Specifically, the

1. Wavelet-based Image Coding: An Overview

55

decoder has access only to quantized coefficients, therefore the estimator of energy at the encoder can only use quantized coefficients. Otherwise, the estimates at the encoder and decoder will vary, resulting in drift problems. This presents the added difficulty of estimating variances from quantized causal coefficients. BMEE incorporates the quantization of the coefficients into the maximum likelihood estimation of the variance. The quantization itself is performed with a dead-zone uniform quantizer (see Figure 15). This quantizer offers a good approximation to entropy constrained quantization of generalized Gaussian signals. The dead-zone and step sizes of the quantizers are determined through a Lagrange multiplier optimization technique, which was introduced in the section on optimal rate allocation. This optimization is performed offline, once each for a variety of encoding rates and shape parameters, and the results are stored in a look-up table. This approach is to be credited for the speed of the algorithm, because no optimization need take place at the time of encoding the image. Finally, the backward nature of the algorithm, combined with quantization, presents another challenge. All the elements in the causal neighborhood may sometimes quantize to zero. In that case, the current coefficient will also quantize to zero. This degenerate condition will propagate through the subband, making all coefficients on the causal side of this degeneracy equal to zero. To avoid this condition, BMEE provides for a mechanism to send side information to the receiver, whenever all neighboring elements are zero. This is accomplished by a preliminary pass through the coefficients, where the algorithm tries to “guess” which one of the coefficients will have degenerate neighborhoods, and assembles them to a set. From this set, a generalized Gaussian variance and shape parameter is computed and transmitted to the decoder. Every time a degenerate case happens, the encoder and decoder act based on this extra set of parameters, instead of using the backward estimation mode. The BMEE coder is very fast, and especially in the low bitrate mode (less than 0.25 bits per pixel) is extremely competitive. This is likely to motivate a re-visitation of the role of side information and the mechanism of its transmission in wavelet coders.

10 Future Trends Current research in image coding is progressing along a number of fronts. At the most basic level, a new interpretation of the wavelet transform has appeared in the literature. This new theoretical framework, called the lifting scheme [41], provides a simpler and more flexible method for designing wavelets than standard Fourier-based methods. New families of nonseparable wavelets constructed using lifting have the potential to improve

56

Geoffrey M. Davis, Aria Nosratinia

coders. One very intriguing avenue for future research is the exploration of the nonlinear analogs of the wavelet transform that lifting makes possible. The area of classification and backward estimation based coders is an active one. Several research groups are reporting promising results [92, 93]. One very promising research direction is the development of coded images that are robust to channel noise via joint source and channel coding. See for example [94], [95] and [96]. The adoption of wavelet based coding to video signals presents special challenges. One can apply 2-D wavelet coding in combination to temporal prediction (motion estimated prediction), which will be a direct counterpart of current DCT-based video coding methods. It is also possible to consider the video signal as a three-dimensional array of data and attempt to compress it with 3-D wavelet analysis. This approach presents difficulties that arise from the fundamental properties of the discrete wavelet transform. The discrete wavelet transform (as well as any subband decomposition) is a space-varying operator, due to the presence of decimation and interpolation. This space variance is not conducive to compact representation of video signals, as described below. Video signals are best modeled by 2-D projections whose position in consecutive frames of the video signal varies by unknown amounts. Because vast amounts of information are repeated in this way, one can achieve considerable gain by representing the repeated information only once. This is the basis of motion compensated coding. However, since the wavelet representation of the same 2-D signal will vary once it is shifted10 , this redundancy is difficult to reproduce in the wavelet domain. A frequency domain study of the difficulties of 3-D wavelet coding of video is presented in [97], and leads to the same insights. Some attempts have also been made on applying 3-D wavelet coding on the residual 3-D data after motion compensation, but have met with indifferent success.

11 Summary and Conclusion Image compression is governed by the general laws of information theory and specifically rate-distortion theory. However, these general laws are nonconstructive and the more specific techniques of quantization theory are needed for the actual development of compression algorithms. Vector quantization can theoretically attain the maximum achievable coding efficiency. However, VQ has three main impediments: computational complexity, delay, and the curse of dimensionality. Transform coding techniques, in conjunction with entropy coding, capture important gains of VQ, 10 Unless the shift is exactly by a correct multiple of M samples, where M is the downsampling rate

1. Wavelet-based Image Coding: An Overview

57

while avoiding most of its difficulties. Theoretically, the Karhunen-Lo´eve transform is optimal for Gaussian processes. Approximations to the K-L transform, such as the DCT, have led to very successful image coding algorithms such as JPEG. However, even if one argues that image pixels can be individually Gaussian, they cannot be assumed to be jointly Gaussian, at least not across the image discontinuities. Image discontinuities are the place where traditional coders spend the most rate, and suffer the most distortion. This happens because traditional Fourier-type transforms (e.g., DCT) disperse the energy of discontinuous signals across many coefficients, while the compaction of energy in the transform domain is essential for good coding performance. The discrete wavelet transform provides an elegant framework for signal representation in which both smooth areas and discontinuities can be represented compactly in the transform domain. This ability comes from the multi-resolution properties of wavelets. One can motivate wavelets through spectral partitioning arguments used in deriving optimal quantizers for Gaussian processes. However, the usefulness of wavelets in compression goes beyond the Gaussian case. State of the art wavelet coders assume that image data comes from a source with fluctuating variance. Each of these coders provides a mechanism to express the local variance of the wavelet coefficients, and quantizes the coefficients optimally or near-optimally according to that variance. The individual wavelet coders vary in the way they estimate and transmit this variances to the decoder, as well as the strategies for quantizing according to that variance. Zerotree coders assume a two-state structure for the variances: either negligible (zero) or otherwise. They send side information to the decoder to indicate the positions of the non-zero coefficients. This process yields a non-linear image approximation rather than the linear truncated KLTbased approximation motivated by our Gaussian model. The set of zero coefficients are expressed in terms of wavelet trees (Lewis & Knowles, Shapiro, others) or combinations thereof (Said & Pearlman). The zero sets are transmitted to the receiver as overhead, as well as the rest of the quantized data. Zerotree coders rely strongly on the dependency of data across scales of the wavelet transform. Frequency-adaptive coders improve upon basic wavelet coders by adapting transforms according to the local inter-pixel correlation structure within an image. Local fluctuations in the correlation structure and in the variance can be addressed by spatially adapting the transform and by augmenting the optimized transforms with a zerotree structure. Other wavelet coders use dependency of data within the bands (and sometimes across the bands as well). Coders based on Trellis Coded Quantization (TCQ) partition coefficients into a number of groups, according to their energy. For each coefficient, they estimate and/or transmit the group information as well as coding the value of the coefficient with TCQ, ac-

58

Geoffrey M. Davis, Aria Nosratinia

cording to the nominal variance of the group. Another newly developed class of coders transmit only minimal variance information while achieving impressive coding results, indicating that perhaps the variance information is more redundant than previously thought. While some of these coders may not employ what might strictly be called a wavelet transform, they all utilize a multi-resolution decomposition, and use concepts that were motivated by wavelet theory. Wavelets and the ideas arising from wavelet analysis have had an indelible effect on the theory and practice of image compression, and are likely to continue their dominant presence in image coding research in the near future.

Acknowledgments: G. Davis thanks the Digital Signal Processing group at Rice University for their generous hospitality during the writing of this paper. This work has been supported in part by a Texas Instruments Visiting Assistant Professorship at Rice University and an NSF Mathematical Sciences Postdoctoral Research Fellowship.

12

References

[1] A. Gersho and R. M. Gray, Vector Quantization and Signal Compression. Kluwer Academic Publishers, 1992. [2] C. E. Shannon, “Coding theorems for a discrete source with a fidelity criterion,” in IRE Nat. Conv. Rec., vol. 4, pp. 142–163, March 1959. [3] R. M. Gray, J. C. Kieffer, and Y. Linde, “Locally optimal block quantizer design,” Information and Control, vol. 45, pp. 178–198, May 1980. [4] K. Zeger, J. Vaisey, and A. Gersho, “Globally optimal vector quantizer design by stochastic relaxation,” IEEE Transactions on Signal Processing, vol. 40, pp. 310–322, Feb. 1992. [5] T. Cover and J. Thomas, Elements of Information Theory. New York: John Wiley & Sons, Inc., 1991. [6] A. Gersho, “Asymptotically optimal block quantization,” IEEE Transactions on Information Theory, vol. IT-25, pp. 373–380, July 1979. [7] L. F. Toth, “Sur la representation d’une population infinie par un nombre fini d’elements,” Acta Math. Acad. Scient. Hung., vol. 10, pp. 299– 304, 1959. [8] P. L. Zador, “Asymptotic quantization error of continuous signals and the quantization dimension,” IEEE Transactions on Information Theory, vol. IT-28, pp. 139–149, Mar. 1982.

1. Wavelet-based Image Coding: An Overview

59

[9] J. Y. Huang and P. M. Schultheiss, “Block quantization of correlated Gaussian random variables,” IEEE Transactions on Communications, vol. CS-11, pp. 289–296, September 1963. [10] R. A. Horn and C. R. Johnson, Matrix Analysis. New York: Cambridge University Press, 1990. [11] N. Ahmed, T. Natarajan, and K. R. Rao, “Descrete cosine transform,” IEEE Transactions on Computers, vol. C-23, pp. 90–93, January 1974. [12] K. Rao and P. Yip, The Discrete Cosine Transform. New York: Academic Press, 1990. [13] W. Pennebaker and J. Mitchell, JPEG still image data compression standard. New York: Van Nostrad Reinhold, 1993. [14] Y. Arai, T. Agui, and M. Nakajima, “A fast DCT-SQ scheme for images,” Transactions of the IEICE, vol. 71, p. 1095, November 1988. [15] E. Feig and E. Linzer, “Discrete cosine transform algorithms for image data compression,” in Electronic Imaging ’90 East, (Boston, MA), pp. 84–87, October 1990. [16] A. Croisier, D. Esteban, and C. Galand, “Perfect channel splitting by use of interpolation/decimation/tree decomposition techniques,” in Proc. Int. Symp. on Information, Circuits and Systems, (Patras, Greece), 1976. [17] P. Vaidyanathan, “Theory and design of M-channel maximally decimated quadrature mirror filters with arbitrary M, having perfect reconstruction property,” IEEE Trans. ASSP, vol. ASSP-35, pp. 476– 492, April 1987. [18] M. J. T. Smith and T. P. Barnwell, “A procedure for desiging exact reconstruction fiolter banks for tree structured subband coders,” in Proc. ICASSP, (San Diego, CA), pp. 27.1.1–27.1.4, March 1984. [19] M. J. T. Smith and T. P. Barnwell, “Exact reconstruction techniques for tree-structured subband coders,” IEEE Trans. ASSP, vol. ASSP34, pp. 434–441, June 1986. [20] M. Vetterli, “Splitting a signal into subband channels allowing perfect reconstruction,” in Proc. IASTED Conf. Appl. Signal Processing, (Paris, France), June 1985. [21] M. Vetterli, “Filter banks allowing perfect reconstruction,” Signal Processing, vol. 10, pp. 219–244, April 1986. [22] P. Vaidyanathan, Multirate systems and filter banks. Englewood Cliffs, NJ: Prentice Hall, 1993.

60

Geoffrey M. Davis, Aria Nosratinia

[23] M. Vetterli and J. Kovaˇcevi´c, Wavelets and Subband Coding. Englewood Cliffs, NJ: Prentice Hall, 1995. [24] T. Berger, Rate Distortion Theory. Englewood Cliffs, NJ: Prentice Hall, 1971. [25] B. Girod, F. Hartung, and U. Horn, “Subband image coding,” in Subband and wavelet transforms: design and applications, Boston, MA: Kluwer Academic Publishers, 1995. [26] P. J. Burt and E. H. Adelson, “The Laplacian pyramid as a compact image code,” IEEE Transactions on Communications, vol. COM-31, Apr. 1983. [27] S. Mallat, “A theory for multiresolution signal decomposition: the wavelet representation,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 11, pp. 674–693, July 1989. [28] G. Strang and T. Q. Nguyen, Wavelets and Filter Banks. Wellesley, MA: Wellesley-Cambridge Press, 1996. [29] C. S. Burrus, R. A. Gopinath, and H. Guo, Introduction to Wavelets and Wavelet Transforms: A Primer. Englewood Cliffs, NJ: PrenticeHall, 1997. [30] I. Daubechies, Ten Lectures on Wavelets. Philadelphia, PA: SIAM, 1992. [31] O. Rioul, “Simple regularity criteria for subdivision schemes,” SIAM J. Math. Analysis, vol. 23, pp. 1544–1576, Nov. 1992. [32] O. Rioul, “Regular wavelets: a discrete-time approach,” IEEE Transactions on Signal Processing, vol. 41, pp. 3572–3579, Dec. 1993. [33] G. Strang and G. Fix, “A Fourier analysis of the finite element variational method,” Constructive Aspects of Functional Analysis, pp. 796– 830, 1971. [34] W. Sweldens and R. Piessens, “Quadrature formulae and asymptotic error expansions for wavelet approximations of smooth functions,” SIAM Journal of Numerical Analysis, vol. 31, pp. 1240–1264, Aug. 1994. [35] M. Unser, “Approximation power of biorthogonal wavelet expansions,” IEEE Transactions on Signal Processing, vol. 44, pp. 519–527, Mar. 1996. [36] J. Kovaˇcevi´c and W. Sweldens, “Interpolating filter banks and wavelets in arbitrary dimensions,” tech. rep., Lucent Technologies, Murray Hill, NJ, 1997.

1. Wavelet-based Image Coding: An Overview

61

[37] M. Antonini, M. Barlaud, and P. Mathieu, “Image Coding Using Wavelet Transform,” IEEE Trans. Image Proc., vol. 1, pp. 205–220, Apr. 1992. [38] M. Vetterli and C. Herley, “Wavelets and filter banks: Theory and design,” IEEE Trans. Acoust. Speech Signal Proc., vol. 40, no. 9, pp. 2207–2232, 1992. [39] J. Villasenor, B. Belzer, and J. Liao, “Wavelet filter evaluation for image compression,” IEEE Transactions on image processing, vol. 2, pp. 1053–1060, Aug. 1995. [40] G. Deslauriers and S. Dubuc, “Symmetric iterative interpolation processes,” Constructive Approximation, vol. 5, no. 1, pp. 49–68, 1989. [41] W. Sweldens, “The lifting scheme: A new philosophy in biorthogonal wavelet constructions,” in Wavelet Applications in Signal and Image Processing III (A. F. Laine and M. Unser, eds.), pp. 68–79, Proc. SPIE 2569, 1995. [42] I. Balasingham and T. A. Ramstad, “On the relevance of the regularity constraint in subband image coding,” in Proc. Asilomar Conference on Signals, Systems, and Computers, (Pacific Grove), 1997. [43] C. M. Brislawn, “Classification of nonexpansive symmetric extension transforms for multirate filter banks,” Applied and Comp. Harmonic Analysis, vol. 3, pp. 337–357, 1996. [44] C. Herley and M. Vetterli, “Orthogonal time-varying filter banks and wavelets,” in Proc. IEEE Int. Symp. Circuits Systems, vol. 1, pp. 391– 394, May 1993. [45] C. Herley, “Boundary filters for finite-length signals and time-varying filter banks,” IEEE Trans. Circuits and Systems II, vol. 42, pp. 102– 114, Feb. 1995. [46] W. Sweldens and P. Schr¨ oder, “Building your own wavelets at home,” Tech. Rep. 1995:5, Industrial Mathematics Initiative, Mathematics Department, University of South Carolina, 1995. [47] H. Gish and J. N. Pierce, “Asymptotically efficient quantizing,” IEEE Transactions on Information Theory, vol. IT-14, pp. 676–683, Sept. 1968. [48] N. Farvardin and J. W. Modestino, “Optimum quantizer performance for a class of non-Gaussian memoryless sources,” IEEE Transactions on Information Theory, vol. 30, pp. 485–497, May 1984.

62

Geoffrey M. Davis, Aria Nosratinia

[49] D. Taubman and A. Zakhor, “Multirate 3-D subband coding of video,” IEEE Trans. Image Proc., vol. 3, Sept. 1994. [50] T. Bell, J. G. Cleary, and I. H. Witten, Text Compression. Englewood Cliffs, NJ: Prentice Hall, 1990. [51] D. L. Duttweiler and C. Chamzas, “Probability estimation in arithmetic and adaptive-Huffman entropy coders,” IEEE Transactions on Image Processing, vol. 4, pp. 237–246, Mar. 1995. [52] Y. Shoham and A. Gersho, “Efficient bit allocation for an arbitrary set of quantizers,” IEEE Trans. Acoustics, Speech, and Signal Processing, vol. 36, pp. 1445–1453, Sept. 1988. [53] P. Moulin, “A multiscale relaxation algorithm for SNR maximization in nonorthogonal subband coding,” IEEE Transactions on Image Processing, vol. 4, pp. 1269–1281, Sept. 1995. [54] J. W. Woods and T. Naveen, “A filter based allocation scheme for subband compression of HDTV,” IEEE Trans. Image Proc., vol. IP-1, pp. 436–440, July 1992. [55] B. A. Wandell, Foundations of Vision. Sunderland, MA: Sinauer Associates, 1995. [56] A. Watson, G. Yang, J. Soloman, and J. Villasenor, “Visual thresholds for wavelet quantization error,” in Proceedings of the SPIE, vol. 2657, pp. 382–392, 1996. [57] S. Mallat and F. Falzon, “Understanding image transform codes,” in Proc. SPIE Aerospace Conf., (orlando), Apr. 1997. [58] R. A. DeVore, B. Jawerth, and B. J. Lucier, “Image Compression through Wavelet Transform Coding,” IEEE Trans. Info. Theory, vol. 38, pp. 719–746, Mar. 1992. [59] W. B. Pennebaker and J. L. Mitchell, JPEG Still Image Data Compression Standard. New York: Van Nostrand Reinhold, 1992. [60] M. Crouse and K. Ramchandran, “Joint thresholding and quantizer selection for decoder-compatible baseline JPEG,” in Proc. ICASSP, May 1995. [61] G. M. Davis, “The wavelet image compression construction kit.” http://www.cs.dartmouth.edu/∼gdavis/wavelet/wavelet.html, 1996. [62] J. Shapiro, “Embedded image coding using zerotrees of wavelet coefficients,” IEEE Transactions on Signal Processing, vol. 41, pp. 3445– 3462, Dec. 1993.

1. Wavelet-based Image Coding: An Overview

63

[63] A. Said and W. A. Pearlman, “A new, fast, and efficient image codec based on set partitioning in hierarchichal trees,” IEEE Trans. Circuits and Systems for Video Technology, vol. 6, pp. 243–250, June 1996. [64] Z. Xiong, K. Ramchandran, and M. T. Orchard, “Space-frequency quantization for wavelet image coding,” preprint, 1995. [65] K. Ramchandran and M. Vetterli, “Best wavelet packet bases in a rate-distortion sense,” IEEE Transactions on Image Processing, vol. 2, no. 2, pp. 160–175, 1992. [66] C. Herley, Z. Xiong, K. Ramchandran, and M. T. Orchard, “Joint space-frequency segmentation using balanced wavelet packet trees for least-cost image representation,” IEEE Transactions on Image Processing, Sept. 1997. [67] Z. Xiong, K. Ramchandran, M. Orchard, and K. Asai, “Wavelet packets-based image coding using joint space-frequency quantization,” Preprint, 1996. [68] R. L. Joshi, H. Jafarkhani, J. H. Kasner, T. R. Fisher, N. Farvardin, and M. W. Marcellin, “Comparison of different methods of classification in subband coding of images,” IEEE Transactions on Image Processing, submitted. [69] S. M. LoPresto, K. Ramchandran, and M. T. Orchard, “Image coding based on mixture modeling of wavelet coefficients and a fast estimation-quantization framework,” in Proc. Data Compression Conference, (Snowbird, Utah), pp. 221–230, 1997. [70] A. S. Lewis and G. Knowles, “Image compression using the 2-d wavelet transform,” IEEE Transactions on Image Processing, vol. 1, pp. 244– 250, April 1992. [71] J. Shapiro, “Embedded image coding using zero-trees of wavelet coefficients,” IEEE Transactions on Signal Processing, vol. 41, pp. 3445– 3462, December 1993. [72] A. Said and W. A. Pearlman, “A new, fast, and efficient image codec based on set partitioning in hierarchical trees,” IEEE Transactions on Circuits and Systems for Video Technology, vol. 6, pp. 243–250, June 1996. [73] G. M. Davis and S. Chawla, “Image coding using optimized significance tree quantization,” in Proc. Data Compression Conference (J. A. Storer and M. Cohn, eds.), pp. 387–396, Mar. 1997.

64

Geoffrey M. Davis, Aria Nosratinia

[74] Z. Xiong, K. Ramchandran, and M. T. Orchard, “Space-frequency quantization for wavelet image coding,” IEEE Transactions on Image Processing, vol. 6, pp. 677–693, May 1997. [75] R. R. Coifman and M. V. Wickerhauser, “Entropy based algorithms for best basis selection,” IEEE Transactions on Information Theory, vol. 32, pp. 712–718, Mar. 1992. [76] R. R. Coifman and Y. Meyer, “Nouvelles bases orthonorm´ees de l2 (r) ayant la structure du syst`eme de Walsh,” Tech. Rep. Preprint, Department of Mathematics, Yale University, 1989. [77] M. V. Wickerhauser, Adapted Wavelet Analysis from Theory to Software. Wellesley, MA: A. K. Peters, 1994. [78] C. J. I. Services, WSQ Gray-Scale Fingerprint Image Compression Specification (ver. 2.0). Federal Bureau of Investigation, Feb. 1993. [79] C. Herley, J. Kovaˇcevi´c, K. Ramchandran, and M. Vetterli, “Tilings of the time-frequency plane: Construction of arbitrary orthogonal bases and fast tiling algorithms,” IEEE Transactions on Signal Processing, vol. 41, pp. 3341–3359, Dec. 1993. [80] J. R. Smith and S. F. Chang, “Frequency and spatially adaptive wavelet packets,” in Proc. ICASSP, May 1995. [81] M. W. Marcellin and T. R. Fischer, “Trellis coded quantization of memoryless and Gauss-Markov sources,” IEEE Transactions on Communications, vol. 38, pp. 82–93, January 1990. [82] G. Ungerboeck, “Channel coding with multilevel/phase signals,” IEEE Transactions on Information Theory, vol. IT-28, pp. 55–67, January 1982. [83] G. D. Forney, “The Viterbi algorithm,” Proceedings of the IEEE, vol. 61, pp. 268–278, March 1973. [84] M. W. Marcellin, “On entropy-constrained trellis coded quantization,” IEEE Transactions on Communications, vol. 42, pp. 14–16, January 1994. [85] P. A. Chou, T. Lookabaugh, and R. M. Gray, “Entropy-constrained vector quantization,” IEEE Transactions on Information Theory, vol. 37, pp. 31–42, January 1989. [86] H. Jafarkhani, N. Farvardin, and C. C. Lee, “Adaptive image coding based on the discrete wavelet transform,” in Proc. IEEE Int. Conf. Image Proc. (ICIP), vol. 3, (Austin, TX), pp. 343–347, November 1994.

1. Wavelet-based Image Coding: An Overview

65

[87] R. L. Joshi, T. Fischer, and R. H. Bamberger, “Optimum classification in subband coding of images,” in Proc. IEEE Int. Conf. Image Proc. (ICIP), vol. 2, (Austin, TX), pp. 883–887, November 1994. [88] J. H. Kasner and M. W. Marcellin, “Adaptive wavelet coding of images,” in Proc. IEEE Int. Conf. Image Proc. (ICIP), vol. 3, (Austin, TX), pp. 358–362, November 1994. [89] W. H. Chen and C. H. Smith, “Adaptive coding of monochrome and color images,” IEEE Transactions on Communications, vol. COM-25, pp. 1285–1292, November 1977. [90] N. S. Jayant and P. Noll, Digital Coding of waveforms. Englewood Cliffs, NJ: Prentice-Hall, 1984. [91] C. Chrysafis and A. Ortega, “Efficient context based entropy coding for lossy wavelet image compression,” in Proc. Data Compression Conference, (Snowbird, Utah), pp. 241–250, 1997. [92] Y. Yoo, A. Ortega, and B. Yu, “Progressive classification and adaptive quantization of image subbands.” Preprint, 1997. [93] D. Marpe and H. L. Cycon, “Efficient pre-coding techniques for wavelet-based image compression.” Submitted to PCS, Berlin, 1997. [94] P. G. Sherwood and K. Zeger, “Progressive image coding on noisy channels,” in Proc. Data Compression Conference, (Snowbird, UT), pp. 72–81, Mar. 1997. [95] S. L. Regunathan, K. Rose, and S. Gadkari, “Multimode image coding for noisy channels,” in Proc. Data Compression Conference, (Snowbird, UT), pp. 82–90, Mar. 1997. [96] J. Garcia-Frias and J. D. Villasenor, “An analytical treatment of channel-induced distortion in run length coded image subbands,” in Proc. Data Compression Conference, (Snowbird, UT), pp. 52–61, Mar. 1997. [97] A. Nosratinia and M. T. Orchard, “A multi-resolution framework for backward motion compensation,” in Proc. SPIE Symposium on Electronic Imaging, (San Jose, CA), February 1995.

Tutorial on Hidden Markov Models

c 1995, 2002 Javier R. Movellan. Copyright This is an open source document. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. Endorsements This document is endorsed by its copyright holder: Javier R. Movellan. Modified versions of this document should delete this endorsement.

1 Notation for discrete HMM S = {S1 , ..., SN }. Set of possible hidden states. N. Number of distinct hidden states. V = {v1 , ..., vM }. Set of possible external observations. M. Number of distinct external observations. o = (o1 , ..., oK ). A sample of sequences of external observations (the training sample). Each element in o is an entire sequence of observations (e.g., a word). K. Number of sequences (e.g., words) in the training sample. o = (o1 , ..., oT ). A sequence of external observations (e.g., a word). If there is more than a sequence I use a superscript l. ot . A variable representing the external observation at time t. If there is more than a sequence of interest, I use a superscript (e.g., o42 = 3 means that in sequence number 4, at time 2 we observe v3 ). T . The number of time steps in the sequence . q = (q 1 , ..., q K ). Collection of sequences of internal states (internal state sequences that may correspond to each sequence in the training sequence). q = (q1 , ..., qT ). A sequence of internal states. If there is more than a sequence of interest I use a superscript to denote the sequence of interest. qt . A variable representing the internal state at time t. If there is more than a sequence I use a superscript (e.g., q24 = 3 means that the system is in state S3 at time 2 in sequence number 4 of the training sample). λ = (A, B, π). A hidden Markov model as defined by its A, B and π matrices. aij = Pλ (qt+1 = j|qt = i). State transition probability for model λ. A = {aij }. The N xN matrix of transition probabilities. bj (k) = Pλ (ot = k|qt = j). Emission probability for observation vk by state Sj . B = {bj (k)}. The MxN matrix of state to observations probabilities. πi = Pλ (q1 = i). Initial state probability for model λ. π = {πi }. The vector of initial state probabilities. ¯ = P Pλ (q|0)log P ¯ (qo). The auxiliary function maximized by the E-M algoQ(λ, λ) λ q ¯ represents the new model under consideration. rithm. λ represents the current model, λ ¯ The E-M function restricted to the sequence ol from the training sample. Ql (λ, λ) αt (i) = Pλ (o1 ...ot qt = i). The forward variable for the sequence o at time t for state i. If there is more than one sequence of interest I use a superscript to denote the sequence. βt (i) = Pλ (ot+1 ...oT |qt = i). The scaled backward variable for the sequence o at time t for state i. If there is more than one sequence of interest I use a superscript to denote the sequence. α ˆ t (i). The scaled forward variable for the sequence o at time t for state i. If there is more than one sequence of interest I use a superscript to denote the sequence. βˆt (i). The scaled backward variable for the sequence o at time t for state i. If there is more

than one sequence of interest I use a superscript to denote the sequence. c1 ...cT . The scaling coefficients in the scaled forward and backward algorithm for the sequence ol . If there is more than one sequence of interest I use a superscript to denote the sequence. γt (i) = Pλ (qt = i|o) If there is more than one sequence of interest I use a superscript to denote the sequence. ξt (i, j) = Pλ (qt = i qt+1 = j|o). If there is more than one sequence of interest I use a superscript to denote the sequence.

2 EM training with Discrete Observation Models In this section we review two methods for training standard HMM models with discrete observations: E-M training and Viterbi training. 2.1 The E-M auxiliary function ¯ represent a candidate model. Our objective is to Let λ represent the current model and λ make Pλ¯ (o) ≥ Pλ (o), or equivalently log Pλ¯ (o) ≥ log Pλ (o). P Due to the presence of stochastic constraints (e.g., aij ≥ and j aij = 1) it turns out to be easier to maximize an auxiliary function Q(·) rather than to directly maximize log P λ¯ . The E-M auxiliary function is defined as follows: ¯ = Q(λ, λ)

X

Pλ (q|o)log Pλ¯ (qo)

(1)

q

¯ ≥ Q(λ, λ) → log P ¯ (o) ≥ log Pλ (o). Here we show that Q(λ, λ) λ ¯ For any model λ or λ it must be true that Pλ¯ (o) =

Pλ¯ (oq) Pλ¯ (q|o)

(2)

or logPλ¯ (o) = logPλ¯ (oq) − logPλ¯ (q|o). Also, logPλ¯ (o) =

X

Pλ (q|o)logPλ¯ (o)

(3)

q

since logPλ¯ (o) is a constant. Thus, from equation 2 it follows that log Pλ¯ (o) =

X

Pλ (q|o)logPλ¯ (oq) −

q

¯ − = Q(λ, λ)

X

Pλ (q|o)logPλ¯ (q|o)

(4)

q

X

Pλ (q|o)logPλ¯ (q|o)

(5)

q

¯ and to (λ, λ), it follows that, Applying the Q(·, ·) function to (λ, λ) ¯ − Q(λ, λ) = log P ¯ (o) − log Pλ (o) − KL(λ, λ) ¯ Q(λ, λ) λ

(6)

where KL(·, ·) is the Kullback-Leibler criterion (relative entropy) of the probability distribution Pλ (q|o) with respect to the probability distribution Pλ¯ (q|o)

¯ = KL(λ, λ)

X

Pλ (q|o)log

q

Pλ (q|o) Pλ¯ (q|o)

(7)

Rearranging terms, ¯ − Q(λ, λ) + KL(λ, λ) ¯ log Pλ¯ (o) − log Pλ (o) = Q(λ, λ)

(8)

and since the KL criterion is always positive, it follows that if ¯ − Q(λ, λ) ≥ 0 Q(λ, λ)

(9)

log Pλ¯ (o)log Pλ (o) ≥ 0

(10)

then

2.2 The overall training sample E-M function We defined the overall E-M function X ¯ = Q(λ, λ) Pλ (q|o)log Pλ¯ (qo)

(11)

q

with o including the entire set of sequences in the training sample. Assuming that the sequences are independent, it follows that ¯ = Q(λ, λ)

X

Pλ (q|o)

q

X

log Pλ¯ (q l ol ) =

l

XX l

(log Pλ¯ (q l ol ))Pλ (q|o)

(12)

q

And since each state sequence q l depends only on the corresponding observation sequence ol it follows that ¯ = Q(λ, λ)

XX X l

(log Pλ¯ (q l ol ))

K Y

Pλ (q m |o)

(13)

m=1

q l q−q l

where q − q l = (q 1 , ..., q l−1 , q l+1 , ..., q K ) represents an entire collection of sequences except for the sequence q l . Thus ¯ = Q(λ, λ)

XX X l

=

XX l

(log Pλ¯ (q l ol ))Pλ (q l |ol )

(log Pλ¯ (q l ol ))Pλ (q l |ol )

X Y

q−q l

and since

K X Y

Pλ (q m |o) =

(14)

m6=l

q l q−q l

ql

K Y

Pλ (q m |o)

(15)

m6=l

Pλ (q m |o) = 1

(16)

q−q l m6=l

it follows that the E-M function can be decomposed into additive E-M functions, one per observation sequence in the training sample: XX X ¯ = ¯ Q(λ, λ) Pλ (q l |ol ) log Pλ¯ (q l ol ) = Ql (λ, λ) (17) l

ql

l

2.3 Maximizing the E-M function For simplicity let us start with the case in which there is a single sequence. The results easily generalize to multiple sequences. Since we work with a single sequence we may drop the l superscript. ¯ = Q(λ, λ)

X

Pλ (q|o)log Pλ¯ (qo)

(18)

q

And since,

aq2 q3 ... Pλ¯ (qo) = π ¯q1 ¯bq1 (o1 )¯ aq1 q2 ¯bq2 (o2 )¯

(19)

it follows that, ¯ = Q(λ, λ)

X

Pλ (q|o)log π ¯ q1 +

(20)

Pλ (q|o)log ¯bqt (ot )+

(21)

q

+

Tl X X t=1

+

q

TX l −1 X t=1

Pλ (q|o)log a ¯qt qt+1

(22)

q

The first term can be expressed as follows X X X Pλ (q|o)log π ¯ q1 = log π ¯j Pλ (q|o)δ(j, q1 ) q

(23)

q

j

where δ(j, q1 ) tells us to include only those cases in which q1 = j. Therefore, X

Pλ (q|o)δ(j, q1 ) = Pλ (q1 = j|o) = γ1 (j)

(24)

q

The second term can be expressed as follows Tl X X t=1

Pλ (q|o)log ¯bqt (ot ) =

q

Tl X X X t=1

i

log ¯bi (j)

X

Pλ (q|o)δ(i, qt )δ(j, ot )

(25)

q

j

where δ(i, qt )δ(j, ot ) tells us to include only those cases for which qt = i and ot = j. Therefore, X

Pλ (q|o)δ(i, qt )δ(j, ot ) = Pλ (qt = i|o)δ(ot , j) = γt (i)δ(ot , j) =

(26)

q

The third term can be expressed as follows TX l −1 X t=1

q

Pλ (q|o)log a ¯qt qt+1 =

TX l −1 X X t=1

i

j

log a ¯ij

X

Pλ (q|o)δ(i, qt )δ(j, qt+1 )

(27)

q

where δ(i, qt )δ(j, qt+1 ) tells us to include only those cases for which qt = i and qt+1 = j. Therefore,

X

Pλ (q|o)δ(i, qt )δ(j, qt+1 ) = Pλ (qt = i qt+1 = j|o) = ξt (i, j)

(28)

q

Putting it together ¯ = Q(λ, λ)

X

γ1 (j) log π ¯j +

(29)

Tl X

(30)

j

+

XX i

log ¯bi (j) (

+

XX i

γt (i)δ(ot , j))+

t=1

j

log a ¯ij (

TX l −1

ξt (i, j))

(31)

t=1

j

When there is more than a sequence, we just need to add up over sequences to obtain the overall Q(·, ·) function ¯ = Q(λ, λ)

X

log π ¯j (

j

+

XX i

+

i

Tl XX

(32)

log a ¯ij (

j

γtl (i)δ(olt , j))+

(33)

t=1

l

XX

γ1l (j))+

l

log ¯bi (j) (

j

X

l −1 X TX

l

ξtl (i, j))

(34)

t=1

¯ function dependent on π Note that the part of the overall Q(λ, λ) ¯j is of the form wj log xj with xj = π ¯j and K X wj = γ1l (j) (35) l=1

with constraints

P

j

xj = 1, and xj ≥ 0.

¯ function dependent of ¯bi (j) is of the form Equivalently, the part of the overall Q(λ, λ) ¯ wj log xj with xj = bi (j) and

wj =

Tl K X X

γtl (j)δ(olt , j)

(36)

l=1 t=1

with constraints

P

j

xj = 1, and xj ≥ 0.

¯ function dependent of a Finally, the part of the overall Q(λ, λ) ¯ij is also of the form wj log xj with xj = a ¯ij and K TX l −1 X wj = ξtl (i, j) (37) l=1 t=1

with constraints

P

j

xj = 1, and xj ≥ 0.

It is easy to show Pthat the maximum of a function of the form wj log xj with constraints that xj ≥ 0 and j xj = 1 is achieved for wj xj = P j wj

(38)

¯ that maximize the overall Q(λ, λ) ¯ function easily The parameters of the new model λ follow:

π ¯i

=

¯bi (j)

=

a ¯ij

=

PK

γ1 (i) K

PK

PTl

l=1

l=1

γt (i)δ(olt ,j) P Tl l t=1 γt (i) l=1 t=1

PK

(39)

PTl −1 t=1 ξt (i,j) Pl=1 PTl −1 l K l=1 t=1 γt (i)

PK

2.4 Obtaining the E-M parameters from the scaled forward and backward algorithms Section under construction 2.5 MM Training MM training (Maximization Maximization) may be seen as an approximation to EM training. In MM training we basically substitute the Expected value operation by a Max operation thus the MM name. Other names for these algorithms are: Viterbi training (because we use the Viterbi algorithm to do the Max operation) and segmented K-means (K-means is a classical clustering method that belongs to the MM family). In Viterbi-based decoding, the degree of match between a model λ and an observation sequence ol is defined as ρ(λ, ol ) = maxql log Pλ (q l ol ) = log Pλ (ˆ q l ol ). We have seen how for a fixed model λ, the Viterbi recurrence can be used to find qˆl and ρ(λ, ol ). When there is more than one training sequences they are assumed independent P l and ρ(λ, o) = K l=1 ρ(λ, o ). Thus, the optimal states can be found by applying Viterbi decoding independently for each of the training sequences. In MM training the objective is to find an optimal point of log Pλ (q l ol ) with the model λ and the state sequence q as optimizing variables. To begin with, assume that Viterbi decoding has found the best sequence of hidden states ¯ such for the current λ model: qˆ = (ˆ q 1 , ..., qˆK ). Once we have qˆ we find a new model λ ˆ that log Pλ¯ (o qˆ) ≥ log Pλ (o qˆ). To do so simply define a dummy model λ such that Pλˆ (qo) = δ(ˆ q , o), thus the dummy model is such that only state sequence qˆ can co-occur with observation sequence o. For such model, Pλˆ (q1 = j|ol ) = δ(i, qˆl ), Pλˆ (qt = i|ol ) = δ(i, qˆt ), and Pλˆ (qt = i qt+1 = l ˆ λ) ¯ = log P ¯ (o qˆ), the function we want j|ol ) = δ(i, qˆt )δ(j, qˆt+1 ). Also note that Q(λ, λ

ˆ parameters in the standard E-M formulas guarantees to optimize. Thus, substituting the λ ˆ λ) ¯ ≥ Q(λ, ˆ λ) or log P ¯ (o qˆ) ≥ log Pλ (o qˆ) that Q(λ, λ Thus, the MM training rules are as follows:

π ¯i

=

¯bi (j)

=

a ¯ij

=

PK

δ(i,ˆ q1l ) K

l=1

PK

l=1

P Tl

δ(i,ˆ qtl )δ(j,olt ) P Tl δ(i,ˆ qtl ) t=1 l=1 t=1

PK

PK

l=1

(40)

PTl −1

l qt+1 ) δ(i,ˆ qtl )δ(j,ˆ PTl −1 l) δ(i,ˆ q t t=1 l=1 t=1

PK

Note that Viterbi decoding maximizes log Pλ (q, o) with respect to q for a fixed model (the first M step) then we maximizes log Pλ (q, o) with respect to λ, for a fixed q (the second M step). Since we are always maximizing with respect to some variables, log P λ (q o) can only increase and convergence to a local maximum is guaranteed.

3 Notation for Continuous Density Models The notation for the continuous case is the same as the discrete case with the following additional terms. P Number of dimensions per observation (e.g. cepstral coefficients): o t = (ot1 ...otP ). M. Number of clusters within a state. V = {v11 , ..., v1M ..., vN 1 ..., vN M }. Set of possible clusters of external observations (M clusters per state). Each cluster vij is identified by a state index i and a cluster index j. mt variable identifying the cluster index of the cluster that occurred at time t. For example if cluster vik occurred at time t then qt = i and mt = k. m = (m1 ...mt ). A sequence of cluster indexes, one per time step. gik = Pλ (mt = k|qt = j). Emission probability (gain) of cluster kvik by state Sj . φ(·). A kernel function (e.g. Gaussian) to model the probability density of a cluster of observations produced by a state. bj (ot ) = Pλ (ot |qt = j). µik = (µik1 , ..., µikP ). The centroid or prototype of cluster vik .   2 σik1 σik12 ... σik1p 2 σik2 ... σik2p   σ Σik =  ik21  The covariance matrix of cluster vik . ... 2 σikp1 σikp2 ... σikp 2 σikn . The variance of the l th dimension (e.g. cepstral) within cluster vik , a diagonal element of Σik .

σiknm , The covariance between dimension l and dimension m within the cluster v ik an off-diagonal elements of Σik . Usually assumed zero.

4 Continuous Observation Models In this section we study the E-M and Viterbi training procedures for continuous observation HMMs. 4.1 Mixtures of Densities In the discrete case the observations are discrete, represented by an integer. In the continuous case the observations at each time step are P-dimensional real-valued vectors (e.g. cepstrals coefficients). In our notation ot = (ot1 , ..., otP ). The continuous observation model produces sequences of observations in the following way: At each time step the system generates a hidden state qt according to a a state to state transition probability distribution aqt−1 qt . Once qt has been generated, the system generates a hidden cluster mt according to a state to cluster emission probability distribution gqt mt . Once the hidden cluster has been determined, an observation vector is produced probabilistically according to some kernel probability distribution (e.g. Multivariate Gaussian). We can think of the clusters as low level hidden states embedded within high level hidden states qt . For example, the high level hidden states may represent phonemes and the low-level hidden clusters may represent acoustic categories within the same phoneme. For simplicity each state is assumed to have the same number of clusters (M) but the set of clusters is different from state to state. Thus, there is a total of NxM clusters, M per state.

We represent cluster k of state Sj as vjk and we use the variable mt to identify the cluster number within a state at time t. Thus if qt = i and mt = k it means that at time t cluster vjk occurred. If we know the cluster at time t, the probability density of a vector of continuous observations (e.g. cepstral coefficients) is modeled by a kernel function, usually a multivariate Gaussian Pλ (ot |vjk ) = Pλ (ot |qt = j mt = k) = φ(ot , µjk , Σjk ) (41) Where φ(·) is the kernel function (e.g. multivariate Gaussian) µjk = (µjk1 , ..., µjkP ) is a centroid or prototype that determines the position of the cluster in P-dimensional space, and  2  σjk1 σjk12 ... σjk1P  σjk21 σ 2 ... σjk2P  jk2  Σjk =  (42)  ...  2 σjkP 1 σjkP 2 ... σjkP is a covariance matrix that determines the width and tilt of the cluster in P-dimensional 2 2 space. The diagonal terms {σjk1 ...σjkP } are the cluster variances for each dimension. They determine the spread of the cluster on each dimension. The off-diagonal elements are known as the cluster covariances and they determine the tilt of the cluster. For the Gaussian case, the kernel function is φ(ot , µjk , Σjk ) =

1 e−d(ot ,µjk ) (2π)P/2 |Σjk |1/2

(43)

where |Σjk | is the determinant of the variance matrix, and d(ot , µjk ) is known as the Mahalanobis distance between the observation and the kernel’s centroid. 1 0 d(ot , µjk ) = (ot − µjk )Σ−1 (44) jk (ot − µjk ) 2 Since the covariance matrices Σjk are symmetric, each kernel is defined by P (P2+3) parameters (P for the centroids and P (P + 1)/2 for the variances). In practice the off-diagonal variances are assumed zero, reducing the number of parameters per kernel to 2P. In such QP 2 case the determinant |Σjk | is just the product of P scalar variances |Σjk | = l=1 σjkl , and the Mahalanobis distance becomes a scaled Euclidean distance: P

d(ot , µjk ) =

1 X (otl − µjkl )2 2 2 σjkl

(45)

l=1

Given a state Sj , the system randomly chooses one of its M possible clusters with state to cluster emission probability P (mt = k|qt = j). This probability is assumed independent of t and thus it can be represented by a parameter with no time index. In our notation P (mt = k|qt = j) = gjk , the gain of the k th cluster embedded in state Sj . Thus, the overall probability density of the observations generated by a state S j is given by a weighted mixture of kernel functions.

Pλ (ot |qt = j) =

M X

P (mt = l|qt = j)P (ot |qt = j, mt = k) =

(46)

l=1

or bj (ot ) =

M X

k=1

gjk φ(ot , µjk , Σjk )

(47)

4.2 Forward and backward variables The un-scaled and the scaled algorithms work the same as in the discrete case. Only now the emission probability terms bj (ot ) are modeled by a mixture of densities. M X

bj (ot ) =

gjk φ(ot , µjk , Σjk )

(48)

k=1

4.3 EM Training In the continuous case the clusters are low level hidden states mt embedded within high level hidden states qt . Thus, the E-M function is defined over all possible q m sequences of high-level and low-level hidden states ¯ = Q(λ, λ)

XX q

Pλ (qm|o)log Pλ¯ (qmo)

(49)

m

and since ¯ q1 m1 ), ..., a ¯ qT mT ) Pλ¯ (qmo) = π ¯q1 g¯q1 m1 φ(o1 , µ ¯ q1 m1 , Σ ¯qT −1 qT g¯qT mT φ(oT , µ ¯ qT mT , Σ (50) it follows that ¯ = Q(λ, λ)

XX q

T −1 X X X q

t=1

+

+

T XX X q

t=1

Pλ (qm|o)log a ¯qt qt+1 +

(52)

Pλ (qm|o)log g¯qt mt

(53)

¯ qt mt ) Pλ (qm|o)log φ(ot , µ ¯ qt mt , Σ

(54)

q

m

m

Since the factors in the first two terms are independent of m they simplify into XX X Pλ (qm|o)log π ¯ q1 = Pλ (q|o)log π ¯ q1 q

(51)

m

T XX X t=1

Pλ (qm|o)log π ¯ q1 +

m

m

(55)

q

and T −1 X X X t=1

q

Pλ (qm|o)log a ¯qt qt+1 =

m

T −1 X X t=1

Pλ (q|o)log a ¯qt qt+1

(56)

q

These terms are identical as in the discrete case and thus the same training rules for initial state probabilities and for state transition probabilities apply here. To find the training formulas for the cluster gains, we focus on the part of Q(·) dependent on the gain terms. This part can be transformed as follows T XX X t=1

q

m

Pλ (qm|o)log g¯qt (mt ) =

(57)

=

T X N X M XX X q

t=1 i=1 k=1

=

Pλ (qm|o)log g¯ik δ(i, qt )δ(j, mt )

(58)

m

T X N X M X

Pλ (qt = i mt = k|o)log g¯ik

(59)

t=1 i=1 k=1

Thus the part of Q(·) that depends on g¯ik is of the form wj log xj with xj = g¯ik and wj =

T X

Pλ (qt = i mt = k|o)

(60)

t=1

P

xj = 1 and xj ≤ 0, with maximum achieved for wj xj = P j wj

(61)

PT PT Pλ (qt = i mt = k|o) t=1 Pλ (qt = i mt = k|o) = g¯ik = PT t=1 PM PT k=1 Pλ (qt = i mt = k|o) t=1 t=1 Pλ (qt = i|o)

(62)

with constraints

j

Thus

To find the learning rules for the centroids and variances we focus on the part of Q(·) that depends on the cluster centroids and variances, which is given by the following expression: T XX X q

t=1

¯ qt mt ) Pλ (qm|o)log φ(ot , µ ¯ qt mt , Σ

(63)

m

This expression can be transformed as follows T XX X t=1

=

q

¯ qt mt ) = Pλ (qm|o)log φ(ot , µ ¯ qt mt , Σ

T X N X M XX X t=1 i=1 k=1 q

=

(64)

m

¯ ik )δ(i, qt )δ(k, mt ) Pλ (qm|o)log φ(ot , µ ¯ik , Σ

(65)

m

T X N X M X

¯ ik ) Pλ (qt = i mt = k|o)log φ(ot , µ ¯ik , Σ

(66)

t=1 i=1 k=1

Thus, at a maximum, T N M ∂ XXX ¯ ik ) = 0 Pλ (qt = i mt = k|o)log φ(ot , µ ¯ik , Σ ∂µ ¯ikn t=1 i=1

(67)

¯ ik ) ∂log φ(ot , µ ¯ik , Σ 1 ) = 2 (otl − µ ¯ikn ) ∂µ ¯ikn σ ¯ikl

(68)

k=1

and since

it follows that at a maximum T X Pλ (qt = i mt = k|o)(otl − µ ¯ikn ) = 0 t=1

(69)

or PT

t=1

µ ¯ikn =

Pλ (qt = i mt = k|o) otn PT t=1 Pλ (qt = i|o)

(70)

2 A similar argument can be made for the diagonal variance σikl . In this case

¯ ik ) 1 (otn − µ ¯ikn )2 ∂log φ(ot , µ ¯ik , Σ = − 2 (1 − ) 2 2 ∂σ ¯ikn 2¯ σikn σ ¯ikn

(71)

Thus, at a maximum T X

Pλ (qt = i mt = k|o)(1 −

t=1

(otl − µ ¯ikn )2 )=0 2 σ ¯ikn

(72)

from which the re-estimation formula easily follows:

σ¯2 ikn =

PT

t=1

Pλ (qt = i mt = k|o)(otl − µ ¯ikn )2 PT t=1 Pλ (qt = i|o)

(73)

Training for the mixture gains, mixture centroids, and mixture variances requires the P (qt = jmt = k|o) terms, for t = 1...T , j = 1..N , k = 1...M . To obtain these terms note the following: P (qt = j mt = k|o) = P (qt = j|o)P (mt = k|qt = j o) =

(74)

= P (qt = j|o)P (mt = k|qt = j ot )

(75)

P (ot mt = k|qt = j) P (ot |qt = j)

(76)

P (mt = k|qt = j)P (ot |qt = jmt = k) P (ot |qt = j)

(77)

gjk φ(ot , µik , Σik ) bj (ot )

(78)

= P (qt = j|o)

= P (qt = j|o) Thus

P (qt = jmt = k|o) = P (qt = j|o)

As in the discrete case, the P (qt = j|o) can be obtained through the scaled feed-forward algorithm. For the case with multiple training sequences the overall Q(·) decomposes into additive Ql (·), one per training sequence. As a consequence we have to add in the numerator and denominator of the training formulas the effects of each training sequence. Summarizing, the E-M learning rules for the mixture of Gaussian densities case with diagonal covariance matrices are as follows:

π ¯i

=

a ¯ij

=

g¯ik

=

µ ¯ikn

=

2 σ ¯ikn

=

PK

Pλ (q1 =j|ol ) K

PK

PTl −1

l=1

l=1

Pλ (qt =i qt+1 =j|ol ) PTl −1 l t=1 Pλ (qt =i|o ) l=1 t=1

PK

P K P Tl P (q =i mt =k|ol ) l=1 PK t=1 PT λ t l l=1 t=1 Pλ (qt =i|o )

(79)

PK

P Tl l l t=1 Pλ (qt =i mt =k|o ) otn P T l) P (q =i|o t λ t=1

PK

P Tl

l=1

l=1

Pλ (qt =i mt =k|ol ) (otn −¯ µikn )2 P K P Tl l t=1 Pλ (qt =i|o ) l=1

t=1

where l = 1, ..., K indexes the training sequence, t = 1, ..., Tl indexes the time within a training sequence, i = 1, ..., N and j = 1, ..., N index the hidden state, k = 1, ..., M indexes the cluster embedded within a state, and n = 1, ..., P indexes the dimension of the continuous vector of observations (e.g. the cepstral coefficient). The P (q t = jmt = k|o), Pλ (qt = i|ol ) and Pλ (qt = i qt+1 = j|ol ) terms are obtained from the scaled forwardbackward algorithms according to the following formulas:

P (qt = i, mt = k|o)

= P (qt = i|o) gik

bi (ot )

=

φ(ot , µik , Σik )

=

PM

k=1

φ(ot ,µik ,Σik ) bi (ot )

gik φ(ot , µik , Σik )

(2π)P/2

1 Q P

n=1

σikn

e−d(ot ,µik ) (80)

PP

otn −µikn 2 ) n=1 ( σikn

d(ot , µik )

=

1 2

Pλ (qt = i|ol )

=

α ˆ l (i)βˆl (i) P t l tˆl ˆ t (i)βt (i) iα

Pλ (qt = i qt+1 = j|ol )

= α ˆt (i)aij βˆt+1 (j)bj (olt+1 )

4.4 Viterbi decoding We can use Viterbi decoding to find the best possible sequence of high level hidden states q and low level clusters m. There are two approaches to this problem. One approach attempts to find simultaneously the best joint sequence of high-level and low-level states. Thus the goal is to find qˆm ˆ = arg maxqm Pλ (qm|o). The second approach first finds the best possible sequence of high level states qˆ = arg maxq Pλ (q|o) and once qˆ has been found, m ˆ is defined as m ˆ = arg maxm Pλ (m|ˆ ˆ q o). The two approaches do not necessarily yield the same results. The second approach is the standard in the literature. For the second, most used version, of Viterbi decoding, the same Viterbi recurrence as in the discrete case applies but using the continuous version of b i (ot ). Once we have qˆt , the desired m ˆt is simply the cluster within Sqˆt which is closest (in Mahalanobis distance) to ot . 4.5 Viterbi training The objective in Viterbi training (also known as segmental k-means) is to find an optimal point (local maximum) of log Pλ (oˆ q m) ˆ with q and λ being the optimizing variables. It does not matter how qˆm ˆ are found as long as a consistent procedure is used ˆ such that throughout training. As in the discrete case we define a dummy model λ l l l l l l l l l Pλˆ (q m |o) = δ(ˆ q , q )δ(m ˆ , m ). For such model, Pλˆ (q1 = j|o ) = δ(i, qˆ1 ), Pλˆ (qtl = l l l l l i|o ) = δ(i, qˆt ), Pλˆ (qt = i qt+1 = j|ol ) = δ(i, qˆtl )δ(j, qˆt+1 ), and Pλˆ (qtl = i mlt = k|ol ) = l l δ(i, qˆt )δ(k, m ˆ t ). ˆ λ) ¯ = log P ¯ (o qˆm), As in the discrete case note that Q(λ, ˆ the function we want to opλ ˆ timize. Thus, substituting the λ parameters in the standard E-M formulas guarantees that ˆ λ) ¯ ≥ Q(λ, ˆ λ), and thus log P ¯ (o qˆm) Q(λ, ˆ ≥ log Pλ (o qˆm) ˆ λ Thus, the Viterbi training rules are as follows:

π ¯i

=

a ¯ij

=

g¯ik

=

µ ¯ikn

=

σ¯2 ikn

=

PK

δ(i,ˆ q1l ) K

l=1

PK

l=1

PTl −1

l δ(i,ˆ qtl )δ(j,ˆ qt+1 ) PTl −1 l qt ) t=1 δ(i,ˆ l=1 t=1

PK

PK PTl δ(i,ˆ qtl )δ(k,m ˆ lt ) l=1 PK t=1 PT qtl ) l=1 t=1 δ(i,ˆ

PK

PTl δ(i,ˆ qtl )δ(k,m ˆ lt ) oltn t=1 PT l) δ(i,ˆ q t t=1

PK

PTl

l=1

l=1

(81)

δ(i,ˆ qtl )δ(k,m ˆ lt )(otn −¯ µikn )2 P K P Tl l qt ) t=1 δ(i,ˆ l=1

t=1

Since Viterbi decoding maximizes Pλ (q, m, o) with respect to q and m and Viterbi training maximizes Pλ (q, m, o) with respect to λ, repeatedly applying Viterbi decoding and

Viterbi training, can only make Pλ (q, m, o) increase and convergence to a local maximum is guaranteed.

5 Factored Sampling Methods for Continuous State Models Many recognition problems can be framed in terms of infering something about q t the internal state of a system, based on a sequence of observations o = o 1 · · · ot . These inferences are in many cases based on estimates of p(qt |o1 · · · ot ). When the states are discrete and countable, these conditonal state probabilities can be obtained using the forwards algorithm. However, the algorithm cannot be used when the states are continuous. In such case, direct sampling methods are appropriate. Here is an example of how these methods work. We start with a sensor model: p(ot |qt ) and a Markovian state dynamics model p(qt+1 |qt ). Our goal is to obtain estimates of p(qt |ot ) for all t. 1. Recursion Assume we have an estimate pˆ(qt |ot ). Our goal is to update that estimate for the next time step p(qt+1 |ot+1 ). (a) First we draw a random sample X from pˆ(qt |ot ). This sample will implicitely define a re-estimation of p(qt |ot ) in terms of a mixture of delta functions: P pˆ(qt |ot ) = N1 N i=1 δ(qt , xi ) (b) For each observation xi we obtain another random observation yi using the state dynamics p(qt+1 |qt = xi ). The new sample Y = {y1 · · · yn } implicitely defines our estimates of p(qt+1 |ot ). PN p(qt |ot ) = N1 i=1 δ(qt+1 , yi ) (c) We know that p(o1 · · · ot ) p(qt+1 |o1 · · · ot+1 ) = p(qt+1 ot+1 |o1 · · · ot ) = (82) p(o1 · · · ot+1 ) p(o1 · · · ot ) p(qt+1 |o1 · · · ot )p(ot+1 |qt+1 ) p(o1 · · · ot+1 ) The fraction is a constant K(ot+1 ) independent of qt+1 , we already have an estimate of p(qt+1 |ot ) so we just need to weight it by p(ot+1 |qt+1 ). P pˆ(qt+1 |ot+1 ) = K N1 N i=1 δ(qt+1 , yi )p(ot+1 |qt+1 ) = N

=

(N )

P

X 1 δ(qt+1 , yi )p(ot+1 |yi ) i p(ot+1 |yi ) i=1

We can now use pˆ(qt+1 |ot+1 ) to estimate parameters like the mean or the variance R of the distribution. More generally, ω ˆ = dqt+1 Q(p(qt+1 |ot+1 ), qt+1 ) 2. Initialization The initialization step is basically the same as the recursion step only that instead of using the state transition probabilities we use the initial state probabilities (a) Obtain a sample of N random states : X= {x1 · · · xN } from the initial state probability function π(·). These N samples will implicitely work as our estimate of the initial state probability. pˆ(q1 ) =

N 1 X δ(q1 , yi ) N i=1

(83)

(b) Weight each observation by the sensor probability p(o1 |q1 = yi ). This defines our initial distribution estimates pˆ(q0 |o1 ) =

1 N

PN

i=1

p(o1 |xi )

δ(q0 , yi )p(o0 |yi )

(84)



  !#"$&%')(  "$*,+-.*

A h

/0

*21

0436587

BDCFEGIHKJLNMOGIPRQFCTSUSVGWHEYX[Z]\#EGIH_^

    9:;12-2 " +-?>@;12
+-,$  * 3 3=< 3 5

L`\&aUJ)PTZ]bYcdGWHfegL EEZ]HKP

Z]\iGIjIPUJEJGIP kIlUmon)prqtsvu`lUw myx?z{k|z?}I~m|€w=p „@…‡†ˆ†
‰‹ŠŒ.Ž,Š Ž;:…‡‘,’8‰”“ •ƒ…—–-?˜™…ƒ•š–™† .›:…‡‘

( -.*ƒ‚@ - 1  ! 3 5 0

œŸž$ ‡¡
¢£¤ *2!¥¥¦-‡§§ ¨]©Ÿª  «¢«¬™­4 ® < +-¯12)1 0 ,"$4*,1.,1.*¥r" 1,+o 3o0‡°  ° "$; 0 91,+-± 043 —;² 12* 0 9³´µ"$!¥ 3 *2" 043  3 |*,1.,-;12-24,"$*2¶ !#" "$!#"$·41," ¹1,+- )—*22" (d¥ " —¹º)² ² 21?³'—;1  .+ " * º³ *  9 *.;²-.4()$i  *2;²-.4(   5i¸ 1,+-  4+T 3  7 t!¥0 .+- "$4K 4 3 4§ 3-04 3 » F)"$*.;-*2*  4123   2¶04" 36¼ 122" "$3 4K;½)!¹² $0 `" )120 4"  7 `3  *.2" (
` 3 2¥4 5¾ r+- º³ 0 3 *   1," 0 *¹.¹ 3 04"$3 ¿»-]°  Wr+- 12+ §W (-4¸  ¥ *2—," (d¥+ V*2 3 ² ² ,1 3 ;0 1 K12.3 4" " ` 0 "$!#² 04 3 —!# 12— 3  `)3" *.;-*.*@" 3  ;1.4" '1,+-K¶0 — ;5] ¸ 4² ² " ¯12.+ 0" ¿-¹+)"$0 .+`"$°* *.0 =1  3 3 *,1.,-;3 1 (d&² 3 .;12" —7 4  §` À!¥ º³ *   1," *rr+ "$.+¯3.  "3 r" 1,+-3 )41. *2+ o3 + Uº ² ² 3 21Á³Á;31 r!¥2+ " *r +   0 04,3 §¥$. 7 0 043 3-043 3 3 5À¸ 0 0 0 0 3 3 ° °  °  3 " 3  3 " 1.  ³´Ã)" !¥ 3 *2" 043 (§` 0 !#²  1," 3  1,+-#³´U)"$!¥ 3 *," 043 9 0 + 0 ! 0  3  0 -*&² 0  § 3-0 !#" 48 3 IÄ4-*2*2"$ 3 ."$4r(-*2"$*¥9ˆ ;1," ¶— ;$* + " $ 2§¼+)"$4+|³´Å)"$!# *2"   $ 2!¥4  §v(  ±"  r9 ¥ 24 "$·412" 3 *2+ 0 *Á1,+-3-40 1r ] 0 .4 "$·41,0 " ²d3 29 .!¥ 043  ²d29 .!¥   3 ¥r043 + " $43 1Á²-.5t *. ¸ 181,+-.° ;½"$*212* 12+  ,§ir3 + "$.04+¯ "$* ›“)0 ŠŽ2Š–-3 {….…ƒÆ 9 3 &º³ * 1,+-.i3 .¹*. .4Á.4-3-!¥0  1.*?0 r+ "$2+¼*2)² ² 0 21?1,+- (-*20—0  I3 + "$4+t 04.3 ;§ 0 9º³ 3 * r+ "$.+`8&2 "$;0 7 *,  12* 9'* !¥K;½²d° 2" !¥ 12*r+ 3 "$.+`2" *2²)"$.0 `(§¯1,+-0 *.&.° 4-!¥ 1.*@.K4$* ² 0 .*. 1.7  @4"  -!¥°  -5*ÁÇ ½!#²  —0 * 0 ¥²- 9È* 96! 3 *,1 9
12+ ¶;§#1,+-3  .!¥* + —2" * ;!#43 1.2" 4  i0 ÉÀ+ ²
3 12+ 41 5 ¸1,+-K.°  3 "   0 ¯12+-41@  3 $=!#040 1.20" 46"$*0 —*,0 1@" #9:.*2+¯ "$0 4+1 5”< 3 3 0  3 ° 3=0 3 5 Ê Ë̋Í&¡d¢4 ª £ º ² ² 0 ,1³';1 0  7 .+ " 3 * º12412"$*,12" —4'Î6 3 " 3  < -+  0 2§ ³´FÏ" !¥ 3 2* " 40 3‹ %”41,12— 3`Ç —  0  3 " 12" 40 3 Ð zÑnÒ8ÓÔ ÕÖ@×ؙÓ)Ù,ÕÁÒ Ú¥ÛÀÜNݔÞÀ߇ÝÁà
á‡Ü`à™âã‡Û”ä á]ݔå™ÝÁÜß]ä{á¥ã«àvÝÀß«à-æ‹ä{çŸÜ=å™èUä è‹ã‡ß«àŸçŸÞ”éã‡àdß«êvêdÜã¯Üë6ã«Üè”á‡ä ædܱã‡ÞŸã«àdß«ä ådìàdèTã«ÛÀÜNí”å
á—ä{é ä{çŸÜå
á¯íÁÜ۔äÈè'çïî6ޔÝÀÝ'à
ߗã`ð?Ü)éã‡à
ß`ñÃådé4ÛÀä èÀÜ)áIò.î6ð±ñ¾á«óô¼Ú¥ÛÀܼíÁà6àdõŸá¼òöðKå™Ý”èÀä õ'÷Kø)ùdù‹ú6ûrðKå™Ý”èÀä õ'÷Kø)ùdù
ü
ó éà
è‹ã«å™ä èÜëÀéÜì ìÈÜè‹ã#çŸÜ)á‡é߇ä ݟã«äÈà
è”á#à™âKî6ð±ñ¾á÷ÁíÀޟã]ã«ÛÀÜêFìÈÜ)å ædÜ ß‡à6à
ýþâöàd߯ådèUådééàdÞÀè‹ãiÿiÛÀà‹á—Ü`ÝÀޔ߇ÝÁà
á‡Ü âöß«àdý ã‡ÛÀÜáã4å™ß‡ãIä{á`ã«àÃã«Üådé4Ûrô
ìÈã‡Û”àd Þ 
ÛÅã‡ÛÀÜá—Þ” í ÜéãtéådèµíÁÜFá«å™ä{çïã‡à ۔å ædÜoá—ã«ådߗã«ÜçVä èÅã«ÛÀÜoì{å-ã«Ü á‡ÜædÜè‹ã‡ä ÜáUò,ðKå™ÝÀèÀä õÁ÷Nø ù -ù‹ó4÷iäÈã|ä á|àdèÀì êOè”à-ÿ_ß«ÜéÜä æ6ä  è  ä è”é߇Ü)ådá‡ä  è ïå™ã—ã«Üè‹ã‡ä àdèr÷iådè”ç á‡àÅã‡ÛÀܾã‡ä ývÜ å™Ý”Ý'Ü)å™ß4áIá‡ÞÀäÈã«ådíÀì Üvâöàdßvå™è ä è
ã«ß‡àŸçŸÞ'éã‡à
߇êïß«Üæ6ä Üÿ=ôµÚ¥ÛÀÜFã‡ÞŸã«àdß«ä ådì#çÀÿ¹ÜìÈì{áWÜè‹ã‡ä ß«Üì ê à
èµã«ÛÀÜFÝ'å-ã—ã«Üß«è ß«Üéà dèÀäÈã«äÈà
èoÝÀß«àdí”ìÈÜýTô#ñÃådè6êoà™â?ã«ÛÀÜNä{çŸÜ)ådá¥ã«ÛÀÜß«ÜWéåd߇߫êoçŸä ß«Üéã‡ì ê|à-ædÜßiã‡à|ã‡ÛÀÜWéådá‡Üáiàdâ?߇ Ü dß«Üá«á—ä àdè Üá—ã‡ä ý|å™ã‡ä àdè¼å™è”ç|ìÈä èÀÜ)å™ß?à
Ý'Üß«å™ã‡àdß&äÈè6æ
Üß4á—ä àdèr÷™íÀޟãiá—Ý”å
éܯéàdè”á—ã‡ß4å™ä è‹ã«áKÝÀ߇Ü)éì ޔçŸÜ)çtã‡ÛÀÜ ÜëŸÝÀì à
ß«å™ã‡ä àdèIà™â ã‡Û”Üá‡Ü=ã‡àdݔä éá#ÛÀÜ߫ܙô Ú¥ÛÀܼã«ÞŸã‡à
߇ä{ådìéàdè‹ã«ådäÈè'á á‡àdývÜIèÀÜÿ ý|å-ã«Üß«ä{å™ì.ô
ì ì?à™â¹ã‡ÛÀÜvÝÀß«à6à™â,á=å™ß«ÜtýIêUà-ÿièïædÜß«á‡ä àdè”á÷ÿiÛÀÜß«Ü

i۔å æ
ÜNÝÀì{ådéÜçUåoá—ã‡ß«àdè|ÜývÝÀÛ'ådá‡ä á]àdèUã«ÛÀÜä ߯íÁÜä èFí'àdã‡Ûïéì Üåd߯å™è”ç¾á—Üì:â ;éà
è‹ã«å™ä èÀÜ)ç8÷Àã«àoý|å™õ
Ü`ã‡Û”Ü ý|å-ã«Üß«ä{å™ìiådáFådééÜá«á—ä íÀì ÜTådávÝÁà
á«á‡äÈí”ìÈÜdôgÚ¥ÛÀä{ávÿ¥ådáoçŸàdèÀÜ å-ã|ã«ÛÀܾÜë6ÝÁÜè'á—ÜÃàdâNá—à
ývÜÜì Ü
 ådè”éÜUådè”ç dÜèÀÜß4å™ì äÈãê±  ÛÀà-ÿ#Üæ
Üßd ÜèÀÜß4å™ì äÈãêFä{á=ޔá‡Þ”ådìÈì êTÜå
á—ä ì êUådçÀçÀÜçïàdè”éÜtã‡ÛÀÜví”å
á—ä{éIä{çŸÜå
á åd߇ÜvéìÈÜ)å™ßô¼Ú¥Û”Ü ì àdè
 Üß#ÝÀß«à‹àdâ,áiå™ß«Ü`éàdì ì Üéã‡Ü)çvä èoã«ÛÀÜ
ÝÀÝÁÜè'çŸäÈëÁô

#êïÿ#å ê à™âiývà™ã«äÈæ-å™ã‡ä àdè÷å™è”çïã«à¾å™ì Ü߇ã=ã‡ÛÀÜF߇Ü)ådçŸÜß=ã‡à¾á‡àdývÜvàdâ¥ã‡ÛÀÜ|ì äÈã‡Üß«å™ã‡ÞÀ߫ܙ÷@ÿ¹Üoá‡ÞÀývý|åd߇ä Ü ‡á àdývÜ]ß«ÜéÜè‹ã#ådÝÀÝÀì ä{éå-ã«ä àdè”áå™è'ç|Üë6ã«Üè”á‡ä àdè”á?àdâ@á‡ÞÀÝÀÝÁàd߇ã¹ædÜ)éã«àdß&ý|å
é4ÛÀä èÀÜáô”àdß¹ã‡ÛÀÜ Ý”å™ã—ã‡Ü߇èo߇Ü)éà èÀäÈã‡ä à
èVéådá‡Ü™÷Kî6ð±ñ¾áW۔å æ
Üví'ÜÜèµÞ'á—Ü)çVâöàdßNä{á‡àdì{å-ã«Üç ۔å™è'çŸÿi߇äÈã‡ã‡ÜèµçÀä
äÈã=ß«Üéà
èÀäÈã‡ä àdèDò¹àd߇ã‡Ü)áWådè”ç ðKå™Ý”èÀä õ'÷¥øù
ù
úŸûKîÀé4 Û àd ì õdà
ݟâ; ÷ #ÞÀß dÜ)áNå™è'çVðKå™ÝÀèÀä õÁ÷¥øùdù‹ú6û&îŸé4 Û à
 ì õdàdÝÀâ; ÷ #ÞÀß dÜáWå™è”çVðKådÝÀèÀä õÁ÷#ø)ùdù Àû #ÞÀß dÜáiådè”çÃîŸé4 Û à
 ì õdàdÝÀâ;÷røùd ù ™ó÷”àd í Ü)éã]ß«Üéà dè”ä:ã«ä àdè ò #ì åd è =Üã±ådì.ô:÷ø)ùd ù 
ó4÷Áá—ÝÁÜådõdÜßiä{çŸÜè‹ã‡ä 'éå™ã‡ä àdè ò2îÀé4ÛÀývä{ç6ã÷Kø)ùdù ‹ó4÷ré4۔åd߇ývÜ)" ç !‹Þ”å™ß«õÃçŸÜã«Üéã‡ä àd è # ÷râ,ådéÜtçÀÜã‡Ü)éã«äÈà
è ä è äÈý|å dÜ)áWò $ á‡ÞÀè”åÀ ÷ Àß«ÜÞÀè'ç ådè”ç % ä ߇à‹á—ä.÷ øùdù -å‹ó4÷#ådè”çDã‡Üë6ãFéå™ã‡Ü 
àdß«ä å-ã«ä àdè ò &™à‹ådé4ÛÀä ý|á÷¯øù
'ù ™ó( ô ”àdß|ã«ÛÀÜÃß«Ü 
߇Ü)á‡á‡ä àdèµÜ)áã«äÈý|å™ã‡ä àdè éå
á—Üd÷î6ð±ñÃá ۔å æ
ÜWíÁÜÜè éà
ývݔå™ß«ÜçTà
è¾íÁÜè”é4۔ý|å™ß«õoã«äÈývÜtá—Ü߇ä Ü)áiÝÀß«ÜçŸä{éã«ä àdèTã‡Ü)áã4á¼ò,) ñ ÞÀ ì ì Üß±Üã`å™ì.ôÈ÷ øù
' ù 6û¹ñUÞÀõ6ÛÀÜ*ß Üܙ+ ÷ $ á‡ÞÀè”å¾ådè”ç % ä ß«à
á‡ä.÷#ø)ùd' ù dó4÷ã‡Û”, Ü #à
á—ã‡à
èµÛÀà
ޔá—ä  è ¾ÝÀ߇à
íÀì Üý *ò -±ß‡Þ'é4õdÜßIÜãvå™ì.ôÈ÷ øù
' ù dó4÷å™è'çOòöà
èïådߗã«ä 'éä{å™ìçÀå™ã«å
ó¯àdè ã‡Û”/ Ü .10&Ú àdÝÁÜß4å-ã‡à
ß±ä è6ædÜß«á‡ä àdèTÝÀß«àdíÀì Üý òöðKå™Ý”èÀä õ'÷ % à
ì à-ÿiä é4Û

å™è'çFî6ývà
ì åÀ÷Áø)ùdù 
ó4ô ;èývà
á—ã¹à™â@ã‡ÛÀÜ)á—Ü=éå
á—Ü)á÷”î6ð ñ 
ÜèÀÜß«ådì ä )å-ã«äÈà
èIÝÁÜ߇âöàdß«ý|å™è'éÜWò,ä2ô ܙô?Ü߇߫àdß#ß«å™ã‡Ü)á àdèFã‡Ü)áã¯á‡Üã4á«ó¹Üä:ã«ÛÀÜߥý|å™ã«é4ÛÀÜ)á¹à
ߥä á¥á‡ädèÀä'éå™è‹ã‡ì êtí'Üã—ã‡Üߥã‡Û”ådèã‡Û”å™ã¥àdâéàdývÝÁÜã«äÈèWývÜã«ÛÀàŸçÀáô?Ú¥Û”Ü Þ”á‡Ü±àdâ@î6ð±ñ¾á&âöà
ßiçŸÜè”á‡äÈãêIÜ)áã«ä ý|å-ã‡ä à
èTò
ïÜ)áã«àdèFÜãiå™ì.ô:÷8øù
' ù dó?å™è”ç
 $]ð
çŸÜ)éà
ývÝ'à‹á—äÈã‡ä à
èò.î‹ã‡äÈã á‡àdè¾Üã=ådì.ô:÷?øù
' ù dói۔å
á±ådì á‡à|íÁÜÜèïá—ã‡Þ”çÀäÈÜ)ç8 ô i Ü 
å™ß4çŸä  è |Üë6ã‡Üè'á—ä àdè'á÷Áã‡ÛÀܼí”å
á—ä{éWîŸð±ñÃá`éàdè‹ã«ådäÈèUè”à ÝÀß«ä àdßiõ6èÀà-ÿiì Ü ç 
Ü=à™âã‡ÛÀÜIÝÀß«àdíÀì Üý ò{âöàdß±ÜëÀå™ývݔìÈÜd÷'å¼ì{å™ß dÜ`éì{å
á‡á¥àdâ¹î6ð±ñÃá]âöàd߯ã‡ÛÀÜNä ý|å dÜ=߇Ü)é à dèÀä ã‡ä à
èݔ߇à
íÀì ÜýYÿ#àdޔì , ç 
äÈæ
Ü ã‡ÛÀÜtá‡ådývÜ`߇Ü)á—ÞÀìÈã4áiäÈâ@ã«ÛÀÜWÝÀäÈëŸÜì{á¥ÿ#Üß«Ü ”ß4áã Ý'Ü߇ýtޟã‡Ü)çß«ådè”çŸàdývì êÃò,ÿiä:ã«Û Üå
é4ÛFä ý|å dܯá‡ Þ 8Üß«äÈ è Wã«ÛÀÜNá‡ådývܱÝÁÜß«ýtޟã«å™ã‡ä àdè'ó÷6ådèå
éã]àdâ@æ-å™è”ç”å™ì ä{á—ý ã‡Û'å-ã]ÿ#àdÞÀì{ç|ì Ü)å ædÜ]ã«ÛÀÜ`íÁÜá—ã ÝÁÜ߇âöàdß«ýväÈ è UèÀÜÞÀß4å™ì¥èÀÜãÿ¹à
߇õŸáIá—ÜædÜß«Üì ê ۔å™è”çÀä éå™ÝÀÝÁÜç'óWådè”çÅýtޔé4Ûµÿ#àdß«õïÛ'ådáIí'ÜÜè çŸàdèÀÜFàdè äÈ è éà
߇ÝÁàdß4å-ã«ä  è WÝÀß«ä àdߥõ6èÀà-ÿiì Ü ç 
ܯä è‹ã‡à|î6ð ñÃáNò2îÀé4 Û àd ì õdà
ݟâ; ÷ #ÞÀß 
Üáiådè”çðKå™ÝÀèÀä õÁ÷røù
ù ÀûÀîÀé4 Û àd ì õdà
ݟâÜã å™ì.ôÈ÷øùdù
üdåÀ û #ÞÀß dÜ)á÷ø)ùdùdü‹ó4ô
ìÈã‡Û”àd Þ 
ÛTî6ð ñÃá¥Û”å æ
 Ü 
à6à6ç dÜèÀÜß4å™ì ä å™ã‡ä àdètÝÁÜ߇âöàdß«ý|å™è”éܙ÷6ã‡ÛÀÜêoéådè íÁÜIå™í6êŸá‡ý|å™ì ì êtá‡ì à-ÿ ä èTã‡Ü)áã¯Ý”Û”ådá‡Ü™÷8åvÝÀß«àdíÀì Üý_ådçÀçÀ߇Ü)á‡á‡Üçä èÅ*ò #ÞÀß dÜá÷@ø)ùd ù Ÿû $ á‡ÞÀè”åvådè”ç % ä ߇à‹á—ä.÷ øù
ùdü‹ó4 ô ]ÜéÜè‹ãNÿ¹à
߇õU۔å
 á dÜè”Üß4å™ì ä Üçã«ÛÀÜví”å
á—ä{éIä çÀÜådátò2îŸývàdì{åŸ÷@îŸé4Û àd ì õdà
ݟâ&å™è”çï) ñ ÞÀ ì ì Üß÷?øù
ùdüdåÀû î6ývà
ì åtå™è'çîŸé4Û àd ì õdà
ݟâ;÷røù
ùdü
ó÷Àá—ÛÀà-ÿièUéàdèÀèÀÜ)éã«äÈà
è”á¹ã‡àvß«Ü 
ÞÀì{å™ß«ä )å-ã«äÈà
ètã‡ÛÀÜàdß«ê¾ò2î6ývà
ì{åŸ÷”îŸé4 Û à
 ìÈõ
àdݟâ å™è'ço) ñ ÞÀ ì ì Üß÷'ø)ùdù
ü™íû % ä ߇à‹á—ä.÷'ø)ùdù
üŸ û
Åå™Û6í”åÀ÷øùdù
ü
ó÷6ådè”çoá‡ÛÀà-ÿiè|ÛÀà-ÿgî6ð±ñ ä{çŸÜå
á&éådèoíÁܱä è”éàdß«Ý'à ß4å-ã‡Ü)çUä èÃå|ÿiä{çŸÜNß4å™ è dÜNàdâ&à™ã«ÛÀÜß å™ì dà
߇äÈã‡Û”ý|á ò.îŸé4 Û à
 ìÈõ
àdݟâ;÷î6ývàdì{åvådè”çÃ) ñ ÞÀ ì ì Üß÷@ø)ùdù
ü™íûrîŸé4 Û à
 ìÈõ
àdݟâ Üã¯ådì.÷8ø)ùdù
üdéóôڥ۔Ü`߇Ü)ådçŸÜß#ý|å ê|ådì{á—à ”è”çFã‡Û”Ü=ã‡ÛÀÜ)á—ä{á#à™âiò2îŸé4Û àd ì õdà
ݟâ;÷øùd ù ™ó&ÛÀÜì ݟâöޔì2ô Ú¥ÛÀÜtÝÀß«àdíÀì ÜýYÿiÛÀä{é4ÛÃçŸß«à-ædÜ`ã«ÛÀÜtäÈè”ä:ã«ä{å™ìrçŸÜædÜì à
ÝÀývÜè‹ã]à™â¹îŸð±ñÃá±àŸééޔ߫á±ä èÃá—ÜædÜß4å™ìdÞÀä{á‡Üá #ã‡Û”Ü íÀä{ådáæ-å™ß«ä ådè”éÜã«ß«å
çŸÜà Tò % Üý|ådèNå™è”ç ¹ä ÜèÀÜè”á—ã‡àŸé4õÁ÷6øùdù dó÷ éå™Ý”å
éäÈãê±éàdè‹ã‡ß«àdìÁò % ÞÀê
àdèNÜãådì2ôÈ÷Ÿøù
ù dó÷ à-ædÜß ”ã—ã‡ä  è Nò2ñUàdè‹ã dàdývÜ߇ê å™è”ç .Üé4õÁ÷Ÿøùdù dó íÀޟãã‡Û”Ü&í”å
á—ä{é?ä{çŸÜ)åiä{áã‡Û”Ü#á«å™ývÜdôiàdÞdÛÀì ê=á—ÝÁÜådõ‹ä è”÷ âöàdß¹å 
ä ædÜèIì Üå™ß«èÀä  è ¯ã«ådá‡õÁ÷
ÿiäÈã‡Û¼å dä æ
Üè 'èÀäÈã‡Ü]å™ývà
ÞÀè‹ãàdâ'ã«ß«ådä èÀä  è  ç”å-ã«åÀ÷™ã‡Û”Ü]í'Ü)áã 
ÜèÀÜß«ådì ä )å-ã«äÈà
è ÝÁÜ߇âöàdß«ý|å™è”éÜWÿiä ì ìíÁÜvådé4ÛÀä ÜædÜçÃä:â¹ã‡ÛÀÜvß«ä dۋã±í'å™ì{å™è”éÜWä{á`á—ã‡ß«Þ”é4õUí'Üãÿ¹ÜÜè ã‡ÛÀÜ|å
ééÞÀß«å
éêUå-ã‡ã«ådäÈè”Üç àdè¼ã«Û”å-ã¥Ý”ådߗã«ä éÞÀì{å™ßã‡ß4å™ä èÀä  è `á‡Üã÷Ÿådè”çtã‡Û” Ü éå™Ý”å
éäÈã ê  à™âã‡Û”ܯý|ådé4۔äÈè”Ü™÷™ã«Û”å-ã#ä á÷dã«ÛÀܱådíÀä ì ä:ãêWà™âã‡Û”Ü ý|ådé4۔äÈè”ܯã«àvìÈÜ)å™ß«èådè6ê|ã‡ß4å™ä èÀä  è tá—Üã¯ÿiäÈã‡ÛÀà
ޟãiÜ߫߇à
ßô
ý|ådé4ÛÀä èÀÜ ÿiäÈã‡Ûã‡à6àvýtޔé4ÛTéå™Ý'ådéäÈãêvä{á¥ì ä õdÜ åí'àdã«ådèÀä{áã`ÿiäÈã‡ÛÅåÝÀÛÀàdã‡à 
ß«ådÝÀÛÀä{éIývÜývàdß«êTÿiÛÀàÀ÷ÿiÛÀÜèVÝÀß«Üá‡Üè‹ã«Üç ÿiä:ã«ÛïåTè”Üÿ[ã‡ß«Üܙ÷éà
è”éì ޔçŸÜ)á ã‡Û'å-ã¯äÈãiä{áièÀà™ã¯åtã«ß‡ÜÜ`í'Ü)éådޔá—Ü`äÈã]۔å
á]åvçŸä 8Üß«Üè‹ãiè‹Þ”ýIíÁÜßiàdâì Ü)å ædÜá¹âöß«àdý_å™è6ê‹ã‡ÛÀä  è ¼á—Û”Ü`۔ådá]á‡ÜÜè íÁÜâöàd߫ܙûåFý|å
é4ÛÀä èÀÜNÿiäÈã‡Û¾ã‡à6àìÈäÈã‡ã‡ì ÜIéå™Ý”å
éäÈãêFä{á ì ä õdÜ`ã«ÛÀܼí'àdã«å™è”ä á—ã $á±ì å êTíÀß«à™ã«ÛÀÜß÷rÿiÛÀàTçŸÜéì åd߇Ü)á ã‡Û'å-ã¥ä:âräÈã  á dß«ÜÜè÷‹äÈã  á#å`ã«ß‡Üܙô  Üä:ã«ÛÀÜß#éåd/ è dÜèÀÜß4å™ì ä Ü¥ÿ#Üì ì.ôÚ¥ÛÀÜ ÜëŸÝÀì àdß4å-ã«ä àdè¼å™è”ç¼âöà
߇ý|ådì ä )å-ã«äÈà
è à™âÁã‡Û”Üá‡Ü]éàdè'éÜÝÀã«á?۔å
á߇Ü)á—Þ”ì:ã«ÜçIä èIà
èÀÜià™âÁã‡ÛÀÜ]á‡ÛÀä èÀä  è  Ý'Ü)å™õŸáà™âÁã‡Û”Ü¥ã‡ÛÀÜàdß«êWà™âá—ã«å-ã«ä{áã«ä éå™ì‹ì Üå™ß«èÀä  è  òöðKådÝÀèÀä õÁ÷rø ù -ù
óô

;èUã«ÛÀÜ`âöà
ìÈì à-ÿiä è ”÷6íÁàdì{çFãê6Ý'Üâ,ådéÜNÿiä ìÈìäÈè'çŸä{éå-ã«Ü=ædÜéã‡à
ß]àdß]ý|å™ã‡ß«äÈë !
Þ'å™è‹ã‡äÈã‡ä Ü)áû”è”àdß«ý|å™ì8ãê6Ý'Üâ,ådéÜ ÿiä ì ìíÁÜtÞ'á—Ü)çUâöàdß=æ
Üéã«àdß å™è”ç¾ý|å-ã‡ß«äÈëTéàdývÝÁàdèÀÜè‹ã«á å™è”çTâöà
ß`á‡éå™ì{å™ß4áô
VÜIÿiä ì ìì{å™íÁÜì?éàdývÝÁàdèÀÜè
ã4á à™â@ædÜéã‡à
ß«á#å™è”çFý|å-ã«ß‡ä{éÜá?ÿiäÈã‡Û % ߇ÜÜõvä è”çŸä{éÜá÷6ådè”ç|ì{å™íÁÜì8æ
Üéã«àdß4á¹ådè”ç|ý|å™ã‡ß«ä{éÜáKã‡ÛÀÜý|á—Üì ædÜá&ÿiäÈã‡Û ià
ý|å™èÅä è”çŸä{éÜ)áô ”ådývä ìÈä{åd߇äÈãêTÿiäÈã‡ÛÅã«ÛÀÜTޔá—ÜFà™ â @å 
ß«åd è dÜvýtÞÀìÈã‡ä ÝÀì ä Üß4á±ã‡àVá—à
ìÈæ
ÜvÝÀß«àdíÀì Üý|áWÿiäÈã‡Û Ü !‹Þ”ådì ä:ãê¼à
ßiäÈè” Ü !‹Þ”å™ì äÈãê¼éàdè'áã«ß«ådäÈè‹ã4á&ä{á¥ådá«á—Þ”ývÜ ç  ô z

 

}IÕÁ×@ÒÖTÕ'ÒÓW€!6Ò"
Ô$#%2Ù'& #ŸÓÙ2Õ'ÒTu‹Ô$(ƒÕ”Ô$)*#”Ò@Ø `Õ+(,#Iu#ŸÓÓ‹Ô-ÒÃm-6Ø
Õ+.ÁÒ@Ù,ÓÙ2Õ'Ò0/12#ŸÔ-Ò,3 Ù,Ò".546#ÀØ7Ù,Ò"

Ú¥ÛÀÜ߇Ü|ä{áNåÃ߇Üý|å™ß«õ ådíÀì ÜNâ,ådýväÈì êÃàdâií'à
ÞÀè”çÀá dà-ædÜ߇è”äÈ è oã«ÛÀÜ|ß«Üì{å-ã«äÈà
è íÁÜãÿ#ÜÜèVã‡ÛÀÜFéå™Ý'ådéäÈãêÃàdâ]å ì Üåd߇èÀä è¥ý|å
é4ÛÀä èÀÜKå™è”çNä:ã4áÝÁÜ߇âöàdß«ý|å™è'éÜ8)ô?Ú¥ÛÀÜ&ã‡ÛÀÜàdß«ê
߇ÜÿVàdޟãà™â”éàdè”á‡ä{çŸÜß4å-ã«äÈà
è”á8à™âÀޔè”çŸÜßÿi۔å™ã éä ß4éÞÀý|á—ã«ådè”éÜ)á÷Àådè”çÛÀà-( ÿ !‹ÞÀä{é4õ6ì ê™÷6ã‡Û”ÜNývÜådèàdâá‡àdývÜ=ÜývÝÀä ߇ä{éåd ì !‹Þ”ådè‹ã‡äÈãêoéà
è6ædÜß dÜ)á#ÞÀèÀäÈâöàdß«ývì ê™÷ ådá]ã‡ÛÀÜtè6ÞÀýtí'Ü߯à™â&ç”å-ã«åoÝ'à
ä è
ã4á]ä è”é߇Ü)ådá‡Üá÷Àã‡àoã‡ÛÀÜIã‡ß«ÞÀÜWývÜ)å™èµòöã‡Û”å™ã±ÿiÛÀä{é4ÛUÿ#àdÞÀì{çTíÁÜtéå™ì{éÞÀì{å™ã‡Üç âöß«àdýYådèä  è 'èÀäÈã‡Ü=å™ývàdޔè
ã¥à™â?çÀå™ã«å
ó ò,ðKå™ÝÀèÀä õÁ÷rø ù -ù‹ó49 ô rÜã]Þ'á]áã4å™ß‡ãiÿiä:ã«ÛFà
èÀÜ=à™â@ã‡ÛÀÜ)á—Ü`íÁàdÞÀè'çÀáô Ú¥ÛÀܾèÀà™ã4å-ã‡ä à
èD۔Üß«ÜUÿiäÈì ìiì{å™ßdÜì ê âöà
ì ìÈà-ÿ ã‡Û”å™ãFà™âvòöðKådÝÀèÀä õÁ÷`øùdù‹údóô î6ÞÀݔÝ'à‹á—ÜÃÿ#ܾådß‡Ü 
äÈæ
Üè;: àdí'á—Ü߇æ-å-ã«ä àdè”áô 0¹ådé4ÛUà
í”á—Ü߇æ-å™ã‡ä àdèÃéàdè”á‡ä{áã4á]à™â#åoݔådä ߯åFædÜ)éã‡à
ß<>=@?5ACBEDGF@Hþø DIIIJDK:#å™è'çUã‡Û”Ü ådá«á‡à6éä{å-ã‡Ü)5 ç 4ã‡ß«ÞŸã‡Û+ML =—÷
äÈæ
Üèã‡àFޔá]í6êUå¼ã«ß‡Þ”á—ã‡Ü)çÃá‡àdÞÀß4éÜdô ;èTã‡ÛÀÜIã‡ß«ÜÜNß«Üéà
èÀäÈã‡ä àdèFÝÀ߇à
íÀì ÜýT÷N<,= ývä dۋã&íÁÜ`åWæ
Üéã«àdߥà™â@ÝÀäÈë6Üì8æ ådì ÞÀÜá òöÜdô ”P ô OQHRdú NâöàdßiåF ø -ëø `ä ý|å dÜ ó4÷6å™è”0 ç L =ÿ¹à
ÞÀì{çvíÁÜtø äÈârã‡Û”Ü ä ý|å 
Ü¥éàdè‹ã«ådä è”á?å=ã‡ß«ÜÜd÷‹ådè”ç «ø¯à™ã‡Û”Üß«ÿiä á‡Ü`òöÿ#ܯޔá‡Ü 4øiÛÀÜß«Ü]ß4å-ã«ÛÀÜß?ã«Û”å™0 è S ã‡àIá‡ä ývÝÀì äÈâöê=á‡ÞÀí”á‡ Ü !‹ÞÀÜè‹ã âöàdß«ýtÞÀì{å™Ü)óô  à-ÿyä:ãä{áFå
á‡á‡ÞÀývÜçDã«Û”å-ãã‡Û”Üܾ߫ÜëŸä{á—ã«áá—à
ývÜÃÞÀèÀõ6èÀà-ÿiè ݔ߇à
í”å™íÀä ì äÈãêOçŸä{á—ã‡ß«äÈí”ÞŸã‡ä àdè


vò<

D LÀóiâöß«àdý ÿiÛÀä{é4ÛÃã‡Û”Üá‡Ü¼ç”å-ã«åå™ß«Ü¼çŸß«å ÿièr÷rä2ô ܙôÈ÷ã‡Û”ܼç”å-ã«åTåd߇ܼådá«á—Þ”ývÜç äÈä{ç¾òöä è”çŸÜÝ'Üè”çŸÜè‹ã«ìÈê çŸß4å ÿiè|å™è”ç|ä çÀÜè‹ã‡ä{éådì ìÈêNçÀä á—ã‡ß«ä íÀޟã«Üç”óô#ò
VÜ ÿiäÈì ìÀÞ'á—Ü
gâöàdߥéÞÀýtÞÀì{å-ã«äÈæ
ܹݔ߇à
í”å™íÀä ì äÈãêNçŸä{áã«ß‡ä íÀÞÀã‡ä àdè”á÷ å™è'çµâöà
ßWã«ÛÀÜä ß¼çŸÜè”á‡äÈã‡ä Üá4ó4ô  à™ã«Ü|ã«Û”å-ãtã‡ÛÀä{átådá«á—Þ”ývݟã‡ä àdèVä{áNývà
ß‡Ü 
ÜèÀÜß«ådìKã«Û”å™èDådá«á—àŸéä{å™ã‡ä èTå ÀëŸÜç LUÿiäÈã‡Û¾ÜædÜ߇ê <  ä:ã`å™ì ì à-ÿ]á¥ã«ÛÀÜß«ÜWã«àFíÁÜvåFçÀä á—ã‡ß«ä íÀޟã«äÈà
èTà™â LTâöà
ß`å 
äÈæ
Üè
ò
ò
Ú¥ÛÀÜ`ÜëŸÝ'Ü)éã«å™ã‡ä àdèFà™âã‡ÛÀÜ=ã«Üá—ã]Ü߫߇à
ß#âöàdß]åIã‡ß4å™ä èÀÜ)ç|ý|ådé4۔äÈè”ܯä{á#ã‡ÛÀÜ߇Üâöàdß«Ü  ¼ò ó1H A

ø  L



ò
 
vò<

Àó

òø ó

DKL

 à™ã‡ÜWã‡Û”å™ã÷ÿiÛÀÜèÃå|çŸÜè'á—äÈãê@ò
Ú¥ÛÀ/ Ü !‹Þ”ådè‹ã‡äÈãê A¼ò ó=ä{á`éådìÈì Ü)çTã‡ÛÀÜoÜëŸÝ'Ü)éã«Üçï߇ä{á‡õÁ÷àdß ޔá—ã`ã‡Û”Ü|߇ä{á‡õÁô¯Üß«Üvÿ¹Ü¼ÿiä ì ìKéådìÈìäÈã`ã‡Û”Ü ådéã‡Þ”ådì@ß«ä{á—õÁ÷Áã‡àÜývÝÀÛ'ådá‡äÜ=ã‡Û”å™ã äÈã ä á¯ã‡Û”Ü !‹Þ”ådè
ã«äÈãêoã«Û”å-ã`ÿ#ÜIå™ß«ÜWޔì:ã«ä ý|å-ã‡Üì ê|ä è
ã«Üß«Üá—ã‡Ü)çTäÈèrô`Ú¥Û”Ü 4ÜývÝÀä ߇ä{éå™ì6߇ä{ᇠõ A”ò óKä{á¹çŸÜ ”è”Üç¼ã«àWíÁÜ ޔá—ã&ã«ÛÀܱývÜ)ådá‡ÞÀß«ÜçIývÜå™èvÜ߇߫àdß&ß«å™ã‡Ü]à
èvã‡ÛÀܯã‡ß4å™ä èÀä è á‡ÜãWò{âöà
ß]å ÀëŸÜç8 ÷ 'èÀäÈã‡Ü=è6ÞÀýIíÁÜߥàdâà
í”á—Ü߇æ-å™ã‡ä àdè”á4!ó 

"#$”ò% ó1H A

( '=*) &  L =+

#

ø  :

ò<,= D ó

ò
 ó

I

 à™ã‡Ü¾ã‡Û”å™ãFèÀàOÝÀ߇à
í”å™í”äÈì äÈãêµçŸä{á—ã‡ß«äÈí”ÞŸã‡ä àdèDå™ÝÀÝÁÜådß«áoÛÀÜ߫ܙô A ݔådߗã«ä éÞÀì{å™ß#é4ÛÀàdä{éÜ à™â, µådè”ç|âöà
ß]åtݔå™ß‡ã‡ä{éޔì ådß&ã«ß«ådä èÀä èIá‡Üã"-7<,=

 ò óvä{áoå /.‹ô

ÀëŸÜç è6ÞÀýtí'Üßvâöàdßå

D L =

Ú¥ÛÀÜ !‹Þ”ådè
ã«äÈãê # L =  ò< = D ó ä{á=éådìÈì Ü)çã‡ÛÀܼì à
á«áô ”àdß ã‡Û”ܼéådá‡ÜtçÀÜá«éß«äÈíÁÜç¾ÛÀÜ߫ܙ÷räÈã`éådè¾àdè”ìÈê    ã«ådõdÜWã‡ÛÀܼæ-å™ì ÞÀÜ)áS å™è”çOø™ô  à-ÿ é4۔à‹à‹á—ÜIá—à
ývÜ0Ãá‡Þ”é4ÛÃã«Û”å-ã S12031 ø™ôNÚ¥ÛÀÜèÃâöàdß=ì à
á«á‡Üáiã4å™õ6ä è ã‡Û”Üá‡Ü`æ ådì ÞÀÜá÷Ÿÿiä:ã«ÛFݔ߇à
í”å™íÀä ì äÈãêFø450Á÷Àã‡Û”ܱâöà
ì ìÈà-ÿiä èWí'à
ÞÀè”çÛÀàdì{çÀá=ò,ðKå™ÝÀèÀä õÁ÷røù
ù
ú
ó  ¼ò ó61 A

Àò ó87:9 ;< A

òöì à”ò

/= < ó>7 ø)ó?ïìÈà”ò@0A=CB‹ó : D

 :

òE‹ó

ÿiÛÀÜß‡Ü ä{á¹åWèÀàd è ƒèÀ Ü 
å-ã«ä ædÜ]ä è‹ã‡ Ü dÜß#éådìÈì Ü)çWã«ÛÀÜ=ðKå™ÝÀè”äÈõ ¹ÛÀÜ߇æ
àdèÀÜèÀõ6ä{á]òöð ¥ó#çŸä ývÜè'á—ä àdèr÷
å™è”çoä{á å ývÜ)ådá‡ÞÀß«< Ü]à™ârã«ÛÀÜ èÀà™ã«äÈà
è|à™âéå™Ý'ådéäÈãêtývÜè‹ã‡ä à
èÀÜç¼å™íÁà-ædÜdô ;èoã‡ÛÀÜ âöà
ìÈì à-ÿiä è ÿ¹Ü ÿiä ì ì'éådì ì”ã«ÛÀܱ߫ädۋã N ۔ådè”çUá—ä{çŸÜ`àdâ101!'ôNòE‹ó#ã‡ÛÀÜ 4ß«ä{á—õ|íÁàdޔè”ç8ô 
VÜWçŸÜݔå™ß‡ã¯ÛÀÜ߇ÜNâö߇à
ý_á‡àdývÜ`ÝÀß«Üæ6ä àdÞ'á¥èÀàdývÜè”éì{å-ã«ÞÀß«Ü  ã‡Û”ÜNå™ÞŸã«ÛÀàdß4á#à™âiò % ÞÀê
àdèÜã¯å™ì.ôÈ÷øùd ù dó¹éådìÈì8äÈã¥ã‡ÛÀ0 Ü  dޔådß«ådè‹ã‡ÜÜ)çvß«ä ᇠõ Ÿ÷ŸíÀޟãiã«ÛÀä{á¥ä{á#á‡àdývÜã‡ÛÀä  è Ià™âå ývä{á—è”àdývÜß÷'á‡ä è”éÜNäÈã ä á¯ß‡Ü)å™ì ì êoå|íÁàdޔè”çÃà
è¾å|ß«ä{á—õÁ÷8èÀà™ã=å|ß«ä á‡õÁ÷8ådè”çUäÈã±Û”àdì{çÀá]à
èÀì êFÿiäÈã«Û¾åoéÜ߇ã«å™ä è ÝÀß«àdí”ådíÀä ì ä:ãêd÷‹ådè”çFá‡à¼ä{á¥èÀà™ã 
ޔå™ß4å™è‹ã‡ÜÜç8ô?Ú¥ÛÀÜNá‡Üéàdè”çoã‡Ü߇ý à
èã‡ÛÀÜ`ß«ä dۋã#Û'å™è”çá—ä{çŸÜ=ä{á¥éådìÈì Ü)çvã‡Û”Ü 4ð  éà
 è 'çŸÜè”éÜdô 
VÜ¥èÀà™ã«Ü&ã«ÛÀ߇Üܹõ
Üê=ÝÁàdä è‹ã«áå™íÁàdޟãã‡ÛÀä{áíÁàdÞÀè”çô   ä ß«á—ã÷-ß«Üý|åd߇õ-å™í”ìÈêd÷äÈãä á@ä è”çŸÜÝ'Üè”çŸÜè‹ãà™âF
vò'
ƒã|ådá«á—Þ”ývÜáIàdèÀì êVã‡Û'å-ãví'àdã‡Û ‡ã ÛÀÜTã«ß«ådä èÀä è¾çÀå™ã«åVå™è”çµã‡ÛÀÜTã«Üá—ã|çÀå-ã4å åd߇ÜTçŸß4å ÿièOä è”çŸÜÝ'Üè”çŸÜè‹ã«ìÈê ådééà
ß«çŸä è¾  ã‡àHGJILKNM
vò'<9D LÀóô 6î Ü)éàdè'ç8÷#äÈã|ä{á¼Þ”á‡Þ”å™ì ì êÅèÀàdã|Ý'à‹á‡á‡ä íÀì Üoã«àµéàdývݔޟã‡ÜTã«ÛÀÜUì Üâ{ã|۔ådè”ç á‡ä çÀܙôNÚ¥ÛÀä ß«ç÷ÁäÈâKÿ#ÜIõ6èÀà-ÿ ÷rÿ¹Ütéådè¾Üå
á—ä ì êFéàdývÝÀÞÀã‡ÜNã«ÛÀÜt߇äd ۋã¯Û'å™è”ç á—ä{çŸÜdô=Ú¥Û6ޔád ä æ
ÜèÃá‡ÜædÜß«ådì

<

çŸä 8Üß«Üè‹ã?ìÈÜ)å™ß«èÀä è¯ý|å
é4ÛÀä èÀÜá#òöß«Üéå™ì ì‹ã‡Û”å™ãC4ì Ü)å™ß«èÀä è]ý|ådé4۔äÈè”Ü]ä{áޔá—ã&å™è”à™ã‡Û”Üß?è”ådývܹâöà
ßKå â,å™ývä ì ê à™â8âöÞÀè'éã‡ä à
è”á ò< D ó—ó4÷Ÿå™è'çvé4ÛÀà6à
á‡äÈè=å ÀëŸÜç8÷Ÿá—Þ
véä Üè‹ã‡ì êWá‡ý|å™ì ì 0Á÷‹í6êIã‡ÛÀÜèvã«ådõ‹ä è ã«Û”å-ã#ý|ådé4ÛÀä èÀÜ ÿiÛÀä{é4ÛTývä èÀä ýväÜá&ã‡ÛÀÜI߇ä
Û
ãi۔ådè”çÃá‡ä{çŸÜ™÷”ÿ#ÜWåd߇ÜWé4۔à‹à‹á—ä èIã‡Û'å-ã±ý|å
é4ÛÀä èÀÜ=ÿiÛÀä{é4Û dä ædÜ)á#ã‡ÛÀÜIìÈà-ÿ#Üá—ã ÞÀÝÀÝÁÜß¼íÁàdÞÀè'çµàdèOã‡ÛÀÜTådéã‡Þ”ådì¹ß«ä{á—õÁôÅÚ¥ÛÀä{á 
äÈæ
ÜáIåÃÝÀß«ä è”éä ÝÀì ÜçïývÜã‡ÛÀàŸçVâöà
ß¼é4۔à‹à‹á—ä  è Ãå¾ì Üå™ß«èÀä  è  ý|ådé4۔äÈè”ÜTâöàdßUå 
äÈæ
Üè ã4ådá‡õ'÷ å™è”ç ä á|ã‡ÛÀÜVÜ)á‡á‡Üè‹ã‡ä{ådì]ä{çŸÜåOà™âWá—ã‡ß«Þ”éã«ÞÀß«ådì±ß«ä{á—õDývä èÀä ývä å-ã«ä àdè ò2á—ÜÜ î6Ü)éã‡ä à
è 6ô 
ó4ô % ä ædÜèU/ å ÀëŸÜçTâ,ådýväÈì ê|àdâKì Üå™ß«èÀä  è ¼ý|ådé4۔äÈè”Üá#ã‡àé4ÛÀà6à
á‡Ü`âöß«àdýT÷”ã«à|ã«ÛÀÜIÜë6ã‡Üè‹ã±ã‡Û”å™ã ã‡Û”ÜIíÁàdÞÀè'çUä{áiã‡ä 
Û
ã¯âöàdß=å-ã±ì Ü)ådá—ã¯àdèÀÜWà™â?ã‡Û”ÜIý|å
é4ÛÀä èÀÜá÷”àdè”ÜWÿiä ì ìrèÀà™ã í'ܼå™í”ìÈÜNã‡àFçÀà|í'Üã—ã«Üß ã‡Û”ådè ã‡Û”ä áô Ú@àUã‡Û”ÜoÜë6ã«Üè‹ãWã«Û”å-ãIã‡ÛÀÜFí'à
ÞÀè”çµä{á`è”à™ãWã«ä dۋãNâöàdßtå™è6ê™÷ã«ÛÀÜoÛÀà
Ý'ÜFä{á`ã‡Û'å-ãWã«ÛÀÜoß«ä dۋãN۔ådè”ç á‡ä çÀÜtá—ã‡ä ì ì dä ædÜ)á¯Þ”á‡Üâöޔì?äÈèÀâöàdß«ý|å-ã‡ä à
èÃådá ã‡àTÿiÛÀä{é4Û ìÈÜ)å™ß«èÀä  è |ý|å
é4ÛÀä èÀÜWývä èÀä ývä Ü)á#ã‡ÛÀÜ|å
éã‡Þ'å™ì?ß«ä á‡õÁô Ú¥ÛÀÜNíÁàdޔè”çTèÀà™ã]íÁÜä  è ¼ã«ä dۋãiâöàdßiã«ÛÀÜNÿiÛÀà
ì Ü=é4۔à
á‡Üèâ,å™ývä ì êvà™âì Üå™ß«èÀä  è tý|ådé4ÛÀä èÀÜ)+ á dä ædÜ)á#é߇äÈã«ä éá¥å ޔá—ã‡ä 'ådíÀì Ü`ã«ådß 
Üã å™ã ÿi۔ä é4ÛÃã‡à ”߇ÜWã‡ÛÀÜä ß éàdývÝÀì{å™ä è‹ã«áô
ã=ÝÀß«Üá‡Üè‹ã÷âöà
߯ã«ÛÀä{á±éådá‡Ü™÷ÿ¹Ütýtޔáã ߇Üì ê àdèÜëŸÝÁÜß«ä ývÜè‹ã¹ã«à¼í'Ü=ã«ÛÀÜ ޔ ç 
ܙô

M  KM G I    Ú¥ÛÀÜ¥ð ÅçÀäÈývÜè”á‡äÈà
è`ä{áå±ÝÀß«àdÝÁÜ߇ãê`à™â8å á‡Üã?àdâ”âöÞÀè”éã‡ä àdè', á - ò ó .Iò,å 
ådäÈèr÷ ÿ#ܥޔá‡6 Ü Uå
áå 
ÜèÀÜ߇ä{é¹á—Üã à™âݔådß«ådývÜã‡Üß«áå¼é4ÛÀà
ä{éÜ à™â, µá—ÝÁÜéä”Üá¥åtݔå™ß‡ã‡ä{éÞÀì{å™ß¹âöÞÀè”éã«ä àdè'ó÷”å™è”çTéådèFíÁÜNçŸÜ”èÀÜçFâöà
ßiæ åd߇ä à
ޔá éì{å
á‡á‡Üá¹à™â@âöÞÀè”éã«ä àdè  ô ]Üß«Ü=ÿ¹Ü=ÿiä ì ì'à
èÀì ê|éà
è”á—ä{çŸÜß&âöޔè”éã«äÈà
è”á#ã‡Û'å-ã¯éà
߇߫Üá‡ÝÁàdè”ç¼ã«à¼ã‡ÛÀÜ ãÿ#à ;éì{ådá«á ݔå™ã—ã‡Ü߇èß«Üéà dèÀäÈã«äÈà
è|éå
á—Üd÷Ÿá‡àIã‡Û”å™ã ò
<

$%'&() M *,+.-4I(*  G/ $10 M  2M 43#!5 M )576   M G8* m:9 î6ÞÀݔÝ'à‹á—ܯã‡Û”å™ã¹ã«ÛÀÜ á‡Ý”å
éÜ]ä è|ÿiÛÀä{é4Û¼ã«ÛÀÜ ç”å-ã«åWì äÈæ
Üiä{á m  ÷Àå™è”ç¼ã«ÛÀÜ=á—Ü4 ã - ò% ó .`éàdè”á‡ä{áã4áKà™ârà
߇ä Üè
ã«Üç —á ã‡ß4å™ädۋãKì ä èÀÜá÷‹á‡àNã«Û”å-ã#âöàdߥå 
äÈæ
Üè¼ì ä èÀܙ÷6å™ì ì'ÝÁàdä è‹ã«á&àdè|à
èÀܱá‡ä{çŸÜ±åd߇ܱådá«á—ädè”ÜçIã‡ÛÀÜ=éì{å
á‡á¯ø™÷Ÿå™è”ç|ådì ì ÝÁàdä è‹ã«á±àdèÃã«ÛÀÜtà™ã‡Û”Üß`á—ä{çŸÜd÷'ã«ÛÀÜvéì{ådá«á`ødôNÚ¥ÛÀÜIà
߇ä Üè
ã4å-ã«äÈà
èTä á á‡ÛÀà-ÿièÃä è ädÞÀß«Ü|øtí6êTådè åd߇߫à-ÿ=÷ á‡Ý'Ü)éäÈâöê6äÈ è  à
è¼ÿiÛÀä{é4Û¼á‡ä{çŸÜ¥à™â8ã‡Û”Ü]ì äÈè”Ü#ÝÁàdä è‹ã«á?åd߇ܥã«à`í'ܱådá«á—ä dè”ÜçWã‡ÛÀÜ]ì{ådí'Üìø™ôP
ÛÀä ì Ü#äÈãKä{áÝÁà
á«á—ä íÀì Ü ã‡à 'è”çã‡ÛÀß«ÜÜ`ÝÁàdä è‹ã«á#ã«Û”å-ã éå™èíÁÜWá—Û'å-ã—ã«Üß«Üçí6ê|ã«ÛÀä{áiá—Üã¯à™ââöÞÀè”éã‡ä àdè”á÷”äÈãiä{á¥èÀà™ã¯ÝÁà
á«á‡äÈí”ìÈܱã‡à 'è”ç âöàdޔßôKÚ¥Û6ޔá¥ã«ÛÀÜ`ð  çŸä ývÜè”á—ä à
è|à™â@ã‡ÛÀÜNá‡Üã]àdâà
߇ä Üè‹ã«Üçvì ä èÀÜá#ä è m  ä{á#ã‡ÛÀß«ÜÜdô

;Á$›“)Ž,…=<> < +-2—² 0 " 3 12*@" 3@?BA *2+-41,12—2 (§ 0 , "$ 3 12  " 3  * 5

Ü ã $á#èÀà-ÿ éà
è”á—ä{çŸÜß&Û6ê6Ý'Ü߇ݔì ådèÀÜáKäÈè m ä{á#äÈèFã‡Û”Ü
ÝÀÝÁÜè”çÀä:ë”ó  B

ôKÚ¥ÛÀܯâöà
ì ìÈà-ÿiä è±ã‡Û”Üàdß«Üý

ÿiä ì ìÀÝÀß«à-ædÜ]Þ'á—ÜâöÞÀì?ò{ã‡Û”ܱÝÀß«à6à™â


    $IAG M GJICKM"GJM  I 5 I(*  G * m B    I ICGJM   #I M I $ M 5 I(*  G  G I( +(*  G   M G  () M M2# I(M  M2  #!5AM)5 6   M G    7 I  6 #1  $ M 5 CI G2  I  M  * M 5 I(*   "CM  I GI * M  MJK  * *,+@5AI*  G   M86 * M  6 # * M 5 M7 M % kIÕ'Ô E Õ %'% #ŸÔ  KÚ¥ÛÀܱð DçÀäÈývÜè”á‡äÈà
è¼à™âã‡Û”Ü á‡Üã¥à™ârà
߇ä Üè‹ã«Üç¼Û6ê6Ý'Ü߇ÝÀì{ådèÀÜá?ä è m B ä{1 á O7Oø™÷Ÿá—ä è”éÜ]ÿ#Ü é ådèÃå™ì ÿ¥å ê6á¥é4ÛÀà6à‹á—Ü O 7 øNÝÁàdä è‹ã«á÷'å™è'çã‡ÛÀÜè¾é4ÛÀà6à
á‡Ü`àdèÀÜWà™âã‡ÛÀÜIÝ'à
ä è
ã4á]ådá¯àdß«ädä è÷Àá—Þ'é4ÛTã‡Û”å™ã¯ã‡Û”Ü ÝÁà
á‡ä:ã«ä àdèÃæ
Üéã«àdß4á¯àdâ&ã«ÛÀܼ߇Üý|å™ä èÀä èMOOÝÁàdä è‹ã«á=åd߇Ütì äÈè”Üå™ß«ì êoä è”çŸÜÝ'Üè”çŸÜè
ã÷íÀޟãWéå™è èÀÜæ
Üß`é4ÛÀà6à
á‡Ü O 7 tá‡Þ”é4ÛTÝ'à
äÈè‹ã4á ò2á—ä è”éÜ=èÀ à O 7 ø=ædÜ)éã‡à
ß«á#ä è m B éå™èí'Ü`ì ä èÀÜ)å™ß«ìÈêtä è”çŸÜÝ'Üè”çŸÜè‹ãó4ô


èDådì:ã«Üß«è”å-ã«ä ædÜvÝÀß«à6à™â]à™â¯ã‡ÛÀÜUéà
߇à
ì ì åd߇êÃéå™è í'ÜFâöàdÞÀè'çOä è ß«ÜâöÜß«Üè'éÜá¹ã«ÛÀÜß«Üä èô

M#"%$ K&MIB-

M  KM G I  7 $ !  

 

ò
è‹ã‡ÛÀà
è6êµå™è”ç ¹ä
á÷¥ø)ùdù‹údó4÷Kådè”ç

KM  MG

Ú¥ÛÀÜÃð _çÀäÈývÜè”á‡äÈà
èµã«Û‹Þ'á 
äÈæ
Üávéàdè”é߇Üã‡Üè”Üá«á¼ã‡àÅã«ÛÀܾè”à™ã‡ä à
èDàdâ=ã‡ÛÀÜïéå™Ý'ådéäÈãêµàdâWå dä æ
Üè á—Üã à™â±âöÞÀè”éã‡ä àdè”áô ;è‹ã‡ÞÀäÈã‡ä æ
Üì ê™÷Kà
èÀÜUývädۋãIíÁÜUì ÜçOã‡àVÜëŸÝÁÜéãvã«Û”å-ãvì Üåd߇è”äÈè¾ý|ådé4۔äÈè”ÜáIÿiä:ã«Û ý|ådè6ê ݔådß«ådývÜã‡Üß«á±ÿ#àdÞÀì{çÃÛ'å ædÜtÛÀädÛ¾ð  çŸä ývÜè'á—ä àdèr÷ÁÿiÛÀä ì ÜtìÈÜ)å™ß«èÀä è|ý|å
é4ÛÀä èÀÜá±ÿiäÈã‡Û âöÜÿ ݔå™ß4å™ývÜã‡Üß4á ÿ#àdÞÀì{ç|Û'å ædÜ=ì à-ÿ ð  çŸä ývÜè”á—ä à
èôÚ¥ÛÀÜ߇Ü=ä{á¥å¼áã«ß‡ä õ6ä  è IéàdÞÀè‹ã‡Ü߇ÜëÀå™ývÝÀì Üiã«à¼ã‡ÛÀä{á÷”çŸÞÀÜ=ã‡à 0]N ô rÜæ6ä è å™è'ç &Àô$îÁô -±ÜèÀõ
ÜßTòöðKådÝÀèÀä õÁ÷]øùdù‹údó 
ì Ü)å™ß«èÀä  è Tý|ådé4ÛÀä èÀÜ|ÿiäÈã«Û ޔá—ã¼à
èÀÜoÝ'å™ß4å™ývÜã«Üß÷?íÀޟãvÿiäÈã‡Û ä  è ”èÀäÈã«Ü ð  çŸä ývÜè'á—ä àdè ò,åtâ,å™ývä ì êvà™â?éì{å
á‡á‡ä ”Üß4á&ä{áiá«å™ä{çoã‡àv۔å æ
Ü ä  è 'èÀäÈã‡Ü=ð  çÀäÈývÜè”á‡äÈà
è|äÈâäÈã]éådè á‡Û”å-ã‡ã‡Üß :#ÝÁàdä è‹ã«á÷èÀàTý|å-ã‡ã‡Üß`ÛÀà-ÿ ì{å™ß d Ü :2óô -±Ü 'èÀܼã‡ÛÀÜ|á—ã‡ÜÝïâöÞÀè”éã«ä àd( è '”*ò )Áó D+) ? m 8-'”,ò )'ó H ø -)/. SŸ0 B û ''*ò )'ó H2`B ø -)1 S .61 ô ¹à
è”á‡ä çÀÜß#ã‡ÛÀÜ`à
èÀÜ ƒÝ”ådß«ådývÜã‡ÜßKâ,ådýväÈì ê¼à™â@âöÞÀè”éã‡ä àdè”á÷”çŸÜ ”èÀÜçí6ê

*ò ),D ó213''ò,á‡äÈèrò% 4)Áó—ó D5),D ?

m I

ò@B6ó

6KàdÞÅé4ÛÀà6à‹á—Üvá‡àdývÜtè‹Þ”ýIíÁÜß

:÷@å™è”çïÝÀß«Üá‡Üè‹ãNývܼÿiäÈã‡Û ã‡ÛÀܼã4ådá‡õ¾àd⠔è'çŸä è ¹ : ÝÁàdä è‹ã«á ã«Û”å-ãWéå™èVíÁÜ á‡Û”å-ã‡ã‡Üß«Üçô ¥é4ÛÀà6à
á‡Ü±ã‡ÛÀÜýYã«àtíÁÜ 

) =

H

øS87 =

DRF"H

ø

:9;9;9DK: I

ò2ú
ó

D

6KàdÞUá‡Ý'Ü)éäÈâöêoådè‹êvì{ådí'Üì á#ê
àdÞìÈä õ
Ü  L  ø D)ø . I # DKL  D:9;9:9 D L D L = ?- ` Ú¥ÛÀÜè ò% ó+
ä ædÜá&ã‡& ۔ä á¥ì{ådí'ÜìÈä èIäÈâ ¹é4۔à‹à‹á—Ü Vã«à¼í'Ü 3 H
2óøS



L =

=

ó I

ò*‹ó

òdó

çÀäÈývÜè”á‡äÈà
è|à™â@ã‡ÛÀä{á¥ý|å
é4ÛÀä èÀܯä{á¥ä è”èÀäÈã«Ü™ô

;è‹ã‡Ü߇Ü)áã«ä èdì ê™÷rÜædÜè ã«ÛÀàdÞdÛïÿ¹Ü|éå™èÅá—Û'å-ã—ã«ÜßNådèVåd߇í”ä:ã«ß«åd߇ä ì êoì{ådß 
ÜIè6ÞÀýtí'Üß=à™â¥Ý'à
äÈè‹ã4á÷ÿ¹Üvéådè å™ì{á‡à ”è”ç ޔá—ã¥âöàdÞÀߥÝÁàdä è‹ã«á&ã‡Û”å™ãiéå™èÀè”à™ã¥í'Ü`á‡Û”å-ã‡ã‡Üß«Üçô?ڥ۔Üê|á‡äÈývݔìÈêt۔å æ
Ü]ã‡àtíÁÜ Ü !‹Þ”å™ì ì êtá‡Ý”å
éÜç÷ å™è'çÃådá«á—ä
èÀÜçFì{å™íÁÜì{á]ådá±á—Û”à-ÿièä è"ädÞÀß«Ü Ÿô¯Ú¥ÛÀä{á¯éådèUí'Ütá—ÜÜè¾å
áiâöà
ìÈì à-ÿ]á
V߇äÈã‡ÜNã‡ÛÀÜIÝÀ۔ådá‡ÜWå™ã ) # åd>á = # H $O < 7@?‹ô]Ú¥ÛÀÜèÃã‡ÛÀÜWé4۔àdä{éÜ=à™â?ì{å™íÁÜì L # H[øWß‡Ü !
ޔäÈß«Üá SAB?CA3<ô¯Ú¥ÛÀÜNݔ۔ådá‡ÜNå-ãD)  ÷ ývàŸQ ç E<÷ä{ á F?‹ûÁã«ÛÀÜè L H H ø G SIA0?JA0<8= Ÿô=î6ä ývä ì{å™ß«ìÈêd÷‹ÝÁàdä è‹D ã ) âöà
ß«éÜ> á ?.K<8=LEŸô ڥ۔Üè¾å™ã ) ÷ <8=CELA0?IA<8= ¼ä ýv ÝÀì ä Üá#ã«Û”å-ã *ò ) D ó H `ø™÷éàdè‹ã«ß«åd߇êFã‡àoã‡ÛÀܼåd8 á«á—ä 
èÀÜçì{å™íÁÜì.ô±Ú¥ÛÀÜ)á—ÜWâöàdÞÀß ÝÁàdä è‹ã«á å™ß«ÜWã«ÛÀÜvå™è”ådìÈà dêd÷”âöàdß ã«ÛÀÜvá—Üã àdâ&âöÞÀè”éã‡ä àdè'á±ä è 0 !'ôF@ò B6ó4÷àdâKã‡ÛÀÜvá‡Üã=à™â¹ã‡ÛÀß«ÜÜtÝ'à
ä è
ã4á¯ì ê6ä  è  å™ì à
 è åTì ä èÀÜd÷âöàdßNà
߇ä Üè‹ã«ÜçÃÛ6ê6ÝÁÜß«ÝÀì{å™èÀÜ)á±ä è m B ô  ÜäÈã«ÛÀÜßNá‡ÜãWéå™èVí'Ü|á‡Û”å™ã—ã‡Ü߇Ü)çïí‹ê¾ã‡ÛÀÜ|é4۔à
á‡Üè â,å™ývä ì ê¼à™â@âöÞÀè”éã«ä àdè”áô

x=0

1

2

3

4

;Á$›“)Ž,…
> ¦ 0 -² 0 " 3 1.*@12+-41 3 3-0 1@(d?*2+-41,1.2—±(§ *2" 3 d2  *,² " 12&" 3) 3 " 12&³´)" !¥ 3 2* " 40 3‹5 1.4

VC Confidence

1.2 1 0.8 0.6 0.4 0.2 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

h / l = VC Dimension / Sample Size

;Á$›“)Ž,… > ³´F 043  )  3 —K"$*! 043-0 1 043 "$K" 3    *  K.*%+ 

M"IE$ 7I# 

*  K4*%+

ädÞÀß«Ü v E á—ÛÀà-ÿ]á¥Û”à-ÿ «ã ÛÀÜNá—Ü)éà
è”çFã‡Üß«ý_àdèã‡ÛÀÜ`ß«ädۋãi۔å™è'çá‡ä{çŸÜ`à™â 01!'ô=òE
ó#æ-åd߇ä Üá#ÿiäÈã‡Û < ÷ 
äÈæ
Üè å|é4ÛÀà
ä{éÜ`à™â&ù‹úYéà
è'çŸÜè”éÜNì ÜædÜì¹ò@0 H SI S
údó¥ådè”çÃå
á‡á‡ÞÀývä èvå¼ã‡ß4å™ä èÀä èvá«å™ývÝÀì Ü=à™â&á‡äÜvøSŸ÷ S S SÀô Ú¥ÛÀÜ`ð  éàdè'çŸÜè'éÜ`ä{á¥åIývà
èÀà™ã«àdèÀä{é¯äÈè'éß«Üådá‡ä è`  âöÞÀè'éã‡ä à
èàdâ ô&Ú¥ÛÀä{á¥ÿiä ìÈì8íÁÜ=ã‡ß«ÞÀÜ âöàdß]ådè‹êoæ ådì ÞÀÜ

<

à™â9:;ô

Ú¥Û6ޔá' ÷ 
ä ædÜèvá‡àdývÜ]á‡Üì Üéã‡ä àdètà™ârì Ü)å™ß«èÀä  è ±ý|å
é4ÛÀä èÀÜáÿi۔à
á‡Ü]ÜývÝÀä ß«ä{éå™ì6ß«ä{á—õIä á Üß«àÀ÷‹àdèÀܱÿ#ådè‹ã«áã«à é4ÛÀà6à
á‡Ü]ã«Û”å-ãiì Üåd߇è”äÈè`ý|ådé4ÛÀä èÀÜ]ÿi۔à
á‡Ü±ådá«á‡à6éä{å-ã‡Ü)çvá—Üãià™ârâöÞÀè'éã‡ä à
è”á&Û'ådá#ývä èÀä ý|å™ìÀð  çÀäÈývÜè”á‡äÈà
èô Ú¥ÛÀä{áÿiä ìÈì6ì Üå
ç`ã«à=å íÁÜã‡ã‡Üß?ޔÝÀÝ'ÜßKíÁàdÞÀè”çtà
èWã«ÛÀÜiådéã«Þ”å™ìŸÜ߇߫àdß ô ;è dÜèÀÜß4å™ì.÷ âöàdßèÀà
è Üß«à=ÜývÝÀä ß«ä éå™ì ß«ä á‡õÁ÷‹à
èÀÜ ÿ#ådè
ã4áKã«àté4۔à‹à‹á—ܯã‡Û”å™ã#ì Ü)å™ß«èÀä  è Ný|ådé4۔äÈè”ÜiÿiÛÀä{é4Ûoývä èÀä ývä Üá@ã«ÛÀÜ=߇ä dۋã&Û'å™è”çFá—ä{çŸÜ±à™ â 0 !'ô ò E‹ó4ô  à™ã‡ÜIã‡Û'å-ã±ä è¾å
çŸàdÝÀã‡ä  è ¼ã‡Û”ä á á—ã‡ß4å-ã‡ Ü dêd÷Áÿ#ÜIåd߇ÜNà
èÀì êFÞ'á—ä  è  0 !'ôtò E‹ó]å
á±å dÞÀä{çŸÜd ô 01!'ô¼ò E
ó dä ædÜ)á öò ÿiäÈã«ÛFá‡àdývܯé4ÛÀà‹á—Üè|ÝÀß«àdí”ådíÀä ì ä:ãêÀóå™èFÞÀÝÀÝÁÜߥíÁàdÞÀè”çoàdèoã‡ÛÀÜ=ådéã‡Þ”ådìÁß«ä{á—õÁôÚ¥ÛÀä{á¹çŸà6Ü)á¹è”à™ã¹Ý”߇ÜædÜè‹ã¥å ݔådߗã«ä éÞÀì{å™ß@ý|ådé4ÛÀä èÀÜ#ÿiäÈã‡ÛIã‡ÛÀÜ]á«å™ývÜ#æ-å™ì ÞÀܹâöàdß?ÜývݔäÈß«ä{éådìdß«ä á‡õÁ÷då™è”çtÿiÛÀà‹á—Ü#âöÞÀè'éã‡ä à
ètá‡ÜãK۔å
áÛÀädÛÀÜß ð  çŸä ývÜè”á‡ä àdè÷‹âöß«àdý_۔å æ6äÈ è IíÁÜã‡ã‡Üß±Ý'Üߗâöà
߇ý|ådè”éÜd1 ô ;èâ,ådéã±å™èTÜëÀådývÝÀì ܱàdâ?åvá‡êŸáã«ÜýYã‡Û”å™ã dä ædÜ)á dà6àŸçTÝÁÜ߇âöàdß«ý|å™è”éÜ=çÀÜá‡ÝÀäÈã‡ÜN۔å æ6ä è ¼ä è ”èÀäÈã«Ü`ð gçŸä ývÜè”á‡ä àdèoä{á dä ædÜèFä èTã‡Û”ÜWèÀÜë‹ã`î6Üéã‡ä àdèrô  à™ã«Ü å™ì{á‡àvã‡Û'å-ã=ã‡ÛÀÜ dß4å™Ý”Û¾á—Û”à-ÿ]á]ã«Û”å-ã=âöàdß =$: . SI E'Ãò,å™è'çÃâöà
" ß 0 H SI S
ú|ådè”ç : Hþø SDKS S S‹ó4÷Áã‡ÛÀܼð  éà
 è 'çŸÜè”éÜ=ÜëŸéÜÜ)çÀá¥ÞÀèÀäÈãê™÷”ådè”çTá—àtâöà
ßi< ÛÀä dÛÀÜß¹æ-å™ì ÞÀÜ)á#ã‡ÛÀÜ`íÁàdޔè”çä + á 
ޔå™ß4å™è‹ã‡ÜÜçoèÀà™ãiã«ä 
ۋãô

 /I 

K 5 6*M G

¹àdè'á—ä{çŸÜßKã‡Û”Ü  ã«ÛFèÀÜ)å™ß«Üá—ã¹èÀÜädÛ6í'à
ÞÀß¹éì{ådá«á—ä”Üß÷‹ÿiä:ã«ÛH ødôKÚ¥ÛÀä{á¹á‡Üãià™ââöÞÀè”éã«ä àdè”á#۔å
á&ä è”è”ä:ã«Ü ð  çŸä ývÜè”á‡ä àdèå™è”ç Üß«àvÜývÝÀä ß«ä éå™ì8ß«ä á‡õÁ÷'á‡äÈè'éÜNå™è6êFè‹Þ”ýIíÁÜß]àdâ?Ý'à
äÈè‹ã4á÷Áì ådí'Üì Üçå™ß«íÀäÈã‡ß4å™ß«äÈì êd÷‹ÿiä ì ì íÁÜWá—Þ'ééÜ)á‡á—âöÞÀì ì ê|ì Üåd߇èÀÜ)çoí6êoã‡ÛÀÜWådìdàdß«äÈã‡ÛÀýfò,ÝÀß«à-æ‹ä{çŸÜ)çoè”à¼ãÿ¹àvÝÁàdä è‹ã«áiàdâ?àdÝÀÝÁà
á‡äÈã‡Ü`éì{ådá«á¥ì ä Ü ß«ädۋã àdèÃã‡àdÝ à™â&Ü)ådé4ÛÃàdã‡ÛÀÜß4óô=Ú¥Û6ޔá±ã«ÛÀÜIíÁàdޔè”çÃÝÀß«à-æ6ä{çŸÜáiè”àoä èŸâöàdß«ý|å-ã«ä àdèô ;è¾â,ådéã÷8âöàdß`å™è6êTéì{ådá«á‡ä 'Üß ÿiäÈã‡Û|ä  è 'èÀäÈã‡Ü¯ð  çŸä ývÜè”á‡ä àdè÷™ã«ÛÀÜ í'à
ÞÀè”çFä á&èÀà™ã¥Üæ
ÜèFæ ådì ä ! ç )ô,¯à-ÿ¹ÜædÜß÷‹ÜædÜè|ã«ÛÀàd Þ 
Ûvã‡Û”Ü íÁàdޔè”ç

ä{á&è”à™ãiæ-å™ì ä{ç8÷
èÀÜ)å™ß«Üá—ã#è”ÜädÛ6íÁàdÞÀß#éì{ådá«á—ä”Üß«á&éå™èáã«ä ìÈì'Ý'Üߗâöà
߇ýþÿ#Üì ì.ô?ڥۋÞ'á&ã«ÛÀä{á 'ß«á—ã#ÜëÀå™ývÝÀì Ü]ä{á#å éådޟã‡ä à
è”å™ß«êtã«å™ì Ü ä  è 'èÀäÈã‡Ü éådݔådéäÈã2 ê IçŸà6Üá¥èÀàdã dޔådß«ådè
ã«ÜÜ Ý'à6à
ßiÝ'Üߗâöà
߇ý|ådè”éÜdô Ü ã $á&âöàdì ì à-ÿÅã‡ÛÀܯã‡ä ývÜiÛÀà
èÀàdÞÀß«ÜçIã«ß«å
çŸäÈã‡ä àdètà™âÞÀè”çÀÜß4áã4å™è”çŸä è=ã«ÛÀä è
áKí‹êIã‡ß«ê6ä è=ã‡àNí”߇Ü)å™õNã«ÛÀÜýT÷ ™å è'ç á‡ÜÜväÈâ¹ÿ#ÜvéådèïéàdývÜtÞÀÝVÿiä:ã«ÛïåUéì{ådá«á—ä”Üß]âöàdß`ÿi۔ä é4Û¾ã‡ÛÀÜvíÁàdޔè”çïä á`á‡ÞÀÝÀÝÁà
á‡Üç¾ã‡àU۔àdì{ç8÷ríÀޟã ÿiÛÀä{é4Ûæ‹ä à
ì å™ã‡Ü)á?ã‡Û”Ü`í'à
ÞÀè”ç8ô
VÜ`ÿ¥å™è‹ã¥ã‡ÛÀÜ`ì Üâ{ã]Û'å™è”çá—ä{çŸÜ=àdâ 0 !”ô]ò%E
ó¹ã‡àvíÁÜNådá¥ì{ådß 
ܱådá¥ÝÁà
á«á‡äÈí”ìÈÜd÷ å™è'çoã«ÛÀÜ`ß«ä 
ۋã#۔ådè”çTá—ä{çŸÜ ã«àví'ÜNå
áiá—ý|ådìÈìå
á¥Ý'à‹á‡á‡ä íÀì ܙô&î6àvÿ#Ü ÿ¥å™è‹ã]åtâ,å™ývä ì ê¼à™â?éì{ådá«á‡ä 'Üß4áKÿiÛÀä{é4Û dä æ
Üáã‡Û”ܯÿ#àdß4áãKÝÁà
á«á—ä íÀì Üiådéã‡Þ”ådì”ß«ä{á—õIà™, â SI ú6' ÷ Üß«à`ÜývݔäÈß«ä{éådì‹ß«ä{á—õIÞÀݼã‡àtá—à
ývÜ¥è‹Þ”ýIíÁÜßKàdâã‡ß4å™ä èÀä  è  àdí'á—Ü߇æ-å-ã«ä àdè”á÷rå™è”ç ÿiÛÀà
á‡Ü¼ð  çŸä ývÜè”á—ä à
èUä á=Ü)ådá‡êTã‡àUéà
ývÝÀޟã«Ü¼ådè”ç ä á ì Üá«á±ã‡Û”ådè :=ò2á—àã‡Û'å-ã`ã‡Û”Ü íÁàdÞÀè”ç ä{á¯è”àdèÃã«ß‡ä æ6ä{å™ìöó4ô
èïÜëÀådývÝÀì ÜNä{á¯ã«ÛÀÜIâöàdì ì à-ÿiäÈ è À÷'ÿi۔ä é4Û ±éådì ìã«ÛÀÜ èÀà™ã«ÜíÁà6àdõÃéì{å
á‡á‡ä ”Üßô  Ú¥ÛÀä{á¼éì{ådá«á‡ä 'Üß¼éàdè'á—ä{á—ã«áIà™â`å è”à™ã‡Üí'à6à
õµÿiäÈã‡ÛDÜèÀà
 Þ dÛ ß‡à6àdý ã‡àVÿi߇äÈã«ÜTçÀà-ÿièOã«ÛÀÜÃéì{ådá«á‡ÜáIà™D â  ã‡ß4å™ä èÀä  è và
í”á‡Üß«æ å™ã‡ä àdè'á÷”ÿi۔Üß«Ü  1 :;ô Ààdß=å™ì ìá—ÞÀí'á—Ü !
ޔÜè‹ã±Ý”å™ã—ã«Üß«è”á÷8ã«ÛÀÜtéì{ådá«á—ä ”Üß]á—ä ývÝÀì êoá«å êŸá ã‡Û'å-ãIådìÈì?Ý'å-ã—ã«Üß«è”áNÛ'å ædÜtã‡ÛÀÜFá‡ådývÜ|éì{ådá«áôTî6ÞÀÝÀÝÁà
á‡Üoå™ì{á‡àoã«Û”å-ãWã‡ÛÀÜoç”å-ã«åÃ۔å ædÜvådáNý|å™è6êÃÝÁà
á‡ä:ã«ä ædÜ ò L H 7Iø)óiå
á]èÀ Ü 
å™ã‡ä ædÜ|'ò L!H `ø)óiÜëÀå™ývÝÀì Ü)á÷'ådè”çTã«Û”å-ã±ã‡ÛÀÜtá‡ådývÝÀì Üáiå™ß«ÜWé4ÛÀà‹á—ÜèUß«ådè”çŸà
ývìÈêdô¥Ú¥Û”Ü èÀàdã‡ÜíÁà6àdõ|éì å
á‡á‡ä ”Üß¹ÿiä ìÈìÁ۔å æ
Ü Üß«àIÜývÝÀä ߇ä{éådìÀß«ä á‡õ¼âöàdߥÞÀÝFã‡& à yà
í”á—Ü߇æ-å™ã‡ä àdè”á û SI ú`ã«ß«ådäÈè”äÈ è NÜ߫߇à
ß âöàdßNådì ìá‡ÞÀí”á‡ Ü !‹ÞÀÜè‹ãNàdí”á‡Üß«æ-å-ã‡ä à
è”á> û SI$úå
éã‡Þ'å™ì?Üß«ß«àdß÷å™è”ç ð  çÀäÈývÜè”á‡äÈà
è H ¾ôFîŸÞÀí”á—ã‡äÈã‡ÞŸã«ä   è  < ã‡Û”Üá‡Ü`æ ådì ÞÀÜá#ä , è 0 !'ôi%ò E
ó÷Àã‡ÛÀÜ`íÁàdޔè”çí'Ü)éàdývÜ)á 

 D B : 1 ìÈèrò  : = Uó 7 4ø 

òø =ETóŸìÈèrò0A= ‹ Bó

ò,ü‹ó

ÿiÛÀä{é4Ûä á¥éÜ߇ã«å™ä èÀì ê¼ývÜã¥âöà
ß]å™ì ì o 0 äÈâ



ò
6ó

H




ÿiÛÀä{é4Ûä á¹ã«ß‡ÞÀÜd÷'á‡äÈè'éÜ

('&  

$   $

7

ÜëŸÝ 



  6  G

#

1

ø D


1 ò  =$:2ó D  S 1
N1

ò
6ó&ä{á¥ývà
èÀà™ã«àdèÀä{éiä è”é߇Ü)ådá‡äÈèÀ÷Ÿå™è”ç



ø



ò,ù‹ó ò
H

ø)ó H

LEÀô

SI 

*  K. ( I

VܯéådèvèÀà-ÿDá‡ÞÀývý|åd߇äÜ&ã‡ÛÀܯÝÀ߇ä è”éä ÝÀì Ü&à™ârá—ã‡ß«Þ”éã«ÞÀß4å™ìÀß«ä{á—õNývä èÀä ýväå™ã‡ä àdèvò.î]ñ¾ó¯òöðKå™Ý”èÀä õ'÷Áøù-ù
óô  àdã‡Üã‡Û'å-ãtã‡ÛÀÜTð þéàdèÁçŸÜè”éÜoã«Üß«ý ä è 01!'ô ò%E
óIçŸÜÝÁÜè'çÀá¼àdèOã‡ÛÀÜUé4ÛÀà‹á—Üè éì{å
á‡áWà™â¯âöޔè”éã«äÈà
è”á÷ ÿiÛÀÜ߇Ü)ådá¹ã‡ÛÀÜ=ÜývÝÀä ߇ä{éådì”ß«ä{á—õvå™è'çFå
éã‡Þ'å™ì8߇ä{á‡õvçŸÜÝÁÜè”çà
èoã«ÛÀÜ=àdèÀÜ=ݔådߗã«ä éÞÀì{å™ß&âöÞÀè”éã«ä àdèFé4۔à
á‡Üèoí6ê ã‡Û”Ü¥ã‡ß4å™ä èÀä  è ]ÝÀß«àŸéÜ)çŸÞÀ߫ܙ> ô
VÜiÿ#àdÞÀì{ç`ì ä õdÜ&ã‡ à ”è”çIã‡Û'å-ãKá—Þ”í”á—Üã?à™âÁã‡ÛÀÜié4ÛÀà‹á—ÜèIá‡ÜãKà™â'âöÞÀè”éã‡ä àdè”á÷dá‡Þ”é4Û ã‡Û'å-ãiã‡Û”Ü ß«ä{á—õvíÁàdޔè”çFâöàdß#ã‡Û”å™ã¯á—Þ”í”á—Üãiä á#ývä èÀä ývä Ü)ç8ô ¹ì Üåd߇ì êtÿ¹Ü`éådèÀèÀàdãiåd߇ß4å™ è dÜ]ã«ÛÀä  è 
á¥á‡àtã‡Û”å™ã ã‡Û”ܱð DçŸä ývÜè”á—ä à
è æ-å™ß«ä Üá&á‡ývà6à™ã«ÛÀì ê™÷
á‡ä è”éܱä:ã#ä{á&å™èoäÈè‹ã«Ü dÜßô ;è”á—ã‡Ü)ådç8÷6ä è
ã«ß‡àŸçŸÞ'éܯå á—ã‡ß«Þ”éã«ÞÀß«Ü  < í6êïçÀäÈæ6ä{çŸä  è ã‡Û”ÜoÜè‹ã«äÈß«Ü|éì{å
á‡á`àdâ¥âöÞÀè”éã«ä àdè”áWäÈè‹ã«àÃèÀÜá—ã‡Ü)çµá‡ÞÀí”á‡Üã«áF*ò ä dÞÀß« Ü B‹óô Àà
ßWÜ)ådé4Ûµá‡ÞÀí”á‡Üã÷ ÿ#ÜNýIÞ'áã¯íÁܼå™íÀì Ü`ÜäÈã«ÛÀÜ߯ã‡àFéàdývÝÀÞÀã‡Ü ÷Áà
ß]ã«à dÜã±å|íÁàdޔè”çUàdè äÈã«á‡ÜìÈâ;ô± î ]ñ ã‡ÛÀÜè¾éà
è”á‡ä á—ã«á]à™â < < ”è”çÀäÈ è Wã«Û”å-ã]á‡ÞÀí”á‡Üã¥à™ââöޔè”éã«äÈà
è”á#ÿiÛÀä{é4Û|ývä èÀä ývä Üáã«ÛÀÜ=í'à
ÞÀè”çFàdèoã‡ÛÀÜ`ådéã‡Þ”ådì8߇ä{á‡õ'ô?Ú¥ÛÀä{á#éå™èíÁÜ çŸà
èÀÜ`í6êá‡ä ývÝÀì êIã«ß«ådä èÀä  è tåvá‡Üß«ä Üá¥àdâý|å
é4ÛÀä èÀÜá÷ŸàdèÀÜ=âöà
߯Üå
é4Ûá‡ÞÀí”á‡Üã÷'ÿiÛÀÜ߇Ü=âöà
߯å dä æ
ÜèTá—ÞÀí'á—Üã ã‡Û”Ü 
à
å™ìà™â&ã‡ß4å™ä èÀä  è vä{á±á‡ä ývÝÀì ê|ã«àFývä èÀä ývä ܱã‡ÛÀܼÜývÝÀä ߇ä{éådì߇ä{á‡õÁô $±èÀÜIã‡Û”ÜèÃã4å™õ
Üá¯ã‡Û”å™ã±ã«ß«ådäÈè”Üç ý|ådé4۔äÈè”ܯä èoã«ÛÀÜNá‡Üß«äÈÜ)á#ÿiÛÀà
á‡Ü á‡ÞÀý à™âÜývÝÀä ß«ä éå™ì'߇ä{á‡õ|å™è”çFð  éàd è ÁçŸÜè”éÜ=ä á#ývä èÀä ý|å™ì.ô


h4

h3

h2

h1

;Á$›“)Ž,…> >@—*,1.±*2 (-*2;1.* 0 9Ÿ9: 3  1," 40 3 * ™0 . 2—¯(§]³´F)" !¥ 3 2* " 40 3‹5

h1 < h2 < h3 ...


VÜv۔å ædܼèÀà-ÿ[ì{å™ä{çTã‡Û”Ü dß«àdÞÀè”çÀÿ¹à
߇õTèÀÜ)éÜá«á«å™ß«êTã‡àTí'Ü dä èïàdÞÀß=ÜëŸÝÀì à
ß«å™ã‡ä àdèÃàdâ#á‡ÞÀÝÀÝÁàd߇ã`ædÜéã‡à
ß ý|ådé4۔äÈè”Üáô


z/&Ù2Ò2#ŸÔTpr×¹Õ”Ô Ó ‹ØdÓÕ'Ô-46#ÀØ7Ù,Ò"  

M & M5

    6*M  

GJM

Vܱÿiä ì ì'á—ã«ådߗã#ÿiä:ã«Ûvã‡Û”Ü á‡ä ývÝÀì Üá—ã?éådá‡Ü ì äÈè”Üå™ß&ý|ådé4ÛÀä èÀÜ)áã‡ß4å™ä èÀÜ)ç¼àdèoá‡Üݔådß«ådíÀì Ü]çÀå-ã4åoò2ådá¹ÿ¹Ü á—Û”ådì ì á‡Üܙ÷Áã‡Û”ÜIå™è'å™ì êŸá—ä{á#âöàd߯ã‡ÛÀÜ dÜèÀÜß4å™ì@éådá‡Ü ¥èÀàdè”ìÈä èÀÜ)å™ßiý|å
é4ÛÀä èÀÜá#ã«ß«ådäÈè”ÜçàdèÃè”àdè ;á—Üݔå™ß4å™í”ìÈÜ`çÀå™ã«å ß«Üá‡ÞÀìÈã«á=ä èÅåTædÜß«ê¾á‡ä ýväÈì{ådß !‹Þ”ådçÀß«å™ã‡ä{éWݔ߇àdß4å™ývývä èvÝÀß«àdíÀì Üýoó4ô

å™ä èïì{å™íÁÜìã‡Û”Üvã‡ß4å™ä èÀä èTçÀå-ã4å m  ô îŸÞÀÝÀÝÁà
á‡Ü|ÿ¹Üo۔å ædÜoá‡àdývܼÛ6ê6Ý'Ü߇ݔì ådèÀÜvÿiÛÀä{é4Û -7< = D L = . D F H ø D:9;9;9DK: D L = ?:- ` ø DCø . D < = ? á‡Üݔådß«å™ã‡Üá&ã‡Û”Ü`Ý'à‹á—äÈã‡ä æ
Üiâö߇à
ýþã‡Û”Ü è”Ü 
å™ã‡ä ædܱÜëÀå™ývݔìÈÜ)á±ò2å á—Üݔå™ß4å-ã«ä  è NÛ6ê6Ý'Ü߇ÝÀì{ådèÀÜ ‹ó4ô?Ú¥ÛÀÜ=Ý'à
äÈè‹ã4á <ÅÿiÛÀä{é4ÛUì ä Ü à
èUã‡Û”ÜWÛ6ê6Ý'Ü߇ÝÀì{ådèÀÜNá‡å™ã‡ä{á—âö ê 9< 7 H SŸ÷8ÿiÛÀÜß« Ü ä{á]èÀà
߇ý|ådì8ã‡àoã‡ÛÀÜIۋê6ÝÁÜß«ÝÀì{å™èÀÜd÷ ç  |ä áWã‡Û”Ü  Þ' =éì ä{ çŸÜvådä{ètá`è”ã«àdÛÀß«Üý Ý'àdÜ߇â ÝÁ Üè”ô çÀrä ÜéãÞÀì{å™ßWçÀò ä á—ã«ó&ådè”í'éܱÜ|ã‡âöÛ”ß‡Ü à
ý á‡ÛÀàdã«ß‡ÛÀã‡ÜÜ)áۋã¥ê6çŸÝÁä{Üáã4ß«ÝÀå™è”ì{å™éèÀÜiÜ|âöß«ã‡àdàÃý ã«ÛÀã‡Û”ÜÜ àdá‡ß«Üä 
ݔä ådèß«÷å™ã‡ådä è”è  0K `Û6ê6Ý'Ü߇ݔì ådèÀÜ 7 ã‡à¼ã«ÛÀÜNéì à‹á—Ü)áã¥Ý'à‹á—äÈã«äÈæ
Ütò,èÀÜ ‹å- ã«äÈæ
Ü) ó&ÜëÀå™ývݔìÈÜdô -¯Ü ”èÀÜ`ã«ÛÀ0 Ü 4ý|ådß 
äÈ+ è `à™â?å¼á—Üݔå™ß4å-ã«äÈ è IÛ6ê6Ý'Ü߇ݔì ådèÀÜ ã‡à`íÁÜ  7  è7 Wô ۋÀê6àdÝÁßÜã«ß«ÛÀÝÀÜiì{å™ì ä èÀèÀܱÜådÿi߇äÈì ã‡ê=Ûá—Üì ådÝ”ß å™
ß4Üå™á—íÀã&ì Ü#ý|éå™ådß á‡dܙä ÷dèã‡ôÛÀÚ¥Ü]ÛÀá‡ÞÀä{á¹ÝÀéÝÁå™àdè߇ãKí'ædÜ Ü)âöéà
ã‡ß‡à
ýtß?å™ÞÀì ì{då-à
ã«Ü߇äÈçvã‡Û”ådý á¹âöá‡à
ä ìÈývì à-ÝÀÿ]ì áê=@ìÈá‡à6ÞÀà
ÝÀõ6ÝÁá@à
âöà
á‡Ü ß ã‡Û”Ü=á‡ÜÝ'å™ß4å-ã‡ä  ã‡Û'å-ã¯å™ì ìÁã‡Û”Ü=ã‡ß4å™ä èÀä  è Iç”å-ã«å¼á«å-ã«ä á—âöê¼ã‡Û”Ü=âöàdì ì à-ÿiäÈ è Wéà
è”áã«ß«ådä è
ã4á 


-9  7  I 7 ø <>=-9

7 1 `   ø <>=

âöàdß âöàdß

7Iø L =H  ø ` L =H

ò—øS‹ó ò—ødø ó

Ú¥ÛÀÜ)á—ÜNéå™èí'ÜNéàdýtíÀä èÀÜç|äÈè‹ã«àtà
èÀÜ`á—Üãià™âäÈè”Ü !‹Þ”å™ì äÈã‡ä Ü)á  ò

L = < =

9 27 ó Dø;S

EF

ò—ø
ó

 à-ÿ éàdè”á‡ä{çŸÜß=ã«ÛÀÜoÝÁàdä è‹ã«á`âöàdßNÿiÛÀä{é4ÛVã«ÛÀÜ|Ü !‹Þ”ådì ä:ãê¾äÈè 0 !”ôµò—øS
ó`ÛÀàdì{çÀá|ò,ß‡Ü !‹ÞÀä ߇ä èFã‡Û”å™ãNã‡Û”Üß«Ü Ü ëŸä{á—ã«ává‡Þ”é4Û åVÝ'à
ä è
ã|ä{ávÜ !‹ÞÀä æ ådì Üè‹ãIã«àOé4ÛÀà6à
á‡ä èVåµá‡éå™ì ÜFâöàdß å™è”ç ó4ô Ú¥ÛÀÜá‡ÜÃÝÁàdä è‹ã«ávì ä ÜFà
è ã‡Û”Ü|Û6ê‹ÝÁÜß«ÝÀì{å™è”Ü! # ;<>=49" 7# H øvÿiäÈã‡ÛVèÀà
߇ý|ådì$ å™è'ç ÝÁÜß«ÝÁÜè”çŸä{éÞÀì{å™ß`çŸä{áã4å™è”éÜtâö߇à
ýfã‡Û”Ü àdß«ä dä è 4 ä ì{å™ß«ì ê™÷™ã«ÛÀÜNÝÁàdä è‹ã«á#âöàdßiÿi۔ä é4ÛFã‡Û”ÜNÜ !‹Þ”ådìÈäÈãê¼ä è,0 !'ô±òø
ø)ó#ÛÀà
ì ç”á¥ìÈä Ü àdèã‡Û”Ü  ø %Ü  = 5 < ô#= î69'ä ýv Û6ê6Ý'Ü߇ÝÀì{ådèÀ&

7( H `ø™÷ÿiäÈã‡Û¾è”àdß«ý|å™ì@å 
ådäÈ è ÷å™è'çÃÝÁÜß«Ý'Üè”çŸä{éÞÀì{ådß±çŸä{á—ã«å™è'éÜ`âöß«àdý  ã‡Û”Ü`àdß«ä 
ä è D ø ) =   ô ¯Üè”éÜ H H ø =  å™è”çFã‡Û”ÜNý|å™ß dä è|ä{áiá‡äÈývݔìÈê  =*  ô  à™ã«Ü ç   ۔å ædÜW 7 ã‡ÛÀܼá‡ådývÜNèÀà
߇ý|ådì{ó¥ådè”çTã«Û”å-ã=èÀàFã‡ß4å™ä èÀä è ¼Ý'à
ä è
ã4á]â,ådì ì ã‡Û'å-+ ã åd߇ ÜIݔå™ß4å™ì ì ÜìKòöã‡ÛÀÜêU # å™è',  íÁÜãÿ#ÜÜèFã‡Û”ÜýTôKÚ¥Û6ޔá¥ÿ¹Ü=éådè 'è”ç|ã«ÛÀÜ=ݔå™ä ߥà™âÛ6ê‹ÝÁÜß«ÝÀì{å™è”ÜáKÿi۔ä é4Û dä ædÜ)áKã«ÛÀÜ=ý|å-ëŸä ýIޔý[ý|ådß 
äÈè í6ê|ývä èÀä ývä ä  è - ,J-÷'á—Þ” í Üéãiã‡àvéà
è”áã«ß«ådä è
ã4á=òø 
ó4ô

Ú¥Û6ޔá=ÿ#ܼÜëŸÝ'Ü)éã=ã‡Û”Ü|á—à
ìÈÞÀã‡ä àdèTâöà
ßNå|ãê6ÝÀä{éådìãÿ¹àTçŸä ývÜè”á‡ä àdè”ådì@éådá‡ÜWã«àÛ'å ædÜIã‡ÛÀÜtâöàdß«ý á‡ÛÀà-ÿiè ä è ädÞÀß«ÜúŸôDÚ¥ÛÀà‹á—ÜFã‡ß4å™ä èÀä è Ý'à
ä è
ã4áNâöà
ßvÿiÛÀä{é4ÛÅã‡Û”ÜÜ !‹Þ”å™ì äÈãêVäÈè)01!'ôgò—ø
óNÛÀà
ì{çÀáTòöä.ô Üdôµã‡Û”à
á‡Ü ÿ ÛÀä{é4ÛTÿiä è”çÞÀÝUì ê‹ä è¼à
èà
èÀÜ`à™âã«ÛÀÜNÛ6ê6Ý'Ü߇ÝÀì{ådèÀÜá. # ÷/  ó÷8ådè”çÿiÛÀà
á‡Ü`߇Üývà-æ-å™ìÿ#àdÞÀì{çé4۔å™è
Ü i ã‡Û”Ütá‡àdì ޟã«äÈà
èoâöà
ÞÀè”ç8÷åd߇ÜIéå™ì ì ÜçTá‡ÞÀÝÀÝÁàd߇ã±æ
Üéã«àdß4áûÁã‡Û”ÜêÃå™ß«ÜNä è”çŸä{éå-ã‡Ü)çäÈè @ä
ÞÀ߇ÜWú|í6êFã«ÛÀÜtÜë6ã‡ß4å éä ß4éì Üáô
VܼÿiäÈì ìèÀà-ÿ á‡ÿiä:ã4é4ÛUã‡àTå!å 
ß«ådèdä{ådèFâöà
߇ýtÞÀì{å-ã«ä àdèà™âKã«ÛÀܼÝÀ߇à
íÀì ÜýTô`Ú¥ÛÀÜܼ߫å™ß«ÜNãÿ#àFß«Üå
á—à
è”á âöàdß çŸàdä è¼ã«ÛÀä{áô Ú¥ÛÀÜ ”ß4á—ã±ä{á¥ã‡Û'å-ã¯ã«ÛÀÜIéàdè”á—ã‡ß4å™ä è‹ã«áWòø
ó¥ÿiä ì ìrí'ÜI߇ÜÝÀì{ådéÜçí6êTéàdè”á—ã‡ß4å™ä è‹ã«á¥à
èUã‡Û”Ü  å dß4å™ è 
Ü¥ýIÞÀìÈã«äÈݔìÈä Üß«árã‡ÛÀÜý|á—ÜìÈæ
Üá÷™ÿiÛÀä{é4Û¼ÿiäÈì ìŸíÁܯýtޔé4Û¼Üå
á—ä Üßã«àN۔å™è'çŸì ܙô?Ú¥ÛÀܯá‡Üéà
è”çtä{áã‡Û”å™ã¹ä è ã‡Û”ä á¹ß«Üâöàdß«ýtÞÀì{å-ã‡ä à
èWà™âã«ÛÀÜ ÝÀ߇à
íÀì ÜýT÷™ã«ÛÀܱã‡ß4å™ä èÀä  è NçÀå™ã«åNÿiä ì ìÀà
èÀì ê¼ådÝÀÝÁÜå™ß òöä èvã«ÛÀÜ å
éã«Þ”å™ì'ã‡ß4å™ä èÀä  è  å™è'çtã‡Üá—ã#ådì dàdß«äÈã‡ÛÀý|á4óä è¼ã«ÛÀÜ]âöà
߇ý à™ârçŸàdã&ÝÀß«à6çÀޔéã4áKíÁÜãÿ#ÜÜèoædÜéã‡à
ß«áôÚ¥ÛÀä{á?ä{á¹åNé߫ޔéä{ådìŸÝÀ߇à
Ý'Üߗãê ÿiÛÀä{é4ÛÿiäÈì ì8å™ì ì à-ÿ ޔá#ã‡à 
ÜèÀÜß«ådì ä Üiã‡Û”Ü`ÝÀ߇àŸéÜçŸÞÀܱ߫ã‡à¼ã‡Û”Ü`èÀàdèÀì ä èÀÜ)å™ß¥éådá‡Ü¼ò2î6Ü)éã‡ä à
è B6ó4ô

Ú¥Û6ޔá÷iÿ¹ÜTä è‹ã‡ß«à6çÀޔéÜTÝÁà
á‡ä:ã«ä ædÜ0@å 
ß«ådèdÜýIޔì:ã«ä ÝÀì äÈÜß«á = D F H ø D;9:9;9JD :;÷¥àdèÀÜâöàdßoÜådé4ÛDà™â ã‡Û”Ü ä èÀÜ !‹Þ”ådì ä:ãêFéàdè'áã«ß«ådäÈè‹ã4áNòø7dó4ô iÜ)éådìÈìã‡Û”å™ã±ã«ÛÀÜt߇ÞÀì ÜNä{á¯ã‡Û”å™ã±âöà
ß éàdè”á—ã‡ß4å™ä è‹ã«á]à™âKã«ÛÀÜWâöà
߇ý10J=RSŸ÷

w H2

-b |w|

H1

Origin Margin

;Á$›“)Ž,…
> ‹Î " 3 @*2² .41," 3 i+§²d2² $  3 —*9 0 1,+-K*.;²-24( $K*. 5Á< +-?*, ² ² 0 2 1 °  ;1 0 2*2;"$.;$ 5 ã‡Û”Üïéàdè”á—ã‡ß4å™ä è‹ã|Ü !‹Þ”å™ã‡ä àdè”áFå™ß«Ü¾ýtÞÀìÈã‡ä ÝÀì ä ÜçOí‹ê 5AI G  *" M @å dß4å™èdÜÃýtÞÀìÈã‡ä ÝÀì ä Üß4áIådè”ç á‡ÞÀíŸã«ß«å
éã‡Ü)ç âöß«àdý[ã‡ÛÀܱàd í Ü)éã‡ä æ
ܹâöޔè”éã«äÈà
è÷‹ã‡à`âöà
߇ý[ã‡ÛÀ Ü å dß4å™è
ä ådèôÀà
ß&Ü !
Þ'å™ì äÈãêNéàdè'áã«ß«ådäÈè‹ã4á÷™ã«ÛÀÜ@å 
ß«ådèdÜ ýtÞÀìÈã‡ä ÝÀì ä Üß4á?å™ß«Ü ޔè”éà
è”áã«ß«ådä èÀÜç8ô?Ú¥ÛÀä{á dä æ
Üá å 
ß«åd è dä{ådè  ø

 1

 ,





 ( & ='L =‡ò'<,= 9 7 ó 7 ( & "= = ) =*) # #

ò—ø ‹ Eó


VÜTýIޔá—ãIè”à-ÿ ývä èÀä ýväÜ ÿiäÈã‡Û ߇Ü)á—ÝÁÜéãWã«à !D- )÷¹ådè”çOá‡ä ýIÞÀìÈã4å™èÀÜàdޔá‡ì êUß‡Ü !
ޔäÈß«Üvã«Û”å-ã¼ã‡Û”Ü çŸÜ߇ä æ-å-ã«äÈæ
Üá à™â ÿiä:ã«ÛÅ߇Ü)á—ÝÁÜéãNã‡à å™ì ìKã«ÛÀÜ "=¯æ-å™è”ä á‡Û÷?å™ì ì&á‡ÞÀíÜ)éãNã«àÃã‡Û”Üoéà
è”áã«ß«ådä è
ã4á =&GS òöì Üã $áKéådìÈì6ã«ÛÀä{áݔå™ß‡ã‡ä{éÞÀì{å™ßá—ÜãKà™âéàdè'áã«ß«ådäÈè‹ã4á # ó4ô  à-ÿ ã‡ÛÀä{áä{á?å`éà
è6ædÜë !
Þ'ådçŸß4å-ã«ä é¹ÝÀß«à
ß«ådývýväÈè ÝÀß«àdíÀì ÜýT÷á‡ä è”éÜoã‡ÛÀÜàd í Ü)éã«äÈæ
ÜvâöÞÀè”éã‡ä àdèOä áIäÈã«á‡ÜìÈâ]éà
è‹æ
Üë8÷Kå™è”çµã‡ÛÀà‹á—ÜFÝ'à
ä è
ã4áWÿi۔ä é4Ûµá«å-ã«ä{áâöêïã‡Û”Ü éà
è”á—ã‡ß4å™ä è‹ã«á]ådì á‡àtâöàdß«ýyåoéà
è6ædÜëTá—Üãtò2å™è6êTìÈä èÀÜ)å™ß]éàdè”á—ã‡ß4å™ä è‹ã¯çŸÜ 'èÀÜá å|éà
è6ædÜëUá‡Üã÷rådè”çÃå|á‡Üã à™â  á—ä ýtÞÀìÈã«ådèÀÜà
ޔá]ì ä èÀÜådß éàdè”á—ã‡ß4å™ä è‹ã«á çŸÜ ”èÀÜá`ã‡ÛÀÜvä è‹ã‡Üß«á‡Üéã‡ä àdèÃàd â  éà
è6ædÜëïá—Üã«á÷@ÿiÛÀä{é4Û¾ä{á`ådì á‡à å¾éàdè6ædÜëµá—Üã4ó4ôµÚ¥ÛÀä{áIývÜådè”á`ã«Û”å-ã¼ÿ¹ÜTéådèµ Ü !‹ÞÀä æ-å™ì Üè
ã«ì ê¾á—à
ì ædÜvã«ÛÀÜoâöà
ì ìÈà-ÿiä  è  çŸÞ”å™ì Uݔ߇à
íÀì Üý  K   K4 M   ÷@á—Þ”í Üéã±ã«àFã«ÛÀܼéàdè'áã«ß«ådäÈè‹ã4á]ã«Û”å-ã ã‡ÛÀÜ 
ß«å
çŸä Üè‹ã¯à™â   ÿiäÈã«Û¾ß«Üá‡Ý'Ü)éã ㇠à få™è”ç æ-å™èÀä{á‡Û÷å™è”çUá‡ÞÀ í Ü)éã`å™ì{á—àtã«à|ã‡Û”Ütéàdè”á—ã‡ß4å™ä è‹ã«á#ã«Û”å-ã ã‡ÛÀ Ü =  SÃòöì Üã $á±éå™ì ì *( ݔådߗã«ä éÞÀì{å™ß]á‡Üã à™â éà
è”á—ã‡ß4å™ä è‹ã« á ó4ôTÚ¥ÛÀä{á`ݔådߗã«ä éÞÀì{å™ß=çŸÞ”ådì?âöàdß«ýIޔì å™ã‡ä àdè¾à™â#ã‡ÛÀÜ|ݔ߇à
íÀì Üýyä{áNéådì ìÈÜ)çUã«ÛÀ Ü
ïà
ìÈâöÜ|çŸÞ”å™ì  *ò ì Üã4é4ÛÀÜß÷?øù
'ü ™óô ƒã ۔å
á]ã«ÛÀܼÝÀß«àdÝÁÜ߇ãêFã«Û”å-ã ã‡Û”ÜIý|å™ë6ä ýtÞÀýYàd â   ÷á‡ÞÀ í Ü)éã ã‡àTéà
è”áã«ß«ådä è
ã4 á ÷  àŸééÞÀß«áiå™ã¥ã‡ÛÀÜNá«å™ývÜ æ-å™ì ÞÀÜá#àdâã‡Û”+ Ü F÷ ¯ådè”ç &÷Áå
á¹ã‡ÛÀÜ`ývä èÀä ýtÞÀý[àd â ¹÷”á—Þ” í Üéãiã‡àvéà
è”áã«ß«ådä è
ã4á # ô

iÜ !‹ÞÀä ߇ä èWã‡Û”å™ãiã‡ÛÀÜ 
ß«å
çŸä Üè‹ã#àdâ 



( H

=

( =



= L =

H



ÿiäÈã‡Ûß«Üá‡Ý'Ü)éã¥ã‡à Y

ådè”ç ]æ-ådèÀä{á—Û dä ædܱã‡ÛÀÜNéàdè”çÀä:ã«ä àdè”á

ò—øJB6ó

= L = < =

ò—ø)ú
ó

SI

î6ä è”éÜ=ã‡ÛÀÜ)á—ÜIå™ß«Ü`Ü !‹Þ”å™ì äÈãê|éà
è”á—ã‡ß4å™ä è‹ã«á¥ä èã‡ÛÀÜWçÀޔå™ìâöà
߇ýtÞÀì{å-ã«ä àdè÷Ÿÿ#ÜNéå™èUá‡ÞÀí”á—ã‡äÈã‡ÞÀã‡Ü`ã«ÛÀÜý_ä è
ã«à

01!'ô]òø
E ó¹ã«à 
ä ædÜ

;H ( =

= 

ø 

(

  =

=  L = L  < = 9 < 

ò—ø ‹ó


 à™ã‡Ü¯ã‡Û”å™ã¹ÿ#ܯ۔å æ
ÜièÀà-ÿ dä æ
Üètã‡ÛÀ Ü å dß4å™ è 
ä ådèIçŸä 8Üß«Üè‹ã&ì{å™íÁÜì{á]%ò
gâöàdß¹ÝÀ߇ä ý|ådì2÷þâöàdß¹çÀޔå™ìöóã«à Ü ývݔ۔ådá‡äÜKã«Û”å-ã?ã«ÛÀÜ¥ãÿ#à±âöà
߇ýtÞÀì{å-ã«ä àdè”áråd߇ÜiçŸä8Üß«Üè‹ã  å™è”ç Oå™ß«ä á‡Ü¹âö߇à
ý ã‡ÛÀÜ]á«å™ývÜ#àdíÜéã«ä ædÜ âöÞÀè”éã‡ä àdè íÀޟã=ÿiäÈã‡Û çÀä 8Üß«Üè‹ã=éàdè'áã«ß«ådäÈè‹ã4áûrå™è'çUã‡Û”Ütá‡àdì ޟã«äÈà
èUä{á]âöà
ÞÀè”ç í‹êUývä èÀä ývää è   à
ß±í6ê ý|å-ëŸä ývä ä  è  ±ô  à™ã‡ÜNådì{á—àtã‡Û'å-ã¯äÈâÿ¹Ü=âöà
߇ýtÞÀì{å-ã«Ü¯ã«ÛÀÜNÝÀß«àdíÀì Üý ÿiäÈã‡, Û @HRSŸ÷'ÿi۔ä é4ÛTå™ývà
ÞÀè‹ã«á#ã«à ß« Ü !‹ÞÀä ߇ä  è =ã«Û”å-ã]ådìÈìÁÛ6ê6Ý'Ü߇ÝÀì{ådèÀÜá¹éàdè‹ã4å™ä èvã‡Û”Ü à
߇ä 
äÈèr÷™ã‡ÛÀÜ`éàdè”á—ã‡ß4å™ä è‹ã±òø údó¹çŸà6Üá#èÀàdãiådÝÀÝÁÜå™ßô?Ú¥ÛÀä{á ä{á¯åoýväÈì{çß«Üá—ã‡ß«ä éã‡ä àdèFâöàdß±ÛÀä dÛ¾çÀäÈývÜè”á‡äÈà
è”å™ìá‡Ý”ådéÜá÷8á‡ä è”éÜWä:ã=å™ývà
ÞÀè‹ã«á¥ã«àoß«ÜçŸÞ”éä  è vã‡Û”ÜWè6ÞÀýtí'Üß à™â?çŸ Ü dß«ÜÜá#àdââöß«ÜÜçÀàdýYí6ê|à
èÀܙô

î6ޔÝÀÝ'à
ߗã ædÜ)éã‡à
ßiã‡ß4å™ä èÀä èTò{âöà
ß]ã«ÛÀÜIá‡Üݔådß«ådíÀì ܙ÷Àì ä èÀÜådßiéådá‡Ü)ó¥ã‡ÛÀÜ߇Üâöàdß«ÜNå™ývàdޔè
ã4á¥ã‡àoý|å-ëŸä ývää è ÿiäÈã‡Û¼ß«Üá‡Ý'Ü)éãKã«à=ã‡ÛÀ Ü =—÷‹á‡ÞÀ í Ü)éãKã«àWéà
è”á—ã‡ß4å™ä è‹ã«áiòø údóådè”ç¼Ý'à‹á—äÈã«äÈæ6äÈãê`à™âÁã‡Û”" Ü "=;÷6ÿiä:ã«Û|á—à
ì ޟã‡ä àdè dä æ
Üèïí‹ê òø B‹óô  à™ã‡ä{éܼã‡Û”å™ãNã‡ÛÀÜ߇Üoä á`å å 
ß«ådè dÜtýtÞÀìÈã‡ä ÝÀì ä Ü" ß "=iâöà
ßNÜæ
Üß«êÃã«ß«ådäÈè”äÈ è FÝÁàdä è‹ãô ;è ã‡Û”Ütá‡àdì ޟã«äÈà
è÷Àã«ÛÀà
á‡ÜWÝÁàdä è‹ã«á¯âöàdß±ÿiÛÀä{é43 Û = . Soåd߇Ütéå™ì ì Ü* ç á—ÞÀݔÝ'à
ߗã¯æ
Üéã‡àdß4Ká Ÿ÷Áådè”çÃì ä Ü`àdè¾àdèÀÜIà™â ã‡Û”ÜFÛ6ê6Ý'Ü߇ݔì ådèÀÜ& á è UÝ'à
äÈè‹ã4áN۔å æ
Ü "= H Sïå™è'çÅì äÈÜoÜäÈã‡ÛÀÜßWà
è # D!  ô
ì ì¹àdã‡ÛÀÜßWã«ß«ådäÈè”äÈ # à
ß , ò ‡ á ” Þ 4 é ¾ Û « 㠔 Û å = ã « ã À Û v Ü Ü ‹ ! ” Þ d å È ì È ä  ã  ê ä è 1 0 ' ! U ô  ò ø 7 dó¯ÛÀàdì{çÀá4ó4÷rà
ß à
è¾ã«Û”å-ãNá‡ä{çŸÜIàdâ d à ß  — á ' Þ 4 é Ã Û ‡ ã ' Û å ` ã ã‡Û”Ü #   á—ã‡ß«ä éãiäÈè” Ü !‹Þ”å™ì äÈãê¼ä , è 0 !”ô òø 
ó#ÛÀàdì{çÀá1 ô Àà
ߥã‡ÛÀÜ)á—Ü`ý|å
é4ÛÀä èÀÜá÷6ã‡ÛÀÜIá—ÞÀݔÝ'à
ߗã]æ
Üéã«àdß4á¥å™ß«Ü±ã«ÛÀÜWé߇äÈã‡ä{éå™ì Üì ÜývÜè‹ã«á#à™âã‡Û”Ü=ã‡ß4å™ä èÀä  è ¼á—Üãô¹Ú¥ÛÀÜêoì ä Ü éì à
á‡Üá—ã¹ã«à¼ã‡ÛÀÜWçÀÜéä{á‡ä àdè|íÁàdÞÀè'çÀå™ß«êÁû”äÈâådìÈì8àdã‡ÛÀÜßiã‡ß4å™ä èÀä  è  ÝÁàdä è‹ã«á#ÿ¹Üß‡Ü ß«Üývà-ædÜ)çUòöà
ß#ývà-æ
Üç|åd߇à
ÞÀè”ç8÷ŸíÀÞÀã¯á—à¼ådá#èÀàdã#ã«àvéß«à
á«á d à ß .  ó ” ÷ ™ å ” è o ç ‡ ã 4 ß ™ å ä À è ä è  W  # ÿ å
á #  ß«ÜÝÁÜå-ã«Üç8÷Ÿã«ÛÀÜNá«å™ývÜ=á—Üݔå™ß4å-ã«äÈ è WÛ6ê6Ý'Ü߇ÝÀì{ådèÀܱÿ#àdޔì çoí'Ü=âöà
ÞÀè”ç8ô

 

$ G   $

M   $%  

$   M $ I   IAG

  

Ú¥ÛÀ

Ü `åd߇Þ'á— Û  =ÞÀÛÀ è ƒÚ@ޔé4õdÜßU ò `Ú]óvéàdè”çŸäÈã«äÈà
è”á¼ÝÀì{å ê åOéÜè‹ã«ß«ådì]߇à
ì ÜUäÈè íÁà™ã«Û ã‡ÛÀܾã‡ÛÀÜàdß«ê ådè”ç ÝÀß4ådéã‡ä{éÜtà™â¥éàdè'áã«ß«ådäÈè”ÜçÃà
ݟã‡ä ýväå™ã‡ä àdèrô Ààdß=ã«ÛÀÜvÝÀß«ä ý|å™ìÝÀ߇à
íÀì Üýfådí'à-æ
ܙ÷rã«ÛÀÜ `Ú éàdè”çŸäÈã«äÈà
è”á ý|å êvíÁÜNáã4å-ã‡Ü)ç ò ìÈÜã«é4ÛÀÜß÷rø)ùd ü ™ó   

5H

  



 ( =

  2 H  (

"= L =*)+=  H

S



H

S

=

= L =

-9 7 ó? ø+ =   =«òL =«ò 9 <,=F 7 ó D  ø)ó H

H

:9;9:9JD  ø D

ò—ø dó

—ò

ò—øü‹ó

L = '<>=

S

FPH S

EF S

EF

ø

:9;9:9JDK: D

ò—øù‹ó ò  S‹ó ò Ÿø ó

Ú¥ÛÀ Ü `Ú éàdè”çÀä:ã«ä àdè”á?åd߇Ü]á«å-ã«ä á ”Ü)çIå™ãKã‡ÛÀÜ á—à
ì ޟã‡ä àdèIà™ârå™è6ê¼éà
è”áã«ß«ådä èÀÜçIàdݟã«ä ývä )å-ã«äÈà
èNÝÀß«àdíÀì Üý ,ò éàdè6ædÜë à
ß|èÀà™ãó4÷¯ÿiäÈã«Û ådè6ê õ6ä è”çDà™âNéà
è”á—ã‡ß4å™ä è‹ã«á÷#ÝÀß«à-æ6ä{çŸÜçOã‡Û”å™ã|ã‡Û”ܾä è‹ã‡Üß4á‡Üéã«ä àdèOàdâ`ã‡ÛÀÜ á—Üã à™â¯âöÜådá‡ä íÀì ÜoçŸä ß«Üéã«ä àdè”á`ÿiäÈã«ÛÅã‡ÛÀÜUá‡Üã¼à™â±çÀÜá«éÜè‹ã¼çŸä ߇Ü)éã‡ä à
è”áWéàdä è”éä{çŸÜ)á`ÿiäÈã‡ÛOã‡ÛÀÜä è
ã«Üß4á—Ü)éã«äÈà
èVà™â ã‡Û”ܾá—Üã|à™â âöÜå
á—ä íÀì ÜTçŸä ߇Ü)éã‡ä à
è” á  I 6 * M   M2@IAG    *  GIÿiäÈã‡Û ã‡Û”ܾá—Üã|à™âNçŸÜ)á‡éÜè‹ãoçŸä ߇Ü)éã«äÈà
è”á ò,á‡ÜÜ @ì Üã«é4ÛÀÜß÷ø)ùd ü ‹û”ñÃé ¹àdß«ývä é4õÁ÷øù
ü E‹ó—ó4ô¥Ú¥ÛÀä{áiß4å-ã‡Û”Üßiã«Üé4ÛÀèÀä{éå™ìß«Ü dޔì åd߇äÈãêvådá«á‡ÞÀývݟã‡ä à
èoÛÀà
ì ç”á âöàdß`ådìÈì?á‡ÞÀÝÀÝÁàd߇ã`ædÜéã‡à
ß±ý|å
é4ÛÀä èÀÜá÷rá—ä è”éÜWã«ÛÀÜvéàdè'áã«ß«ådäÈè‹ã4á±åd߇Ütå™ì ÿ#å êŸá¯ìÈä èÀÜ)å™ßô Àޔߗã«ÛÀÜß«ývàd߫ܙ÷8ã‡Û”Ü ÝÀß«àdíÀì Üý âöà
ß¼îŸð±ñÃá¼ä{átéàdè6æ
Üë ò,åïéàdè6æ
ÜëÅàd í Ü)éã«äÈæ
ÜvâöÞÀè”éã‡ä àdèr÷&ÿiäÈã‡Û éàdè”á—ã‡ß4å™ä è‹ã«áNÿiÛÀä{é4Û dä ædÜFå éà
è6ædÜëOâöÜådá‡ä íÀì Üoß«Ü 
äÈà
è'ó4÷¥å™è”ç âöà
ßoéà
è‹æ
ÜëµÝÀß«àdí”ìÈÜý|áTòöäÈâ±ã«ÛÀÜUß«Ü dޔì åd߇äÈãêVéà
è”çŸäÈã‡ä àdè ÛÀàdì{çÀá4ó4÷#ã‡Û”Ü `Úgéàdè'çŸäÈã‡ä àdè”áiåd߇: Ü  M  M GG  #  7 G $M  âöà
 ß !D D" Oã‡à|íÁÜWå|á‡àdì ޟã«ä àdè ò ìÈÜã«é4ÛÀÜß÷øù
' ü ™óô Ú¥Û6ޔá á—à
ìÈæ6ä  è tã‡Û”ܼî6ð ñ ÝÀß«àdí”ìÈÜý ä{á¯ Ü !‹ÞÀä æ-å™ì Üè‹ãiã‡à ”è”çÀäÈ è oåoá‡àdì ޟã«äÈà
èã‡à|ã«ÛÀ Ü `Úgéàdè”çÀä:ã«ä àdè”áô Ú¥ÛÀä{á&â,ådéã¥ß«Üá‡ÞÀìÈã«á¹äÈèFá—ÜædÜß4å™ìÁå™ÝÀݔ߇à‹ådé4ÛÀÜ)á?ã‡à ”è”çÀäÈ è Nã‡Û”Ü á‡àdì ޟã«äÈà
èÃò{âöà
ß#ÜëÀå™ývÝÀì ܙ÷
ã‡ÛÀÜ ÝÀß«ä ý|å™ì ƒçÀޔå™ì ݔå™ã‡Ûâöàdì ì à-ÿiä  è `ývÜã«ÛÀàŸçoývÜè
ã«ä àdèÀÜ)ç|ä èTîŸÜéã«ä àdèú
ó4ô


á¯å™èUä ývývÜ)çŸä{å-ã‡Ü å™ÝÀݔìÈä{éå-ã‡ä à
è÷‹è”à™ã‡ÜNã‡Û”å™ã÷'ÿi۔äÈì Ü _ä á¥ÜëŸÝÀì ä éäÈã‡ì êvçŸÜã«Üß«ýväÈè”ÜçFí‹êFã‡Û”Ü`ã‡ß4å™ä èÀä è  Ý ß«àŸéÜçÀÞÀ߇Üd÷”ã«ÛÀÜNã«ÛÀ߇Ü)á—Û”àdì{ç =ä{áièÀà™ã÷8ådìÈã‡ÛÀà
ÞdÛTäÈã¯ä{á]ä ývÝÀì ä{éäÈã‡ì ê¼çŸÜã«Üß«ývä èÀÜç8ô¯à-ÿ¹ÜædÜß ä á¯Üå
á—ä ì ê À âöàdޔè”çOí6êµÞ'á—ä è ã‡ÛÀÜ `Ú éà
ývÝÀì ÜývÜè‹ã4å™ß«ä:ãêéà
è”çŸäÈã‡ä àdèr÷ 01!'ôgò Ÿø ó4÷¹í6êOé4۔à‹à‹á—ä èïå™è6ê F`âöà
ß

ÿiÛÀä{é4Û = H
S|ådè”çUéà
ývÝÀޟã«äÈè ¼ò,èÀà™ã«ÜNã«Û”å-ã¯äÈã¯ä{áiè6ÞÀývÜ߇ä{éå™ì ì êtá«å-âöÜß]ã‡à|ã«å™õ
Ü=ã‡ÛÀÜIývÜådèæ-ådìÈޔÜ`à™â ¯ß‡Ü)á—ÞÀìÈã«äÈèNâöß«àdý ådì ì8á‡Þ”é4ÛÜ !‹Þ”å™ã‡ä àdè”á4ó4ô  à™ã‡ä{éÜoã«Û”å-ãoådìÈì#ÿ#Ü  ædÜUçŸàdè”Üá‡àïâ,å™ß¼ä{áWã«àÅéådá—ãtã‡ÛÀÜUÝÀß«àdí”ìÈÜý äÈè‹ã«àïådè à
ݟã‡ä ývä å™ã‡ä àdèVÝÀß«àdíÀì Üý ÿ ÛÀÜ߇ÜWã‡ÛÀÜvéà
è”á—ã‡ß4å™ä è‹ã«á]åd߇ÜIß«å™ã‡ÛÀÜß±ývà
߇ÜIý|å™è”ådÜådíÀì Ü ã‡Û”ådèÃã‡Û”à
á‡ÜIä è 0 !‹áô|òøS
ó4÷&ò—ødø ó4ô ä è”çŸä è i ã‡Û”Ütá‡àdì ޟã«äÈà
èoâöà
߱߫Üådìÿ#àdß«ì{çFݔ߇à
íÀì Üý|á¥ÿiä ì ìޔá‡Þ”å™ì ì ê|ß«Ü !‹ÞÀä ß«Ü`è‹Þ”ývÜß«ä éå™ìrývÜã‡ÛÀàŸçÀáô@
VÜIÿiä ì ìr۔å æ
Ü ývàdß«Ütã‡àÃá«å êÃàdèïã«ÛÀä{á`ì{å-ã‡Üßô ¯à-ÿ¹ÜædÜß÷ì Üã $ á ”ß«á—ãNÿ#àdß«õÃàdޟãtåTß«åd߇ܼéå
á—ÜvÿiÛÀÜ߇ܼã«ÛÀÜvÝÀß«àdíÀì Üýfä{á èÀà
è
ã«ß‡ä æ6ä{å™ìKòöã‡ÛÀÜtè6ÞÀýIíÁÜß±à™â&çŸä ývÜè”á—ä à
è”á¥ä{á±åd߇íÀäÈã«ß«åd߇êd÷'å™è”çUã‡ÛÀÜtá—à
ìÈÞÀã‡ä àdèUéÜߗã4å™ä èÀì êoè”à™ã±à
í6æ‹ä à
ޔá«ó÷ íÀޟã]ÿi۔Üܱ߫ã«ÛÀÜNá‡àdì ޟã‡ä à
èoéådèíÁÜ=âöàdޔè”çFådè”å™ì ê‹ã‡ä{éå™ì ì ê™ô

0 5   K  673#!5 M )576  




!

M GB   K 576*M

ÛÀä ì Ü=ã‡ÛÀÜIý|å™ä èUå™ä ýYàdâ?ã‡ÛÀä{á¯îŸÜéã«ä àdèUä{á¥ã‡àFÜëŸÝÀì àdß«Ü`å|èÀà
 è .ã‡ß«äÈæ6ä{å™ì8ݔå™ã—ã«Üß«èU߇Ü)éà 
èÀäÈã‡ä àdèÝÀß«àdíÀì Üý ÿiÛÀÜ߇ܼã«ÛÀÜoá‡ÞÀÝÀÝÁàd߇ãNædÜéã‡à
ßNá—à
ìÈÞÀã‡ä àdèïéå™èVíÁÜvâöàdޔè”çÅå™è”ådì ê
ã«ä{éå™ì ì ê™÷Áã‡Û”Üv߇Ü)á—ÞÀìÈã4á`çŸÜß«ä ædÜç ÛÀÜ߇Üvÿiä ì ì å™ì{á‡à|íÁÜtÞ'á—ÜâöÞÀì?ä è¾åoì{å™ã‡Üß ÝÀß«à6à™â;ô ”àdß±ã‡ÛÀܼÝÀß«àdíÀì Üý éàdè”á‡ä{çŸÜß«Üç÷ÁÜædÜß«êã‡ß4å™ä èÀä  è |ÝÁàdä è‹ã±ÿiä ì ì@ã‡ÞÀß«è àdÞÀãiã‡à¼í'ÜNå¼á‡ÞÀÝÀÝÁàd߇ã]ædÜ)éã‡à
ß÷ŸÿiÛÀä{é4Ûä{á¹à
èÀÜ=߇Ü)ådá‡àdèFÿ¹Ü`éåd, è ”è”çFã‡ÛÀÜNá‡àdì ޟã«ä àdèoådè”å™ì ê‹ã‡ä{éådì ìÈêdô

¹à
è”á—ä{çŸÜ ß O 7 øá—ê6ývývÜã‡ß«ä éå™ì ì êFݔì å
éÜ)çVÝ'à
äÈè‹ã4áNì ê6äÈèà
èOå¾á‡ÝÀÛÀÜß‡Ü p B 7 # àdâ]ß«å
çŸä ޔá A |ývàdß«Ü ÝÀß«Üéä á‡Üì êd÷Ÿã‡ÛÀÜIÝ'à
äÈè‹ã4á¥âöàdß«ýYã«ÛÀÜWæ
Ü߇ã‡ä{éÜ)á¥à™âKå™è O ;çŸä ývÜè'á—ä àdè'å™ìá—ê6ývývÜã‡ß«ä é á‡äÈývݔìÈÜë8ô ƒã¯ä{á¯éàdè6ædÜ èÀä Üè‹ãiã«àvÜýtí'Ü)çoã«ÛÀÜNÝÁàdä è‹ã«á¥ä è m B  # ä èTá‡Þ”é4ÛUå¼ÿ¥å êvã‡Û”å™ã]ã‡Û”ÜêTå™ì ìì äÈÜ=ä èFã‡ÛÀÜNÛ6ê6ÝÁÜß«ÝÀì{å™èÀÜ=ÿiÛÀä{é4Û Ý”å
á‡á‡Üá@ã‡Û”߇à
 Þ dÛIã‡ÛÀÜ¥à
߇ä dä èWådè”çIÿiÛÀä{é4ÛWä{áÝ'Ü߇ÝÁÜè”çÀä éÞÀì{å™ß@ã‡à ã‡Û”ÜNò O7Uø ó ƒædÜéã‡à
ß#ò—ø D ø DII IDø ó&ò,ä èWã«ÛÀä{á âöàdß«ýtÞÀì{å-ã‡ä à
è÷-ã‡ÛÀÜ=ÝÁàdä è‹ã«á&ìÈä Ü]à
è p B 7 # ÷Ÿã‡ÛÀÜê|á—Ý'å™è m B ÷Ÿådè”çoåd߇ܱÜýtí'Ü)çÀçŸÜ)çvä è m B  # ó41 ô 0KëŸÝÀì ä éäÈã‡ì ê™÷ ß«Üéådì ìÈä  è ã«Û”å-ã¼ædÜ)éã‡à
ß«áWã‡Û”Üý|á‡Üì ædÜ)áWå™ß«Üoì{ådí'ÜìÈÜ)çVí6* ê iàdý|ådèµä è”çŸä{éÜ)áWå™è'çÅã‡Û”Üä ßvéà6àdß4çŸä è”å-ã«Üá`í6ê % ß«ÜÜõÁ÷Ÿã«ÛÀÜNéà6àdß4çŸä è”å™ã‡Üá#å™ß«Ü dä ædÜèoí6 ê 

)N= 2 H Nòø4 ?= Àó 9

A

KòO O

7

ø)ó

7 ?=   O

AO

7

ø

ò  
ó

ÿiÛÀÜ߇ܼã«ÛÀÜ =ß«àdèÀÜ)é4õdÜß=çÀÜìÈã«åÀ÷ ? =  ÷ä á`çÀÜ ”èÀÜ)çVí6 ê ? =  HyøväÈ â HGF‡9 ÷ STà™ã«ÛÀÜß«ÿiä{á—ÜdôFÚ¥Û6ޔá÷@âöà
ß ÜëÀådývÝÀì ܙ÷‹ã‡ÛÀÜ`æ
Üéã«àdß4á¹âöàdߥã‡Û”߇ÜÜ`Ü !‹ÞÀä{çŸä{áã4å™è‹ã¹ÝÁàdä è‹ã«á¥à
èoã«ÛÀÜ`ÞÀèÀäÈã]éä ß4éì ÜIò2á—ÜÜ ä
ÞÀß«ÜIø7dó#å™ß«Ü  H ò # Ñ <

H 

Ñò H 8

E

`ø

Ñò

<

<



 D



`ø D



`ø



D



 D







`ø 

E



D

D



`ø ó

`ø ó





E





ò LE‹ó ó

$±èÀÜTéà
è”á‡ Ü !‹ÞÀÜè'éÜ|àdâiã‡ÛÀÜTá—ê6ývývÜã‡ß«ê¾ä{á`ã«Û”å-ãtã‡ÛÀÜTå™èdì Üví'Üãÿ¹ÜÜèDå™è6ê Ý'å™ä ßWàdâ¯ædÜ)éã‡à
ß«áNä áWã‡Û”Ü á«å™ývܼò,å™è'çFä{á#Ü !‹Þ”ådì8ã‡à|ådß«ééà
áò`ø =$Oó—ó 



<,=

 

-9 

H

<>= < H

A

ò CB6ó 

@A  7= O

ò d ú
ó

à
ß÷Àývà
߇Ü=á—Þ'ééä è”éã‡ì ê™÷

-9  B H ?= 4 òø4 ?= )ó A 

ø

<>= <

O

I

ò ‹  ó


á‡á‡ä dèÀä è µåµéì{ådá«ávì{å™íÁÜ ì ? - 7Iø D `Cø . å™ß«íÀäÈã‡ß4å™ß«äÈì êïã‡à Üå
é4Û ' Ý à
äÈè‹ã÷¯ÿ¹Ü¾ÿiä á‡Û ã‡à ”è”ç ã‡Û”å™ã Û6ê6Ý'Ü߇ÝÀì{ådèÀÜIÿi۔ä é4Ûïá—Üݔå™ß4å-ã«Üá±ã‡Û”ÜIãÿ#àUéì{ådá«á‡Üá±ÿiä:ã«Û ÿiä{çŸÜ)áã`ý|ådß 
Èä èrôIÚ¥Û6ޔá=ÿ¹Ü¼ýtޔá—ã ý|å™ë6ä ýväÜ

ä è 0 ' ! ôïòø
 ó÷á‡ÞÀí Üéã`ã‡à =  SUådè”çVå™ì{á—àTá‡ÞÀíÜéãNã‡àTã«ÛÀÜvÜ !‹Þ”ådì ä:ãêÃéàdè”á—ã‡ß4å™ä è‹ã÷0 !'ô òø údó4ô $±ÞÀßiá—ã‡ß4å-ã‡Ü dêtä{á&ã«àtá‡ä ývÝÀì êIá‡àdì ædÜ]ã‡ÛÀÜ=ÝÀß«àdíÀì Üý å
áKã«ÛÀàdÞdÛ|ã«ÛÀÜß«Ü ÿ¹Üß‡Ü èÀàtäÈè”Ü !‹Þ”å™ì äÈãêWéàdè”á—ã‡ß4å™ä è‹ã«áô

ƒâ?ã‡Û”ÜN߇Ü)á—Þ”ì:ã«ä èv  á—à
ìÈÞÀã‡ä àdèçŸà6Üá]ä èTâ,å
éã±á«å-ã«ä á—âöê =  S.EF«÷”ã«ÛÀÜèUÿ#ÜNÿiä ì ì۔å ædÜ=âöà
ÞÀè”çTã«ÛÀÜ 
ÜèÀÜß«ådì á‡àdì ޟã‡ä à
è÷”á—ä è”éÜ=ã‡ÛÀÜIådéã«Þ”å™ì@ý|å-ëŸä ýIޔý àdâ  ÿiä ìÈìrã‡ÛÀÜèUìÈä Ü=ä èã‡ÛÀÜWâöÜådá‡ä íÀì Ü=ß‡Ü d ä àdè÷Àݔ߇à-æ6ä{çŸÜçoã‡Û”Ü Ü ‹ ! ޔådì ä:ãêÃéàdè”á—ã‡ß4å™ä è‹ã÷ 01' ! ôÅòø)ú
ó4÷ä{áNå™ì{á‡àývÜãô ; èÅàdß4çŸÜß=ã‡àÃä ývÝÁà
á‡ÜIã«ÛÀÜ|Ü
! Þ'å™ì äÈãê¾éà
è”áã«ß«ådä è
ã=ÿ#Ü ä è‹ã‡ß«à6çÀޔéÜ=å™èUådç”çŸäÈã‡ä àdè”ådìE@å
 ß«ådèd ܱýIÞÀìÈã«äÈݔìÈä Üß
ô¹Ú¥Û6ޔá¥ÿ#Ü`á—ÜÜõvã«àvý|å-ëŸä ývä Ü



B  #  1 ( = 

)

=

( # =  ) #

ø

B



#

=

=

B  #    (

*) =

ÿiÛÀÜ߇Ü=ÿ#Ü`۔å ædÜ ä è‹ã‡ß«à6çÀޔéÜ)çvã‡Û”Ü]Üá«á‡ä ådè

 1;L = L  =

î6Üã—ã«äÈè

9 

<

H

= <

#

ò ™ ü‹ó I

dä ædÜ)á

S

 

?ó = 7 NL =H

òl

ò  dó

= L = D

EF ø

ò ™ ù‹ó

 à-ÿ l ۔å
áWåUæ
Üß«ê á‡ä ývÝÀì ܼá—ã‡ß«Þ”éã‡ÞÀß«Ü tã‡ÛÀÜoà  ƒçŸä{å dàdè'å™ì?Üì ÜývÜè‹ã«á`åd߇Ü L ='L A  =$O÷?ådè”çïã‡Û”Ü ç ä{å 
àdè”ådì8Üì ÜývÜè
ã4á¥å™ß«ÜCA  ô¥Ú¥ÛÀÜ=â,å
éã]ã«Û”å-ã¯ådì ì8ã‡ÛÀÜNà  ;çŸä{å 
àdè”ådì'Üì ÜývÜè‹ã4áiçŸäÁÜßiàdèÀì ê|í6ê|â,å
éã«àdß4á Ÿ à™âPL =?á‡Þ
Üá—ã«á#ì à6àdõ6ä è`âöàdß]åvá‡àdì ޟã«ä àdè|ÿiÛÀä{é4ÛF۔ådá#ã«ÛÀÜ=âöàdß«ý 

="H ; ø 7* L = D 7 ; 4ø   ÿiÛÀÜ߇Ü

ò%E S‹ó

D

å™è”ç å™ß«Ü±ÞÀè”õ‹è”à-ÿiè”áô1.&ì Þ
äÈè`ã‡ÛÀä{á#âöà
߇ý

;

L =

7

7 D ;  D  ø

O O

 ; 7  D

L = O

ø4L H

A

ä è 01!'ô]ò d ù
ó dä ædÜ)á  ò%EÀø ó =



ÿiÛÀÜ߇Ü6Uä{á¥çŸÜ 'èÀÜçí6ê

#

B I1 (  = )

ò%E 
ó

L = I

#

Ú¥Û6ޔá

7

ò%E ‹ Eó

$O

H A

ò'O7 

ø)ó

ådè”çTá—ÞÀí'áã«ä:ã«ÞŸã‡ä èIã‡Û”ä á¥ä è‹ã‡àIã‡Û”Ü`Ü !‹Þ”å™ì äÈãêvéà
è”áã«ß«ådä è
ã 01!'ô]òø údó&ã‡à 'è”ç

O H

7

ò

A

 O



;ø4

ø)ó

O

7 Fø D D

H

O A

ò

 'O

7

ø ó

;ø7

 O

7 ø

D

÷


äÈæ
Üá ò%ELB6ó

ÿiÛÀä{é4Û,dä ædÜ)áKâöà
ߥã‡ÛÀÜNá‡àdì ޟã«ä àdè

=

H

; 4ø 

O

A

ò

7

ø)ó

 'O


ì á‡àÀ÷ òl  ó

=PH

ø4 O

L =

7

 ø

I

O

L =



7 Aø D

ò%E
ú
ó

ò%E‹ó

L ¯Üè”éÜ (  # =  L = L  < = 9 <  =  ) # B  # ( =$; ø4 L =  O 7 øAD = ) # B

 , 

H

H


H

B H

l



( # =*) # =

H



A

O  

ø4

  ; O7 øAD

ò%E'dó

 à™ã‡Ü=ã«Û”å-ãiã«ÛÀä{á¥ä{á¹à
èÀÜ`à™â@ã‡ÛÀà‹á—Ü`éådá‡Üá¥ÿiÛÀÜß‡Ü ã‡Û”Ü å 
ß«ådèdܱýIޔì:ã«ä ÝÀì äÈÜß Uéå™è߇Üý|å™ä è|ÞÀè”çÀÜã‡Üß ývä èÀÜçÅò2å™ìÈã‡Û”àdÞ
Û¾çŸÜã«Üß«ývä èÀä èväÈã±ä{á]ã«ß‡ä æ6ä{å™ìöó4ô
ïÜt۔å ædÜIèÀà-ÿ á‡àdì ædÜ)çã‡ÛÀÜtÝÀß«àdíÀì ÜýT÷8á—ä è”éÜIådìÈì@ã‡Û”Ü "=¯å™ß«Ü|éì Ü)å™ß«ìÈêTÝÁà
á‡äÈã‡ä ædܼàdß Üß«àÅòöä èVâ,ådéãNã‡Û”Ü =¯ÿiä ìÈì?à
èÀì êÃíÁÜ Ü߇àTäÈâ]å™ì ìã‡ß4å™ä èÀä èÝÁàdä è‹ã«á`۔å æ
Ü ã‡Û”ÜNá‡ådývܱéì{å
á‡á4ó4ô  àdã‡Ü ã‡Û'å-ã   çŸÜÝÁÜè'çÀáiàdè”ìÈêvà
èoã«ÛÀÜ  $ KCM¹à™â@ÝÁà
á‡äÈã‡ä ædÜtòöèÀÜ 
å™ã‡ä ædÜ óKÝÁàdì{å™ß«äÈãê ÝÁàdä è‹ã«á÷
å™è”çtèÀàdã?àdè¼ÛÀà-ÿµã«ÛÀܯéì å
á‡áì{å™íÁÜì{áå™ß«Üiådá«á—ä dè”Üç`ã«à=ã‡ÛÀÜiÝÁàdä è‹ã«áä è 01!'ôiò  
ó4ô?Ú¥ÛÀä{áä á?éì Üå™ß«ì ê èÀàdãiã‡ß«ÞÀÜ=à™$ â äÈã«á‡ÜìÈâ;÷ŸÿiÛÀä{é4ÛFä{+ á dä ædÜèoí6ê

(  # ; L =>  A  òO 7 ø)ó = ) O 7 Fø D <,= # Ú¥ÛÀÜ`ý|ådß 
ä è÷ H  * =   ÷'ä á¹ã«Û‹Þ'á dä ædÜèoí6ê



B

O

H

 H

 A

¼òø4Dò  Ÿ = ò'O O

7

ø ó—ó  ó

I

ò%Edü‹ó

ò%Edù‹ó

Ú¥Û6ޔá±ÿiÛÀÜèÃã‡ÛÀÜNè6ÞÀýtíÁÜ߯àdâ?Ý'à
äÈè‹ã4á@O 7 øNä{áiÜæ
Üè÷Áã‡Û”ÜWývä èÀä ýIޔýþý|å™ßdä èFàŸééÞÀß4á]ÿiÛÀÜè H S òöÜ ‹ ! ޔå™ì¹è‹Þ”ýIíÁÜß4á=à™âiÝÁà
á‡ä:ã«ä ædÜvå™è”çVè”Ü
å™ã‡ä ædܼÜëŸådývÝÀì Üá4ó4÷ä èVÿiÛÀä{é4ÛÅéå
á—ܼã«ÛÀÜoý|ådß 
äÈè ä{á  = B H ô ƒP â O 7 ø`ä{á¥à6ç”ç8÷Àã‡Û”ÜNývä èÀä ýIÞÀý ý|ådß 
äÈèoà6ééÞÀß4áiÿiÛÀÜN è !H Iø™÷”ä èÿiÛÀä{é4ÛTéådá‡Ü  = B H  A= O+  A¼' ò O7ïø) ó =Ÿ'ò O O 7 
ó4ô ;èvíÁà™ã‡Û|éådá‡Üá÷dã«ÛÀÜ]ý|å-ëŸä ýtÞÀý ý|å™ß dä èWä{á 
äÈæ
Üè¼í6 ê   H A¼ò O7ïø ó =$Oô Ú¥Û6ޔá÷âöàdß=ÜëÀå™ývݔìÈÜd÷'âöà
߯ã«ÛÀÜtãÿ¹àTçŸä ývÜè'á—ä àdè'å™ìrá‡äÈývݔìÈÜëTéàdè”á‡ä{áã«ä  è và™âKã«ÛÀß«ÜܼÝ'à
ä è
ã4á¯ì ê6ä  è |à
è p # ò,ådè”ç|á‡Ý”å™è”èÀä  è  m  ó4÷Àå™è'ç|ÿiäÈã‡Ûvì{å™íÁÜì ä  è Ná‡Þ”é4Ûvã«Û”å-ã#èÀà™ãiå™ì ìÀã«ÛÀß«ÜܯÝÁàdä è‹ã«á&۔å ædÜ]ã‡ÛÀÜ=á‡ådývÜ]ÝÁàdì{å™ß«äÈãê™÷ ã‡Û”Ü`ý|å-ëŸä ýIÞÀý å™è'çFývä èÀä ýtÞÀý ý|å™ß dä èoå™ß«Ü í'àdã‡ Û E A= Fò,á‡ÜÜ ä dޔ߇ܼò7ø dó—óô ð 



à™ã‡Ü¯ã‡Û”å™ã¹ã«ÛÀܯ߫Üá‡ÞÀìÈã«áKà™âã‡Û”ä á#î6Üéã‡ä àdè|ådývàdÞÀè‹ãKã«àWå™èFå™ìÈã‡Ü߇è”å™ã‡ä ædÜd÷déà
è”áã«ß‡Þ'éã‡ä æ
ÜiÝÀ߇à6àdâÁã«Û”å-ã#ã‡Û”Ü çŸä ývÜè”á‡ä àdèvà™âàdß«ä Üè‹ã‡Ü)ç|á‡Üݔådß«å™ã‡ä  è WÛ6ê6ÝÁÜß«ÝÀì{å™èÀÜ)á¹ä è m B ä áiå™ãiìÈÜ)ådá—ã O 7 ø™ô

 

MG

- 

GM

$±è”éܱÿ#Ü=۔å ædܯã‡ß4å™ä èÀÜçFåtîŸÞÀÝÀÝÁàd߇ã¥ð?Ü)éã«àdß]ñÃå
é4ÛÀä èÀܙ÷6ÛÀà-ÿ éå™èÿ¹Ü ޔá‡Ü äÈã 0
ïÜ`á‡äÈývݔìÈê¼çŸÜã‡Üß«ývä èÀÜ àdèvÿi۔ä é4Û|á‡ä{çŸÜ]àdâÁã«ÛÀܱçÀÜéä{á‡ä àdè¼í'à
ÞÀè”çÀåd߇êTò{ã«Û”å-ã#Û6ê‹ÝÁÜß«ÝÀì{å™è”ÜiìÈê6ä è=۔ådì:âÿ¥å êWíÁÜãÿ#ÜÜè # å™è'ç-  å™è'çÃݔådß«ådìÈì Üìrã‡àFã‡Û”Üýoó±å 
äÈæ
ÜèUã«Üá—ã`ݔå-ã‡ã‡Üß«è < ì äÈÜ)á¯ådè”ç å
á‡á‡ä dèTã«ÛÀÜvéàdß«ß«Üá‡Ý'à
è”çŸä è|éì{å
á‡á±ì{å™íÁÜì.÷ ä.ô Üdô?ÿ¹Ü ã«ådõdÜ ã‡Û”ÜNéì{ådá«á#à™ â < ã‡àvíÁ Ü  OKò 9< 7 ó4ô  

M"I(  & M 5

    6 M  

GM

Ú¥ÛÀÜvå™íÁà-ædÜtå™ì dà
߇äÈã«ÛÀý âöà
ß=á‡ÜÝ'å™ß4å™íÀì ÜWçÀå™ã«åÀ÷rÿiÛÀÜèVå™ÝÀݔìÈä Ü)çTã‡àèÀàdè ƒá‡Üݔådß«ådíÀì ÜNçÀå-ã4åŸ÷ÿiä ì ì 'è”ç¾è”à âöÜå
á—ä íÀì Ü]á‡àdì ޟã«äÈà
è rã«ÛÀä{áKÿiä ì ìÀí'Ü Üæ6ä{çŸÜè”éÜ)ç¼í‹êIã‡Û”ܱà
íÜéã‡ä ædÜ¥âöÞÀè'éã‡ä à
è¾òöä.ô ܙôã‡ÛÀÜ=çŸÞ”ådì+ådß4å™è
ä{å™è'ó dß«à-ÿiä è Iåd߇í”ä:ã«ß«åd߇ä ì êWì{ådß 
ܙôKî6àvÛÀà-ÿ éå™èÿ¹Ü`Üë6ã‡Üè'çoã«ÛÀÜá‡Ü`ä{çŸÜå
á¹ã‡àv۔ådè”çŸì Ü=èÀàdè ;á‡Üݔådß«ådíÀì ܯçÀå-ã4å
VÜWÿ#àdÞÀì{çTì ä õdÜ ã‡àF߇Üì å™ëoã«ÛÀÜIéàdè”á—ã‡ß4å™ä è‹ã«á`ò— ø S
ó¯å™è”çOòødø ó4÷8íÀޟã àdèÀì êÿiÛÀÜèÃè”ÜéÜ)á‡á«å™ß«ê™÷'ã‡Û”å™ã±ä{á÷'ÿ#Ü ÿ#àdÞÀì{ç|ì ä õdܯã‡à¼ä è‹ã‡ß«à6çÀޔéÜ=åIâöÞÀߗã«ÛÀÜß±éà
á—ãNòöä.ô ܙôKå™èä è”éß«Üå
á—ܱäÈèFã‡Û”Ü`ÝÀ߇ä ý|ådì'à
 í Üéã‡ä ædܯâöÞÀè”éã«ä àdè'ó¹âöà
ß çŸà
äÈ è Fá‡àÀôtڥ۔ä á=éå™è¾íÁÜ|çŸà
èÀÜIí6êÃä è‹ã‡ß«àŸçŸÞ”éäÈ è |Ý'à‹á—äÈã«äÈæ
ÜWá‡ì å
é4õTæ åd߇ä{ådíÀì Ü á = D FH ø D;9:9;9D : ä èÃã‡Û”Ü éà
è”á—ã‡ß4å™ä è‹ã«á ò ¹àd߇ã‡Ü)áiå™è”çFðKå™Ý”èÀä õ'÷rø)ùdù‹údó4÷ŸÿiÛÀä{é4ÛFã‡ÛÀÜèTí'Ü)éàdývÜ 

-9  7  I 7 ø4  = -9  7 1 `  ø7  =

<>=

=

4 S

7Iø `ø

âöà
ß âöà
ß

<>=

òB S‹ó òB”ø ó òB 
ó

L =H L =H

FKI

Ú¥Û6ޔá÷Àâöàdß]ådèTÜ߫߇à
ß&ã«àvà6ééÞÀß÷Ÿã‡Û”ÜNéàdß«ß«Üá‡Ý'à
è”çŸä  è  = ýIÞ'áãiÜëÀéÜÜçFÞÀèÀäÈãêd÷”á—à
=  = ä{á¥å™èTÞÀÝÀÝÁÜß í àdÞÀè”ç àdè ã«ÛÀܾè6ÞÀýtí'Üßoàdâ`ã‡ß4å™ä èÀä èVÜ߫߇à
ß«áô ]Üè”éÜVåÅè”å™ã‡ÞÀß4å™ì±ÿ#å ê ã‡àDådá«á—ä
è ådè Üë6ã‡ß4åOéà
á—ã Á âöàdßWÜ߫߇à
ß«á ä{á=ã‡àÃé4Û'å™è
ܼã‡ÛÀÜ|à
íÜéã‡ä ædÜIâöÞÀè”éã‡ä àdèVã«àUí'ÜoýväÈè”äÈýväÜçâöß«àdý  ,  = oã‡à   =  7

Dò
= =.ó 6÷8ÿi۔Üß«Ü ä{á±åoݔå™ß4å™ývÜã‡Üßiã‡àFí'ܼé4ÛÀà
á‡ÜèUí6êã‡Û”ÜIޔá‡Üß÷rå|ì{å™ß dÜß þéà
߇߫Üá‡ÝÁàdè”çŸä ètã«à ådá«á‡ä 
èÀä  è |åÛÀä dÛÀÜß±ÝÁÜè'å™ìÈãêTã‡àTÜß«ß«àdß4áô
á=äÈã=á—ã«ådè”çÀá÷rã‡ÛÀä{á ä{á=åFéàdè6ædÜëÃÝÀ߇ à dß4å™ývývä  è ¼ÝÀß«àdíÀì Üý âöàdß]ådè6êvÝ'à‹á—äÈã‡ä æ
ܯä è‹ã‡Ü 
Üß ÁûŸâöàd ß MH Iådè”ç H ø=ä:ãiä{á¥å™ì{á‡àIå !
Þ'ådçŸß4å-ã«ä é±ÝÀß«à 
ß«ådývýväÈ è  ݔ߇à
íÀì ÜýT÷ å™è'çµã«ÛÀÜUé4ÛÀàdä{éÜ  H ø۔ådáIã«ÛÀÜTâöÞÀ߇ã‡ÛÀÜß¼å
çŸæ-å™è‹ã«å dÜFã‡Û”å™ãvèÀÜäÈã‡Û”ÜßIã‡ÛÀÜ  = ÷¹è”àdßtã‡ÛÀÜä  ß @å 
ß«åd è dÜ ýtÞÀìÈã‡ä ÝÀì ä Üß4á÷
ådÝÀÝÁÜå™ß¥ä èFã‡ÛÀC Ü
VàdìÈâöÜ`çŸÞ”ådìÝÀ߇à
íÀì ÜýT÷ŸÿiÛÀä{é4ÛoíÁÜéàdývÜ)á 

  K. M



 1 ( =

G $ JM   I  SN1 = 1 (



=  L = L  < = 9 < 

  =

òB ‹ Eó

òB 6 Bó

D

= L =>H =

( ø

= 

òB‹ú
ó

SI

Ú¥ÛÀÜNá‡àdì ޟã«ä àdè|ä{áiå ‹å™ä è/
äÈæ
ÜèFí‹ê

(





H  =

)

#

òB‹ó

= L = < = I

ÿiÛÀÜß‡Ü  ïä{á ã‡ÛÀÜvè6ÞÀýtí'Üß`à™â¥á—ÞÀݔÝ'à
ߗãNæ
Üéã«àdß4áô|Ú¥Û6ޔá ã«ÛÀÜ|àdè”ìÈêÃçÀä 8Üß«Üè'éÜtâö߇à
ýyã‡Û”Üvàdݟã«äÈý|ådì Û6ê6Ý'Ü߇ÝÀì{ådèÀÜ]éå
á—Ü]ä{áã«Û”å-ã¹ã‡ÛÀÜ = èÀà-ÿ ۔å æ
Ü]å™èvÞÀÝÀÝÁÜß#íÁàdÞÀè”ç|à™â IôKÚ¥ÛÀܱá‡äÈã‡Þ”å™ã‡ä àdètä{á&á‡ÞÀývý|å™ß«äÜ)ç á«é4ÛÀÜý|å™ã‡ä{éådìÈì êNä è ä 
ÞÀß« Ü Àô
VÜTÿiäÈì ì¹èÀÜÜ)çµã«ÛÀÜ `åd߇Þ'á—Û  =ÞÀÛÀè ƒÚ@ޔé4õdÜßWéà
è”çŸäÈã‡ä à
è”á`âöà
ßWã«ÛÀÜTÝÀ߇ä ý|ådìKÝÀß«àdíÀì ÜýTôÅÚ¥ÛÀÜTÝÀß«ä ý|å™ì ådß4å™è
ä{å™èvä{á 

  H

ø

 ,





7

(

=  ( =

= $- L =

ò

= '< =

9 7 ó  ø 7  = ." (

=

=

 =

òBdó

ÿiÛÀÜ߇Ü=ã«ÛÀÜ ,=¹å™ß«Ü±ã«ÛÀÜ@å dß4å™èdÜ ýIÞÀìÈã«äÈݔìÈä Üß«á?ä è‹ã‡ß«àŸçŸÞ”éÜçFã‡àvÜèŸâöàdß4éÜ`ÝÁà
á‡äÈã‡ä æ6ä:ãê¼àdâã«ÛÀÜ=—ô¥Ú¥Û”Ü éàdè”çÀä:ã«ä àdè”á¹âöàdߥã«ÛÀÜ`ÝÀß«ä ý|å™ì8ÝÀß«àdíÀì Üý ådß‡Ü ã«ÛÀÜß«Üâöà
߇Üvòöè”à™ã‡Ü F¹ß‡Þ”è”á#âö߇à
ý ø ã‡à¼ã«ÛÀÜ`è6ÞÀýtí'Üßià™â ã‡ß4å™ä èÀä  è IÝ'à
ä è
ã4á÷Àådè” ç |âöß«àdý ø±ã«à¼ã‡ÛÀÜNçÀäÈývÜè”á‡äÈà
èvà™âã‡ÛÀÜNç”å-ã«å‹ó `Ú



 



 H

 

 ( =

 2 H  (

"= L =*)+=  H

S

òB
ü‹ó



H

S

òB
ù‹ó

=

= L =

 

ò

L = '<

 

= =

:  => ,=>H S 9 7 ó? ø7  = ;S

ò.ú ‹ S ó

H

ò.úŸø ó ò. ú 
ó ò.Lú E‹ó ò.Cú B6ó ò.údú
ó ò.ú ‹ó

= ;  S =$;S

,=$;  S

=!-7L =—ò<=-9 7 ó?Dø7 = .CH

,=

=H

S S


áFíÁÜâöà
߇Üd÷±ÿ#Ü éå™è ޔá‡ÜÃã«ÛÀÜ `ÚyéàdývݔìÈÜývÜè‹ã«åd߇äÈãêµéàdè”çŸäÈã«äÈà
è”á÷ 0 !6áôyò2ú
údó|ådè”ç ò.ú ‹ó4÷]ã«à çŸÜã‡Üß«ývä èÀÜ]ã«ÛÀÜ ã‡ÛÀß«Üá‡ÛÀà
ì ç )ô  à™ã«Ü¯ã«Û”å-ã 0 !”ô]ò.ú S
ó¹éà
ýIí”äÈè”Üç¼ÿiäÈã‡Û,01!'ô]ò2ú ‹ó#á—ÛÀà-ÿ]á&ã‡Û'å-ã=H SIäÈâ "= A Iô¹Ú¥Û6ޔáiÿ#ÜNéå™èTá‡äÈývݔìÈêtã4å™õdÜNådè6êvã‡ß4å™ä èÀä èIÝ'à
äÈè‹ã¥âöàd߯ÿiÛÀä{é4Û!S&A =2A ã‡àvÞ'á—Ü 0 !”ô¯ò.údú
ó òöÿiäÈã«Û =PHRS‹ó#ã‡àoéàdývÝÀÞÀã‡& Ü )ô`ò
á]íÁÜâöàd߫ܙ÷ÁäÈã¯ä{á¥è‹Þ”ývÜß«ä éå™ì ì êIÿiä{á‡Üߥã«àvã«ådõdÜ=ã‡Û”ÜWå ædÜß«ådÜ à-ædÜß]ådì ì á‡Þ”é4Ûoã«ß«ådä èÀä  è WÝÁàdä è‹ã«áô$ó

w -b |w| −ξ |w| ;Á$›“)Ž,…
> ‹Î " 3 @*2² .41," 3 i+§²d2² $  3 —*9 0 1,+- 3-043 » *2;²-.4(  &*2 5 (

 M    *  

6 B    6 I +#

¹àdè'á—ä{çŸÜß&ã«ÛÀÜ`éådá‡Ü=ä èoÿiÛÀä{é4Û|ã«ÛÀÜ`çÀå-ã4å¼å™ß«Ü±ä è m  ô¹î6ޔÝÀÝ'à‹á—Ü ã«Û”å-ã¥ã‡Û”Ü`ä  ã‡Ûá—Þ”ÝÀÝ'à
ߗãiæ
Üéã«àdߥÜëŸÜߗã4á åFâöàdß4éÜ=@H "='L = f  à
èïåTá—ã‡äµá—ÛÀÜÜã`ì ê6äÈèFå™ì à
è|ã«ÛÀÜ|çŸÜéä{á—ä àdèÃá‡ÞÀ߇â,ådéÜòöã‡ÛÀÜ çŸÜéä á‡ä àdèÃá‡ÛÀÜÜãK‹ó òö۔Üß« Ü  çŸÜèÀà™ã«ÜáNã‡ÛÀÜ|ޔèÀäÈãWædÜ)éã‡à
ß`ä è ã«ÛÀÜoçŸä ß«Üéã‡ä àdè |ó4ôÃÚ¥ÛÀÜèVã«ÛÀÜoá‡àdì ޟã«äÈà
è òB
óNá‡å™ã‡ä{á 'Üá ã‡Û”Ü éà
è”çŸäÈã‡ä à
è”á&àdâývÜ)é4۔å™è”ä éå™ì' Ü !‹ÞÀä ì äÈí”߇ä ÞÀý 

( (

(

”àdß4éÜ)á Úà
ß!‹ÞÀÜ)á H

H

 H ='L = G

(

=

=

ò.údó S

 = ïò "= L = | ó  H

 

H

SI

ò.ú™ü‹ó

%ò ]Üß«Ü ã‡ÛÀÜ  = å™ß«Ü¯ã«ÛÀÜ`á—ÞÀݔÝ'à
ߗãiædÜ)éã«àdß4á÷Ÿå™è”

ç µçÀÜèÀàdã‡Üá¹ã«ÛÀÜ=ædÜéã‡à
ß#ݔ߇àŸçŸÞ”éãô ó ÀàdßiçÀå™ã«åIä è m B ÷ é ì Ü)å™ß«ìÈêFã‡Û”Ü|éàdè'çŸäÈã‡ä àdèUã«Û”å-ã`ã«ÛÀÜ|á‡ÞÀýfà™â¹âöàdß4éÜ)á=æ ådèÀä{á—Û ä{á á—ã‡ä ì ìývÜãô $±è”ܼéå™èïÜå
á—ä ì êá‡ÛÀà-ÿ ㇔ Û å™ã ã‡Û”Ü=ã‡àdß !‹ÞÀÜ`å™ì{á‡àWæ-å™èÀä{á‡ÛÀÜáô  Ú¥ÛÀä{á@ývÜé4۔ådèÀä{éådìdå™è”ådì à dê±çŸÜÝÁÜè”ç”á@à
èÀì ê±à
èNã«ÛÀÜ&âöà
߇ý àdâÀã‡ÛÀÜ¥á‡àdì ޟã‡ä à
è¼òB‹ó4÷™å™è”çNã‡ÛÀÜ߇Üâöàdß«ÜKÛÀà
ì ç”á âöàdßWí'àdã‡Ûïã‡ÛÀÜFá—Üݔå™ß4å™íÀì Ütå™è”ç ã‡Û”Ü|èÀàdè ƒá‡Üݔådß«ådíÀì ÜIéådá‡Üáô/ ;è â,å
éãNã«ÛÀä{á`å™è”ådì àdêTÛÀà
ì{çÀá=ä è 
ÜèÀÜß«ådì

òöä.ô ܙôÈ÷8ådì{á—à¼âöà
߯ã«ÛÀÜWèÀà
èÀì ä èÀÜådß]éådá‡ÜIçÀÜá«éß«äÈíÁÜçíÁÜì à-ÿ¯ó4ô=Ú¥ÛÀÜIådè”å™ì  à dê|ÜývÝÀ۔ådá‡ä Ü)á#ã‡ÛÀÜtä è
ã«Üß«Üá—ã‡ä  è  ÝÁàdä è‹ã¹ã«Û”å-ã#ã‡ÛÀÜ-ývà
á—ã¹ä ývÝ'à
ߗã4å™è‹ãK`ç”å-ã«åIÝÁàdä è‹ã«á#å™ß«Ü]ã‡Û”Ü=á‡ÞÀÝÀÝÁàd߇ã¥ædÜéã‡à
ß«á¹ÿiäÈã‡Ûo۔ä
ÛÀÜá—ã¹æ-ådìÈޔÜá¹à™â &÷”á—ä è”éÜ]ã‡Û”Üê|Üë6Üߗã#ã‡Û”ܱÛÀä 
ÛÀÜá—ã¹âöà
ß«éÜá¹àdèoã‡ÛÀÜ`çŸÜ)éä{á—ä à
è|á—ÛÀÜÜãô ”àdß#ã‡Û”Ü è”àdè ;á—Üݔå™ß4å™í”ìÈÜ]éådá‡Ü™÷6ã‡Û”Ü ÞÀÝÀÝÁÜß#íÁàdÞÀè”ç = 1 éà
߇߫Üá‡Ý'à
è”çÀáã«àWå™èoÞÀÝÀÝÁÜß&íÁàdޔè”çvàdè|ã‡ÛÀܯâöàdß4éܯådè‹ê dä ædÜè¼Ý'à
äÈè‹ã&ä{áKådì ìÈà-ÿ#Üç ã‡àTÜë6ÜߗãNà
è¾ã«ÛÀÜ|á‡ÛÀÜÜãôoÚ¥ÛÀä{á=å™è”ådì à dêUådì á‡àÝÀ߇à-æ6ä{çŸÜ)á±åß«Üådá‡àdèDò,å
á dà6àŸç å
á`å™è6êÃà™ã«ÛÀÜßó¯ã‡àUéådì ì ã‡Û”Üá‡Ü`ݔå™ß‡ã‡ä{éޔì ådß&æ
Üéã«àdß4á á‡ÞÀÝÀÝÁàd߇ã¥æ
Üéã‡àdß4Ká '#
ô

   K 5 6*M G #.- *   $  M G

 

ä dÞÀß« Ü Wá—ÛÀà-ÿ]á&ãÿ#àWÜëÀådývÝÀì Üá?àdâ@å`ãÿ#à ;éì{ådá«á?ݔå™ã—ã«Üß«èoß«Üé à dèÀäÈã«äÈà
èIݔ߇à
íÀì ÜýT÷
à
èÀÜ=á—Üݔå™ß4å™íÀì Ü]ådè”ç àdè”Ü è”à™ãô&ڥ۔Ü=ãÿ¹à¼éì{ådá«á‡Üá¥å™ß«Ü çÀÜèÀàdã‡Üçí6êoéäÈß4éì Ü)á¹ådè”çFçÀä á‡õŸá#߇Ü)á—ÝÁÜéã‡ä ædÜìÈêdôî6ޔÝÀÝ'à
ߗã]æ
Üéã‡àdß4á¥å™ß«Ü ä{çŸÜè‹ã«ä 'ÜçtÿiäÈã‡Ûvå™èvÜë‹ã«ß«å`éä ß«éìÈÜdô@Ú¥ÛÀÜ]Ü߇߫àdß&äÈètã«ÛÀÜ]èÀà
 è ƒá‡ÜÝ'å™ß4å™íÀì Ü¥éådá‡Ü]ä{á?ä{çŸÜè
ã«ä ”ÜçIÿiäÈã‡Û|åNé߇à‹á‡áô Ú¥ÛÀÜ ß‡Ü)ådçŸÜß&ä{á¹äÈè6æ6äÈã‡Ü)çtã‡àtޔá—C Ü rÞ'éÜè‹ã  áiî6ð ñ
ݔÝÀì Üã`*ò #ÞÀß dÜá÷ =èÀä ß«á«é4Û|å™è”ç ¯ådß«å™ã«á«é4Û÷Áø)ùd ù 
óã«à ÜëŸÝÁÜß«ä ývÜè‹ã#ådè”çTéß«Üå-ã«Ü=ÝÀä{éã‡Þ”߇Ü)á¹ì ä õdܯã‡ÛÀÜ)á—Ü|ò,äÈâ@ÝÁà
á«á—ä íÀì ܙ÷‹ã‡ß«êoÞ'á—ä  è  ø tàd@ ß LBtí”ä:ã]éàdì àdßó4ô

; $›“)Ž,…> +-K " *. *.;²-24( $ $;9ˆ1   3  3-0 1 2" 4+1  5< +-?(-.¶ 0  3 N 0  0 -?*,+ 0 *12+-&*2+ 4²
 0 9”12+- Á   ;"$*2" *2-< ,9È 3 043 5

¹Õ”Ô Ó!C6ØdÓÕ'Ô046#ÀØ7Ù,Ò"

z UÕÁÒ%2Ù,Ò" #ÀÔFpr×  

]à-ÿ

éå™èFã‡Û”Ü=ådí'à-æ
ܯývÜã‡ÛÀàŸçÀá#íÁ Ü 
ÜèÀÜß«ådì ä Üçtã«àWã«ÛÀÜ`éådá‡Ü ÿiÛÀÜ߫ܯã«ÛÀÜNçŸÜ)éä{á—ä à
è¼âöÞÀè”éã«ä àd è ##iä{á#èÀàdã å ì ä èÀÜådߥâöÞÀè”éã‡ä àdèUàdâã«ÛÀÜIç”å-ã«å  ò*#à‹á—Üß÷ % ÞÀêdà
èÃå™è”çTðKådÝÀèÀä õÁ÷@øù
ù 
ó4÷Áá‡ÛÀà-ÿ#ÜçFã‡Û'å-ã å|ß«å™ã‡ÛÀÜß]àdì{ç v ã‡ß«ä{é4õïò
äÜ߇ý|ådè÷@ø)ù B‹óiéå™èÃíÁÜIÞ'á—Ü)çTã‡àådééàdývÝÀì ä{á‡Ûoã«ÛÀä{á¯ä è¾ådè¾ådá—ã‡à
èÀä{á—ÛÀä è
ì ê|áã«ß«ådä
ۋã—âöàdß«ÿ¥å™ß4ç ÿ¥å ê™ô @ä ß4áãKè”à™ã‡ä{éÜiã‡Û”å™ã&ã«ÛÀܯà
èÀì êWÿ¥å êIä èvÿiÛÀä{é4Ûtã‡ÛÀÜ çÀå-ã4åWådÝÀÝ'Ü)å™ß4á?ä è¼ã‡ÛÀܯã‡ß4å™ä èÀä  è =ÝÀß«àdíÀì ÜýT ÷ 01!6áô @ò B E
+ó `@ò B6údó÷Áä{áiä èTã‡Û”ÜNâöàdß«ý à™âKçŸàdã±ÝÀß«àŸçŸÞ”éã4á÷ <,= 9< ™ô  à-ÿ á—ÞÀݔÝ'à‹á—ÜIÿ¹Ü 'ß«á—ã¯ý|å™Ý”Ý'Ü)çTã‡ÛÀÜtçÀå-ã4å ã‡àtá—à
ývÜià™ã«ÛÀÜß òöÝÁà
á«á—ä íÀì êWä  è 'èÀäÈã‡Ü]çŸä ývÜè'á—ä àdè'å™ìö ó 0&ޔéì ä{çŸÜ)å™è¼á—Ý”å
é

Ü Ã÷‹Þ”á‡ä  è Wå`ý|ådÝÀÝÀä  è `ÿiÛÀä{é4Û¼ÿ#Ü ÿiä ì ì8éå™ì ì   

 m 



I

ò.ú™ù‹ó

Ú¥ÛÀÜèWàdâ”éàdޔ߫á‡Ü?ã‡Û”ÜKã‡ß4å™ä èÀä  è iådì 
àdß«äÈã‡ÛÀý ÿ#àdޔì ç=à
èÀì ê±çŸÜÝ'Üè”çNà
è`ã‡Û”ܹçÀå™ã«åiã«ÛÀß«àdÞ
ÛNçŸà™ãÝÀ߇àŸçŸÞ'éã«á ä è U÷Áä.ô ܙô¥àdèTâöÞÀè'éã‡ä à
è”áià™âã«ÛÀÜNâöà
߇ý ò<>=ƒó 9 ò< )ó4ô  à-ÿ ä:âã‡Û”Üß«ÜNÿ¹Ü߇ÜNå 4õdÜ߇è”Üì8âöÞÀè”éã‡ä àdè+ á‡Þ”é4Û¼ã‡Û”å™ ã O'ò <,= DK< )ó H 'ò <>=28 ó 9 ò < )ó4÷6ÿ¹Ü]ÿ#àdޔì çtà
èÀì êNèÀÜÜ)ç¼ã‡àWޔá—Ü fäÈètã«ÛÀÜ]ã«ß«ådäÈè”äÈè ådì
àdß«äÈã‡ÛÀýT÷ å™è'çoÿ#àdÞÀì{çFèÀÜæ
ÜߥèÀÜÜçFã‡à¼ÜëŸÝÀì ä{éäÈã«ìÈêtÜæ
Üèõ6èÀà-ÿ ÿi۔å™ ã  ä{á1 ô $±è”Ü ÜëÀå™ývÝÀì ܯä{á

Oò'<,= DK<

)ó1H
7 < 7

    <

 

ò S‹ó

$I

;èoã‡ÛÀä{áKÝ'å™ß‡ã‡ä{éÞÀì{ådß?ÜëÀå™ývÝÀì Üd÷ ä{áKä è'èÀäÈã‡Ü¯çŸä ývÜè”á—ä à
è”å™ì.÷™á—àIäÈã&ÿ¹à
ÞÀì{ç¼èÀà™ã#í'Ü ædÜ߇êtÜå
á—êIã‡àtÿ#àdß«õ ÿiäÈã‡Û  ÜëŸÝÀì ä éäÈã‡ì ê™ô]à-ÿ#Üæ
Üß÷'äÈâKàdè”ÜWß«ÜÝÀì{ådéÜá <= 9$< Wí6ê  ò<>= D < )ó]Üæ
Üß«ê‹ÿi۔Üß«ÜNä èTã‡ÛÀÜIã‡ß4å™ä èÀä è å™ì 
àdß«ä:ã«ÛÀýT÷Àã«ÛÀÜvå™ìdà
߇äÈã‡Û”ýYÿiä ì ì۔ådÝÀÝÀä ì êFÝÀß«àŸçŸÞ”éÜtåá—Þ”ÝÀÝ'à
ߗã=ædÜ)éã«àdß ý|ådé4ÛÀä èÀÜNÿi۔ä é4Û¾ìÈä æ
Üá]ä è¾ådè ä  è ”èÀäÈã«Ü¼çÀäÈývÜè”á‡äÈà
è”å™ì?á‡Ý”ådéܙ÷å™è”çïâöÞÀ߇ã‡Û”Üß«ývàdß«ÜtçŸà á—àUä èV߇à
 Þ d۔ìÈêTã«ÛÀÜoá«å™ývÜ|ådývàdÞÀè‹ã`à™â#ã‡ä ývÜväÈã ÿ#àdÞÀì{ç¼ã4å™õdܱã‡àtã«ß«ådäÈèoàdè|ã«ÛÀÜ=ÞÀ è ƒý|å™ÝÀÝÁÜçFçÀå-ã4åŸô
ì ì8ã«ÛÀÜ`éàdè'á—ä{çŸÜß«å™ã‡ä àdè”áàdâã«ÛÀÜ=ÝÀ߇Üæ6äÈà
ޔá#á—Ü)éã‡ä à
è”á ÛÀà
ì ç÷”á—ä è”éܱÿ#Ü`å™ß«Ü=á—ã‡ä ì ìçŸàdä  è ¼åtì äÈè”Üå™ß¥á‡Üݔådß«å™ã‡ä àdè÷6íÀޟã]ä èTå¼çŸä ÁÜ߇Üè
ã]á‡Ý”ådéܙô #ޟã]ÛÀà-ÿgéådèÿ#Ü=ޔá‡Ü±ã«ÛÀä{á¥ý|ådé4ÛÀä èÀÜ 
{â ã‡Ü߯å™ì ì.÷6ÿ¹Ü=èÀÜÜç ÷'å™è”çoã‡Û'å-ã]ÿiä ì ìÁì ä ædܱä è yå™ì{á‡àFò2á—ÜÜ 01!'ô¾òB
ó‡ó4ô #ޟãNä èïã‡Ü)áãNݔ۔ådá‡Üvå™èµî6ð±ñ {ä á=ޔá‡Üçïí6ê éàdývÝÀޟã«ä èçÀà™ãNÝÀß«àŸçŸÞ”éã4á=à™âiå dä ædÜèÃã«Üá—ã ÝÁàdä è‹ã <ïÿiäÈã‡Û ÷”àdߥývàdß«Ü=á—ÝÁÜéä Áéå™ì ì êtí‹êéà
ývÝÀޟã‡ä èW  ã‡ÛÀÜNá‡äd èFà™â



ò<@ó H

(   = L =  ò  =ƒó 9 = ) #

 ò'<@ó

7 H

(    ='L = Oò  = DK<ó 7 = ) #

òÀø ó

ÿiÛÀÜ߇Ütã«ÛÀÜ  = åd߇Ütã«ÛÀÜ|á—Þ”ÝÀÝ'à
ߗã`æ
Üéã«àdß4áô|î6àUå ‹å™ä è¾ÿ#Üvéådèïå æ
àdä{çÃéà
ývÝÀޟã«äÈè  ò<ó ÜëŸÝÀì ä{éäÈã‡ì ê å™è'çoޔá‡Ü=ã‡Û”Ü Oò  = DK<ó1H  ò  = ó49  ò'<@ó#ä è”á—ã‡Üå
ç8ô





Üãiޔáiéådì ì'ã«ÛÀÜ`á—Ý'ådéÜ ä èoÿiÛÀä{é4Ûoã‡ÛÀÜ`çÀå™ã«åIìÈä æ
ܙ÷ ]ô¯ò¯Üß«Ü`å™è”çFí'Üì à-ÿ ÿ¹Ü=ޔá‡Ü å
áiåWývèÀÜývàdèÀä{é âöàdßC4ì à-ÿµçŸä ývÜè”á‡ä àdè”ådì Ÿ÷)ådè”ç âöàdßÛÀädÛIçÀäÈývÜè”á‡äÈà
è”å™ì ÁäÈã?ä{áޔá—Þ'å™ì ì ê=ã‡ÛÀÜ]éådá‡Ü¥ã‡Û”å™ã?ã‡ÛÀÜiß4å™è
Ü¥à™â  ä á¥àdâ?ýIޔé4ÛÛÀä d۔ÜßiçŸä ývÜè”á‡ä àdèvã«Û”å™èUäÈã4áiçŸàdý|ådäÈèÁó4ô  àdã‡Ü=ã«Û”å-ã÷'ä èå
çÀçŸäÈã‡ä à
è|ã‡à|ã‡ÛÀÜ=â,å
éã]ã«Û”å-ã

ì ä ædÜáä è U÷‹ã‡ÛÀÜ߇ܯÿiäÈì ìÀä è dÜèÀÜß4å™ì”í'ܱèÀàNædÜ)éã«àdßKä è OÿiÛÀä{é4Û¼ý|å™Ý'á÷‹æ‹ä{å ã«ÛÀÜ]ý|å™Ý ¯÷‹ã‡& à Fô ƒâ8ã‡Û”Üß«Ü ÿ#Ü߫ܙ÷ ò <óiä è 0 !'ôI*ò ”ø)ó]éàdÞÀì{çTíÁÜIéàdývÝÀޟã«ÜçUä èUà
èÀÜIá—ã‡ÜÝ÷8å æ
àdä{çŸä  è tã‡Û”ÜIá‡ÞÀý ò,å™è'çUý|å™õ6ä  è ¼ã‡Û”Ü



éà
߇߫Üá‡ÝÁàdè”çŸä  è ¼î6ð±ñ  ã‡ä ývÜ)á&â,å
áã«Üß÷ÀÿiÛÀÜß«Ü  ä á#ã«ÛÀÜ`è6ÞÀýtí'Üßià™â?á—ÞÀݔÝ'à
ߗã]æ
Üéã«àdß4á«ó1 ô -¯Ü)á—Ý”ä:ã«Ü ã‡Û”ä á÷rä{çŸÜå
á=å™ì àd è |ã‡ÛÀÜ)á—ܼì ä èÀÜá éådè¾íÁÜvޔá—Ü)çUã«àTá‡ä dèÀä 'éå™è‹ã‡ì êá—ÝÁÜÜ)ç¾ÞÀÝ ã‡Û”ÜIã«Üá—ãNÝÀ۔å
á—Ütà™âiî6ð±ñÃá *ò #ÞÀß dÜ)á÷iø)ùdù ‹ó4ô  à™ã«ÜFådì á‡àTã‡Û'å-ã¼ä:ãtä{áNÜå
á—ê ã‡à ”è”çµõdÜß«èÀÜì áFò{âöà
ßIÜëŸådývÝÀì ܙ÷õdÜ߇èÀÜì{á`ÿiÛÀä{é4Ûµå™ß«Ü âöÞÀè”éã‡ä àdè'á¥à™âã‡ÛÀÜNçŸàdã]ÝÀß«à6çÀޔéã4áià™âã‡ÛÀÜ <,=KäÈè #ó]á—Þ”é4ÛFã‡Û'å-ã]ã«ÛÀÜ=ã‡ß4å™ä èÀä  è tå™ì dà
߇äÈã‡Û”ý ådè”çTá—à
ì ޟã‡ä àdè âöàdޔè”çTå™ß«Ü ä è”çÀÜÝÁÜè”çŸÜè‹ã¥àdâ@ã«ÛÀÜNçŸä ývÜè'á—ä àdèvàdâíÁà™ã«Û å™è”ç Uô







;èïã‡Û”ÜvèÀÜë6ãIî6Ü)éã‡ä à
è¾ÿ#Ütÿiä ì ì?çŸä{á‡éޔá«á±ÿi۔ä é4Û âöÞÀè”éã‡ä àdè' á  åd߇Üvå™ì ì à-ÿ¥å™íÀì ÜNå™è'ç¾ÿiÛÀä{é4Ûïå™ß«Ütè”à™ãô rÜã¯Þ”á±Üè”çã«ÛÀä{á±î6Ü)éã«äÈà
èTÿiä:ã«ÛUåvædÜ߇êá—ä ývÝÀì Ü=ÜëÀå™ývݔìÈÜ=àdâ?ådèÃå™ì ì à-ÿ¹Ü)çvõdÜ߇è”Üì.÷Àâöàdß±ÿiÛÀä{é4ÛTÿ¹ÜI   éà
è”á—ã‡ß«Þ”éã¥ã‡Û”Ü`ý|å™ÝÀÝÀä è ¯ô 

î6ޔÝÀÝ'à‹á—ÜIã‡Û'å-ã`êdà
ÞÀß ç”å-ã«åFå™ß«ÜWæ
Üéã«àdß4á¯ä è m  ÷ådè”çÃê
àdÞ¾é4ÛÀà6à‹á—Ü Oò<,= D < )óH ò'<>=9$< )ó  ôWÚ¥ÛÀÜè äÈã$á#Üådá‡ê¼ã‡à ”è”çå¼á—Ý'ådéÜ U÷”ådè”ç|ý|ådÝÀÝÀä è Dâö߇à
ý m  ã‡à U÷”á‡Þ”é4Û|ã«Û”å-ãNò'<J9  ó  H ò'<ó9  ò  ó  ÿ#Ü=é4۔à‹à‹á—Ü H m 8 å™è”ç



 ò<@ó H

) 





#

) #)

)N

ò 
ó





ò,èÀà™ã«ÜNã‡Û'å-ã¯ÛÀÜ߇Ü=ã«ÛÀÜIá‡ÞÀí”á«éß«ä ݟã«á¥ß«ÜâöÜß]ã«àvædÜéã‡à
߯éà
ývÝ'à
èÀÜè‹ã«á4ó4ô Àà
߯çÀå™ã«åvä è çŸÜ”èÀÜ)çTàdèTã‡Û”Ü á !
Þ'å™ß«Ü *`ø D)ø `ø Dø @? m  ò,åãê6ÝÀä{éådì&á—äÈã«Þ”å-ã«äÈà
è÷râöà
ß dß«Üê ìÈÜædÜì?äÈý|ådܼçÀå-ã4å
ó÷@ã«ÛÀܾòöÜè‹ã‡ä ß‡Ü ó ä ý|å 
Ü]à™â  ä{á¥á—Û”à-ÿièoä è,ädÞ”ß‡Ü üŸô&Ú¥ÛÀä{á ä
ÞÀß«Ü ådì á‡àWä ì ì ޔá—ã‡ß4å-ã‡Ü)á?ÛÀà-ÿ ã‡àtã‡ÛÀä èÀõoà™âã«ÛÀä{á#ý|å™ÝÀÝÀä è ã‡Û”Ütä ý|å dÜNàdâ  ý|å êTì ä ædÜNä èïåFá‡Ý”ådéÜIàdâ¹æ
Üß«ê۔ä
Û çŸä ývÜè”á—ä à
è÷Áí”ÞŸã`ä:ã=ä{á ޔáãNåVòöÝÁà
á«á‡äÈí”ìÈêædÜ߇ê éà
è‹ã‡àd߇ã‡Ü)ç”ó#á—ޔߗâ,å
éÜ=ÿiÛÀà
á‡Ü äÈè‹ã«ß‡ä è”á‡ä{é¯çŸä ývÜè'á—ä àdè #  ä á ޔá—ãiã‡Û'å-ã]à™â ]ô

  



à™ã‡Ü¯ã‡Û”å™ã¹èÀÜäÈã‡ÛÀÜßKã‡Û”ܯý|å™Ý”ÝÀä è  èÀàdß&ã‡ÛÀÜ á—Ý”å
éÜ Ü !‹Þ”ådì ìÈê¼ÿ#Üì ì8۔å ædÜ`é4ÛÀà‹á—Üè _ã‡àvå
å™ä èFí'Ü m 8 ådè”ç 

 ò<@ó H





ø 

ò,)

 ) ) #)  ,ò ) # 7 )  





ó

#

ó

å™ß«Ü]ÞÀèÀä !‹ÞÀÜiâöà
ß¹å 
äÈæ
Üè¼õdÜ߇è”Üì.ô>
VܱéàdÞÀì{ç

ò ‹ Eó

1 0.8 0.6 0.4 0.2 0

0.2

0.4

0.6

;Á$›“)Ž,…
> É{!¥ " 3=-0 9Ÿ1,+-K*.¿-.  ƒ  à
ß

ã«à¼í'Ü m

å™è”ç



) ) # )#  ) #) ) 





 ò<@ó H

 





0.8

   ƒ 

1

-1

0 -0.5

0.5

1

?BA  3  r12+ &!#4² ² " 3  5

òLB6ó I

Ú¥ÛÀÜ`ì äÈã‡Üß«å™ã‡ÞÀß«Ü]à
èTî6ð ñÃáiޔá‡Þ”ådìÈì êt߇ÜâöÜß4á¹ã‡à¼ã‡Û”Ü=á‡Ý”å
éÜ _å
á¥å¯ä ìÈíÁÜ߇ãiá—Ý”å
éÜd÷Ÿá‡à¼ì Üã$á#Üè”çFã«ÛÀä{á î Ü)éã‡ä à
è|ÿiäÈã‡ÛoåNâöÜÿ èÀàdã‡Ü)á¹à
èvã‡Û”ä á¹ÝÁàdä è‹ãô 6KàdÞéå™è|ã‡ÛÀä èÀõ¼à™âå]ä ì íÁÜ߇ã#á‡Ý”ådéܱådá#å 
ÜèÀÜß«ådì ä)å-ã«äÈà
è 6 à™â 0KÞ'éì ä{çŸÜådèIá‡Ý”ådéܹã«Û”å-ãKíÁÜÛ'å ædÜáä èvådÜè
ã«ì Üý|å™è”ìÈê â,å
á—ÛÀä à
èôî6ÝÁÜéä'éådìÈì êd÷ äÈã?ä{á?å™è6êNì ä èÀÜådß?á—Ý”å
éÜd÷ ÿiäÈã‡ÛOådèµä èÀèÀÜßWÝÀß«àŸçŸÞ”éã¼çŸÜ 'èÀÜç8÷Kÿi۔ä é4ÛÅä{áIå™ì{á—à éàdývݔìÈÜã‡ÜvÿiäÈã‡ÛÅß«Üá‡ÝÁÜéãIã‡à¾ã‡ÛÀÜTéà
߇߫Üá‡Ý'à
è”çŸä è èÀà
߇ý ò{ã«Û”å-ãvä{á÷&å™è6ê #å™Þ”é4Û6êVá—Ü !
ޔÜè”éÜFàdâ¯Ý'à
ä è
ã4áWéàdè6ædÜß 
Üá`ã«à å Ý'à
ä è
ãtä èÅã‡ÛÀÜTá‡Ý”ådéÜ)ó4ôOî6à
ývÜ å™ÞÀã‡ÛÀà
ß«á=òöÜdô ”ô¯ ò =à
ìÈývà dà
߇à-æÁ÷Àø ù $S‹ó—ó#å™ì{á‡àWß« Ü !‹ÞÀä ߇ܯã‡Û”å™ã]äÈã¥íÁÜNá‡Üݔådß«ådíÀì ÜIò{ã«Û”å-ã]ä{á÷ÀäÈã¥ýtޔá—ã#Û'å ædÜ=å éà
ÞÀè‹ã«ådíÀì ÜKá—ÞÀí'á—ÜãÿiÛÀà
á‡ÜKéì à‹á—ÞÀß«Ü?ä{á8ã‡Û”Ü&á—Ý'ådéÜ&ä:ã4á—ÜìÈâ«ó4÷-å™è”ç`á‡àdývÜ]ò,ܙô Àô ¯å™ì ývà‹á÷dø)ù ™ó8çŸà
è  ãô ƒã  á@å dÜèÀÜß4å™ì ä å™ã‡ä àdè ý|å™ä èÀì ê±í'Ü)éå™Þ'á—Ü#äÈã«áä èÀèÀÜßÝÀß«àŸçŸÞ”éã?éådèWíÁÜ   #¹ä èÀèÀÜßÝÀß«àŸçŸÞ”éã÷™èÀàd ã Þ'áãã‡Û”Ü¥á«éådì{å™ß ò çŸà™ã 
ó=ݔ߇àŸçŸÞ”éãWޔá‡ÜçVÛÀÜ߇ÜÃò2å™è”çVä è 0&ޔéì ä{çŸÜådè á‡Ý”ådéÜá`ä è 
ÜèÀÜß«ådì{óô ƒã  áWäÈè‹ã«Üß«Üá—ã‡ä  K è Fã‡Û”å™ãWã‡Û”Ü àdì{çŸÜߥý|å-ã‡Û”Üý|å-ã«ä{éå™ì'ì ä:ã«Üß4å-ã«ÞÀ߇ÜtòöÜdô ”ô =à
ì ývà dà
߇à-æÁ÷Àø)' ù $S‹ó&ådì{á—àtß‡Ü !‹ÞÀä ߇Ü)ç¼ã‡Û”å™ã ]ä ì í'Üߗãiá‡Ý”ådéÜá¥íÁÜ ä  è ”èÀäÈã«Ü¯çŸä ývÜè”á‡ä àdè”ådì.÷-ådè”ç¼ã«Û”å-ã#ý|å-ã‡Û”Üý|å-ã«ä{éä{å™è”áådß‡Ü !‹ÞÀäÈã«Ü]۔å™Ý”Ý‹êvçŸÜ ”èÀä  è Nä  è 'èÀäÈã‡Ü¯çŸä ývÜè”á—ä à
è”å™ì 0KÞ'éì ä{çŸÜådè á‡Ý”ådéÜáô ]Üá‡Üå™ß4é4ÛVàdè ]ä ì íÁÜ߇ãWá—Ý”å
éÜ)áNéÜè
ã«Üß4á`àdèÅà
Ý'Üß«å™ã‡à
ß«á=ä èVã‡Û”à
á‡Ü|á—Ý”å
éÜ)á÷?á—ä è”éÜ ã‡Û”Ü|í”ådá‡ä{éWݔ߇à
Ý'Üߗã«äÈÜ)á ۔å ædܼì àd è Uá‡ä è”éܼíÁÜÜèVÿ#àdß«õdÜ)çÃàdÞÀãôTîŸäÈè'éÜvá—à
ývܼÝ'ÜàdÝÀì ܼÞÀè”çŸÜß«á—ã«ådè”çÀådíÀì ê íÀì{å™è'é4Ûïå™ã`ã‡ÛÀÜvývÜè
ã«ä àdè¾àd6 â ¯ä ìÈíÁÜ߇ãNá—Ý'ådéÜ)á÷

çŸÜ)éä{çŸÜ)çÃã«àUޔá—ܼã«ÛÀܼã‡Ü߇ý 0Kޔéì ä çÀÜå™è¾ã‡ÛÀß«àd Þ dÛÀà
ޟã ã‡Û”ä á#ã«ÞŸã‡à
߇ä{ådì2ô

 M M G:$I   I Àà
ß]ÿiÛÀä{é4ÛTõdÜ߇èÀÜì{á#çÀà‹Ü)áiã«ÛÀÜß«Ü`ÜëŸä{áã¯å¼Ý'å™ä ß"- D 4.‹÷”ÿiäÈã«ÛFã«ÛÀÜNÝÀß«àdÝÁÜ߇ã‡ä Ü)áiçŸÜá«éß«ä í'Ü)çådí'à-æ
ܙ÷”ådè”ç âöàdßÿiÛÀä{é4ÛtçŸà6Üá@ã‡ÛÀÜ߇ܥèÀàdã Ú¥ÛÀÜiå™è'á—ÿ#Üßä{á 
ä ædÜèWí‹êNñÃÜß4éÜß$áéàdè”çÀä:ã«ä àdè|òöðKådÝÀèÀä õÁ÷Àøù
ù
ú6û'¹à
ÞÀß4å™è‹ã å™è'ç ¯äÈì íÁÜ߇ã÷rø)ù
Lú E‹ó ?Ú¥ÛÀÜß«Ü=ÜëŸä{áã4á¥åIý|ådÝÀÝÀä è ådè”çTå™èÜëŸÝ”å™è'á—ä àdè Oò'
 ó

H

(

 ò'<@ó =  ò =

 ó

=

ò
ú
ó

äÈâ?å™è”çFàdèÀì êväÈâ;÷Ÿâöàdß]ådè6ê 8  ò'<ó¥á—Þ'é4ÛFã«Û”å-ã

 ò<ó  

ò‹ó

ä{á ”èÀäÈã‡Ü <

ã«ÛÀÜè



ò
 ó  ò<ó ò  ó  <   ;SI

ò'dó

 à™ã‡Üoã‡Û”å™ãNâöàdßtá—ÝÁÜéä'évéå
á—Ü)á÷äÈãNý|å ê¾èÀàdãWí'ÜFÜå
á—ê¾ã‡à¾é4ÛÀÜ)é4õ¾ÿiÛÀÜã‡ÛÀÜßIñUÜß«éÜß áWéàdè'çŸäÈã‡ä àdèïä{á á«å-ã‡ä{á ”Üçô 0 !”ô]ò'™ó&ýtޔáã¥ÛÀà
ì çvâöàdßM" M# vÿiäÈã‡Û ”èÀäÈã‡Ü  èÀà
߇ýfò,ä2ô ܙô?ÿiÛÀä{é4Ûoá«å-ã«ä{á ”Ü)á 0 !'ôiò‹ó—ó4ô ]à-ÿ#Üæ
Üß÷Áÿ#ÜIéå™è Üådá‡ä ì êoÝÀß«à-ædÜ`ã«Û”å-ã  ã‡Û”Ütéàdè”çŸäÈã«äÈà
èUä{á±á«å- ã«ä á ”Ü)çâöàdß=Ý'à‹á—äÈã«äÈæ
Ü`ä è
ã«Ü
ß«ådìÝÁà-ÿ¹Üß«á¯à™â ã‡Û”ÜNçŸà™ã]ÝÀß«àŸçŸÞ”éã  O'ò <9D  ó H ò
  ò( ) = ) #

= L =



ó   ò<ó  ò  ó

Ú ÛÀÜNãê‹Ý”ä éå™ì8ã«Üß«ý ¥ âöàdß«ý

#  

:9;9:9ò  

<  

òdü‹ó

SI

ä èFã‡ÛÀÜNýtÞÀìÈã‡ä èÀà
ývä{å™ì”ÜëŸÝ”å™è”á‡ä àdèFà™â]ò




=*)


#

)N= L =ƒó 

éàdè‹ã‡ß«ä íÀޟã«Üáiåtã‡Ü߇ý

   9;9:9 ó  )#  )  9:9;9 L# L   9;9;9 8ò'<ó ò  ó  <  

#

à™âã‡Û”Ü òdù‹ó

ã«à¼ã‡ÛÀÜ`ì Üâ{ãi۔ådè”çTá—ä{çŸÜ à™â 01!'ô]ò*™ó÷ÀÿiÛÀä{é4Ûoâ,å
éã‡à
߇äÜá H

# 



;9:9;9)ò 



#

   9:9;9 ó ò/ ) #  )   9:9;9 8ò'<ó 

@ó <



;SI

ò $S‹ó

$±èÀܹá—ä ývÝÀì Üéàdè'á—Ü !
ޔÜè”éÜ?ä áÁã«Û”å-ãå™è6ê¯õ
Üß«èÀÜìdÿiÛÀä{é4Û éå™è`íÁÜKÜëŸÝÀß«Üá«á—Ü)ç=ådá  ò
 ä è”å™ì ì ê™÷ÀÿiÛ'å-ã¯Û”ådÝÀÝÁÜè”á±äÈâà
èÀÜ`ޔá‡Üá±åvõdÜ߇èÀÜìÿiÛÀä{é4ÛUçŸà6Üá¯èÀà™ã á‡å™ã‡ä{áâöêñUÜß4éÜß$á]éàdè”çŸäÈã«äÈà
è  ;è  dÜèÀÜß4å™ì.÷ ã‡Û”Üܹ߫ý|å ê=ÜëŸä{á—ãçÀå™ã«å á—Þ'é4ÛWã«Û”å-ãã‡Û”Ü6¯Üá«á—ä{ådèWä{áä è”çÀÜ ”èÀäÈã«Ü™÷då™è'çNâöà
ßÿi۔ä é4ÛWã‡ÛÀÜ !‹Þ”å
çŸß«å™ã‡ä{é ÝÀß«à
 ß«ådývýväÈèWÝÀß«àdíÀì ÜýYÿiä ì ìr۔å ædÜWèÀàFá‡àdì ޟã«äÈà
èïòöã‡ÛÀܼçŸÞ”ådì@à
íÜéã‡ä ædÜ=âöޔè”éã«äÈà
è¾éådèUí'Ü)éà
ývÜWå™ß«íÀä ã‡ß4å™ß«ä ì ê|ì{å™ßd Ü)óô ]à-ÿ#Üæ
Üß÷'Üæ
ÜèTâöà
ß±õ
Üß«èÀÜì{á¥ã«Û”å-ã=çŸàoèÀàdã á«å-ã«ä á—âöêFñÃÜß4éÜß$á¯éà
è”çŸäÈã‡ä à
è÷”àdè”ÜWýväd ۋã á—ã‡ä ì ì'è”çNã«Û”å-ã&å
 äÈæ
Üè`ã«ß«ådä èÀä è±  á‡Üã?ß«Üá‡ÞÀìÈã«áä ètå ÝÁà
á‡ä:ã«ä ædܹá‡Üývä{çŸÜ”èÀäÈã‡Ü]Üá«á‡ä ådè÷™ä èIÿi۔ä é4Ûtéå
á—Ü#ã‡Û”Ü ã‡ß4å™ä èÀä èo  ÿiäÈì ìéàdè6æ
Üßd ÜWÝÁÜ߇âöÜéã‡ì êTÿ¹Üì ì2ô ; è¾ã‡ÛÀä{á=éå
á—Üd÷rÛÀà-ÿ#ÜædÜß÷Áã‡Û”Ü
 Üà
ývÜã‡ß«ä{éådìrä è
ã«Üß«ÝÀß«Üã«å™ã‡ä àdè çŸÜ)á‡é߇ä íÁÜçoådí'à-æ
Ü=ä á#ì{ådé4õ6ä è”  ô

$%'& ICKM#"I  M GI

 7

ñUÜß«éÜß  áNéàdè”çÀä:ã«ä àdèïã‡ÜìÈì{á`Þ'áNÿiÛÀÜã‡ÛÀÜßIà
ßWèÀàdãtåUÝÀß«à
á‡ÝÁÜéã«ä ædÜvõdÜ߇è”ÜìKä{áWå
éã«Þ”å™ì ì ê¾åÃçŸàdãIÝÀß«àŸçŸÞ”éã ä è¾á‡àdývÜNá‡Ý”ådéܙ÷íÀޟã=äÈã±çŸà6Üá±èÀà™ã ã‡Üì ìޔá ÛÀà-ÿgã‡àTéà
è”á—ã‡ß«Þ”éã  àd߯ÜædÜèÃÿi۔å-ã ä{áô]à-ÿ#Üæ
Üß÷Áå
á ÿiäÈã‡Ûtã‡Û”Ü]ÛÀàdývà dÜèÀÜà
ޔá¥ò{ã‡Û'å-ã&ä{á÷dÛÀàdýv à dÜèÀÜà
ޔáäÈètã«ÛÀÜ]çŸà™ã¹ÝÀ߇àŸçŸÞ'éãKä è #ó !‹Þ”å
çŸß«å™ã‡ä{é#Ý'à
ì ê‹è”àdývä{å™ì õdÜ߇è”ÜìÀçŸä{á«éޔá«á—Ü)çWådí'à-æ
ܙ÷™ÿ#Ü]éå™ètÜëŸÝÀì ä{éäÈã«ìÈê=éà
è”á—ã‡ß«Þ”éãã‡Û”Üiý|å™ÝÀÝÀä  è ±âöàdßKá‡àdývÜ#õdÜ߇è”Üì{áô ;èvîŸÜéã«ä àdè Ÿô øWÿ#ÜIá‡ÛÀà-ÿ ÛÀà-ÿ 0 !'ôvò  
ó]éådè¾íÁÜIÜë6ã‡Üè'çŸÜçÃã‡àFåd߇í”ä:ã«ß«åd߇êFÛÀàdývà dÜèÀÜà
ޔáiÝÁàdì ê6èÀàdývä{å™ìõ
Üß«èÀÜì{á÷  7 # ôtÚ¥Û6ޔá âöà
ß å™è'çÃã«Û”å-ã=ã‡Û”ܼéàd߫߇Ü)á—ÝÁàdè'çŸä  è |á—Ý'ådéÜ ä á=, å 0Kޔéì ä çÀÜå™èÃá‡Ý”å
éÜtà™â¹çŸä ývÜè”á—ä à
 è      ÜëÀådývÝÀì ܙ÷-âöàdßKå`çÀÜ dß«ÜÜ 0H B=Ý'à
ìÈê6èÀà
ývä{å™ì.÷ ådè”çIâöàdßKçÀå™ã«å`éàdè”á‡ä{áã«äÈ è ¯à™â?ø ±í6êF ø  ä ý|å dÜ)á¹ò2+ ç HCdú 
ó÷ çŸä ýÃò ó&ä{á øü EŸ÷ øü”ø™÷ E' Àô



±á—Þ”ådì ìÈêd÷
ý|ådÝÀÝÀä è `êdà
ÞÀß¹ç”å-ã«åNã‡àtå0«âöÜ)å-ã‡Þ”ß‡Ü á—Ý”å
éÜ`ÿiäÈã‡Û|å™èoÜèÀà
߇ývà
ޔá?è6ÞÀýtí'Üß&à™âçÀäÈývÜè”á‡äÈà
è”á ÿ#àdÞÀì{çÃíÁàŸçŸÜvä ì ì@âöà
ß ã‡ÛÀÜ 
ÜèÀÜß«ådìÈä)å-ã‡ä à
èÝÁÜ߇âöàdß«ý|å™è'éÜWàdâ¹ã«ÛÀܼ߇Ü)á—Þ”ì:ã«ä èoý|å
é4ÛÀä èÀܙô
â{ã‡ÜßWå™ì ì.÷ã‡Û”Ü




á‡ÜãFàdâWå™ì ì¯Û6ê6ÝÁÜß«ÝÀì{å™èÀÜ)á - 0D J.Åå™ß«ÜÃݔå™ß4å™ývÜã‡Ü߇äÜ)çOí6ê çŸä ýÃò ó 7IøÃè6ÞÀýtí'Üß«áô[ñUà‹áãoÝ'å-ã—ã«Üß«è ß«ÜéàdèÀäÈã«äÈà
èWá‡ê6á—ã‡Üý|áÿiäÈã‡Û¼íÀä ì ìÈä à
è”á÷ àdß&ÜædÜèvå™è¼ä è”èÀäÈã‡Üd÷dè6ÞÀýtí'Üß?à™âݔå™ß4å™ývÜã‡Üß«á@ÿ¹à
ÞÀì{çWè”à™ã&ý|ådõdÜ äÈã¯Ý”å
áãiã«ÛÀÜIá—ã«ådߗã ‹å-ã‡Üd ô ]à-ÿ éà
ývÜ`îŸð±ñÃá çŸàoá‡àvÿ¹Üì ì  $±èÀÜNývä
Û
ã]ådß 
ÞÀÜ=ã‡Û”å™ã÷ dä ædÜèoã«ÛÀÜ`âöà
߇ý à™âá‡àdì ޟã«ä àdè÷™ã«ÛÀÜܱ߫åd߇ܱå-ã&ývà‹áã : 7µø¯å
ç ޔá—ã«å™í”ìÈܯݔå™ß4å™ývÜã‡Üß4áiòöÿi۔Üß« Ü :ä{á?ã«ÛÀܯè6ÞÀýtíÁÜß&àdâã‡ß4å™ä èÀä  è  á«å™ývÝÀì Üá4ó4÷
íÀޟã&ã«ÛÀä{á&á‡ÜÜý|áã‡àIí'ܱí' Ü dä  è =ã«ÛÀÜ !‹ÞÀÜ)áã«ä àd è # 8 ô ƒã¹ýtޔá—ã&í'Ü á—à
ývÜã«ÛÀä  è  ã‡àWçŸàIÿiäÈã‡ÛvàdÞ”ß ß« Ü !‹ÞÀä ߇ÜývÜè‹ãKà™$ â K   K $ K K  +*IÛ6ê6ÝÁÜß«ÝÀì{å™èÀÜ)á?ã‡Û'å-ã¥ä á¹á‡å æ6ä  è =ã«ÛÀÜ ç”å ê™ô
á#ÿ#ܱá‡Û”å™ì ì'á‡Üܱí'Üì à-ÿ=÷ å¼áã«ß‡à
 è véå
á—Ü=éådèFíÁÜ`ý|ådçŸÜ âöà
ߥã‡ÛÀä{áiéì{ådäÈýTô





î6ä è”éÜtã‡ÛÀÜ|ý|ådÝÀÝÁÜçïá—ÞÀ߇â,ådéܼä á àdâ#ä è‹ã‡ß«ä è”á—ä{éNçÀäÈývÜè”á‡äÈà
è¾çŸä ýÃò ¥ó4÷ÞÀèÀì Üá«á=çŸä ýÃò #ó HYçŸä ýÃò Fó÷äÈã ä{á`àdí6æ6ä àdޔá ã«Û”å-ãIã‡ÛÀÜFý|å™ÝÀݔäÈèTéå™èÀè”à™ãWíÁÜoà
è
ã«àµò,á‡ÞÀßÜéã«ä ædÜ)óô ƒãtå™ì{á‡àTèÀÜÜ)çÅèÀà™ãIí'ÜFàdèÀÜvã«àÃàdè”Ü òöí”ä Ü)éã«äÈæ
Ü)ó &éàdè'á—ä{çŸÜß ) #  ) # D )   )  ä è 01!'ô±*ò  dóô#Ú¥ÛÀÜ`ä ý|å 
ܱàdâ  èÀÜÜçTèÀàdã]ä:ã4á—ÜìÈâ@íÁÜ åNæ
Üéã‡àdߥá‡Ý”ådéÜ ?å 
ådä è÷‹éàdè”á‡ä{çŸÜß«ä  è  ã‡ÛÀÜ=å™íÁà-ædܯá‡ä ývÝÀì Ü !‹Þ”ådçŸß4å-ã«ä{éiÜëÀå™ývݔìÈÜd÷dã«ÛÀܱæ
Üéã‡àd" ß  ò <ó&ä{á èÀàdã#ä èvã«ÛÀܱä ý|å dÜià™â  ÞÀèÀì Üá«1 á
ä è

ò'<>=-9<

)ó1H

7

ø)ó

ò 6ø ó 

ÿiä ì ì™éà
è‹æ6ä è”éÜê
àdÞ ã‡Û'å-ãrã«ÛÀÜ&éà
߇߫Üá‡ÝÁàdè”çŸä è  éå™è=ý|ådݯãÿ#àiædÜ)éã‡à
ß«áÁã«Û”å-ãå™ß«Ü?ìÈä èÀÜ)å™ß«ì ê#çŸÜÝ'Üè”çŸÜè
ã à
è‹ã‡àtãÿ¹àtædÜ)éã«àdß4á&ã«Û”å-ã¯åd߇Ü=ì ä èÀÜåd߇ì êtä è”çŸÜÝÁÜè'çŸÜè‹ãiä è Uô



î6à¾â,å™ßIÿ¹Ü|Û'å ædÜ|éà
è”á‡ä çÀÜß«ÜçVéå
á—Ü)á`ÿiÛÀÜß«Ü  ä{áWçŸà
èÀÜ|ä ývÝÀì ä{éäÈã‡ì êdô $±èÀÜoéå™èÅÜ !‹Þ”ådì ìÈê¾ÿ¹ÜìÈìKã‡ÞÀß«è ã‡Û”äÈè
á¥å™ß«àdÞÀè'çFådè”ç G     ÿiäÈã‡Û ¯÷”å™è'çoã«ÛÀÜèUéàdè”á—ã‡ß«Þ”éã¥ã«ÛÀÜ`éàdß«ß«Üá‡Ý'à
è”çŸä èWõ
Üß«èÀÜì.ô Àà
ßiÜëÀå™ývÝÀì Ü òöðKådÝÀèÀä õÁ÷?øùdù 
ó÷ÁäÈâ RH m # ÷ã«ÛÀÜèïå ”àdÞÀß«ä Ü߯ÜëŸÝ”å™è”á‡ä àdèTä èUã‡Û”ÜtçÀå™ã«åL)÷éޟã à Vå™â{ã‡Üß  ã‡Üß«ý|á÷ ۔å
á#ã‡ÛÀÜ=âöà
߇ý



*ò )Áó1H  7 (  ò #  ) #

éà‹áò  Á ) ó87



á‡ä èrò  ' ) ó‡ó

ò
 ó




då è”ç ã‡ÛÀä{áéå™è=í'Ü&æ‹ä Üÿ¹Ü)ç±å
áråiçŸà™ãrݔ߇àŸçŸÞ”éãrí'Üãÿ¹ÜÜè=ãÿ#à¥ædÜéã‡à
ß«á8ä è m   #  # H ò D DIIIJD   # #

#  å™è'ç|ã«ÛÀÜ=ý|å™ÝÀÝÁÜç  ò*)'ó H ò # D4éà
áò*)Áó D«éà
á)ò F)'ó DIIID«á‡ä èrò*)Áó D«á‡ä èò F)'ó DIII$óô&Ú¥ÛÀÜèFã‡Û”Ü=éàd߫߇Ü)á—ÝÁàdè'ç   ä  è T*ò -±äÈß«ä{é4ÛÀì ÜãóKõ
Üß«èÀÜìéådèTí'ÜNéàdývÝÀޟã«ÜçoäÈèTéì à‹á—Ü)çvâöàdß«ý 

*) = ó49  ò*) 

 ò

ó

ò*) 

H

)  ó

= D

H

á—ä èrò—ò  7 øC= d óò*)+=+() ) ó—ó Ká—ä èò—ò,) =  )  ó = 
ó

Ú¥ÛÀä{á¥ä{á#Üå
á—ä ì êvá—ÜÜèUådá#âöà
ìÈì à-ÿ]áì Üã‡ã‡ä è&?1K)

*) = ó49  ò*) 

 ò

7 ( ø

ó H

)



H

 H



#

7 ( ø



ø

)

éà
áò 

) =

óŸéà
áò 

éà
áò  ™? ó

2 H

()  =

÷

)  >ó 7Oá‡äÈèrò  ) 

ø

7 A

óŸá‡ä èrò  =

L- ( 

7*A L-Ÿò—ø4 =   #  ó =Àòø4  HÑò,á‡ä èrò—ò  7 øC= dó?™ó—ó = ?á‡ä èÁò,? = dó I 

ò CE‹ó I

) =



 ó

=



) ó

.

.

ä è”å™ì ì ê™÷räÈã`ä{á`éì Üådß ã‡Û”å™ã`ã‡ÛÀÜ|ådí'à-æ
ÜIä ývÝÀì ä{éäÈã¯ý|ådÝÀÝÀä è Fã‡ß«ä é4õUÿiä ì ìÿ¹à
߇õTâöà
ß   #tådì dàdß«äÈã‡ÛÀý ä è ÿiÛÀä{é4Û|ã«ÛÀÜNçÀå™ã«åIàdèÀì ê|å™Ý”Ý'Ü)å™ßiådáiçŸàdãiÝÀ߇àŸçŸÞ'éã«á`òöâöàdßiÜëŸådývÝÀì ܙ÷‹ã‡ÛÀÜ=èÀÜ)å™ß«Üá—ãièÀÜädÛ6íÁàdߥå™ìdà
߇äÈã«ÛÀýoó4ô Ú¥ÛÀä{á=â,å
éãW۔å
á`íÁÜÜèÅÞ'á—Ü)çïã‡àÃçŸÜ߇ä ædÜvåUè”àdèÀì ä èÀÜådß±æ
Üß4á—ä à
è¾à™âiÝÀß«ä è”éä ݔådì?éàdývÝÁàdè”Üè‹ãWå™è”ådì ê6á‡ä{á¯í6ê ò2îÀé4 Û àd ì õdà
ݟâ;÷@î6ývà
ì åFå™è”çV) ñ ޔ ìÈì Üß÷?øù
ùdüdí'ó4ûä:ãNá‡ÜÜý|á±ì ä õdÜì êoã«Û”å-ã`ã«ÛÀä{á ã‡ß«ä é4õTÿiä ì ì?éàdè‹ã«äÈè6ÞÀÜIã‡à 'è”ç ޔá‡Üá¥Üì á‡Üÿi۔Ü߫ܙô

DIII

ó÷

!'& ICKM    K 5 6*M GID"I(

6 * M   &

 G

Ú¥ÛÀÜ ”ß4á—ã¥õ
Üß«èÀÜì{á¹ä è‹æ
Üá—ã‡ä
å™ã‡Ü)ç¼âöàdߥã«ÛÀÜ`ݔå-ã‡ã‡Ü߇è߇Ü)éàdèÀäÈã‡ä à
èvÝÀ߇à
íÀì Üý Oò'
 ó

H

Oò'
 ó

H

Oò'
 ó

H

ò
 7

7 < 7 

ø)ó



   

ã«ådèÀÛò
N
ÿ#Üܱ߫ã‡ÛÀÜ=âöà
ìÈì à-ÿiä è ò 6 Bó

 

ò ™ú
ó

 (?™ó

ò  ‹ó

0 !”ô]ò  B‹ó¹ß«Üá‡ÞÀìÈã«á¹ä èFå¼éì{ådá«á‡ä 'Üß?ã«Û”å-ãiä{á¥åWÝÁàdì ê6èÀàdývä{ådì”à™âçŸÜ
߇ÜÜTä è|ã«ÛÀÜ`çÀå-ã4åŸû01!'ô]òdúdó1dä ædÜ)á å % ™å ޔá«á—ä{ådè|ß«å
çŸä{å™ìÁí”ådá‡ä{áKâöޔè”éã«äÈà
èFéì å
á‡á‡ä”Üß÷‹ådè”ç 0 !'ôiò  ‹ó1
ä ædÜá¹åtݔå™ß‡ã‡ä{éޔì ådßKõ6ä è”ç|àdâãÿ#à ƒì å ê
Üß á‡ä 
ývàdä{çÀådìKèÀÜޔ߫ådìièÀÜãÿ#àdß«õÁô ”àdß¼ã‡Û”ÜQ   éå
á—Üd÷&ã«ÛÀÜUè6ÞÀýIíÁÜß¼àdâ=éÜè‹ã‡Üß«áUò  ä è)0 !'ô òÀø ó—ó4÷ ã‡Û”ÜéÜè‹ã‡Üß«áWã‡ÛÀÜý|á—Üì ædÜáoò{ã‡Û”Ü  =.ó÷Kã‡Û”ÜFÿ#Üä dۋã4áoò "=ƒó4÷¹ådè”çVã‡Û”Üoã«ÛÀ߇Ü)á—Û”àdì{ç ò óIå™ß«ÜFådì ì&ÝÀß«àŸçŸÞ”éÜ)ç è |ådè”" ç 
äÈæ
ÜNÜëÀéÜì ì Üè‹ã]߇Ü)á—Þ”ì:ã4á±éàdývݔåd߇Ü)çFã‡àTéì{ådá«á—ä{éå™, ì   á÷  $  ICK ( *  6*6 #tí6êã‡ÛÀÜvî6ð±ñ ã‡ß4å™ä èÀä  âöàdßWã‡ÛÀÜFéådá‡Ü|àdâ % ådޔá«á—ä{å™5 è   áoò.îŸé4 Û àd ì õ
àdݟâ¥ÜãIådì.÷#ø)ùd' ù dó4ô Àà
ßNã‡Û”Ü|èÀÜޔ߫ådì&èÀÜãÿ¹à
߇õïéådá‡Ü™÷ã‡Û”Ü ”ß4áãtì{å êdÜßNéàdè”á‡ä{áã4á`à™â  á‡Üã4áNà™â]ÿ#Üä 
Û
ã4á÷Üå
é4Ûµá—ÜãIéà
è”á‡ä á—ã‡ä  è Tàdâ  òöã‡ÛÀÜçŸä ývÜè”á‡ä àdèïà™âiã‡Û”Ü çÀå™ã«å
ó]ÿ¹Üä dۋã«á÷Áådè”çTã‡Û”Ütá‡Üéàdè”çTì{å êdÜ߯éà
è”á‡ä á—ã«á]à™â  ÿ#Üä dۋã«áIò{ã«ÛÀ Ü " =2ó÷á—à|ã‡Û”å™ã=ådèÃÜæ-å™ì ޔå™ã‡ä àdè á‡äÈývݔìÈê`ß« Ü !‹ÞÀä ß«Üáã4å™õ6ä  è Nå`ÿ#Üä dۋã‡Ü)ç¼á—ÞÀý à™âá‡ä dývà
ä ç”á÷ ã‡Û”Üý|á‡Üì ædÜ)áÜæ-å™ì ޔå™ã‡Üçtà
è|çŸà™ã&ݔ߇àŸçŸÞ”éã«á&à™â ã‡Û”ÜNã‡Ü)áã çÀå-ã4åvÿiäÈã‡ÛTã‡Û”ÜIá‡ÞÀÝÀÝÁàd߇ã¯ædÜ)éã‡à
ß«áôiڥۋÞ'á]âöà
ßiã‡ÛÀÜIèÀÜޔ߫ådìèÀÜãÿ¹à
߇õéådá‡Ü™÷'ã‡ÛÀÜIå™ß4é4ÛÀäÈã‡Üéã‡ÞÀß«Ü òöè6ÞÀýtíÁÜßià™âÿ¹Üä dۋã«á4óKä{áiçŸÜã‡Ü߇ývä èÀÜ)çví6êî6ð ñ ã«ß«ådä èÀä  è Àô  à™ã‡Üd÷#۔à-ÿ¹ÜædÜß÷?ã‡Û'å-ã¼ã‡ÛÀÜTÛ6ê6Ý'Ü߇íÁàdì ä{é¼ã4å™ è 
Üè‹ãtõ
Üß«èÀÜì¥à
èÀì êµá‡å™ã‡ä{á 'ÜátñUÜß4éÜß $áIéàdè”çÀä:ã«ä àdèÅâöà
ß é Üߗã4å™ä èÅæ-å™ì ÞÀÜ)á`à™â]ã«ÛÀÜoÝ'å™ß4å™ývÜã«Üß4á å™è”ç ?Åò2å™è”çOà™âiã«ÛÀÜFçÀå™ã«å  <   ó4ôÅÚ¥ÛÀä{áNÿ¥ådá ”ß«á—ãIè”à™ã‡ä{éÜç ÜëŸÝÁÜß«ä ývÜè‹ã«ådìÈì êòöðKå™Ý”èÀä õ'÷rø)ùdù‹údó4ûŸÛÀà-ÿ#Üæ
Üßiá‡àdývÜ èÀÜéÜá«á‡åd߇ê|éàdè”çŸäÈã«äÈà
è”á¹àdèFã‡ÛÀÜ)á—Ü`ݔådß«ådývÜã«Üß4áKâöà
ß ÝÁà
á‡ä:ã«ä æ‹äÈãê¼å™ß«Ü=èÀà-ÿ õ6èÀà-ÿi è #™ô

 äd ÞÀß«Ü`ùvá—Û”à-ÿ]á¥ß‡Ü)á—ÞÀìÈã4á#âöàdߥã‡Û”ÜWá‡ådývÜ=ݔå-ã‡ã‡Üß«èT߇Ü)éàd èÀäÈã‡ä à
è|ÝÀß«àdíÀì Üý ådá#ã‡Û'å-ã±á‡ÛÀà-ÿièäÈè ädÞÀß«Ü  ‹÷Kí”ÞŸãtÿi۔Üß«Ü|ã‡Û”ÜoõdÜ߇è”Üì#ÿ#å
áWé4ÛÀà‹á—ÜèVã‡à¾íÁÜTå¾éޔíÀä{évÝ'à
ì ê‹è”àdývä{å™ì.ô  àdã‡ä{éÜvã«Û”å-ã÷&Üæ
ÜèVã«ÛÀàdÞ
Û ã‡Û”ÜNè6ÞÀýIíÁÜ߯à™âKçŸÜ
 ߇ÜÜá¥àdââöß«ÜÜ)çŸàdý ä{áiÛÀäd ÛÀÜß÷”âöàdßiã«ÛÀÜNì ä èÀÜåd߇ì êvá—Üݔå™ß4å™í”ìÈÜ`éådá‡Ü|òöì Üâ{ã]ݔådèÀÜìöó÷”ã‡Û”Ü á‡àdì ޟã‡ä à
èµä{áIß«àdÞd ÛÀì êÅì ä èÀÜådß÷Kä è”çŸä{éå-ã‡ä èà  ã‡Û”å™ãvã‡ÛÀÜÃéå™Ý”å
éäÈãêVä á¼íÁÜä èV  éà
è‹ã‡ß«àdì ì Üç8ûiå™è”çOã‡Û'å-ãvã‡Û”Ü ì ä èÀÜåd߇ì êtèÀàdè; á‡Üݔådß«ådíÀì ܯéå
á—Üvòöß«äd ۋã¥Ý”å™èÀÜìöó&Û'ådá¥íÁÜéà
ývÜ á‡ÜÝ'å™ß4å™íÀì ܙô

;Á$›“)Ž,…> ρ2—  ² 0  § 3-0 !¹"$4Ÿ¶ 3 ; 5Á< + (-.¶ 0  3   0  0 -*,+ 0 *r12+-?*,+-4²d 0 9Ÿ12+-?)—;"$*," 043 *2 29:— 5 ä è”å™ì ì ê™÷™è”à™ã‡Ü¥ã«Û”å-ã#å™ìÈã‡ÛÀà
ÞdÛIã‡Û”ܱî6ð±ñyéì{ådá«á—ä”Üß«áçŸÜá«éß«ä í'Ü)çIådí'à-æ
Ü]å™ß«Ü¥íÀä è”å™ß«êWéì{å
á‡á‡ä”Üß4á÷ ã«ÛÀÜê å™ß«Ü¼Üå
á—ä ì êÃéà
ýIíÀä èÀÜ)çUã«àU۔å™è”çÀìÈÜtã«ÛÀÜ|ýtÞÀìÈã‡ä{éì{ådá«á¯éådá‡Ü™ô
á—ä ývÝÀì ܙ÷rÜÁÜ)éã«äÈæ
ÜvéàdýtíÀä è”å™ã‡ä àdèUã«ß«ådä è”á



 à
èÀÜ ƒædÜß«á‡Þ”á .ß«Üá—ã¥éì{ådá«á—ä ”Üß«á=ò,á«å ê™1÷ 4à
èÀÜ ¼ÝÁà
á‡äÈã‡ä ædÜdP÷ 4ß«Üá—Kã ¼è”Ü 
å™ã‡ä ædÜ ó¹âöàd߯ã‡ÛÀÜ  ƒéì{ådá«áiéådá‡ÜWådè”ç ã«ådõdÜ)áNã«ÛÀÜUéì{ådá«áNâöà
ßtå¾ã‡Ü)áãvÝÁàdä è‹ãWã«àïíÁÜFã‡Û”å™ãvéàd߫߇Ü)á—ÝÁàdè'çŸä èTã‡àïã‡Û”ÜFì{å™ßdÜ)áãIÝ'à‹á—äÈã‡ä æ
ÜoçŸä{áã4å™è”éÜ *ò #à‹á—Üß÷ % ÞÀê
àdèTå™è”çFðKå™Ý”èÀä õ'÷rø)ùd ù dó4ô




*I  

6

 ;$AM

 6 & I 6 $  IAG  7

 



M G G

ÛÀÜèFä{á¹ã‡ÛÀÜ`á‡àdì ޟã«äÈà
è¼ã‡àIã‡ÛÀÜ`á‡ÞÀÝÀÝÁàd߇ãiædÜéã‡à
ß&ã«ß«ådäÈè”äÈèN  ÝÀß«àdíÀì Üý
 ìÈà
í”å™ì.÷6å™è”çoÿiÛÀÜèä{á¹äÈã¥ÞÀèÀä !‹ÞÀÜ  #ê  
ìÈà
í”å™ì À÷'ÿ#ÜtývÜ)å™èUã«Û”å-ã=ã«ÛÀÜܼ߫ÜëŸä{áã4á¯èÀàTàdã‡ÛÀÜß ÝÁàdä è‹ã=äÈè¾ã‡ÛÀÜtâöÜ)ådá‡äÈí”ìÈÜIß‡Ü dä àdèÃå™ã`ÿiÛÀä{é4ÛÃã‡Û”Ü àdíÜéã«ä ædÜâöÞÀè”éã‡ä àdè=ã4å™õ
Üáråiì à-ÿ#Üßæ-å™ì ÞÀÜdô,
VÜ&ÿiä ì ìdådçÀçÀ߇Ü)á‡á8ãÿ#àiõ6ä è”çÀáràdâŸÿ#å êŸáä è`ÿiÛÀä{é4Û=ÞÀèÀä !‹ÞÀÜè”Üá«á ý|å ê=èÀà™ã?ÛÀà
ì{ç   á‡àdì ޟã«äÈà
è”áâöà
ßÿiÛÀä{é4ÛN- !D J.]åd߇ܹã‡Û”Üý|á‡Üì ædÜ)áÞÀè”ä‹! ÞÀÜd÷™íÀޟã?âöà
ß?ÿiÛÀä{é4ÛNã‡Û”Ü#ÜëŸÝ”å™è”á‡ä àdè à™â$ _äÈè 0 ” ! ô ò@B‹ ó#ä{áièÀà™ãûÁådè”çTá—à
ìÈÞÀã‡ä àdè”á¹ÿiÛÀà‹á—Ü -' 0D J.IçŸäÁÜßô #  à™ã‡ÛUåd߇Ü=à™â?ä è‹ã‡Ü߇Ü)áã K  ÜædÜèTä:âã‡Û”Ü ݔådäÈß - 0D .Uä{áIÞÀèÀä ‹ ! ÞÀܙ÷&äÈâ]ã«ÛÀÜ =`åd߇ÜFèÀà™ã÷&ã‡Û”Üß«ÜFý|å êïí'ÜTÜ
! ޔäÈæ-ådìÈÜè‹ãWÜëŸÝ”ådè”á—ä à
è”áNàdâ ÿiÛÀä{é4Û ß«Ü ‹ ! ÞÀä ߇Ü]âöÜÿ#Üߥá—ÞÀݔÝ'à
ߗã#ædÜéã‡à
ß«á±ò,åWã‡ß«äÈæ6ä{å™ì”ÜëÀå™ývÝÀì Üiàdârã‡ÛÀä{á¹ä{á d ä æ
ÜèvíÁÜì à-ÿ¯ó4÷Ÿå™è”ç|ÿiÛÀä{é4Ûvã‡ÛÀÜ߇Üâöàdß«Ü ß«Ü ‹ ! ÞÀä ߇ܱâöÜÿ#Üßiä è”á—ã‡ß«Þ”éã‡ä àdè”á#çŸÞÀß«ä èI  ã‡Üá—ã]ÝÀ۔å
á—Üdô


ƒãvã«ÞÀ߇è'áIà
ޟã¼ã‡Û”å™ãvÜædÜ߇êVì àŸéådì¥á‡àdì ޟã«ä àdèÅä{á¼å™ì{á—à dì àdí”ådì.ôÅڥ۔ä átä{á¼å¾Ý”߇à
Ý'ÜߗãêÅàdâ å™è6êµéàdè6ædÜë ÝÀß«à
ß«ådývýväÈè ÝÀß«àdí”ìÈÜý òìÈÜã«é4ÛÀÜß÷tøù
ü'™óô ”ÞÀ߇ã‡ÛÀÜ߇ývà
߇Üd÷]ã‡ÛÀÜVá‡àdì ޟã‡ä à
è ä á dޔådß«ådè‹ã‡ÜÜ)ç ã‡àDíÁÜ ÞÀèÀ ä !‹ÞÀÜUäÈâ=ã‡ÛÀܾàdíÜ)éã‡ä æ
ÜFâöޔè”éã«äÈà
è ò*0 !”ô òB E
ó‡ó¼ä{á|á—ã‡ß«ä éã‡ì êµéà
è6ædÜë8÷iÿiÛÀä{é4Û ä èDà
ÞÀßoéådá‡ÜÃývÜådè”á ã‡Û'å-ã=ã‡ÛÀÜ ¯Üá«á—ä{ådè¾ýtޔáã=íÁÜtÝÁà
á‡äÈã‡ä ædÜtçŸÜ 'èÀäÈã‡ÜUòöèÀàdã‡Ütã‡Û'å-ã`âöàdß !‹Þ”å
çŸß«å™ã‡ä{éNà
 í Üéã‡ä ædÜIâöÞÀè”éã‡ä àdè'á t÷ ã‡Û” Ü ¯Üá«á—ä{å™èUä{á±Ý'à‹á—äÈã«äÈæ
ÜWçŸÜ ”èÀäÈã‡ÜIä:â&ådè”çÃà
èÀì êFäÈâ  ä á±áã«ß‡ä{éã‡ì êéàdè6ædÜë8ûÁã«ÛÀä{á¯ä{á±èÀà™ã ã‡ß«ÞÀÜWâöàdß=èÀàd è !‹Þ”ådçÀß«å™ã‡ä{é  ¹ã‡ÛÀÜ߇Üd÷Áå¼ÝÁà
á‡äÈã‡ä ædÜ=çŸÜ 'èÀäÈã‡ Ü ]Ü)á‡á‡ä{å™èä ývÝÀì äÈÜ)á¹åvá—ã‡ß«ä{éã«ìÈêFéàdè6æ
Üëoà
 í Üéã‡ä ædܱâöÞÀè”éã‡ä àdè÷ íÀޟã¼èÀàdãtæ6ä{éÜ|æ
Üß4á‡åÅò2éà
è”á—ä{çŸÜß  H ) -ó|*ò ì Üã4é4ÛÀÜß÷]øù
' ü dó—ó4ô ]à-ÿ#Üæ
Üß÷ÜædÜèµäÈâ#ã«ÛÀ Ü ]Üá«á‡ä ådèÅä{á ÝÁà
á‡ä:ã«ä ædÜvá—Üývä{çŸÜ ”è”ä:ã«Ü™÷@ã‡ÛÀÜá—à
ìÈÞÀã‡ä àdèVéådèµáã«äÈì ìKíÁÜ|ÞÀè”ä !‹ÞÀÜ ¼éàdè”á‡ä{çŸÜßNãÿ¹àUÝÁàdä è‹ã«áIå™ì àd è ã«ÛÀÜoß«Üådì ì ä èÀܱÿiäÈã«ÛTéà6àdß4çŸä è”å-ã«Üá ) # H øNå™è'I ç )  HR6÷'ådè”çÿiä:ã«ÛFÝÁàdì{åd߇äÈã‡ä Ü) á 7 å™è”5 ç = ô ¯Üܱ߫ã«ÛÀ Ü ¯Üá«á—ä{å™èFä{á ÝÁà
á‡ä:ã«ä ædÜ=á—Üývä çÀÜ ”èÀäÈã«Ü™÷ŸíÀޟã]ã«ÛÀÜNá‡àdì ޟã‡ä à
è ò H 2D @HED ="HRS¼ä , è 0 !6áô=@ò B S‹ó4÷@ò B”ø ó4÷@ò B dó‡ó#ä{á ÞÀèÀ ä !‹ÞÀÜdô ƒã¥ä{á¹ådì{á—àNÜ)ådá‡êtã‡à ”è”çoá‡àdì ޟã«ä àdè”á?ÿi۔ä é4ÛFå™ß«Ü¯èÀàdã#ÞÀè”ä !‹ÞÀÜ ä èvã‡Û”Ü á‡Üè”á‡Ü±ã‡Û”å™ã¹ã«ÛÀÜ =@ä è|ã‡Û”Ü ÜëŸÝ”ådè”á‡äÈà
è|à™â Yå™ß«Ü èÀà™ãiÞÀèÀ ä !‹ÞÀÜ  âöàdߥÜëŸådývÝÀì ܙ÷Ÿéà
è”á—ä{çŸÜß&ã«ÛÀÜ`ÝÀß«àdíÀì Üý àdââöà
ÞÀß]á—Üݔå™ß4å™í”ìÈܯÝ'à
äÈè‹ã4á àdèUåvá !‹Þ”å™ß«Ü=äÈè m   ) # H ˆ ø Dø 2 ÷ ) H ` ø Dø 2 ÷ ) 8 H ` ø D `ø å™è”/ ç ) H ˆ ø D `ø .÷ÀÿiäÈã‡ÛTÝÁàdì{å™ß«äÈã‡ä Üá 7D CD CD 7 ¯ß«Üá‡Ý'Ü)éã‡ä æ
Üì ê™ô $±èÀÜ á‡ àdì ޟã«ä àdèDä  á H ˆ ø DKS 2÷ H SŸ÷ H SI d2 ú D SI d ú DKSI d ú DKSI 
ú .û å™è”à™ã‡Û”Ü߯Û'ådáiã«ÛÀܼá‡ådýv+ Ü ådè”ç )÷8íÀޟã H SI 2 ú D SI 2 ú D SDKS &òöèÀàdã‡ÜWã‡Û”å™ã±íÁà™ã‡Ûïá—à
ì ޟã‡ä àdè'áiá‡å™ã‡ä{áâöêFã‡Û”Ü éà
è”á—ã‡ß4å™ä è‹ã« á = . SIådè”ç
= = L = H S
óP ô
ÛÀÜèTéådè|ã‡Û”ä á#àŸééÞÀß#äÈè dÜèÀÜß4å™ì  % äÈæ
Üèoá‡àdývÜ á—à
ì ޟã‡ä àdè &÷”é4ÛÀà6à
á‡Ü±å™è rÿiÛÀä{é4Û|ä{á&ä è|ã«ÛÀÜ=è6ÞÀì ìÁá‡Ý”ådéܱàdârã‡Û”Ü ¯Üá«á—ä{åd è L = L  < = 9 <  ÷”å™è”çoß‡Ü !‹ÞÀä ߇ܯã‡Û”å™ã = H íÁÜNàd߇ã‡Û”à dà
è”å™ì8ã«à¼ã‡ÛÀÜIædÜ)éã‡à
ß]å™ì ìràdâÿi۔à
á‡ÜNéàdývÝÁàdè”Üè‹ã«áiåd߇Üvødô¥Ú¥ÛÀÜèÃådçÀçÀäÈè  rã‡à OäÈ" è 0 !'ô @ò B E
ó¥ÿiä ì ìì Üå æ
Ü  ޔè”é4۔å™è dÜç+ô ƒ1â S1 = 7 = 1 ådè” ç á«å-ã‡ä{á ”Üá 0 !'ô=ò B‹údó÷”ã«ÛÀÜ3 è 7 @ä{á å™ì{á‡àtå¼á‡àdì ޟã«äÈà
 è #ô





 











    





  







¯à-ÿ ådí'à
ޟãWá—à
ì ޟã‡ä àdè'á±ÿi۔Üܼ߫ã‡ÛÀÜ -' 0D J.Få™ß«Ü¼ã‡ÛÀÜý|á—Üì ædÜá±èÀàdãWÞÀèÀä !‹ÞÀÜ Yò
VÜ|Üývݔ۔ådá‡äÜIã‡Û”å™ã ã‡Û”ä ávéå™è àdè”ìÈêO۔å™ÝÀÝÁÜè ä èDÝÀ߇ä è”éä ÝÀì ÜoäÈâ±ã«ÛÀÜ ]Ü)á‡á‡ä{å™è ä á¼è”à™ã|ÝÁà
á‡ä:ã«ä ædÜTçŸÜ ”è”ä:ã«Ü™÷iå™è”çDÜædÜèDã«ÛÀÜè÷ ã‡Û”ÜNá—à
ìÈÞÀã‡ä àdè”á¥åd߇Ü=èÀÜ)éÜá«á«å™ß«äÈì ê dì àdí'å™ìöó4ôKÚ¥ÛÀÜ=âöà
ìÈì à-ÿiä èNæ
Üß«êoá‡äÈývݔìÈܯã‡ÛÀÜàdß«Üý_á—ÛÀà-ÿ]á#ã«Û”å-ã]äÈâèÀàdè ÞÀèÀ ä !‹ÞÀܼá—à
ì ޟã‡ä àdè'á]àŸééÞÀß÷ã«ÛÀÜè ã‡Û”ܼá‡àdì ޟã«ä àdèÃå-ã`à
èÀÜIà
ݟã‡ä ý|ådì@ÝÁàdä è‹ã ä{á éà
è
ã«ä è‹Þ”àdޔá‡ì êçŸÜâöàdß«ý|å™íÀì Ü ä è‹ã‡àtã‡ÛÀÜNá‡àdì ޟã«ä àdèFå™ãiã‡ÛÀÜWà™ã‡Û”ÜßiàdÝÀã‡ä ý|å™ì8ÝÁàdä è‹ã÷”ä èTá—Þ”é4ÛUå¼ÿ¥å êvã‡Û'å-ã¯å™ì ìä è‹ã‡Ü߇ývÜ)çŸä{å-ã‡Ü±Ý'à
ä è
ã4áiå™ß«Ü å™ì{á‡àtá‡àdì ޟã«ä àdè”áô



     

     

   

  ?    "1    87   ? B B


    8M  * M4"    6*M G    > I( * M 5  *I"    6*M G - !D J.  8M  * M3M GG2    I( * M 57 I  6 M KHM5 ICG2  *"CMGJM K. M    M ?GJI $ ( $ M IFJM  *" M  $ -  I  GDI "CM   M   7 M  /I 5AI *  G  /  *  * MIFJM  *" M  $ -  I ()) * G  G K.*  K  6 "  6 $AM   M $ M MM  G  G # 5 (* H ¾ò ”ó H ò—ø ”ó 7 ?  SDø  $G $   $ ( ¾ò ”ó8 G  GJI(6 $  I(+ I(  6*6  # D u=Ô-ÕrÕE( rÜã ã‡Û”ÜIývä èÀä ýtÞÀý æ-å™ì ÞÀÜIà™âKã«ÛÀÜIà
íÜéã‡ä ædÜNâöÞÀè”éã«ä àdèÃíÁÜ = ô`ڥ۔Üè¾í6êÃå
á‡á‡ÞÀývݟã«äÈà
è÷ vò ó H vò # ó H 8 = B ô #êFéàdè6ædÜë6äÈãê|àdâ t÷ vò ïò ”ó‡ó òø ”ó vB ò ó vò # ó H = ô B Àޔߗã«ÛÀÜß«ývàd߫ܙ÷Ÿí6êoì ä èÀÜåd߇äÈãêd÷dã‡Û”Ü ò ”óiá‡å™ã‡ä{á—âöê¼ã‡ÛÀÜWéàdè”á—ã‡ß4å™ä è‹ã«á+01!'ô±ò S‹ó4÷@ò ”ø)ó KÜëŸÝÀì ä{éäÈã‡ì êTò,å
å™ä è éà
ýIí”äÈè”äÈ è Ní'àdã‡ÛUéà
è”á—ã‡ß4å™ä è‹ã«á¹äÈè‹ã«àtà
èÀÜ)ó 

 

L =

ò


#9< = 7 dó H



ò‡òø4 ”óò òø ”óò—ø

L =







 =

ó



9< =  7 ó 7 rò # 9< = 7 # ó—ó 7 ròø   = ó H ø  =



ò dó


ì:ã«ÛÀàd Þ dÛ|á‡äÈývݔìÈÜd÷-ã‡ÛÀä{áKã‡ÛÀÜàdß«Üý ä á1!
ޔä:ã«Ü]ä è”á—ã‡ß«Þ”éã«äÈæ
ܙô ”àdß¹ÜëÀå™ývÝÀì Üd÷dàdèÀܯývä
ۋãã«ÛÀä èÀõIã‡Û”å™ã&ã‡Û”Ü ÝÀß«àdíÀì Üý|áçŸÜÝÀä{éã‡Ü)çWä è @ä
ÞÀß‡Ü øS±Û”å æ
Ü#á‡Üæ
Üß4å™ì6çŸäÁÜ߇Üè
ã?à
ݟã‡ä ý|å™ì6á‡àdì ޟã‡ä à
è”á&òöâöàdßã‡Û”Ü¥éådá‡Ü¥à™â'ì ä èÀÜ)å™ß á‡ÞÀÝÀÝÁàd߇ã¯ædÜ)éã‡à
ß]ý|ådé4۔äÈè”Üá4ó4ô4]à-ÿ#ÜædÜß÷”á‡ä è”éÜ`à
èÀÜWéå™èÀèÀàdã±á‡ývà‹àdã‡ÛÀì ê|ývà-æ
Ü ã‡ÛÀÜIۋê6ÝÁÜß«ÝÀì{å™èÀÜ âöß«àdý àdè”ÜWÝÀß«àdÝÁà
á‡ÜçÃá‡àdì ޟã«ä àdèã‡àå™èÀàdã‡ÛÀÜß±ÿiäÈã‡Û”àdޟ ã dÜè”Üß4å-ã‡ä  è vÛ6ê6ÝÁÜß«ÝÀì{å™èÀÜ)á]ÿiÛÀä{é4Û¾åd߇ÜNè”à™ã`á—à
ìÈÞÀã‡ä àdè”á÷ ÿ#Üoõ6èÀà-ÿ ã‡Û'å-ãtã‡ÛÀÜ)á—ÜÝÀß«àdÝÁà
á‡Üçµá‡àdì ޟã«äÈà
è”áNåd߇ÜFäÈèµâ,ådéãtèÀà™ãvá‡àdì ޟã‡ä à
è”áNå-ãvå™ì ì.ô ;èOâ,ådéã÷Kâöà
ßIÜ)ådé4Û à™â&ã‡ÛÀÜ)á—Ütéådá‡Üá÷Áã‡ÛÀÜtàdݟã«ä ý|å™ìޔèÀ ä !
ޔÜWá—à
ì ޟã‡ä àdèTä{á å-ã H6SÀ÷ÿiä:ã«Û¾åoá‡ÞÀäÈã«ådíÀì ÜNé4ÛÀàdä{éÜNà™â |òöÿiÛÀä{é4Û Û”å
á ã‡ÛÀÜ|Ü ÁÜ)éãNàdâ¥å
á‡á‡ä dèÀä  è oã‡ÛÀÜ|á«å™ývܼì{å™íÁÜìã«àTådì ìã‡ÛÀÜvÝÁàdä è‹ã«á4ó4ô  à™ã«Ü¼ã‡Û”å™ã`ã‡ÛÀä{á=ä{á`åTÝ'ÜߗâöÜ)éã«ìÈê
S‹ó ådééÜݟã«ådíÀì ܼá‡àdì ޟã«ä àdèÃã«àTã‡ÛÀÜFéì{ådá«á—ä 'éå-ã«äÈà
èUÝÀß«àdíÀì Üý Nådè‹ê¾ÝÀ߇à
Ý'à‹á—Ü)ç¾Û6ê6ÝÁÜß«ÝÀì{å™èÀÜUòöÿiäÈã«Û HG ÿiä ì ì8éå™Þ”á‡Ü ã‡ÛÀÜ`ݔ߇ä ý|å™ìÁàd í Üéã«ä ædܱâöÞÀè”éã«ä àdèFã‡àtã«ådõdÜ`åtÛÀä dÛÀÜß#æ-ådìÈޔܙô

;Á$›“)Ž,…=<> <  0 -²  0 ( $!¥* r" 12+ ²- 0 ² 0 *. " 3  0 2.;1  3-043 »  3 "$¿-¹* 0   ,1 " 40 3 * 5 ä è”å™ì ì ê™÷6èÀàdã‡Ü ã‡Û”å™ã¥ã‡ÛÀÜ â,ådéãiã‡Û”å™ã¯î6ð±ñ ã‡ß4å™ä èÀä è tå™ì ÿ#å êŸá ”è”çÀáiå dì à
í”å™ì8á—à
ì ޟã‡ä àdèvä{á#ä èFéàdè‹ã‡ß4ådá—ã ã‡àtã«ÛÀÜNéå
á—Ü=à™âèÀÜÞÀß«ådìèÀÜãÿ#àdß«õŸá÷ŸÿiÛÀÜ߇Ü=ý|ådè‹êvì àŸéådì8ýväÈè”äÈý|åNޔá—Þ'å™ì ì êtÜëŸä á—ãô



z46‹ÓÕrÖ I  ÕE(±pÕ+%2×Ó)Ù,ÕÁÒ

Ú¥ÛÀÜ`á‡ÞÀÝÀÝÁàd߇ã#æ
Üéã«àdß#àdݟã«ä ývä )å-ã«äÈà
èIݔ߇à
íÀì ÜýþéådèoíÁÜ=á‡àdì æ
Üçvå™è'å™ì ê‹ã‡ä{éådìÈì êNà
èÀì ê¼ÿiÛÀÜèoã‡ÛÀÜ=è6ÞÀýtí'Üß à™â@ã‡ß4å™ä èÀä ètçÀå-ã4åIä{á¥æ
Üß«ê|á—ý|ådìÈì.÷‹àdߥâöàdߥã«ÛÀÜNá‡Üݔådß«ådíÀì ܱéå
á—Ü ÿiÛÀÜèäÈãiä{á#õ‹è”à-ÿièFíÁÜâöà
߇Ü۔å™è”çFÿiÛÀä{é4Û à™âã«ÛÀÜ]ã«ß«ådä èÀä  è `çÀå-ã4å`í'Ü)éà
ývܯá—Þ”ÝÀÝ'à
ߗã#ædÜ)éã‡à
ß«á¯ò,ådá¹ä èoî6Üéã‡ä àdè'$ á EŸô ENå™è'/ ç Ÿô dóô  à™ã‡Ü¯ã‡Û”å™ã&ã«ÛÀä{á&éådè ۔ådÝÀÝ'ÜèVÿiÛÀÜè ã‡ÛÀÜvݔ߇à
íÀì Üýy۔å
á`á—à
ývÜtá‡ê6ývývÜã‡ß«êÅò2î6Ü)éã«äÈà
5 è EÀô E‹ó4÷í”ÞŸã`ã‡Û”å™ãNäÈã=éå™èVå™ì{á—à۔ådÝÀÝ'Üè ÿiÛÀÜèFäÈãiçŸà6Üá#èÀà™ãNò.î6Üéã‡ä àd, è Ÿô dó4ô ”àdß#ã‡ÛÀÜ 
ÜèÀÜß«ådìÁådè”å™ì ê‹ã‡ä{é]éådá‡Ü™÷6ã‡ÛÀÜ=ÿ#àdß4áã¥éå
á—Ü=éà
ývÝÀޟã«å™ã‡ä àdè'å™ì éà
ývÝÀì ÜëŸäÈãêFä{á=àdâ¹àdß4çŸÜß Q 8 òöä è6ædÜß«á‡ä àdèUàdâ¹ã«ÛÀÜ ]Ü)á‡á‡ä{å™è'ó÷ÿiÛÀÜß« Ü  ä{á±ã‡Û”Üvè‹Þ”ýIíÁÜß`àdâ#á‡ÞÀÝÀÝÁàd߇ã ædÜ)éã«àdß4á÷Àå™ìÈã«ÛÀàd Þ 
Û|ã«ÛÀÜ=ãÿ¹à¼ÜëŸådývÝÀì Üá dä ædÜèoíÁà™ã«ÛÛ'å ædÜ=éàdývݔìÈÜëŸä:ãêtà™1 â $tòø ó4ô

¯à-ÿ¹ÜædÜß÷äÈèVývà‹áãWß«ÜådìKÿ#àdß«ì çïéå
á—Ü)á÷ 0 !
Þ'å-ã‡ä à
è”ávò@B E
óvò,ÿiä:ã«ÛµçŸà™ãtÝÀß«à6çÀޔéã4áNß«ÜÝÀì{ådéÜç í6ê õ
Üß èÀÜì á4ó4÷ò@B B‹ó4÷å™è'çÅò@B‹ú
ó¥ýtޔáã±í'Ütá—à
ìÈæ
ÜçFè6ÞÀývÜß«ä{éå™ì ì ê™ô+Ààdß á—ý|ådì ìÝÀ߇à
íÀì Üý|á÷”å™è6ê,dÜè”Üß4å™ìrÝÀޔ߇ÝÁà
á‡Ü àdÝÀã‡ä ývä å-ã«ä àdèWÝ'ådé4õ-å 
Ü¥ã‡Û”å™ã¥á‡àdì ædÜ)áì äÈè”Üå™ß«ì êNéà
è”áã«ß«ådä èÀÜç¼éà
è6ædÜë !‹Þ”ådçÀß«å™ã‡ä{é¥ÝÀß«àdß4å™ý|áÿiä ì ì”çŸà”ô
dà6àŸçÃá‡ÞÀß«ædÜêFà™âKã«ÛÀÜIå æ-ådäÈì{ådíÀì Ü á‡àdì ædÜß«á÷'ådè”çUÿiÛÀÜ߇ÜWã‡à 
Üã±ã‡ÛÀÜýT÷8éå™èÃíÁÜNâöà
ÞÀè”ç #  ä èOò2ñUàdß ÜNådè”ç
V߇ä dۋã÷8ø)ùdù E
ó4ô



”àdßtì ådß 
Üß`ÝÀß«àdí”ìÈÜý|á÷åUß4å™è 
Ü|à™âiÜëŸä{á—ã‡ä è Tã‡Ü)é4ÛÀèÀ ä !‹ÞÀÜáWéå™èOí'ÜFíÀ߇à
ÞdۋãNã«à¾íÁÜådßô
âöÞÀì ì¹Üë ÝÀì àdß4å-ã«ä àdèTà™â¹ã‡ÛÀÜt߇Üì{å-ã‡ä æ
ÜNývÜß«äÈã«á]àdâKã‡Û”Üá‡ÜtývÜã‡ÛÀàŸçÀá±ÿ#àdÞÀì{ç ”ì ì?å™è”à™ã‡Û”Üß ã‡ÞŸã«àdß«ä{å™ì.ô ¯Üß«ÜIÿ#Ü ޔáã çŸÜ)á‡é߇ä íÁÜ`ã‡ÛÀÜ dÜèÀÜß4å™ìä{á‡á‡ÞÀÜ)á÷å™è”çUâöàdß=éàdè'éß«Üã‡ÜèÀÜá«á÷ 
ä ædÜNåFíÀ߇ä ÜâÜëŸÝÀì{å™è”å™ã‡ä àdèTà™â&ã‡ÛÀÜIã‡Ü)é4ÛÀèÀä !‹ÞÀÜ ÿ#Ü¥éÞÀ߫߇Üè
ã«ì ê`ޔá—Üdô #ÜìÈà-ÿ=÷då 4â,ådéÜ ¯ývÜ)å™è”áå`á‡Üã?à™â8ÝÁàdä è‹ã«áì ê‹ä  è  àdèIã‡Û”Üií'à
ÞÀè”çÀåd߇êNàdâ”ã«ÛÀÜ¥âöÜå
á—ä íÀì Ü ß«Ü dä à
è÷å™è”çÅådè ådéã‡ä ædܼéà
è”á—ã‡ß4å™ä è‹Kã |ä{á`åUéàdè”á—ã‡ß4å™ä è‹ã±âöà
ßNÿiÛÀä{é4Û ã‡ÛÀÜ|Ü !‹Þ”å™ì äÈãêUÛÀàdì{çÀáô Àà
ßNývàdß«Ü

àdèèÀà
èÀì äÈè”Üå™ß¹ÝÀß«àdß4å™ývývä è=ã«Üé4ÛÀè”ä!‹ÞÀÜ)á¥á—ÜܼòìÈÜã«é4ÛÀÜß÷ø)ùdü'6ûÀñÃå™è‹ådá«å™ß«ä{å™è÷Áøù
ùŸûŸñÃé ¹à
߇ývä{é4õÁ÷ øù
ü E‹ó4ô Ú¥ÛÀÜWí”ådá‡ä{é±ß«Üéä Ý'Ü=ä{á¥ã‡àÃò—ø)ó¥è”à™ã‡ÜNã‡ÛÀÜNà
ݟã‡ä ý|ådìÈäÈãêT ò `Ú]ó¥éàdè”çŸäÈã«äÈà
è”á#ÿiÛÀä{é4Ûã‡ÛÀÜWá‡àdì ޟã«ä àdèoýtޔá—ã «á å-ã‡ä{á—âöê™÷iò 
ó`çŸÜ 'èÀÜFåÃá—ã‡ß4å-ã«Ü
êÃâöàdßIå™ÝÀÝÀß«à
å
é4ÛÀä èàdݟã«äÈý|ådì ä:ãêTí6êïÞÀèÀäÈâöàdß«ývì êÃä è”éß«Üå
á—ä èoã‡ÛÀÜçŸÞ”å™ì àdíÜéã«ä ædÜ âöÞÀè”éã‡ä àdèUá‡ÞÀíÜéã]ã‡à¼ã«ÛÀÜWéàdè”á—ã‡ß4å™ä è‹ã«á÷Àå™è”çÅò%E
ó¥çŸÜ)éä{çŸÜ`àdèUåvçŸÜ)éà
ývÝ'à‹á—äÈã‡ä à
èoå™ìdà
߇äÈã«ÛÀý á‡à|ã‡Û'å-ã=àdèÀì êÝ'à
ߗã«äÈà
è”áiàdâKã‡ÛÀÜIã‡ß4å™ä èÀä  è vçÀå™ã«åoèÀÜÜ)çÃíÁÜI۔ådè”çŸì ÜçUå™ã=å dä ædÜèã‡ä ývÜo*ò #à‹á—Üß÷ % ÞÀê
àdè å™è'çUðKå™ÝÀè”äÈõÁ÷øù
ù 6 û $ á‡ÞÀè”åŸ÷ Àß«Üޔè”ç¾ådè”ç % ä ߇à‹á—ä.÷ø)ùd ù -å
ó@ ô
VÜ dä æ
ÜWåvíÀß«ä Üâ&çŸÜ)á‡é߇ä ݟã«äÈà
èFàdâ¹á‡àdývÜ à™â#ã‡Û”Ü|ä{á‡á‡ÞÀÜá ä è6ædà
ìÈæ
Üç8ô $±èÀÜ|éå™èïæ6äÈÜÿ ã«ÛÀÜ|ÝÀß«àdíÀì Üý ådá=ß« Ü !‹ÞÀä ß«äÈ è |ã‡Û”Üoá—à
ì ޟã‡ä àdè à™âiåUá—Ü !‹ÞÀÜè”éÜ à™âiÜ !
Þ'å™ì äÈãêÃéàdè'áã«ß«ådäÈè”ÜçÃݔ߇à
íÀì Üý|áô
dä ædÜè Ü !‹Þ”å™ì äÈãêÃéà
è”áã«ß«ådä èÀÜç ÝÀß«àdíÀì Üý éå™èVíÁÜoá‡àdì ædÜ)ç¾ä è àdè”Ü=á—ã‡ÜÝTí‹êFޔá‡äÈ è Wã«ÛÀÜ  Üÿ¥ã‡à
èývÜã‡ÛÀàŸçïò,ådì:ã«ÛÀàd Þ dÛ|ã«ÛÀä{áiß« Ü !‹ÞÀä ߇Ü)á¹á—ã‡à
ß«å dܱâöàdß]åtâ,å
éã‡à
߇ä )å-ã‡ä à
èvà™â ã‡Û”ÜWÝÀß«à Ü)éã‡Ü)ç ]Ü)á‡á‡ä{å™è'ó÷'à
߯ä èÃå-ã ývà
á— ã :Ká—ã‡ÜÝ'á¯Þ”á‡ä  è oéàdè  Þ ‹å-ã«Ü dß4ådçŸä Üè
ã¯å
á‡éÜè‹ãIò .Kß«Üá«á]Üã=å™ì.ôÈ÷ øù
ù 
ó&ò,ÿiÛÀÜß«@ Ü :8ä{áã‡ÛÀÜiè6ÞÀýtí'Üß?à™âçÀå-ã4å Ý'à
äÈè‹ã4áâöàdß?ã«ÛÀÜiÝÀß«àdíÀì Üý[éÞÀ߫߇Üè‹ã‡ì ê`í'Üä  è `á—à
ìÈæ
Üç rè”à`Üë6ã‡ß4å á—ã‡àdß4å 
ܼä á=ß« Ü !‹ÞÀä ß«Üç”óôTî6à
ývÜvå™ì dà
߇äÈã‡Û”ý|á¯ývà-æ
ÜIÿiäÈã‡Û”äÈèÅå 
äÈæ
Üè â,ådéÜvÞÀè‹ã‡ä ì¹åTèÀÜÿþéàdè'áã«ß«ådäÈè‹ã=ä{á Üè'éàdޔè
ã«Üß«Üç8÷Ÿä èÿiÛÀä{é4ÛFéå
á—ܱã‡Û”Ü=ådì dàdß«äÈã‡ÛÀý ä{á#߇Ü)áã4å™ß‡ã‡ÜçoÿiäÈã‡Û|ã«ÛÀÜ`èÀÜÿ éà
è”á—ã‡ß4å™ä è‹ã#ådç”çŸÜçoã‡àtã‡Û”Ü ì ä{áãKàdâ Ü !‹Þ”ådìÈäÈãêWéàdè”á—ã‡ß4å™ä è‹ã«áôڥ۔ä á&ývÜã«ÛÀàŸçv۔ådá&ã‡Û”ܱçŸä{á«ådçŸæ-å™è‹ã4å dÜ¥ã«Û”å-ã#àdèÀì êtàdèÀܱèÀÜÿ éàdè”á—ã‡ß4å™ä è‹ã ä{á=ý|ådçŸÜvå
éã‡ä æ
ܼå™ãWåã‡ä ývܙô  .Kß«à Ü)éã«äÈà
èÃývÜã«ÛÀàŸçÀKá T۔å ædÜvå™ì{á‡àFíÁÜÜèµéà
è”á—ä{çŸÜ߇Ü)çOò2ñUàdß Ü™÷#øù
ùÀø)ó÷ ÿiÛÀÜ߇ܥå Ý'à
äÈè‹ãàdޟã4á—ä{çŸÜ¹ã‡Û”ܹâöÜ)ådá‡ä íÀì Ü&ß«Ü dä à
èNä{áéàdývÝÀÞÀã‡Üç÷då™è”çWã‡ÛÀÜèIì ä èÀÜ#á—Ü)å™ß4é4ÛÀÜáå™è'çNÝÀß«à Ü)éã‡ä à
è”á å™ß«Ü±çÀàdèÀÜ=á—àWã‡Û”å™ã¥ã‡ÛÀÜ=ådéã‡Þ”ådìÁývà-æ
Ü]ß«Üý|å™ä è”áKäÈè'á—ä{çŸÜiã«ÛÀܱâöÜådá‡ä íÀì Ü]ß«Ü dä à
èôÚ¥ÛÀä{á#å™ÝÀݔ߇à‹ådé4Ûoéå™èoå
çÀç á‡ÜædÜß«ådìèÀÜÿ éà
è”áã«ß«ådä è
ã4á=å-ãNàdè'éܙô  à™ã«Ü¼ã‡Û”å™ãWä è íÁà™ã«Ûµå™ÝÀݔ߇à‹ådé4ÛÀÜ)á÷á‡Üæ
Üß4å™ì?ådéã‡ä ædܼéà
è”áã«ß«ådä è
ã4á éådè íÁÜéàdývÜ *    *" MNä è¾àdè”ܼá—ã‡ÜÝô ;èÅå™ì ì?å™ì dà
߇äÈã‡Û”ý|á÷ÁàdèÀì êTã‡Û”ÜtÜ)á‡á‡Üè‹ã«ä ådìݔå™ß‡ã`à™â&ã«ÛÀÜ ]Ü)á‡á‡ä{å™è ò{ã«ÛÀÜFéàdì ÞÀývè”á`éàd߫߇Ü)á—ÝÁàdè'çŸä  è ã‡3 à "= H
S‹ó=èÀÜÜ)çÅíÁÜFéà
ývÝÀޟã«Üç ò,ådì:ã«ÛÀàd Þ dÛÅá—à
ývÜ|å™ì dà
߇äÈã«ÛÀý|á±çÀà éà
ývÝÀޟã«Ü¯ã«ÛÀܱÿi۔àdì  Ü ¯Üá«á—ä{ådè'ó4ô Àà
ß¹ã«ÛÀÜ  Üÿ¥ã‡àdèå™ÝÀݔ߇à‹ådé4Û÷6àdèÀÜ=éådèFådì á‡àNã4å™õ
ܱådçÀæ ådè‹ã«å 
ܯàdâã‡Û”Ü â,ådéã±ã«Û”å-ã ã‡Û” Ü ¯Üá«á—ä{å™èUä{á¯Ý'à‹á—äÈã‡ä æ
ÜNá—Üývä çÀÜ ”èÀäÈã«Ü`í‹êUçŸä{å dàdè'å™ì ä ä  è WäÈã ÿiä:ã«ÛUã‡Û”Ü #ÞÀè”é4 Û `ådޟâöý|å™è å™ì 
àdß«ä:ã«ÛÀý ò #ÞÀè”é4Û å™è”ç `å™ÞŸâöý|ådè÷¯ø ù 6+ û #ÞÀè”é4ÛDå™è”ç `å™ÞŸâöý|ådè÷]øù
ü S‹ó|òöäÈâ]ã«ÛÀ Ü ]Ü)á‡á‡ä{å™èµÿ#Üß«Ü ä è”çŸÜ ”èÀäÈã‡Üd÷'äÈã éàdÞÀì{çUá—ã‡ä ì ìrí'ÜtÜå
á—ä ì ê|ß«ÜçŸÞ'éÜçã«! à - ë víÀì à6é4õTçŸä{å 
àdè”ådì8âöàdß«ý ÿiä:ã«ÛUã‡Û”ä á±å™ì dà
߇äÈã«ÛÀýoó4ô

;èµã«ÛÀä{áIådì dàdß«äÈã‡ÛÀýT÷@ÿiÛÀÜèDåÃèÀÜÿ éàdè'áã«ß«ådäÈè‹ãWä{áWý|ådçŸÜådéã«ä ædÜvàdßtä è”ådéã‡ä ædÜd÷@ã«ÛÀÜFâ,ådéã«àdß«ä å-ã«ä àdèïà™â ã‡Û”ÜWÝÀß«à Ü)éã‡Ü) ç ]Üá«á‡ä ådèTä á]Üådá‡ä ì êvÞÀÝÁç”å-ã‡Ü)çµò,å
á]à
ÝÀÝ'à‹á—Ü)çFã‡à|ß«ÜéàdývÝÀޟã«ä  è tã‡ÛÀÜWâ,ådéã«àdß«ä å-ã«ä àdèvâöß«àdý á«éß4å-ã«é4ÛÁó4ô @ä è”ådì ìÈêd÷Áä èïä è
ã«Üß«ä àd߯ÝÁàdä è‹ã=ývÜã‡Û”à6ç”á÷rã«ÛÀÜvæ-å™ß«ä{å™íÀì Üá±å™ß«ÜIÜ)á‡á‡Üè‹ã«ä ådì ìÈêF߇Ü)á‡éå™ì Üç¾á—àUådá±ã«à å™ì ÿ¥å êŸá߇Üý|å™ä èIä è”á‡ä{çŸÜiã«ÛÀÜ]âöÜ)ådá‡äÈí”ìÈÜiß«Ü 
ä àdèô
è|ÜëŸådývÝÀì Üiä{á?ã‡Û” Ü   $
$`å™ì dà
߇äÈã«ÛÀý à™â&ò,ðKå™è”çÀÜß íÁÜä.÷Køù
Lù B
åÀûðKå™è”çŸÜ߇íÁÜä.÷?øù
Lù B
í'ó4÷ÿiÛÀä{é4ÛÃä{á=åoݔ߇ä ý|å™ì ;çŸÞ”ådìݔå-ã«Û¾âöà
ìÈì à-ÿiä  è vå™ì 
àdß«ä:ã«ÛÀýTô ڥ۔ä á±ì{ådá—ã ývÜã«ÛÀàŸçtä áKìÈä õ
Üì ê ã‡àNíÁÜ]ޔá‡ÜâöÞÀì'âöàdß&ÝÀ߇à
íÀì Üý|áÿiÛÀÜ߇ܥã«ÛÀܯè6ÞÀýtí'Üß?à™ârá‡ÞÀÝÀÝÁàd߇ã&ædÜ)éã‡à
ß«á?å
áKå=âöß4ådéã‡ä àdè à™âã‡ß4å™ä èÀä  è Iá‡ådývÝÀì Ü á‡ä Ü ä á#Üë6ÝÁÜéã‡Ü)çoã«àví'Ü`ì{ådß 
ܙô




VܼíÀ߇ä  Ü ”êçÀÜá«éß«äÈíÁÜNã«ÛÀܼéàdß«ÜWà
ݟã‡ä ýväå™ã‡ä àdèývÜã«ÛÀàŸçUÿ#ÜtéÞÀ߇߫Üè‹ã«ìÈêޔá‡Ü # -ô ƒã=ä{á±ådè ådéã‡ä ædÜIá—Üã ývÜã«ÛÀàŸçWéàdýtíÀä èÀä è dß4ådçŸä Üè
ãå™è”çtéà
è Þ
å™ã‡Ü dß4ådçŸä Üè‹ãådá«éÜè‹ãô>
ÛÀÜèÀÜæ
Üßã‡ÛÀÜ¥à
íÜéã‡ä ædÜ&âöÞÀè”éã«ä àdè ä{á±éàdývÝÀÞÀã‡Üç÷á‡àoä{á]ã‡ÛÀÜ dß4ådçÀäÈÜè‹ã÷Áå™ã æ
Üß«êFì äÈã—ã«ì ÜNÜë6ã‡ß4åoéà
á—ãô ;è¾Ý”Û”ådá‡Üoød÷Áã«ÛÀÜvá—Ü)å™ß4é4ÛÃçŸä ߇Ü)éã«äÈà
è”á  å™ß«Ü¼å™ì àdè oã‡ÛÀÜ 
ß«å
çŸä Üè‹ãôtÚ¥ÛÀܼèÀÜåd߇Ü)áã â,ådéܼådì àdè oã‡ÛÀÜ|á‡Üådß«é4Û¾çÀäÈß«Üéã‡ä àdè¾ä á±âöà
ÞÀè”ç8ô ƒâ#ã‡ÛÀÜvçŸàdã ÝÀß«àŸçŸÞ”éã à™â?ã«ÛÀÜ 
ß«å
çŸä Üè‹ãiã‡ÛÀÜ߇ÜIÿiäÈã‡Û  äÈè'çŸä{éå-ã«Üá¥ã«Û”å-ã±ã‡ÛÀÜtý|å-ëŸä ýtÞÀý å™ì àd è   ì ä ÜáiíÁÜãÿ#ÜÜèUã‡Û”Ü éޔ߇߫Üè‹ãKÝÁàdä è‹ã&å™è”çIã‡Û”Ü]èÀÜåd߇Ü)áã?â,å
éÜd÷
ã«ÛÀÜ]àdÝÀã‡ä ý|å™ì6ÝÁàdä è‹ã&å™ì àd è  ã«ÛÀܯá‡Üå™ß4é4Û¼çŸä ߇Ü)éã‡ä à
èWä{áKéà
ývÝÀޟã«Üç å™è'å™ì ê‹ã‡ä{éådìÈì ê=òöèÀàdã‡Ü&ã‡Û”å™ãã‡Û”ä á@çŸà6ÜáèÀà™ãß‡Ü !
ޔäÈß«ÜKåiì ä èÀÜKá‡Üå™ß4é4Û'ó÷ ådè”ç=ÝÀ۔å
á—Ü ]ä{áÜè‹ã‡Ü߇Ü)ç8ô $¯ã«ÛÀÜß«ÿiä{á—Üd÷ ÿ#Ü ÞÀývÝã‡à¼ã‡Û”Ü`èÀÜÿgâ,ådéÜNå™è”çß«ÜÝÁÜå™ã¥Ý”Û”ådá‡Üvø™1 ô ;èUݔ۔ådá‡Ü 6÷ .à
ì åd' õ iä íÀä Ü߇Ü]éàdè  Þ ‹å-ã« Ü 
ß«å
çŸä Üè‹ã ådá«éÜè‹ã=ò .Kß«Üá«á&Üãiådì.ô:÷8øù
ù dóKä{á#çÀàdèÀÜd÷6ޔè
ã«ä ì8åIèÀÜÿ â,ådéܱä{á¹Üè”éàdÞÀè‹ã«Üß«Üç¾ò,ä èoÿiÛÀä{é4Ûoéådá‡Ü±ÝÀÛ'ådá‡ÜIø ä{á ß«Ü ƒÜè‹ã‡Ü߇Ü)ç”óKà
ߥã‡ÛÀÜNá—ã‡à
ÝÀÝÀä  è ¼éß«äÈã‡Üß«ä àdè|ä{á#ývÜãô  àdã‡Ü ã‡ÛÀÜ=âöà
ì ìÈà-ÿiä  è 



î6Üådß«é4ÛµçŸä ß«Üéã‡ä àdè”á`åd߇ÜFå™ì ÿ#å êŸá ÝÀß«à Ü)éã‡Ü)çµá‡àUã‡Û”å™ãWã«ÛÀÜ = éàdè‹ã‡ä è6ÞÀܼã«à á«å-ã‡ä{á—âöêÃã«ÛÀÜFÜ !‹Þ”å™ì äÈãê éàdè'áã«ß«ådäÈè‹ã 01!'ô ò@B6údó4ô  à™ã«ÜFã«Û”å-ãvã«ÛÀÜÃéà
è Þ‹å-ã‡Ü 
ß«å
çŸä Üè‹ã¼ådìdàdß«äÈã‡ÛÀý ÿiä ì ì¥á—ã‡ä ì ì¥ÿ¹à
߇õÁûiÿ#Ü å™ß«Üá‡ä ývÝÀì êïá‡Üådß«é4ÛÀä è ä è åVá—Þ”í”á—Ý'ådéÜdô ¯à-ÿ¹ÜædÜß÷KäÈãvä{áIä ývÝÁàd߇ã«ådè
ãIã‡Û”å™ãvã‡ÛÀä{átÝÀß«à Ü)éã‡ä à
èµä{á äÈývݔìÈÜývÜè‹ã‡Ü)çäÈèïá—Þ'é4Û åFÿ#å êã‡Û'å-ã=èÀà™ã`àdè”ìÈêTä{ á 01!'ôF@ò B6údó¯ývÜãvòöÜ)ådá‡êŸó÷ríÀޟãNå™ì{á‡àoá‡àoã«Û”å-ã=ã‡Û”Ü å™ è 
ìÈÜ&í'Üãÿ¹ÜÜèIã‡ÛÀÜ#߇Ü)á—Þ”ì:ã«ä  è ¯á‡Üå™ß4é4ÛWçŸä ß«Üéã‡ä àdè÷-å™è'ç`ã‡Û”Ü¥á‡Üådß«é4ÛIçŸä ߇Ü)éã‡ä à
è`ÝÀ߇ä à
ßrã‡à ÝÀß«à Ü)éã‡ä à
è÷ ä á#ývä èÀä ývä Ü)çUòöèÀàdã !‹ÞÀäÈã«Ü=á‡à¼Üå
á—êÀó4ô

ïÜoådì{á—àޔá‡Üoå á—ã‡ä{é4õ6êTâ,ådéÜáKUå™ìdà
߇äÈã‡Û”ý ±ÿiÛÀÜèÀÜæ
ÜßWå 
äÈæ
Üè¾â,ådéÜoä á=۔ä:ãNývà
߇Ütã«Û”å™èÅà
è”éÜd÷ ã‡ÛÀÜtá—Ü)å™ß4é4ÛTçÀäÈß«Üéã‡ä àdè'áiå™ß«ÜWådç Þ'áã«ÜçÃá‡àvã‡Û'å-ã ådì ì@á‡ÞÀí”á‡Ü !‹ÞÀÜè‹ã±á‡Üådß«é4۔Üá]åd߇ÜWçÀàdèÀÜNÿiäÈã«ÛÀä èTã‡Û”å™ã â,ådéÜdô
ì  ì á—ã‡ä{é4õ6êoâ,å
éÜKá |åd߇ÜI߇Ü)á—Üã¼ò,ý|ådçŸÜ 4èÀà
è ƒá—ã‡ä{é4õ6ê
ó#ÿiÛÀÜèÃã«ÛÀÜIß4å-ã«ÜWàdâ&ä è”éß«Üå
á—ÜNàdâKã‡Û”Ü àd í Ü)éã‡ä æ
ܯâöÞÀè'éã‡ä à
èoâ,ådìÈì{á¹í'Üì à-ÿ åIã‡ÛÀß«Üá‡ÛÀà
ì çô 


Ú¥ÛÀÜ]å™ì dà
߇äÈã«ÛÀý áã«àdݔá?ÿi۔Üè¼ã‡Û”Ü¥âöß«å
éã‡ä à
è”å™ìŸß4å-ã«Ü#àdâÁä è”é߇Ü)ådá‡Ü¥à™âÁã‡ÛÀÜià
íÜéã‡ä ædÜ#âöÞÀè”éã‡ä àdè  â,å™ì ì{á í'ÜìÈà-ÿþåÃã‡àdì Üß«ådè”éܾòöãê6ÝÀä{éå™ì ì êµøÜ 4øSŸ÷âöà
ßtçŸà
ÞÀíÀì Üoݔ߇Ü)éä{á—ä à
è'ó4ô  à™ã«Üoã«Û”å-ã¼àdèÀÜéådè å™ì{á‡àUޔá‡Ü ådá=áã«àdÝÀݔäÈ è é߇äÈã‡Ü߇ä à
èTã‡ÛÀÜ|éàdè”çŸäÈã«äÈà
èUã‡Û'å-ã=ã‡ÛÀÜ|á‡äÜtà™â¹ã‡ÛÀܼÝÀß«à Ü)éã‡Ü)ç á‡Üådß«é4ÛïçŸä ߇Ü)éã«äÈà
èTâ,å™ì ì{á í'ÜìÈà-ÿ åWã«ÛÀß«Üá‡ÛÀàdì{ç8$ ô ]à-ÿ#Üæ
Üß÷
ã«ÛÀä{áiéß«äÈã‡Üß«ä àdèoçÀà‹Ü)á¥èÀà™ã]Û'å™è”çŸì Ü=á‡éå™ì ä  è Wÿ#Üì ì.ô 



;èµýIê àdÝÀä èÀä à
è¾ã«ÛÀÜo۔ådß«çÀÜá—ãNã‡ÛÀä èTã«à 
ÜãWß«ädۋãNä{á`۔å™è'çŸì äÈèݔ߇Ü)éä{á—ä à
è ÝÀß«àdí”ìÈÜý|á`éà
߇߫Üéã‡ì ê ÜædÜ߇ê6ÿiÛÀÜ߇Üdô ƒ âã«ÛÀä{á?ä{áKèÀàdã¹çŸàdè”Ü™÷‹ã‡ÛÀÜ å™ìdà
߇äÈã‡Û”ý ý|å êWèÀàdã#éà
è6ædÜßdÜd÷™àdß¹ý|å êWíÁܯýtޔé4Ûvá—ì à-ÿ#Üß

ã‡Û”ådèäÈãièÀÜÜçÀá#ã«àví'Üdô


dà6àŸç|ÿ¥å êtã‡à¼é4ÛÀÜé4õtã«Û”å-ãiêdà
ÞÀߥå™ìdà
߇äÈã‡Û”ý ä{á&ÿ#àdß«õ6ä è`ä{áKã«à¼é4۔Üé4õtã‡Û”å™ã¹ã«ÛÀÜ`á—à
ì ޟã‡ä àdè|á«å-ã«ä{á ”Ü)á å™ì ì?ã«ÛÀÜ `åd߇Þ'á—Û  =ÞÀÛÀè ƒÚ@ޔé4õdÜß`éàdè'çŸäÈã‡ä àdè”á âöà
ßNã«ÛÀÜ|ÝÀß«ä ý|å™ì?ÝÀß«àdíÀì ÜýT÷á‡äÈè'éܼã‡Û”Üá‡Üoå™ß«Ü|èÀÜ)éÜá«á«å™ß«ê å™è'çÃá— Þ
véä Üè‹ã±éà
è”çŸäÈã‡ä à
è”á¥ã‡Û'å-ã¯ã«ÛÀܼá—à
ìÈÞÀã‡ä àdèí'ÜtàdÝÀã‡ä ý|å™ì.ô]Ú¥ÛÀÜ `Ú éàdè”çŸäÈã«äÈà
è”á]ådß‡Ü 0 !6áôIòB
ü‹ó ã‡Û”߇à
 Þ dÛ ò.ú 
ó÷?ÿiä:ã«ÛµçŸà™ãIÝÀ߇àŸçŸÞ'éã«áWí'Üãÿ¹ÜÜèOç”å-ã«åÃædÜéã‡à
ß«á`ß«Üݔì å
éÜ)ç¾í6êVõdÜ߇èÀÜì{á=ÿiÛÀÜß«Üæ
Üß`ã«ÛÀÜê å™Ý”Ý'Ü)å™ß]òöè”à™ã‡. Ü [ýIÞ'áãKíÁÜiÜëŸÝ”ådè”çŸÜç¼å
áäÈè 0 !”ô¥@ò B‹ü
ó ”ß4áã÷‹á‡ä è”éÜ ä{áèÀàdãKä è 
ÜèÀÜß«ådì6ã‡ÛÀÜiý|ådÝÀÝÀä  è  à™âNåÅÝÁàdä è‹ã|ä è #óô ڥۋÞ'ávã‡à é4ÛÀÜé4õOã‡ÛÀ

Ü `Ú éàdè”çŸäÈã«äÈà
è”á÷¹äÈãoä{á|á‡Þ
|éä Üè‹ãvã‡à é4ÛÀÜé4õOã‡Û”å™ãoã‡Û”Ü "=±á«å-ã«ä á—âöê S 1 =1 I÷?ã‡Û”å™ãWã«ÛÀÜFÜ !‹Þ”å™ì äÈãê¾éà
è”á—ã‡ß4å™ä è‹ãvò B
ù‹ó`ÛÀà
ì ç”á÷ã‡Û”å™ãtådìÈì¹Ý'à
ä è
ã4á`âöà
ßWÿiÛÀä{é4Û S 1: = A

á‡å™ã‡ä{áâö, ê 01!'ôWò2úÀø)ó]ÿiä:ã«Û  = H SÀ÷Áå™è'çTã‡Û”å™ã ådìÈìÝ'à
äÈè‹ã4á]ÿiäÈã‡Û = H á«å-ã‡ä{á—âöê 0 !”ôWò.úŸø ó âöàdß`á‡àdývÜ  = 6SŸôvÚ¥ÛÀÜá‡Üvå™ß«Ütá‡Þ
|éä Üè‹ã`éà
è”çŸäÈã‡ä àdè'áiâöàdßNådì ì@ã«ÛÀÜ `Ú éà
è”çŸäÈã‡ä à
è”á]ã«àTÛÀàdì{ç  èÀà™ã«Ü ã‡Û'å-ãNí6ê çŸà
ä  è oã«ÛÀä{á=ÿ¹Ü¼èÀÜædÜß`۔å ædÜtã‡àTÜë6ݔìÈä{éä:ã«ì êéàdývÝÀޟã«Ütã‡ÛÀÜ  = àd ß = ÷å™ìÈã«ÛÀàd Þ 
Û çŸà
ä  è á‡àTä{á ã‡ß«ä æ6ä ådì.ô




( $ ICK

576 M    #



&   6   *6   #



  .-    6*6 M6    *6   #

î6ÞÀݔÝ'à
ߗã¯æ
Üéã‡àd߯ý|ådé4ÛÀä èÀÜ)á¥Û”å ædÜ ã‡Û”ÜNâöàdì ì à-ÿiä  è Wæ
Üß«êFá—ã‡ß«ä õ‹ä  è tÝÀß«àdÝÁÜ߇ãê™ô #à™ã«ÛFã«ß«ådä èÀä èvå™è”çã«Üá—ã âöÞÀè”éã‡ä àdè'á]çŸÜÝÁÜè'çUàdèã‡Û”ÜWçÀå-ã4åvàdè”ìÈê|ã‡ÛÀß«àdÞ
Ûã‡ÛÀÜNõ
Üß«èÀÜìrâöÞÀè”éã«ä àdè”á  ò<,= D < )óô 0&ædÜèFã«ÛÀàdÞdÛUäÈã éà
߇߫Üá‡ÝÁàdè”çÀá±ã«àTåTçŸàdã`ÝÀ߇àŸçŸÞ'éãNä èïåTá‡Ý”ådéÜIàdâ¥çÀäÈývÜè”á‡äÈà
è ÷@ÿi۔Üß«Ü éå™èïí'Üvæ
Üß«êUì ådß 
ÜWà
ß ä  è ”èÀäÈã«Ü™÷6ã‡ÛÀÜNéàdývÝÀì Üë6äÈãêtà™â?éà
ývÝÀޟã‡ä  è   éå™èíÁÜ=â,å™ß]á‡ý|å™
ì ì Üßô Àà
ßiÜëÀå™ývÝÀì Üd÷
âöà
ßiõdÜß«èÀÜì á¹à™âã‡Û”Ü  âöàdß«ý  H ò <>=9 < )ó  ÷‹å=çŸàdãKÝÀ߇àŸçŸÞ'éã?ä è ÿ#àdÞÀì{çNß« Ü !‹ÞÀä ß«Ü&àdâÁà
ß«çŸÜß     7 #  àdÝÁÜß4å-ã«ä àdè”á÷-ÿiÛÀÜß«Üå
á ã‡Û”ÜNéàdývݔޟã«å™ã‡ä àdè|àdâ  ò <,= D < )ó#ß« Ü !‹ÞÀä ß«Üá¹àdèÀì ê¼ò ó#à
Ý'Üß«å™ã‡ä àdè”á òöß«Üéå™ì ì   ä{á#ã‡ÛÀÜNçŸä ývÜè”á—ä à
èvè à™ â ã‡Û”ÜWçÀå-ã4å
ó+ ô ƒã¯ä{á#ã«ÛÀä{á¥â,ådéã]ã‡Û”å™ã±å™ì ì à-ÿ]á¹Þ”áiã«àoéà
è”á ã«ß‡Þ'éã]Û6ê6Ý'Ü߇ÝÀì{ådèÀÜá#ä è㇠ÛÀÜ)á—ÜNæ
Üß«ê|ÛÀä dÛUçŸä ývÜ á‡äÈà
è”å™ì?á‡Ý”ådéÜá=ê
ÜãNáã«ä ìÈìíÁÜvì Üâ{ãNÿiäÈã‡ÛVåFã‡ß4ådéã4å™íÀì ܼéà
ývÝÀޟã«å™ã‡ä àdèrô¼Ú¥Û6ޔáWîŸð±ñÃáNéä ß«éÞÀýIæ
Üè‹ã±íÁà™ã«Û âöàdß«ý|á]àdâ?ã‡ÛÀÜ éÞÀß«á‡ÜNàdâ&çŸä ývÜè”á‡ä àdè”ådìÈäÈã ê ã«ÛÀÜIݔ߇à
ìÈäÈâöÜß«å™ã‡ä àdè|àdâKݔå™ß4å™ývÜã‡Üß«á]éå™Þ”á‡ä  è vä è‹ã‡ß4ådéã4å™íÀì Ü éà
ývÝÀì ÜëŸäÈãê™÷6å™è”çFã‡Û”Ü`ÝÀ߇à
ì ä:âöÜß«å™ã‡ä àdè¼àdâ@ݔådß«ådývÜã‡Üß«á#éådޔá‡äÈ è Ià-æ
Üß Àã—ã«äÈ è Àô

((   * *,+ ”àdßéà
è”éß«Üã«ÜèÀÜ)á‡á÷-ÿ¹Ü¹ÿiäÈì ì dä ædÜKß«Üá‡ÞÀìÈã«árâöàdßã‡Û”Ü#éàdývÝÀޟã4å-ã«äÈà
è”å™ì
éàdývݔìÈÜëŸä:ãê¯à™â àdè”ܯã«ÛÀÜ=ã‡ÛÀÜ`ådí'à-æ
Ü]ã‡ß4å™ä èÀä èIå™ìdà
߇äÈã‡Û”ý|á]ò*#ÞÀè”é4Û `ådޟâöý|å™è'ó#  ò `ådޟâöý|å™è÷ø)ùdùdü‹ó4ô?Ú¥ÛÀÜá‡Ü=߇Ü)á—Þ”ì:ã4á ådá«á‡ÞÀývÜiã‡Û'å-ã#çÀä 8Üß«Üè‹ã¥áã«ß«å™ã‡Ü 
ä Üá?å™ß«Ü¯Þ'á—Ü)çvä èoçŸä ÁÜ߇Üè‹ã¹á—äÈã«Þ”å-ã«äÈà
è”áô>
Vܱéà
è”á‡ä çÀÜßKã«ÛÀܯݔ߇à
íÀì Üý à™â ã‡ß4å™ä èÀä  è Wàdè ޔáã]à
èÀÜ é4Û6ÞÀèÀ õ Fò,á‡ÜÜ=í'Üì à-ÿ¯ó4ô
‹å™ä èoì Üã :í'Ü ã«ÛÀÜ`è6ÞÀýIíÁÜߥàdâã‡ß4å™ä èÀä  è WÝ'à
äÈè‹ã4á÷ 

ã‡Û”ÜNè6ÞÀýIíÁÜ߯à™â?á—ÞÀݔÝ'à
ߗã¯æ
Üéã‡àdß4á`ò2îŸð±á4ó4÷å™è”ç ã‡ÛÀÜWçÀäÈývÜè”á‡äÈà
è|à™âã«ÛÀÜNä èÀÝÀޟã çÀå-ã4åŸô ;èTã‡ÛÀÜWéådá‡Ü  ÿiÛÀÜ߇ÜNývà‹áã±îŸð±á å™ß«ÜNèÀà™ã å-ã]ã«ÛÀÜWÞÀݔÝ'Üß±íÁàdÞÀè'ç8÷8ådè”ç  = :A%A ø™÷Áã‡Û”ÜWè6ÞÀýtí'Üß]à™â?àdÝÁÜß4å-ã«ä àdè” á ä{á¼ò Q 8 7Dò   ó : 7 

:.ó4ô ƒâä è”á—ã‡Ü)ådç  F= :  ø™÷6ã«ÛÀÜè Ãä{á¼ò Q 8 7  +: 7 

:.ó¯ò,í”ådá‡ä{éå™ì ì ê   í6êÃáã4å™ß‡ã‡ä  è |ä èUã«ÛÀÜtäÈè‹ã«Üß«äÈà
ßià™âKã«ÛÀÜIâöÜådá‡ä íÀì ÜNß«Ü dä à
è'ó4ô Àà
߯ã«ÛÀÜtéådá‡ÜWÿiÛÀÜ߇ÜIývà
á—ã=î6ð á=å™ß«ÜIå-ã ã‡Û”Ü ÞÀÝÀÝÁÜß=íÁàdÞÀè'ç8÷@å™è'ç  =$: A%A ød÷ã‡ÛÀÜè µä{á ¼ò Q  7 

 :2óô äÈè'å™ì ì ê™÷8ä:âKývà‹áãNî6ð á åd߇Ütå-ã=ã‡Û”Ü ÞÀÝÀÝÁÜ߯í'à
ÞÀè”ç8÷'ådè”ç  =$: ød÷Àÿ¹Ü=۔å æ
Ü ¾à™â ¼ ò  :  ó4ô  

”àdßNì{å™ßdÜ߱ݔ߇à
íÀì Üý|á÷Áãÿ#àÃçŸÜ)éàdývÝÁà
á‡äÈã‡ä àdèÃådìdàdß«äÈã‡ÛÀý|á¯Û”å ædÜtíÁÜÜèÅݔ߇à
Ý'à‹á—Ü)çÃã«àÃçÀå-ã«Ü™ô ;èïã‡Û”Ü é4Û6ÞÀè”õ‹ä è  oývÜã‡ÛÀàŸç ò¹  à‹á—Üß÷ % ÞÀêdà
èµå™è”çïðKådÝÀèÀä õÁ÷#ø)ùdù 
ó4÷àdèÀÜoá—ã«ådߗã4á`ÿiäÈã‡ÛÅåÃá‡ý|å™ì ì.÷å™ß«íÀäÈã‡ß4å™ß«ê

 á‡ÞÀí”á‡ÜãKà™âÁã‡Û”Ü]çÀå-ã4å`å™è”çWã‡ß4å™ä è”áà
èWã«Û”å-ãô?ڥ۔Üi߇Ü)áãKàdâ”ã‡Û”Üiã‡ß4å™ä èÀä è çÀå-ã4å=ä á@ã«Üá—ã‡Üç¼à
èWã«ÛÀÜ]ß«Üá‡ÞÀìÈã‡ä è éì{å
á‡á‡ä ”Üß÷‹ådè”çFåtì ä{áã¥à™â@ã‡ÛÀÜ`Ü߇߫àdß4á¹ä áiéàdè”á—ã‡ß«Þ”éã‡Üç÷”á—à
ߗã«Üç|í6ê|۔à-ÿ â,å™ßià
èoã«ÛÀÜ`ÿiß«àd è ¼á—ä{çŸÜ à™âã‡Û”Ü ý|å™ß dä èvã«ÛÀÜêì ä Ütò,ä2ô ܙô#ÛÀà-ÿgÜ dß«Ü 
ä àdޔá‡ì êWã«ÛÀÜ `Ú éàdè”çÀä:ã«ä àdè”á¥å™ß«Ü=æ6äÈà
ì{å-ã‡Ü)ç”ó4ô&ڥ۔ÜNèÀÜë6ã¯é4Û6ÞÀèÀõFä{á éà
è”á—ã‡ß«Þ”éã«Üçtâöß«àdý ã«ÛÀÜ 'ß«á—ã  à™â8ã‡Û”Üá‡Ü™÷ŸéàdýtíÀä èÀÜ)çWÿiäÈã‡Ûtã«ÛÀ Ü  |á‡ÞÀÝÀÝÁàd߇ã&ædÜ)éã‡à
ß«á?ådì ߇Ü)ådçŸê`âöà
ÞÀè”ç8÷ ÿiÛÀÜß‡Ü  7  ä{á]çŸÜéä{çŸÜçÛÀÜÞÀ߇ä{á—ã‡ä{éådì ìÈêTò,åoé4ۋޔèÀõFá‡ä Ü=ã‡Û'å-ã¯ä{á]ådìÈì à-ÿ#Üç¼ã«/ à dß«à-ÿ ã«à6à !‹ÞÀä{é4õ‹ì êvà
ß ã‡à6àoá‡ì à-ÿiì êvÿiäÈì ìrß«Üá‡ÞÀìÈã]ä èUá—ì à-ÿ à-æ
Üß4å™ì ìréà
è6ædÜß dÜè”éÜ ó4ô  à™ã«ÜNã«Û”å-ã¯æ
Üéã‡àdß4á]éå™èUíÁÜWçŸß«àdÝÀÝÁÜçâöß«àdý åvé4Û6ÞÀèÀõÁ÷”å™è'çoã«Û”å-ã á—ÞÀݔÝ'à
ߗã]æ
Üéã«àdß4á#äÈèTà
èÀÜ`é4ۋޔèÀõoý|å êvèÀàdã¯å™ÝÀÝÁÜådßiä èFã«ÛÀÜ ”è”ådìá‡àdì ޟã‡ä à
èôKÚ¥ÛÀä{á ÝÀß«àŸéÜá«á#ä{áiéàdè‹ã«äÈè6ÞÀÜ)ç|ÞÀè‹ã«äÈìrådì ì8ç”å-ã«å¼ÝÁàdä è‹ã«á¥ådß‡Ü âöàdޔè”çFã‡àvá‡å™ã‡ä{á—âöê¼ã‡ÛÀÜ `Ú éàdè'çŸäÈã‡ä àdè”áô Ú¥ÛÀÜ¥å™íÁà-ædÜ&ývÜã‡Û”à6çNß‡Ü !
ޔäÈß«Üáã‡Û”å™ãã‡Û”ܹè6ÞÀýtí'Üßàdâ”á—ÞÀݔÝ'à
ߗãædÜ)éã‡à
ß«

á  í'Ü#á—ý|ådì ì™ÜèÀà
ÞdÛIá—à¯ã‡Û”å™ã å ¯Üá«á—ä{ådèvà™â@á‡äÜ í6ê ÿiä ì ìÀã¥ä è|ývÜývà
߇êdô
èTå™ìÈã«Üß«è”å-ã«äÈæ
ÜiçŸÜéàdývÝÁà
á‡ä:ã«ä àdè¼å™ìdà
߇äÈã‡Û”ý ۔å
á  íÁÜÜèTÝÀß«àdÝÁà
á‡ÜçFÿiÛÀä{é4Ûà-ædÜß4éà
ývÜáKã‡ÛÀä{áiì ä ýväÈã«å-ã«ä àdèUò$ á‡ÞÀè”åÀ÷Àß«ÜÞÀè”çTådè”ç % äÈß«à
á‡ä.÷Áø)ùdù í'óô

ådäÈèr÷ ä èWã«ÛÀä{áKå™ì 
àdß«ä:ã«ÛÀýT÷à
èÀì êWå`á‡ý|å™ì ì
ÝÁàd߇ã‡ä à
èIàdâ'ã«ÛÀÜ¥ã‡ß4å™ä èÀä  è  çÀå-ã4å=ä áã«ß«ådä èÀÜçWàdèvå-ã&å 
äÈæ
ÜèWã‡ä ývܙ÷dådè”ç âöÞÀ߇ã‡ÛÀÜ߇ývà
߇Üd÷dàdè”ìÈêtåWá‡ÞÀí”á‡Üã&àdâã‡ÛÀÜ á—Þ”ÝÀÝ'à
ߗã¹ædÜéã‡à
ß«á?è”ÜÜç|í'ܱäÈè¼ã«ÛÀ Ü ÿ¹à
߇õ6ä  è =á—ÜKã |ò,ä.ô Üdô@ã«Û”å-ã¥á—Üã à™âÝÁàdä è‹ã«á¥ÿiÛÀà‹á— Ü 1 á¯å™ß«Ü=ådì ìÈà-ÿ#Üçvã‡àvæ-å™ß«êÀó4ô¹Ú¥ÛÀä{á¥ývÜã‡ÛÀàŸçF۔ådáiíÁÜÜèUá—ÛÀà-ÿièã«àví'ÜNådíÀì ܱã«àvÜå
á—ä ì ê ۔ådè”çŸì Ü|åUÝÀß«àdíÀì ÜýfÿiäÈã‡Û ødø SŸ÷ S S S|ã«ß«ådäÈè”äÈ è FÝÁàdä è‹ã«áNådè”ç ø S SŸ÷ S S SFá‡ÞÀÝÀÝÁàd߇ãWædÜ)éã«àdß4á ô ]à-ÿ#Üæ
Üß÷@äÈã ýtޔáã¹í'Ü èÀà™ã«Üç|ã‡Û”å™ã&ã‡Û”Ü á‡Ý'ÜÜçvàdârã‡Û”ä á¹å™ÝÀݔ߇à‹ådé4Ûvß«Üì ä Üáàdèoý|å™è6êIàdâã‡ÛÀÜ=á‡ÞÀÝÀÝÁàd߇ã¹ædÜ)éã«àdß4áK۔å æ6ä è éà
߇߫Üá‡ÝÁàdè”çŸä  è  å dß4å™ è 
ܯýtÞÀìÈã‡ä ÝÀì ä Üß4, á = å™ã¥ã‡ÛÀÜ`ÞÀݔÝ'Üß]í'à
ÞÀè”ç8÷ = H Iô Ú¥ÛÀÜ)á—ܹã‡ß4å™ä èÀä  è ]ådì dàdß«äÈã‡ÛÀý|á8ý|å ê±ã«ådõdÜ¥ådçŸæ-ådè
ã4å 
ÜKà™â”Ý'å™ß4å™ì ì Üì™ÝÀß«àŸéÜ)á‡á‡ä  è iä èIá‡ÜædÜß«ådìdÿ¥å êŸáô ä ß4áã÷ ™å ì ìKÜì ÜývÜè‹ã4á=à™âiã‡Û”Ü ]Ü)á‡á‡ä{å™èVäÈã«á‡ÜìÈâ¥éå™èÅíÁÜFéàdývÝÀޟã«ÜçVá‡äÈýtÞÀìÈã«ådèÀÜà
ޔá‡ìÈêdôî6Ü)éàdè'ç8÷Ü)ådé4ÛÅÜìÈÜývÜè‹ã à™â{ã«Üèïß‡Ü !‹ÞÀä ߇Ü)áiã‡Û”ܼéàdývÝÀÞÀã«å-ã«ä àdèUà™â¥çŸà™ã`ÝÀß«àŸçŸÞ”éã«á à™â¹ã‡ß4å™ä èÀä è|çÀå™ã«åŸ÷@ÿiÛÀä{é4Û¾éà
ÞÀì{ç¾å™ì{á‡à|íÁÜtÝ'å™ß å™ì ì Üì ä Ü)ç8ôIÚ¥ÛÀä ß4ç8÷ã«ÛÀÜ|éà
ývÝÀޟã4å-ã‡ä à
è¾à™â¹ã‡ÛÀÜoàd í Ü)éã‡ä æ
ÜWâöÞÀè'éã‡ä à
è÷@à
ß dß4ådçŸä Üè‹ã÷rÿiÛÀä{é4Û ä{á=åUá—ÝÁÜÜ)ç íÁà™ã—ã«ì ÜèÀÜ)é4õ'÷-éådè`í'ܹݔå™ß4å™ì ì Üì ä Ü)çNòöäÈãß« Ü !‹ÞÀä ß«Üáåiý|å™ã‡ß«äÈë¯ýIޔì:ã«ä ÝÀì ä éå-ã«äÈà
è'ó4ô äÈè'å™ì ì ê™÷à
èÀÜ&éådè`Üè6æ6ä{á—ä àdè ݔådß«ådìÈì ÜìÈä äÈ è Iå-ã`åFÛÀä dÛÀÜß±ì Üæ
Üì.÷”âöà
ß=ÜëÀå™ývÝÀì ÜNí6êTã«ß«ådäÈè”äÈ è |àdèïçŸä ÁÜ߇Üè
ã=é4Û6ÞÀèÀõŸá=á—ä ýtÞÀìÈã«å™è”ÜàdÞ'á—ì ê™ô îŸé4ÛÀÜývÜá#á—Þ'é4Ûå
á&ã«ÛÀÜá‡Ü™÷”éàdýtíÀä èÀÜç|ÿiä:ã«Û|ã«ÛÀÜ`çŸÜéàdývÝÁà
á‡ä:ã«ä àdèvå™ì dà
߇äÈã«ÛÀý[àdâ#ò $ á—ÞÀè'åŸ ÷ Àß«ÜÞÀè'çFådè”ç % ä ߇à‹á—ä.÷‹øùdù  í'ó÷-ÿiäÈì ì
í'Ü¥èÀÜÜçŸÜ)ç`ã‡à±ý|å™õ
ÜKædÜ߇ê=ì{å™ß dÜKݔ߇à
íÀì Üý|áKò,ä2ô ܙô .%. ø S SŸ÷ S S Siá‡ÞÀÝÀÝÁàd߇ãædÜéã‡à
ß«á÷ ÿiäÈã‡Ûý|å™è6êvèÀàdã¯å-ãií'à
ÞÀè”ç”ó÷”ã«ß«å
éã«ådíÀì ܙô

($ M G  *,+ ;è ã‡Üá—ãTÝÀ۔ådá‡Ü™÷±àdèÀÜ ýIޔá—ãá‡ä ývÝÀì êOÜæ ådì ޔå-ã«Ü 01!'ôyò*Àø ó4÷ ÿiÛÀä{é4Û ÿiäÈì ì±ß‡Ü !
ޔäÈß«Ü  ò   ó#à
Ý'Üß«å™ã‡ä àdè'á÷‹ÿiÛÀÜß«Ü ä{á&ã«ÛÀÜ=è6ÞÀýIíÁÜß#à™âàdÝÁÜß4å-ã«äÈà
è”á¹ß‡Ü !
ޔäÈß«Üçvã‡àtÜæ-å™ì ޔå™ã‡Ü±ã‡ÛÀÜ=õdÜ߇è”Üì.ô ¼ Àà
ß±çŸàdã±ÝÀß«àŸçŸÞ”éã å™è'çQ  DõdÜß«èÀÜì á÷  ä{á ¼ò  ó÷8ã‡ÛÀÜtçŸä ývÜè”á‡ä àdèà™â?ã‡Û”ÜIçÀå™ã«åvæ
Üéã«àdß4áô

ådäÈèr÷  íÁà™ã‡Ûã«ÛÀÜ`Üæ-å™ì ޔå™ã‡ä àdèoà™âã‡ÛÀÜ`õ
Üß«èÀÜìå™è'çFàdâã«ÛÀÜNá—Þ”ý å™ß«Ü ÛÀädÛÀì êvݔådß«ådìÈì ÜìÈä)å™íÀìÈÜ¥ÝÀ߇àŸéÜçŸÞÀß«Üáô

;èFã‡ÛÀÜ=å™í'á—Üè”éÜ à™âÝ'å™ß4å™ì ì ÜìÀ۔ådß«çÀÿ#åd߇Üd÷
àdè”Ü éå™èFá—ã‡ä ì ì'á‡ÝÁÜÜçoÞÀÝFã‡Üá—ã¥ÝÀ۔ådá‡Ü í‹ê|åWì{å™ßdÜiâ,å
éã‡à
ß÷Àå
á çŸÜ)á‡é߇ä íÁÜç|ä èUî6Üéã‡ä àdèTùŸô


z q



ok Ù')*6Ò Ù2Õ'Ò Õ+(±p×&Õ'Ô ÓC6ØdÓÕ'Ô-46#ÀØ7Ù,Ò"

"

VÜvèÀà-ÿ á‡ÛÀà-ÿ ã«Û”å-ã`ã«ÛÀܼð  çŸä ývÜè”á—ä à
èUà™â¥îŸð±ñÃáNéå™èïí'Üvæ
Üß«êUì{å™ßdÜFò,Üæ
ÜèïäÈè”èÀäÈã‡Ü ó4ô
ïÜvÿiä ì ì ã‡Û”Üè ÜëŸÝÀì àdܾ߫á—ÜædÜß«ådì±å™ßdޔývÜè‹ã«áFådáoã‡àDÿiۋêd÷±ä è á‡ÝÀäÈã‡Ü à™â`ã‡Û”ä á÷=îŸð±ñÃáޔá‡Þ”å™ì ì êOÜëŸÛÀä íÀäÈã dà6àŸç dÜèÀÜß4å™ì ä å™ã‡ä àdèVÝÁÜ߇âöàdß«ý|å™è”éܙô ]à-ÿ#Üæ
ÜßtäÈãoá—ÛÀà
ÞÀì{çµíÁÜÃÜývݔ۔ådá‡äÜçÅã‡Û”å™ãvã‡ÛÀÜ)á—ÜÃåd߇ÜTÜá«á‡Üè‹ã‡ä{å™ì ì ê ÝÀì{å™Þ'á—ä íÀä ì äÈãêïådß 
ÞÀývÜè‹ã«áô ¹ÞÀ߫߇Üè
ã«ì êÅã‡ÛÀÜß‡Ü ÜëŸä{áã4ávèÀàÅã«ÛÀÜà
߇êDÿiÛÀä{é4Û +F$      M M Gtã‡Û”å™ãå 
äÈæ
Üè â,å™ývä ì ê¼à™â?î6ð±ñÃá]ÿiäÈì ì8۔å æ
ܱÛÀä 
Ûå
ééޔ߫å
éêvàdèTå dä ædÜèoÝÀß«àdí”ìÈÜýTô



VÜIÿiä ì ì@éå™ì ìådè‹êTõ
Üß«èÀÜìã‡Û”å™ã=á«å-ã«ä á ”Ü)á]ñUÜß4éÜß$á¯éàdè”çÀä:ã«ä àdèUå|ÝÁà
á‡äÈã‡ä ædÜ`õ
Üß«èÀÜì.÷Áådè”çTã«ÛÀÜtéàd߫߇Ü á‡Ý'à
è”çŸä è¯á‡Ý”ådéÜ ã‡ÛÀÜ#Üýtí'Ü)çÀçŸä è¯á‡Ý”å
éܙô
ïÜ#ÿiä ì ì
å™ì{á‡à¯éådì ì
å™è6ê=ÜýtíÁÜçÀçŸä è±á—Ý”å
éܹÿiä:ã«ÛNývä èÀä ý|å™ì çŸä ývÜè'á—ä àdè|âöàdß]å dä ædÜèoõ
Üß«èÀÜìåQ4ývä èÀä ý|å™ì'Üýtí'Ü)çÀçŸä è¼á—Ý'ådéÜŸô"
ïÜ`Û'å ædÜ ã‡ÛÀÜ=âöà
ì ìÈà-ÿiä è




    8M   M  5AI G  *" M  M M6 /  *  I M G)5AI7CG  I  . K *  K  66M KCM2*,+3G5  M 1  M  $ M  ( KNM GI I $ MJI M G)5 I(7(*%+G $ 55 I(  " M   I( K    * M /  M M $ M M I 5AM   6  # *      G  6*6*I/M2  I )  M  6*6 "  6 $AM G   G F Vò ó>7 ø  





ƒâã‡Û”ܯývä èÀä ý|å™ì6ÜýtíÁÜçÀçŸä èNá‡Ý”å
éÜi۔å
á&çŸä ývÜè'á—ä àdè

 ÷‹ã‡ÛÀÜè  ÝÁàdä è‹ã«á&äÈètã«ÛÀܯä ý|å 
Ü¥à™â é å™è|íÁܱâöàdÞÀè”çoÿiÛÀà
á‡Ü¯ÝÁà
á‡äÈã‡ä àdè¼æ
Üéã«àdß4áKä è Yåd߇ܱì äÈè”Üå™ß«ì ê`ä è”çŸÜÝÁÜè'çŸÜè‹ãô Àß«àdý Ú¥ÛÀÜà
߇Üýød÷#ã«ÛÀÜá‡ÜUædÜ)éã«àdß4ávéå™è íÁܾá‡Û”å-ã‡ã‡Üß«ÜçDí6êDۋê6ÝÁÜß«ÝÀì{å™èÀÜ)átäÈè Uô Ú¥Û6ޔáví6êOÜäÈã‡ÛÀÜß ß«Üá—ã‡ß«ä éã‡ä èàdÞÀß4á‡Üì ædÜ)á ã‡à î6ð ñÃáWâöàdßNã«ÛÀÜoá‡Üݔådß«ådíÀì Üvéådá‡Ü¾ò.î6Üéã‡ä àdè EŸô ø)ó÷?àdß`âöà
ßWÿiÛÀä{é4Ûïã‡ÛÀÜFÜ߫߇à
ß ÝÁÜè”ådì:ãê ä{á#å™ì ì à-ÿ¹Ü)çNã«àWã4å™õ
ܱå™ì ìÁæ-å™ì ÞÀÜá±ò,á‡àWã«Û”å-ã÷ÀäÈâã‡ÛÀÜ Ý'à
ä è
ã4á¹åd߇ܱì äÈè”Üå™ß«ì êWá‡Üݔådß«ådíÀì ܙ÷‹å éådè íÁÜIâöà
ÞÀè”çVá‡Þ”é4Û¾ã«Û”å-ã=ã«ÛÀÜvá—à
ìÈÞÀã‡ä àdè¾çÀà‹Ü)á=ä è”çŸÜÜ)ç¾á‡Üݔådß«å™ã‡ÜIã‡ÛÀÜýoó4÷rã«ÛÀÜtâ,å™ývä ì êFàdâ#á‡ÞÀÝÀÝÁàd߇ã`ædÜéã‡à
ß ý|ådé4۔äÈè”ÜáÿiäÈã‡Û¼õdÜ߇èÀÜì yéådèvå™ì{á—à`á‡Û”å-ã‡ã‡Üß?ã‡ÛÀÜ)á—ܯÝ'à
äÈè‹ã4á÷
ådè”ç¼ÛÀÜè”éÜi۔å
áKð OçŸä ývÜè”á‡ä àdè 
7ïødô u=Ô-ÕrÕE(

DÞÀè”çŸÜß&ã«ÛÀܱý|ådÝÀÝÀä è



Üã$áiì à6àdõ|å™ã#ãÿ#à¼ÜëÀå™ývݔìÈÜ)áô 

M  KM G I    

¹àdè'á—ä{çŸÜߥådèTîŸð±ñ Oò'<

#

DK< 

ó1H

 I-4I(6 # ILK4  6 M M6 G

ÿiäÈã‡ÛÛÀàdývàdÜè”ÜàdÞ'á&ÝÁàdì ê6èÀàdývä{ådì”õdÜ߇è”Üì.÷Àådéã«ä èIà
èç”å-ã«åtä è m   ò< #

9<  ó 

D

<

#

D@< 

?

m- 

ò -ü‹ó


áNä èVã‡ÛÀÜFéådá‡ÜvÿiÛÀÜè   H Tå™è”çïã«ÛÀÜ|õdÜ߇è”ÜìKä{á !‹Þ”å
çŸß«å™ã‡ä{éTò2î6Ü)éã«äÈà
è B‹ó4÷?à
èÀÜ|éådèÅÜëŸÝÀì ä{éäÈã‡ì ê é à
è”á—ã‡ß«Þ”éãKã«ÛÀܯý|ådÝ ¯ô rÜã‡ã‡ä è
7=HB) # =,)  =‡÷Ÿá—à`ã«Û”å-ã Oò'< # D <  ó H ò
# 7 9:9;9 7
ó  ÷‹ÿ#ܱá‡ÜÜ]ã‡Û”å™ã  Üå
é4ÛçÀäÈývÜè”á‡äÈà
èvà™â éàd߫߇Ü)á—ÝÁàdè'çÀá¹ã‡àvåIã‡Üß«ý ÿiäÈã‡Û,dä æ
ÜèoÝÁà-ÿ#Üß4á¹àdâã«ÛÀÜ
7=äÈèFã‡Û”Ü`ÜëŸÝ”å™è'á—ä àdèoà™â Å ô ;èUâ,ådéã±äÈâKÿ#ÜWé4ÛÀà6à
á‡Ü`ã‡àFì{å™íÁÜìrã«ÛÀÜIéàdývÝÁàdèÀÜè‹ã«á]àdâ  ò<@ó]ä èTã‡ÛÀä{á]ý|ådèÀèÀÜß÷8ÿ¹ÜIéå™è¾ÜëŸÝÀì ä{éäÈã‡ì ê ÿiß«ä:ã«Ü¯ã«ÛÀÜ`ý|å™Ý”ÝÀä  è Wâöà
ß]å™è6 ê Uå™è'ç 





 


  

ò<@ó H

9 ;



)  ) 9:9;9)     :9  #   ;9;9   /D #   D

(   = = ) # H

,D  = 

Ú¥ÛÀä{á¥ì Üå
çÀá¹ã‡à

ò -ù‹ó S




   * * MG)5   M * /  *  $ M ) 6 *" M  G KMAG2I(      M  H m    * M  KM  G I  I $ M K.*  K  6 MJK&M2(*%+NG)5  M  I  ILKNI!+ M MI $ G 5AI6 # ICK.  6  M M 6 G I  M +( M M,  Oò'< # D <  ó H ò< # 9 <  ó  D < # D<  ? m    G      7 #  

 

ò2Ú¥ÛÀÜvÝÀß«à‹àdâ&ä{á=ä èÃã‡Û”Ü
ÝÀÝÁÜè”çŸäÈë”ó4ô|Ú¥Û6ޔá ã«ÛÀÜvð  çŸä ývÜè”á—ä à
èUà™â¥îŸð±ñÃá=ÿiäÈã‡Û ã‡Û”Üá‡Ütõ
Üß«èÀÜì{á±ä{á 7 ødô
á¥èÀàdã‡ÜçTådí'à-æ
ܙ÷6ã‡ÛÀä{á 
Üã«á¥æ
Üß«ê|ì{å™ßdܱædÜß«ê/!‹ÞÀä{é4õ6ìÈêdô

    7 # 

M  KM G I $%   

 I 

   6

  G  G $    I(   M M6 G


     $IAG M * M%6  GG I  M M  M M6 G  I /  *  Oò< # D <  ó S  GJ< # -<    7  (I @/  *  Oò'
m : 7 M  * M   K.*6 #ID6  G G MGDIAG G  *%+NI4G $ 55AI  "CM  IK    * M G>$ G*,+ * M GJM  M M6 G    (I 8/  *  * MM I 5AM  6  #  G  6*6 I/M2  I    M  6*6 "  6 $AM G   G*    M   KMAGI 





 







u=Ô-ÕrÕE( Ú¥ÛÀÜ`õdÜ߇èÀÜìrý|å-ã«ß‡äÈë8÷  =  1 Oò'< = DK<  ó4÷Áä áiå % ß4å™ý ý|å-ã«ß‡äÈëÃò,å¼ý|å™ã‡ß«ä:ë|àdâçÀà™ã]ÝÀß«àŸçŸÞ”éã4á  á‡Üܵò ¯àdß«è÷Nøù
ü
ú
ó—óIä è Ãô ¹ì Üåd߇ì êÅÿ#ÜÃéådè é4ÛÀà6à
á‡ÜFã«ß«ådä èÀä èïç”å-ã«åÅá‡Þ”é4Û ã«Û”å-ãFådì ìià  ƒçŸä{ådàdè'å™ì Üì ÜývÜè‹ã«á  =)   éå™è¼íÁÜ]ý|ådçÀÜ¥åd߇í”ä:ã«ß«åd߇ä ì ê±á‡ý|å™ì ì.÷™å™è”çtí6êIå
á‡á‡ÞÀývݟã«ä àdèWådìÈìÀçÀä ådà
è”å™ì6Üì ÜývÜè‹ã«á  = )  å™ß«Ü]àd â $tòø)óôڥ۔ܯý|å-ã«ß‡äÈ ë  ä{áKã‡ÛÀÜè|à™ââöÞÀì ìÁß«ådèÀõÁû‹Û”Üè”éÜ]ã‡Û”ܱá‡Üã¹àdâædÜéã‡à
ß«á÷
ÿiÛÀà‹á—Ü çŸà™ã¹ÝÀ߇àŸçŸÞ'éã«á ä è âöàdß« ý ï÷‹åd߇Ü#ì ä èÀÜåd߇ì ê¯ä è”çÀÜÝÁÜè”çŸÜè‹ãi%ò ]à
߇è÷'øùdü‹údóûdÛÀÜè'éܙ÷
í‹êIÚ¥ÛÀÜà
߇Üýþø™÷™ã«ÛÀÜiÝ'à
ä è
ã4áéå™ètíÁÜ á‡Û”å-ã‡ã‡Üß«Üç|í‹êvÛ6ê6ÝÁÜß«ÝÀì{å™èÀÜ)á?ä  è U÷”å™è'çvÛÀÜè'éÜ=å™ì{á—à`í6ê|á‡ÞÀÝÀÝÁàd߇ã¹æ
Üéã«àdß#ý|ådé4ÛÀä èÀÜ)á?ÿiäÈã‡Ûoá‡Þ
|éä Üè
ã«ì ê ì{å™ß dÜ Ü߫߇à
ßiÝ'Üè”å™ìÈãêdô#î6ä è”éÜ=ã«ÛÀä{áiä{á#ã‡ß«ÞÀÜ`âöà
߯å™è6/ ê 'èÀäÈã‡Ü`è6ÞÀýtí'Üßià™âÝ'à
äÈè‹ã4á÷Àã«ÛÀÜNð  çŸä ývÜè'á—ä àdèoà™â ã‡Û”Üá‡ÜNéì{ådá«á—ä ”Üß«á&ä{á¹ä  è 'èÀäÈã‡Ü™ô

 à™ã‡Ü=ã«Û”å-ãiã«ÛÀÜWå
á‡á‡ÞÀývݟã«äÈà
è”á¹äÈèFã‡Û”Ü=ã‡ÛÀÜàdß«ÜýYåd߇Ü`á—ã‡ß«àd è 
Üß#ã‡Û”ådèTèÀÜéÜá«á‡åd߇êÃòöã‡ÛÀÜêoÿ#Üß«Ü=é4۔à
á‡Üè ã‡àý|å™õ
ÜNã«ÛÀÜvéàdè”èÀÜéã‡ä àdèUã«àTß«å
çŸä{å™ìí'ådá‡ä á¯âöÞÀè”éã‡ä àdè'á éì Üå™ßó4ô ;è â,ådéã äÈã=ä{á±à
èÀì êTèÀÜéÜá«á‡åd߇êã‡Û'å-ã : ã‡ß4å™ä èÀä  è tÝÁàdä è‹ã«áiéådèTí'ÜWé4۔à
á‡ÜèUá—Þ'é4ÛFã«Û”å-ãiã«ÛÀÜNß4å™èÀõFà™âã‡ÛÀÜWý|å-ã‡ß«äÈë  =  ä è”é߇Ü)ådá‡Üá¹ÿiäÈã‡ÛÀà
ޟã]ì äÈýväÈã ådá :@ä è”éß«Üå
á—Ü)áô ÀàdߥÜëŸådývÝÀì ܙ÷
âöàdß % å™Þ”á«á‡ä ådè   Åõ
Üß«èÀÜì{á÷dã‡Û”ä á¥éå™èFådì{á—àIí'Ü`ådééà
ývÝÀì ä á‡ÛÀÜ)çFò,ÜædÜè m   ó=í6ê é4ÛÀà6à
á‡ä  âöàdßNã‡ß4å™ä èÀä  è TçÀå-ã4åT߇Ü)áã«ß‡ä{éã«Üç¾ã‡àÃì ä ܼä èVåTíÁàdޔè”çŸÜçÅá‡ÞÀí”á‡ÜãIà™â  è Tá—ý|ådìÈì?ÜèÀàd Þ dÛ   Ãÿiä{ç6ã«Û”áô ¯à-ÿ¹ÜædÜßä è dÜèÀÜß4å™ì6ã«ÛÀÜið µçÀäÈývÜè”á‡äÈà
èWàdâî6ð ñ    éì å
á‡á‡ä ”Üß4áéå™èvéÜߗã4å™ä èÀì ê`íÁÜ ”èÀäÈã«Ü™÷
ådè”çWä è”çÀÜÜç÷-âöàdßKçÀå™ã«å±ß‡Ü)áã«ß‡ä{éã«ÜçNã‡à=ì äÈܹä èIå í'à
ÞÀè”çŸÜ)çIá‡ÞÀí”á‡Üã?àdâ m   ÷‹é4ÛÀà6à‹á—ä  è ]ß«Üá—ã‡ß«ä{éã«äÈà
è”á àdèFã‡Û”Ü   µÿiä{ç6ã‡Û'á¥ä áiå 
à‹àŸçFÿ#å ê¼ã«àvéàdè‹ã«ß‡à
ì8ã‡ÛÀÜ`ð  çŸä ývÜè”á‡ä àdèô

Ú¥ÛÀä{á`éådá‡Ü 
ä ædÜá ޔáWåá‡Üéàdè”çïà
ÝÀÝ'à
ߗã«ÞÀèÀäÈãêTã‡àUݔ߇Ü)á—Üè
ãNåUá‡äÈã‡Þ”å™ã‡ä àdèÃÿi۔Üܼ߫ã‡ÛÀÜoî6ð ñ á—à
ì ޟã‡ä àdè éådèÅí'Üéà
ývÝÀޟã‡Ü)çÅå™è”ådì ê
ã«ä{éå™ì ì ê™÷ÿiÛÀä{é4Ûµådì á‡àUå™ývà
ÞÀè‹ã«á=ã«à åÃá‡Üéàdè”ç8÷Kéàdè”á—ã‡ß«Þ”éã«ä ædܼÝÀß«à‹àdâià™âiã‡Û”Ü Ú¥ÛÀÜàdß«ÜýTô Àà
ßvéàdè'éß«Üã‡ÜèÀÜá«átÿ¹Üÿiä ì ì&ã4å™õdÜFã‡Û”ÜTéådá‡Üoâöà
ß % å™Þ”á«á—ä{ådè*   õdÜ߇èÀÜì{áWàdâ]ã‡ÛÀÜâöà
߇ý O'ò < DK< ó H 7 <  7 <   ô rÜã±Þ”á¯é4ÛÀà6à‹á—Ü ã‡ß4å™ä èÀä è tÝÁàdä è‹ã«á]á‡Þ”é4Ûã‡Û”å™ã]ã‡Û”ÜWá—ý|ådì ìÈÜ)áã¥çŸä{áã4å™è”éÜ #  íÁÜãÿ#ÜÜèå™è6êvݔådäÈß#à™â@Ý'à
äÈè‹ã4á¹ä{á¹ýIÞ'é4Û|ì{å™ß dÜß&ã‡Û”ådè|ã«ÛÀÜ=ÿiä çŸã‡Û ô+¹à
è”á‡ä çÀÜß¹ã‡ÛÀÜ`çŸÜ)éä{á—ä à
è¼âöÞÀè”éã«ä àdè Üæ-ådìÈÞ'å-ã‡Ü)ç|à
èFã«ÛÀÜNá‡ÞÀÝÀÝÁàd߇ã¥æ
Üéã‡àdß   

   



ò  )ó H

( =




"=L = 7  7        7 I

ò2ü S‹ó



Ú¥ÛÀÜNá‡ÞÀý à
èoã«ÛÀÜ`ß«ä 
ۋã#۔ådè”çá—ä{çŸÜ ÿiä ìÈìÁã«ÛÀÜèTíÁÜ ì{ådß 
Üì êtçÀàdývä è”å-ã«Üçví6êvã«ÛÀÜ=ã‡Ü߇ý F"H ÀûÀä èFâ,ådéã ã‡Û”Ü`ß«å™ã‡ä àWàdâã‡Û'å-ãiã‡Ü߇ý ã‡àtã‡ÛÀÜNéàdè‹ã‡ß«ä íÀޟã«äÈà
èvâöß«àdý ã‡Û”Ü`߇Ü)áãià™â@ã‡ÛÀÜNá‡ÞÀýYéå™èí'Ü`ý|å
çŸÜ=å™ß«íÀäÈã‡ß4å™ß«äÈì ê ì{å™ß dÜ=í6êFé4ÛÀà6à‹á—ä  è tã‡Û”ÜNã‡ß4å™ä èÀä ètÝÁàdä è‹ã«á¥ã«à|í'ÜIå™ß«íÀäÈã‡ß4å™ß«äÈì êtâ,å™ß±å™Ý”ådߗãô ;èUà
ß«çÀÜßiã«à ”è”çTã«ÛÀÜtî6ð ñ á‡àdì ޟã‡ä à
è÷&ÿ#ÜTå 
ådäÈè ådá«á—ÞÀývÜFâöàdß¼ã«ÛÀÜUývàdývÜè‹ãIã«Û”å-ã|ÜædÜß«êVã‡ß4å™ä èÀä  è  Ý'à
ä è
ãvíÁÜéàdývÜá¼åVá‡ÞÀÝÀÝÁàd߇ã ædÜ)éã«àdß÷@å™è”çïÿ#Üvÿ¹à
߇õ¾ÿiä:ã«Ûµî6ð±ñÃá`âöàdß`ã«ÛÀÜoá‡Üݔådß«ådíÀì ÜIéå
á—ÜÃò.î6Üéã‡ä àdè EŸô ø)ó¼òöã‡ÛÀÜFá‡ådývÜtådß 
ÞÀývÜè‹ã ÿiä ì ì¹ÛÀà
ì{çÅâöàdß|î6ð ñÃá¼âöà
ßIã«ÛÀÜTèÀàd è ƒá‡Üݔådß«ådíÀì Üoéå
á—ÜäÈâ fäÈ) è 01!'ô @ò B B6óNä{á¼å™ì ì à-ÿ#Üç ã‡àïã«ådõdÜì{å™ß dÜ Üè”àd Þ 
Û æ-å™ì ÞÀÜá4ó4ôþî6ä è”éÜ å™ì ì±ÝÁàdä è‹ã«áFå™ß«Ü¾á‡ÞÀÝÀÝÁàd߇ãFædÜ)éã«àdß4á÷iã«ÛÀÜ Ü !‹Þ”å™ì äÈã‡ä Üá¼ä è 0 !6áô òø S‹ó4÷Iò—ødø ó ÿiä ì ìÛÀà
ì çUâöàdß ã‡ÛÀÜýTô rÜã ã‡Û”Üß«ÜIíÁ Ü  ò  7 ó±Ý'à‹á—äÈã«äÈæ
Ü|òöèÀ Ü 
å™ã‡ä ædÜ óiÝÁàdì{å™ß«äÈãêoÝÁàdä è‹ã«á ô
VÜIâöÞÀߗã«ÛÀÜß  ådá«á‡ÞÀývÜWã«Û”å-ãNådìÈìÝÁà
á‡äÈã‡ä ædÜòöèÀ Ü 
å-ã«ä ædÜ)ó¯ÝÁàdì{å™ß«äÈãêFÝÁàdä è‹ã«á ۔å æ
ÜWã«ÛÀÜ|á‡ådývÜIæ-ådìÈޔ Ü ò ó]âöà
ß=ã‡ÛÀÜä ß % 7   å dß4å™ è 
ܱýtÞÀìÈã‡ä ÝÀì ä Üßô]ò
VÜWÿiä ì ìõ6èÀà-ÿ ã«Û”å-ã¯ã‡ÛÀä{á]å
á‡á‡ÞÀývݟã«äÈà
èFä{áiéàd߫߇Ü)éã]äÈâäÈã¯çŸÜì äÈæ
Üß4á¥å|á—à
ì ޟã‡ä àdè ÿiÛÀä{é4ÛTá‡å™ã‡ä{á 'Üá¥å™ì ìÁã‡ÛÀÜ `Ú éà
è”çŸäÈã‡ä à
è”á¹ådè”çTéà
è”áã«ß«ådä è
ã4á«óôڥ۔Üè 01!6áô¯òøù‹ó4÷'ådÝÀÝÀì ä Üç¼ã«à|å™ì ì8ã‡Û”Ü ã‡ß4å™ä èÀä  è tçÀå-ã4åŸ÷”ådè”ç|ã«ÛÀÜ` Ü !‹Þ”ådì ä:ãêvéàdè”á—ã‡ß4å™ä è‹ã 0 !'ôiò—øü‹ó4÷ÀíÁÜéà
ývÜ

 7  7 7      7 7

Ñø H

`ø

H

H S

ò2üÀø ó

ÿiÛÀä{é4ÛTå™ß«Ü`á‡å™ã‡ä{á 'Üç|í6ê

 H

7 H

H

  7  7 7      7 7       7  7 7  

ò2ü 
ó

Ú¥Û6ޔá÷'á‡äÈè'éܱã‡ÛÀÜ`ß«Üá‡ÞÀìÈã‡ä  è  = åd߇Ü=å™ì{á‡àIÝÁà
á‡äÈã‡ä ædÜd÷6ådì ì'ã«ÛÀÜ `Ú éàdè”çÀä:ã«ä àdè”á#å™è”çTéàdè”á—ã‡ß4å™ä è‹ã«á#å™ß«Ü «á å-ã‡ä{á ”Üç÷Ÿådè”çFÿ¹Ü`ýtޔá—ã#۔å æ
Ü âöàdÞÀè”çoã‡Û”Ü dì àdí”ådìá—à
ìÈÞÀã‡ä àdè¾ò,ÿiäÈã‡Û,Üß«àIã‡ß4å™ä èÀä èWÜ߇߫àdß4á4ó4ôKî6ä è”éÜ ã‡Û”Ü è6ÞÀýtí'Üß]à™â?ã«ß«ådäÈè”äÈètÝÁàdä è‹ã«á÷Áådè”çã‡Û”Üä ß]ì{å™íÁÜì ä èÀ÷Àä{á±å™ß«íÀäÈã‡ß4å™ß«ê™÷”å™è'çã‡ÛÀÜêTåd߇ÜNá‡ÜÝ'å™ß4å-ã‡Ü)çTÿiäÈã‡ÛÀà
ޟã Üß«ß«àdß÷‹ã«ÛÀÜ`ð  çÀäÈývÜè”á‡äÈà
è|ä{á#äÈ è ”èÀäÈã‡Üdô Ú¥ÛÀÜNá‡äÈã‡Þ”å™ã‡ä àdèoä áiá‡ÞÀývý|åd߇äÜ)çvá‡é4۔Üý|å-ã«ä{éå™ì ì êNä è @ä
ÞÀ߇Ütødødô

Ä 4-*.*,"$ ¦ º³ * 9?*,
i;"$ 1, §*.!¥4 r"$)1,+T ;$*2*2" 9ˆ§| 2( " 12.,"  §o$. -!K(d 9 Å ;Á$›“)Ž,…4<<>  3 3 3 0 12.4" " i² " 21 * .23T 1, ǧ /  ¯  12+7 -*r+-0  ?" " 3 12&³´F)" !¥ *2" 3 3 0 3 0 3 ° 3 Â)3 3 043  à-ÿ ÿ#Ü]å™ß«Ü¥ìÈÜâ{ã?ÿiä:ã«Ûvå`áã«ß‡ä õ6ä  è ±éàdè6ÞÀè”çŸß«ÞÀýTô 0KædÜèWã«ÛÀàdÞdÛtã‡ÛÀÜä ß?ð µçŸä ývÜè”á—ä à
èNä{á@ä è'èÀäÈã‡Ü`òöäÈâ ã‡Û”ÜIçÀå™ã«åvä{á¯ådì ìÈà-ÿ#Üç|ã‡àoã«å™õ
ÜWå™ì ìræ-ådìÈޔÜáiä è m  ó4÷î6ð ñ   á±éå™èU۔å æ
ÜNÜëÀéÜì ìÈÜè‹ã¥ÝÁÜ߇âöàdß«ý|å™è'éÜ ò2îÀé4 Û àd ì õdà
ݟâÜã¯å™ì.÷øùd ù ™óô
á‡ä ýväÈì{ådß&áã«àdß«ê|ÛÀà
ì ç”á&âöà
ßiÝ'à
ì ê‹è”àdývä{å™ì8î6ð±ñÃáô]à-ÿgéà
ývÜ 

 z

q

"

€!6Ò‹Ô$#%2Ù'& #ÀÓÙ,ÕÁÒ u‹Ô$(ƒÕ”Ô$)*#ÀÒØ ¾Õ+(±p

04 

;è`ã«ÛÀä{á@î6Ü)éã‡ä à
è`ÿ¹Ü#éàdì ì Üéãæ åd߇ä à
ޔárå™ßdޔývÜè‹ã«áådè”ç`íÁàdÞÀè”ç”áß«Üì{å-ã«ä è#ã‡à]ã«ÛÀÜ1
ÜèÀÜß«ådì ä)å-ã«äÈà
è]ÝÁÜ߇âöàdß ý|å™è'éÜ¥à™âî6ð ñÃáô"
Vܯáã4å™ß‡ãKí6êWÝÀß«Üá‡Üè‹ã‡ä è`  å â,ådýväÈì ê=à™âî6ð ñ ƒ ìÈä õ
Ü#éì å
á‡á‡ä”Üß4áâöàdßKÿiÛÀä{é4Û¼áã«ß‡Þ'éã‡Þ”ß«ådì ß«ä á‡õ¼ývä èÀä ývä) å-ã«äÈà
èWéådèoíÁÜ=߇äd à
߇à
ޔá‡ìÈêWäÈývݔìÈÜývÜè‹ã‡Ü)ç8÷‹ådè”ç|ÿi۔ä é4ÛFÿiä ìÈì d ä ædܱޔá¥á—à
ývܯä è”á‡äd ۋã¹ådá¹ã«à ÿiÛ6ê|ý|å-ëŸä ývä ä è

 ã‡Û”Ü`ý|å™ßd ä è|ä{á¥á—à¼ä ývÝ'à
ߗã4å™è‹ãô 



  KM G I

I



 5 I 6*M     6  GG2 MG

m  óNÿiÛÀä{é4Ûïÿ#Üoÿiä ì ìKéådì ì ‹å™Ý ¹àdè'á—ä{çŸÜßWåTâ,å™ývä ì êUà™â¯éì{ådá«á—ä”Üß«átòöä.ô ܙô¾åÃá‡Üãtà™â¥âöÞÀè”éã‡ä àdè”á [àdè ã‡à
ì Üß4å™è‹ã]éì{ådá«á—ä”Üß«áô 
ݔådߗã«ä éÞÀì{å™ßiéì{å
á‡á‡ä”Üß =5?  ä{á]á‡ÝÁÜéä”Ü)çTí6êoã«ÛÀÜWì àŸéå™ã‡ä àdèå™è”çÃçÀä ådývÜã«Üß à™âKåví'å™ì ìrä è m  ÷8ådè”çTí6ê|ãÿ#à|Û6ê‹ÝÁÜß«ÝÀì{å™è”Üá÷Àÿiä:ã«ÛUݔå™ß4å™ì ì Üì8èÀà
߇ý|ådì{á÷Àå™ì{á‡à¼äÈè m  ô #ådìÈìrã‡ÛÀÜWá‡Üã±à™â ÝÁàdä è‹ã«á¹ìÈê6ä  è `íÁÜãÿ#ÜÜè÷ŸíÀޟã¥èÀà™ã¥àdè÷‹ã‡Û”ܱÛ6ê6Ý'Ü߇ݔì ådèÀÜáKã‡ÛÀÜ 4ý|ådß 
ä ètá‡Üãô ïÚ¥ÛÀÜ=çŸÜéä á‡ä àdètâöÞÀè”éã‡ä àdè'á =¼å™ß«Ü&çŸÜ ”èÀÜ)çWådárâöàdì ì à-ÿ]á ÝÁàdä è‹ã«árã‡Û”å™ã@ì ä ÜKä è”á—ä{çŸÜã«ÛÀÜ#í”å™ì ì.÷íÀÞÀãèÀàdãä è=ã«ÛÀܹý|ådß 
ä è á‡Üã÷då™ß«Ü&ådá«á—ä 
èÀÜç éì{å
ᇠá - Iø .6÷‹çŸÜÝ'Üè”çŸä  è Nà
è|ÿiÛÀä{é4Ûvá—ä{çŸÜ]àdâã‡ÛÀܱý|å™ß dä è¼á—Üã&ã‡Û”Üêtâ,å™ì ì.ô
ì ì”à™ã«ÛÀÜß¹Ý'à
ä è
ã4á&åd߇ܯá‡ä ývÝÀì ê çŸÜ ”èÀÜçÃã‡àFí'Q Ü éàd߫߇Ü)éKã À÷'ã«Û”å-ã ä{á÷8ã‡Û”ÜêÃå™ß«ÜNèÀàdã=å
á‡á‡ä dèÀÜ)çTåoéì{å
á‡á¯í‹êã«ÛÀÜtéì{ådá«á—ä ”Üß÷'ådè”ç¾çÀàoèÀàdã éà
è‹ã‡ß«äÈí”ÞŸã‡ÜWã‡àTå™è6êT߇ä{á‡õ'ôNÚ¥ÛÀܼá‡ä:ã«Þ”å-ã«ä àdèUä{á±á‡ÞÀývý|åd߇ä Ü)ç8÷”âöà
ß  H 6÷rä è ä dÞÀß«Ü|7ø 6ôWÚ¥ÛÀä{á±ß«å™ã‡ÛÀÜß àŸçÀçoâ,å™ývä ì êtàdâéì å
á‡á‡ä ”Üß4á÷dã‡à dÜã‡ÛÀÜß#ÿiäÈã«Ûå¼éà
è”çŸäÈã‡ä à
è|ÿ¹Ü=ÿiä ì ì'ä ývÝÁà
á‡Ü¯à
èFÛÀà-ÿ ã‡ÛÀÜêoå™ß«Ü±ã‡ß4å™ä èÀÜç÷ ÿiä ì ì'ß«Üá‡ÞÀìÈã¹ä èFá‡êŸáã«Üý|á¹ædÜ߇êvá‡äÈývä ì{å™ßã«à¼îŸð±ñÃá÷”å™è”çoâöàdߥÿiÛÀä{é4Ûoá—ã‡ß«Þ”éã‡ÞÀß4å™ì8ß«ä á‡õ¼ývä èÀä ývä )å-ã«äÈà
èWéådè íÁÜNçŸÜývà
è”áã«ß«å™ã‡Ü)ç8ô
ß«ä dà
߇à
ޔá&çŸä{á«éޔá«á‡äÈà
è|ä{á dä ædÜè|ä èoã«ÛÀÜ
ݔÝ'Üè”çŸäÈë8ô @å™íÁÜìrã«ÛÀÜNçŸä{å™ývÜã‡Üß#àdâ@ã«ÛÀÜNí”ådì ì yå™è”çFã‡Û”Ü`Ý'Ü߇ÝÁÜè”çÀä éÞÀì{å™ßiçŸä{á—ã«å™è'éÜ=í'Üãÿ¹ÜÜèTã«ÛÀÜ=ãÿ¹àvÛ6ê6ÝÁÜß ÝÀì{å™è”Üá  ôڥ۔Ü#ð OçŸä ývÜè'á—ä àdè`ä{á?çŸÜ”èÀÜ)çIå
áí'Üâöàdܹ߫ã«à`í'Ü¥ã«ÛÀÜ¥ý|å-ëŸä ýIޔýgè6ÞÀýtíÁÜßà™âÁÝ'à
äÈè‹ã4áã‡Û”å™ã éådè¾íÁÜvá—Û”å™ã—ã«Üß«ÜçUí6êTã‡Û”ÜIâ,ådývä ìÈêd÷'íÀÞÀã=í6ê á—Û”å™ã—ã«Üß«Üçoÿ¹ÜtývÜådèUã‡Û”å™ã±ã«ÛÀܼÝ'à
ä è
ã4á±éå™è à6ééÞÀß`å
á M IGKä è|å™ì ì'Ý'à‹á‡á‡ä íÀì Üiÿ#å êŸá]ò2á—ÜÜ]ã‡Û”Ü
ÝÀÝ'Üè”çŸäÈë¼âöàdß&âöÞÀߗã«ÛÀÜߥçŸä{á«éޔá«á—ä à
è'ó4ô ¹ìÈÜ)å™ß«ì êNÿ¹Ü éådè|éàdè‹ã«ß‡à
ì ã‡Û”ÜFð  çÀäÈývÜè”á‡äÈà
è àdâ±åUâ,ådývä ìÈê à™âiã«ÛÀÜá‡Üéì{ådá«á—ä ”Üß«á=í6êÅéà
è
ã«ß‡à
ì ìÈä  è Fã‡Û”ÜFývä èÀä ýtÞÀý ý|ådß 
äÈè  å™è'ç¼ý|å-ëŸä ýIÞÀý[çŸä{å™ývÜã«Üß  ã«Û”å-ã#ývÜýtí'Üß«áàdâ8ã‡ÛÀܯâ,å™ývä ì êWå™ß«Ü¯å™ì ì à-ÿ#Üç`ã«àWådá«á‡ÞÀývܙô Ààdß¹ÜëÀå™ývݔìÈÜd÷ éà
è”á‡ä çÀÜߥã‡Û”Ü`â,å™ývä ì ê|à™â ‹å™Ýã‡à
ìÈÜß«ådè‹ãiéì{ådá«á—ä ”Üß«á¹ä è m  ÿiäÈã‡ÛÃçŸä{ådývÜã‡Ü ß  H 6÷8á—Û”à-ÿièä " è ä dÞÀß«Ü ø Ÿô¥Ú¥Û”à
á‡ÜNÿiäÈã‡ÛTý|å™ß dä èTá‡å™ã‡ä{áâöê6ä   è   1 E = ¼éå™èÃá‡Û”å™ã—ã‡Üßiã‡Û”߇ÜÜNÝ'à
ä è
ã4áûÁäÈâ E = &A  A Ÿ÷”ã«ÛÀÜê éådè¾á‡Û”å-ã‡ã‡Üß ãÿ#àÀûådè”çUäÈâ   Ÿ÷Áã«ÛÀÜêÃéå™èïá—Û”å™ã—ã«Üß àdèÀì êTàdèÀÜdô 0&ådé4Û¾à™âKã«ÛÀÜá‡ÜWã«ÛÀß«ÜÜNâ,ådývä ìÈä Ü)á¥à™â




éì{å
á‡á‡ä”Üß4á`éà
߇߫Üá‡Ý'à
è”çÀá=ã«à¾à
èÀÜoàdâiã‡ÛÀÜTá—Üã«áIà™â¯éì å
á‡á‡ä”Üß4á ä è ä
ÞÀß«Ü À B ÷?ÿiäÈã‡Û Þ'áãtã‡ÛÀß«ÜÜFèÀÜá—ã‡Ü)ç á‡ÞÀí”á‡Üã«á¥àdâ@âöޔè”éã«äÈà
è”á÷”å™è”çFÿiäÈã‡Û # H ø™÷ H Ÿ÷”å™è”ç H À E ô 8

<

<



<

Φ=0 Φ=1

D=2

M = 3/2

Φ=0 Φ=−1

Φ=0

;Á$›“)Ž,…=<
> ¾4² 1 0 $. 3 1;$*.*,"    40 3  412#" 38? A5 Ú¥ÛÀÜ)á—ÜFä{çŸÜå
áNéådèÅí'ÜFޔá—Ü)çVã‡à á—ÛÀà-ÿ ÛÀà-ÿ ‹å™ÝVã‡à
ì Üß4å™è‹ãWéì{ådá«á‡ä 'Üß4á±ä ývÝÀì ÜývÜè‹ã=á—ã‡ß«Þ”éã‡ÞÀß4å™ì¹ß‡ä{á‡õ ývä èÀä ýväå-ã«ä àdèô Ú¥ÛÀܾÜë‹ã«Üè”á‡ä àdè àdâ`ã‡ÛÀÜVå™íÁà-ædÜÃÜëÀå™ývÝÀì Üã‡àDá—Ý”å
éÜ)áoàdâWå™ß«íÀäÈã‡ß4å™ß«êOçŸä ývÜè”á‡ä àdèDä{á Üè'éå™Ý'á—ÞÀì{å™ã‡Üçoä èåTòöývàŸçŸä”Ü)ç”ó&ã‡ÛÀÜàdß«Üý àdâ#ò,ðKå™ÝÀèÀä õÁ÷røù
ù
ú
ó 





m  $ M   KMAG2I(
   
8I: ) *  I+  5  I6*M     6  GG MG I K4*  K+$ K K  +*   = B  7K   K+$ K   KM  M N : G> I $ <7 !M  I"CM #  # K.* -     =   = B D  . 7 ø 

”àdß ã‡ÛÀÜtÝÀß«à6à™â&ÿ#Ütå
á‡á‡ÞÀývÜ`ã«ÛÀÜtâöàdì ì à-ÿiä è¼ìÈÜývý|åŸ÷ÁÿiÛÀä{é4ÛÃä èOòöðKådÝÀèÀä õÁ÷Køù'™ù
ó¥ä{á±ÛÀÜì{çÃã‡àFâöàdì ì à-ÿ âöß«àdýYá‡ê6ývývÜã‡ß«êvå™ßdÞÀývÜè‹ã«á  



/ ) )*#  ¹à
è”á‡ä çÀÜ1 ß O51  7Oø¯ÝÁàdä è‹ã«á¹ì ê‹ä è`ä èoåWí”å™ì ì ? m  ô1Üã¹ã‡ÛÀÜ Ý'à
ä è
ã4á&íÁÜ=á—Û”å™ã—ã«Üß4å™íÀì Ü í6ê ‹å™Ýtã‡à
ìÈÜß«ådè‹ãéì å
á‡á‡ä”Üß4á@ÿiä:ã«ÛIý|ådß 
ä è  ô?Ú¥ÛÀÜè¼ä è¼àdß4çŸÜßâöà
ß  ã«à`í'Üiý|å™ëŸäÈýväÜç8÷)ã‡ÛÀܯÝ'à
äÈè‹ã4á ý ޔáãiì ä Ü=àdèã‡ÛÀÜWædÜ߇ã‡ä{éÜá#à™â?å™èÅòODø)ó ƒçŸä ývÜè”á—ä à
è”å™ì8á—ê6ývývÜã‡ß«ä é±á—ä ývÝÀì ÜëÁ÷Ÿå™è'çFýtޔá—ã¯å™ì{á‡àIì ä ܱà
è t ã‡Û”ÜNá—ÞÀ߇â,ådéÜ=à™âã«ÛÀÜ`í”ådìÈì.ô



u=Ô-ÕrÕE(
ïÜWèÀÜÜ)çFà
èÀì êFéàdè”á‡ä{çŸÜß#ã‡Û”ÜWéå
á—Ü`ÿiÛÀÜ߇Ü=ã«ÛÀÜNè6ÞÀýtí'Üßià™â?ÝÁàdä è‹ã«á OÅá‡å™ã‡ä{á ”Üá O 1 7 ø™ô òO(. 7 VøiÝÁàdä è‹ã«á?ÿiä ì ìŸèÀà™ã¹í'Ü á—Û”å™ã—ã«Üß4å™íÀì ܙ÷‹á—ä è”éÜ¥ã‡ÛÀܱð OçÀäÈývÜè”á‡äÈà
èWàdâ8à
߇ä Üè
ã«ÜçtÛ6ê6Ý'Ü߇ÝÀì{ådèÀÜáä è m  ä á 7Ãø™÷6å™è”ç¼ådè‹êNçÀä á—ã‡ß«ä íÀޟã«äÈà
èNà™â8ÝÁàdä è‹ã«áÿiÛÀä{é4Ûtéå™ètí'Ü]á‡Û”å™ã—ã‡Ü߇Ü)çWí6êWå‹å™ÝIã‡àdì Üß«ådè
ãéì{ådá«á‡ä 'Üß éådèFådì  á‡àWíÁÜ=á—Û”å™ã—ã«Üß«Üç|í6êoådèoà
߇ä Üè‹ã«ÜçvÛ6ê6Ý'Ü߇ݔì ådèÀܙû‹ã‡Û”ä á¥ådì á‡àWá‡ÛÀà-ÿ]á&ã‡Û”å™ã < 1  7Dø ó4ô
 ‹å™6ä ô"è|
Vÿ#ÜÜ éà
è”á‡ä çÀÜßKÝÁàdä è‹ã«á?à
èoåNá‡ÝÀÛÀÜ߇Ü]àdâçŸä{å™ývÜã‡Üß U÷6ÿi۔Üß«Üiã‡ÛÀÜ á—Ý”ÛÀÜß«Ü]äÈã«á‡ÜìÈâä{á?à™âçŸä ývÜè”á—ä à
è  ÿiä ì ì‹è”ÜÜçIãÿ#à±ß«Üá‡ÞÀìÈã«á@âöß«àdý î6Üéã‡ä àd è EŸô EŸ÷dè”ådývÜì êvòø óä:E â OFä{áÜæ
Üè÷
ÿ¹Ü]éå™è ”è”ç¼å=çŸä{áã«ß‡ä íÀޟã«ä àdèNàd+ â O ÝÁàdä è‹ã«á#ò{ã‡Û”Ü#æ
Ü߇ã‡ä{éÜ)áàdâÀã‡Û”Ü='ò O|ø ó ;çŸä ývÜè”á‡ä àdè”ådì™á—ê6ývývÜã«ß‡ä{é?á‡ä ývÝÀì Üë”óÁÿiÛÀä{é4ÛIéå™èWíÁÜiá—Û'å-ã—ã«Üß«Üç`í6ê 
ådÝtã‡àdì Üß«ådè
ãKéì å
á‡á‡ä ”Üß4á@ä:â     =   = B H OÃø™÷6å™è”çTò 
óäÈâ Oä{áàŸçÀç8÷6ÿ#ܯéådè 'è”çvåNçŸä{á—ã‡ß«äÈí”ÞŸã‡ä àdè à™P â O¾ÝÁàdä è‹ã«á¥ÿi۔ä é4ÛéådèFíÁÜNá—àvá‡Û”å-ã‡ã‡Ü߇Ü)çoäÈâ -   =   = H ò O  ø ó 
ò O 7 ø ó =$O, ô B

ƒâPO ä{á#Üæ
Üè÷Áå-ãiývà
á—ã ï O ÝÁàdä è‹ã«áiéådèFíÁÜNá‡Û”å-ã‡ã‡Üß«ÜçFÿiÛÀÜèÀÜæ
Üß

 O

ø

1 

  =   =

B

A

OPI

ò2ü ‹ Eó

Ú¥Û6ޔáNâöàdß

-  

=  

O =


B

Üæ
Üèïã‡ÛÀÜvý|å™ëŸäÈýtÞÀý 7 ødô

è6ÞÀýtí'ÜßNà™â¥Ý'à
ä è
ã4á ã‡Û”å™ãIéådèVí'Üoá‡Û”å™ã—ã‡Ü߇Ü)ç¾ý|å ê¾í'Ü|ÿiß«äÈã—ã«Üè

ƒâ Å O ä á±àŸçÀç8÷å™ã ývà‹áãOOÝ'à
äÈè‹ã4á±éå™èÃíÁÜtá‡Û”å™ã—ã‡Ü߇Ü)çÃÿiÛÀÜè     =  ]à-ÿ#Üæ
Üß÷6ã‡ÛÀÜ ‹! ޔådè
ã«äÈãêvàdèFã‡ÛÀÜ`ß«äd ۋã#Û'å™è”çá—ä{çŸÜ`á«å-ã«ä á ”Ü)á

5+A O

òO



ø ó  'ò O

7

ø ó

=7O  A O

 =

B

YòO H

Dø

ø ó  'ò O g 7 ø)ó =7O

ô 

ò2üLB6ó

âöà
ß±å™ì ìrä è‹ã‡ Ü dÜß O . ø™ô]Ú¥Û6ޔá]âöàdßOVàŸçÀçã«ÛÀÜWì{å™ßdÜ)áãiè6ÞÀýtí'Üß]à™â?ÝÁàdä è‹ã«á¥ã«Û”å-ã=éådèUí'Ütá—Û”å™ã—ã«Üß«Üç ä{áNéÜߗã4å™ä èÀì êUíÁàdÞÀè”çÀÜçÅå™íÁà-ædܼí6ê    =   = 7 ød÷ådè”çïâö߇à
ýfã«ÛÀÜoå™íÁà-ædÜtã«ÛÀä{á`í'à
ÞÀè”çÅä{á`ådì á‡à B á«å-ã‡ä{á ”Üç ÿiÛÀÜè ODä{á=Üæ
Üèô ¯Üè”éÜväÈè 
ÜèÀÜß«ådì@ã«ÛÀÜ|ð  çŸä ývÜè”á‡ä àdè < à™â ‹å™Ýïã‡à
ì Üß4å™è‹ã=éì{ådá«á—ä”Üß«á ýtޔáã]á«å-ã«ä á—âöê



    7
1  <   = B 



ø I

ò2ü
ú
ó

< 1  7 ø™÷Áéàdè'éì ޔçŸÜ)á&ã«ÛÀÜ`ÝÀß«à6à™â;ô G G MG &   $   $   6  G   *  K. ( I

Ú¥ÛÀä{á¥ß«Üá‡ÞÀìÈã÷6ã‡àdÜã‡ÛÀÜßiÿiä:ã«Û 




$%

 5 (I 6*M     6 

 



  &

 G

rÜ ã $áiá—Üܱ۔à-ÿ ÿ¹Ü=éådèFçŸàvá—ã‡ß«Þ”éã«ÞÀß4å™ì8߇ä{á‡õIývä èÀä ýväå™ã‡ä àdèIÿiäÈã‡Û ‹å™Ýoã‡àdì Üß«ådè
ã¥éì{ådá«á‡ä 'Üß4áô,
VÜ=èÀÜÜ)ç àdè”ìÈê=éà
è”á‡ä çÀÜßã«Û”å-ã?á—Þ”í”á—ÜãàdâŸã‡ÛÀÜ ¯÷déådì ìdäÈã  r÷-âöàdß@ÿiÛÀä{é4Û`ã‡ß4å™ä èÀä è á—Þ”ééÜÜçÀáKŸ÷ ÿiÛÀÜß«Ü&í6ê`á‡Þ”ééÜá«á ÿ#ÜWývÜådèTã‡Û”å™ã=ådì ìrã‡ß4å™ä èÀä  è vçÀå™ã«åFå™ß«ÜIådá«á‡ä 
èÀÜçTåoì ådí'Üì1? - IøC.òöèÀàdã‡ÜIã‡Û”å™ã±ã«ÛÀÜá‡ÜWì{å™íÁÜì{á]çÀàoèÀàdã ۔å æ
ÜWã«àéàdä è”éä{çŸÜIÿiäÈã‡ÛÃã«ÛÀÜ|ådéã‡Þ”ådìì ådí'Üì{á÷ä.ô ÜdôNã‡ß4å™ä èÀä  è FÜ߫߇à
ß«á±å™ß«Ütådì ìÈà-ÿ#Üç'ó4ô
äÈã‡ÛÀä è  ÷ 'è”ç ã‡Û”ÜNá—ÞÀí'á—Üã]ÿiÛÀä{é4Û 
äÈæ
Üá¹ã‡Û”Ü=âöÜÿ#Üá—ãiã‡ß4å™ä èÀä  è WÜ߇߫àdß4á ¹éådì ì'ã«ÛÀä{á¥è6ÞÀýIíÁÜßiàdâÜ߇߫àdß4 á  = B ô
äÈã‡ÛÀä è ã‡Û'å-ãiá—ÞÀí'á—Üã ÷ ”è”çoã‡ÛÀܱâöÞÀè”éã‡ä àdL è =TÿiÛÀä{é4/ Û dä æ
Üá?ý|å™ë6ä ýtÞÀý ý|ådß 
ä èTò2å™è”çoÛÀÜè'éܱã‡ÛÀÜ ì à-ÿ¹Ü)áã&íÁàdޔè”ç àdè¾ã‡ÛÀܼð  çÀäÈývÜè”á‡äÈà
è'ó4ô  àdã‡ÜIã‡ÛÀܼæ-å™ì ÞÀÜIà™â¹ã‡ÛÀÜt߇Ü)á—Þ”ì:ã«ä  è |ß«ä á‡õí'à
ÞÀè”çDò{ã‡Û”Ütß«ä dۋã±Û'å™è”ç á—ä{çŸÜIà™â 01!'ôiò E‹ó4÷ŸÞ”á‡äÈ è =ã‡ÛÀÜ í'à
ÞÀè”çvà
è¼ã‡ÛÀÜ ð  çŸä ývÜè”á‡ä àdètä èvÝÀì{ådéܯà™â8ã‡ÛÀÜ ð  çŸä ývÜè”á‡ä àdè'óô  Üë6ã÷6ÿiäÈã‡ÛÀä è  ÷ ”è'çTã‡Û”å™ã á‡ÞÀí”á‡Üã¯ÿi۔ä é4Û 
ä ædÜ á   = B 7 ø`ã«ß«ådä èÀä  è ¼Ü߫߇à
ß«áô

ådäÈèr÷'ÿiäÈã‡ÛÀä èã‡Û'å-ã á‡ÞÀí”á‡Üã÷ 'è”ç ã‡Û”C Ü =µÿiÛÀä{é4Û 
äÈæ
Üá¯ã‡ÛÀܼý|å™ë6ä ýtÞÀý_ý|å™ß dä è÷å™è”ç èÀà™ã«Ütã‡ÛÀÜ|éàd߫߇Ü)á—ÝÁàdè”çÀäÈ è |߇ä{á‡õTí'à
ÞÀè”ç8ô ƒã‡Üß«å™ã‡Üd÷ å™è'ç|ã4å™õdÜ ã«Û”å-ã¯éì å
á‡á‡ä ”Üß¹ÿiÛÀä{é4Û 
ä ædÜá&ã‡Û”Ü`à-ædÜß4å™ì ìÁývä èÀä ýIÞÀý ß«ä á‡õvíÁàdÞÀè”çô


è¼ådìÈã‡Üß«è”å™ã‡ä ædÜ&ådÝÀÝÀß«à
å
é4ÛWä{á@ã‡àNçŸä æ6ä{çŸÜKã«ÛÀÜ¥âöÞÀè”éã‡ä àdè”á Åä è
ã«à±è”Üá—ã‡Üç¼á‡ÞÀí”á‡Üã4á  = D F ? D F  ø™÷ då á¯âöàdì ì à-ÿ]á¥ådìÈì =*?  =¹Û”å ædÜ- D  .¼á‡å™ã‡ä{á—âöê‹ä è   =  1 F«ô=Ú¥Û6ޔá±ã‡Û”ÜNâ,å™ývä ì êoàdâKâöÞÀè”éã‡ä àdè'á ä è  =K۔å
á]ð  çŸä ývÜè”á‡ä àdèFí'à
ÞÀè”çŸÜ)çÃå™íÁà-ædÜ`í6êýväÈèò'F D ó?7 ø™ô  àdã‡ÜWådì{á—àtã‡Û'å-ã  = = # ô±î]ñ   ã‡Û”ÜèÅÝÀß«àŸéÜÜ)çÀá`í6ê ã«å™õ6ä èã‡Û'å-ã&=µâöàdßWÿiÛÀä{é4Û ã«ß«ådä èÀä èá‡Þ”ééÜÜ)çÀáNä èVÜå
é4ÛÅá—ÞÀí'á—ÜãIå™è”çïâöà
ßNÿiÛÀä{é4Û ã‡Û”ÜIÜývÝÀä ߇ä{éådì8߇ä{á‡õFä{á]ývä èÀä ývä Ü)ç|ä èUã«Û”å-ã=á—ÞÀí'á—Üã÷rå™è”ç¾å 
ådä è÷Áé4ÛÀà6à‹á—ä  è ¼ã«Û”å-ã = ÿiÛÀä{é4" Û dä ædÜ)á¥ã‡Û”Ü ì à-ÿ¹Ü)áã¥à-ædÜß«ådì ì”ß«ä á‡õvíÁàdÞÀè”çô





 à™ã‡Ü&ã‡Û”å™ã@äÈãä{áÜ)á‡á‡Üè‹ã‡ä{ådì-ã‡àiã«ÛÀÜá‡Ü¹ådß 
ÞÀývÜè‹ã4áÁã‡Û”å™ãã‡Û”Ü&íÁàdÞÀè”ço ò E
órÛÀàdì{çÀáâöàdß   #Ké4ÛÀà‹á—ÜèNçŸÜéä{á—ä àdè âöÞÀè”éã‡ä àdèr÷
èÀàdãޔá—ãKã‡Û”ܯàdè”Üiã‡Û”å™ã&ývä èÀä ýväÜáã‡Û”Ü]ÜývÝÀä ß«ä{éå™ì6ß«ä{á—õoò,à™ã«ÛÀÜß«ÿiä{á—ÜiÜìÈä ývä è”å™ã‡ä è¯á‡àdì ޟã«äÈà
è”á âöàdßiÿi۔ä é4ÛTá‡àdývܱã‡ß4å™ä èÀä  è NÝÁàdä è‹ã <ïá«å-ã«ä á ”Ü)á ='ò <@1 ó H Stÿ#àdÞÀì{çFä è‹æ-ådìÈä{çÀå™ã‡Üiã«ÛÀÜNå™ß dޔývÜè‹ã4óô

Ú¥ÛÀÜF߇Ü)á—Þ”ì:ã«ä  è  
å™ÝVã«àdì Üß4å™è‹ãWéì å
á‡á‡ä ”Üß=ä{áNä èVâ,å
éãtå¾á—ÝÁÜéä ådì&õ6ä è”çVà™â]á‡ÞÀÝÀÝÁàd߇ãIæ
Üéã«àdßIý|ådé4ÛÀä èÀÜ ÿiÛÀä{é4Ûvá—ä ývÝÀì ê`çŸà6Üá&èÀà™ã&éàdÞÀè‹ã¹ç”å-ã«å â,ådì ìÈä è±àdޟã4á—ä{çŸÜ¥ã«ÛÀܯá‡ÝÀÛÀÜ߇Ü]éàdè‹ã«ådäÈè”äÈè±å™ì ìŸã«ÛÀÜiã«ß«ådäÈè”äÈè çÀå™ã«åÀ÷ àdß¼ä è”á‡ä{çŸÜ|ã«ÛÀÜUá—Üݔå™ß4å-ã«äÈ è Ãý|å™ß dä è÷KådávådèOÜ߇߫àdßô ƒã|á—ÜÜý|áIædÜ߇êÅß«Üå
á—à
è”å™íÀì Üvã«àïéàdè”éì ޔçŸÜFã‡Û”å™ã á‡ÞÀÝÀÝÁàd߇ã#æ
Üéã«àdߥý|ådé4۔äÈè”Üá÷
ÿiÛÀä{é4Ûoåd߇ܱã‡ß4å™ä èÀÜ)çvÿiäÈã‡Ûoæ
Üß«êvá—ä ývä ì{å™ß?àd í Üéã«ä ædÜá÷Ÿå™ì{á—à 
å™ä èoåtá‡äÈývä ì{å™ß õ6ä è”çVà™â±éå™Ý”å
éäÈãê éàdè‹ã‡ß«àdì&âö߇à
ý ã«ÛÀÜä ßIã‡ß4å™ä èÀä  è ”3 ô ]à-ÿ#ÜædÜß÷?" å ‹å™ÝVã«àdì Üß4å™è‹ãWéì{å
á‡á‡ä ”Üß`ä{áNè”à™ãtådè î6ð ñV÷Áådè”çUá—à¼ã«ÛÀÜWådß 
ÞÀývÜè‹ã]çŸà6Üá¯èÀà™ã¯éàdè”á—ã‡äÈã‡ÞÀã‡ÜWå¼ß«ä dàdß«àdÞ'á¹çŸÜývàdè”á—ã‡ß4å-ã«ä àdèoàdâKáã«ß‡Þ”éã‡ÞÀß4å™ì߇ä{á‡õ ývä èÀä ývä å-ã«ä àdè±âöàdß¹î6ð ñÃáô?ڥ۔Üiàdß«ä 
ä è”å™ì‹å™ß dÞÀývÜè
ãâöà
ßKáã«ß‡Þ”éã‡ÞÀß4å™ìŸß«ä{á—õWýväÈè”äÈývä )å-ã‡ä à
è¯âöà
ß¹î6ð±ñ¾á?ä{á õ6èÀà-ÿiètã‡à`íÁÜ 'å ÿ#Üç8÷‹á—ä è”éÜ¥ã‡ÛÀÜ]á—ã‡ß«Þ”éã«ÞÀß«Ü¥ã‡ÛÀÜ߇Ü]ä{á?çŸÜã‡Ü߇ývä èÀÜ)çWí6êNã«ÛÀܯçÀå™ã«åvò2á—ÜÜNòöðKå™Ý”èÀä õ'÷Áøù
ù
údó÷ î6Ü)éã‡ä à
è¾ú6ô ødø)ó ô iíÁÜì ä Üæ
Ü`ã‡Û'å-ã ã‡ÛÀÜ߇ÜIä á±åvâöÞÀ߇ã‡Û”Üß`á—Þ”íŸã‡ì ÜNÝÀß«àdí”ìÈÜý ÿiäÈã«ÛUã‡ÛÀÜtàdß«ä dä è”ådìrå™ß dÞÀývÜè
ãô Ú¥ÛÀܯá—ã‡ß«Þ”éã‡ÞÀß«Üiä{á?çŸÜ ”è”Üçvá‡à=ã‡Û”å™ã&èÀà=ã‡ß4å™ä èÀä  è  ÝÁàdä è‹ã«áKåd߇ܥývÜýtíÁÜß4á@àdâ8ã‡ÛÀÜ]ý|ådß 
ä èWá—Üã$ ô ]à-ÿ#Üæ
Üß÷ àdè”ÜKýIÞ'áã@á—ã‡ä ì ìdá‡Ý'Ü)éäÈâöê=ÛÀà-ÿïã‡Ü)áãÝ'à
äÈè‹ã4áã‡Û'å-ãâ,ådì ìdä è‹ã‡à¥ã‡Û”Ü&ý|å™ß dä è=å™ß«Ü?ã‡à±í'ܹì{å™íÁÜì Üç8ô ƒâ'àdèÀÜ&á‡ä ývÝÀì ê



ådá«á‡ä 
è”á@ã‡ÛÀÜ á‡ådývܙ÷ ”ë6Ü)çvéì{ådá«áã‡à=ã«ÛÀÜýyò2á‡å ê7Iø)ó4÷
ã‡ÛÀÜè¼ã‡ÛÀܯð OçÀäÈývÜè”á‡äÈà
èWÿiä ì ìŸí'ܯÛÀädÛÀÜß  #&ã‡Û”ådè ã‡Û”Ü`í'à
ÞÀè”ççŸÜß«ä ædÜçoä èFÚ¥ÛÀÜàdß«Üý Ÿô¯à-ÿ¹ÜædÜß÷‹ã‡ÛÀÜNá«å™ývܱä{á&ã«ß‡Þ”Ü`ä:â@àdèÀÜ=ì{å™íÁÜì{á&ã‡ÛÀÜý å™ì ìådá#Ü߇߫àdß4á ò,á‡Üܹã‡Û”Ü
ÝÀÝÁÜè”çŸäÈë”ó4ô ƒâ'à
èÀÜ#ì ådí'Üì{áã‡Û”Üý ådìÈì6ådá@éàdß«ß«Üéã Ÿ÷ àdèÀÜ¥å™ß«ß«äÈæ
Üárå™ã 
å™ÝNã‡àdì Üß«ådè
ãéì{ådá«á‡ä 'Üß4áô

$±èFã‡ÛÀÜ`àdã‡ÛÀÜߥ۔å™è”ç÷Àä:ãiä{á#õ6èÀà-ÿièoÛÀà-ÿ ã‡àvçŸàváã«ß‡Þ'éã‡Þ”ß«ådìÁß«ä{á—õ¼ývä èÀä ýväå™ã‡ä àdèIâöàdßiá‡ê6á—ã‡Üý|á#ÿiÛÀÜß«Ü ã‡Û”Üá—ã‡ß«Þ”éã«ÞÀß«ÜçŸà6Ü)átçÀÜÝÁÜè”çOàdèµã«ÛÀÜTçÀå-ã4å ò.î6۔å ÿ#Ü ;Ú@å ê6ì àdß`Üã¼ådì2ôÈ÷]øù
ù
åŸû#î6Û'å ÿ¹Ü ƒÚå ê6ìÈà
ß`Üãvå™ì.ôÈ÷ øù
ù dí'ó4ô ¯èŸâöàd߇ã‡Þ”è”å-ã«Üì êÃã«ÛÀÜv߇Ü)á—Þ”ì:ã«ä  è FíÁàdޔè”çÀáWåd߇ܼýtޔé4ÛVì à‹à‹á—Üß ã‡Û”ådè ã«ÛÀÜ|ð  íÁàdÞÀè”ç”áNå™íÁà-ædÜd÷ ÿiÛÀä{é4ÛTå™ß«ÜNå™ì ߇Ü)ådçŸê¼æ
Üß«êoì à6à
á‡Ütò,ÿ¹Ü`ÿiä ì ìÜëÀå™ývä èÀÜ=åIãê6ÝÀä{éå™ìréådá‡Ü`í'Üì à-ÿ ÿiÛÀÜß‡Ü ã‡Û”ÜNð  íÁàdޔè”çTä{á åoâ,ådéã«àdß=à™â±ø S SoÛÀä dÛÀÜ߯ã«Û”å™è¾ã‡ÛÀܼývÜ)ådá‡ÞÀ߇Ü)çã‡Ü)áã`Üß«ß«àdßó4ôWÚ¥Û6ޔá=å-ã ã‡ÛÀܼývà
ývÜè‹ã±á—ã‡ß«Þ”éã«ÞÀß«ådì߇ä{á‡õ ývä èÀä ývä å-ã«ä àdè¯ådìÈà
èÀܹçÀà‹Ü)áèÀà™ãÝÀß«à-æ‹ä{çŸÜ¹å  + I I $ GÜëŸÝÀì{ådè”å-ã«äÈà
è`ådá@ã‡à ÿiÛ6êWî6ð±ñÃáàdâ{ã‡Üèt۔å æ
1 Ü dà6àŸç dÜèÀÜß4å™ì ä å™ã‡ä àdè±Ý'Üߗâöà
߇ý|ådè”éÜdô ]à-ÿ#Üæ
Üß÷)ã«ÛÀÜ¥å™íÁà-ædܹådß 
ÞÀývÜè‹ã4áráã«ß‡à
è dì ê=á— Þ dÜ)áãã‡Û”å™ãådì 
àdß«äÈã‡ÛÀý|á ã‡Û'å-ã±ývä èÀä ývä  Ü   =  éådèUí'ÜIÜëŸÝÁÜéã«Üçã«à dä æ
Ü íÁÜã‡ã‡Üß 
ÜèÀÜß«ådìÈä )å-ã‡ä à
èvÝ'Üߗâöà
߇ý|ådè”éÜdô ”ÞÀߗã«ÛÀÜß Üæ6ä{çŸÜè”éÜ?âöà
ßã«ÛÀä{áä{árâöàdÞÀè”çWä è=ã‡ÛÀܹâöàdì ì à-ÿiä  è &ã«ÛÀÜà
߇Üý à™â@ò,ðKå™ÝÀèÀä õÁ÷Ÿøù
ùdü
ó÷)ÿi۔ä é4ÛWÿ¹+ Ü !
ޔà™ã‡Ü¹ÿiäÈã‡ÛÀà
ޟã ÝÀß«à6à™â K 


   
8 II25   K

 6  #!5AM)576  

5  G G*,+ *  IE$+  *

MG



MI +* / M

 "

M

 
vò    ó  1    =  ò2ü‹ó /  M M6
vò    ó G * M57 I   *6   #I M I("I $ M  M G  GJM   * MM 5AM   ( II * M 6*M*   G"I(" M  6*6    * *%+ GJM  GIG M ïø   7 * MM 5 M  ) I(I( $ M +   GI"CM  6*6    * *%+NG M  GIG2 M 



:

:

:



¯à-ÿ¹ÜædÜß÷Ÿä èFà
ß«çŸÜß#âöàdߥã‡Û”Üá‡Ü`àdí”á‡Üß«æ-å-ã‡ä à
è”áKã«àví'Ü`Þ'á—ÜâöÞÀìâöàdßiß«ÜådìÝÀß«àdíÀì Üý|á÷6ÿ¹Ü=èÀÜÜçTåtÿ#å ê¼ã«à é à
ývÝÀޟã«Ü=ã‡ÛÀÜtçŸä{å™ývÜã«Üߥà™â?ã«ÛÀÜNývä èÀä ý|å™ìÁÜè'éì à
á‡ä èvá—ÝÀ۔Üß«ÜNçŸÜá«éß«ä í'Ü)çådí'à-æ
ܙ÷Àâöà
ß±ådè‹êFè6ÞÀýIíÁÜ߯à™â ã‡ß4å™ä èÀä èIÝ'à
ä è
ã4á¥å™è”çFâöàdß]ådè6ê|õdÜ߇èÀÜì8ý|å™ÝÀÝÀä è”ô 

I $ ICK 5 $  M

3I/ 

!

M 

*

   6*I G*,+

*  K  6

& 5  M 

M


‹å™ä èïìÈÜã  íÁÜvã‡Û”Ü|ý|å™ÝÀݔäÈè|ã‡àTã«ÛÀÜ|ÜýIíÁÜçÀçÀäÈèá‡Ý”ådéÜ Uô
VÜ|ÿiä{á—Û ã‡àÃéàdývÝÀÞÀã‡Ütã‡ÛÀÜoß«å
çŸä ޔá à™â¹ã‡ÛÀÜvá‡ý|å™ì ì Üá—ã¯á—Ý”ÛÀÜß«ÜIä è ÿiÛÀä{é4ÛÃÜè”éì à‹á—Ü)á]ã«ÛÀܼý|å™ÝÀÝÁÜçÃã‡ß4å™ä èÀä èoçÀå™ã«å±ã«Û”å-ã`ä{á÷ÿ#ܼÿiä á‡ÛUã«à ývä èÀä ývä @ Ü A  á—Þ” í Üéã#ã«à



 ò,) = ó



 k

1 

 

A

ò2ü'dó F

ä á#ã«ÛÀÜoò,ÞÀèÀõ6èÀà-ÿiè'ó¥éÜè‹ã‡Üßià™âã‡ÛÀÜWá‡ÝÀÛÀÜ߇Üdô¹Ú¥Û6ޔá¥ä è‹ã‡ß«à6çÀޔéä ètÝ'à‹á—äÈã«äÈæ
Ü@å 
ß«ådèdÜ ‡÷Ÿã‡Û”Ü`ÝÀ߇ä ý|ådìE@å dß4å™èdä{å™èvä{á

ÿiÛÀÜß‡Ü k ýtÞÀìÈã‡ä ÝÀì ä Üß4á

? N=

5H

 A

 (

«ò



N= 'A =

#  ò'<,=2ó 

ó k

ò2üdü‹ó I

Ú¥ÛÀä{áiä{á]å
ådäÈèTå|éàdè6ædÜë !‹Þ”ådçŸß4å-ã«ä{é±Ý”߇àdß4å™ývývä èNÝÀß«àdí”ìÈÜýT÷”á—àvÿ#ÜWéå™èTä è”áã«Üå
çFý|å™ëŸäÈýväÜ]ã‡Û”Ü VàdìÈâöÜ`çŸÞ”å™ì


 

( H

+=  =

ò<>=

.ó

D <>=

 (

  =

+=



Oò'<,= DK<

ò,ÿiÛÀÜß«Ü`ÿ¹Ü=۔å æ
Ü å
ådäÈèF߇ÜÝÀì{ådéÜç

 ò'<,=.ó

( = =



N=

ø H

S

9

 ò<

)ó )ó#í6ê 

ò2üdù‹ó ò<,=

D <

)ó‡ó¥á—ÞÀíÜéãiã«à ò2ù S‹ó ò2ùÀø ó

ÿiäÈã‡ÛUá‡àdì ޟã«äÈà
è/dä æ
Üè|í6ê k H

(

ò<,=ƒó

N= 

ò2ù 
ó I

=

Ú¥Û6ޔá]ã‡ÛÀÜNݔ߇à
íÀì ÜýYä{á¥æ
Üß«êFá‡äÈývä ì{å™ß&ã‡àvã«Û”å-ã±à™âKá—Þ”ÝÀÝ'à
ߗã]æ
Üéã‡àdß]ã‡ß4å™ä èÀä èÀ÷Àådè”çTä èFâ,å
éã]ã«ÛÀÜWéà6çÀÜ âöàdß ã«ÛÀÜIì{å™ã—ã‡Üß±ä{á±Üå
á—ä ì êoývàŸçŸä”Ü)çTã‡àTá—à
ì ædÜ`ã«ÛÀÜvå™íÁà-ædÜNÝÀß«àdí”ìÈÜýTô  à™ã«ÜWã«Û”å-ã=ÿ#ÜIÿ#Üß«ÜWä è åá‡Üè”á‡Ü 4ì ޔé4õ6 ê Ÿ÷í'Ü)éådޔá—Ü|ã‡ÛÀÜoådí'à-æ
Ü|å™è”ådì ê6á‡ä{á=á—ÛÀà-ÿ]áNޔáNã«Û”å-ãWã‡ÛÀÜß‡Ü M  G  G`å™èµÜëŸÝ”å™è”á‡ä àdè ò,ù dó âöà
ßNã‡Û”Ü éÜè‹ã‡ÜßûŸã‡Û”Üß«Ü ä{á#èÀà  5 Iß«Üå
á—à
è|ÿiÛ6ê|ÿ¹ÜNá‡ÛÀà
ÞÀì{ç|ÜëŸÝÁÜéã#ã«Û”å-ãiã«ÛÀÜNéÜè‹ã‡Üßiàdâã«ÛÀÜNá—Ý”ÛÀÜܱ߫ä è á‡ÛÀàdÞÀì{çTíÁÜIÜëŸÝÀ߇Ü)á‡á‡ä íÀì Ü`ä èTã‡Üß«ý|á]àdâKã‡Û”ÜIý|ådÝÀÝ'Ü)çTã‡ß4å™ä èÀä  è vçÀå™ã«åoäÈèÃã‡ÛÀä{á±ÿ#å êdô Ú¥ÛÀܼá‡ådývÜWéådèÃíÁÜ á«å™ä{çà™â?ã‡Û”ÜIá‡àdì ޟã‡ä à
èoâöà
߯ã«ÛÀÜIá‡ÞÀÝÀÝÁàd߇ã¯ædÜ)éã«àd߯ݔ߇à
íÀì ÜýT÷ 0 !”ôWò B‹ó4ôI%ò ¯ådçÃÿ¹Üté4ÛÀà
á‡ÜèUá‡àdývÜNàdã‡ÛÀÜß dÜàdývÜã‡ß«ä éå™ìÁéàdè'áã«ß‡Þ”éã‡ä àdèr÷Àÿ¹Ü=ývä dۋã¥èÀà™ã]۔å æ
Ü=í'ÜÜèÃá‡àIâöà
ߗã«ÞÀè”å™ã‡Ü™ô ¹àdè”á‡ä{çŸÜß#ã‡Û”ÜNá—ý|ådìÈì Ü)áã¥å™ß«Üå Ü !‹ÞÀä ì{å-ã«Üß4å™ì@ã‡ß«ä{å™è 
ìÈܼéà
è‹ã«å™ä èÀä è oãÿ¹à 
ä ædÜè Ý'à
ä è
ã4á=ä è m  , ô ƒâ#ã«ÛÀÜ|ÝÁàdä è‹ã«á rÝ'à‹á—äÈã«äÈà
è¾æ
Üéã«àdß4á`å™ß«Ü ì ä èÀÜåd߇ì ê¼çŸÜÝÁÜè”çÀÜè‹ã÷Àã«ÛÀÜNéÜè‹ã‡Üßiàdâ@ã«ÛÀÜ=ã‡ß«ä{å™ è 
ì ܯéådèÀèÀà™ãiíÁÜ`ÜëŸÝÀß«Üá«á—Ü)çoä èFã‡Ü߇ý|á#à™âã‡Û”ÜýTô ó 

  

"IE$

 2 

ILK

M

 "CM 0  M  0

$ 

òöðKådÝÀèÀä õÁ÷røù
ù
údó1dä æ
Üá#å™èUå™ìÈã‡Ü߇è'å-ã‡ä æ
Ü]í'à
ÞÀè”çàdèFã‡Û”ÜNådéã«Þ”å™ìß«ä{á—õvàdâá‡ÞÀÝÀÝÁàd߇ãiædÜ)éã‡à
ߥý|ådé4ÛÀä èÀÜ)á 

 
vò    ó  1   ÞÀýtÞÀýtíÁÜíÁßiÜßià™â@à™â?ã‡ß4á‡å™ÞÀä ÝÀèÀÝÁä èàdt߇ã¥á‡ådæ
ývÜéÝÀã‡ì àdÜß4á á ò2ù E‹ó ÿiÛÀÜ߇Ü
vò    ó=ä{á±ã‡ÛÀÜvådéã‡Þ”ådì?߇ä{á‡õâöàdß`åoý|å
é4ÛÀä èÀÜNã«ß«ådä èÀÜçUà
è 8 ø¼ÜëÀå™ývݔìÈÜ)á÷  
vò    ó  ä{áFã‡ÛÀÜÅÜë6ÝÁÜéã«å™ã‡ä àdègà™âIã‡ÛÀܵå
éã‡Þ'å™ì`ß«ä á‡õ à-ædÜßUå™ì ì`é4ÛÀàdä{éÜ)áà™âWã«ß«ådä èÀä è á‡ÜãÃàdâtá‡äÜ 6 ø™÷Nådè”ç   ÞÀýtíÁÜßià™âá—Þ”ÝÀÝ'à
ߗã]æ
Üéã‡àdß4áä{á¥ã‡ÛÀÜIÜëŸÝÁÜéã4å-ã«äÈà
èTà™âã‡ÛÀÜIè6ÞÀýIíÁÜ߯à™â&á‡ÞÀÝÀÝÁàd߇ã¯ædÜ)éã‡à
ß«á]à-ædÜß]ådì ì 



D

:

:



é4ÛÀà
ä éÜá¹à™ârã«ß«ådä èÀä èWá‡Üã«á¥àdâ@á‡äÜ:ô ƒã$á¥Üådá‡ê¼ã‡à¼á‡ÜÜ=ÛÀà-ÿ ã«ÛÀä{á¹íÁàdޔè”çFåd߇ä{á‡Üáéàdè”á‡ä{çŸÜß&ã‡ÛÀÜ=ãê6ÝÀä{éådì á‡ä:ã«Þ”å-ã«ä àdèoå™â{ã‡Üß]ã‡ß4å™ä èÀä  è Wà
èå 
äÈæ
Üèoã‡ß4å™ä èÀä èIá‡Üã÷”á—Û”à-ÿièFä è @ä
ÞÀ߇Ütø EÀô

; $›“)Ž,… < > º  ² ² 21 —;1 . * ;"$.;$*    3 (d 0 !#. 0 . *  0 2* *  49ˆ12—Á.! 0‡° 4d 3 ]. » 12.4" 3 " 3  1,+- 0 1,12—] " 3  Á   12*@1,+- ; Ã0  ° " *2" 0 *,-2:9   3-0 3 043 5
VÜiéå™è dÜãå™èIÜá—ã‡ä ý|å-ã«Ü?à™âÀã«ÛÀÜ&ã«Üá—ãÜ߇߫àdß@í6ê ß«Üývà-æ6ä  è ¥àdè”ܹàdâŸã‡ÛÀÜ#ã‡ß4å™ä èÀä  è iÝÁàdä è‹ã«á÷-߇Ü 2ã«ß«ådä èÀä  è À÷ ™å è'ç`ã‡Û”Üètã‡Ü)áã«äÈè¯àdèIã‡Û”Ü#ß«Üývà-æ
Üç=ÝÁàdä è‹ãû
ådè”ç`ã«ÛÀÜètß«ÜÝÁÜå-ã«ä è]ã«ÛÀä{á÷-âöà
ßådì ì
ã«ß«ådä èÀä è]ÝÁàdä è‹ã«áôÀß«àdý ã‡Û”ܯá—Þ”ÝÀÝ'à
ߗãKæ
Üéã‡àdßKá‡àdì ޟã«ä àdèWÿ#Üiõ6èÀà-ÿ ã‡Û”å™ãK߇Üývà-æ‹ä è å™è6êNã«ß«ådä èÀä è¯ÝÁàdä è‹ã«áã«Û”å-ã&åd߇Üiè”à™ã&á‡ÞÀÝÀÝÁàd߇ã ædÜ)éã«àdß4ávò{ã‡Û”Üoì{å-ã‡ã‡ÜßWäÈè'éì ޔçŸÜtã‡Û”ÜoÜß«ß«àdß4á«ó±ÿiä ì ì?۔å ædÜvèÀà¾JÜ 8ÜéãIàdèVã‡Û”Ü|Û6ê‹ÝÁÜß«ÝÀì{å™è”ܼâöàdÞÀè”çôUÚ¥Û6ޔá ã‡Û”Ütÿ#àdß4á—ã±ã«Û”å-ãNéådè¾Û”ådÝÀÝÁÜèïä á±ã«Û”å-ã`Üæ
Üß«êUá—ÞÀݔÝ'à
ߗã`ædÜ)éã«àdß ÿiäÈì ìí'Ü)éà
ývÜtådè¾Üß«ß«àdßôIÚ@ådõ6äÈ è |ã‡Û”Ü ÜëŸÝÁÜéã«å-ã«ä àdèïà-ædÜß`ådìÈì?á‡Þ”é4Û ã‡ß4å™ä èÀä  è Fá‡Üã4á=ã‡Û”Üß«Üâöàdß«Ü dä ædÜ)á±ådèVÞÀÝÀÝÁÜßNíÁàdÞÀè'ç à
è¾ã«ÛÀÜoå
éã‡Þ'å™ì?ß«ä á‡õÁ÷ âöàdߥã«ß«ådä èÀä  è Iá‡Üã4á¥à™â?á—ä C Ü : Dø™ô


ì:ã«ÛÀàd Þ dÛNÜìÈ Ü 
ådè
ã ÷ rÛ'å ædÜKê
Üãã«à ”è'çNå]ޔá‡ÜKâöà
ßã«ÛÀä{áríÁàdÞÀè'ç8ôڥ۔Üß«Ü&á—ÜÜýgã‡à±í'Ü#ý|å™è6ê=á—äÈã«Þ”å-ã«äÈà
è”á ÿ ÛÀÜ߇Ütã‡Û”ܼå
éã«Þ”å™ì?Ü߇߫àdß ä è”éß«Üå
á—Ü)á]Üæ
Üè ã‡ÛÀà
ÞdÛ ã‡Û”Ütè6ÞÀýtíÁÜß=à™â¥á—ÞÀݔÝ'à
ߗã`ædÜ)éã«àdß4á±çŸÜ)éß«Üå
á—Ü)á÷á‡à i ã‡Û”ÜIä è‹ã‡ÞÀäÈã«äÈæ
Ü`éàdè'éì ޔá‡äÈà
èÅò,á‡ê6á—ã‡Üý|áiã«Û”å-ã dä ædÜ=âöÜÿ¹Üß á‡ÞÀÝÀÝÁàd߇ã±æ
Üéã‡àdß4á 
ä ædÜ`íÁÜã—ã«Üß=ÝÁÜ߇âöàdß«ý|å™è”éÜ)ó à™â{ã«ÜèUá‡ÜÜý|á¹ã«àIâ,ådä ì2ô Àޔߗã«ÛÀÜß«ývàd߫ܙ÷Àå™ìÈã«ÛÀàdÞ
Ûvã‡Û”Ü`í'à
ÞÀè”çTéådèFíÁÜ=ã‡ädۋã«Üß#ã‡Û”ådèã‡Û”å™ãiâöàdÞÀè”çFޔá‡ä è ã‡Û”ÜIÜ)áã«äÈý|å™ã‡Ü`àdâKã‡ÛÀÜIð  çŸä ývÜè'á—ä àdèTéà
ýIí”äÈè”ÜçÿiäÈã‡Û 0 !”ôvò E‹ó4÷äÈã éå™è¾å™ã¯ã‡Û”Ütá«å™ývÜ`ã«ä ývÜNí'Ütì Üá«á ÝÀß«ÜçŸä{éã‡ä ædÜd÷6å
á¥ÿ¹Ü`á‡Û”å™ì ìá—ÜÜ`ä èoã«ÛÀÜ`èÀÜë6ã¯î6Ü)éã‡ä à
èô









&

 I$

7

G

 7 *

M D   $

 6 

G 

rÜãIޔáIÝÀޟãtã‡Û”Üá‡Üoàdí'á—Ü߇æ-å-ã«ä àdè”á ã«à á‡àdývÜ|ޔá‡Ü™ô
áIývÜè‹ã‡ä à
èÀÜçVå™íÁà-ædÜd÷@ã«ß«ådä èÀä èÃå™èOî6ð ñ    éì{å
á‡á‡ä”Üßvÿiä ì ì±ådޟã‡à
ý|å-ã‡ä{éå™ì ì ê 
ä ædÜUæ-å™ì ÞÀÜ)ávâöàdßFã‡Û”Ü    ÿ#Üädۋã«á÷]è6ÞÀýtí'Üßoà™âWéÜè‹ã‡Üß«á÷±éÜè‹ã‡Üß ÝÁà
á‡ä:ã«ä àdè”á÷å™è”ç¾ã‡Û”߇Ü)á—ÛÀà
ì{ç8/ ô Ààdß % ådޔá«á—ä{å™è   á÷ã«ÛÀÜß«Üvä{á àdèÀì êÃà
èÀܼݔå™ß4å™ývÜã‡Üß ì Üâ{ã `ã‡ÛÀÜ    ÿiä{ç6ã‡Ûgò äÈè 01!'ô ò, ü S
ó—óFòöÿ#ÜTådá«á—ÞÀývÜoÛÀÜß«Üoà
èÀì êïà
èÀÜ    ÿiä çŸã‡Ûµâöà
ßWã«ÛÀÜTÝÀß«àdíÀì Üýoóô #ådèµÿ#Ü ”è”ç|ã‡ÛÀܱàdݟã«ä ý|å™ì”æ-ådìÈޔÜiâöàdß¹ã‡Û”å™ã&ã«à‹à”÷Ÿí‹êvé4ÛÀà6à‹á—ä  è =ã«Û”å-ã UÿiÛÀä{é4Ûvývä èÀä ývä Üá   =   ä dޔ߇ÜNø B á‡ÛÀà-ÿ]áå]á‡Üß«ä ÜáàdâŸÜëŸÝ'Ü߇ä ývÜè
ã4árçŸà
èÀÜKàd è dü-N ë ™ü  —îŸÚÅçŸä däÈãçÀå™ã«åÀ÷ ÿiäÈã«Ûvø SŸ÷ S S S#ã‡ß4å™ä èÀä  è ¥ÝÁàdä è‹ã«árådè”ç  SÀ÷ S S S]ã‡Üá—ã?Ý'à
ä è
ã4áôÚ¥ÛÀÜ#ã‡à
ÝtéÞÀ߇æ
ܹä èWã‡ÛÀÜiì Üâ{ã۔ådè”çWÝ'å™èÀÜìÀá—ÛÀà-ÿ]á@ã«ÛÀÜ¥ð VíÁàdÞÀè”çTò,ä2ô ܙô@ã‡ÛÀÜiíÁàdޔè”ç ß«Üá‡ÞÀìÈã‡ä  è ±âö߇à
ý[å™Ý”ÝÀ߇à ëŸä ý|å-ã«ä  è iã‡Û”Üið OçÀäÈývÜè”á‡äÈà
èNä è 01!'ô¥%ò E
ó  8 í6ê 01!'ô¥ò2ü
ú
ó—ó4÷
ã‡ÛÀܯývä ç”çŸì Ü#éÞÀ߇æ
Ü á‡ÛÀà-ÿ]á¹ã‡ÛÀÜ`íÁàdÞÀè'ç|âöß«àdý ì Ü)å ædÜ ƒàdè”Ü ƒàdޟã±ò 01!'ô]ò,ù E
ó—ó÷'å™è'ç|ã«ÛÀÜ=í'àdã—ã‡à
ý éÞÀ߇æ
Ü á‡ÛÀà-ÿ]á¹ã‡Û”Ü`ývÜådá‡ÞÀß«Üç ã‡Ü)áã¥Üß«ß«àdßô ¹ì Üåd߇ì ê™÷‹ä èvã‡ÛÀä{á#éå
á—Üd÷
ã«ÛÀܱíÁàdޔè”çÀá¥å™ß«Ü¯æ
Üß«êIì à6à
á‡Ü™ô?Ú¥ÛÀÜ ß‡ä dۋã&Û'å™è”çvÝ'å™èÀÜì8á‡ÛÀà-ÿ]á ޔáã ã‡Û”Ütð  í'à
ÞÀè”çDò{ã‡Û”ÜIã«àdÝVéÞÀß«ædÜd÷8âöàdß  .  S S
ó÷8ã‡à 
Üã«ÛÀÜß ÿiäÈã‡ÛÃã«ÛÀÜtã‡Ü)áã`Üß«ß«àdß÷8ÿiäÈã«ÛÃã‡Û”ÜIì{å-ã‡ã‡Üß á«éå™ì Ü)ç¾ÞÀÝÅí6êïåâ,ådéã«àdßNàdâ ø S SVòöè”à™ã‡Ü|ã‡Û”å™ãNã‡ÛÀÜ|ãÿ¹àUéޔ߇æ
ÜáNé߇à‹á‡á4ó4ô ƒãIä á`á—ã‡ß«ä õ6äÈ è oã«Û”å-ãNã«ÛÀÜvãÿ#à éޔ߇æ
Üá`۔å æ
Üvývä èÀä ý|å|ä èïã‡ÛÀÜoá«å™ývÜvݔì å
éÜ Nã‡Û6ޔáWäÈèïã«ÛÀä{áNéå
á—Üd÷@ã«ÛÀÜoð  íÁàdÞÀè”ç÷?ådìÈã‡ÛÀà
 Þ dÛïì à‹à‹á—Üd÷ á‡ÜÜý|á?ã‡àWíÁÜ èÀÜæ
Ü߇ã‡ÛÀÜìÈÜ)á‡áݔ߇Ü)çŸä{éã‡ä æ
ܙ ô 0KëŸÝ'Ü߇ä ývÜè‹ã4áàdèoçŸä 
ä:ã4á =ã«ÛÀ߇à
 Þ dÛ|ùIá—Û”à-ÿ¹Ü)çtã‡Û”å™ã&ã‡Û”ܯð  íÁàdÞÀè”ç ‹å ædÜiå=ývä èÀä ýIޔý âöà
ßKÿiÛÀä{é4Û  ÿ¥ådá?ÿiäÈã‡Û”äÈè¼å â,ådéã‡à
ßKà™â8ãÿ#à àdâ8ã‡Û”å™ã&ÿiÛÀä{é4ÛIývä èÀä ývä Ü)ç=ã‡Û”Ü ã‡Ü)áã#Ü߫߇à
ß]ò,çŸä 
ä:ã]ø¯ÿ#å
á?ä è”éà
è”éì ޔá‡ä ædÜ)óô ;è‹ã‡Üß«Üá—ã‡ä  è 
ì ê™÷™ä è¼ã‡ÛÀà‹á—ܱéådá‡Üá?ã‡ÛÀܱð µíÁàdޔè”ç|éà
è”á‡ä á—ã‡Üè‹ã‡ì ê 
å æ
Üiå=ìÈà-ÿ#ÜßÝÀß«ÜçÀä éã‡ä àdèWâöàdß  ã‡Û”ådè¼ã‡Û”å™ã&ÿiÛÀä{é4ÛIývä èÀä ývä Ü)ç=ã‡Û”Ü¥ã‡Üá—ã&Ü߇߫àdßô $±ètã‡Û”Ü]à™ã«ÛÀÜßKÛ'å™è”ç8÷ ã‡Û”ÜIì Üå æ
Ü ƒàdèÀÜ .à
ޟã¥í'à
ÞÀè”ç8÷ådì:ã«ÛÀàd Þ dÛã‡ä dۋã«Üß÷8çŸà6Ü)á¯èÀàdã á‡ÜÜý ã«àoíÁÜIݔ߇Ü)çŸä{éã‡ä æ
ܙ÷'á‡ä è”éÜIä:ã±Û”ådçUè”à ývä èÀä ýIޔý âöà
ߥã‡ÛÀÜ`æ-å™ì ÞÀÜ)á¥à™â  ã‡Üá—ã‡Ü)ç8ô

0.7

0.7

0.6

0.65

VC Bound : Actual Risk * 100

Actual Risk : SV Bound : VC Bound



0.5 0.4 0.3 0.2 0.1 0

0.6 0.55 0.5 0.45 0.4 0.35

0

100 200 300 400 500 600 700 800 900 1000 Sigma Squared

   r+-  .*  ;Á$›“)Ž,…=<> < +-³´F( 0  3   3 (d²-2)"$;12" ° ¹ ° 3 3 00 5

0

100 200 300 400 500 600 700 800 900 1000 Sigma Squared



z/&Ù µ ) ÙöÓ#ŸÓÙ2Õ'Ò

.Ü߫۔ådݔá]ã«ÛÀܼíÀä
Üá—ã±ì ä ývä:ã4å-ã«äÈà
èFàdâ&ã«ÛÀÜvá—ÞÀݔÝ'à
ߗã=ædÜ)éã‡à
ß`å™ÝÀÝÀß«à
å
é4ÛÃì ä Üá]ä èïé4ÛÀàdä{éÜIà™â¹ã‡ÛÀܼõdÜ߇è”Üì.ô $ è”éÜ|ã‡Û”ÜoõdÜ߇è”Üì&ä{á ÀëŸÜç8÷¹î6ð ñéì å
á‡á‡ä”Üß4á ۔å ædÜoàdèÀì êïàdèÀÜFޔá—Üß ; é4ÛÀà‹á—ÜèïÝ'å™ß4å™ývÜã«Üß|òöã‡ÛÀÜÜ߫߇à
ß ± ÝÁÜè”ådì:ãêÀó÷‹íÀÞÀã¹ã«ÛÀܱõ
Üß«èÀÜì'ä{á¹åWædÜß«êtíÀäN  ߇ÞN  ÞÀè”çŸÜߥÿiÛÀä{é4Û¼ã‡àIá‡ÿ#ÜÜÝoݔå™ß4å™ývÜã‡Üß4áôî6àdývܱÿ¹à
߇õt۔å
á íÁÜÜèFçŸàdèÀܯàdèvì ä ývä:ã«ä è¯  õ
Üß«èÀÜì{á?ޔá‡ä è`  ÝÀß«ä àdß?õ6èÀà-ÿiì Üç
 Ü`ò2îŸé4Ûà
 ì õdàdÝÀâÁÜã#å™ì.ôÈ÷'ø)ùdùdü
åŸû#  ÞÀßd Ü)á÷Áøù
ùdü
ó÷ íÀޟãiã«ÛÀÜ`íÁÜá—ã¯é4ÛÀàdä{éÜ à™âõdÜ߇è”ÜìÁâöàdß]å d ä æ
Üèoݔ߇à
íÀì Üýþä{áiáã«ä ìÈìåtß«Üá‡Üå™ß4é4Û|ä{á«á—ÞÀÜdô
á‡Üéàdè”çDì ä ýväÈã«å-ã«ä àdèÅä{á|á‡Ý'ÜÜç å™è'çDá‡ä Üd÷#íÁà™ã«Û äÈèDã«ß«ådäÈè”äÈèï  ådè”çDã‡Ü)áã«äÈèÀ ô
۔äÈì Üã‡Û”ܾá—ÝÁÜÜ)ç ÝÀß«àdíÀì Üý ä èOã‡Üá—ã¼ÝÀ۔ådá‡Üä áIì{å™ßd ÜìÈêïá—à
ì ædÜçVä ègò*#  ÞÀßd Üá÷¯ø)ùdù‹ ó4÷?ã«ÛÀä{áIá—ã‡ä ì ì¹ß«Ü ‹! ÞÀä ß«ÜáNãÿ¹à ã‡ß4å™ä èÀä è ݔå
á‡á‡ÜáôÚ@ß«ådä èÀä èi  âöàdßæ
Üß«ê`ì ådß
 Ü&çÀå™ã«ådá‡Üã4á¥òöývä ì ìÈä à
è”á8à™â8á—ÞÀݔÝ'à
ߗã?ædÜ)éã«àdß4á«órä áådèWÞÀè”á‡àdì æ
ÜçNݔ߇à
íÀì ÜýTô - ä{á‡é߇Üã‡Ü±ç”å-ã«åIÝÀß«Üá‡Üè‹ã«á#å™è”à™ã‡Û”Üß#ÝÀ߇à
íÀì ÜýT÷6å™ìÈã‡ÛÀà
Þd Ûoÿiä:ã«Ûoá‡ÞÀäÈã«å™í”ìÈܯ߇Ü)á‡éå™ì ä è` ±  ÜëÀéÜì ìÈÜè‹ã&߇Ü)á—Þ”ì:ã4á ۔å æ
ÜIèÀÜædÜߗã«ÛÀÜì Üá«á±í'ÜÜèVàdíŸã4å™ä èÀÜ)çOòd& à
å
é4ÛÀä ý|á÷øù
ù'd ó4ô @  ä è”ådì ìÈêd÷rå™ìÈã‡ÛÀà
Þd Û á—à
ývÜIÿ#àdß«õU۔ådá=íÁÜÜè çŸà
èÀÜvàdèVã«ß«ådäÈè”äÈè  åTýtÞÀìÈã‡ä{éì{å
á‡á=î6ð±ñ ä èVàdè”Ü|áã«ÜÝ  d÷ã‡Û”Ü|àdݟã«ä ý|å™ìKçŸÜ)á—äd è âöàdßWýIÞÀìÈã«ä éì{ådá«á î6ð ñ éì{å
á‡á‡ä”Üß4áKä{á¥åtâöÞÀ߇ã‡ÛÀÜß]å™ß«ÜåWâöàdßiß«Üá‡Üå™ß4é4Ûô
z w

Ó 6Ò)Ù,ÕÁÒ

< 



Vܱæ
Üß«êtí”߇ä Ü 'ê¼çÀÜá«éß«äÈíÁÜiãÿ#àIàdârã‡Û”Ü á‡ä ývÝÀì Üá—ã÷
ådè”ç|ývà‹áã#ÜJ8Üéã«ä ædܙ÷6ývÜã«ÛÀàŸçÀá¹âöàdß#ä ývÝÀ߇à-æ6ä è ã‡Û”Ü ÝÁÜ߇âöàdß«ý|å™è”éܱàdâî6ð ñÃáô


Ú¥ÛÀܾæ‹ä ߇ã‡Þ”ådì]á—Þ”ÝÀÝ'à
ߗãFædÜéã‡à
ßvývÜã‡Û”à6ç ò.îŸé4 Û à
 ìÈõ
àdݟâ;÷ #ÞÀß dÜávådè”ç ðKå™Ý”èÀä õ'÷=øù
ùÀû #ÞÀß 
Üávådè”ç î é4Ûà
 ì õdàdÝÀâ;÷@øù
ù'dó4÷å-ã—ã«Üývݟã4á]ã«àoä è”éàdß«Ý'à
ß«å™ã‡Ü`õ6èÀà-ÿièUä è6æ åd߇ä{ådè”éÜ)á¥à™âKã«ÛÀÜWÝÀß«àdí”ìÈÜý òöâöàdß ÜëÀå™ývݔìÈÜd÷ Ÿ ã‡ß4å™è'á—ì{å-ã«ä àdèïäÈè6æ-å™ß«ä{å™è”éÜtâöàdßNã«ÛÀÜoä ý|ådÜvß«ÜéàdèÀäÈã«äÈà
è¾ÝÀß«àdí”ìÈÜýoó í6ê 'ß«á—ãWã«ß«ådäÈè”äÈèTå á—êŸá—ã‡ÜýT÷ådè”ç ã‡Û”Üèoé߇Ü)å-ã‡ä  è `è”Üÿ çÀå™ã«å`í6êvçŸä{áã«àd߇ã‡ä  è  ã‡Û”ܯ߫Üá‡ÞÀìÈã‡ä  è `á—Þ”ÝÀÝ'à
ߗã#ædÜ)éã‡à
ß«á¯ò{ã‡ß4å™è'á—ì{å-ã«ä è ã‡ÛÀÜýT÷‹ä è¼ã‡Û”Ü éå
á—ÜWývÜè‹ã‡ä àdè”Üç”ó÷Áå™è', ç ”è”å™ì ì ê|ã«ß«ådä èÀä  è våvèÀÜÿ á‡êŸáã«Üý àdèTã«ÛÀÜIçÀä á—ã‡à
ߗã«ÜçÅò2å™è”çTã«ÛÀÜWޔè”çŸä{áã«àd߇ã‡Ü)ç”ó çÀå™ã«åŸô Ú¥ÛÀÜTä{çŸÜå ä{átÜådá‡êVã‡àVä ývÝÀì ÜývÜè‹ãIå™è”çDá‡ÜÜý|áIã«àïÿ#àdß«õÅíÁÜã—ã«Üß¼ã«Û”å™è à™ã«ÛÀÜß¼ývÜã‡ÛÀàŸçÀátâöà
ß ä è”éà
߇ÝÁàdß4å-ã«ä  è Nä è6æ åd߇ä{ådè”éÜ)áKÝÀß«àdÝÁà
á‡ÜçFá—àtâ,å™ßô Ú¥ÛÀÜvß«ÜçÀޔéÜ)ç á‡ÜãNývÜã‡ÛÀàŸçD*ò #ÞÀß dÜá÷&ø)ùdù Àû #ÞÀß dÜá=ådè”çÅîŸé4 Û àd ì õ
àdݟâ;÷?øù
' ù dó]ÿ¥ådá=ä è‹ã‡ß«àŸçŸÞ”éÜ)çUã«à ådç”çŸß‡Ü)á‡á]ã‡ÛÀÜtá—ÝÁÜÜ)çUà™âKá‡ÞÀÝÀÝÁàd߇ã±æ
Üéã«àdß]ý|å
é4ÛÀä èÀÜá]äÈèTã«Üá—ã±ÝÀÛ'ådá‡Ü™÷å™è”çUådì á‡à|á—ã«å™ß‡ã«á¯ÿiäÈã‡ÛÃå¼ã«ß«ådäÈè”Üç î6ð ñVô6ڥ۔Ü]ä{çŸÜå=ä{á?ã«àNß«ÜÝÀì{ådéÜ#ã‡ÛÀÜ á—ÞÀý ä / è 01!'ôi@ò B‹óí6êtåWá—ä ývä ì{å™ßá—ÞÀýT÷‹ÿiÛÀÜß«Ü]ä è”á—ã‡Ü)ådçtà™ârá‡ÞÀÝÀÝÁàd߇ã ædÜ)éã«àdß4á÷
éàdývÝÀÞÀã‡Üçtæ
Üéã«àdß4á]òöÿi۔ä é4Ûvåd߇ÜièÀàdã&Üì ÜývÜè‹ã«áà™â8ã«ÛÀÜ]ã«ß«ådäÈè”äÈ è  á‡Üãó?å™ß«Üiޔá‡Üç8÷Ÿå™è”çtä è”á—ã‡Ü)ådç à™â¯ã‡ÛÀ Ü =—÷¹å çŸä ÁÜ߇Üè
ã¼á—Üãtàdâ¯ÿ#Üä dۋã«áWå™ß«ÜFéà
ývÝÀޟã«Üç8ôµÚ¥ÛÀÜè6ÞÀýIíÁÜßIà™â¯Ý'å™ß4å™ývÜã«Üß4á`ä{áIé4۔à
á‡Üè íÁÜâöàdß«ÜÛ'å™è”çVã«à 
ä ædÜvã«ÛÀÜá‡ÝÁÜÜçÀÞÀÝDçÀÜá‡äÈß«ÜçôVÚ¥ÛÀÜß«Üá‡ÞÀìÈã‡ä  è UædÜ)éã«àdßWä{átáã«ä ìÈì#å¾æ
Üéã‡àdßIä è U÷¹ådè”ç ã‡Û”ÜWݔådß«ådývÜã‡Üß«á¥åd߇Ü`âöà
ÞÀè”çTí6êýväÈè”äÈývä äÈ è `ã‡ÛÀÜ 0Kޔéì ä çÀÜå™èFèÀàdß«ý_à™âã‡ÛÀÜtçŸä ÁÜ߇Üè”éÜ`íÁÜãÿ#ÜÜèUã‡Û”Ü àdß«ä dä è”ådìÀædÜ)éã‡à
ß Yå™è”çoã‡Û”Ü=ådÝÀÝÀß«à ëŸäÈý|å™ã‡ä àdèIã‡à¼äÈãôKÚ¥ÛÀÜ`á«å™ývܱã‡Ü)é4ÛÀèÀ ä !‹ÞÀÜ éàdÞÀì{çoí'Ü=ޔá‡Üçoâöàd߯î6ð ñ ß«Ü dß«Üá«á‡äÈà
è¾ã«" à ”è'çµýtޔé4ÛÅývàdß«Ü|Ü
|éäÈÜè‹ãNâöÞÀè”éã‡ä àdèµß«Üݔ߇Ü)á—Üè
ã4å-ã«äÈà
è”á|ò,ÿiÛÀä{é4Ûµéàdޔì çVíÁÜޔá—Ü)ç8÷Kâöà
ß ÜëÀådývÝÀì ܙ÷6äÈèTçÀå™ã«å¼éà
ývÝÀ߇Ü)á‡á‡ä àdè'óô

¹à
ýIíÀä èÀä èNã‡ÛÀÜ)á—ܱãÿ¹àtývÜã«ÛÀàŸçÀá+
å æ
ܱåIâ,ådéã‡àdß#à™â?ú Stá—ÝÁÜÜ)çŸÞÀÝVòöÿiÛÀä ì Ü]ã«ÛÀÜ=Ü߫߇à
ß¹ß4å-ã«Ü±ä è”é߇Ü)ådá‡Üç âöß«àdýfødô S [ã‡àUødôÈø Ió¹à
èoã«ÛÀÜ  —îŸÚgçŸädäÈã4á ò#ÞÀß 
Üá¥å™è'çTîÀé4Ûàd ì õdà
ݟâ;÷8øùdù™óô Ð z

kIÕÁÒ@Ø %,×  Ù2Õ'Ò



î6ð ñÃávÝÀß«à-æ6ä çÀÜFåïèÀÜÿ å™Ý”ÝÀ߇à‹ådé4ÛOã‡àïã‡ÛÀÜUݔ߇à
íÀì Üý à™â ݔå-ã‡ã‡Ü߇è ß«Üéàd èÀäÈã«äÈà
è ò{ã«àd Üã‡ÛÀÜßvÿiäÈã‡ÛD߇Ü dß«Üá«á‡äÈà
è|Üá—ã‡ä ý|å™ã‡ä àdèoådè”çì äÈè”Üå™ß#àdÝÁÜß4å-ã«àdßiä è6ædÜß«á‡ä àdè'óÿiäÈã«ÛTéì Üå™ßiéàdèÀèÀÜ)éã«äÈà
è”á¹ã‡à¼ã«ÛÀÜNÞÀè”çÀÜß«ìÈê6ä è á—ã«å-ã«ä{áã«ä éå™ìŸì Üåd߇è”äÈè¯  ã«ÛÀÜà
߇êdôÚ¥ÛÀÜêtçÀä 8Ü߹߫å
çŸä{éådìÈì ê âöß«àdý[éàdývݔådß«ådíÀì Ü#ådÝÀÝÀß«à
å
é4ÛÀÜá?á‡Þ”é4Û|ådá&èÀÜޔ߫ådì èÀÜãÿ¹à
߇õŸá¥  î6ð ñ ã‡ß4å™ä èÀä èv  å™ì ÿ¥å êŸá+”è”çÀá å d ì à
í”å™ìrývä èÀä ýtÞÀýT÷Ÿå™è”çTã«ÛÀÜä ߯á‡ä ývÝÀì Ü d Üà
ývÜã«ß‡ä{é ä è
ã«Üß ÝÀß«Üã4å-ã‡ä à
è|ÝÀß«à-æ‹ä{çŸÜ)á?âöÜߗã«äÈì Ü
 ߇à
ÞÀè”ç¼âöà
ß#âöÞÀߗã«ÛÀÜßiä è6ædÜ)áã«ä
 å-ã«ä àdèô
èUî6ð ñ ä{á¹ì ådß
 Üì êIé4Û'å™ß4ådéã«Üß«ä Üç í6êã‡ÛÀܼé4ÛÀàdä{éÜ`à™âKäÈã«á¯õdÜß«èÀÜì2÷8å™è”ç î6ð±ñÃá±ã«Û‹Þ'á¯ì ä èÀõFã‡ÛÀÜIÝÀ߇à
íÀì Üý|á¥ã«ÛÀÜêUåd߇ÜWçÀÜá‡ä
 èÀÜçâöà
ß±ÿiäÈã«ÛÃå ì{å™ßd ÜKíÁàŸçŸê`à™â'ÜëŸä{áã«äÈè]  ÿ¹à
߇õ àdèIõdÜ߇èÀÜì
í'ådá‡ÜçNývÜã‡ÛÀàŸçÀáô 

ÛÀàdÝÁÜ&ã«Û”å-ãã«ÛÀä{árã«ÞŸã‡à
߇ä{å™ì‹ÿiä ì ìdÜè'éàdޔ߫åd Ü á‡àdývܱã‡à¼ÜëŸÝÀì àdß«Ü`î6ð±ñÃáiâöàdߥã‡Û”Üý|á‡Üì ædÜ)áô

 oØ
rÒÕ 

%'6Ö.E)*6Ò8Ó 

æ
Üß«ê 
ß«å™ã‡ÜâöޔìÀã‡à .ô =èÀä ß4á‡é4Ûr ÷ ¯ô  àdÛÀì. ÷ 0i ô $ á‡ÞÀè”åÀ ÷ 0] ô iä Üã‡ý|å™èr ÷ ±ôŸîŸé4Û àd ì õdà
ݟâ;÷ 6WôÀî6ä  è 
Üß÷
ô î ývà
ì åÀ÷¯ô8î‹ã«Üè”ådß«ç÷8ådè”çTðNôÁðKå™ÝÀèÀä õÁ÷”âöà
ß]ã«ÛÀÜä ߯éàdývývÜè‹ã4á¥àdèTã‡Û”ÜNý|å™è6ޔá«éß«ä ݟãôiڥ۔å™è”õ6á¯å™ì{á—àtã«à 6 ã‡Û”ܯ߫Üæ6ä Üÿ#Üß4á÷
ådè”çtã‡àWã‡ÛÀÜ0&çŸäÈã«àdß÷ `ô”å ê6ê
å
ç8÷dâöà
ß&Üë‹ã«Üè”á‡ä ædÜd÷
ޔá‡ÜâöÞÀì8éà
ývývÜè‹ã«áôîŸÝ'Ü)éä{å™ìŸã«Û”å™è”õ6á å™ß«ÜWçŸÞ”ÜNã‡àFðNôðKå™Ý”èÀä õ'÷8ÞÀè”çÀÜß ÿiÛÀà
á‡ÜNݔå™ã‡ä Üè‹ã 
ÞÀä{çÀå™è”éÜ iì Üåd߇èÀÜ)çFã‡ÛÀÜt߇à
Ý'Ü)áûÁã‡à
ôî6ývà
ì å¼ådè”ç ±ô8îŸé4Û àd ì õdà
ݟâ;÷Àâöàdß±ý|å™è6êoä è‹ã‡Ü߇Ü)áã«ä è vå™è”çâöß«ÞÀäÈã—âöÞÀìçŸä{á«éޔá«á—ä à
è”áû”ådè”çã‡à &Ÿôî6۔å ÿ#Ü ;Ú@å ê6ì àdߥå™è'" ç -Nô îŸé4Û6ÞÀÞÀß«ý|å™è'á÷6âöàdßiæ-ådìÈÞ'å™íÀì ܱçÀä á«éÞ'á‡á‡ä àdè”á&àdèUáã«ß‡Þ'éã‡Þ”ß«ådìÁß«ä{á—õvývä èÀä ývä å™ã‡ä àdèrô

ý

- 16ÒÖ@Ù < 

u=Ô ÕÕ+( v  ÕE(]q

Uz Ð z 

‹Õ'Ô$2)%

"

Vܯéà
ìÈì Ü)éã?ÛÀÜß«Üiã«ÛÀÜiã‡Û”Üàdß«Üý|á?á—ã«å™ã‡Ü)ç¼äÈè¼ã«ÛÀÜiã«Üë6ã÷‹ã‡àdÜã«ÛÀÜß&ÿiä:ã«Û¼ã‡ÛÀÜä ßKÝÀß«à‹àdâ,áô?Ú¥ÛÀÜÜývý|å ۔å
á å|á‡ÛÀàd߇ã‡Üß]ÝÀß«à6à™âޔá—ä è|å Ú¥ÛÀÜàdß«ÜýYàdâã‡ÛÀÜ
ìÈã‡Ü߇è”å™ã‡ä ædÜd÷ ò,ñ¾å™è‹ådá«å™ß«ä ådè÷røùdù‹ó&í”ÞŸã±ÿ#ÜNÿiä{á—Û”Üç ã‡à¼õ
ÜÜÝã«ÛÀÜ`ÝÀß«à‹àdâ,á¥ådáiá‡ÜìÈ â ƒéàdè‹ã«ådä èÀÜç|å
á¥Ý'à‹á‡á‡ä íÀì ܙô


> /I3G M  G I85 I(*  G * m B K  #(M GJM 5 *  M G M   I(5I $ M*+I( "CM   $ 6*6 8 G  GMJK 5  #  



  ( 2M (#   #!5 M )576  

M 

  (I  6 #   *

M

u=Ô-ÕrE Õ (
VÜIå™ì ì à-ÿ ã«ÛÀÜNèÀàdã‡ä àdè'áià™âKÝÁàdä è‹ã«áiä è m B ÷8ådè”çTÝ'à‹á—äÈã«äÈà
èFæ
Üéã‡àdß4áiàdâã«ÛÀà
á‡ÜNÝÁàdä è‹ã«á÷”ã«àoíÁÜ Þ”á‡Üçïä è‹ã‡Üß4é4۔ådèdÜ)å™íÀì êoä è ã‡ÛÀä{á=ÝÀß«à6à™â;ô-Üã &÷ gí'Ü|ã‡ÛÀÜ|éàdè6ædÜëÃۋޔìÈì{á±à™â#ãÿ¹àUá‡Üã4á=à™â¥Ý'à
äÈè‹ã4á  ÷ ä è m B ô rÜã   çÀÜèÀàdã‡ÜUã«ÛÀÜïá‡ÜãFà™âNÝÁàdä è‹ã«á|ÿi۔à
á‡ÜÃÝ'à‹á—äÈã«äÈà
è ædÜéã‡à
ß«á|ådß‡Ü dä ædÜèDí6ê #  D # ?  D *? yò,èÀà™ã«Ü ã‡Û”å™ã   çŸà6Üá¥èÀàdã]éàdè‹ã4å™ä è|ã«ÛÀÜ=àdß«ä 
ä è'ó4÷Ÿå™è”çFì Üã   ۔å æ
Ü ã‡Û”Ütéàd߫߇Ü)á—ÝÁàdè”çÀäÈ è vývÜå™è”äÈ è vâöàdß±ã‡ÛÀܼéà
è6ædÜëUÛ6ÞÀì ì{áô=Ú¥ÛÀÜèïá—Û”à-ÿiäÈ è vã‡Û”å™ã  å™è'ç _å™ß«ÜWì ä èÀÜåd߇ì ê á‡Üݔådß«ådíÀì Ü ò2á—Üݔå™ß4å™íÀì Ü¥í6êtå=Û6ê6ÝÁÜß«ÝÀì{å™èÀÜ óä áÜ !‹ÞÀä æ ådì Üè‹ãã‡àWá‡ÛÀà-ÿiä  è  ã‡Û”å™ãKã‡Û”ܯá—Üã   ä{á?ì ä èÀÜåd߇ì ê á‡Üݔådß«ådíÀì Ü`âö߇à
ý ã‡ÛÀܼà
߇ä dä è Iô Àà
ß=á‡ÞÀÝÀÝÁà
á‡ÜIã‡ÛÀܼì{å-ã‡ã‡Ü ß ]ã«ÛÀÜ è  ? m B D ? m D+ CA Sá‡Þ”é4Û ã‡Û'å-ã <9 7 J. S E< ?   |ô  à-ÿ ÝÀä{é4õ¾á‡àdývÜ  ? v÷?å™è”çVçŸÜèÀà™ã«Ü¼ã‡ÛÀÜ|á‡ÜãNàdâ¥ådì ì?Ý'à
äÈè‹ã4á #  7  D # ?  D *? í6ê   7  ôKÚ¥ÛÀÜè < 9 H7 >.  9 E< ?   7  ÷Àå™è”ç|éì Üå™ß«ì ê  9 7 A  9 ÷Àá‡à`ã‡ÛÀÜ=á‡Üã«á   7  ådè”ç  åd߇ܯìÈä èÀÜ)å™ß«ì êWá—Üݔå™ß4å™í”ìÈÜdô iÜÝ'Ü)å-ã‡ä è Wã‡ÛÀä{á&ÝÀ߇àŸéÜá«á á‡ÛÀà-ÿ]áiã«Û”å-ã   ä{á]ì ä èÀÜåd߇ì êoá‡Üݔådß«ådíÀì Ü=âö߇à
ý_ã‡ÛÀÜtàdß«ä dä èFäÈâ&ådè”çUàdè”ìÈêäÈâ  ådè”ç _å™ß«ÜNì ä èÀÜåd߇ì ê á‡Üݔådß«ådíÀì ܙô























3







ä á¯ì äÈè”Üå™ß«ì êvá—Üݔå™ß4å™íÀì Ü=âöß«àdý ã‡ÛÀÜIàdß«ädä èô ä{áFéàdè6ædÜë8÷±á—ä è”éÜ E< # H # #  # D<  H #    D ? SDø D # # D #  ?  D # D  ? i÷™ÿ#Ü&۔å æ
ܱò—A ø  Áó < # 7 N<  H # ò—ò—ø  Áó # # 7 ó , ò‡òø Áó # 7   ó?   ]ô ]Üè”éÜIä:ã±ä á¯á—Þ
|éäÈÜè‹ãiã‡àFá‡ÛÀà-ÿ ã‡Û”å™ã ådè‹ê éà
è6ædÜëVá—Üã#÷?ÿiÛÀä{é4ÛÅçŸà6ÜáIèÀàdãIéà
è
ã4å™ä è I÷Kä{á`ì äÈè”Üå™ß«ì êÃá—Üݔå™ß4å™í”ìÈÜtâöß«àdý Iô Üã <> = B ? íÁÜ ã‡Û'å-ã¯ÝÁàdä è‹ã]ÿiÛÀà
á‡Ü 0&ޔéì ä{çŸÜådèFçŸä{á—ã«ådè”éÜ=âöß«àdý I÷ J<> = B -÷'ä{áiývä èÀä ý|å™ì.ô]ò  à™ã‡ÜWã‡ÛÀÜ߇ÜNéå™èUíÁÜNàdè”ìÈê àdè”ÜNá—Þ”é4ÛTÝÁàdä è‹ã÷”á‡ä è”éÜ=äÈâ@ã«ÛÀÜß«Ü`ÿ#Üܱ߫ãÿ#àÀ÷”ã‡ÛÀÜNé4ÛÀà
ß«ç à
äÈè”äÈ è Wã«ÛÀÜýT÷”ÿi۔ä é4ÛTå™ì{á‡àIì ä Üá#ä  è #÷'ÿ#àdÞÀì{ç éà
è‹ã«å™ä èïÝ'à
ä è
ã4á`éì à
á‡Üß=ã«à Iô$ó
VÜ|ÿiä ìÈì?á‡ÛÀà-ÿ[ã‡Û”å™ ã E
= #ô ]Üè”éÜ`ã«ÛÀÜ=ã‡Û”߇ÜÜ`Ý'à
äÈè‹ã4á I+÷ < å™è'ç <  = B âöàdß«ý ådèÃàdíÀã‡Þ”á‡ÜFò,àdß ß‡ä dۋãó#ã‡ß«ä{å™ è 
ìÈÜd÷”ÿiäÈã‡ÛÃà
íŸã‡Þ'á—Üòöàdß ß«ä 
ۋã4óiåd è dì Ü`àŸééÞÀ߇߫ä  è |å-ã ã‡Û”Ü ÝÁàdä è‹ã Iô -¯Ü ”èÀÜ Ò  1þò <3 <  = B ó =* < <  = B  ô=Ú¥ÛÀÜè¾ã‡ÛÀÜtçŸä{áã4å™è”éÜ`âöß«àdý_ã‡ÛÀܼéì à‹á—Ü)áã]ÝÁàdä è‹ã±ä è  ã«à  ä{& á J<> = B    ò <  = B 9 Ò  ó  ÷8ÿiÛÀä{é4ÛUä á¯ì Üá«á¥ã‡Û”åd è  <> = B    ô ]Üè”éÜ
ç Dä{á ì ä èÀÜåd߇ì êNá‡Üݔådß«ådíÀì ܹâöß«àdý Iô&Ú¥Û6ޔá   Åä{á?ì ä èÀÜåd߇ì êNá‡Üݔådß«ådíÀì Ü#âö߇à
ý I÷Àå™è'ç   I  I   ä{á#ìÈä èÀÜ)å™ß«ì êtá‡Üݔådß«ådíÀì Ü]âöß«àdý I÷'å™è”çoã‡Û6ޔá  ä á#ì ä èÀÜåd߇ì ê¼á—Üݔå™ß4å™í”ìÈܯâö߇à
ý vô VÜIè”à-ÿ


¹ì Üåd߇ì ê 



á‡ÛÀà-ÿ



ã‡Û”å™ã÷äÈâ 

 

;H6÷Áã‡ÛÀÜè 

_çŸà6ÜáèÀà™ãÃéàdè‹ã«ådäÈè ã«ÛÀÜVàdß«ädä èô

 



ÀÞÀ߇ã‡ÛÀÜ߇ývà
ß‡Ü 

: 









ƒã¥ß‡Üý|å™ä è”áã«àIá‡ÛÀà-ÿDã‡Û”å™ã÷ŸäÈâã‡ÛÀܱãÿ#àIá‡Üã«á¹à™ârÝÁàdä è‹ã«á  ÷ [å™ß«Ü¯ì ä èÀÜåd߇ì êNá‡Üݔådß«ådíÀì ܙ÷™ã«ÛÀܱä è‹ã‡Üß«á‡Üé ã‡ä à
èFàdâã‡Û”Üä ßiéàdè6æ
Üë|Û6ÞÀì ì{á¹äÈâÜývݟãêdô #êoå
á‡á‡ÞÀývݟã«ä àdèvã«ÛÀÜß«Ü=ÜëŸä{áã4á#å¼Ý'å™ä ß ? m B D ?!AI÷”á‡Þ”é4Û ã‡Û'å-ã  # = ?  D 9 # = 7 >. S=ådè” ç  =9? MD 9 "= 7 >A SÀô ¹à
è”á—ä{çŸÜß?å 
ÜèÀÜß«ådì6Ý'à
äÈè‹ãP<5? ¹ô ƒã



ý|å êNíÁÜ]ÿiß«ä:ã‡ã‡Üè < H
= = # = D
= î6ä ývä ì{å™ß«ìÈêd÷âöàdßvÝÁàdä è‹ã«á  ?  ÷. 9 ÿ#àdÞÀì{ç í'Üïå™íÀì Üã‡à 'è”ç V å Ý'à
ä è
ã
ø D1S 1 ødôڥ۔Üè 9 <7 H
= = -' 9 # = 7, J. . SŸô = 1  7 A SÀô]Üè”éÜ    H 6÷]á‡ä è”éÜTà™ã«ÛÀÜß«ÿiä{á—Üÿ#Ü  ÿiÛÀä{é4Û á—ä ýtÞÀìÈã«ådèÀÜà
ޔá—ì êVá‡å™ã‡ä{á ”Üá¼íÁà™ã«Û äÈè”Ü !‹Þ”å™ì äÈã‡ä Ü)áô

q "6ՔÔ$2) Ð  ¹à
è”á—ä{çŸÜßIá‡àdývÜoá‡Üã¼à™âD ÝÁàdä è‹ã«áWä è m B ô)¹ÛÀà6à‹á—Üå™è6êVàdèÀÜà™â]ã«ÛÀÜÝ'à
äÈè‹ã4áIå
á àdß«ädä èô Ú¥ÛÀÜè ã‡ÛÀÜ Ý'à
äÈè‹ã4á|éådèDíÁܾá‡Û”å-ã‡ã‡Ü߇Ü)ç í6ê àdß«äÈÜè‹ã‡Üç Û6ê6Ý'Ü߇ÝÀì{ådèÀÜátäÈâ=ådè”ç àdè”ìÈêOä:â ã‡Û”Ü ÝÁà
á‡ä:ã«ä àdè|æ
Üéã«àdß4á#à™âã‡ÛÀÜ`ß«Üý|ådäÈè”äÈèNÝ'à
ä è
ã4á¥å™ß«Ü ì ä èÀÜ)å™ß«ìÈêtä è”çŸÜÝ'Üè”çŸÜè‹ãô



u=Ô-ÕrE Õ ( ådí'Üìrã«ÛÀÜWà
߇ä dä è I÷ådè”çUådá«á‡ÞÀývÜ ã‡Û”å™ã¯ã‡Û”Ü   øNÝÁà
á‡äÈã‡ä àdèædÜ)éã«àdß4áià™âã‡Û”ÜWß«Üý|å™ä èÀä è ÝÁàdä è‹ã«á=å™ß«ÜIì ä èÀÜ)å™ß«ìÈêFä è”çŸÜÝÁÜè'çŸÜè‹ãô ¹àdè'á—ä{çŸÜß ådè‹êUÝ'å™ß‡ã‡äÈã‡ä àdè¾à™â&ã«ÛÀÜ& Ý'à
ä è
ã4á äÈè‹ã«àoãÿ#àá‡ÞÀí”á‡Üã4á÷  ß  # å™è'ç   ß«Üá‡Ý'Ü)éã‡ä æ
Üì ê™÷8á—àoã‡Û”å™ã# # 7@  H ¾ô rÜã # íÁÜWã«ÛÀܼá—ÞÀí'á—Üã # å™è”ç   ÷àdâ&àdß4çŸÜD éà
è‹ã«å™ä èÀä  è  Iô#Ú¥ÛÀÜèTã‡ÛÀÜNéàdè6ædÜëoÛ6ÞÀì ì à™â  ä{á#ã‡Û”å™ã¯á—Üã¯à™âÝÁàdä è‹ã«á¥ÿiÛÀà‹á—Ü=ÝÁà
á‡ä:ã«ä àdèoæ
Üéã‡àdß4á < # # á«å-ã‡ä{á—âöê

 (  = = ) #

<

H

#

 (  = ) #

= D

ø H

=

= ;  S D

ò
ôÈø ó

ÿiÛÀÜ߇ÜNã‡ÛÀÜ  # =Kå™ß«Ü=ã‡ÛÀÜWÝ'à‹á—äÈã‡ä à
èFæ
Üéã«àdß4á¥à™âã‡Û”Ü  # Ý'à
äÈè‹ã4áiä è # òöä è”éìÈÞ'çŸä èWã«ÛÀÜNè6ÞÀì ìrÝÁà
á‡äÈã‡ä àdè ædÜ)éã«àdß&àdâã‡ÛÀܱàdß«ädä è'ó4ô?î6ä ývä ì åd߇ì êd÷ã«ÛÀܱéàdè6ædÜëtÛ6ÞÀì ì  ™à â   ä{á?ã‡Û'å-ãiá—Üã¹à™ârÝÁàdä è‹ã«á&ÿiÛÀà
á‡Ü]ÝÁà
á‡äÈã‡ä àdè ædÜ)éã«àdß4á


H

( =

)




=

#





= D

(

*) =




ø D

=H


=



ò
ô 
ó S

#

ÿiÛÀÜ߇ÜNã‡ÛÀÜ  = åd߇Ü`ã«ÛÀÜNÝÁà
á‡äÈã‡ä àdèTædÜ)éã«àdß4áià™âã‡Û”Ü  ÝÁàdä è‹ã«áiä è  ô  à-ÿ á—Þ”ÝÀÝ'à‹á—ÜNã‡Û”å™ã # ådè”ç   

ä è‹ã‡Üß«á‡Üéãô?Ú¥ÛÀÜèvã«ÛÀÜß«Ü]Üë6ä{á—ã«á¹å™èM<5? m B ÿiÛÀä{é4Û|á—ä ýtÞÀìÈã«ådèÀÜà
ޔá—ì ê=á‡å™ã‡ä{á ”Üá 0 !'ôiò
ô ø)ó&å™è”ç/0 !'ô  ò
ô 
ó4ô&î6ÞÀíÀã‡ß4ådéã«ä  è  ã‡ÛÀÜ)á—Ü]Ü !‹Þ”å-ã«äÈà
è”á dä æ
Üá?å=ì ä èÀÜådßéàdýtíÀä è”å-ã«ä àdèNàdâÁã«ÛÀÜ 2Ãø¥èÀà
è .è6ÞÀì ìŸÝÁà
á‡äÈã‡ä àdè ædÜ)éã«àdß4á±ÿi۔ä é4Ûïæ-å™èÀä{á‡ÛÀÜá÷rÿiÛÀä{é4ÛVéàdè‹ã«ß«å
çŸä{éã«á¯ã‡Û”Ü|ådá«á—ÞÀývÝÀã‡ä àdèÃàdâ#ì ä èÀÜ)å™ß äÈè'çŸÜÝÁÜè”çÀÜè”éܙ/ ô #êÃã‡Û”Ü ì Üývý|åÀ÷'á—ä è”éÜ # å™è”ç  çŸàFèÀà™ã=ä è
ã«Üß4á—Ü)éã÷'ã«ÛÀÜß«ÜWÜë6ä{á—ã«á±åoÛ6ê6Ý'Ü߇ݔì ådèÀÜNá‡Üݔådß«å™ã‡ä  è   # å™è”ç   ô î6ä è”éܱã«ÛÀä{á¥ä{á&ã«ß‡Þ”Ü=âöàdß]å™è6êoé4۔àdä{éܱà™âݔå™ß‡ã‡äÈã‡ä à
è÷
ã«ÛÀ% Ü fÝÁàdä è‹ã«á¥éådèFíÁÜNá‡Û”å-ã‡ã‡Üß«Üçô

ƒãKß«Üý|ådäÈè'árã«àNá—ÛÀà-ÿµã‡Û”å™ãKä:âÁã‡Û”Ü :Tø&èÀà
è ƒè‹Þ”ìÈìŸÝÁà
á‡äÈã‡ä àdèWædÜéã‡à
ß«áå™ß«Ü¥èÀà™ã?ì ä èÀÜåd߇ì ê ä è”çŸÜÝÁÜè'çŸÜè‹ã÷ ã‡Û”ÜèIã‡ÛÀÜ þÝÁàdä è‹ã«áéådèÀèÀà™ãíÁÜ¥á‡Û”å™ã—ã‡Ü߇Ü)çWí6ê`à
߇ä Üè‹ã«Üç`Û6ê6Ý'Ü߇ݔì ådèÀÜáô ƒâ'ã‡ÛÀÜ HFø&ÝÁà
á‡ä:ã«ä àdèNæ
Üéã«àdß4á å™ß«Ü=èÀà™ãiì ä èÀÜåd߇ì êtä è”çŸÜÝÁÜè'çŸÜè‹ã÷Ÿã«ÛÀÜèTã«ÛÀÜß«Ü=ÜëŸä{áã  Dø=è6ÞÀýIíÁÜß4á÷ = ÷”á—Þ”é4Ûã«Û”å-ã

 7# (  == = ) #

H

ò
ô E‹ó S

 

ƒâ@ådì ìÀã‡Û”Ü2=@åd߇ܱà™âã‡Û”ܱá«å™ývܯá‡ädè÷
ã‡ÛÀÜèoÿ#ܱéådè|á‡éå™ì Üiã«ÛÀÜý á‡à`ã‡Û'å-ã =P? SDø Áådè”ç
=  ="H ø™ô 01!'ô]ò
ô E‹ó¹ã‡ÛÀÜèUáã4å-ã‡Ü)á¹ã‡Û”å™ãiã‡ÛÀÜ=àdß«ädä èvì ä Üá#ä è|ã‡Û”ÜNéàdè6æ
ÜëvÛ6ÞÀì ìà™âã«ÛÀÜ=߇Üý|å™ä èÀä èNÝÁàdä è‹ã«áûŸÛÀÜè'éܙ÷ í6êoã«ÛÀÜtìÈÜývý|åŸ÷Ÿã«ÛÀÜWà
߇äd ä èéå™è”èÀà™ã í'Ütá—Üݔå™ß4å-ã«ÜçFâö߇à
ý_ã‡ÛÀÜI߇Üý|å™ä èÀä èW  ÝÁàdä è‹ã«á¯í6êåoۋê6ÝÁÜß«ÝÀì{å™èÀÜd÷ å™è'ç|ã«ÛÀÜ`Ý'à
ä è
ã4áiéådèÀèÀà™ãiíÁÜNá—Û'å-ã—ã«Üß«Üç8ô

ƒâã‡ÛÀÜ =

ådß‡Ü èÀà™ã¯ådì ìÁàdâ@ã«ÛÀÜNá«å™ývÜ=á—ädèr÷6ݔì å
éÜ=å™ì ìÁã‡ÛÀÜ=ã«Üß«ý|á¥ÿiäÈã‡ÛèÀÜ‹å-ã‡ä æ
Ü

(    H (           



=

àdèFã‡ÛÀÜ`ß«ädۋã ò
ô B6ó



ÿ ÛÀÜ߇Ü # ÷   å™ß«ÜKã‡Û”ܹä è”çŸä{éÜáàdâÀã‡ÛÀÜ¥éà
߇߫Üá‡ÝÁàdè”çŸä èiÝ'å™ß‡ã‡äÈã‡ä àdè=àdâ   ò,ä.ô Üdô@à™âŸã«ÛÀÜ¥á—Üã Ãÿiä:ã«Û`ã‡Û”Ü i àdß«ädä è|ß«Üývà-æ
Üç”óô  à-ÿgá‡éå™ì ܱã‡ÛÀä{á¥Ü !‹Þ”å™ã‡ä àdèá—àtã«Û”å-ã]Üä:ã«ÛÀÜß
   H ø`å™è”ç
     1 ø™÷     àdß
   1 ø|å™è”ç
  H_ødôî6ÞÀݔÝ'à‹á—ÜvÿiäÈã‡Û”àdޟãNì à
á«á à™â dÜèÀÜß4å™ì äÈãêã‡Û”å™ã`ã‡ÛÀÜvì{å-ã‡ã‡Üß     â 0 !'ôUò
ô B‹ó ä{á±ã‡ÛÀÜvÝÁà
á‡äÈã‡ä àdè¾ædÜéã‡à
ß`à™â#åÝÁàdä è‹ã`ìÈê6ä èFäÈè ã‡Û”Ü ÛÀà
ì ç”áôvÚ¥ ÛÀÜèïã«ÛÀÜvì Üâ{ã`۔å™è'ç  á‡ä{çŸÜ¼à™+





éà
è6ædÜëtÛ6ÞÀì ìÀà™â8ã‡Û”Ü]Ý'à
ä è
ã4á-
   .  ò,àdß÷
äÈâ8ã‡ÛÀÜ¯Ü !‹Þ”å™ì äÈãêNÛÀà
ì{çÀá÷
à™â8ã‡ÛÀܯÝ'à
äÈè‹ã4á-
   .™ó4÷  à
äÈè‹ã4á å™è'çWã«ÛÀÜiß«ä 
ۋã۔ådè”çtá—ä{çŸÜ¥ä{áã‡ÛÀÜiÝÁà
 á‡äÈã‡ä àdèWædÜéã‡à
ß?à™âÁå=ÝÁàdä è‹ã?ì ê‹ä è±ä èWã«ÛÀÜ]éà
è‹æ
ÜëNÛ6ÞÀì ìŸà™âÁã‡Û”ÜiÝ'    ÷rá—àvã«ÛÀܼéàdè6æ
ÜëFÛ6ÞÀì ì{áià-æ
Üß«ì ådÝ÷'ådè”çUí6êoã«ÛÀÜtìÈÜývý|åŸ÷Àã«ÛÀÜNãÿ#àoá‡Üã4á¯àdâKÝ'à
äÈè‹ã4á¯éådèÀèÀàdã±íÁÜ  ådß«å™ã‡Ü á‡Üݔ çoí6êFåtÛ6ê‹ÝÁÜß«ÝÀì{å™è”Ü™ô?ڥۋÞ'á#ã‡ÛÀ% Ü fÝÁàdä è‹ã«á¥éå™èÀèÀàdãií'ÜNá‡Û”å-ã‡ã‡Ü߇Ü)ç8ô





q "6Ք$Ô 2)  " ƒâ ã‡ÛÀÜUçÀå™ã«åïä{á ;çŸä ývÜè'á—ä àdè'å™ì±ò,ä.ô Üdô m  ó4÷#ã‡ÛÀÜÃçŸä ývÜè”á—ä à
èµà™â±ã‡ÛÀÜUývä èÀä H  ý|å™ìÜýtíÁÜçÀçŸä èá—Ý'ådéÜd÷8âöàdß=ÛÀàdývàdÜèÀÜà
ޔáiÝÁàdì ê6èÀàdývä{å™ìõdÜß«èÀÜì á¯à™â¹çÀÜdß«ÜÜ ò Oò'< # DK< óH ò'< # 9   m  ó÷”ä{á     7 #  ô < ó  D < # D@< ? 









u=Ô-ÕrÕE( ä ß4áã¯ÿ#ÜIá‡ÛÀà-ÿgã‡Û'å-ã ã‡ÛÀÜWã‡ÛÀÜtè6ÞÀýIíÁÜß±à™â&éàdývÝÁàdèÀÜè
ã4á]àdâ  ò<ó¯ä á    7 #  ô å™íÁÜì@ã‡Û”Ü éà
ývÝ'à
èÀÜè‹ã4á`à™â[ådá`ä è 0 !”ôµò -ù
óôÃÚ¥ÛÀÜèµåÃéàdývÝÁàdèÀÜè‹ãNä áNÞÀèÀä !‹ÞÀÜì êÃä{çŸÜè
ã«ä”Üçïí6êÃã«ÛÀÜFé4ÛÀà
ä{éÜ à™â#ã‡Û”Ü ä è
ã«Ü 
Üß4á  = SŸ÷
= )  =@H2rô  à-ÿ[éàdè”á‡ä{çŸÜß6ÅàdíÜéã4á`çŸä{áã«ß‡ä íÀÞÀã‡Üç å™ývà
è
á—ã  ø # ݔådߗã«ä:ã«ä àd è”á¼òöè6ÞÀýtíÁÜß«Üç ø¼ã‡ÛÀß«àd Þ 
Û  ø)ó÷á‡Þ”é4Ûïã‡Û'å-ãNàd í Üéã4áNå™ß«Üvå™ì ì à-ÿ¹Ü)çTã‡àUíÁÜvã«àTã‡ÛÀÜ|ì Ü â{ãNà™â  å™ì ì?ݔådߗã«äÈã‡ä àdè”á÷ràdß=ã«àTã‡ÛÀÜo߇ä dۋã`à™âiå™ì ì?ݔådߗã«äÈã‡ä àdè”áôoî6ÞÀÝÀÝÁà
á‡C Ü  à
 í Üéã«á=â,ådìÈì&í'Üãÿ¹ÜÜèVݔå™ß‡ã‡äÈã«äÈà
è”á   d å ” è ç 7 ™ ø ô r   Ü ± ã « ã À Û { ä ± á  é d à « ß ‡ ß ) Ü — á Á Ý d à ' è  ç ‡ ã F à | å ‡ ã  Ü ‡ ß ý ) ä U è ‡ ã À Û t Ü À Ý « ß Ÿ à Ÿ ç ” Þ éã ä è 01!'ôvò -ù‹ó4ôNî6ä ývä ì åd߇ì êd÷    # àd í Üéã4áiâ,ådìÈì ä  è Iã‡àoã‡ÛÀÜNì Üâ{ã¯à™â&ådìÈìݔå™ß‡ã‡äÈã‡ä à
è”áiéà
߇߫Ü á‡Ý'à
è”çÀá#ã«àFå¼ã«Üß«ý )  ÷å™è'/ ç  àd í Üéã4áiâ,ådìÈì ä  è  # ã‡à¼ã«ÛÀÜNß«ä dۋãià™âKå™ì ìݔådߗã«ä:ã«ä àdè”á¥éà
߇߫Üá‡ÝÁàdè”çÀá¹ã«àoåtã‡Ü߇ý )  ô#Ú¥Û6ޔá]ã‡ÛÀÜ`è6ÞÀýtí'Üß]à™â?çŸä{á—ã‡ä è”éã¥ã«Üß«ý|á  à™â ã‡Û”ÜTâöàdß«ý )  )  9;9;9)  D
= )  =MH ,D  =  Sïä átã«ÛÀÜUè6ÞÀýtí'Üßtàdâ ÿ¥å êÅà™â`çŸä{á—ã‡ß«äÈí”ÞŸã‡ä  è  #  #  á#å™ývà
è 
á—ã¹ã«ÛÀÜý|á‡Üì æ
Üá÷6ývàŸçŸÞ”ìÈàtÝÁÜß«ýIޟã4å-ã«äÈà
è”á¹à™âã‡ÛÀÜNÝ'å™ß‡ã‡äÈã‡ä àdè'á¹ådè”ç ã‡Û”Ü`àd í Ü)éã«á¯å™è”çݔådߗã«ä:ã«ä àdè”  ÝÁÜß«ýIޟã4å-ã«äÈà
è”á¹à™âã‡ÛÀÜ`à
 í Üéã«á÷ÀÿiÛÀä{é4Ûoä{á    7 #  ô





 Üë6ã?ÿ¹Ü¥ýtޔáãá—Û”à-ÿÅã‡Û”å™ãã‡ÛÀÜiá‡Üã?à™â”æ
Üéã‡àdß4á@ÿiä:ã«ÛtéàdývÝÁàdèÀÜè‹ã«á      ò<@óá‡Ý”å™èWã‡ÛÀÜiá‡Ý”ådéÜ Uô Ú¥ÛÀä{á¹âöàdì ì à-ÿ]áKâöß«àdý ã‡ÛÀÜ=â,å
éã¥ã‡Û”å™ãiã‡ÛÀÜNéàdývÝÁàdèÀÜè‹ã«á#à™â  ò<@ó#å™ß«Ü=ì äÈè”Üå™ß«ì êWä è”çŸÜÝ'Üè”çŸÜè
ãiâöޔè”éã«äÈà
è”áô Àà
ß`á—ÞÀݔÝ'à‹á—ÜIäÈè'áã«ÜådçUã‡Û”å™ã ã«ÛÀÜtäÈý|ådÜNàdâ  å
éã‡ä èFàdè < ? ä{á±åá‡ÞÀí”á‡Ý”ådéÜWà™â UôIÚ¥ÛÀÜèÃã‡Û”Üß«Ü ÜëŸä{á—ã«á¥å ÀëŸÜçèÀàd è Üß«àtædÜéã‡à
ß  ? yá‡Þ”é4Ûoã«Û”å-ã



çŸä ý  

(

*) =



= 

ò @ó

= <

H

S

#

 <



?

ò
ô$ú
ó

@I

±á—ä ètã‡Û”Ü`ì ådí'Üì äÈèNä è
ã«ß‡àŸçŸÞ'éÜçFå™íÁà-ædÜd÷”éàdè'á—ä{çŸÜß#åtݔådߗã«ä éÞÀì{å™ß¥éà
ývÝ'à
èÀÜè‹ã¥à™â 





ò<ó   D

(   =PH ,I = ) #

 

)

ó

#

9;9;9 ò   ) 

 ò
ô ‹ó

î ä è”éÜ/01!'ôÅò
ô údó`ÛÀàdì{çÀá âöà
ßWå™ì ì9




ä è 01!'ôïò -ù‹ó=éÜ߇ã«ådä èÀì êU۔ådáNådì ì

ó 

ò
ô dó

ã«à 0 !'ôFò
ô$ú
ó4÷ÿiÛÀä{é4ÛÃÿiä ì ìÝÀä{é4õFã«Û”å-ã`à
èÀÜWã«Üß«ýyÿiäÈã‡Û éàd߫߇Ü)á—ÝÁàdè'çŸä è|ÝÁà-ÿ¹Üß«á±à™âKã«ÛÀÜC)N=¥ä è 0 !'ô ò™ù
ó÷dä æ6äÈè 



  

H

ò
ô ü‹ó

SI

î6ä è”éÜNã«ÛÀä{á]ä{áiã«ß‡ÞÀÜIâöàdß å™ì ì@é4۔àdä{éÜ)áià™â  #

 ýtޔáãiæ-å™è”ä á‡Ûô,]Üè”éÜ=ã«ÛÀÜ`ä ý|å dܱà™â Uz  z 

€!#



 D:9;9;9D   á‡Þ”é4Ûã‡Û'å-ã
=*) #  å
éã‡ä ètàdè!<5? á‡Ý”ådè”á Uô



Hr÷Üæ
Üß«êéàdývÝÁàdèÀÜè
ã±à™â

=1H

 q#Õ+%
Ô$#ÀÒ8Óok %'# )Ù P‹Ô #ÀÒÖ(ok }tÕ'×Ò@Ö 

Ú¥ÛÀÜ¥âöà
ìÈì à-ÿiä è]ÝÁàdä è‹ã?ä{áéÜè‹ã‡ß4å™ì6ã«à=ã‡ÛÀÜ]ådß 
ÞÀývÜè‹ãô $±èÀÜ¥èÀàdß«ý|å™ì ì ê±ã‡ÛÀä èÀõŸáà™âå=éà
ìÈì Ü)éã‡ä à
è`à™â8Ý'à
äÈè‹ã4á ådáíÁÜä  è Má‡Û”å-ã‡ã‡Üß«ÜN ç ]í6êWå á—Üãàdâ”âöÞÀè”éã‡ä àdè'á÷™äÈâÀâöà
ßådè6ê=é4۔àdä{éܹà™â'ì{ådí'Üì árâöàdßã‡Û”Ü#ÝÁàdä è‹ã«á÷™å±âöÞÀè”éã«ä àdè

âöß«àdýgã‡ÛÀÜ¥á‡Üãéå™èNíÁܹâöàdÞÀè”çWÿiÛÀä{é4ÛNådá«á—ä
è”á8ã‡Û”à
á‡Ü&ì{å™íÁÜì{áã«à]ã«ÛÀܹÝÁàdä è‹ã«áô@Ú¥ÛÀÜ#ð ïçŸä ývÜè'á—ä àdè à™â”ã‡Û”å™ã á‡Üã¥à™ârâöÞÀè'éã‡ä à
è”á&ä{á&ã‡ÛÀÜèFçŸÜ ”èÀÜ)çFå
áKã«ÛÀܱý|å™ëŸäÈýtÞÀý[è‹Þ”ýIíÁÜß#à™ârÝÁàdä è‹ã«á&ã‡Û”å™ã¥éå™èFí'Ü=á—à¼á‡Û”å-ã‡ã‡Üß«Üçô ]à-ÿ#Üæ
Üß÷réàdè'á—ä{çŸÜß åá—ì ä dۋã«ìÈêTçŸä 8Üß«Üè‹ã=çÀÜ ”èÀäÈã«äÈà
èô ÜãNåFá‡Üã`àdâ¹ÝÁàdä è‹ã«á±í'Ü|á‡Û”å-ã‡ã‡Ü߇Ü)çÃí6ê¾åá—Üã à™â?âöޔè”éã«äÈà
è”áiäÈâ?âöàdß å™è6êFé4۔àdä{éÜ`àdâKì ådí'Üì{á#âöàdß]ã«ÛÀÜWÝÁàdä è‹ã«á÷Áå|âöÞÀè”éã‡ä àdèTâöß«àdý ã‡ÛÀÜtá—Üã éå™èUíÁÜNâöàdޔè”ç ÿiÛÀä{é4ÛTådá«á—ä dè'áKã«ÛÀ: Ü * I M  ì{å™íÁÜì{á¹ã‡àvådìÈì8ã«ÛÀÜ`ÝÁàdä è‹ã«áô

ådäÈèì Üã¥ã‡Û”Ü`ð  çŸä ývÜè”á—ä à
èvà™âã‡Û”å™ã¯á—Üã à™ââöÞÀè”éã‡ä àdè'á¹íÁÜNçŸÜ ”èÀÜçTå
á#ã‡ÛÀÜ`ý|å™ë6ä ýtÞÀý è6ÞÀýIíÁÜߥàdâÝÁàdä è‹ã«á¹ã‡Û'å-ã¯éådèFíÁÜNá‡à¼á‡Û”å™ã—ã‡Ü߇Ü)ç8ô

ƒãoä{átä èOâ,å
éãvã«ÛÀä{á|á—Ü)éà
è”ç çÀÜ ”èÀäÈã«äÈà
è ò,ÿiÛÀä{é4Û ÿ#ÜÃådçŸà
ݟã¼âö߇à
ý ÛÀÜ߇ÜUà
è'óIã‡Û”å™ã|Üè‹ã‡Üß«átã«ÛÀÜUð  íÁàdÞÀè”çµÝÀ߇à6àdâ,á|òöðKådÝÀèÀä õÁ÷#ø)ù'-ùÀû -¯Üæ‹ß«à-êdÜd÷ % ê àd ß µå™è”ç Þdà‹á—ä.÷#øùdù
óô"$¯â]éà
ÞÀß4á—ܼâöà
ßNâöÞÀè”éã‡ä àdè'á ÿiÛÀà‹á—Ü]ß4å™ è dÜiä{ á - Iø .tòöä.ô ܙô?å™ì ì'ç”å-ã«åNÿiäÈì ìÀíÁܱå
á‡á‡ädèÀÜ)çtÜäÈã‡ÛÀÜßKÝÁà
á‡ä:ã«ä ædÜiàdß&èÀÜ‹å-ã‡ä æ
Ü]éì{ådá«á«ó÷dã«ÛÀÜ]ãÿ#à çŸÜ ”èÀäÈã‡ä àdè'á¹ådß‡Ü ã‡Û”Ü=á«å™ývÜdô ¯à-ÿ¹ÜædÜß÷ŸäÈâ@å™ì ì8ÝÁàdä è‹ã«á¹â,å™ì ì ä  è Nä èá‡àdývܱ߇ Ü dä àdèFå™ß«Ü=á‡ä ývÝÀì êtçÀÜÜývÜ)çvã«à íÁ0 Ü 4Üß«ß«àdß4á À÷‹à
ß éàd߫߇Ü)éKã À÷
ã«ÛÀÜ`ãÿ#àvçŸÜ ”è”ä:ã«ä àdè”á¥å™ß«Ü`çŸä ÁÜ߇Üè‹ãô
á]å¼éàdè'éß«Üã‡Ü`ÜëÀå™ývÝÀì ܙ÷Ÿá‡ÞÀÝÀÝÁà
á‡Ü ÿ#Ü çŸÜ ”èÀÜ  ‹å™Ý|ä è‹ã‡à
ìÈÜß«ådè‹ã&éì{ådá«á—ä ”Üß«Ká Ÿ÷
ÿiÛÀä{é4Ûoå™ß«Ü¯ì ä õdÜ ‹å™Ývã«àdì Üß4å™è‹ã¹éì{ådá«á—ä ”Üß«á÷díÀޟãiÿiÛÀä{é4Û|ì{ådí'Üì å™ì ì6ÝÁàdä è‹ã«áì ê‹ä  è ¯äÈèIã‡Û”Ü#ý|ådß 
ä è`àdßàdޟã4á—ä{çŸÜ¹ã‡Û”Ü¥á‡ÝÀÛÀÜ߇Üiå
6 á M I(G«ô ¹à
è”á—ä{çŸÜßå 
ådäÈèNã‡ÛÀÜiá‡ä:ã«Þ”å-ã«ä àdèNä è ä dÞÀß«ÜNø Ÿ÷ŸíÀޟã]ådá«á—ä 
ètÝÁà
á‡äÈã‡ä ædܱéì{ådá«á?ã«àtådìÈì'ã‡Û”߇ÜܱÝÁàdä è‹ã«áôڥ۔ÜèFå 
ådÝ|ä è‹ã‡àdì Üß«ådè
ã&éì{ådá«á—ä ”ÜßKÿiäÈã‡Û ý|å™ß dä èÃÿiä{ç6ã«Û dß«Üå-ã«Üß ã‡Û'å™èïã‡ÛÀÜoí”å™ì ìKçŸä{å™ývÜã‡Üß éå™èÀèÀàdãWá—Û'å-ã—ã«Üß=ã«ÛÀÜ|ÝÁàdä è‹ã«á=äÈâ¹ÿ#Üvޔá‡Ü¼ã‡ÛÀÜ ”ß4á—ã çŸÜ ”èÀäÈã‡ä àdèFà™ â á‡Û”å-ã‡ã‡ÜKß À÷6í”ÞŸã¯éådèá‡Û”å-ã‡ã‡Üߥã‡ÛÀÜ`ÝÁàdä è‹ã«á¥äÈâ@ÿ#Ü=ޔá‡Ü±ã«ÛÀÜNá‡Üéà
è”çïò,éà
߇߫Üéã4ó#çŸÜ ”èÀäÈã‡ä àdèrô
äÈã‡Û¼ã«ÛÀä{á&éå ædÜå™ã?äÈè¼ývä è”ç8÷‹ÿ¹Üiè”à-ÿDà
ޟã‡ì ä èÀÜ¥ÛÀà-ÿ ã‡ÛÀܱð Oí'à
ÞÀè”çÀá¹éå™è|ådÝÀÝÀì ê`ã«à`âöÞÀè”éã‡ä àdè”á&ÿiäÈã‡Û ß4å™è
Ü- Iø DKS .‹÷rÿiÛÀÜ߇ܼã«ÛÀÜvì{å™íÁÜì SývÜådè”á ã‡Û'å-ã`ã‡Û”ÜvÝ'à
äÈè‹ã`ä{á=ì{å™íÁÜì Üç;éà
߇߫Üéãô Yò2Ú¥ÛÀÜví'à
ÞÀè”çÀá ÿiä ì ìå™ì{á‡à¼ådÝÀÝÀì ê|ã«à|âöÞÀè'éã‡ä à
è”á]ÿiÛÀÜß‡Ü Soä{á]çŸÜ 'èÀÜçTã«à|ývÜådè 4Üß«ß«àdß Ÿ÷”íÀÞÀã¯ã‡Û”ÜIéà
߇߫Üá‡ÝÁàdè”çŸä ètð  çŸä ývÜè'á—ä àdèIÿiä ìÈì”í'Ü]۔ä 
ÛÀÜß÷dÿ#ÜådõdÜè”äÈè±ã«ÛÀܯíÁàdޔè”ç8÷Ÿå™è”ç¼ä è¼àdÞÀß#éå
á—Üd÷
ý|å™õ6ä è äÈã&ޔá‡Üì Üá«á«óô>
Vܯÿiä ì ì âöàdì ì à-ÿ ã‡Û”Ü`èÀà™ã4å-ã‡ä à
èoà™â¥*ò -±Üæ6ß«à-êdܙ÷ % ê à
 ß Uå™è”! ç r Þ dà
á‡ä.÷øùd ù 
óô

¹à
è”á—ä{çŸÜßiÝ'à
ä è
ã4á <5?QA  ÷Áådè”çì Üã$@ò'<óiçŸÜèÀà™ã«ÜNåvçŸÜè”á‡äÈãê|àdè A  ô rÜãD=ÃíÁÜWåtâöÞÀè'éã‡ä à
èà
è A  ÿiäÈã‡ÛÅß4å™è
Ü - Iø D S .6÷å™è'çVìÈÜã  í'ÜåÃá—ÜãWà™â]á‡Þ”é4ÛVâöÞÀè”éã‡ä àdè'áô rÜãIÜå
é4Û5< ۔å ædÜ|ådèVå
á‡á‡àŸéä{å-ã«Üç ì{å™íÁÜì L  ?3- IøC.‹ô9rÜã-7< # D:9;9:9JD < B .WíÁÜNå™è6ê ”è”ä:ã«Ü=è‹Þ”ýIíÁÜßiàdâ@ÝÁàdä è‹ã«á¥ä è!A  ?ã‡ÛÀÜèTÿ¹Ü=ß«Ü !‹ÞÀä ß‡Ü  ã‡àvÛ'å ædÜ ã‡ÛÀÜ`ݔ߇à
Ý'Üߗãêvã«Û”å-ãiã«ÛÀÜß«Ü`ÜëŸä{á—ã«á¥å-ã]ì Üå
áãià
èÀÜ = ?  á—Þ'é4Ûoã«Û”å-ãD=ò<"=2ó ?3- Iø .@)N=—ô Àà
ß dä æ
ÜL è =÷'çÀÜ ”èÀÜ=ã«ÛÀÜNá‡Üãià™âÝ'à
ä è
ã4á  í‹ê

:- ), L H  H

ø D =@ò,)'ó1H

`ø .
-E), LH2`ø D = ò*)'ó

VÜ`ß‡Ü !‹ÞÀä ߇ܯã‡Û”å™ãiã‡ÛÀÜ U = íÁÜNá‡Þ”é4Ûoã«Û”å-ã]ådìÈìá‡Üã4á 



!

ÒÙöÓ)Ù,ÕÁÒ 0Üã<>=

 B

ò/-$<>= D

=+.™ó1H

ø D;9:9;9JDKO

DKF H

-7<,= D =+.Nã‡àvíÁÜ

òø =$Oó

( =

B

)



#

<



í'Ü!O

!

,= H H
vò<5?

 ó

ò
ô øS‹ó



!

à™ã«Ü=ã‡Û”å™ã#ã«ÛÀÜ`ÜývݔäÈß«ä{éådì”ß«ä á‡õvä{á Ü߇àtäÈâ  = ò< = ó



8Q< = ô S

ò
ô ødø ó



 

<

âöàdßiÿiÛÀä{é4Û @ = ò'<@ó

;9:9;9D <

ÒÙöÓ)Ù,ÕÁÒ  Àà
ß ÀëŸÜçVò< #

- -$< # ;D 9:9;9D < B .

H

I

à™ã‡ÜNådì{á—àIã‡Û”å™ã¥ã‡ÛÀà‹á—Ü`ÝÁàdä è‹ã«á 

çŸÜè”à™ã‡Ü ã«ÛÀÜNá—Üãià™âådì ì

ÝÁàdä è‹ã«áô5
ïÜUçŸÜ”èÀÜFã‡ÛÀÜÜývÝÀä ß«ä{éå™ì&߇ä{á‡õ¾âöà
ßIã«ÛÀÜTá—Üã

ÒÙöÓ)Ù,ÕÁÒ P
VÜNçŸÜ ”è”Ü=ã‡ÛÀÜNå
éã‡Þ'å™ìß«ä á‡õ¼âöàdߥã«ÛÀÜ=âöÞÀè”éã‡ä àdè T = ã«àví'Ü ò 8ó

ò
ô ù‹ó

1I

ÿiÛÀÜ߇Üvä{á#ã«ÛÀÜ`ä è”çŸä{éå™ã‡àdß&âöÞÀè”éã‡ä àdèô



øC.

å™ß«Ü±ývÜ)ådá‡ÞÀß«ådíÀì ܙôÜã




 ô

H

D

 . ?

B

ó ?

A

H

tçÀàtè”à™ã¯éà
è‹ã‡ß«äÈí”ÞŸã‡Ü±ã‡à¼ã«ÛÀÜNådéã‡Þ”ådì߇ä{á‡õ'ô S

 ÷”ì Üã í'Ü=ã«ÛÀÜ`è6ÞÀýtí'Üß#àdâçÀä 8Üß«Üè‹ãiá—Üã«áiä è ò
ô ø
ó




ÿiÛÀÜß‡Ü ã«ÛÀÜNá—Üã«á 

‹ò DKOó1H <

åd߇Ü=çŸÜ”èÀÜçTådí'à-æ
ܙôKÚ¥ÛÀÜ`è .ã‡ÛTá—Û'å-ã—ã«Üß]éà‹Ü
|éä Üè‹ãià™â [  ä{áiçŸÜ”èÀÜ)ç ý|å-ë     <



VÜ|å™ì{á‡àçŸÜ”èÀܼã«ÛÀÜvð  ÿiÛÀä{é4Û 
ò D ”ó H   ô

 


±ò'< # :D 9;9;9D <

ó B

ò
ô ø ‹ Eó I



çÀäÈývÜè”á‡äÈà
èTâöàdß=ã«ÛÀÜ|éì{ådá«á

ã‡àUíÁܼã‡ÛÀÜvý|å™ë6ä ýtÞÀý_äÈè‹ã«ÜdÜß )

    

 

ø¼âöà
ß


       5  M2 2 ICK  M"  I(# M # I   7 $+ I G      M I( MJK    *"CM  B ò -7<,= D =+.™ó ò Á = ó  7‹ò D Oó M  M2  I"CM  7+*" M O 5 I(*  G ò< DIII DK< ó ?!A  6*M   M I  M * G $   GJM  I  G $   *  6*6 = ?  G (  G 2#+=ò<=.ó ?3- IøC.8 <,# =    B  G8 M G  *  I5K  # M " M/M2  G 5    I * M    * *,+  6 + I( $ K   M  I   #NG $   =




vò   B ò 7- <,= D =+.™ó 

,=  .4ó1 ü ‹ò D Oó6ÜëŸÝ 7

ò 8ó

B 





 

ò
ô øJB6ó

 8 

Ú¥ÛÀÜtÝÀß«à6à™â?ä{á]ÜëÀå
éã‡ì êFã‡Û”å™ã±àdâ±ò*-±Üæ6ß«à-êdܙ÷ % ê àd ß å™è”ç rÞ
à
á‡ä2÷@øùdù
ó÷î6Ü)éã«äÈà
è”áNø76ô EŸ÷ø6ô Boådè”ç ø Ÿô$úŸ÷Ú¥ÛÀÜàdß«Üý|á¼ø76ô úTå™è”ç ø76ô Ÿô!
ïÜF۔å ædÜ|çŸß«àdݔÝ'Ü)ç¾ã«ÛÀÜ5á—Þ”ÝNã‡àÃÜývÝÀ۔å
á—äÜtã«Û”å-ãNã«ÛÀä{á`ÛÀà
ì ç”á  âöàdß±å™è6ê|à™âã‡Û”Ü`âöÞÀè”éã‡ä àdè”á =ô ;èTݔå™ß‡ã‡ä{éޔì ådß÷6äÈã]ÛÀàdì{çÀá#âöà
ßiã‡Û”à
á‡Ü%=¾ÿiÛÀä{é4ÛýväÈè”äÈýväÜiã‡ÛÀÜWÜývÝÀä ß«ä éå™ì Üß«ß«àdßNå™è'ç¾âöà
ßNÿiÛÀä{é4ÛÅå™ì ìã‡ß4å™ä èÀä èTçÀå™ã«åã«ådõdܼã«ÛÀÜ|æ-å™ì ÞÀÜ)á- Iø .6ô  àdã‡ÜvÛÀà-ÿ#Üæ
Üß=ã«Û”å-ãNã«ÛÀÜ|ÝÀß«à6à™â àdè”ìÈêFÛÀàdì{çÀáiâöàdßiã«ÛÀÜIá‡Üéàdè”çUçŸÜ ”èÀäÈã‡ä à
èFàdâ?á‡Û”å™ã—ã‡Ü߇ä  è  
ä ædÜèTå™íÁà-ædÜdô @ä è”ådì ìÈêd÷ŸèÀà™ã«Ü`ã‡Û”å™ã¯ã«ÛÀÜNޔá‡Þ”å™ì âöàdß«ý à™â8ã‡ÛÀܯð µíÁàdޔè”çÀáKä{á?Ü)ådá‡ä ìÈêNçÀÜß«äÈæ
ÜçWâö߇à
ý 0 !'ô¥ò
ô Jø B‹ó?í‹êtޔá‡ä  è ‹ò DKO ó 1gò O8= ó vòöÿi۔Üß«Ü < è Wâöàdß ô ä{á¹ã‡ÛÀÜ`ð  çŸä ývÜè'á—ä àdèÁóiòöðKådÝÀèÀä õÁ÷røùdù‹údó÷Ÿá‡Üã‡ã‡ä  è N0H  ü ‹ò D Oó6ÜëŸÝ 7 B   8  ÷'ådè”çá—à
ìÈæ6ä 



<

¹ì Üåd߇ì êVã«ÛÀÜá‡ÜÃß«Üá‡ÞÀìÈã«ávå™Ý”ÝÀì êÅã‡àµà
ÞÀß 
ådÝDã‡àdì Üß«ådè
ã|éì å
á‡á‡ä”Üß4áIà™âWî6Üéã‡ä àdè ‹ô ø™ô Àà
ßvã‡ÛÀÜýT÷]å ݔådߗã«ä éÞÀì{å™ß`éì{å
á‡á‡ä”Üß =R?[ä{áNá‡Ý'Ü)éä”ÜçVí6êVåÃá‡ÜãIàdâ]ݔå™ß4å™ývÜã‡Üß«á- D! D  .‹÷?ÿiÛÀÜß«Ü ä áWå í”ådìÈìä è A  ÷  ? m ä{á]ã«ÛÀÜvçŸä{å™ývÜã‡Üß à™â v÷ ä{á å  øvçŸä ývÜè”á‡ä àdè”ådìràdß«ä Üè‹ã‡Ü)çUÛ6ê6Ý'Ü߇ÝÀì{ådèÀÜWä è  ã‡Û”Ü|ý|å™ß dä èm  ÷?å™è”ç   m ä{áNåUá«éå™ì{ådß±ÿiÛÀä{é4Ûïÿ¹ÜvÛ'å ædÜ|éådì ìÈÜ)çT ? ô äÈã«á‡ÜìÈâ¹ä{á`á‡Ý'Ü)éä ”Üçïí6ê¾äÈã4á èÀà
߇ý|ådìò,ÿiÛÀà
á‡ÜNçŸä ß«Üéã«ä àdèFá‡ÝÁÜéä ”Ü)á¥ÿiÛÀä{é4ÛFÝÁàdä è‹ã«á ò ó]å™ß«Ü ì{ådí'ÜìÈÜ)çoÝÁà
á‡äÈã‡ä ædܼòöè”Ü 
å™ã‡ä ædÜ óKí6ê 7 ã«ÛÀܼàdß«ä 
ä èô Ààdß`å dä ædÜ  ã«ào ã‡Û”ÜIâöޔè”éã«äÈà
è'ó4÷ådè”ç í‹êTã«ÛÀܼývä èÀä ý|å™ìçŸä{á—ã«ådè”éÜWâö߇à
ý è = ? ¯÷rã‡Û”Ü ý|å™ß dä èvá‡Üã  ä{á¹çÀÜ ”èÀÜ)çoå
á&ã«ÛÀÜ`á—Üã¥éàdè”á‡ä{áã«ä  è `à™â@ã‡ÛÀà‹á—Ü Ý'à
ä è
ã4á&ÿi۔à
á‡Ü¯ývä èÀä ý|å™ì”çÀä á—ã«ådè”éܯ㇠à ä{áiì Üá«á¥ã‡Û'å™è := Ÿô -¯Ü ”èÀ Ü 1   | ÷    1 ÷8ådè” ç    1 ô¯Ú¥Û”Ü`âöÞÀè”éã‡ä àdè = 7 7   ä{á¹ã‡ÛÀÜèÃçŸÜ ”è”ÜçTådá¹âöàdì ì à-ÿ]á 









=ò<@ó1H

øE<

?



5=ò<ó D

ådè”çFã‡ÛÀÜNéàd߫߇Ü)á—ÝÁàdè”çÀäÈèWá—Üã«á 

H

`ø

E<

?

7 D

=ò'<ó H

Iàdã‡ÛÀÜ߇ÿiä{á‡Ü S

å
á¥ä è,0 !”ô]ò
ô$ù
ó4ô



UÕ'Ó

 5  -5 7 )  $ %'2" ° 41.¹´ 0 !¥!K 3 "$41," 043 +-]. ¹" + !g12+)"$*¥; "$;" 12*]`*2" ¶" N9:—; " W" *¹-2—¼1 *21,-)§ º1.2  "$#$%'& ¦À 12.+- ! 5]< 3 0 +-.K"$*i*,"$!#² 3 $¹3  !¥;1.,3 "$4'" 12—,²-.;1240 12" 9'Î6. 3 ¹!K  1," ²  " —2*+@41  "$#()& "$*2+ ² "$"$* 0  ..*,² 5@<)" K1 K*2" 4$ *,1.0 24" 1 1,+-3 .)"$ 1 9‹041,3t ?(  /  ,§i +-r0 9: 1," 3 (d" #;½1.2—!¹"$·— !K-*20 1?3 (d¥²-240  $;1 04123 +-i3 20)"$ 1 3 91,+-04¥3 9ˆ ;123 " + *.¥ 3 1 0 -2*#*,²
;3 " 9ˆ§N1204+-3 i( 3  )2§ r1& 3 " 0 12.*.;1," 3 9'04 3 *212.0 4" 12* 043 1,+-0 ¹.)"$ 1K!& *21(d0 &3²-.4  5”1  (   ,§` .2—*,² 0 )" ]1 1,+-¹ 041," #3 " 1,+-K043¼  " 0 —3  !K( 0 " 412" 043 3 *. 0 9'04" 3 ¿-4 3 " 1{§N *212.4" 12* 3  9”12+ ¹2)"$ 12* 9”120 +  39ˆ ;12" 0 *@r+ 3 *2?04 3 1 3-04-3 .» *@3 *2²d" 9:° §]1,+-3 ?(   20 § 3 043 3 0 3 0 3 4 0 3 0 4 0 3 0 0 3 5  ,2 — " W!¥.+)" / .`"  (d] *.I9 ¹ §W9: 1," *21,"$!¥41," 4$  5 É 3 1,+ "$*¹²-4²d— 12+ ¯² + .*.2" 12+-! ,21.24" " $ .&9 À1,+-²-.!¥;3 123 8*21,"$!¥41,3 " ²- —-. ,21.0 *,12" $ 3 .¹9 ”3 1,+-r04 3 !#²  12412" 043 9d12+ 04 » 0 ,2²d—,9 2!¥ /.¯9 1,+-¥04 3 .40  "$·412" ;-23 § 0 "  —2 0 241.¹* 043

9ˆ ;12" 4 3 - 3  0 1.0 *,1K*.;1 5 5 043 0 *2" ·—3 12 043I  *° 1 " 3 " 1{§   $0 *.* 312+-," 0 *.K*21241. 3 3 0 3  3 3 0 5

ò
ô ø)ú
ó



Ä "  12+- !¥ ,21.*218*.;1 .&²d—,+-4²-*Á8*2+  $]4$* -*. ,212.4" *.;1 & .&( )1812+ + ( (§" *212* 1Á12+-2  5 @ 3 0 0 3 0 0 2*2° 1 3  @-5 *2@1,+-@12.! , ,"$ 12±+§²d2²   /.¹1 !#² + *2"$·1,+-4181,+-!¥41,+-!¥41,"$4 ( ;1r *2"$ 2—
* 5i"$¸ *À12+ ²-4"      0 r+-3 . O" *”1,+-*.;13 9™² 0 " 1.*'+)"$.+¥ " " 12+-r+§²d2²   @  0  "$*'²-043 21,"$; $ 0 0 3 3 3 3 .+ " —?9 12+-? " 1 .!¥4 +-*          .?" d—2 1 2"$ 1.=+§²d2²)$ * 0 0 3 3 0 5”< 3 3 0 3 3 5 º-2+=¥*21 9  ² " 1.* r+ "$2+`*2²-     )"$!¥ *," 4À*, (-*2² — 9À& " *,²-  2?*.4"$ % 5 1 (
 ,2"  0 —24-² 0 3*2" 1," .   ! 3   3 3+-r 043 ;½¥+   9d?0 *21 93 O² " 12*Á"  .4 "( ² 0 *2" 1," 3   3 * 0   043  )"$!¥ 0 *," 0 0 4‹0—*2°
"$!#²)$;½ 5”12+ <  2041,"$3° * 9
r+ "$0 .+¯.r1,0 +-² " 0 1.3 *81,+-3 !¥*.;3  * 0 +-)04—3 ," 41, " 3 043 *Á1,+-4112+ !#² "$° ,"$462" *2¶¥ 0  0  3 §i1 12+ ;12° -4 5 9‹3 1,+-(  ¯*.3 *,-!¥ .* " 9 .!# ( 5]<2" *2¶K*À12+-°  -04!K 3 (d0 — 9-12.40 " 3 "  (-*. 41," *Á" —2*.* ³Á4² " ¶ 043° 3 0 0 "( " 5  3 —*2*.2§& 3 ¹*2
 ¥"  3 1 3 0 3 3 0 ° 4 0 3 3 3  )" 1," 9 1,+ "$*?"$*?1,+-41? "$ !   È  r+-2  "$*?1,+- -!K(
 9812.4" " =*2!#² $*K   043 : " *041,3 +-0 ³´F 3 12 0 ²§ 0 9”1,+-?*21 0 9À ;"$*," 043 9: 3 ;12" 043 * ³Á4² 3 " 3 ¶  "( "'& 0 ³'4² 3 " 3 ¶ 3  "$"$* 5 ¦ 0 r 3 3 § *.;1 9-9ˆ ;12" *'r" 1,+¥" " 12³´I)" !¥ *2" 1,+-8³´t 12 ²§K"$*    ! +
+- r9 Ÿ12+ —*2r *.*2" .* 3 * 0 ;" 12+ —0  *r12+ 3 ( 0    1,+-?0 .¿ 3 "$. 043  " 9 .!O3  Â)3 . # 3 * 043‹1r+ $  ± 3 0 043° 3 0 3-0 0 3 0&3 0 0 3 5 +-.&"$*? " —# !#122" ¥" 122²-2;1.41," 9 12+ ¥)-4”²- ()$! +" 1?"$*( *2"$4  § "  1,+-#1{ # 5]<; *.*,1@² " 3 1.* 9À 0 ;½ +  3  * 9Ÿ1,+-?1{ 043 *.;0 12* º 0  3 3 0 /  3 3 1,1 3  / .  3 *,1.;" 3 — "$"$# 5 0 ? 0 )3 ? 0 12+-0431 ° .¿-?1 (d0  0 5 " 5 @3 ! 3 Â)3 0 0

"$# &% % % # (
' *) # % % % #
 #
+' , #

 r+-.].;²d41.¼"




)"$*i.]*2-!#!¥  21 "$*,§)!#!¥;1.,"$±12 3 * &" 12+ ).  3 !#²  12*  +-3 ?*,0 -! 9”1 .¿% % -% / * 0 043 3 58< 0 0 043

(

0

) # 1% % % #
2 0 #
+'  , 0 #




(

0

 1,+-¯2"$4+1#+  v*2"$)  tr+-2 ) "$*¹1,+-]1 124  § 0‡° 043 3 3 0  5 Ç 4 r1,+-41]Ä2—;¶W" 3 )" —*i2] *.t1 0   -3 0 1.]12 3 * 0  ,1 +-K " *2" *,+-1"$*1,+- + 043 3

Í4 # ) # 1% % % # 3
2 0 #
+'   0 Ì 056




7) # .% % % # $
8 # (
'  8 4 #


9

 5  %

 5  ( 

& 5 É 3 1,+- 0 ,"$4" 3 4Ÿ9 0 .!K $41," 043 ³Á4² 3 " ¶ "( " 1,+-;§±24 $ ,.;½12.!¥ ° ;1 0 2* 5 .   5ö/ § ,. ;"$*2" 043 9: 3 ;12" 043 .=¥!¥ 3 ¯9: 3 ;12" 043;: =<Ÿ r+ 0 *2i*2"$ 3 .;²-2—*2 3 12*?12+-i;$*2*¹*2*2"$ 3 —W1 0  412#² " 1 < 0 3 5 ! 5ö/ § ,2" 3 122" 3 *2"$)"$!¥ 3 *," 043 .K@!#— 3 1,+- 3  !&(d 0 9d²-.!¥;12.*Á2¿ " .i1 0 *,²
;" 9ˆ§i² 0 " 3 1 043 12+  !¥ " 9 $ 3 0 5 « 5 @ 1. 3 412" ° ; § 043 ¹ 3 24-&1,+-41 4" °  3 12+-K9 0 .! 0 9”1,+-K* 0   1," 0436 12+ K² 0 *2*2" (   Í !K-*21 "$K" 3  *2)(-*2²- 9”)"$!# *2"  0 3 043 5 ‡ 5{¸W0 2¶]" 3 ²-.;²-2412" 043‹5 * 5ö< +- 3 ¶*1 0  5 º! 0  ¥9 0 ² 0 " 3 12" 3 ]1,+ "$* 0  1 5 % 5,7  3 §¯1,+- 3 ¶*r1 0i043  0 9Ÿ1,+-K. ° "$;.*r9 0 ² 0 " 3 12" 3 i12+ " * 0  1 5  ( 5ö< +-] 0 .]¿  .41,"$ 0 ² 1,"$!#" ·—¥"$*¹4( 0  1 (>  " 3 —* 0 9´ ?@? 5N< +-]+ "$4+-#$ ° ; 0   1 0 +- 3 $ .+ "  9À 1²  )-;1.* .+ ¶"  É ;12  "$*¿)" 1.K !#²  ½± = *,"$ .4() § $2 3 0 0 0 3 3 ¹ 0 3 043 5 # 5ö< +- 3 ¶*1 0 Î 5 ?4 9:!¥ 3 9 0 ²- 0—° "$" 3 ]!¥r" 1,+=1,+-*.K.*2) 1.* 5 — * ,.*.!¥4 $*21" 1.r.412—81,+- 8¿-4™1 .N@ * 12+-2 " 5öÇ —4 d1,+-4181,+- ,.;"  " 3 $ .¥*2"$ B ADC 4!# "$*¹1{§² " 1,+-K;12-4Ÿ9 .!K3 $¹ "  3 " ³'4² " ¶  3 "$"$* r+ " .+`ÉÁ+- 3¯ ?0  ..;12—±0 +-5 . 0 0 3 0 ° 3 3 3 ° 0 5 !> > 12 9 @½!#²   1,+-411,+-¹"$*212 #(d;1{  2§=² 4"$ 9 —,12" —* 9”12+ #*2§!¥!¥;122"$&*2" !#² $;½ 5 "$*Á0 1,+-*20!¥$ +À*.F 3 3 ° 0 ° 0  E'¿ !$% 5 H5 G@0  °  ¹,"$ 0  0  *8²- 00 9‹"$* 3 —   3 ]*89È'*ɔ¶ 3-0 "$*8$2¶" 3  5 +- ¶*1 º+ ‡ —§ 9 ² " 12" i12+ " *  1 ! 5ö< 3 0 I-5 6 »{< 0 0 0 3 3 0 5 !$! ³ ³Á4² " ¶ %Á," 412¹´ !#!& "$412" 5 5 3 ° 0 3 043‹5 +-.?"$*@ 4 1. 412" #(   &!#"$4+1-*2 !¥; §=1,+-41 2.*2² )" ¯1 12+ ¹*21 9”1 124  § ! 5ö(<    3 3 ° 0 3 4 0 3 412" ¹9: 1," * EÁ¿-41," È -3 !$# " ³'4² " ¶ 0  "$"$*04,3  3 80   9 @ 0 *.*0 9ˆ  5 +)"$G*0 (  ° ="$*0  0 *2—1,+-3  » 1," 0 3 *124¶" 3-04±3 1,»È+-3  4 -° ¹·— 3 04 3   `" 049'3 1,+-¹5 —!¹² "$23 " —4'2" *23 ¶¯"$*· 12 043 3 ° 0¯0 043 R 3 0 0 3 00 3 . R Y 1,+-41@" EÁ¿  r+-  K  JMLDN O PQL A JTSVU  SW$N O PQLDX S Z   % r+ "$.+=" *1,+-?*.?+-. 3 5 3 ° 5 $ · %'2" 41.¹´ !¥!K "$41," !4 ³ 5 5)/ 3 ° 0 3 043 m-2( ‹Ô$‹ÒØ   "$·.!¥ E .  —2!¥  7& !¥5 ;1,5 + ±" ²-4361,12— 5 7t. 5À /  ° " 1," $36 t 0 3 3 0 3 40 3 3

FÎ É ·  +- .;12" —49   412" * 912+  ² 1. 12"$49:  1," 3"  5 ” 5 Ç0 043 5Y< 0 0 3 043 + 0 0 3 3 40 3 3 \ 5 [ )“ {>]#ŠÈÈ–IŠ–™Æ^r….]¥{…!_”–- Ž,† !$* #$! )#( "$%4 5

 1,+ §v v> "$* %À¯$ " W  ;-24 ;1{ 2¶* É ˜‹š …
?Š–™ÆŒ2‡ti„Ž,Šˆ–V˜‹š)…ƒŽIŠ–™Æ 7t5 3 043 3 0 5 3 K…—“)Ž,Š† K…— '3 Ž 4‘ ²-56/ * %$5 "4«)%$"( "$"$3 * 3 5 3 3  %  ;121=  E 2  *212;"  Ä !¥;1.,§T" $ "  É ²-W1 4² ²d 5 5*2/ + " 3 43 1 3 412+-3 !¥41,5 "$4”0 *2* " 412" 3 9À@3 !¥3 2"$5  3 …ƒ>]¥…‡ÈŽFŠ=Ž  0 Ï 3 ´ 5"$/ "$# ¸ 3 4 0 ‹ 3 5 $ 5 Á 5 7 0 4 0 N 3 0 5 ´ " *2+ ² ?…‡“Ž2Š†?…‡ 'Ž ‘;ŽrŠ {…‡Ž;–K^…ƒ•ƒ.›–-ˆ È– ´Á$2  %Á2*.* ½9 . "$"$* ³ 5 7t5)$ / · 0 5 º.+  ¶ ² 9 ) 12+  ´ -2—* ³ ³'45 ² " ¶  3  043 ³'1,12+— 0 ´ !#²-,"$5* 9 "$; (-*. 5”( /
;3 1¹28/ 5  " 12" 0  0 4  G2¹ / *&  043F0 º° ;$  " 1,+-5À!¥ -*20" `.45” / "$*,12"$  t! 5  ;$* 3 É ´ 3 _ [   ²-* !$*K !$*$% -/ , " 36 0  6 3 3  / 5 3 0 0 5[  "$"$% º²-2" @Î6;1,-.K> 12*" ´ !#²  12º;"$  ³   ! E 5 *2 3 É 567t5 Ä@ § 043‹ 0  3 W³ 3 5 ³'0 4² 3 " ¶ 5 µ12.3 4" 3 " 3 `0 45 $ 0 ," 1,+-5 !g9 0  0 ² 12" !¥4!¥24" 3 ;$*2*2"  2* 5 É 3 ;Á ƒ:š /5 56/0 –-–-“ Š † = Ž ‘2š) v; – _”Q ]™“){Š :–™Š†6‰‹…ƒŠŽ;–-ˆ–›I˜‹š ….Ž  %'" 1,1.*,( -24+  "$"$! 5 ´ 7t5 [ !¥*  .+W `Î
"  ]?4 9:!¥ º !¥&*2124( $¹!¥;12+  *9 @4$;  412" =" 21,"$¯ `*  " ¯*2§!¥!¥;122"$ I  " —@ÇK*25-§/*212!¥ 3 * 3 3 365 0 0 0 3 3 3 0 ° 3  Š ˆ  š … . ¥ ]  Š  :  ‡ • K ‘   &  . •  Q  ] ™  ) “ 5  =΋" )±?4 9:!¥  {Š  :!¹– ²  124 ö12«"  ( 4+ 8 %!¥‡ ;1, + ( " = 9 "($r( 125 +-K" ) " 12¥¿  .41,"$¹²  .!#!#"  !¥3 *  .+N I ²- ( $ÇK! 5™/ ‰”3 ˆ–™…ƒŠŽ 3 † ›…«Œ—3 Ž,Š±Š–™Æ¯: ‘ 36!5 ™† :•ƒŠ 0 È–-‘ 4 + 4043  (  "$0 #> 0 3 Â)3 0 3 0 5 [ [ 5 ´ ´  .*@  º2+  ¶ ² 9 É{!#²  " ¥1,+-K;-.;§± =*2²d 9À*2)² ² 21 —;1 @$ " ]!¥.+)" * 3 /. 5  0   0 5 %À12*.2+-0‡° 3 )" 1 .* Æ#"4Š–™•ƒ…‡‘K3 ˆ–K…—“)Ž,Š0 †$;–%;Ž ]¥ 0 Š :° –&0 Ž,‡•ƒ…—‘ƒ‘ƒ3 ˆ–3 ›('$‘ƒ{….]#3 ‘  5 É M5 I-5 5 / · 3²-` t 7 ) 5 = 7 0 -7 t 5  I 0 ‹ 3 3 K < 5 0 [ *  ( *‡$ # ´!K(-,"$    "$"( É %'.*.* ´ ´ -2* º"$!#²  " =*2 d7²)² ,1 ;1 57  <;"$*2" 2 5  —* É Î . ·¥º4" 1,1. )" 1   ?ˆš …¯˜‹š:Ž ) 5{I-…ƒ…—5 –-:5š* / ;–- …‡Ž;–™5 Š :–™Š† _”–%…—Ž,…‡–™0 •ƒ…¥° – 0 Š•šˆ–™…&04‰‹3 …ƒŠŽ;–-:–› 5 ²-3 0 * (3 '($( 2" É 1.4 0 § r"$Ž,"$—% •ƒ…ƒ…ƒÆˆ–.›‘K  ?4 9È!# ´ ´ -.*  " $)" ¯ 4  §`" 2"$ 1¶ ;$* É rŽ,—•ƒ…ƒ…ƒÆˆ–›‘#  K-/:š)= … <   (+',=5Á7= Ž 0‘2š) v3 – 'd“ !)36Ž;5 5 .dI-….5 •‡{5 /Ž Š•š5¥ 3 5 3 ˆ–™/ …‡0 ‘ /ö ±3 Š !)0 ….Š2Ž 1  "$3"$° # 5 3 ´ ´ -.* %  " .*.2+   241.*2.+ º)² ² 21 ;1 ;(=²-$+8+121,² + 3#3«* ! 2*...+ (
;  $4(-*  ! 5 I-5 .+ 5 / " —4À2;² ,51 ΋3 - 1 3 .+ ÇK+5  G 4" —* 5 0 ° 0 ° 5 5 » 5 0 5  "$"$% ´ < ´ 23 12*@ ±³ 0 ³Á4² " ¶ 3 º<  ² ² 3-0 21 0 ;1  ;1{5 2¶*  Š ; •  š :  ™ – & … ‹ ‰ ƒ …  Š ; Ž – ˆ   – › !> + !(4‡)  !$"(  "$"$* 5 ´ 0 -. 13  =Ï 5 0:š)‡Æ° ‘& 0 3 Šˆš ….]¥ 0 Š :•ƒ 5 Š† 8š4‘ƒ:•‡‘ É 12.*.;"$  5 " 3  (
,5 1 ÇK 0   3 § 3 6 5  …—W 5 :•I3 ˜‹š)…ƒŽ =3  8 Š"$ * …‡ Ž;5 – ^…ƒ•ƒ.›–-ˆ :– º²-,"  ΋-5 ¹Ï Î 5 *.·;5 7 G5 Ä@§  0 Ä 45 ( ?Î
- *2" r  , Ž   . Œ  Š — Œ ˆ  † ˆ  ƒ ‘  0 ² ²  "$41," 0 * 9 0  Â641,+-3 !¥41,"$*0 ³  0  ; 5  "$[ "$% 5 3 ³'—,$ °  4 0 3 0 7 0 5 5 ´ ´ -.* Î ?4)9È!¥  º! $  W³ ³'4² " ¶ º ² ² ,1 ;1 ¹2—2*.*2" !¥.+)" * G¹5 Ï#Æ "4,Š-–™.•ƒ¶…—‘? :& 5 I-5 5d/ 5 – Ž,‡•ƒ3‹…—‘ƒ‘ƒˆ–5 ( 0 ° 0 043 3 5 › '$0 ‘ƒ{.… ]#‘ 3 " +  *$5 *‡  % 3  "$5 "( 5 [ ¦Ÿ$;1.2+- – r?Ž,Š…‡•‡“) Ž,:•ƒŠŠ$† † ƒ%– …—: Ž š)]#‡ŠÆÈ‘KÈ 9 + " $  ; 

§  `  º * É  ±  —    " 2 1 " (:;™  ]# <‡Š :– 5 I0 3=¸ !  "$#( ÇK º1,5 -21Ä!¥5   !E' "  "   *21 .¶ >@;-.4 1{  2¶* ¯12+-?3 ( " *8043 3 23 "$ 5$ K3 )"  —!#!¥04 36 ?…‡“Ž2Š5 †_”>]™“) Š!) 3 3 / 3 3 0 5 3 0 3 ° 3 5  :–  + ) *$#  "$"$! ¦ Ä@"$ *2"  ¿ " 45   —(d1{8— *,²-.*284² ²- ½"$!¥41,"  #*, ² ² 21 ;1 '!¥.+ " —* K…—“)Ž,Š † _'> ]d“{Š :– 5 /ö{¯Š0 !)5…ƒŠŽ2173 =@_Ÿ„ _6° ‰ 3  …1]# < 3 —’ ˜ 0 043 3 0 ° 0 3 5  "$"$# [ 5 É Ä@)§ ³ ³Á4² " ¶ *. Î 1,1   Tº  º  $ º122-;1,-.42"$*,¶|!#" "$!¹"$·412" 9 i.+-212 5 .  0436" 1," 5 #Æ "43 Š–™•ƒ/…—‘? :5'& /?0 …‡“)Ž,Š$† ƒ58%– / 0 Ž ]#0 ŠÈÈ& 3 rŽ,‡•.5 …‡‘ƒ5 ‘ƒˆ–0 ( 5 ‘ƒ{.… ]#‘ + 3 043 0 –     – › $ '     ( ( "  "$"$! 0 34$! 0436*  5 [ 5 % 5 Ç&5 — G  0 5 [  =:† ´ÁŒ.…‡+-Ž;>2'?$* Š•ƒ…@rŽ,+ Œ—† * …1] „@—Š È5 Ž;B Ï A 5 ³Á–™ Š3 † > ‘ƒ:0 ‘ *,1.2´8 3 !&`(-´ ,"$0  !# ²- ¤ 3 § " É 3 . *2"5$1{§= "$%Á%2( 5 *.*  "$#$* Ç0 .+ 5 " !¥ G* 0 3 ;3 ½1&—412 2"$Ç& 5 >-!K(
 ·45 1,"I0 3 "041236+t50*2 ² ² ,1 [ ;1 ¹!¥5 2+ " * —2+ 3 "$° 4.;² 21 ÎÀº=³É,É2É? !   !    `º ³ ¦ !#" ƒ–- Ž,‡Æ“ •—{Ž5 ^…ƒŠ† –™Š† ‘ƒˆ‘ %Á2 1,"$ 4  É   "( 5 Î 5 0  0.0 *,"$0— ° 3 K–-5 † :–™5 …ƒŠ0  3 5$ 5  E 3 2¶  » "$G %$" Ž r36Ž,.5 ›Ž,>Š ]\]#ˆ–› 5¹7 ‡Ä.— [ G "   >;5 F ¹5 5 7 3 6 3 5 0 5 Ä21,+N% ‡´ .!¹"$.¶ K–W‰”ˆ–™…ƒŠ6 Ž Ž,ƒ›Ž2>Š ] ]#ˆ–4› G=˜‹š)…ƒŽ «’ [ † ›Ž;:ˆš ]¹‘#Š–™Æ [ ?™† È•ƒŠ :–-‘ 5 I0 + 3`¸ " $;§N 3  5 º * É 5‹7  "$0 # 3 125$ !¥2§¯ 5   E  %À.¶ ;–-ÈŽ2—Æ“ •— È–W{#‰”:–™….ŠF Ï ´ 043 + " $;§  =º * É  Ž ^…ö›Ž,…—‘ƒ‘ƒ:– [ –™Š† ‘ƒˆ‘  5 5)7=043 0 3 5 5 5 5 I0 3±¸ 3 043 3 5$ ! ±—" 12"  "$"$! 3 4 4 0 6 3 5 5   2"$4+1 :;dÈ=]¹B<‡Š :– “):Æ… ºÉ{ "$" 7=0 3 ¸ 4 @ 5  3 ±5 Ä.. 0¹<Ÿ0 24$ 0 5 @3 5 1,+-7t *   1," 5 96 .¿- 2412"$²  .!#!#" K²- ()$!¥*" 12+¯(   I0  . *212I™.5)47= 0 0 0 0 3 0 0 3 " 12* 'H >:;™  ]# <‡Š :–  ö4 + "‡ « 043=  "$" º 043  ¶+-3 
 5 E [JI *2   `¦ Ä@"$ *2" >  " ²-.)" 5 1," 9Á2+- 12" ¹1,"$!¥&*.2"$* *2" ]i*, ² ² 21 —;1  5d!¥7 .+)"  3 3 Ž ‘,š   M 3 Ž,‡•ƒ…—‘ƒ‘ƒˆ– É 5 rŽ,‡•.…ƒ3 …ƒÆ:–›3 ‘N `5 :š) … 0K>K+5 KL04= – K04…‡3I “Ž,0 Š † K…‡N 0 Á Ž 8 ‘  Ž 'd ›–™ŠO †  › 0  ° ²-0 * 3 5 3 *)  * " !# "$iÉ{*2$ 3  ¦ÀÎ  "$"( 5    $  º! $ Ä 41.*2.+ º.+  ¶ ² 9  + $! .  N³ ³'4² " ¶ %'.)"$;1," =1,"$!¥¥*2—,"$* 5
7 *,  ² ² 21 5 ;1 & 0 !#.+ 5‹" Ç  * É Ÿ/ Ž25 —•ƒ…ƒ….Æ0  ˆ–›0 ‘, r5 »{ÇK " 1,+t ’ Iƒ–-™{5 …‡Ž;–™0 Š :–™0 Š† _”36%– ;3 …‡Ž,…—–™•ƒ5 …]– 3 [ Ž;È5  Á•—ÈŠ; † K…‡“3 Ž,Š; † K…— ' Ž 4‘ 0 ° 0 3 5 3 ²- "$"$" º²-," Î6;1,-.K> 12*" ´ !#²  12º"  —  "$"( EÁ) *2 5 3  ÁÇ3 0 (d—,1i¦-.; 3  0  3 o¦-3  0 ,"$ 0 Ä@"$ 0 *2" 5 3  3 "$!#²- 5 0‡° v1.24" 3 " 3 t4$ 0 ," 12+ ! 9 0 #*, ² ² 0 21 ° —;1 0  !¥.+)" * É rŽ,‡•.…ƒ…ƒÆ:–›‘] ¥ˆš B … <   (PK+K>KQ= Ž ‘2š)  – ?…‡“Ž2ŠR † K…‡N Á Ž S ‘ ;@ Ž 'd$›–™ŠT † rŽ,—•ƒ…‡‘.‘ƒ:–›« ’ KrÆ‘ > >I TrŽ;ˆ3 –™•‡ 5 …’”3 7 ‰ > ˆ† …‡‘,> ’  > ŽÈ›Š–)+ ’ K > ˆ† ‘;– ² —* !( %& !$#$* @!¥; "$iÉ{*,$ 3  ¦ÀÎ  "$"( 5 EÁ) *, 3  ŸÇ0 (d21K¦-.; 3   3 W¦-—)—,"$ 0 Ä@"$ 0 *," 5¾< .4" 3 " 3 =*, ² ² 0 21 ° ;1 0 &!¥.+ " 3 *+& 3 4²)²  "$41," 043 1 0 9:— ;12—;1," É Ž .Ÿ:‘ƒ:–`Š–™6 Æ Š  …‡Ž;– ^…ƒ•ƒ.›–-ˆ :– ²-* «> ]«$ %  "$"( 5 04365 3 PK+K>K _”%– …—Ž,…‡–™•ƒ…K– _”> ]™“){…—(



EÁ)

*2 W o¦- 2"$

Ä@"  *2"

)-;" I12+- 2

12"$!#  !#² $;½" 1{§

9?*2 ²)² ,1

;1 ]!¥.+)" *

É

0 ° 0 3 5 3 ƒ–-{…‡Ž;–™Š :3 –™Š†3 _”–%;…‡Ž,…‡–™•ƒ…#0 –r0 ŠÈ{5 …‡Ž;– Ç ^…ƒ•ƒ.›3 –-ˆ È– /ö‘ƒ“ 3 Œ.» ]#ˆ {….Æ71 0 "$"$# 5 0 "   "$! %Á2*.* .4" % ¦Ÿ$ 2§ º4 ' ; ¶ $*2¶§   "   "$! ³';12122"  “ ]¥…—Ž;È•.Š†”Ž,…ƒ•‡ …‡‘#ˆ– ¸ 5 _>ÀG ˆš …#G¹ ŠŽ;8 ?  ‘;•‡:
/…‡–-È Á 3•¹•ƒ>5 ]™ “)3 ˆ3 –› 5 ´!K(-,"$ 5d<  ¤ 3 0" ° —2*2" 1{§ 3 %Á2—¸ *2* ! 3 =
ˆ–-:–› @É8%'.*.* 7  3  0 %À,¶ ´ "$"$* 5 º.+  ¶ 5 ² 9 ´ -2—*  ]³ ³'4² " ¶ É  2² .41," ¥" ,"$ *" *, ² ² 21 ;1 $ " ¥!#.+ " * É /´5 0   0  4$5*,/ ( -. 3 3 3 5 3 º5   3 5 ´ 3 ³ 0 2( 0  -3  3°  3 º )+3  )0 " 1 2° * 0 Ž;  Á•‡:Š3 †#K …—“)Ž,Š†4K…— 'Ž 4‘ ‡ 5  ° 4 0 3 7 ‡¸ — 5  ° 4 0 3 ‹ 3 I 5 5 0  6 3 3  / 5 3 0 0 [ ²     —  * —  ,   "  º ² ,  "       6 Î     , 1  2  &  > . 1   r * " ´ # ! ² )  . 1      º ;  $ "     ³  L_ [    (K *$! "$"$% ! 5 º.+  ¶ ²)9 % º"$!#.  º! )/$  36&³ ³Á45 ² " ¶ 3 %'2" Ÿ¶ r$ 0 " *23  ²)² 0 ,1 1 '¶3  ;$* 0 É 5 2  /5 ?0   0 *  5 ]º º  $ 5 )" 1 0 .* Æ#3 "«Š–™•.5 …‡‘r:– 3 ?…‡5 “)Ž,Š†40 ƒ–%3-Ž 0 ]¥Š :–(3 Ž2—•ƒ…—‘ƒ‘ƒˆ0 –›6'$° ‘ƒ{0 …1]¹‘ < 3 ´85 !K3¥(-7t 2"  5&I 0  36 7t5 3 É %Á3 2—*2* 5 É 0 ²-.*2* 0 M[ )7  "$"$# º.+ 5Á 7 ¶ ² < 9  º!5 3 $  `5    $ >  " K !#²  1 4 §*2"$*K*i¶ ;”;"$ 4 -¥²- ( $! /K 5 …—“)Ž,0 Š † _” 0 > ]™“)5 {Š :0 – 3 5 »{Ç&5-7  5 043 3 0 043 3 3 3 3° 0 5  "$"$# É ²-2*.* 5 3 5 º2+  ¶ ² 9  º!      $ ´ ´ -2—*  =³ ³Á4² " ¶ º ² ² 21 1 !¥;1,+  *"  — "  / 5 ¯9È0  41,0 -.?;½51.21,0 " 5 I-5 – 5-/ _”–›Ž,…‡‘ƒ‘K3 –&K5 …—“)Ž,Š† 3 K…—5  ÁŽ 4‘00 /ö{¯° Š !0 …ƒŠŽ21 "$"$0 # 3 3 3 É 5 »{Ç&5-7  3 º2+  ¶ ² 9  º 04 365 ´ 3 -:.–-ˆš * [ ¦ “‘ƒ Ä@Ž,Š"  † :Š*29 " % >@" § 4" % 4"  t³ ³Á4² " ¶ ´ !¹²-52" `*2 ² ² ,1 /5 ;1 0   0 !#.+ " 5 *r" 3 1,+N4-5Ÿ*./ *,"$ ¶ ;5 $*1 0 .)" 45 Ÿ(-*20"$*@9:Á<  ?5 1," 0 ;$0 *.*," 3 .* 5 PK+K>K 3 ˜-Ž,5 Š–-‘ >R0 'd ›–>;3Ž,‡•ƒ…—‘ƒ‘ƒˆ0–› ° + 0 3 3 3 0 3 043  5  * !( *$#¹   !( %$*  "$"( 5 + º+ ‡ —§  %À;12—±Î ,12 1,1 (
,1`´ "   "$!¥*   ,12"  12+ §  9:.!# 2¶|9  I0 *21223  1,-24
,»{<"$*2¶¥!#0 " "$!¹"$·412" 58/ É Ž,‡•ƒ….Džƒ 0 –›‘2’ :š 5Á¸ –-–-“)Š† _”043‹–%;…‡Ž,…—3 –™•ƒ…¹7 – _'3 >]d3 “{Š04 3 :–™5 Š†
‰‹…ƒŠŽ;–-ˆ–›N˜‹0 š)…ƒŽ0   Æ :  043‹5 3 [ ²-* %$#‡' ( %  "$"$% 3 5 ´ "   "$!#*   21,"  1,+ § º122-;1,-.4'2" *2¶`!#" " I0 !#+ "$3 ·4º1,+-" ‡8 » < —Ÿ‡§) 40 1.  %”);²d12 & Î  5
/ 1'+),"$12 .1,.1 2ŸÇ+ @"$0 * (d21¹ 3 » .+ 5™"$¸ 4-.;² 2041 36>@ 3  ´ 7 Î 3 .3 + "$043 4 5 ² ,1À>?´ "$% $* 4 0 i 3 — 0 ° » 3 3 r 5 < 3 0 0 ¼ < < 3 Ç 0 { » r <  Ç » »  "$"$%  º! 5 $¯  º.+  ¶ ² 9 i¶ ; (-*.N!¥;12+ `9 @²-41,1. 2  " 12" .2—*2*2" 4²)²- ½"$!#412" 3 /ö{» ¯Š!)…ƒŠŽ21 0 0 3 0 3 0436 043‹ 0 043  5  0 ²
243 1 r/" 5 —2*2" 0  0 5 † ›@3Ž;:ˆš ]¹È•.Š9  "$"$# 3 0 0  3 ° 4 0 ‹ 3 5  [ 5  º! $ º.+  ¶ ² 9  `   $ Ä .4À *21@9: ;12" *9 *2 ²)² ,1 ;1 ..*.*," É ?ˆ–-:š 5 “)‘ƒÈ0Ž2Š† Èd/Š 0 ° 0 04365 3 – 5 _'–›0  Ž2…—‘ƒ0 ‘K& B – 3 K…‡“Ž,Š5 »  † Ç&?5 …‡7  '  Ž 0 ‘ 5 /ö ¯3 Š !…ƒŠ2Ž 1 0  "$"$3# 5 043 0 @[ ½ º! $  +-2Tº.+  ¶ ² 9  F 4-* (d21   $ +-  1," (d;1{ .4 $,"$·412" /*, ² ² 3 21 ;1 0  ¶—0  ;$* 3  3 043 ²d.I-45 1 .*0 

† ?»{Ç …‡ 0 ' Ž  ‘ 7 /ö ± Š !)5 ….Š2Ž < 1  "$"$04# 3)5 3 043 0 0º12" 1.* 3  Ä0 !¥!¥° 2!¥0  ³ 3 ³Á45 ² ?" …‡¶ “Ž2Š³  ³ ¶ ´ 41,¶" *   *21 º ² ² 21 1 ¥  7t 5 ¹5 5 .+ " —4”2² 36 ,1 5 §4 3   5 —0‡§N ° ´  $5‹¸  ¸ ´rºÏ 04365 "( !$0! "$° "( 0 3-0—°  !#² *2" 041,36" ;² 3 21 3 -!&I-(d56@ 3 ( 2"$ #%Á.*2»{*
> ' _  >  ] d  %) {4 + $  !‡4 ÇKM5 I-5 3 5 3 0 0 3 0 0 3 0 043 5 [ I  "$"4 ³Á5  )—,(d" Î +- " 122" ² " 1  K9 ¿  .41,"$¹²  .!#!#"  .+ "$4Á2² ,1 %' .!O" ÇKº5 I™125 412" *21,3 "$* 5 ²d.41,@" * 3 *.3 .2+ 0 %Á,0" 3 ;1 0 ¤ 0 " 2*2" 1{§  "$"4 0 3 5]< 3 0 0 3 4 0 3 Ç 3 4 0 3 3 ° 5 ³ ³Á4² " ¶ @ > 4   4 ¶  . *   K‘ƒ ]¥Š :–F &…P)…—–™Æ…‡–™•ƒ…—‘i„@Š‘;….Ƽ– K]™:Ž;:•ƒŠ† &Š{Š {ˆ– ^“)‘ƒ‘ƒ:Š– 5  "( " Á7=0 0 5 5 E 3 4 " 3 *2+`5 12. 3 *,$412" 043 +rº² 2" 3 ³Á2  >; E 0 2¶  "$#$! 5 ³ ³Á4² " ¶ ˜6š)@ º²-,"  ³'2$ >@; E ,¶  "$"$* … ?   Š  ) “ , Ž ¥ …     d '   Š È  :  ƒ ‘ È  È  ƒ •  Š
† ‹ ‰ ƒ …  Š ; Ž – ˆ   – I › ‹ ˜ ) š ƒ …   Ž   ³ 5 ³Á4² 3 " ¶ 5 'd{Š :‘. È•.Š†d‰6….ŠŽ;–-ˆ–›I˜‹š)…ƒŽ + " $;§= 5 `º 3 * É »  >@;FE ,¶ 0 " ²-2;²-.5 41," 5 3 5 5 @  I 0 = 3 ¸ ³ ³'4² " ¶ º Ä  r"$.+  v º! $ º)² ² 21 ;3 1 i04!#3 1,+ 3 o5$ 9 ¹9ˆ ;0 1," 43 ² ²- ½" !¥41204" 365 ..*.*," 5 *21,"$!¥43 1," 5 ±0 *20 "$ 4‹²- 3 —*2*2" 5  0 !Æ "«5 Š–™•ƒ…‡‘?ˆ& – 0 K…‡° “Ž,Š† 0 ;%– ; Ž ]¥Š0  :& – 0 rŽ,—•ƒ3 …‡‘.‘ƒ:04–3 ( › '$‘.{1…0 ]¹‘ " + !$04#36) !$#(  "$"$04% 3 5 4 0 6 3 3 3 0 3 5  [ Ä. 4+(- º ²)² ,1 —;1 r!¥.+ " —* .;²- -;" ¥¶ ;
+ "  (d21*2² —* ]12+-? É rŽ,‡•.…ƒ…ƒÆ:–›‘&  °d5 3 :š)… < ¸    5+',=0 Ž ‘2š)°  ¼0& – 'd“ !3 Ž; .™…ƒ•‡ 0Ž Š•;3 š:–™…—0 ‘ 3 /ö{¯Š !…ƒŠ2Ž 1  "$"$# 5 3 *21  Ä!#!¥.!¥ º1," 12* ³ ³'4² " ¶ ³ ³ ¶  N´ 412¶" * Ï *," 1{§=*,12"$!#412" -*,"  I-5 *2)¸ ² ² 0421 36 —;1 5 !¥.+ " * 36-7t 0‡° ´   —3  5 ;¸ ² 21 3 -!K 5 (d3 ´rºÏ 043 3 —2+ (5 ¹5"$4
.;² 043621 5 §4 3   5 ‡§¯ 0 ° 0 3 5@< 3 0 Ç0 G@0 0 0 )Ç 0 3 »{