Dictionar Roman Englez

Dictionar Roman-Englez

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Dictionar Roman-Englez de Termeni Matematici AĂÂBCDEFGHIÎJKLMNOPQRSŞTŢUVWXYZ

A abscisă absolut aceleaşi semne acurateţe admisibil aduna adunare afirmaţie algebră algoritm algoritm de divizare algoritmul lui Euclid alternanţă alternare amplasare amplitudine analiza analiză analiză matematică analogic analogie anula aplicată aplicaţie arbitrar arc arccosinus

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abscissa absolute like signs accuracy admissible to add addition statement algebra algorithm algorithm for division, division algorithm Euclidean algorithm alternance alteration disposition amplitude to analyze analysis calculus analogous, analogical analogy to vanish applicate map, mapping arbitrary arc anticosine, arccosine

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2/17/05

Dictionar Roman-Englez arcsinus arctangentă argument argument al funcţiei arie arie a suprafeţei asemănare asimetric asimetrie asimptotă asimptote ale hiperbolei axa x axă axă de coordonate axă de rotaţie axă orizontală axă principală axă verticală axiomă

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antisine, arcsine arctangent argument argument of a function area surface area similarity, similitude asymmetric(al), unsymmetric asymmetry asymptote asymptotes of a hyperbola x-axis axis coordinate axis pivot horizontal axis major axis, principal axis vertical axis axiom

B bază bază ortogonală bază ortonormată bilion binar bine definit binom bisectoare biunivoc

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base, basis orthogonal base orthonormal basis billion binary well-defined binomial bisector, bisectrix one-to-one

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calculus, computation to count computable evaluation of determinant

C calcul calcula calculabil calculul determinantului

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Dictionar Roman-Englez canonic determinarea extremului câmp cât cel mai mare divizor comun cel mai mic multiplu comun cel mai mic numitor comun centru centru de rotaţie centru al unei circumferinţe exînscrise centrul circumferinţei circumscrise centrul circumferinţei înscrise cerc cercuri concentrice cifră cilindru circumferinţă înscrisă circumferinţă unitară circumscris coardă coeficient coeficient binomial coeficient unghiular coincident coliniaritate combinaţie comparare comparaţie con concavă condiţie condiţie iniţială condiţie necesară şi suficientă congruenţă conjugat consecinţă constantă constantă de integrare continuitate convex convexitate

Page 3 of 20 - canonical - hill climbing - field - quotient, ratio - greatest common divisor, greatest common factor - lowest common multiple - lowest common denominator - centre, center - pivot - excentre, centre of the escribed circle - circumcenter - incenter - circle - concentric circles - digit - cylinder - incircle, inscribed circle - unit circle - escribed - chord - coefficient - binomial coefficient - slope - coincident - collinearity - combination - comparation - comparation - cone - concave - condition - initial condition - necessary and sufficient condition - congruence - adjoint, conjugate - corollary - constant - integration constant - continuity - convex - convexity

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Dictionar Roman-Englez coordonată coordonate afine coordonate carteziene coplanar corolar cosinus director creşte creştere criteriu criteriu de divizibilitate cuantor cuantor existenţial cuantor universal cub cunoscut cuplu cuplu ordonat curbă curbă concavă curbă convexă curbă cubică curbă integrală

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coordinate affine coordinates Cartesian coordinates coplanar corollary direction cosine to increase increment criterion criterion for divisibility quantifier existential quantifier universal quantifier cube known pair ordered pair curve concave curve convex curve cubic, cubic curve integral curve

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if and only if data decision deduction to define everywhere defined definition degenerate dividend multiplicand dilatation, stretching to prove proof dependent differentiability

D dacă şi numai dacă date decizie deducţie defini definit peste tot definiţie degenerat deîmpărţit deînmulţit delatare demonstra demonstraţie dependent derivabilitate

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Dictionar Roman-Englez derivată derivată în punct descazut descompunere descompunere binomială descompunere în factori descompunere în factori primi descompunere în fracţii elementare descompunere a unei fracţii descreşte determina determinant deviaţie deviere dezvoltare dezvoltare a determinantului diagonală diagonală principală diametru diferenţă diferenţiabilitate diferit dimensiune direct proporţional direcţie discontinuu discriminant dispărea distanţă distinct divizare divizibil {prin} divizor divizor comun divizor propriu domeniu domeniu de definiţie dreaptă dreaptă numerică drept drepte neconcurente

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derivative derivative at a point minuend decomposition, expansion binomial expansion factoring, factorization prime factorization partial-fraction expansion decomposition of a fraction to decrease to determine determinant deviation deviation expansion expansion of a determinant diagonal main diagonal, principal diagonal diameter difference differentiability unequal dimension directly proportional direction discontinuous discriminant to vanish distance, way distinct division divisable {by} divisor common divisor, common factor proper factor domain, range domain of definition line number line straight skew lines

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Dictionar Roman-Englez dreptunghi drum dual

Page 6 of 20 - rectangle - way - dual

E echilateral echivalent ecuaţie ecuaţie algebrică ecuaţie a liniei drepte ecuaţie a unei drepte cu coeficient unghiular ecuaţie bipatrată ecuaţie de ordinul întâi ecuaţie exponenţială ecuaţie liniară ecuaţie neliniară ecuaţie omogenă ecuaţie de ordinul doi(patrată) element element {al multimii} element de arie element de arie (al suprafeţei) element diagonal eliminare eliminare prin substituţie elipsă eroare eroare de rotunjire esenţial estimaţie evaluare exact exactitate excepţie exemplu exînscris exponent exponenţial

- equilateral - equivalent - equation - algebraic equation - equation of a straight line - slope-intercept form of a straight line equation - biquadratic equation - first-order equation - exponential equation - linear equation - non-linear equation - homogeneous equation - quadratic equation - element - member {of a set} - differential of area, element of area - surface element - diagonal element - elimination - elimination by substitution - ellipse - error - rounding error - essential - estimation - evaluation - exact - accuracy - exclusion - example - escribed - degree, exponent, grade, power - exponential

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Dictionar Roman-Englez expresie expresie algebrică expresie de sub integrală extremal extremum absolut

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expression algebraic expression integrand extremal absolute extremum

F factor familie fascicul fascicul de drepte faţă laterală figură circumscrisă figură congruentă figuri congruente flux focar formă generală formă parametrică formă standardă formula binomului formula coeficientului unghiular formula lui Leibniz formula rădăcinilor ecuaţiei patrate formulă formulele jumătăţii unui unghi fracţie fracţie elementară fracţie ordinară fracţie subunitară fracţie supraunitară fracţie zecimală fracţie zecimală finită fracţie zecimală infinită fracţie zecimală periodică fracţii cu numitori egali funcţie funcţie concavă

- factor - family - pencil - pencil of lines - leg - circumscribed figure - equivalent figure - congruent figures - flux - focus - general form - parametric form - standard form - binomial formula - slope formula - Leibniz rule - quadratic formula - formula - half-angle formulas - fraction - partial fraction - common fraction - proper fraction - improper fraction - decimal fraction - terminating decimal fraction - infinite decimal fraction, non-terminating decimal - periodic decimal fraction - similar fractions - function - concave function

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2/17/05

Dictionar Roman-Englez funcţie continuă funcţie de mai multe variabile funcţie de o variabilă funcţie de variabilă complexă funcţie descrescătoare funcţie diferenţiabilă funcţie discontinuă funcţie impară funcţie în scară funcţie liniară funcţie monoton crescătoare funcţie monoton descrescătoare funcţie monotonă funcţie pară funcţie periodică funcţie primitivă funcţie raţională funcţie signum funcţie simetrică funcţie trigonometrică fundamental

Page 8 of 20 - continuous function - function of several variables - function of a single variable - function of a complex variable - decreasing function - differentiable function - discontinuous function - odd-function - inter-stage function, jump function, step function - linear function - monotone increasing function - monotone decreasing function - monotonic function - even function, symmetric function - periodic function - primitive - rational function - signum function - symmetric function - trigonometric function - fundamental

G general geometrie geometrie euclidiană grad grad al polinomului

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general geometry Euclidean geometry degree, grade degree of a polinomial

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hexahedron, hexaeder hexagon hyperbola hyperbolic

H hexaedru hexagon hiperbolă hiperbolic

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Dictionar Roman-Englez

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I identitate imagine impar implicaţie incertitudine include incluziune incompatibil incorect independent independenţă liniară inducţie inducţie completă inducţie matematică inecuaţie inegal infinit inflexiune iniţial integrală integrare prin părţi interdependenţă intersecţie interval interval deschis interval închis interval mărginit interval semiînchis invers invers proporţional ipotenuză ipoteză ireductibilitate

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identity image, map, mapping odd implication uncertainty to include inclusion inconsistent incorrect independent linear independency induction complete induction mathematical induction inequality unequal infinite inflexion initial integral integration by parts interdependency intersection interval open interval closed interval bounded interval semiclosed interval inverse, vice versa inversely proportional hypotenuse hypothesis irreductibility

Î http://math.ournet.md/formule/dic/roman.html

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Dictionar Roman-Englez împărţi în două {în jumătate} împărţire fără rest în întregime înălţime înălţime a triunghiului închis înclinare înlocui înmulţi înmulţire înmulţitor înscrie

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to halve exact division whole height altitude of a triangle closed slope to replace to multiply multiplication multiplier to inscribe

K kilogram kilometru

- kilogram(me) - kilometre

L latură {a unui unghi} latură adiacentă lege lege asociativă lege comutativă lege de compozitie lege distributivă legea terţului exclus lemă limită limită inferioară limite de integrare linie de fracţie liniar liniaritate linie dreaptă linie frântă linie punctată

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side {of an angle} adjacent side law associative law commutative law law of composition distributive law law of the excluded middle lemma limit lower limit limits of integration bar, vinculum linear linearity straight-line brocken line dotted line

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Dictionar Roman-Englez linie tangentă loc geometric al punctelor local logaritm logaritm zecimal logaritm natural lungime lungimea arcului

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tangent line geometric locus local logarithm common logs natural logarithm length arc-length

mărime măsură măsurabil matematică matrice matrice de transformare matrice diagonală matrice inversă matrice nulă maxim maximum mediană medie aritmetică medie geometrică

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metoda eliminării metoda inducţiei complete metru minim minimum modul monom monoton muchie mulţime mulţime de soluţii mulţime de valori mulţime universală mulţime vidă

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magnitude measure measurable mathematics matrix matrix of the transformation diagonal matrix reciprocal matrix null-matrix maximum maximum median {of a triangle} mean, arithmetic mean geometric average, geometric mean, mean proportional elimination method principle of complete induction meter minimim minimum module monom monotonous edge set solution set range universal set empty set, null-set

M

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Dictionar Roman-Englez multiplicare multiplicator multiplu multiplu comun mutual mutual disjuncte

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multiplicaton factor, multiplier multiple, multiplex common multiple mutually mutually disjoint

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unknown undefined non-degnerate non-decreasing indeterminancy, uncertainty indefinite, undetermined negative negligible non-linear unbounded inhomogeneous non-periodic non-symmetric node normal normal to the surface number complex number integer number irrational number natural number even number prime number rational number real number transcendental number decimal number ordinal number numerator term of a fraction consecutive integers

N necunoscută nedefinit nedegenerat nedescrescător nedeterminare nedeterminat negativ neglijabil neliniar nemărginit neomogen neperiodic nesimetric nod normală normală la suprafaţă număr număr complex număr întreg număr iraţional număr natural număr par număr prim număr raţional număr real număr transcendent număr zecimal număr ordinal numărător numărător al fracţiei numere întregi consecutive

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Dictionar Roman-Englez numere reciproc prime numitor numitor comun numitori diferiţi numitori identici

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coprime numbers, relatively prime numbers denominator common denominator unlike denominators like denominators, same denominators

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oblique octagon homogeneous degree, exponent order of derivative order of equation order ordinate origin {of coordinates} horizontal oval

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even parabola parabolic parallel parallelepiped parallelogram parameter braces square brackets, brackets parentheses quadrate, square tetragon quadratic pentagon perimeter period

O oblic octagon omogen ordin ordin al derivatei ordin al ecuaţiei ordine ordonată origine de coordonate orizontal oval

P par parabolă parabolic paralel paralelepiped paralelogram parametru paranteze figurate paranteze patrate paranteze rotunde patrat patrulater pătratic pentagon perimertu perioadă

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Dictionar Roman-Englez perpendiculară picior al perpendicularei piramidă plan plan tangent plan xOy planimetrie plus poliedru poligoane asemenea poligoane congruente poligon poligon convex poligon înscris poligon regulat polinom posibilitate pozitiv precedent precizie preimagine presupunere primă derivată principal principiul dualitatii prizmă procent produs produs scalar produs vectorial progresie progresie aritmetică progresie geometrică proiecţie proporţie propoziţie proprietate proprietate asociativă punct punct de discontinuitate punct de extrem

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normal, perpendicular foot {of a perpendicular} pyramid plane tangent plane xy-plane plane geometry, planimetry plus sign polyhedron similar polygons congruent polygons polygon convex polygon inscribed polygon equilateral polygon, regular polygon polynomial possibility positive preceding accuracy pre-image assumption first derivative principal duality principle prism per cent product dot product cross-product, vector product progression arithmetic progression geometric progression projection proportion proposition, statement property associative property dot, point point of discontinuity extremum point

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Dictionar Roman-Englez punct de inflexiune punct diametral opus punct focal punct izolat putere

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point of inflexion diametrically opposite point focal point acnode degree, grade, power

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radian radical, radical sign branch rank of a matrix ratio, relation common difference common ratio rational radius radius vector root double root repeated root square root simple root conjugate roots real vice versa to reduce combine similar terms reducible rule parallelogram law, parallelogram rule relation relative multiplex to represent remainder, rest restriction union to solve to square

R radian radical ramură rang al unei matrici raport raţie a unei progresii aritmetice raţie a unei progresii geometrice raţional rază rază vectoare rădăcină rădăcină dublă rădăcină multiplă rădăcină patrată rădăcină simplă rădăcini conjugate real reciproc reduce reducere termenilor asemenea reductibil regulă regula paralelogramului relaţie relativ repetat reprezenta rest restricţie reuniune rezolva ridica la patrat

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Dictionar Roman-Englez ridica la putere romb rotaţie rotunji

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to raise to a power lozenge, rhombus, rhomb rotation to round

salt satisface scalar scădere scăzător schemă bloc schema lui Horner schimb de variabilă secantă secţiune secţiune conică secţiune conică centrală secţiune conică degenerată secţiune conică nedegenerată

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sector circular segment segment de pe axa x segment liniar segmente congruente semicerc semidreaptă semisferă semn semnul adunării semnul rădăcinii semnul scăderii sferă simbol simetric simetrie centrală simplifica simplificare

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jump to satisfy scalar subtraction subtrahend flow chart Horner's scheme, synthetic division change of variable secant cut conic, conic section central conic degenerate conic non-degenerate conic, non-degenerate conic section sector of a circle closed interval, intercept, segment x-intercept line segment, straight-line segment congruent segments semi-circle ray hemisphere sign addition sign, plus sign radical, radical sign minus {sign} sphere symbol symmetric central symmetry to cancel simplification

S

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Dictionar Roman-Englez simplu sinusoidă sistem sistem cartezian de coordonate sistem cu mai multe necunoscute sistem de coordonate sistem de ecuaţii liniare sistem neortogonal de coordonate sistem omogen sistem ortogonal de coordonate sistem rectangular de coordonate soluţie admisibilă soluţie aproximativă soluţie completă soluţie exactă soluţie generală soluţie nenulă soluţie netrivială soluţie nulă soluţie periodică soluţie unică soluţiona soluţionare {a unei probleme} spaţiu spaţiu euclidian strict strict monoton substitui substituţie substituţie inversă suma sumă suprafaţă suprafaţă de rotaţie submulţime

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simple sine curve system Cartesian coordinate system many-variable system coordinate system system of linear equation non-orthogonal coordinate system homogeneous system orthogonal coordnate system rectangular coordonate system permissible solution, admissible solution approximate solution complete solution exact solution general solution non-zero solution non-trivial solution zero solution periodic solution unique solution to solve solution {of a problem} space Euclidean space strict strongly monotonic to replace substitution back-substitution to add sum surface surface of revolution subset

T tangentă teorema lui Pythagoras

- tangent - Pythagorean theorem

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Dictionar Roman-Englez teoremă teorema fundamentală a aritmeticii teorema lui Bezout teorema lui Thales teorema sinusurilor teoria mulţimilor teorie termen termen al sumei termen general termeni asemenea tetraedru tot transforma transformare de coordonate translare translaţie tranzitivitate trapez trigonometric trigonometrie trisecţia unui unghi triunghi triunghi ascutunghic triunghi asemenea triunghi dreptunghic triunghi echilateral triunghi isoscel triunghi obtuzunghic triunghi scalen triunghiular

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theorem fundamental theorem of arithmetic factor theorem intercept theorem law of sines, sine rule set theory theory term addend, summand general term similar terms tetrahedron whole to transform transform of coordinates translation bridging transitivity trapezoid, trapezium trigonometric trigonometry trisection of the angle triangle acute triangle similar triangle right triangle equilateral triangle isosceles triangle obtuse triangle scalene triangle triangular

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angle adjacent angle adjacent angle acute angle complementary angle

U unghi unghi adiacent unghi alăturat unghi ascuţit unghi complementar {până la 90o}

- explementary angle http://math.ournet.md/formule/dic/roman.html

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Dictionar Roman-Englez unghi complementar până la 360o unghi de înclinare unghi diedru unghi drept unghi exterior {al triunghiului} unghi interior unghi înscris unghi întins unghi la bază a triunghiului unghi la centru unghi obtuz unghi plin unghi suplimentar până la 360o unghiuri adiacente suplimentare unghiuri alterne unghiuri alterne externe unghiuri alterne interne unghiuri congruente unghiuri corespondente unghiuri opuse la vârf unghiuri suplimentare {până la 180o} unic unicitate unitate univoc

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- slope angle - dihedral angle - right angle - exteriar angle {of a triangle} - interior angle - inscribed angle - straight angle - base angle {of a triangle} - central angle - obtuse angle - straight ahgle - conjugate angle - adjacent supplementary angles - alternate angles - alternate exterior angles - alternate interior angles, opposite interior angles - congruent angles - corresponding angles - vertex angles - supplementary angles - single - uniqueness - unit - unambiguous

V valabil valid valoare valoare absolută valoare absolută a numărului complex valoare iniţială valoare medie valoare limită variabilă variabilă independentă

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valid valid value absolute value absolute value of a complex number initial value average, average value limit value variable independent variable

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Dictionar Roman-Englez vârf vector vector coplanar vector de bază vector normal vector nul vector unitar vector tangent unitar veridicitate verifica vertical viteză volum

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vertex vector coplanar vector base vector normal vector null-vector unit vector unit tangent vector truth to verify vertical speed, velocity volume

Z zecimal zero

- decimal - zero

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Dictionar Englez-Roman

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Dictionar Englez-Roman de Termeni Matematici ABCDEFGHIJKLMNOPQRSTUVWXYZ

A abscissa absolute absolute extremum absolute value absolute value of a complex number accuracy acnode acute angle acute triangle add addend addition addition sign adjacent angle adjacent side adjacent supplementary angles adjoint admissible admissible solution affine coordinates algebra algebraic equation algebraic expression algorithm algorithm for division alternance alteration http://math.ournet.md/formule/dic/engl.html

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abscisă absolut extremum absolut valoare absolută valoare absolută a numărului complex precizie, exactitate, acurateţe punct izolat unghi ascuţit triunghi ascuţitunghic a aduna, a suma termen al sumei adunare semnul adunării unghi adiacent, unghi alăturat latură adiacentă unghiuri adiacente suplimentare conjugat admisibil soluţie admisibilă coordonate afine algebră ecuaţie algebrică expresie algebrică algoritm algoritm de divizare alternanţă alternare 2/17/05

Dictionar Englez-Roman alternate angles alternate exterior angles alternate interior angles altitude of a triangle amplitude analogous analogical analogy analysis analyze angle anticosine antisine applicate approximate solution arbitrary arc arccosine arc-length arcsine arctangent area argument argument of a function arithmetic mean arithmetic progression associative law associative property assumption asymmetric(al) asymmetry asymptote asymptotes of a hyperbola average average value axiom axis

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unghiuri alterne unghiuri alterne externe unghiuri alterne interne înălţime a triunghiului amplitudine analogic analogic analogie analiză a analiza unghi arccosinus arcsinus aplicată soluţie aproximativă arbitrar arc arccosinus lungimea arcului arcsinus arctangentă arie argument argument al funcţiei medie aritmetică progresie aritmetică lege asociativă proprietate asociativă presupunere asimetric asimetrie asimptotă asimptote ale hiperbolei valoare medie valoare medie axiomă axă

B http://math.ournet.md/formule/dic/engl.html

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Dictionar Englez-Roman back-substitution bar base base angle {of a triangle} base vector basis billion binary binomial binomial coefficient binomial expansion binomial formula biquadratic equation bisector bisectrix bounded interval braces brackets branch bridging brocken line

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substituţie inversă linie de fracţie bază unghi la bază a triunghiului vector de bază bază bilion binar binom coeficient binomial descompunere binomială formula binomului ecuaţie bipatrată bisectoare bisectoare interval mărginit paranteze figurate paranteze patrate ramură translaţie linie frântă

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analiză matematică, calcul a simplifica canonic sistem cartezian de coordonate coordonate carteziene unghi la centru secţiune conică centrală simetrie centrală centru centru unei circumferinţe exînscrise centru schimb de variabilă coardă cerc centrul circumferinţei circumscrise figură circumscrisă

C calculus cancel canonical Cartesian coordinate system Cartesian coordinates central angle central conic central symmetry centre centre of the escribed circle center change of variable chord circle circumcenter circumscribed figure

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Dictionar Englez-Roman closed closed interval coefficient coincident collinearity combination combine similar terms common denominator common difference common divisor common factor common fraction common logs common multiple common ratio commutative law comparation complementary angle complete induction complete solution complex number computable computation concave concave curve concave function concentric circles condition cone congruence congruent angles congruent figures congruent polygons congruent segments conic conic section conjugate conjugate angle conjugate roots consecutive integers constant

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închis interval închis, segment coeficient coincident coliniaritate combinaţie reducere a termenilor asemenea numitor comun raţie a unei progresii aritmetice divizor comun divizor comun fracţie ordinară logaritm zecimal multiplu comun raţie a unei progresii geometrice lege comutativă comparaţie, comparare

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unghi complementar {până la 90o} inducţie completă soluţie completă număr complex calculabil calcul concavă curbă concavă funcţie concavă cercuri concentrice condiţie con congruenţă unghiuri congruente figuri congruente poligoane congruente segmente congruente secţiune conică secţiune conică conjugat

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unghi suplimentar până la 360o rădăcini conjugate numere întregi consecutive constantă

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Dictionar Englez-Roman continuity continuous function convex convex curve convex polygon convexity coordinate coordinate axis coordinate system coplanar coplanar vector coprime numbers corollary corresponding angles count criterion criterion for divisibility cross-product cube cubic cubic curve curve cut cylinder

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continuitate funcţie continuă convex curbă convexă poligon convex convexitate coordonată axă de coordonate sistem de coordonate coplanar vector coplanar numere reciproc prime consecinţă, corolar unghiuri corespondente a calcula criteriu criteriu de divizibilitate produs vectorial cub curbă cubică curbă cubică curbă secţiune cilindru

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date zecimal fracţie zecimală număr zecimal decizie descompunere descompunere a unei fracţii a descreşte funcţie descrescatoare deducţie a defini definiţie degenerat

D data decimal decimal fraction decimal number decision decomposition decomposition of a fraction decrease decreasing function deduction define definition degenerate

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Dictionar Englez-Roman degenerate conic degree degree of a polinomial denominator dependent derivative derivative at a point determinant determine deviation diagonal diagonal element diagonal matrix diameter diametrically opposite point difference differentiability differentiable function differential of area digit dihedral angle dilatation dimension direction direction cosine directly proportional discontinuous discontinuous function discriminant disposition distance distinct distributive law dividend divisable {by} division division algorithm divisor domain domain of definition dot

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secţiune conică degenerată putere, ordin, grad, exponent grad al polinomului numitor dependent derivată derivată în punct determinant a determina deviaţie, deviere diagonală element diagonal matrice diagonală diametru punct diametral opus diferenţă derivabilitate, diferenţiabilitate funcţie diferenţiabilă element de arie cifră unghi diedru delatare dimensiune direcţie cosinus director direct proporţional discontinuu funcţie discontinuă discriminant amplasare distanţă distinct lege distributivă deîmpărţit divizibil {prin} divizare algoritm de divizare divizor domeniu domeniu de definiţie punct

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Dictionar Englez-Roman dot product dotted line double root dual duality principle

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produs scalar linie punctată rădăcină dublă dual principiul dualităţii

-

muchie element element de arie eliminare eliminare prin substituţie metoda eliminării elipsă mulţime vidă ecuaţie ecuaţie a liniei drepte echilateral poligon regulat triunghi echilateral echivalent figură congruentă eroare circumscris, exînscris esenţial estimaţie algoritmul lui Euclid geometrie euclidiană spaţiu euclidian evaluare calculul determinantului par funcţie pară număr par definit peste tot exact împărţire fără rest soluţie exactă exemplu

E edge element element of area elimination elimination by substitution elimination method ellipse empty set equation equation of a straight line equilateral equilateral polygon equilateral triangle equivalent equivalent figure error escribed essential estimation Euclidean algorithm Euclidean geometry Euclidean space evaluation evaluation of determinant even even function even number everywhere defined exact exact division exact solution example

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Dictionar Englez-Roman excentre exclusion existential quantifier expansion expansion of a determinant explementary angle exponent exponential exponential equation expression exterior angle {of a triangle} extremal extremum point

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centru al unei circumferinţe exînscrise excepţie cuantor existenţial descompunere, dezvoltare dezvoltare a determinantului

-

unghi complementar până la 360o exponent, ordin exponenţial ecuaţie exponenţială expresie unghi exterior {al triunghiului} extremal punct de extremum

-

factor, multiplicator teorema lui Bezout descompunere în factori descompunere în factori familie câmp primă derivată ecuaţie de ordinul întâi schemă bloc flux punct focal focar picior al perpendicularei formulă fracţie funcţie funcţie de variabilă complexă funcţie de o variabilă funcţie de mai multe variabile fundamental teoremă fundamentală a aritmeticii

F factor factor theorem factoring factorization family field first derivative first-order equation flow chart flux focal point focus foot {of a perpendicular} formula fraction function function of a complex variable function of a single variable function of several variables fundamental fundamental theorem of arithmetic

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G general general form general solution general term geometric average geometric locus geometric mean geometric progression geometry grade greatest common divisor greatest common factor

-

general formă generală soluţie generală termen general medie geometrică loc geometric al punctelor medie geometrică progresie geometrică geometrie grad, exponent, putere cel mai mare divizor comun cel mai mare divizor comun

-

formule ale jumătăţii unui unghi a împărţi în două {în jumătate} înălţime semisferă hexagon hexaedru hexaedru determinarea extremului omogen ecuaţie omogenă sistem omogen orizontal axă orizontală schema lui Horner hiperbolă hiperbolic ipotenuză ipoteză

H half-angle formulas halve height hemisphere hexagon hexaeder hexahedron hill climbing homogeneous homogeneous equation homogeneous system horizontal horizontal axis Horner's scheme hyperbola hyperbolic hypotenuse hypothesis

I http://math.ournet.md/formule/dic/engl.html

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Dictionar Englez-Roman

identity if and only if image implication improper fraction incenter incircle include inclusion inconsistent incorrect increase increment indefinite independent independent variable indeterminancy induction inequality infinite infinite decimal fraction inflexion inhomogeneous initial initial condition initial value inscribe inscribed angle inscribed circle inscribed polygon integer number integral integral curve integrand integration by parts integration constant inter-stage function intercept intercept theorem

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identitate dacă şi numai dacă imagine implicaţie fracţie supraunitară centrul circumferinţei înscrise circumferinţă înscrisă a include incluziune incompatibil incorect a creşte creştere nedeterminat independent variabilă independentă nedeterminare inducţie inecuaţie infinit fracţie zecimală infinită inflexiune neomogen iniţial condiţie iniţială valoare iniţială a înscrie unghi înscris circumferinţă înscrisă poligon înscris număr întreg integrală curbă integrală expresie de sub integrală integrare prin părţi constantă de integrare funcţie în scară segment teorema lui Thales 2/17/05

Dictionar Englez-Roman interdependency interior angle intersection interval inverse inversely proportional irrational number irreductibility isosceles triangle

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interdependenţă unghi interior intersecţie interval invers invers proporţional număr iraţional ireductibilitate triunghi isoscel

J jump jump function

- salt - funcţie în scară

K kilogram(me) kilometre known

- kilogram - kilometru - cunoscut

L law law of composition law of sines law of the excluded middle least common denominator least common multiple leg Leibniz rule lemma length like denominators like signs limit limit value

-

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lege lege de compoziţie teorema sinusurilor legea terţului exclus cel mai mic numitor comun cel mai mic multiplu comun faţă laterală formula lui Leibniz lemă lungime numitori identici aceleaşi semne limită valoare limită

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Dictionar Englez-Roman limits of integration line line segment linear linear equation linear function linear independency linearity local logarithm lower limit lowest common denominator lowest common multiple lozenge

Page 12 of 22 -

limite de integrare dreaptă segment liniar liniar ecuaţie de gradul întâi (liniară) funcţie liniară independenţă liniară liniaritate local logaritm limită inferioară cel mai mic numitor comun cel mai mic multiplu comun romb

-

mărime diagonală principală axă principală sistem cu mai multe necunoscute imagine, aplicaţie imagine, aplicaţie metru inducţie matematică matematică matrice matrice de transformare maximum, maxim medie aritmetică medie geometrică măsurabil măsură mediană element {al mulţimii} minimum, minim descăzut semnul scăderii modul monom

M magnitude main diagonal major axis many-variable system map mapping meter mathematical induction mathematics matrix matrix of the transformation maximum mean mean proportional measurable measure median {of a triangle} member {of a set} minimum minuend minus {sign} module monom

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Dictionar Englez-Roman monotone decreasing function monotone increasing function monotonic function monotonous multiple multiplex multiplicand multiplication multiplier multiply mutually mutually disjoint

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funcţie monoton descrescătoare funcţie monoton crescătoare funcţie monotonă monoton multiplu multiplu, repetat deînmulţit înmulţire, multiplicare înmulţitor, multiplicator a înmulţi mutual mutual disjuncte

-

logaritm natural număr natural condiţie necesară şi suficientă negativ neglijabil nod nedescrescător nedegenerat secţiune conică nedegenerată secţiune conică nedegenerată neliniar ecuaţie neliniară sistem neortogonal de coordonate neperiodic nesimetric fracţie zecimală infinită soluţie netrivială soluţie nenulă normală, perpendiculară normală la suprafaţă vector normal vector nul matrice nulă mulţime vidă număr

N natural logarithm natural number necessary and sufficient condition negative negligible node non-decreasing non-degnerate non-degenerate conic non-degenerate conic section non-linear non-linear equation non-orthogonal coordinate system non-periodic non-symmetric non-terminating decimal non-trivial solution non-zero solution normal normal to the surface normal vector null-vector null-matrix null-set number

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Dictionar Englez-Roman number line numerator

Page 14 of 22 - dreaptă numerică - numărător

O oblique obtuse angle obtuse triangle octagon odd odd-function one-to-one open interval opposite interior angles order order of derivative order of equation ordered pair ordinal number ordinate origin {of coordinates} orthogonal base orthogonal coordnate system orthonormal basis oval

-

oblic unghi obtuz triunghi obtuzunghic octagon impar funcţie impară biunivoc interval deschis unghiuri alterne interioare ordine ordin al derivatei ordinul ecuaţiei cuplu ordonat număr ordinal ordonată origine de coordonate bază ortogonală sistem ortogonal de coordonate bază ortonormată oval

-

cuplu parabolă parabolic paralel paralelepiped paralelogram regula paralelogramului regula paralelogramului parametru formă parametrică

P pair parabola parabolic parallel parallelepiped parallelogram parallelogram law parallelogram rule parameter parametric form

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Dictionar Englez-Roman parentheses partial fraction partial-fraction expansion pencil pencil of lines pentagon per cent perimeter period periodic decimal fraction periodic function periodic solution permissible solution perpendicular pivot plane plane geometry planimetry plus sign point point of discontinuity point of inflexion polygon polyhedron polynomial positive possibility power pre-image preceding prime factorization prime number pimitive principal principal axis principal diagonal principle of complete induction prism product progression projection

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paranteze rotunde fracţie elementară descompunere în fracţii elementare fascicul fascicul de drepte pentagon procent perimertu perioadă fracţie zecimală periodică funcţie periodică soluţie periodică soluţie admisibilă perpendiculară axă de rotaţie, centru de rotaţie plan planimetrie planimetrie semnul adunării, plus punct punct de discontinuitate punct de inflexiune poligon poliedru polinom pozitiv posibilitate putere, exponent preimagine precedent descompunere în factori primi număr prim funcţie primitivă principal axă principală diagonală principală metoda inducţiei complete prizmă produs progresie proiecţie

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Dictionar Englez-Roman proof proper factor proper fraction property proportion proposition prove pyramid Pythagorean theorem

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demonstraţie divizor propriu fracţie subunitară proprietate proporţie propoziţie a demonstra piramidă teorema lui Pythagoras

Q quadrate quadratic quadratic equation quadratic formula quantifier quotient

-

patrat pătratic ecuaţie de gradul al doilea (patrată) formula rădăcinilor ecuaţiei de gradul al doilea (ecuaţiei patrate) - cuantor - cât

R radian radical radical sign radius radius vector raise to a power range rank of a matrix ratio rational rational function rational number ray real real number reciprocal matrix

-

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radian radical, semnul rădăcinii radical, semnul rădăcinii rază rază vectoare a ridica la putere mulţime de valori, domeniu rang al unei matrici cât, raport raţional funcţie raţională număr raţional semidreaptă real număr real matrice inversă

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Dictionar Englez-Roman rectangle rectangular coordonate system reduce reducible regular polygon relation relative relatively prime numbers remainder repeated root replace represent rest restriction rhomb rhombus right angle right triangle root rotation round rounding error rule

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dreptunghi sistem rectangular de coordonate a reduce reductibil poligon regulat relaţie, raport relativ numere reciproc prime rest rădăcină multiplă a substitui, a înlocui a reprezenta rest restricţie romb romb unghi drept triunghi dreptunghic rădăcină rotaţie a rotunji eroare de rotunjire regulă

S same denominators satisfy scalar scalene triangle secant sector of a circle segment semi-circle semiclosed interval set set theory side {of an angle} sign signum function

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numitori identici a satisface scalar triunghi scalen secantă sector circular segment semicerc interval semiînchis mulţime teoria mulţimilor latură {a unui unghi} semn funcţia signum

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Dictionar Englez-Roman similar fractions similar polygons similar terms similar triangle similarity similitude simple simple root simplification sine curve sine rule single single root skew lines slope slope angle slope formula slope-intercept form of a straight line equation solution {of a problem} solution set solve space speed sphere square square brackets square root standard form statement step function straight straight angle straight-line straight-line segment stretching strict strongly monotonic subset substitution subtraction subtrahend http://math.ournet.md/formule/dic/engl.html

Page 18 of 22 - fracţii cu numitori egali - poligoane asemenea - termeni asemenea - triunghi asemenea - asemănare - asemănare - simplu - rădăcină simplă - simplificare - sinusoidă - teorema sinusurilor - unic - rădăcină simplă - drepte neconcurente - înclinare, coeficient unghiular - unghi de înclinare - formula coeficientului unghiular - ecuaţie a unei drepte cu coeficient unghiular - soluţie {a unei probleme} - mulţime de soluţii - a soluţiona, a rezolva - spaţiu - viteză - sferă - patrat, a ridica la patrat - paranteze patrate - rădăcină patrată - formă standardă - propoziţie, afirmaţie - funcţie în scară - drept - unghi întins, unghi plin - linie dreaptă - segment liniar - delatare - strict - strict monoton - submulţime - substituţie - scădere - scăzător 2/17/05

Dictionar Englez-Roman

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sum summand supplementary angles surface surface area surface element surface of revolution symbol symmetric symmetric function synthetic division system system of linear equations

- sumă - termen al sumei -

unghiuri suplimentare {până la 180o} suprafată arie a suprafeţei element de arie {al suprafeţei} suprafaţă de rotaţie simbol simetric funcţie simetrică, funcţie pară schema lui Horner sistem sistem de ecuaţii liniare

T tangent tangent line tangent plane term term of a fraction terminating decimal fraction tetragon tetrahedron theorem theory transcendental number transform transform of coordinates transitivity translation trapezium trapezoid triangle triangular trigonometric trigonometric function trigonometry trisection of the angle truth

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tangentă linie tangentă plan tangent termen numărător al fracţiei fracţie zecimală finită patrulater tetraedru teoremă teorie număr transcendent a transforma transformare de coordonate tranzitivitate translare trapez trapez triunghi triunghiular trigonometric funcţie trigonometrică trigonometrie trisecţie a unui unghi veridicitate 2/17/05

Dictionar Englez-Roman

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U unambiguous unbounded uncertainty undefined undetermined unequal union unique solution uniqueness unit unit circle unit tangent vector unit vector universal quantifier universal set unknown unlike denominators unsymmetric

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univoc nemărginit incertitudine, nedeterminare nedefinit nedeterminat inegal, diferit reuniune soluţie unică unicitate unitate circumferinţă unitară vector tangent unitar vector unitar cuantor universal mulţime universală necunoscută numitori diferiţi asimetric

-

valid, valabil valoare a se anula, a dispărea variabilă vector produs vectorial viteză a verifica vârf unghiuri opuse la vârf vertical axă verticală invers, reciproc linie de fracţie

V valid value vanish variable vector vector product velocity verify vertex vertex angles vertical vertical axis vice versa vinculum

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Dictionar Englez-Roman volume

Page 21 of 22 - volum

W way well-defined whole

- drum, distantă - bine definit - tot, în întregime

X x-axis x-intercept xy-plane

- axa x - segment de pe axa x - plan xOy

Y y-axis

- axa y

Z z-axis zero zero solution

- axa z - zero - soluţie nulă

Bibliografie

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Dictionar Englez-Roman

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Fraze uzuale

Page 1 of 70

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Dictionar Matematic Roman-Englez de Fraze Uzuale AĂÂBCDEFGHIÎJKLMNOPQRSŞTŢUVWXYZ

A abia

The superscripts are hardly (or scarcely) visible.

abordare

A more careful approach is needed.

absolut analog

In complete analogy with the result, we may prove that ...

accentua atenţie se accentuiază atenţie asupra In this section, emphasis is placed (or attention is focussed) on autonomous systems. accepta o ipoteză

to accept a hypothesis

aceiaşi

The numbers +8 and +3 have like signs. No two polynomials can have all their roots alike (or the same) unless they differ by a numerical factor.

acelaşi

The preceding discussion may be summed up by saying that the arithmetic for complex numbers is much the same as that for real numbers.

acum

We now want to let j→∞. For the present we content ourselves with demonstrating the assertions just made concerning the sets G and H.

arăta acum este uşor de arătat că

We can now easily show that ...

adecvat

The model fits the process under study adequately.

adeveri se adevereşte că s-a adeverit a fi

Thus the martingale ψ(t) turns out to be a process with independent increments. This proved to be a bigger problem than originally anticipated.

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Page 2 of 70 This value was found to be cos(π/17).

admite la fel admite adresa ne adresăm la aduce demonstraţia este adusă aduna adunăm afirma afirmă

afirmaţie afirmaţie mai generală afirmaţie necesară afirmaţie reciprocă

The function η(t) also admits decomposition of the form (4). We now turn to the task of considering the continuous-parameter supermarlingales. The proof of the theorem is given in [8]. Now write down these inequalities for n, n+1, ..., n+m and add them to obtain the estimation (6). The fundamental theorem of arithmetic asserts that every integer greater than 1 can be factored in one and only one way, apart from order, as the product of positive prime numbers. Bohm [2] claims that there is such a thing as the primacy of metaphysics in connection with biology. A more general statement can be found in [15]. Approaching the limit as N→∞ we obtain the required assertion. The converse of (5) is false. The converse of this statement is known to be false. However, if we are concerned only with versions of a given process, the converse holds in the following sense.

afla a afla ajunge ajungem la ... ajutor cu ajutorul

We are now in a position to determine (or to find) the value of x*. Thus we arrive at the following conclusion [definition, equation, theorem, assertion, result]. By means of the equivalence relation we get the decomposition of the set of all oriented segments into disjoint classes such that segments in the same class are equivalent and segments in different classes are not equivalent. When formalizing the notions of probability theory, the first fundamental assumption is that the results of experiments can be described by means of a certain

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Page 3 of 70 set Ω. The data were obtained with the help (or with the aid) of a digitizer. Both problems are solved by constructing a right-hand inverse matrix Sr-1 of S.

alt altă

Another way to approach the problem is to take a Taylor series expansion of g(t). Any term may be transposed from one member of equation to the other, provided its sign is changed. The random variable η may have a different distribution.

altfel

This definition also makes use of the abstraction principle, but in another way.

altfel zis

Otherwise speaking [fn(x)] = [gn(x)] if and only if {fn(x)}~{gn(x)}.

amănunţit mai amănunţit ambele

ambele părţi

These operations are discussed in more detail in next section. Both estimates are quite rough. But the functions f and g are both continuous, and thus the two functions coincide. Multiplying the two sides of (4) by cosmx and integrating, we obtain ... The two sides of Young's inequality are equal if and only if b=ϕ(a).

ambiguu aminti amintim că

analiza pentru a analiza este analizată

analogic

This ambiguity does not lead to misunderstanding. We recall that the compensator γn of the measure of jumps µn is non-negative. Recall that L is generated by all processes whose trajectories are right-continuous and have right limits. To analyse the equation, it is convenient to go over to non-dimensional variables. To analyse the equation, it is convenient to go over to non-dimensional variables. The sum S is analysed analogously. The proof of (11) is completely analogous to that of (10).

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Page 4 of 70 We shall introduce distributions in a way analogous to that used in the Cantor theory when generalizing rational to real numbers.

analogie

The analogue of the Fermat's theorem is also valid.

anume

In this particular case only one side of the triangle, namely BC, has traces within the limits of the drawing. If S consists of a single element x, then R(x), namely the smallest subring containing x, is commutative. The extended solutions have the property II, namely of uniform continuity.

aparatul matematic

mathematical apparatus (or tools / techniques) a body of mathematics

aparţine aparţine la

care nu aparţine apărea apare aplica să aplicăm pot fi aplicate se poate de aplicat aproape aproape de aproape peste tot aproape toate aproape toţi aproximativ acelaşi

The process X(t) falls into one of the following three classes depending on its asymptotic behaviour. This equation falls among the so-called renewal equations. This equation contradicts the assumption that G contains points not belonging to D. The problem arises of finding the conditions under which the distribution p(u) is unimodal. Let us apply this result to the case of an increasing process At. Formulas (1) and (2) may be employed on condition that the contour C has no loops. Since p is of the form 1+4m, Corollary 2.5 is applicable and hence p=x2+y2. The temperature was observed to be near zero. The angle θ is closed to 45 deg. The Lebesque-integrable functions are equal if and only if they have the same values almost everywhere. Nearly all scientific problems deal with quantities and relations among them. It is easy to verify that f(x)
aproximaţie în primă aproximaţie As (or in) a first approximation, x(t) may be chosen http://math.ournet.md/formule/fraze/fraze.html

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Page 5 of 70 to be t2.

aranja în ordine crescătoare [descrescătoare] sunt aranjaţi în ordine crescă- The jump moments t1, t2, ..., tn are arranged in toare [descrescătoare] ascending [descending] order. arăta arată vom arăta este arătat pentru a arăta că

cum este arătat mai sus

Fig.12 depicts the slope of the line ∆x versus ∆t. We shall show that in this case p must divide b. The slope of the line ∆x versus ∆t is shown schematically in Fig.12. To show that q and r are unique, suppose that q' and r' is any pair of integers for which a=bq'+r', 0 ≤ r' < b. As has been shown above, the process X t is a continuous L2-martingale.

arbitrar

Let ξ(t) be an arbitrary quasi-potential. Suppose this is true for an arbitrary n.

argument

This is a convincing reason for the study of stochastic equations of the form. This reason suggests that the solution to Eqn (17) is unique. There are doubtful points in the above reasoning.

argumente aritmetică

asigura asigură

asociat asociată cu (referitoare la)

asociat la

Arithmetic is the elementary branch of mathematics dealing with the properties of numbers and their operations. Property 1o ensures the feasibility of the addition. The MIS furnishes (or supplies) data for management. The error is associated with (or is due to) the inaccuracy in computations. The study is concerned with linear functionals of supercritical processes. The first problem is directly related to partial differential equations. Further extension of the notion of a stochastic integral is connected with the extension of the notion of convergence. The polynomial R(t) corresponding to this function by (6) is almost everywhere nonzero. The space, S2k-1, and the projective metric, L(r,s),

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Page 6 of 70 associated with it will play an important part in the future. It is convenient to replace k by a new factor µ related to k by the relation µ=4πk.

astfel astfel încât

astfel ... ca astfel ... încât

Thus the following theorem has been proved. The factors µ and k are related as follows: µ=4πk. Let X be a space such that from every open cover we can select a finite cover. We break up the component a into several components in such a way that A and B now belong to different components. We select a basis in the underlying space such that ... Specifically, the parameter can be chosen so that the distribution is unimodal. The model is so constructed as to allow further modification and improvement. We take a point on the line such that ... We take a point on the line such that ... Applying Lemma 20.3 k times, we infer that there exists a periodic function G0(x) with G0(k)=g(x).

aşa numit

In applications we often come across the so-called Heaviside function.

aşa ... pentru care

We take a point on the line such that ...

aşadar

Thus, the integral on the right-hand side of (5) can be analytically continued to the whole domain. The last formula becomes p(la+mq)=b, so that p|b. Every element in T has norm at least equal to 1, so there must be one of them, γ , which is of least positive norm. Since p is of the form 1+4m, Corollary 2.5 is applicable and hence p=x2+y2. Therefore P(x,s,t,A) saitsfies the ChapmanKolmogorov equation. Consequently,

ataşa la poate fi ataşat la

atinge atinge maximum [minimum]

The process X(t) may be placed (or put) into one of the following three classes depending on its asymptotic behaviour.

The function ψ(x) reaches its maximum [minimum] at x*=3h/2. Let b1(s,w) coincide with the z for which the

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Page 7 of 70 minimum of the function | B(s,w) z |, | z |=1, is attained and let b2(s,w) coincide with z for which the above function attains its maximum.

atinge scopul de a atinge scopul este atins scopul

atinge se atinge

atrage atenţia să atragă atenţia

se atrage atenţia

atribui sens pentru a atribui sens

se atribuie sens

atribui valoarea atribuim valoarea valori pot fi atribuite

In order to attain the purpose, we shall not avail ourselves of the methods of functional analysis. This aim is attained (or objective is accomplished) by introducing non-dimensional variables x' = x/x0 and y' = y/y0. Evidently, if the inf in (1.5) is attained at a certain point , then the inf Φm(x,um) is also attained at that point. The reader should take notice of the fact that the scale parameter a does not enter into the expression for G (u). It is necessary to pay special attention to the behaviour of F(x) at infinity. Recently much attention (or thought / consideration) has been given to the study of the diffusion processes. To attach a meaning to the solution of this stochastic equation, we must define the measure µ more accurately. Analogous meaning is attached to the notation D [0,∞).

We assign the value 0 to this state. A variable is a quantity to which an unlimited number of values can be assigned in an investigation.

automat

It follows automatically that these Rm-semisimple rings are strongly Rm-semisimple.

auxiliar

As a preliminary to the proof of the theorem, we shall discuss several auxiliary assertions.

avantaj avantaje avantaj principal

To appreciate the merits (or the advantages) of this theory, it suffices to observe that ... The main (or principal / major) advantage of this

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Page 8 of 70 approach is that it enables the results of the previous chapter to be extended to the case under consideration.

avea avem nu are

Since a satisfies (1), we have Y 1=Y 2. For instance, the function has not a derivative at

poate avea nu poate avea avea de-a face avem de-a face avea intenţia vom avea intenţia

avea loc această nu poate avea loc această are loc în cazul avea maxim [minim] are maxim [minim] avea neajunsul au neajunsul are neajunsul avea nevoie de vom avea nevoie de avea o formă ... are o formă ... avea o mare importanţă are o mare importanţă

the point 0. The prime number p possess no factors other than ± 1 and ± p. It follows that the solution can have only a finite number of zeros in D. It follows that the solution cannot have more than a finite number of zeros in D. In this section we deal only with rational integers. In this section we are going to modify our definition of distribution so as to obtain a notion equivalent to that of S. Sobolyev and L. Schwartz. This cannot be, for l/p is not a rational number. This happens to be the case for those equations in which S increases. The function ψ(x) has a maximum [minimum] at the point x*=3h/2. These equations possess a fundamental defect (or demerit / draw back / disadvantage). His hypothesis has limitations of its own. In order to achieve this, we shall need some material from elementary number theory. Equation (3.2) is of the following form ... In applications the following theorem is of great importance.

avea sens

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dacă are sens The symbol

nu are sens avea soluţie unică are soluţie unică avea un caracter are un caracter avea valoarea are valoarea

, if it is meaningful, always

represents a number. If the common part is empty, the difference has no sense. Equation (7) has a unique solution. Noise ω(t) is stochastic in character (or in nature). Noise ω(t) is of a stochastic character (or nature). The derivative F'(x) has the value 0 at the point 0.

B baza pe se bazează pe

The proof is based on the ideas of Kunita and Watanabe [36]. The previous analysis proceeds from the assumption that the class M* contains the class M2,∞.

bineînţeles

It is clear that the definitions and properties given below can be extended to σ-algebras.

biunivocitate

Hence from (6) and the remark concerning the one-to-one-ness of the correspondence, relation (8) yields F=0.

C ca de obicei

As usual, we use continuous versions of (4).

ca şi în

As in the example discussed above, we can easily prove the following proposition.

ca şi în cazul ...

In the same manner as in the case of convergence to a Wiener process, one can easily deduce the following result. In the case of distributions, as in the case of functions,

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Page 10 of 70 the following properties hold true.

ca urmare

Such a solution does not exist in consequence (or on account) of (6) and Remark 4.

calcula sunt calculate

As is easy to calculate ... The coefficients of expansion (9) are evaluated (or computed) from formulas (5). as is easy to calculate as is easy to evaluate

cum este uşor de calculat calcula integrală dea-lungul curbei

to take the integral along a curve

calcula integrală pe domeniu

to take the integral over a domain

care

In classical analysis there have been considered several sequences of functions which are distributionally convergent to δ(x). Choose in D2 any function that tends to infinity as cărui al cărui care este

x→0. A variable is a quantity to which an unlimited number of values can be assigned in an investigation. Let A be a strongly positive operator whose domain of definition does not depend on t. Consider the function λ(y), which is a periodical distribution coinciding with δ(y) for | y | < 1/2.

caz caz particular cazul considerat

căuta vom căuta sunt căutate căutată

care va fi căutată

This formula can be considered as a particular case of formula (14). In the case under consideration the process α(t) is continuous. Let us generalize equation (6) to the case under discussion. We shall seek a solution to (12) in the form z(t,u) = x(t) y(u). It is convenient, in solving problems, to use letters for the numders which are sought. The sought-for quantity M is involved in the equation reciprocally. The desired value of m can be determined from conition (4). Consider the equation (3), where f(t) is the function to be found. Here v* is the value of v to be determined.

căutare

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Fraze uzuale în căutarea câteva

câteva modalităţi

Page 11 of 70 In looking (or in search) for the optimal solution we discovered that ... To see this, we shall prove several lemmas. A few words should be said about physical meaning of this parameter. Some other examples of fundamental sequences will be given after the following lemma. There is more than one way to approach the problem.

ce

For instance, the Dirac delta distribution cannot be identified with any continuous function, which will be proved in Section 8.

cel mai bine

The nature of this problem can best be made plain by examples.

cel mai mare divizor comun (CMMDC)

The greatest common divisor (GCD) of two integers h and k is the largest positive factor common to both h and k.

cel mai mic multiplu comun (CMMMC)

Twelve is the least common multiple (LCM) of 3 and 4.

cel mult

The set of the points of discontinuity of f(t) is at (the) most countable.

cel puţin

Formula (7) is true under the hypothesis that at least one of the integrals on the left-hand side is meaningful.

cercetare cercetare profundă cercetare ulterioară

the in-depth study To determine whether these ideals coincide, we must make a further study of the polynomials.

cere se cere se cere întrebarea

cerinţă mai puternică

It is required to estimate in the best possible manner the value of a random variable ξ. It is therefore reasonable to ask for which classes of "integers" the fundamental theorem holds, and for which it does not. The process Mt, satisfies the conditions of (1) with the stronger requirement that martingale.

chiar în cazul când

ci

be an L2-

This example shows that the ordinary derivative does not always coincide with the distributional derivative, even when the ordinary derivative wrists everywhere except at one point. Thus we need not 8, but 9 parameters to describe the system.

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Fraze uzuale ciocni cu ne ciocnim cu

Page 12 of 70

In applications we often come across the so-called Heaviside function.

cititor

The reader will find it a useful exercise to supply the details of the proof.

clasic

In classical analysis there have been considered several sequence of functions which are distributionally convergent to δ(x).

clar este clar că

intuitiv este clar

coeficientul pe lângă coeficientul pe lângă termenul superior coincide coincide cu

complet satisfăcător completa este completată

Clearly, the sum, difference and product of such integers are of the same form. It is clear that a and -a are both in the set A. It is clear intuitively that every collection, finite or infinite, of non-negative integers contains a smallest one. the coefficient of smth. the coefficient multiplying smth. A polinomial is monic if its leading coefficent is 1.

For an arbitrary function F(x) satisfying conditions 1) 6) there exists a unique probability measure whose distribution function coincides with F(x). The characteristic through A is the same as the one through B. Completely satisfactory solution of the functional problem is given in [6]. It is assumed that σ-algebra F0 is completed (or augmented) by all P-null sets.

complicat

The problem of optimization is, in general, more complicated. Integration of equation (3.2) represents a rather involved boundary-value problem.

conclude

From a>b and b>c we conclude that a>c.

condiţie condiţia asupra

condiţiile conform cărora

This result was established in another way under somewhat weaker conditions on the diffusion coefficients. The problem arose of finding the conditions under which the distribution p(t) is unimodal.

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Fraze uzuale condiţie de convergenţă la condiţie de existenţă şi unicitate a soluţiei condiţii exacte pentru condiţie mai fină condiţii sunt verificate

alte condiţii fiind aceleaşi

în condiţia că

în condiţiile teoremei

Page 13 of 70 The condition for convergence to a Wiener process is stated in the next section. Let the coefficients of equation (9) satisfy the condition for the existence and uniqueness of the solution. The exact conditions for equilibria to exist and be stable are examined. The tangent bundle of the manifolds provides a sharper necessary condition. Assume that conditions (5) and (20) are satisfied. If the conditions of Theorem 10 are satisfied and, moreover, A(ϕ,t) is continuously differentiable, then ... Let the conditions of Theorem 4 be satisfied and the function be twice continuously differentiable. The distinction between (14) and (15) is that the variable x entering into (15) depends on ω, other things being equal. Thus, the function f(x) is uniquely determined, provided that it exists everywhere. Any term may be transposed from one member of equation to the other, providing its sign is changed. The abstraction principle can be applied to sets of arbitrary elements provided there exists an equivalence relation ~. These functions are locally Riemann-integrable under (or on) the condition that their values are equal at all points of continuity. Under the conditions of Theorem 6.1 the limit exists and is differentiable.

conduce conduce la ecuaţia

This reasoning yields a simple equation.

confirma ipoteza

to verify a hypothesis

conform conform algoritmului conform criteriului conform definiţiei conform ipotezei inducţiei conform lemei

The classes can be formed by the FLAMENCO algorithm. By the Eisenstein's criterion, the polynomial xn-2 is irreducible over the field of rational numbers. Then, by definition, ... By the induction hypothesis, the theorem is true for f (x). By Lemma 2.3, the sequence {gn(x)} is fundamental.

conform presurunerii By assumption, E(1)=0. By the assumption of the lemma, the solution to (1) is positive. http://math.ournet.md/formule/fraze/fraze.html

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conform proprietăţii

conform teoremei consecinţă consecinţă din este o consecinţă

consecutiv

considera considerăm poate fi considerată să considerăm

sunt considerate

considerată

poate fi considerat ca

în lucrare se consideră considera acum considerăm acum considera de la început vom considera de la început

Page 14 of 70 Since ... by hypothesis, (4) is a bounded sequence. By property (4) of norms, every element in T has norm at least equal to 1, so there must be one of them, γ, which is of least positive norm. By Theorem 1.1, we can find q and r so that b=dq+r, 0 ≤ r < d. To complete the proof of the lemma, we use the corollary of Lemma 2. The inequality f(x)≤0 is a direct consequence of the system fi(x)≤0, i=l, 2, ..., n. When an expression contains parentheses within parentheses, they may be removed in succession, beginning with the innermost. Consider the rational number a/b. The problem can be approached in many different ways. Let us now consider the case a>1. Consider now the case a>1. Several other functions, very useful from the practical point of view, are considered in the following section. Some properties of the operator L are dealt with in the next chapter. An integral part of the creative approach to the problem under consideration (or under study) is the analysis of its similarity to and difference from the analogous problems. C (R+) can be regarded as a complete separable metric space under the topology of uniform convergence. The paper deals with the theory of distributions of finite order. We now consider the process η(t). Consider now the process η(t). Consider first the simplest case where a=0. We will first confine our attention to discreteparameter martingales.

considera util

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Page 15 of 70 We find it useful to define a new concept of conjugacy and to discuss its relation to the old.

considerabil mai simplu

Theorems IV and VI are considerably simpler than the analogous theorems in the differential calculus.

consideraţie

We now turn to the task of considering the continuous-parameter stochastic processes. The consideration (or discussion) of this problem in more detail shows that ... We must now subject the concept of equivalence to a somewhat closer scrutiny. Further inspection (or examination) of Fig.1 discloses that there are two points of inflection on the curve.

consta constă în ...

construcţie construi pentru a construi

continuu continuă din dreapta după t

The abstraction principle consists (or lies) in the identification of objects (mathematical entities) which have a common property. By a slight change in the method of (7), the construction of such functions is easily carried out. The abstraction principle is often applied in mathematics to construct new notions. In the case under consideration the process α(t) is continuous at every point of the interval (c,d). Ft is continuous from the right (or right-continuous) in t if for all t ≥ 0

contrazice ce contrazice

.

So π is a unit, contrary to the hypothesis. So p|L, contradicting the fact that L is prime.

această contrazice

This contradicts r ≥ 0.

contrazice presupunerii

This equation contradicts the assumption that G contains a point not belonging to D. This contradicts the hypothesis that k→∞ and that (6) holds for all l
contribui pentru a contribui la răspândirea

conţine se conţine

This series of expository presentations of the best thoughts and keenest researches in pure and applied mathematics is intended to contribute to the dissemination of mathematical knowledge. The subgroup Pn is contained in the centralizer of the element p in B.

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Fraze uzuale conţine proprietăţi convenabil convenabilă

Page 16 of 70 The following list contains the fundamental properties of the norm. In some cases the following complex form of trigonometrical series

mai convenabil este convenabil va fi convenabil deseori este convenabil conveni vom conveni converge converge absolut converge aproape peste tot converge în medie converge în probabilitate converge uniform convinge ne convingem că

a se convinge în trebuie să se convingă că pentru a se convinge în ...

pentru a se convinge în aceasta

is advantageous.

It is more convenient to break up the experiment into three phases. It is convenient to call two Gaussian integers associates if α=εβ, where ε is a unit. It will be convenient to define the convergence of fn (x) in the following sense. It is often convenient to establish the position of points by giving their coordinates. We shall agree to write A(x,t)∈S if A(x,t) is a process with independent increments. The series (1) converges absolutely. to converge almost everywhere to converge in (the) mean to converge in probability Thus the series (4) converges uniformly. Make (or be) sure that the coefficients have no common factors other than +1. To complete the proof, check that ... Before applying Theorem 10.7, check to see that there is an α in A such that ... To prove the convergence X n→M, it suffices to make sure that ... The investigator must assure himself that the errors are within the admissible limits. To see that if θ satisfies an equation anxn + an-1xn-1 + ... + a0 = 0, then there is a rational r such that rθ is an algebraic rational, observe that anθ satisfies (3). To be certain that c is the required element, check that D[c,A]>0. To see this, notice that Powell's frequency function is identical with our function Φb(x).

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Fraze uzuale se poate de convins că este uşor de convins că coordona se coordonează

Page 17 of 70 Then, similarly as has been done in [1], one ascertains that the limit function satisfies the equation (4). It is easy to make sure that ϕ(x) is convex on [c,∞).

The theory checks well (or agrees / is in good agreement) with experiment. The hypothesis fits the experimental data. This conjecture is in line with that of Seidenberg.

corect

This definition is correct; since the sequence {pn(m) (x)} is fundamental.

corectitudinea definiţiei

To verify the correctness of this definition, we must prove that ...

corespondenţă biunivocă

The one-to-one correspondence established above makes it unnecessary to distinguish between functions f(x) and distributions [f(x)].

crescător crescătoare strict crescător creşte creşte monoton după t

There is an increasing sequence of integers km such that ... Introduce a strictly increasing funstion g(x) defined on [0,∞). Ft monotonically increases in t, that is Fs⊂ Ft, for s
cu ... ce tinde ...

Each summand on the right of (12) becomes vanishingty small with (or as) n→∞.

cub al numărului

When a is used three times as a factor, the product is called the third power of a, or the cube of a.

cunoaşte cum noi deja cunoaştem cunoscut cu

este cunoscut că

Since, as we already know, all these trigonometrical integrals vanish, we obtain formula (5). The reader acquainted with the Lebesgue integrals can interpret formula (1) in a still wider sense, viz. as the equality almost everywhere. It is well known that for an arbitrary function F(x) satisfying conditions 1) - 6) there exists a unique probability measure whose distribution function coincides with F(x).

cum este cunoscut As is well known, the nth partial sum of the series is nothing else

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Page 18 of 70

than the Dirichlet kernel este bine cunoscut

.

It is well known that there exist sequences {Tn} and {Sn} such that ηn=Tn-Sn.

este cunoscută ca cuvânt cu alte cuvinte

This relation is known as the Biot-Savart law. To put it in another way, the gross weight W can be represented as W=Wst+We+Wp.

D da dă

dăm da posibilitate dă posibilitate

da restul de la împărţire dă restul de la împărţire dacă

dacă acesta este cazul dacă acesta există

The repeated application of relation (2.1) yields ... The following theorem provides the description of the everywhere dense set E. We give (or present) below a synthesis of various approaches available. The martingale approach enables us (or permits) to suggest a new method for constructing solutions of stochastic differential equations. Knowledge of the numerical values of a and b offers a means of (or makes possible) determining ... This method makes it possible to tackle the boundaryvalue problem under discussion. The main advantage of this approach is that it enables the results of the previous chapter to be extended to the case under consideration. Every integer a leaves a remainder r, 0 ≤ r < m, on division by m. In the course of the proof of the theorem we have described the method for determining the functions Φn (x0, ..., xn) if X is locally compact. If this is the case, the deterministic approach is quite suited for this purpose. Hence, it follows that the representation of f(x) in the form (4), if it exists, is unique.

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Fraze uzuale dacă este necesar dacă n-ar fi dacă nu

dacă numai

Page 19 of 70 The integer k can be, if necessary, replaced by any greater integer. Were is not for the discontinous coefficient,the problem would be much simpler. If there is no such ζ, we write max f = ∞. Unless otherwise stated, condition (2) will always be assumed to be in force. Thus, the function f(x) is uniquely determined, provided (that) it exists everywhere.

dacă şi numai dacă dacă şi numai dacă există

R is a realization of η(t) if and only if ... The martingale ξ(t) is uniformly integrable if and only if there exists a random variable η such that E |η| < ∞ and ξ(t) = E {η | Ft}.

dacă ..., atunci

Evidently, if the inf in (1.5) is attained at a certain point , then the inf Φm(x,um) is also attained at that point.

dacă ... nu

Unless u is a power of 2, this can always be done.

dar

Element ξ belongs to V, whereas (or and) ζ belongs to W. This value is negative — not positive. But then we could proceed from the pair (x, y) to a still smaller in the same way as from (x1, y1) to (x2, y2).

dar atunci dar nu

With time the number of particles increases, rather than remains constant.

date iniţiale

the initial (or the input) data

datorită

Delacherie and Meyer are well known for their book 'Stochastic Processes and Potentials'. Mathematics, due to (or owing to / because of) its abstract nature, finds wide application in most spheres of human activity.

de aceea

For this reason the points belonging to Ω are called elementary events. Conversely, if N=1, we have a+b=1, so that either a=0 or b=0. The last formula becomes p(la+mq)=b, so that p|b. Every element in T has norm at least equal to 1, so there must be one of them, g, which is of least positive norm.

de asemenea

Observe also that |Dn(s)| ≤ C logn. From equation (3) we find, as well, . If α and β belong to F, so do α+β, α−β, αβ, and α/β if

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Page 20 of 70 β≠0.

de fapt

In fact, this follows from the consideration that the sequence gn(x) is bounded for x>0. Since these are continuous processes, we actually have ...

de fiecare dată când

In stating rules, letters are used to represent not only the numbers to be found, but also the numbers that must be given whenever the rule is applied.

de k ori

Applying Lemma 20.3 k times, we infer that there exists a periodic function G0(x) with G0(k)(x)=g(x). Differentiating formula (8) k times, we obtain the equality ...

de la

The Arab notation has been in use since the XVI century. Prof. Romanov is giving a course of lectures on chaos and bifurcations in room 406, commencing October 1. In order to start from the very beginning, we introduce first the basic notions and concepts of the theory of measurements. From this time on, the density function decreases. We shall start with the basic notions and concepts of the theory of measurements.

de unde

Setting (*) we obtain (**), whence, in view of inequality (1), ... The author does not explain from whence the estimate (7) emerges.

deduce deducem

deducem ecuaţia

defini definim definită

Applying Lemma 20.3 k times, we infer (or deduce) that there exists a periodic function G0(x) with G0(k)=g (x). We derive (or deduce) the equation for the distribution of cell size at division and the distribution of generation time. Define unit and prime just as we did for the Gaussian integer. The function h(x) is defined at every point of the interval [c,d]. The variable, whose value is determined once the value of the independent variable is given, is called the dependent variable.

bine definită

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The integral

definit anterior pozitiv definit

definire în definirea definiţie din definiţie definiţie echivalentă definiţie exactă demonstra demonstrează pentru a demonstra este demonstrată

is well defined in

accordance with the above definition. Using the Haar family {gn} defined above, we introduce functions {hn}. A complex-valued function f(x) is called positive definite if for any n and complex numbers zj

In defining K(x) we have made use of our assumption that b(x)=0 for x
To prove this, interchange the order of integration in the double integral in (6), after which all the integrals except (8) can be evaluated exactly by appropriate change of variables. To prove the theorem, note first that neither a nor b can be zero. If p|a, there is nothing to prove. nu este ce demonstra se poate de demonstrat că It may be proved that if ... de la început vom demonstra We shall first prove the following two lemmas. înainte de a demonstra Before proving the theorem we illustrate its applications. teorema trebuie să demonstrăm că We must prove that the two factorizations differ at most in the order in which the primes appear.

pentru a demonstra

demonstrare în procesul demonstrării

In the course of the proof of this theorem we have made use of the following lemma.

demonstraţie

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Page 22 of 70 We shall assume without proof the following theorem.

denumire

The name 'distribution' is due to L. Schwartz, who developed the theory in 1945 and later.

deoarece

This cannot be, for 1/p is not a rational number. In the theory of distributions a distinction between the trigonometrical series and the Fourier series is unnecessary since each convergent trigonometrical series is an expansion of a periodic distribution. Because (3), we have ... All numbers amn can be arranged in a sequence {bk} convergent to b0, for otherwise it would suffice to neglect a finite number of initial terms a1n, a2n, ..., apn to get a contradiction to property (5). Since the functions w are solutions of this equation, the difference of any two of them may be regarded as the solution of a certain linear differential equation.

deoarece în caz contrar

deoarece ..., atunci

deosebi se deosebeşte prin

The function f(x) differs from F(x)(k-1)by a constant. We must prove that the two factorizations differ at se deosebesc prin most in the order in which the primes appear. se deosebesc una de alta prin The equation f'(x) = g'(x) holds if and only if the distributions f(x) and g(x) differ from each other by a constant function.

deosebire spre deosebire de ...

The absolute value of a number a, as distinguished from its algebraic value, is represented by |a|. As opposed to (or as distinct from / in contrast to) the real case, any polynomial always has a root in the field of complex numbers. In fundamental contrast to M. R. Stein, we have preferred the algebraic approach.

deosebit este deosebit de importantă The case of a linear substitution is especially important (or is of particular importance) for us. depinde care nu depinde de nu depinde de alegere depinde esenţial de depinde monoton de t

To this end, we establish first that there exists a constant L not depending on s and n such that ... The distribution (1) does not depend on the choice of the indefinite integral ψ(x). In this new definition the fact that the choice of the integer k depends on the interval [a,b] is essential. Let (2) be a family of finite stopping times which are

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Page 23 of 70 monotonically dependent on t and right-continuous in t.

deriva deriva ... după x deriva ... în raport cu x derivând derivând de k ori derivând succesiv derivata după x deschide deschide posibilităţi

To differentiate v(t,x) with respect to x. To take the derivative of v(t,x,u) with respect to x. In fact, by differentiating (3) we obtain ... Hence, by successive differentiation we obtain ... Differentiating formula (8) k times, we obtain the equality ... Hence by successive differentiation we obtain ... A derivative of u(t,x,u) with respect to x. The interpretative approach opens perspectives in that direction.

deschiderea parantezelor

the removal of parentheses

descompune în factori

to expand the number into factors to factor the number to present smth in a factor form to factor smth into components

descompune în serie descompunem în serie Expand function ϕ(x) in a Taylor series. descompunem în serie în Expand function u(t,x,u) into a Taylor series about x. raport cu x poate fi descompusă în serie Under certain conditions a continuous function can be expanded into a Fourier-Bessel series. Function f(x) can be expanded as an infinite power series. descompunere descompunere în descompunere în fracţii elementare descompunere în serie

Any supermartingale ζ(t) has exactly one decomposition of the form ζ(t)=µ(t)-α(t). 6 = 2*3 = 3*2 are the only factorizations of 6 into positive prime factors. The decomposition of a fraction into partial fractions. Hence, we obtain the expansion of ctgx: ctgx = 2(sin2x + sin4x + sin6x + ...).

descrescător

Let {an} be a decreasing sequence and {bn} an increasing sequence of numbers.

descriere formală

A formal description of the basic notions is usually based on the set-theoretical model of probability theory developed by A. N. Kolmogorov in 1929.

descris

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Fraze uzuale descris anterior descrisă de ecuaţie desemna desemnează

desen pe desen din desen se vede că în limitele desenului

deşi

detaliat mai detaliat detaliile

determina determină determinat cu exactitate de

deveni devine

dezacord

Page 24 of 70 We now show how the method described above can be extended to move the existence of µ. We will consider only those systems whose evolution is given by the equation of form (8). The symbol Fn(x)→ will denote that the sequence {Fn(x)} converges almost uniformly to a function. In Fig.11 the planes π1 and π2 are shown as coinciding. In Fig.10 it is seen that ... In this particular case only one side of the triangle, namely BC, has traces within the limits of the drawing. It is possible that the ordinary derivative does not exist, although the distributional derivative has a value at this point. It is necessary to consider this problem in greater detail. The details can be found in most texts on the theory of equations, for example in the book of Thomas listed in bibliography. The Introduction gives details concerning the application of stochastic differential equations in natural sciences. Two sequences {fn(x)} and {gn(x)} determine the same distribution if and only if they are equivalent. The indefinite integral is determined up to a constant. Integrating f(x) k times, we obtain a locally integrable function F(x), which is determined up to a polynomial of degree less than k. The relation of equivalence is then transformed into the usual equality. The last formula becomes p(la+mq)=b, so that p|b. With this terminology Theorem 1.5 becomes meaningful for the Gaussian integers. This explains the discrepancy in behaviour of the two functions.

diferenţă

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Fraze uzuale unică diferenţă între diferi diferă de diferite de

dificil este mai dificil

dificultate dificultate poate fi trecută

Page 25 of 70 The only difference among them is the order of appearance of the factors. The function f(x) differs from F(k-1)(x) by a constant. The proof is not like that of Theorem 5.6. Consider the numbers which are different from 0 and ±1. The prime numbers p possess no factors other than ±1 and ± p. It is slightly more difficult to see that

This difficulty can be remedied by a slight restatement of the fundamental theorem to include negative numbers.

din nou

Now we can, once again, use the arguments analogous to those presented in Section II.

direct

This inequality can be proved directly (or immediately) using (7) and Lemma 9 in Section 1.

direcţie în aceea direcţie discontinuitate de speţa întâia [a doua] discuta discutăm nu va fi discutată distinge pentru a distinge

The interpretative approach opens perspectives in that direction. With probability 1 ξ(t) has no discontinuities of the first [second] kind. In Chapter 8 we discuss the filtering problem. The validity of this assumption will not be debated here. The one-to-one correspondence established above makes it unnecessary to distinguish between functions f(x) and distributions [f(x)].

distingere

In the theory of distributions a distinction between the trigonometrical series and the Fourier series is unnecessary since each convergent trigonometrical series is an expansion of a periodic distribution.

distribuţie comună

the joint (or simultaneous) distribution

divide se divid Then the numbers 1, 2, ..., p-1 fall into pairs such that http://math.ournet.md/formule/fraze/fraze.html

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se divide se divide prin divizibil sunt divizibile prin domeniu de definiţie

Page 26 of 70 the product of any pair is congruent to 1 modulo p. The procedure is then broken down (or divided / subdivided) into the three steps: ... Nine is a multiple of three. All the terms on the right are divisible by p. Let A be a strongly positive operator whose domain of definition does not depend on G.

dori dorit este de dorit după ce

Passing to the limit in (6) we obtain the desired result. This is exactly the wanted decomposition. It is desirable in many applications that the function Xt be also measurable. To prove this, interchange the order of integration in the double integral in (6), after which all the integrals except (8) can be evaluated exactly by appropriate change of variables.

E ecuaţie din ecuaţia ecuaţie cu n necunoscute ecuaţie de gradul n ecuaţie iniţială ecuaţie în raport cu (necunoscuta) x ecuaţie rezolvabilă în raport cu x editarea

efectua pot fi efectuate de a efectua transformarea

From equation (2) we find that ... the equation in n unknowns the equation of degree n the input equation the equation in (the unknown) x the equation solvable for x The publication of this monograph was made possible due to ... He has over 70 published works. The calculations can be carried out (or performed) in the same way as in Section 15. On many occasions it is useful to take the Laplace transform of equation (15).

egal sunt egali

The vectors

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and

are equal in magnitude.

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elibera să ne eliberăm de

Page 27 of 70

The statistical theory has been developed quite recently. The solution of equation (5) is obtained in an elementary manner by expanding function x(t) in a Fourier series. But here we need only free ourselves of the term V(x) *µ.

enunţul problemei

The most general statement of the problem can be formulated as follows.

evident

A more exact answer to the question requires apparently (or evidently) a refinement of the definition of a system. It is obvious (or evident) that Xt is progressively measurable. Obviously (or evidently / apparently), the above theorem hot index set is taken to be a finite interval [0,T ] instead of R+.

este evident

eveniment eveniment sigur

We note that any subset of Ω is called an event. the sure event

evident

The proof of this property is obvious since b=0. Applying Theorem 12.3 to the obvious equality (*) we get ...

evita pentru a evita confuzii pentru a evita neînţelegeri

pentru a evita repetări în viitor exact exact unul mai exact exactitate cu exactitate de

To avoid possible confusion, the continuity condition will be restated in each case. To avoid misunderstanding, we may call the first kind of intervals (including end-points) the closed intervals and the second kind the open intervals. To avoid repetitions in future, we present here basic concepts and definitions. Any prime Π divides exactly one positive rational prime p. More precisely, we shall assume that the following condition is satisfied.

Then we can write for the division-size density function (up to a normalization factor) . The factors in product (4) are determined uniquely except for order.

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Page 28 of 70 Factorization of any positive integer is unique to within the order of factors. The fundamental theorem of arithmetic asserts that every integer greater than 1 can be factored in one and only one way, apart from as the product of positive prime numbers.

excepţie cu excepţia

exclude se exclude exclude posibilitate

exemplu

de exemplu

prin exemple

exemplu interesant exemplu tipic unele exemple în calitate de exemplu exista există limita

existenţă

The function ν possesses the property (1) for all ω and t except for some sets of measure 0. To prove this, interchange the order of integration in the double integral in (6), after which all the integrals except (8) can be evaluated exactly by appropriate change of variables. The second of these possibilities is excluded, for r(x) is in T. This approach excludes the possibility of considering the equations having solutions which belong to a more general class of numbers. Another example of the abstraction principle is the Cantor definition of a real number. Let us first look at some simple examples. For instance (or for example), the coefficient of variation, CV, of size at division is typically 10% and the CV of age at division is typically 20% or more. As an example, let α=5 and β=1+2i. The nature of this problem can best be made plain by examples. We shall explain it by means of examples. We shall construct an interesting example of such a process in this chapter. The polynomials Bn(t) are typical examples of generalized Brownian functionals. Some other examples of fundamental sequences will be given after the following lemma. As an example, consider the primitive polynomials x2+3 and 3x2+7x-11. With probability 1 there exists the limit . The existence of the Wiener process follows easily from the Kolmogorov's criterion.

explica

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Page 29 of 70 We shall explain the essence of the problem by giving several examples. The variations are greater than can be accounted for by chance. Chance is unlikely to account for so great variations.

exponentul puterii

A figure or a letter placed a little above and to the right of a number is called an index, or an exponent, of the power thus indicated.

expresie algebrică

An expression composed of numbers, letters and signs of algebraic operations and aggregation is called an algebraic expression.

exprima exprima ... prin ...

să exprim recunoştină expunere

extinde va fi extinsă extindem ... asupra de extins în cazul

extinde rezultatele de a extinde rezultatele

rezultatele pot fi extinse extinderile noţiunii de

To express the solution of equation (2) in terms of w (t), it is sufficient to observe that w(t) satisfies equation (3). I wish to express deep gratitude to O. A. Oleinik for her valuable advice and her interest in my work. The exposition of mathematical subject is set forth in a manner comprehensible not only to teachers and students specializing in mathematics, but to all those taking interest in mathematics and its applications. The definition of a stochastic integral will be further extended to cover the alternating functions. We now extend the definition of a stochastic integral to a wider class of random processes. In §21 we show how to extend the basic definitions and theorems to the case of distributions of infinite order. The main purpose of this section is to extend the results of the preceding section to the processes of a continuous argument. These results can be extended to all other cases. As it is currently being recognized, similar extensions of the notion of entropy are needed in the field of biology.

extrage rădăcină patrată [cubică] To extract the square [cube] root of a number. extragere a rădăcinei extremitate cu extremităţi

Evolution may be defined as the process of finding a root of a number, or as the inverse of involution. In every closed subinterval with ends a and b there is at most a finite number of poles.

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Page 30 of 70

F face de a face

This helps to make the proofs of all the theorems elementary and very simple.

făcând ...

Making (or letting) n→∞ along the subsequence nk, we have ... Before presenting the main results, we want to make a few comments on the notion of entropy. The one-to-one correspondence established above makes it unnecessary to distinguish between function f(x) and distribution [f(x)].

să facem câteva observaţii referitor face de prisos

face referinţă de a face referinţă la facem referinţă la poate să facă referinţă la

To complete the proof, it is sufficient to refer to the Cantor theorem on uniform continuity. For the proof of this theorem, we refer the reader to Theorem 9. For further information the reader is referred to Chapter VI.

fals

That such an impression is erroneous, we demonstrate by exhibiting another simple class of "integers" for which Theorem 8 is false.

familie (set)

M is a collection (or family / set) of processes mt , t∈T.

fi poate fi fi de prisos este de prisos

fi în relaţie sunt în aşa relaţie încât

fi în stare suntem în stare

In case of right-continuity the condition (2) may be replaced by the requirement (i). In the theory of distributions a distinction between the trigonometrical series and the Fourier series is unnecessary since each convergent trigonometrical series is an expansion of a periodic distribution. When two variables are so related that the value of the first variable is determined when the value of the second variable is given, then the first variable is said to be a function of the second. We are now in a position to prove the fundamental

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Page 31 of 70 theorem stated in the opening paragraph of the chapter.

fi recunoscător ... pentru ... sunt recunoscător ... pentru ... fie

acum fie fie că este dat fie dat

I am particularly indebted to Miss Leila R. Raines for her painstaking assistance in the revision and preparation of the manuscript for publication. Put w=5.7. Set v=t. Let d be the smallest positive number in the set A. Let G denote the set of all Gaussian integers, and J the set of all rational integers. . Now set Let there be given a continuous increasing process ξ (t). Given point A,construct point B symmetrical to the given point A in relation to plane π. Given any ε rel="nofollow">0, we fix an index n0 such that | fn(x) - f(x) |<ε for all n>n0.

fiecare fiecare dintre care

pentru fiecare din fix acelaşi

Let ξ1, ξ2, ..., ξn be a sequence of random variables each of which is the sum of certain number of mutually independent random variables. M is the set of processes mt for each of which there exists a random variable m such that ... Precisely the same proof is valid for Gaussian primes provided only that we can find one prime.

fixa vom fixa

We fix some t∈S. Fix k points -1 ≤ x1 < x2 < ... < xn ≤ 1.

fixat este fixată

Indeed, let c>0 be an arbitrary fixed number. A quantity whose value is fixed in any investigation is called a constant.

folosi

se foloseşte

In the order to construct the solution to (2), we can use (or utilize / make use of / apply / employ) relation (12). The reader can draw on the data from [4]. The Arab notation has been in use since the XVI century. This property is utilized (or exploited) in numerous technical applicarions. The term "information" is sometimes used (or applied) to mean "data".

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ne vom folosi acum de ... font aldin forma formează a forma ecuaţia formalizare în formalizarea

formă de formă

în formă în formă explicită într-o formă concisă într-o formă mai simplă formula să formulăm formulare formulă formula schimbului de variabilă formulă iniţială funcţie funcţii absolut continue

funcţie caracteristică funcţie inversă

Page 32 of 70 Let us now make use of the fact that the trajectory of process is right-continuous. The title should be printed in bold type. The numbers 0, 1, 2, ..., p-1 form a complete residue system modulo p. to set up (or to formulate / to form / to write) the equation When formalizing the notions of probability theory, the first fundamental assumption is that the results of the experiments can be described by means of a certain set Ω. The function η(t) also admits decomposition of the form (4.3). An integral of this form can be evaluated by substitution. Write equation (3.2) as (or in the form) ... To represent the solution of equation (12) explicitly, it is necessary to solve equation (8) for u(t). An equation that states a rule in a brief form is called a formula. More recently some of results were obtained in a simplified form. Let us formulate the conditions under which the statement is valid. The new statement of the Fundamental Theorem is this.

Apply now the change of variables formula to (1). the assumption formula The class of indefinite Lebesgue integrals of locally integrable functions is the class of absolutely continuous functions. EΩ(x) is the characteristic function of set Ω, equal to unity inside Ω and to zero outside Ω. the inverse function

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Page 33 of 70

G garanta garantează

This condition ensures (or assures) the uniqueness of the solution.

găsi găsim se poate găsi greutăţi mari genera generează generată general mai general

generaliza pentru a generaliza poate fi generalizat poate uşor fi generalizat generaliza ... pentru cazul

From (5), (6) and (7), we find ... The facts used below can be found in [5]-[7]. The estimation of ν runs into severe (or great) difficulties. Here w(t) denotes the process which generates the current {Ft}. Fs is a σ-algebra generated by the events {ξ(s)∈B}. More generally, if a sequence {fn(x)} converges almost everywhere, then it converges also in a distributional sense. If the function f(x) is sectionally continuous, or, more generally, if it is Riemann-integrable, equation (1) is also true except of the points of discontinuities of f(x). To generalize the theorem, it is necessary to modify the definition of convolution. We now show how the method described above can be extended to prove the existence of µ. Theorem 6 can be easily generalized. Let us generalize equation (6) to the case under discussion.

poate fi generalizată pentru This theorem may be generalized to the case of multiplicative error. cazul generalizare

Distributions are a generalization of functions.

grad de gradul n grupa grupăm termenii asemenea

pn(x) is a polynomial of degree n. To solve the equation 2x+x=12, unite (or group / collect) similar terms in the first member to obtain 3x

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Page 34 of 70 and then divide both members by 3 to find the value of x to be 4.

I imediat

imita vom imita

implica această implică

Using Lemma 2 we can prove at once that G(u) satisfies condition (ii). It follows at once from 12.4 that ... In proving that Theorem 1.5 holds for the Gaussian integers we shall imitate as far as possible the proof already given for rational integers. This implies the completeness of {hn} in L01[0,∞).

important foarte important

In applications the following theorem is of great importance. una din cele mai importante The concept of hierarchy is among the most important in the theory of general systems.

importanţă

The importance of the primes is due to the fact that, together with 0 and ±1, all the other integers can be constructed from them.

impresie falsă

That such an impression is erroneous, we demonstrate by exhibiting another simple class of "integers" for which Theorem 8 is false.

impune să impunem condiţii asupra In order that the solutions converge, we must impose the following conditions on the coefficients of equations (2). include include (ind. prez. 3 sg.)

sunt incluse inclusiv

It is obvious that the set F includes the set of rational numbers. Equation (8) incorporates normal losses. The dictionary contains (or includes) some information about the appropriate use of collocations. Some elements of artificial intelligence are now incorporated in system. The function f(x,y) possesses bounded partial derivatives up to the third order inclusive. Let Ft be a σ-algebra of the events observed up to and

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Page 35 of 70 including the moment t.

independent independent de

mutual independente

indica indică că indiferent este indiferent imposibil este imposibil

inducţie după inducţie inducţie după k infinit la infinit până la infinit infinit mic

By a slight change in the method of [3] the reader can easily prove the theorem by himself. Let θ be a random variable independent of the σalgebra. This is true irrespective of whether E is a Hausdorff space or not. We can always do this, no matter how many points from D are contained in the sphere S. After each moment of regeneration the process X(t) evolves independently of its past. The distance between two points is the same irrespective of the direction in which it is measured. The function ex remains unaltered regardless of the number of times it is differentiated. Let ξ1, ξ2, ..., ξn be a sequence of random variables each of which is the sum of certain number of mutually independent variables. These results indicate (or suggest) that ... It makes no difference whether we say that t is a stopping time relative to {Ft} or {F't}. But this is impossible for any pair of integers a and b is in J. It is impossible to represent 12 as m2 + n2. Many of these questions are difficult if not impossible to answer. We shall prove the theorem by induction. Define the random variables Mn inductively. We shall prove the relation by induction in k. The value of a function f(x) at infinity, written f(∞), is the limit of this function as x →∞. to infinity the infinitesimal (or the infinitesimal quantity / the infinitely small quantity) The limit distribution functions are determined by the fact that they are the limits for sums of an increasing number of independent random variables such that a single summand becomes vanishingly small (or

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Page 36 of 70 infinitesimal) with n→∞.

iniţial

The distribution of the initial vector converges weakly to a limit as u→0. The reader should be reminded that the initial value of ν was chosen to be ν0=7.18.

instructiv

This example is instructive because it displays the structure of the process µ(P).

integra prin părţi

to integrate by parts

integrală integrală definită integrală improprie

the definite integral If g(x) is a locally integrable function and the improper integral

integrală nedefinită integrant integrare după integrare integrare prin părţi interesant este interesant de observat că intra intră în care intră în ...

introduce introducem introducând a fost introdusă sunt introduse

exists, then g(∞)=0

provided g(∞) exists. By an indefinite integral of a function f(x) we understood each function ψ(x) such that ψ'(x) = f(x). the integrand (or the expression under the integral sign) Replacing x by x+t in (1), we have, after integration, ... the integration by parts It is interesting to observe that the above formula remains true if the summation is understood according to the method of arithmetic means. This term appears in (or enters into) equation (9) in parentheses. The coefficient a*(x,s) entering into (or involved in) the expression for I t can be assumed to be constant. Introduce the sequence {an}n ∈ N of positive numbers. Allow sinu within the differential sign. Introducing 2n new variables ξi and ζi we deduced 2n equations. The notion of distribution was introduced to mathematics by S. Sobolyev in 1936. Data are entered (or fed) into the computer from the keyboard.

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Page 37 of 70

intuitiv

The model emerged from the intuition that the notions of hierarchy, complexity, stability, and entropy were to be very important.

inutil pentru

If the coefficients of equation (2) do not satisfy the Lipschitz condition, then the method of successive approximations is unsuitable for the construction of solutions.

invers

Conversely, formula (6) defines a measurable linear functional for any sequence of numbers (z,ek).

ipoteza despre

Observe that if the hypothesis on the existence of values of F(x) at a and b is not fulfilled, the equality may be wrong. The proof might have been arranged in a different way if we had made no hypothesis about the value of G.

Î în în acest scop

în afară în aplicaţii în aşa mod încât

în concordanţă cu în conformitate cu

To this end, we establish first that there exists a constant L such that ... For this purpose (or with this object in view / with this in mind), an attempt will be made to generalize the notion of entropy. EΩ(x) is the characteristic function of set Ω, equal to

unity inside Ω and to zero outside Ω. In applications we often come across the so-called Heaviside function. For the new distributions one can define addition, subtraction, multiplication by a number or by a function in such a manner that the fundamental properties are preserved. A variable x is said to vary continuously through an interval [a,b] when x increases from a to b in such a manner as to assume all the values between a and b in the order of their magnitudes. It follows from (1), (2) and (3) that the whole set is decomposed into abstraction classes so that two elements are in the same abstraction class if and only if they are equivalent. In accordance with the law of large numbers, a ≈ mξ = x. In accordance with (or by) the law of large numbers, a ≈ mξ = x.

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Page 38 of 70

în continuare

Next, define M∞ as the limit of Mt , as t→∞.

în fond

Basically, the difficulty with establishing inequality (6) is that ... The cards may be processed singly or grouped (or in groups). EΩ(x) is the characteristic function of set Ω, equal to

în grupe în interior

în întregime

în loc de această în mare măsură în realitate în sfârşit în spiritul în totalitate

în totalitate

în vigoare în virtutea

în caz în caz contrar

unity inside Ω and to zero outside Ω. All points of Ω lie in the interior of y2
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în caz general în caz particular

în caz special în cazul acesta în cazul când

în cazul considerat în cazul dacă în cazul limită în cazul mai sumplu în acest caz particular în cel mai bun caz în multe cazuri în mod în mod analogic

Page 39 of 70

Moreover Np=0, for otherwise p≠0 so that Π=α, contrary to assumption. This derivative is, in general, not a function, but a distribution. In the special (or particular) case where α=1 equation (3) can be solved explicitly. In the specific case of Fig.12 the function B(u) has two extrema. In the special case where the L2-martingale is continuous, the variation <M> will also be continuous. In this case ... In case where (or in the case when) a = 4k+3 equation (2) assumes a more simple form. In the case when measure µ is transferred from F into a smaller space F0, the space F0 often turns out to be a linear space. In the case under consideration the process α(t) is continuous. In the case of a negative determinant a quadratic equation has no roots. In the limiting case the stochastic process ν(t) becomes a constant. In the simpler case where α=1 this relation takes the form p=he2/3. In this particular case only one side of the triangle, namely BC, has traces within the limits of the drawing. This problem has received at best only fragmentary treatment. For many purposes (or in many cases / on numerous occasions) we shall need the following assumption. Analogously (or similarly / likewise), it is sometimes useful consider only positive solutions to Eqn (17). This theorem can be proved in much the same way (or manner) as Theorem 1. A similar argument yields ... The plane passing through the vertex A and the vertex C can likewise be constructed.

în mod arbitrar The first term can be made arbitrarily small by an appropriate choice of ε. The variable, to which the values may be assigned at http://math.ournet.md/formule/fraze/fraze.html

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Page 40 of 70 pleasure, is called the independent variable. The initial values of ξ and ζ may be chosen arbitrarily (or at will).

în sens în sens intuitiv în sens larg

în sens uzual

în sensul

în sensul că

în sensul larg în sensul nou în sensul următor în sensul vechi

într-un anumit sens

înainta a fost înaintată

înainta o ipoteză încă

Distributions may be, in an intuitive sense, approximated by functions. The reader acquainted with the Lebesgue integrals can interpret formula (1) in a still wider sense, viz. as the equality almost everywhere. A sequence of constant functions converges in the distributional sense if and only if it converges in the usual sense. More generally, if a sequence {fn(x)} converges almost everywhere, then it converges also in the distributional sense. This process is unique in the sense that if {Bt} is any process with properties (1) through (4), then At=Bt for every t. This process is unique in the sense that if {Bt} is any process with properties (1) through (4), then At=Bt for every t. First of all, probabilistic methods are preferable in biological studies in that they allow to take into consideration the variability of the live matter. in a broad (or wide / loose) sense So α has n conjugate in the new sense, but m in the old. The converse holds in the following sense. Sequences fundamental in the earlier sense are equivalent in the new sense if and only if they are equivalent in the earlier sense. So α has n conjugates in the new sense, but m in the old. A minimal, in some sense, current is obtained if one observes nothing else, but the process ξ(t) itself. Here the solution is a non-decreasing function of the drift coefficients and, in a sense, of the jump functions. A new theory has been put forward this year. We were informed that a new idea had been advanced at the closing session. to advance a hypothesis Many roadblocks must still be overcome before all the subgroups of G are described.

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Fraze uzuale încă mai mult

încă unul începe vom începe cu se începe început de la început încheia demonstrarea pentru a încheia demonstrarea îndoială fără îndoială

Page 41 of 70 These estimates could be further improved by including additional data in the computation. In order to improve the estimates still further, additional data should be included in the computation. Let us formulate one further assumption. To conclude, consider one more example. To find the sum of the series, we start from the known formula (1). Each command begins with a forward slash (/). We first establish the necessity of the condition. First let us define ...

To complete the proof, it suffices to observe that ...

Undoubtedly (or without a doubt), the introduction of the theory ...

îndrepta atenţia vom îndrepta atenţia asupra Let us take note of the fact that the portions of the straight lines A1A2 and A3A2 are equal. înlocui este înlocuit poate fi înlocuit prin înlocuit ... prin înlocui a prin b

se poate de înlocuit putem înlocui prin însemna înseamnă va înseamna

F∞ is replaced by Ft if t∈[0,T ]. The integer k can be, if necessary, replaced by any greater integer. Replacing in (2) α by β and afterwards x by αx we obtain ... to replace a with (or by) b to substitute b for a to change a for b In case of right-continuity the condition (2) may be replaced by the requirement (3). In the definition of the mth derivative we can replace the polynomials pn(x) by functions fn(x). A(C) denotes the completion of B(C) with respect to µν. The letter M, with subscripts, will denote positive constants. 'ln' signifies the natural logarithm.

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această înseamnă că

întâmpina dificultăţi întâmpinăm dificultăţi

întoarce ne întoarcem între

întreaba se întreabă întrebare considerată întrece care nu întrece

Page 42 of 70 The expression (a+b)c signifies that the sum of a and b is to be multiplied by c. In this equation X stands for the velocity to be found. The subscript i identifies the number of the group. This means that a is different from any of a1, a2, ..., an. If we try to apply the principle of unique factorization to negative integers, we encounter an obvious difficulty in the possible presence of minus signs in the factors. We turn to the proof of the theorem. We can restrict the variable so that it takes on only values lying between a and b. Denoting by p(x) the polynomial within (or between) the signs | | on the left-hand side of (3), we can write ... It is asked what properties the limit distribution function must possess. These ideas can bring advantages in attacking the problem under consideration, over the usual theories. The distribution δ2π(x) is the derivative of the function E(x/2π), where the symbol E(a) denotes the greatest integer not exceeding a.

înţelege înţelegem prin ... vom înţelege ...

prin ... se înţelege înţelegere profundă

Practically it suffices to understand integrability in the sence of Riemann or of Lebesgue. By the integral of the P-measurable function f we mean the Lebesgue-Stiltjes integral. By the product of a distribution f(x) = [fn(x)] by a number λ we understand the distribution [λfn(x)]. By an indefinite integral of f(x) we mean a function of the form (2). A current is understood to be an increasing family of σ-algebras such that ...

This information is needed for a fundamental understanding of (or a deep insight into) the properties of ...

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Page 43 of 70

J jos mai jos de mai jos juca un rol va juca un rol decisiv joacă un rol important

The operations introduced below will be identical with the usual operations on functions. The definitions given below are for the processes defined on R+. The following fact will play a decisive role in the proof. The class of normal functionals plays an important role from the viewpoint of the causal calculus. Let us formulate the conditions under which the statement is valid (or true).

just

L la fel la fel ca în cazul la fel ca şi

This is proved in the same manner as in the case of discrete-argument submartingales. But then we could proceed from the pair (x,y) to a still smaller pair in the same way as from (x1, y1) to (x2, y2).

lăsa lăsăm va fi lăsată cititorului

The rest we leave to the reader. The slight change in the original proof which is needed here will be left to the reader.

este legat de

The estimation of ν runs into (or entails) great difficulties.

lega

limita ne vom limita la Not striving for the greatest generality, we shall confine ourselves to he study of martingales with continuous characteristics. In this section, we shall restrict our attention to integration over processes µ(t) with continuous characteristics. For the present we shall content ourselves with http://math.ournet.md/formule/fraze/fraze.html

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este limitată de ne vom limita la cazul

Page 44 of 70 demonstrating the assertions just made concerning the sets G and H. We will first confine our attention to discrete-parameter supermartingales. We will restrict ourselves to the simplest case of λ=0. Accuracy is limited only by that of the original data. We shall restrict ourselves, in this paper, to the case ...

limită în limită In the limit, as n→∞, the last term in (8) vanishes. limită inferioară de integrare the lower limit of an integral limită superioară de integrare the upper limit of an integral literal

In the term 4x2y the numerical multiplier of the literal part is the coefficient.

lua în consideraţie luăm în consideraţie

If we consider (or take into consideration) that the series is a series of positive terms, we have ... The analysis has been made without reference (or fără a lua în consideraţie ... without regard) to heat losses. Regardless of the last term which is sufficiently small, equation (9) may be solved in radicals for x. The statement becomes apparent when it is dacă se ia în consideraţie că considered (or if it is remembered) that ... The statement becomes apparent when it is considered (or taken into account) that ... trebuie de luat în consideraţie One should keep in mind (or it should be kept in că mind / it should be remembered) that the partial sums of the series (2.2) are the Fourier transforms of functions Bn(t).

lua în paranteze rotunde [patrate, figurate]

to put (or to enclose) in parentheses [brackets, braces]

M mai mai ales mai mic decât mai mult ca atât

This chapter deals mainly (or chiefly / predominantly) with semigroups. Now p is in T and has norm less than N(π). Moreover, every solution of (9) may be represented in the form (3).

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margine margine inferioară exactă margine superioară exactă matematică matematică aplicată matemitică finită matematică pură material material bogat despre

Page 45 of 70 Over (or more than) two thousand scientists took part in the 1st World Congress. The set B contains much more elements than the set A. The set B contains twice as many elements as the set A. the greatest lower bound the least upper bound applied mathematics Finite mathematics is considered as superior tool to use, over differential equations. pure mathematics In order to achieve this, we shall need some material from elementary number theory. This volume presents a wealth of material on stochastic integrals.

matrice inversă

the inverse matrix

maxim

We denote by xmax the point at which the function f(x) has a maximum (or is at its maximum).

maximum

Any polinomial of degree n ≥ 1 has at most n distinct roots. We shall make the most (or the best use) of the notation proposed by Dynkin in [8].

mărgini putem mărgini mărginit

We can restrict the variable so that it shall take on only values lying between a and b. The sequence gn(x)=1/(1+e-nx ) is bounded by number 1.

mărimea erorii

The investigation of the magnitude of this error is connected with profound properties of the geometry of numbers.

membru al ecuaţiei (parte a ecuaţiei)

Any term may be transposed from one member (or side) of equation to the other, provided its sign is changed.

menţiona în primul rând trebuie de menţionat menţionăm

The first thing to notice is a wide variety of concepts of entropy in the scientific literature. We note that any arbitrary subset of Ω is called an event.

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Page 46 of 70 Observe that (ξt ,Ft) is a potential.

vom menţiona că

It should be observed here that this definition of equality is usually adopted in the theory of the Lebesgue integral.

metoda aproximărilor succesive

the method of successive approximations

minuţios

The results reported by Dr. Evans need thoroughly be checked.

modifica puţin modificând modificând puţin metoda modificare a metodei moment din acest moment

până la momentul

până la acest moment în momentul t în punctul x

We shall modify slightly the definition of ... By a slight change in the method of (3) the reader can easily prove the theorem by himself. This is a modification (or an adaption) of the method of steepest descent. From now on, we assume B to be a complete separable metric space. From this time on, the density function decreases. Define the σ-field of events prior to a moment (or a time / an instant) τ as Let Ft be a σ-algebra of events observed up to and including the moment t. Up till now no assumptions have been imposed concerning the convergence of ψ(t). Let a(t,x) be a velocity of the particle at time t at point x.

multiplu multiplu întreg

Twelve is a multiple of three. 12.6 is a whole (or an integer) multiple of 4.2.

mulţime mulţime de măsură 0 mulţime dintr-un singur element

The σ-algebra A is generated by the P-null sets. We will assume in the future that the σ-algebra, N, contains all single-element sets.

N neajuns

To appreciate the demerits (or disadvantages) of the theory, to observe that ...

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Page 47 of 70 In general, φe and φ* need not be identical.

Let α(t) be an increasing (but not necessarily adapted) integrable process. necesar necesară este necesar este necesar ca

necesar pentru ca

Hence one of the factors of f(x) necessarily has the same degree as g(x). We are going to show that the function F(x) has the required property. In some problems of the filtering theory, it is necessary to work with Wiener processes on R. In order that two triangles be congruent, it is necessary that two angles of the first triangle be equal, correspondingly, to those of the second triangle. G(x) is the time needed for a cell to grow from size a to size x.

necesitate

By Theorem 2.1 it suffices to prove the necessity.

necunoscut

the unknown the unknown quantity the sought-for (or the desired) quantity A number whose value is to be found is called an unknown number. Thus we obtained an equation in two unknowns.

cu două necunoscute nedescrescător

The comparison theorem for diffusion equations asserts that the solution of these equations is a non-decreasing function of the drift coefficient.

nici nici o ... a cărei

Let

be p-dimensional cycles, no linear combination

nici un nici ...

of which is homologous to zero. No s∈D is an element of Γ. is homologous to zero. No linear combination of

nici ..., nici ...

The number 0 is neither positive nor negative.

vom nota

We denote by Λ the set of all points of discontinuity of F(x). Let us denote by C the completion of S in this norm and by C* the corresponding conjugate space. Denote the points by P1, P2, ..., Pn.

nota

să notăm prin

este notat prin sunt notate prin pentru a nota

The area of the triangle FGH is denoted by S. This volume is designated as V. Variables are denoted usually by the later letters of the alphabet. The symbol f(x) is used to denote a function of x.

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Page 48 of 70 Let us introduce the following notation where

notaţie standardă noţiune

noţiuni şi definiţii de bază nu

nu ..., ci ...

ψ'(x) = f(x). The Arab notation has been in use since the XVI century. The following designations are used: α=1/a, β=c/a, and γ=b/a. the standard notation When formalizing the notions of probability theory, the first fundamental assumption is that the results of experiments can be described by means of a certain set Ω. basic concepts and definitions The line L does not intersect A or B. Last time we were not appealing to Theorem 7. But nothing in this definition indicates from whence the class [x] is obtained. This shows that this method cannot give all solutions of the equation. It is not a mere academic exercise to pick out the cases of equality in such an inequality as Gronwall's. Never did mathematicians worry much about the foundations in algebra. If there is no such ξ, we write max f = ∞. Thus we need not 8, but 9 parameters to describe the system.

numeraţie roman

The Roman system uses seven capital letters to express numbers, viz.: I(1), V(5), X(10), L(50), C(100), D(500), and M(1000).

numai

We must prove that the two factorizations differ at most in the order in which the primes appear. Only then we can proceed to the limit in (14). It was not until the beginning of this century that the Euclidean geometry became completely axiomatized. This fact was not known until 1947. The process X(t) depends solely on u(t) and is independent of µ(t) and ω(t). Algebraic means alone is not sufficient to prove the fundamental theorem of algebra. In stating rules, letters are used to represent not only the numbers to be found, but also the numbers that must be given whenever the rule is applied. These factorizations differ not merely in the order of the factors, but in the factors themselves.

nu numai

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Page 49 of 70

A unit or an aggregate of units is called a whole number, or an integer.

număr la cub

A number a3 is read 'the cube of a', or 'the third power of a'.

număr la pătrat

A number a2 is read 'the square of a', or 'the second power of a'.

numi vom numi se numeşte

se va numi

Let 1 and -1 be called units. We shall call r the radius of the set D. Such an instant of time is called (or is termed) a random time. Occasionally it is referred to as a random variable independent of the future, as a Markov moment or as a stopping time. Occasionally a stopping time is referred to as a random time. The method will be termed the method of imbedding. This portion of the curve will be called (or be named) the stable section.

O oare

We don't know whether the limit law exists for such a process. If f(x) and g(x) are locally integrable functions, then the symbol f(x)+g(x) has the same meaning whether we interpret them as functions or as distributions.

obişnuit

In the case where the distribution is a function, this definition coincides with the usual one.

observa observaţi că se observă că

Notice that U is not unique. It may be noted that the problem lies actually in finding an α such that M(t) ≤ Cα for all t. To prove 2o, observe that ... Observe that s is not uniquely determined. . Note that for s
este util de observat că

de la început observăm că

It is useful to remark that the integer k which occurs in the definition of fundamental sequences can be, if necessary, replaced by any greater integer. Observe first that ...

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obţine obţinem

se obţine obţine forma obţinem în rezultat obţine valori ocupa ocupă poziţie în spaţiu

ocupă un loc special

Page 50 of 70 It is worth noticing that, in accordance with the law of large numbers, a ≈ mξ = x. It should be noted that R(t) is almost everywhere nonzero. Identifying equivalent oriented segments we obtain the notion of free vector. Hence, by the induction hypothesis, we infer that a1=...=an=0. Thus we arrive at a factorization m=p1p2...pn, where each pi is positive and prime. This function is obtained by application to expression (4) a nonlinear differential operator Π. In this case equation (7) becomes (or takes the form / assumes the form) ... After k similar steps we finally obtain ... Let f(x,t) take on (or assume) values in Rm. Two lines may occupy the following relative positions in space: 1) they may be parallel, 2) they may be intersect or, 3) not intersect. The prime numbers occupy a special position among integers.

omite putem omite

omitem

este omis operaţie operaţii aritmetice asupra

operaţie inversă

ordine în ordine aleatoare ordinea integrării

Since and n > 200, we may drop the last term on the right of (19). It would take us too far afield to give a proof of this theorem, so we omit it and content ourselves with demonstrating some applications of the theorem. The proof is omitted here. It follows immediately from the definition of convergence that the arithmetic operations on limits of sequences of distributions can be performed in the same way as on sequences of functions. Evolution may be defined as the process of finding a root of a number, or as the inverse of involution. Division is the inverse of multiplication. Numbers may be added in any order. To prove this, interchange the order of integration in the double integral in (6).

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Page 51 of 70

oricare n-ar fi

Whatever (or no matter what) t (may be), one can always find a sufficiently large N to satisfy the inequality ζN>t.

oricât ar dori

Let ε be an arbitrary however small value.

orice pentru orice

For each ω∈Ω, the function τ→Ξτ(ω) is called the sample path. For every number ε>0 there exists an index n0 such that | am - an |<ε for all m, n>n0.

P pagină la pagina

Powell's carrier distribution, C(τ) on page 245, is our distribution fm(T).

paranteză în paranteze

When numbers are collected by the sign of aggregation ( ), they are commonly said to be in parentheses. The expression between (or in) the brackets is precisely the Laplace transform of the function ϕ*(t). paranteze rotunde [patrate, The signs of aggregation are the parentheses (), the figurate] brackets [] and the braces {}. When an expression contains parentheses within paranteze pot fi deschise parentheses, the parentheses may be removed in succession, beginning with the innermost.

parcurge parcurge valorile

parte parte componentă

parte dreaptă parte întreagă parte stângă partea fracţionară

If x goes through the values 1, 2, ..., p-1, then, by Theorem 2, ax goes through a complete residue system modulo p, excepting 0. An integral part of the creative approach to the problem under consideration is the analysis of its similarity to and difference from the analogous problems. The right-hand side of equation (12). The integer (or whole) part of a real number is the greatest integer not exceeding the number. The left-hand side (or member) of equation (12). The fractional part of 5.6 is 0.6.

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Page 52 of 70 The term on the right-hand side (or on the right) of (12) vanishes as k→∞. The term on the left-hand side (or on the left) of Eqn (12) vanishes as k→∞. On the other hand, if [fn(x)] = f(x), then the first two conditions of Theorem 1 are satisfied.

partiţie

Let tk, k =1, ..., m, be a partition of the interval (s,t) into subintervals of equal length.

patrat al numărului

When a is used twice as a factor, the product is the second power of a, or the square of a.

păstra păstrează vom păstra

sunt păstrate

primi fi primit

pentru pentru aceasta

pentru ca

pentru care

pentru comoditate

pentru nici un

Absolute constants retain the same value in all problems. Since distributions are a generalization of the notion of functions, we shall retain for them the usual notation of functions, i.e., f(x), g(x), etc. For the new distributions one can define addition, subtraction, multiplication by a number or by a function in such a manner that the fundamental properties are preserved. It should be observed that this definition of equality is usually adopted in the theory of the Lebesgue integrals. The height of the triangle is taken to be 12m. For this purpose we select first the Gaussian integers. To this end it suffices for the first term in (1) to have the distribution F(x) and for the others to be zero with probability 1. In order that the last property be valid, it is sufficient that ... In order to achieve this, we shall need some material from elementary number theory. To formulate the corresponding result, we consider the following conditions which are the strengthened versions of conditions of Theorem 8. It is therefore reasonable to ask for which classes of "integers" the fundamental theorem holds, and for which it does not. For convenience we define a continuous locally increasing process to be an adapted process which has properties (1) through (4). The function cannot be constructed for any value of x.

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Page 53 of 70 For the sake of simplicity, consider the concentration of particles at height x above a horizontal barrier.

periodică cu perioada

We say that a function f(x) is periodic with a period α, if f(x+α) = f(x).

peste

By the Eisenstein's criterion the polynomial xn-2 is irreducible over the field of rational numbers.

peste tot peste tot densă

The function H(p(x)) is everywhere equal to 0. To determine IT (Ft), it is sufficient to describe the everywhere dense set IT2(Ft).

principial nou probabilitate cu probabilitatea

polinom de poseda posedă

posedă o proprietate

We will describe a radically (or fundamentally) new approach to the problem. With probability 1 there exists the limit . By a polinomial in x1, x2, ..., xn we mean a sum of the form (6). The derivative F'(x) has the value 0 at the point 0. The stochastic integral (4) possesses finite moments up to the (2m)th order inclusive. The function ν possesses the property (1) for all ω and t. We are going to prove that the limits exist and the function F(x) has the required property.

posibil posibilă

este posibil ca

este întotdeauna posibil de făcut practic precedent precedentă

To avoid possible confusion, we shall use this notation ... The error, if any, will be allowed for by visual estimation. It is possible that the ordinary derivative does not exist, although the distributional derivative has a value at this point. This we can always do by choosing s and t as rational integers nearest to A and B respectively. Practically it suffices to understand integrability in the sense of Riemann or of Lebesgue.

On account of the foregoing remark, the distributions in the earlier sense can be identified with distributions of finite order.

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nivele precedente

Page 54 of 70 This follows from the previous theorem. Here X(t) is the number of paths, by which we go from the preceding levels of a hierarchy to the new level t.

precum şi

Using this technique we can calculate the phase as well as the amplitude of the process at time t.

preliminar

I think that my project is only tentative and subject to continuous revision as the work makes a progress.

presupune se presupune că

vom presupune că

presupunem contrariul

It is assumed that σ-algebra F0 is completed by all Pnull sets. All stochastic processes considered in this book will be assumed to be measurable. {Ft} is assumed to be right-continuous. We assume that F*G=H, the operation * being defined by F*G=RS, where R=TV. Suppose Xt is a continuous process. Suppose that the theorem is true for k-1. Suppose the conclusion of the theorem to be true for all f(x) of degree between 0 and n-1 inclusive. Suppose the contrary.

presupunere presupunerea că

In defining K(x) we have made use of our assumption that b(x)=0 for x
prezenta în forma poate fi prezentat în forma prezenta un interes prezintă un interes

va prezenta un interes prezintă un interes aparte care prezintă un mare

Any potential ζt can be represented in the form ... Of interest also are the induced statistics. The results have encouraged (or aroused) a more serious interest in precision methods of control. I hope this paper will be of interest to you. To prove the existence of t, the following lemma, which is of interest in its own right, will be required. Another theoretical approach of great interest consists

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Page 55 of 70 in looking at the evolution toward equal phenotypic fitness.

prin prin alegere probabil puţin probabil

This we can always do by choosing s and t as rational integers nearest to A and B respectively. These conditions are unlikely to occur in practice.

probabilist

There is presented below a glossary translating settheoretic notions into probabilistic notions.

problemă de bază

We are now in a position to state the basic problem of algebraic number theory.

proces coordonat

A random process ξ(t), t ≥ 0, is said to be adapted to a current of σ-algebras {Ft}t 0 if for every t the random ≥

variable ξ(t) is Ft-measurable. provine provine de la provin de la punct în punct

în puncte

The word 'school' has its origin in the Greek word 'schole' meaning 'leisure'. The arithmetic symbols were derived from the Arabs and the Hindus. The function

has no a

derivative at the point 0. Such an identification can be extended to a wider class of functions which have the poles at some points and are locally integrable elsewhere.

punct ce se mişcă dea-lungul In the equation of straight line

x and y are

the variable coordinates of a point moving along the line. punct de vedere din punct de vedere logic

din punct de vedere practic punct de referinţă

pune în corespondenţă operatorul pune în

From the logical point of view, this generalization is exactly of the same type as the generalization of rational numbers in the Cantor theory. Several other functions, very useful from the practical point of view, are considered in the following section. Here the starting point is the notion of fundamental sequences of rational numbers. Approximation is our starting point for the definition of distributions. The operator θ, assigns to (or associates with) any

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Page 56 of 70

corespondenţă

function from D a function from D(−∞,0].

este pusă în corespondenţă

A probability is assigned to each outcome.

purta numele poartă numele putea nu putem nu se poate

It is instructive to study the Ito's method of defining the stochastic integral that now bears his name. If n=1, we cannot impose this condition. If n=1, this condition cannot be discarded. As regards the solution of equation (3), it may not be argued that it satisfies Condition 2.

puterea a n-a a numărului

If a is used n times as a factor, the product is called the nth power of the number.

puţin

This definition will be somewhat generalized below. The assumption under which formula (10) is valid can be somewhat weakened.

R raţionament

raţiona raţionând analogic raţionând ca

rădăcină rădăcină cubică din rădăcină de ordinul n din rădăcină din număr rădăcină patrată din rămânea rămâne

There are doubtful points in the above reasoning. Following the argument of N. N. Bautin, we can now easily show that ... Reasoning similarly we convince ourselves that the solution to Eqn (7) is also unique. Reasoning as in the proof of Theorem 7, we can prove, under suitable assumptions, that relation (1) holds for s>Nα. 2 is the third, or cube, root of 8. indicates the nth root of a. When the factors of a number are all equal, any of the factors is called the root of the number. 5 is the square root of 25. It therefore remains only to verify that At is uniformly integrable. Much remains to be done in this area. This relation remains valid when one passes from functions to their differentials.

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rămâne în vigoare

răspuns reaminti reamintim reaminteşte că

trebuie de reamintit că reciproc şi reciproc reduce reduce la

Page 57 of 70 Theoretically, it still remains to prove for each particular system considered that an equilibrium actually exists and is stable. Theorem 4 and Lemma 9.5 remain true. This result holds true if we proceed from the group G to any of its subgroups. The answer to the question must be postponed until later. Because of its great importance in this book, we recall the definition of a normal distribution. The reader is reminded that the conjugates over F of a number α algebraic over F are the roots of the minimal polynomial of α over F. It should also be recalled that R(t) is almost everywhere nonzero. Equal chords cut off equal arcs, and the converse. This fact reduces the construction of the solution to (2) to that of a process such that is a martingale.

se reduce la

reduce ecuaţia la formă reduce termenii asemenea reducem termenii asemenea

referi se referă la nu se referă la care se referă ceea ce se referă

Under these conditions equation (3) reduces to (4). This investigation amounts (or boils down) to finding the operator Uγ such that Uγµ=UγUµ. to reduce the equation to the form ... To solve the equation 2x+x = 12, unite (or group / collect) similar terms in the first member to obtain 3x and then divide both members by 3 to find the value of x to be 4. The Henry's Law refers only to the effect of pressure. Evidently the Einstein's restriction does not apply to the case under consideration. We now state a number of results concerning ordinary differential equations. As regards (or in regard to / in respect to) the solution of equation (3), it may not be argued that it satisfies Condition 2o.

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Page 58 of 70 As for property (ii), suppose first that a is a unit.

regulă

The rule for differentiation of a product. The rule for the change of variables in integral calculus.

relativ

As regards (or in regard to / in respect to) the solution of Eqn (3), it may not be argued that it satisfies Condition 2. The process ξ(t-) is a supermartingale with respect to the family of σ-algebras (Ft-) t≥0. Given point A, construct point B symmetrical to the given point A in relation to plane Π. For the present we content ourselves with demonstrating the assertions just made concerning the sets G and H. In relation to Ω, an experiment is completely characterized by a certain class of events.

relativ la remarca este cazul de a remarca repeta poate fi repetată repetată reprezenta reprezentat în formă

It is relevant to remark that the diffusion corresponds to the increase of entropy. This procedure can be repeated with q2, q3, ... until all the prime factors on one side are exhausted. The repeated application of this procedure proves the assertion. Any potential ζt can be represented in the form (3).

poate fi reprezentată în formă By the well-known Weierstrass approximation theorem, every distribution can be represented in the form [pn(x)], where pn(x) are polynomials. de a reprezenta ... ca ...

To prove this statement, it suffices to present the integral

as a limit of .

respectiv

The largest Π-factors are ν1 and ν2 respectively.

respinge o ipoteză

to reject the hypothesis

restrânge generalitate fără a restrânge generalitate

We shall also assume, without restricting the generality, that z0=s0=0. We can assume, without any restriction of

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Page 59 of 70 generality, that all numbers amn can be arranged in a sequence {bk} convergent to b0. In our discussion we may assume, without loss of generality, that M is a subset of the set of natural numbers.

restricţie

With this additional restriction the problem just posed received an exhaustive solution in the papers of Bawty and Khinchin.

a rezolva ecuaţia în raport cu x explicit [implicit]

to solve the equation for x explicitly [implicitly]

rezolva în raport cu x

to solve for x

rezolvare la rezolvare problemei rezulta rezultă din

rezultă imediat rezultă nemijlocit această rezultă din din cele spuse rezultă că

din ... rezultă că

de aici rezultă că

nemijlocit rezultă din

ridica ridica la puterea n ridica un număr la cub ridica un număr la patrat

It is convenient, in solving problems, to use letters for the numbers which are sought. The result follows from the induction hypothesis. The term R/n results from the Bohr theory. Much of the utility of the theorem stems from the properties of the binomial coefficients. The proof follows immediately if we observe that ... Since u(t) is continuous, it follows at once that x(u (t),t) satisfies condition (ii). This results from the following lemma. This follows from the fact that e πi/2 = i. From what has been said it follows that ... It follows from what has been said that T>0 almost surely on the set B. It follows from (1), (2), and (3) that the whole set is decomposed into abstraction classes without common elements so that two elements are in the same abstraction class if and only if they are equivalent. From this it follows that A is consistently defined except on a P-null set. This implies that ... It follows from this that ... It follows that R(s) is a commutative ring. It follows directly from the definition that ... This assertion follows immediately from relation (2). to raise to the nth power to cube a number to square a number

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ridicarea la putere

Page 60 of 70

The process of finding any required power of a number is called involution.

S satisface se satisfac condiţiile sau ..., sau schimb schimb de variabilă

schimb de variabilă la integrare schimba schimbăm ordinea de integrare fără a schimba este schimbat semnul schimbare schimbare mică schimbarea ordinii de integrare fără schimbări

Condition (4) is equivalent to the following conditions being satisfied. Since a satisfies (1), we have that either a=1 or a=p1. Relation (8) follows from the general rule of the change (or the substitution) of variables in integral calculus. the change of a variable in integration

To prove this, interchange the order of integration in the double integral in (6). The introduction of locally integrable allows us to strengthen Lemma 6.8 without changing its proof. Any term may be transposed from one member of equation to the other, provided its sign is changed. A slight change in the original proof will be left to the reader. The main point of the proof is to justify the interchange of the order of integration in (8.4). With this transform the first term in (4) vanishes, whereas the last one remains unaltered (or unchanged).

scoate în afară parantezelor scoatem în afară parantezelor Now factor out of the expression in parentheses the largest positive factor common to all the bi. Now take the largest positive factor out of the brackets. scop (obiectiv)

The purpose (or objective) of this monograph is to make available elementary parts of classical algebraic number theory. The aim of this section is to define the stochastic integral for a wider class of predictable integrands

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scopul acestei lucrări scopul constă în ...

Page 61 of 70 and, at the same time, to relax the restrictions on M. The purpose of this paper is to present the elementary exposition of the theory of distributions. The purpose of introducing real numbers is to make it always possible to carry out certain operations, e.g., the evaluation of roots or logatithms.

scrie se poate de scris în formă putem scrie

scriind vom scrie scriem ... în formă scriere în scrierea semn cu semn opus semnul opus sub semnul sub semnul integralei

The function F(x) can be written in the form ... Then we can write for the division-size density function (up to a normalization factor) . Writing down these inequalities for n, n+1, ..., n+m and adding them we obtain the estimation ... Write down the coordinates of the points A, B and C. If there is no such G, we write max f = 0. Write the equation as (or in the form) (2). In writing numbers, the figures are separated into groups of three figures each, called periods. Subtracting any number is equivelent to adding it with its sign changed (or with reversal of its sign). The numbers +2 and -2 have unlike signs. The number -2 is opposite in sign to +2. The products under the sign of Σ are the products of continuous functions. under the integral sign In the last formula the symbol 0 has two different meanings: on the left-hand side it denotes the number zero and on the right-hand side the zero distribution.

sens

serie absolut [condiţional] convergentă

the absolutely [conditionally] convergent series

sfat

I wish to express gratitude to O. A. Oleinik for her valuable advice.

similar

Progressive measurability of an S-valued process is defined similarly. Reasoning similarly, we convince ourselves that the solution to Egn (17) is also unique. The definition is the same as the one given above, but with R instead of [0,T ].

similar ca simplifica simplificăm simplifică

Simplify 4a-(2a-2b). Thus the introduction of distributions makes the

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pentru a simplifica poate fi simplificată simplifica fracţie prin a simplifica prin putem simplifica prin sistem sistem de notaţii sistem zecimal

situa situate aproape

Page 62 of 70 calculation easier. To simplify the equation, make the substitution u=2x2-1. This equation can be simplified by the substitution u=2x2-1. To reduce the fraction

by the factor a.

We can divide out (p-1)! from each side to obtain the conclusion. the system of notation The Arabic system, which is a decimal system, employs ten figures to express numbers, viz.: naught, one, two, three, four, five, six, seven, eight, and nine. This quantity may take on any values, however closely spaced.

slăbi putem slăbi pentru a slăbi restricţiile referitor ... soluţie soluţie unică spaţiu spaţiu de stări spaţiu metric separabil complet spaţiu-fază

specifica cum a fost specificat mai sus dacă nu este specificat contrariul

Hence, we can weaken condition ϕ in the following way. The aim of this section is to define the stochastic integral for a wider class of predictable integrands and, at the same time, to relax the restrictions on M. Equation (7) has a unique solution. A function ξ:Ω→S is called an S-valued stochastic process and S is called the state space of ξ. C (R+) can be regarded as a complete separable metric space under the topology of uniform convergence. The values of ξ(t) are sometimes called the states of a stochastic system Σ and the space S, the phase space of the system Σ. As has been stated (or specified) above, the functions pm(x) defined by (1.6) are bounded. We agree to assume, unless otherwise specified, that all processes are adapted. Unless otherwise stated, condition (2) will always be

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Page 63 of 70 assumed to be in force.

spune spune (ind. prez. 3 sg.)

se spune vom spune vom spune că ... spunând spunând mai exact din cele spuse mai sus stabili stabileşte stabilim a fost stabilit a fost stabilită de a stabili condiţii

strict studia studiat

substitui substituim substituind ... în

The principle of mathematical induction says: "If P is true ..." The second law states (or reads): "If the same current flows ..." The theory holds (or states) that ... Stated in a simple form, the hypothesis runs as follows. If conditions (i) and (ii) are also fulfilled, then the current is said to satisfy the 'ordinary' conditions. If the members of a pair are equal, say to a, then a2=1 (mod p). We shall say that two integers a and b are relatively prime if they share no factors except ± 1. The function F is, so to say, the optimal cost of the continuation of control. To put it more exactly, ... From what has been outlined (or said) above the following conclusions may be drawn. This requirement places (or sets) the upper limit on the admissible error. We first establish the necessity of the condition. This fact was ascertained (or established) from experience. The average value was found to be 16 ± 3. Theorem 1.6 allows to lay down the conditions under which the solutions to equations (6) are continuous in parameter z. Approximations, strictly defined, is our starting point for the definition of distributions. The experiments studied in probability theory are carried out when a certain set of conditions Y is satisfied. For this we make the substitution t-x=s. Substituting these numbers into (or in) (1), we find that ...

substitui a cu b

to make the substitution of b for a to carry out the change of a for b

substituţia b cu a

the substitution of b for a

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Fraze uzuale suficient suficient de mare

suficient de mic

este suficient este suficient de arătat că ... este suficient de spus că ...

este suficient pentru ca

pentru această este suficient ca suma parţială

Page 64 of 70 This equation determines a sufficient condition. Whatever t may be, one can always find a sufficiently large N such that ... Now ECt < ε/2, if n is taken large enough. The term

can be made sufficiently small

by makingthe partition of the interval [0,T ] fine enough. By theorem 2.1, it suffices to prove the necessity. It suffices to show that if α is a Gaussian integer, so is β and hence α+β. Suffice it to say that ... It suffices to say that ... It is enough to say that ... In order that the series (1) of independent random variables converge, it is sufficient that the series (2) and (3) converge. To this end it suffices for the first term in (1) to have the distribution F(x) and for the others to be zero with probability 1. A series

is said to be convergent if the

sequence of its partial sums fn(x) = g1(x) + ... + gn(x) converges. sumar supune se supune legii

se supune ecuaţiei

A survey of the general theory of stochastic processes can be found in [4]. The distribution of errors adheres to the law of probability. The kinetics of the system obey (or follow) the ideal gas law. It the coefficients on the right of (2) satisfy the Lipschitz condition, then the function w(t) obeys the equation (3.1). The equilibrium distribution of particles in a gravitational field adheres to (or is governed by) the Boltzmann equation.

sus mai sus

It is easy to see that the relation of equivalence defined above has the following properties.

de mai sus The integral

is well-defined in

accordance with the above definition. The definition is the same as the one given above, but http://math.ournet.md/formule/fraze/fraze.html

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Page 65 of 70 with R instead of [0,T].

T tăia taie târziu mai târziu teoremă teorema despre valoare medie teorema probabilităţii totale teoremă preliminară teoremă reciprocă teoretic

teorie teoria jocurilor teoria mulţimilor teoria numerilor teoria probabilităţilor teoria proceselor stocastice teorie unitară teoria ce va fi dezvoltată termen termen cu termen termina termină am terminat terminologie în terminologia

Equal chords cut off equal arcs, and the converse. We shall show later on that this generalization contains also class of non-continuous functions.

the theorem of the mean the theorem of total probability Some preliminary theorems will be established first. The converse theorem in not true. Theoretically, it still remains to prove for each particular system considered that an equilibrium actually exists and is stable. the theory of games the theory of numbers (or the set theory) the theory of numbers (or the number theory) the theory of probability the theory of stochastic processes the unified theory Continuous functions on a fixed interval are the starting point for the theory to be developed here. Any term may be transposed from one member of equation to the other, provided its sign is changed. Roughly speaking, each convergent series can be differentiated term by term (or termwise). This completes (or concludes) the proof of Lemma 4 and also that of Theorem 2. Then r=s and we are done. {Ut} is the 'dual predictable projection' in the terminology of Dellacherie and Meyer [42].

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Page 66 of 70

timp în timp ce

tinde la când ... tinde la când ... tinde la infinit

Thus 5 and 9 are relatively prime, whereas 6 and 9 are not. As the point x tends to a, the function g(x) approaches the value g(a). Each summand on the right of (12) becomes vanishingty small with (or as) n→∞. to let smth approach (or to let smth go to) infinity

tinde la infinit

to tend to (or to approach) infinity

tinde la zero (ind. prez. 3 sg.)

The limit distribution functions are determined by the fact that they are the limits for sums of an increasing number of independent random variables such that a single summand becomes vanishingly small with n→∞.

toate după toate

Summing up these conditional probabilities over all the newborn particles in the sample, we find that ...

tot (în întregime)

Thus the integral on the right side of (5) can be analytically continued to the whole domain. The class of functionals of degree n covers the entire space Hn(-n).

toţi

When the factors of a number are all equal, any of the factors is called a root of the number. Every succeeding (or subsequent) member an must be less than aN.

toţi următorii nu toţi

pentru toţi

However, not all distributions can be represented in the form [f(x)], i.e., not all distributions can be identified with continuous functions. The function ν possesses the property (1) for all ω and t.

transforma a transforma ecuaţia să transformăm la formă a se transforma în infinit

to rearrange the equation Let us transform (1) to the form (2). to become infinite

transformare inversă

the inverse transform

trebui trebuie

The problem must be solved as soon as possible. To ensure the existence of a solution to (14), additional assumptions need be made. Much needs to be done to complete the theory. It should be particularly emphasized that condition (*)

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care trebuie

ceea ce trebuia ceea ce trebuia de demonstrat

trece trecând la limită

de a trece la limita poate fi trecut dintr-o parte a ecuaţiei în altă trecere trecerea de la ...

trecerea la limită la trecerea la limită

Page 67 of 70 is not essential and was introduced solely for the sake of convenience. To comprehend the main idea of my talk, it is essential (or necessary) that the students be familiar with the generalized Galois theory. In stating rules, letters are used to represent not only the numbers to be found, but also the numbers that must be given whenever the rule is applied. But p is of the form 1+4n, so that a p=l+im, as required. It follows that the points C1 and C2 lie on the line of

intersection of the planes α and β, which is what had to be proved. However, by condition m>n/2 one has q>2, q.e.d.

Approaching the limit as k→∞ and utilizing Fatou's lemma we obtain (8). Proceeding (or passing) to the limit in equality (5) we convince ourselves that this limit is a solution of equation (2). To obtain equation (3), it is necessary to pass to the limit in relation (1). Any term may be transposed from one member of equation to the other, provided its sign is changed.

The identification of equivalent elements consists in the passage from the elements of the set in question to the abstraction classes. This equation is obtained from relation (3) by passing to the limit. Inequality (7) becomes the exact equality if one passes to the limit. When passing to the limit, inequality (7) becomes the exact equality.

Ţ ţine minte ţinând seamă

Considering (or taking into account) that the series is a series of positive terms, we have ...

trebuie să ţinem minte că It is to be remembered (or it should be borne in http://math.ournet.md/formule/fraze/fraze.html

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Page 68 of 70 mind) that the specific properties of the limit distribution functions are determined by the fact that ...

U unul şi numai unul

Hence there is one and only one x which satisfies the congruence.

unde

We assume that F*G=RS, where R=TV.

uneori

Occasionally a stopping time is referred to as a random time. Sometimes a current is understood to be an increasing family of σ-algebras.

unic

The element q is unique. If S consists of a single element x, then R(x) is commutative.

univoc

This set of conditions uniquely determines the results of the experiment. Thus, the function f(x) is uniquely determined provided it exists everywhere.

univoc determinată urma această urmează din

This follows from the Fundamental Theorem of arithmetic. cum urmează din cele spuse As appears from the above, the cycle C determines a curve L.

următorul următoare

Such an element is unique. Making the change of variables x=acosϕ and ϕ=ωt+θ the following relation may be obtained .

este următoare

The new statement of the Fundamental Theorem is this.

este uşor de ...

It is not difficult to establish the existence of an infinite number of rational primes.

uşor

util

utiliza putem utiliza metoda

This property of martingales may be useful when proving the existence theorems for stochastic differential equations. The following result will be very helpful in proving the theorems of this chapter.

In order to construct such a process, we can take such

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va fi utilizat permanent utilizate de obicei

Page 69 of 70 an approach. We shall prove below a very useful result which will be constantly used in our work. Using condition 2) of the theorem and commonly employed estimates we assure ourselves that ...

V valoare

The value of a function f(x) at x=x0 will be denoted by f(x0).

cu valori

The problem of interpolation consists in approximate evaluation of a function f(x) at a given point x1<x*<xn from its values on a finite set of points x1, x2, ..., xn.

cu valori în

Let β(ϕ,t) be a random function with values in Rm.

valoare în punct

The Dirac function δ(x) has the value 0 at each point x≠0 and has no value at the point x=0.

valoare absolută după valoarea absolută variabile aleatoare cu acelaşi repartiţie vedea vedeţi este uşor de văzut că

cum este uşor de văzut verifica se verifică

sunt verificate

The absolute value of b is greater that of a. These numbers are equal in absolute magnitude. Let θj , j=1, 2, ..., n, be mutually independent identically distributed random variables. See, for example, the deduction of inequality (11) in the proof of Theorem 1. It is easy to see that {gn} is complete in L1[0,1].

It will readily be seen that that for any δ>0 the sequence Z(nδ,ω) forms a Galton-Watson process. As is easily seen, there exists a Borel function θs such that ...

In the case of distributions the following identities are valid. Since inequality (4.8) holds true, the function ϕ(x) is non-negative. Since the conditions (1), (2) and (3) are fulfilled,the set of all fundamental sequences gets decomposed into disjoint abstraction classes.

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Page 70 of 70

verifică cetinţe pentru a verifica

The functionals L and M satisfy (or meet) the aforementioned requirements. To verify that h(t) is the desired solution, observe that ...

It is easily verified that the polynomial xn-2 is irreducible over the field of rational numbers. The experiments studied in probability theory are dacă o serie de condiţii este carried out when a certain set of conditions F is verificată satisfied. verifică condiţia Assume that the function B(ϕ,t) satisfies the local Lipschitz condition. We shall prove that the function F(x) satisfies (or verifică ecuaţie fulfils) equation (1). vom verifica ipoteza despre ... We shall test the hypothesis that ... este uşor de verificat că

viitor în viitor

In what follows the notation 'm|n' means 'm divides n'. Here and in the sequel the ai denote positive constants. Henceforth we shall adhere to the following model of the probability theory. In the future σn will always be assumed to be bounded. In the future, when continuous local martingales are discussed, the σ n will always be assumed to take values from T.

voluminos

This method is too cumbersome (or awkward / unwieldy).

vorbind în general

The results of the experiments will, generally, be different. Generally, Eqn (10) may have no solution.

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