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Dictionar Roman-Englez de Termeni Matematici
A Ă Â B C D E F G H I Î J K L M N O P Q R S Ş T Ţ U V W X Y Z
A
abscisă - abscissaabsolut - absoluteaceleaşi semne - like signsacurateţe - accuracyadmisibil - admissibleaduna - to addadunare - additionafirmaţie - statementalgebră - algebraalgoritm - algorithmalgoritm de divizare - algorithm for division, division algorithmalgoritmul lui Euclid - Euclidean algorithmalternanţă - alternancealternare - alterationamplasare - dispositionamplitudine - amplitudeanaliza - to analyzeanaliză - analysisanaliză matematică - calculusanalogic - analogous, analogicalanalogie - analogyanula - to vanishaplicată - applicateaplicaţie - map, mappingarbitrar - arbitraryarc - arcarccosinus - anticosine, arccosine
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arcsinus - antisine, arcsinearctangentă - arctangentargument - argumentargument al funcţiei - argument of a functionarie - areaarie a suprafeţei - surface areaasemănare - similarity, similitudeasimetric - asymmetric(al), unsymmetricasimetrie - asymmetryasimptotă - asymptoteasimptote ale hiperbolei - asymptotes of a hyperbolaaxa x - x-axisaxă - axisaxă de coordonate - coordinate axisaxă de rotaţie - pivotaxă orizontală - horizontal axisaxă principală - major axis, principal axisaxă verticală - vertical axisaxiomă - axiom
B
bază - base, basisbază ortogonală - orthogonal basebază ortonormată - orthonormal basisbilion - billionbinar - binarybine definit - well-definedbinom - binomialbisectoare - bisector, bisectrixbiunivoc - one-to-one
C
calcul - calculus, computationcalcula - to countcalculabil - computablecalculul determinantului - evaluation of determinant
Page 2 of 20Dictionar Roman-Englez
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canonic - canonicaldeterminarea extremului - hill climbingcâmp - fieldcât - quotient, ratio
cel mai mare divizor comun - greatest common divisor, greatest common factor
cel mai mic multiplu comun - lowest common multiplecel mai mic numitor comun - lowest common denominatorcentru - centre, centercentru de rotaţie - pivotcentru al unei circumferinţe exînscrise - excentre, centre of the escribed circlecentrul circumferinţei circumscrise - circumcentercentrul circumferinţei înscrise - incentercerc - circlecercuri concentrice - concentric circlescifră - digitcilindru - cylindercircumferinţă înscrisă - incircle, inscribed circlecircumferinţă unitară - unit circlecircumscris - escribedcoardă - chordcoeficient - coefficientcoeficient binomial - binomial coefficientcoeficient unghiular - slopecoincident - coincidentcoliniaritate - collinearitycombinaţie - combinationcomparare - comparationcomparaţie - comparationcon - coneconcavă - concavecondiţie - conditioncondiţie iniţială - initial conditioncondiţie necesară şi suficientă - necessary and sufficient conditioncongruenţă - congruenceconjugat - adjoint, conjugateconsecinţă - corollaryconstantă - constantconstantă de integrare - integration constantcontinuitate - continuityconvex - convexconvexitate - convexity
Page 3 of 20Dictionar Roman-Englez
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coordonată - coordinatecoordonate afine - affine coordinatescoordonate carteziene - Cartesian coordinatescoplanar - coplanarcorolar - corollarycosinus director - direction cosinecreşte - to increasecreştere - incrementcriteriu - criterioncriteriu de divizibilitate - criterion for divisibilitycuantor - quantifiercuantor existenţial - existential quantifiercuantor universal - universal quantifiercub - cubecunoscut - knowncuplu - paircuplu ordonat - ordered paircurbă - curvecurbă concavă - concave curvecurbă convexă - convex curvecurbă cubică - cubic, cubic curvecurbă integrală - integral curve
D
dacă şi numai dacă - if and only ifdate - datadecizie - decisiondeducţie - deductiondefini - to definedefinit peste tot - everywhere defineddefiniţie - definitiondegenerat - degeneratedeîmpărţit - dividenddeînmulţit - multiplicanddelatare - dilatation, stretchingdemonstra - to provedemonstraţie - proofdependent - dependentderivabilitate - differentiability
Page 4 of 20Dictionar Roman-Englez
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derivată - derivativederivată în punct - derivative at a pointdescazut - minuenddescompunere - decomposition, expansiondescompunere binomială - binomial expansiondescompunere în factori - factoring, factorizationdescompunere în factori primi - prime factorizationdescompunere în fracţii elementare - partial-fraction expansiondescompunere a unei fracţii - decomposition of a fractiondescreşte - to decreasedetermina - to determinedeterminant - determinantdeviaţie - deviationdeviere - deviationdezvoltare - expansiondezvoltare a determinantului - expansion of a determinantdiagonală - diagonaldiagonală principală - main diagonal, principal diagonaldiametru - diameterdiferenţă - differencediferenţiabilitate - differentiabilitydiferit - unequaldimensiune - dimensiondirect proporţional - directly proportionaldirecţie - directiondiscontinuu - discontinuousdiscriminant - discriminantdispărea - to vanishdistanţă - distance, waydistinct - distinctdivizare - divisiondivizibil prin - divisable bydivizor - divisordivizor comun - common divisor, common factordivizor propriu - proper factordomeniu - domain, rangedomeniu de definiţie - domain of definitiondreaptă - linedreaptă numerică - number linedrept - straightdrepte neconcurente - skew lines
Page 5 of 20Dictionar Roman-Englez
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dreptunghi - rectangledrum - waydual - dual
E
echilateral - equilateralechivalent - equivalentecuaţie - equationecuaţie algebrică - algebraic equationecuaţie a liniei drepte - equation of a straight lineecuaţie a unei drepte cu coeficient unghiular
- slope-intercept form of a straight line equation
ecuaţie bipatrată - biquadratic equationecuaţie de ordinul întâi - first-order equationecuaţie exponenţială - exponential equationecuaţie liniară - linear equationecuaţie neliniară - non-linear equationecuaţie omogenă - homogeneous equationecuaţie de ordinul doi(patrată) - quadratic equationelement - elementelement al multimii - member of a setelement de arie - differential of area, element of areaelement de arie (al suprafeţei) - surface elementelement diagonal - diagonal elementeliminare - eliminationeliminare prin substituţie - elimination by substitutionelipsă - ellipseeroare - erroreroare de rotunjire - rounding erroresenţial - essentialestimaţie - estimationevaluare - evaluationexact - exactexactitate - accuracyexcepţie - exclusionexemplu - exampleexînscris - escribedexponent - degree, exponent, grade, powerexponenţial - exponential
Page 6 of 20Dictionar Roman-Englez
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expresie - expressionexpresie algebrică - algebraic expressionexpresie de sub integrală - integrandextremal - extremalextremum absolut - absolute extremum
F
factor - factorfamilie - familyfascicul - pencilfascicul de drepte - pencil of linesfaţă laterală - legfigură circumscrisă - circumscribed figurefigură congruentă - equivalent figurefiguri congruente - congruent figuresflux - fluxfocar - focusformă generală - general formformă parametrică - parametric formformă standardă - standard formformula binomului - binomial formulaformula coeficientului unghiular - slope formulaformula lui Leibniz - Leibniz ruleformula rădăcinilor ecuaţiei patrate - quadratic formulaformulă - formulaformulele jumătăţii unui unghi - half-angle formulasfracţie - fractionfracţie elementară - partial fractionfracţie ordinară - common fractionfracţie subunitară - proper fractionfracţie supraunitară - improper fractionfracţie zecimală - decimal fractionfracţie zecimală finită - terminating decimal fraction
fracţie zecimală infinită - infinite decimal fraction, non-terminating decimal
fracţie zecimală periodică - periodic decimal fractionfracţii cu numitori egali - similar fractionsfuncţie - functionfuncţie concavă - concave function
Page 7 of 20Dictionar Roman-Englez
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funcţie continuă - continuous functionfuncţie de mai multe variabile - function of several variablesfuncţie de o variabilă - function of a single variablefuncţie de variabilă complexă - function of a complex variablefuncţie descrescătoare - decreasing functionfuncţie diferenţiabilă - differentiable functionfuncţie discontinuă - discontinuous functionfuncţie impară - odd-function
funcţie în scară - inter-stage function, jump function, step function
funcţie liniară - linear functionfuncţie monoton crescătoare - monotone increasing functionfuncţie monoton descrescătoare - monotone decreasing functionfuncţie monotonă - monotonic functionfuncţie pară - even function, symmetric functionfuncţie periodică - periodic functionfuncţie primitivă - primitivefuncţie raţională - rational functionfuncţie signum - signum functionfuncţie simetrică - symmetric functionfuncţie trigonometrică - trigonometric functionfundamental - fundamental
G
general - generalgeometrie - geometrygeometrie euclidiană - Euclidean geometrygrad - degree, gradegrad al polinomului - degree of a polinomial
H
hexaedru - hexahedron, hexaederhexagon - hexagonhiperbolă - hyperbolahiperbolic - hyperbolic
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I
identitate - identityimagine - image, map, mappingimpar - oddimplicaţie - implicationincertitudine - uncertaintyinclude - to includeincluziune - inclusionincompatibil - inconsistentincorect - incorrectindependent - independentindependenţă liniară - linear independencyinducţie - inductioninducţie completă - complete inductioninducţie matematică - mathematical inductioninecuaţie - inequalityinegal - unequalinfinit - infiniteinflexiune - inflexioniniţial - initialintegrală - integralintegrare prin părţi - integration by partsinterdependenţă - interdependencyintersecţie - intersectioninterval - intervalinterval deschis - open intervalinterval închis - closed intervalinterval mărginit - bounded intervalinterval semiînchis - semiclosed intervalinvers - inverse, vice versainvers proporţional - inversely proportionalipotenuză - hypotenuseipoteză - hypothesisireductibilitate - irreductibility
Î
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împărţi în două în jumătate - to halveîmpărţire fără rest - exact divisionîn întregime - wholeînălţime - heightînălţime a triunghiului - altitude of a triangleînchis - closedînclinare - slope înlocui - to replaceînmulţi - to multiplyînmulţire - multiplicationînmulţitor - multiplierînscrie - to inscribe
K
kilogram - kilogram(me)kilometru - kilometre
L
latură a unui unghi - side of an anglelatură adiacentă - adjacent sidelege - lawlege asociativă - associative lawlege comutativă - commutative lawlege de compozitie - law of compositionlege distributivă - distributive lawlegea terţului exclus - law of the excluded middlelemă - lemmalimită - limitlimită inferioară - lower limitlimite de integrare - limits of integrationlinie de fracţie - bar, vinculumliniar - linearliniaritate - linearitylinie dreaptă - straight-linelinie frântă - brocken linelinie punctată - dotted line
Page 10 of 20Dictionar Roman-Englez
2/17/05http://math.ournet.md/formule/dic/roman.html
linie tangentă - tangent lineloc geometric al punctelor - geometric locuslocal - locallogaritm - logarithmlogaritm zecimal - common logslogaritm natural - natural logarithmlungime - lengthlungimea arcului - arc-length
M
mărime - magnitudemăsură - measuremăsurabil - measurablematematică - mathematicsmatrice - matrixmatrice de transformare - matrix of the transformationmatrice diagonală - diagonal matrixmatrice inversă - reciprocal matrixmatrice nulă - null-matrixmaxim - maximummaximum - maximummediană - median of a trianglemedie aritmetică - mean, arithmetic meanmedie geometrică - geometric average, geometric mean,
mean proportionalmetoda eliminării - elimination methodmetoda inducţiei complete - principle of complete inductionmetru - meterminim - minimimminimum - minimummodul - modulemonom - monommonoton - monotonousmuchie - edgemulţime - setmulţime de soluţii - solution setmulţime de valori - rangemulţime universală - universal setmulţime vidă - empty set, null-set
Page 11 of 20Dictionar Roman-Englez
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multiplicare - multiplicatonmultiplicator - factor, multipliermultiplu - multiple, multiplexmultiplu comun - common multiplemutual - mutuallymutual disjuncte - mutually disjoint
N
necunoscută - unknownnedefinit - undefinednedegenerat - non-degneratenedescrescător - non-decreasingnedeterminare - indeterminancy, uncertaintynedeterminat - indefinite, undeterminednegativ - negativeneglijabil - negligibleneliniar - non-linearnemărginit - unboundedneomogen - inhomogeneousneperiodic - non-periodicnesimetric - non-symmetricnod - nodenormală - normalnormală la suprafaţă - normal to the surfacenumăr - numbernumăr complex - complex numbernumăr întreg - integer numbernumăr iraţional - irrational numbernumăr natural - natural numbernumăr par - even numbernumăr prim - prime numbernumăr raţional - rational numbernumăr real - real numbernumăr transcendent - transcendental numbernumăr zecimal - decimal numbernumăr ordinal - ordinal numbernumărător - numeratornumărător al fracţiei - term of a fractionnumere întregi consecutive - consecutive integers
Page 12 of 20Dictionar Roman-Englez
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numere reciproc prime - coprime numbers, relatively prime numbersnumitor - denominatornumitor comun - common denominatornumitori diferiţi - unlike denominatorsnumitori identici - like denominators, same denominators
O
oblic - obliqueoctagon - octagonomogen - homogeneousordin - degree, exponentordin al derivatei - order of derivativeordin al ecuaţiei - order of equationordine - orderordonată - ordinateorigine de coordonate - origin of coordinatesorizontal - horizontaloval - oval
P
par - evenparabolă - parabolaparabolic - parabolicparalel - parallelparalelepiped - parallelepipedparalelogram - parallelogramparametru - parameterparanteze figurate - bracesparanteze patrate - square brackets, bracketsparanteze rotunde - parenthesespatrat - quadrate, squarepatrulater - tetragonpătratic - quadraticpentagon - pentagonperimertu - perimeterperioadă - period
Page 13 of 20Dictionar Roman-Englez
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perpendiculară - normal, perpendicularpicior al perpendicularei - foot of a perpendicularpiramidă - pyramidplan - planeplan tangent - tangent planeplan xOy - xy-planeplanimetrie - plane geometry, planimetryplus - plus signpoliedru - polyhedronpoligoane asemenea - similar polygonspoligoane congruente - congruent polygonspoligon - polygonpoligon convex - convex polygonpoligon înscris - inscribed polygonpoligon regulat - equilateral polygon, regular polygonpolinom - polynomialposibilitate - possibilitypozitiv - positiveprecedent - precedingprecizie - accuracypreimagine - pre-imagepresupunere - assumptionprimă derivată - first derivativeprincipal - principalprincipiul dualitatii - duality principleprizmă - prismprocent - per centprodus - productprodus scalar - dot productprodus vectorial - cross-product, vector productprogresie - progressionprogresie aritmetică - arithmetic progressionprogresie geometrică - geometric progressionproiecţie - projectionproporţie - proportionpropoziţie - proposition, statementproprietate - propertyproprietate asociativă - associative propertypunct - dot, pointpunct de discontinuitate - point of discontinuitypunct de extrem - extremum point
Page 14 of 20Dictionar Roman-Englez
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punct de inflexiune - point of inflexionpunct diametral opus - diametrically opposite pointpunct focal - focal pointpunct izolat - acnodeputere - degree, grade, power
R
radian - radianradical - radical, radical signramură - branchrang al unei matrici - rank of a matrixraport - ratio, relationraţie a unei progresii aritmetice - common differenceraţie a unei progresii geometrice - common ratioraţional - rationalrază - radiusrază vectoare - radius vectorrădăcină - rootrădăcină dublă - double rootrădăcină multiplă - repeated rootrădăcină patrată - square rootrădăcină simplă - simple rootrădăcini conjugate - conjugate rootsreal - realreciproc - vice versareduce - to reducereducere termenilor asemenea - combine similar termsreductibil - reducibleregulă - ruleregula paralelogramului - parallelogram law, parallelogram rulerelaţie - relationrelativ - relativerepetat - multiplexreprezenta - to representrest - remainder, restrestricţie - restrictionreuniune - unionrezolva - to solveridica la patrat - to square
Page 15 of 20Dictionar Roman-Englez
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ridica la putere - to raise to a powerromb - lozenge, rhombus, rhombrotaţie - rotationrotunji - to round
S
salt - jumpsatisface - to satisfyscalar - scalarscădere - subtractionscăzător - subtrahendschemă bloc - flow chartschema lui Horner - Horner's scheme, synthetic divisionschimb de variabilă - change of variablesecantă - secantsecţiune - cutsecţiune conică - conic, conic sectionsecţiune conică centrală - central conicsecţiune conică degenerată - degenerate conicsecţiune conică nedegenerată - non-degenerate conic,
non-degenerate conic sectionsector circular - sector of a circlesegment - closed interval, intercept, segmentsegment de pe axa x - x-interceptsegment liniar - line segment, straight-line segmentsegmente congruente - congruent segmentssemicerc - semi-circlesemidreaptă - raysemisferă - hemispheresemn - signsemnul adunării - addition sign, plus signsemnul rădăcinii - radical, radical signsemnul scăderii - minus signsferă - spheresimbol - symbolsimetric - symmetricsimetrie centrală - central symmetrysimplifica - to cancelsimplificare - simplification
Page 16 of 20Dictionar Roman-Englez
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simplu - simplesinusoidă - sine curvesistem - systemsistem cartezian de coordonate - Cartesian coordinate systemsistem cu mai multe necunoscute - many-variable systemsistem de coordonate - coordinate systemsistem de ecuaţii liniare - system of linear equationsistem neortogonal de coordonate - non-orthogonal coordinate systemsistem omogen - homogeneous systemsistem ortogonal de coordonate - orthogonal coordnate systemsistem rectangular de coordonate - rectangular coordonate systemsoluţie admisibilă - permissible solution, admissible solutionsoluţie aproximativă - approximate solutionsoluţie completă - complete solutionsoluţie exactă - exact solutionsoluţie generală - general solutionsoluţie nenulă - non-zero solutionsoluţie netrivială - non-trivial solutionsoluţie nulă - zero solutionsoluţie periodică - periodic solutionsoluţie unică - unique solutionsoluţiona - to solvesoluţionare a unei probleme - solution of a problemspaţiu - spacespaţiu euclidian - Euclidean spacestrict - strictstrict monoton - strongly monotonicsubstitui - to replacesubstituţie - substitutionsubstituţie inversă - back-substitutionsuma - to addsumă - sumsuprafaţă - surfacesuprafaţă de rotaţie - surface of revolutionsubmulţime - subset
T
tangentă - tangentteorema lui Pythagoras - Pythagorean theorem
Page 17 of 20Dictionar Roman-Englez
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teoremă - theoremteorema fundamentală a aritmeticii - fundamental theorem of arithmeticteorema lui Bezout - factor theoremteorema lui Thales - intercept theoremteorema sinusurilor - law of sines, sine ruleteoria mulţimilor - set theoryteorie - theorytermen - termtermen al sumei - addend, summandtermen general - general termtermeni asemenea - similar termstetraedru - tetrahedrontot - wholetransforma - to transformtransformare de coordonate - transform of coordinatestranslare - translationtranslaţie - bridgingtranzitivitate - transitivitytrapez - trapezoid, trapeziumtrigonometric - trigonometrictrigonometrie - trigonometrytrisecţia unui unghi - trisection of the angletriunghi - triangletriunghi ascutunghic - acute triangletriunghi asemenea - similar triangletriunghi dreptunghic - right triangletriunghi echilateral - equilateral triangletriunghi isoscel - isosceles triangletriunghi obtuzunghic - obtuse triangletriunghi scalen - scalene triangletriunghiular - triangular
U
unghi - angleunghi adiacent - adjacent angleunghi alăturat - adjacent angleunghi ascuţit - acute angleunghi complementar până la 90o - complementary angle
- explementary angle
Page 18 of 20Dictionar Roman-Englez
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unghi complementar până la 360o
unghi de înclinare - slope angleunghi diedru - dihedral angleunghi drept - right angleunghi exterior al triunghiului - exteriar angle of a triangleunghi interior - interior angleunghi înscris - inscribed angleunghi întins - straight angleunghi la bază a triunghiului - base angle of a triangleunghi la centru - central angleunghi obtuz - obtuse angleunghi plin - straight ahgleunghi suplimentar până la 360o - conjugate angleunghiuri adiacente suplimentare - adjacent supplementary anglesunghiuri alterne - alternate anglesunghiuri alterne externe - alternate exterior angles
unghiuri alterne interne - alternate interior angles, opposite interior angles
unghiuri congruente - congruent anglesunghiuri corespondente - corresponding anglesunghiuri opuse la vârf - vertex anglesunghiuri suplimentare până la 180o - supplementary anglesunic - singleunicitate - uniquenessunitate - unitunivoc - unambiguous
V
valabil - validvalid - validvaloare - valuevaloare absolută - absolute valuevaloare absolută a numărului complex - absolute value of a complex numbervaloare iniţială - initial valuevaloare medie - average, average valuevaloare limită - limit valuevariabilă - variablevariabilă independentă - independent variable
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vârf - vertexvector - vectorvector coplanar - coplanar vectorvector de bază - base vectorvector normal - normal vectorvector nul - null-vectorvector unitar - unit vectorvector tangent unitar - unit tangent vectorveridicitate - truthverifica - to verifyvertical - verticalviteză - speed, velocityvolum - volume
Z
zecimal - decimalzero - zero
Bibliografie
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Dictionar Englez-Roman de Termeni Matematici
A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
A
abscissa - abscisăabsolute - absolutabsolute extremum - extremum absolutabsolute value - valoare absolutăabsolute value of a complex number - valoare absolută a numărului complexaccuracy - precizie, exactitate, acurateţeacnode - punct izolatacute angle - unghi ascuţitacute triangle - triunghi ascuţitunghicadd - a aduna, a sumaaddend - termen al sumeiaddition - adunareaddition sign - semnul adunăriiadjacent angle - unghi adiacent, unghi alăturatadjacent side - latură adiacentăadjacent supplementary angles - unghiuri adiacente suplimentareadjoint - conjugatadmissible - admisibiladmissible solution - soluţie admisibilăaffine coordinates - coordonate afinealgebra - algebrăalgebraic equation - ecuaţie algebricăalgebraic expression - expresie algebricăalgorithm - algoritmalgorithm for division - algoritm de divizarealternance - alternanţăalteration - alternare
Page 1 of 22Dictionar Englez-Roman
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alternate angles - unghiuri alternealternate exterior angles - unghiuri alterne externealternate interior angles - unghiuri alterne internealtitude of a triangle - înălţime a triunghiuluiamplitude - amplitudineanalogous - analogicanalogical - analogicanalogy - analogieanalysis - analizăanalyze - a analizaangle - unghianticosine - arccosinusantisine - arcsinusapplicate - aplicatăapproximate solution - soluţie aproximativăarbitrary - arbitrararc - arcarccosine - arccosinusarc-length - lungimea arculuiarcsine - arcsinusarctangent - arctangentăarea - arieargument - argumentargument of a function - argument al funcţieiarithmetic mean - medie aritmeticăarithmetic progression - progresie aritmeticăassociative law - lege asociativăassociative property - proprietate asociativăassumption - presupunereasymmetric(al) - asimetricasymmetry - asimetrieasymptote - asimptotăasymptotes of a hyperbola - asimptote ale hiperboleiaverage - valoare medieaverage value - valoare medieaxiom - axiomăaxis - axă
B
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back-substitution - substituţie inversăbar - linie de fracţiebase - bazăbase angle of a triangle - unghi la bază a triunghiuluibase vector - vector de bazăbasis - bazăbillion - bilionbinary - binarbinomial - binombinomial coefficient - coeficient binomialbinomial expansion - descompunere binomialăbinomial formula - formula binomuluibiquadratic equation - ecuaţie bipatratăbisector - bisectoarebisectrix - bisectoarebounded interval - interval mărginitbraces - paranteze figuratebrackets - paranteze patratebranch - ramurăbridging - translaţiebrocken line - linie frântă
C
calculus - analiză matematică, calculcancel - a simplificacanonical - canonicCartesian coordinate system - sistem cartezian de coordonateCartesian coordinates - coordonate cartezienecentral angle - unghi la centrucentral conic - secţiune conică centralăcentral symmetry - simetrie centralăcentre - centrucentre of the escribed circle - centru unei circumferinţe exînscrisecenter - centruchange of variable - schimb de variabilăchord - coardăcircle - cerccircumcenter - centrul circumferinţei circumscrisecircumscribed figure - figură circumscrisă
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closed - închisclosed interval - interval închis, segmentcoefficient - coeficientcoincident - coincidentcollinearity - coliniaritatecombination - combinaţiecombine similar terms - reducere a termenilor asemeneacommon denominator - numitor comuncommon difference - raţie a unei progresii aritmeticecommon divisor - divizor comuncommon factor - divizor comuncommon fraction - fracţie ordinarăcommon logs - logaritm zecimalcommon multiple - multiplu comuncommon ratio - raţie a unei progresii geometricecommutative law - lege comutativăcomparation - comparaţie, compararecomplementary angle - unghi complementar până la 90ocomplete induction - inducţie completăcomplete solution - soluţie completăcomplex number - număr complexcomputable - calculabilcomputation - calculconcave - concavăconcave curve - curbă concavăconcave function - funcţie concavăconcentric circles - cercuri concentricecondition - condiţiecone - concongruence - congruenţăcongruent angles - unghiuri congruentecongruent figures - figuri congruentecongruent polygons - poligoane congruentecongruent segments - segmente congruenteconic - secţiune conicăconic section - secţiune conicăconjugate - conjugatconjugate angle - unghi suplimentar până la 360o
conjugate roots - rădăcini conjugateconsecutive integers - numere întregi consecutiveconstant - constantă
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continuity - continuitatecontinuous function - funcţie continuăconvex - convexconvex curve - curbă convexăconvex polygon - poligon convexconvexity - convexitatecoordinate - coordonatăcoordinate axis - axă de coordonatecoordinate system - sistem de coordonatecoplanar - coplanarcoplanar vector - vector coplanarcoprime numbers - numere reciproc primecorollary - consecinţă, corolarcorresponding angles - unghiuri corespondentecount - a calculacriterion - criteriucriterion for divisibility - criteriu de divizibilitatecross-product - produs vectorialcube - cubcubic - curbă cubicăcubic curve - curbă cubicăcurve - curbăcut - secţiunecylinder - cilindru
D
data - datedecimal - zecimaldecimal fraction - fracţie zecimalădecimal number - număr zecimaldecision - deciziedecomposition - descompuneredecomposition of a fraction - descompunere a unei fracţiidecrease - a descreştedecreasing function - funcţie descrescatoarededuction - deducţiedefine - a definidefinition - definiţiedegenerate - degenerat
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degenerate conic - secţiune conică degeneratădegree - putere, ordin, grad, exponentdegree of a polinomial - grad al polinomuluidenominator - numitordependent - dependentderivative - derivatăderivative at a point - derivată în punctdeterminant - determinantdetermine - a determinadeviation - deviaţie, devierediagonal - diagonalădiagonal element - element diagonaldiagonal matrix - matrice diagonalădiameter - diametrudiametrically opposite point - punct diametral opusdifference - diferenţădifferentiability - derivabilitate, diferenţiabilitatedifferentiable function - funcţie diferenţiabilădifferential of area - element de ariedigit - cifrădihedral angle - unghi diedrudilatation - delataredimension - dimensiunedirection - direcţiedirection cosine - cosinus directordirectly proportional - direct proporţionaldiscontinuous - discontinuudiscontinuous function - funcţie discontinuădiscriminant - discriminantdisposition - amplasaredistance - distanţădistinct - distinctdistributive law - lege distributivădividend - deîmpărţitdivisable by - divizibil prin division - divizaredivision algorithm - algoritm de divizaredivisor - divizordomain - domeniudomain of definition - domeniu de definiţiedot - punct
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dot product - produs scalardotted line - linie punctatădouble root - rădăcină dublădual - dualduality principle - principiul dualităţii
E
edge - muchieelement - elementelement of area - element de arieelimination - eliminareelimination by substitution - eliminare prin substituţieelimination method - metoda eliminăriiellipse - elipsăempty set - mulţime vidăequation - ecuaţieequation of a straight line - ecuaţie a liniei drepteequilateral - echilateralequilateral polygon - poligon regulatequilateral triangle - triunghi echilateralequivalent - echivalentequivalent figure - figură congruentăerror - eroareescribed - circumscris, exînscrisessential - esenţialestimation - estimaţieEuclidean algorithm - algoritmul lui EuclidEuclidean geometry - geometrie euclidianăEuclidean space - spaţiu euclidianevaluation - evaluareevaluation of determinant - calculul determinantuluieven - pareven function - funcţie parăeven number - număr pareverywhere defined - definit peste totexact - exactexact division - împărţire fără restexact solution - soluţie exactăexample - exemplu
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excentre - centru al unei circumferinţe exînscriseexclusion - excepţieexistential quantifier - cuantor existenţialexpansion - descompunere, dezvoltareexpansion of a determinant - dezvoltare a determinantuluiexplementary angle - unghi complementar până la 360o
exponent - exponent, ordinexponential - exponenţialexponential equation - ecuaţie exponenţialăexpression - expresieexterior angle of a triangle - unghi exterior al triunghiuluiextremal - extremalextremum point - punct de extremum
F
factor - factor, multiplicatorfactor theorem - teorema lui Bezoutfactoring - descompunere în factorifactorization - descompunere în factorifamily - familiefield - câmpfirst derivative - primă derivatăfirst-order equation - ecuaţie de ordinul întâiflow chart - schemă blocflux - fluxfocal point - punct focalfocus - focarfoot of a perpendicular - picior al perpendiculareiformula - formulăfraction - fracţiefunction - funcţiefunction of a complex variable - funcţie de variabilă complexăfunction of a single variable - funcţie de o variabilăfunction of several variables - funcţie de mai multe variabilefundamental - fundamentalfundamental theorem of arithmetic - teoremă fundamentală a aritmeticii
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G
general - generalgeneral form - formă generalăgeneral solution - soluţie generalăgeneral term - termen generalgeometric average - medie geometricăgeometric locus - loc geometric al punctelorgeometric mean - medie geometricăgeometric progression - progresie geometricăgeometry - geometriegrade - grad, exponent, puteregreatest common divisor - cel mai mare divizor comungreatest common factor - cel mai mare divizor comun
H
half-angle formulas - formule ale jumătăţii unui unghihalve - a împărţi în două în jumătateheight - înălţimehemisphere - semisferăhexagon - hexagonhexaeder - hexaedruhexahedron - hexaedruhill climbing - determinarea extremuluihomogeneous - omogenhomogeneous equation - ecuaţie omogenăhomogeneous system - sistem omogenhorizontal - orizontalhorizontal axis - axă orizontalăHorner's scheme - schema lui Hornerhyperbola - hiperbolăhyperbolic - hiperbolichypotenuse - ipotenuzăhypothesis - ipoteză
I
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identity - identitateif and only if - dacă şi numai dacăimage - imagineimplication - implicaţieimproper fraction - fracţie supraunitarăincenter - centrul circumferinţei înscriseincircle - circumferinţă înscrisăinclude - a includeinclusion - incluziuneinconsistent - incompatibilincorrect - incorectincrease - a creşteincrement - creştereindefinite - nedeterminatindependent - independentindependent variable - variabilă independentăindeterminancy - nedeterminareinduction - inducţieinequality - inecuaţieinfinite - infinitinfinite decimal fraction - fracţie zecimală infinităinflexion - inflexiuneinhomogeneous - neomogeninitial - iniţialinitial condition - condiţie iniţialăinitial value - valoare iniţialăinscribe - a înscrieinscribed angle - unghi înscrisinscribed circle - circumferinţă înscrisăinscribed polygon - poligon înscrisinteger number - număr întregintegral - integralăintegral curve - curbă integralăintegrand - expresie de sub integralăintegration by parts - integrare prin părţiintegration constant - constantă de integrareinter-stage function - funcţie în scarăintercept - segmentintercept theorem - teorema lui Thales
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interdependency - interdependenţăinterior angle - unghi interiorintersection - intersecţieinterval - intervalinverse - inversinversely proportional - invers proporţionalirrational number - număr iraţionalirreductibility - ireductibilitateisosceles triangle - triunghi isoscel
J
jump - saltjump function - funcţie în scară
K
kilogram(me) - kilogramkilometre - kilometruknown - cunoscut
L
law - legelaw of composition - lege de compoziţielaw of sines - teorema sinusurilorlaw of the excluded middle - legea terţului exclusleast common denominator - cel mai mic numitor comunleast common multiple - cel mai mic multiplu comunleg - faţă lateralăLeibniz rule - formula lui Leibnizlemma - lemălength - lungimelike denominators - numitori identicilike signs - aceleaşi semnelimit - limitălimit value - valoare limită
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limits of integration - limite de integrareline - dreaptăline segment - segment liniarlinear - liniarlinear equation - ecuaţie de gradul întâi (liniară)linear function - funcţie liniarălinear independency - independenţă liniarălinearity - liniaritatelocal - locallogarithm - logaritmlower limit - limită inferioarălowest common denominator - cel mai mic numitor comunlowest common multiple - cel mai mic multiplu comunlozenge - romb
M
magnitude - mărimemain diagonal - diagonală principalămajor axis - axă principalămany-variable system - sistem cu mai multe necunoscutemap - imagine, aplicaţiemapping - imagine, aplicaţiemeter - metrumathematical induction - inducţie matematicămathematics - matematicămatrix - matricematrix of the transformation - matrice de transformaremaximum - maximum, maximmean - medie aritmeticămean proportional - medie geometricămeasurable - măsurabilmeasure - măsurămedian of a triangle - medianămember of a set - element al mulţimiiminimum - minimum, minimminuend - descăzutminus sign - semnul scăderiimodule - modulmonom - monom
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monotone decreasing function - funcţie monoton descrescătoaremonotone increasing function - funcţie monoton crescătoaremonotonic function - funcţie monotonămonotonous - monotonmultiple - multiplumultiplex - multiplu, repetatmultiplicand - deînmulţitmultiplication - înmulţire, multiplicaremultiplier - înmulţitor, multiplicatormultiply - a înmulţimutually - mutualmutually disjoint - mutual disjuncte
N
natural logarithm - logaritm naturalnatural number - număr naturalnecessary and sufficient condition - condiţie necesară şi suficientănegative - negativnegligible - neglijabilnode - nodnon-decreasing - nedescrescătornon-degnerate - nedegeneratnon-degenerate conic - secţiune conică nedegeneratănon-degenerate conic section - secţiune conică nedegeneratănon-linear - neliniarnon-linear equation - ecuaţie neliniarănon-orthogonal coordinate system - sistem neortogonal de coordonatenon-periodic - neperiodicnon-symmetric - nesimetricnon-terminating decimal - fracţie zecimală infinitănon-trivial solution - soluţie netrivialănon-zero solution - soluţie nenulănormal - normală, perpendicularănormal to the surface - normală la suprafaţănormal vector - vector normalnull-vector - vector nulnull-matrix - matrice nulănull-set - mulţime vidănumber - număr
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number line - dreaptă numericănumerator - numărător
O
oblique - oblicobtuse angle - unghi obtuzobtuse triangle - triunghi obtuzunghicoctagon - octagonodd - imparodd-function - funcţie imparăone-to-one - biunivocopen interval - interval deschisopposite interior angles - unghiuri alterne interioareorder - ordineorder of derivative - ordin al derivateiorder of equation - ordinul ecuaţieiordered pair - cuplu ordonatordinal number - număr ordinalordinate - ordonatăorigin of coordinates - origine de coordonateorthogonal base - bază ortogonalăorthogonal coordnate system - sistem ortogonal de coordonateorthonormal basis - bază ortonormatăoval - oval
P
pair - cupluparabola - parabolăparabolic - parabolicparallel - paralelparallelepiped - paralelepipedparallelogram - paralelogramparallelogram law - regula paralelogramuluiparallelogram rule - regula paralelogramuluiparameter - parametruparametric form - formă parametrică
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parentheses - paranteze rotundepartial fraction - fracţie elementarăpartial-fraction expansion - descompunere în fracţii elementarepencil - fasciculpencil of lines - fascicul de dreptepentagon - pentagonper cent - procentperimeter - perimertuperiod - perioadăperiodic decimal fraction - fracţie zecimală periodicăperiodic function - funcţie periodicăperiodic solution - soluţie periodicăpermissible solution - soluţie admisibilăperpendicular - perpendicularăpivot - axă de rotaţie, centru de rotaţieplane - planplane geometry - planimetrieplanimetry - planimetrieplus sign - semnul adunării, pluspoint - punctpoint of discontinuity - punct de discontinuitatepoint of inflexion - punct de inflexiunepolygon - poligonpolyhedron - poliedrupolynomial - polinompositive - pozitivpossibility - posibilitatepower - putere, exponentpre-image - preimaginepreceding - precedentprime factorization - descompunere în factori primiprime number - număr primpimitive - funcţie primitivăprincipal - principalprincipal axis - axă principalăprincipal diagonal - diagonală principalăprinciple of complete induction - metoda inducţiei completeprism - prizmăproduct - produsprogression - progresieprojection - proiecţie
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proof - demonstraţieproper factor - divizor propriuproper fraction - fracţie subunitarăproperty - proprietateproportion - proporţieproposition - propoziţieprove - a demonstrapyramid - piramidăPythagorean theorem - teorema lui Pythagoras
Q
quadrate - patratquadratic - pătraticquadratic equation - ecuaţie de gradul al doilea (patrată)quadratic formula - formula rădăcinilor ecuaţiei de gradul al doilea
(ecuaţiei patrate)quantifier - cuantorquotient - cât
R
radian - radianradical - radical, semnul rădăciniiradical sign - radical, semnul rădăciniiradius - razăradius vector - rază vectoareraise to a power - a ridica la putererange - mulţime de valori, domeniurank of a matrix - rang al unei matriciratio - cât, raportrational - raţionalrational function - funcţie raţionalărational number - număr raţionalray - semidreaptăreal - realreal number - număr realreciprocal matrix - matrice inversă
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rectangle - dreptunghirectangular coordonate system - sistem rectangular de coordonatereduce - a reducereducible - reductibilregular polygon - poligon regulatrelation - relaţie, raportrelative - relativrelatively prime numbers - numere reciproc primeremainder - restrepeated root - rădăcină multiplăreplace - a substitui, a înlocuirepresent - a reprezentarest - restrestriction - restricţierhomb - rombrhombus - rombright angle - unghi dreptright triangle - triunghi dreptunghicroot - rădăcinărotation - rotaţieround - a rotunjirounding error - eroare de rotunjirerule - regulă
S
same denominators - numitori identicisatisfy - a satisfacescalar - scalarscalene triangle - triunghi scalensecant - secantăsector of a circle - sector circularsegment - segmentsemi-circle - semicercsemiclosed interval - interval semiînchisset - mulţimeset theory - teoria mulţimilorside of an angle - latură a unui unghisign - semnsignum function - funcţia signum
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similar fractions - fracţii cu numitori egalisimilar polygons - poligoane asemeneasimilar terms - termeni asemeneasimilar triangle - triunghi asemeneasimilarity - asemănaresimilitude - asemănaresimple - simplusimple root - rădăcină simplăsimplification - simplificaresine curve - sinusoidăsine rule - teorema sinusurilorsingle - unicsingle root - rădăcină simplăskew lines - drepte neconcurenteslope - înclinare, coeficient unghiularslope angle - unghi de înclinareslope formula - formula coeficientului unghiularslope-intercept form of a straight line equation
- ecuaţie a unei drepte cu coeficient unghiular
solution of a problem - soluţie a unei problemesolution set - mulţime de soluţiisolve - a soluţiona, a rezolvaspace - spaţiuspeed - vitezăsphere - sferăsquare - patrat, a ridica la patratsquare brackets - paranteze patratesquare root - rădăcină patratăstandard form - formă standardăstatement - propoziţie, afirmaţiestep function - funcţie în scarăstraight - dreptstraight angle - unghi întins, unghi plinstraight-line - linie dreaptăstraight-line segment - segment liniarstretching - delatarestrict - strictstrongly monotonic - strict monotonsubset - submulţimesubstitution - substituţiesubtraction - scăderesubtrahend - scăzător
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sum - sumăsummand - termen al sumeisupplementary angles - unghiuri suplimentare până la 180osurface - suprafatăsurface area - arie a suprafeţeisurface element - element de arie al suprafeţeisurface of revolution - suprafaţă de rotaţiesymbol - simbolsymmetric - simetricsymmetric function - funcţie simetrică, funcţie parăsynthetic division - schema lui Hornersystem - sistemsystem of linear equations - sistem de ecuaţii liniare
T
tangent - tangentătangent line - linie tangentătangent plane - plan tangentterm - termenterm of a fraction - numărător al fracţieiterminating decimal fraction - fracţie zecimală finitătetragon - patrulatertetrahedron - tetraedrutheorem - teoremătheory - teorietranscendental number - număr transcendenttransform - a transformatransform of coordinates - transformare de coordonatetransitivity - tranzitivitatetranslation - translaretrapezium - trapeztrapezoid - trapeztriangle - triunghitriangular - triunghiulartrigonometric - trigonometrictrigonometric function - funcţie trigonometricătrigonometry - trigonometrietrisection of the angle - trisecţie a unui unghitruth - veridicitate
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U
unambiguous - univocunbounded - nemărginituncertainty - incertitudine, nedeterminareundefined - nedefinitundetermined - nedeterminatunequal - inegal, diferitunion - reuniuneunique solution - soluţie unicăuniqueness - unicitateunit - unitateunit circle - circumferinţă unitarăunit tangent vector - vector tangent unitarunit vector - vector unitaruniversal quantifier - cuantor universaluniversal set - mulţime universalăunknown - necunoscutăunlike denominators - numitori diferiţiunsymmetric - asimetric
V
valid - valid, valabilvalue - valoarevanish - a se anula, a dispăreavariable - variabilăvector - vectorvector product - produs vectorialvelocity - vitezăverify - a verificavertex - vârfvertex angles - unghiuri opuse la vârfvertical - verticalvertical axis - axă verticalăvice versa - invers, reciprocvinculum - linie de fracţie
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volume - volum
W
way - drum, distantăwell-defined - bine definitwhole - tot, în întregime
X
x-axis - axa xx-intercept - segment de pe axa xxy-plane - plan xOy
Y
y-axis - axa y
Z
z-axis - axa zzero - zerozero solution - soluţie nulă
Bibliografie
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Dictionar Matematic Roman-Englez de Fraze Uzuale
A Ă Â B C D E F G H I Î J K L M N O P Q R S Ş T Ţ U V W X Y Z
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 Thus the martingale ψ(t) turns out to be a process
with independent increments. s-a adeverit a fi This proved to be a bigger problem than originally
anticipated.
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This value was found to be cos(π/17).
admite la fel admite The function η(t) also admits decomposition of the
form (4).
adresa ne adresăm la We now turn to the task of considering the
continuous-parameter supermarlingales.
aduce demonstraţia este adusă The proof of the theorem is given in [8].
aduna adunăm Now write down these inequalities for n, n+1, ..., n+m
and add them to obtain the estimation (6).
afirma afirmă 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.
afirmaţie afirmaţie mai generală A more general statement can be found in [15]. afirmaţie necesară Approaching the limit as N→∞ we obtain the
required assertion. afirmaţie reciprocă 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 We are now in a position to determine (or to find)
the value of x*.
ajunge ajungem la ... Thus we arrive at the following conclusion
[definition, equation, theorem, assertion, result].
ajutor cu ajutorul 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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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 Another way to approach the problem is to take a Taylor series expansion of g(t).
altă 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 These operations are discussed in more detail in next
section.
ambele 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 ...
ambele părţi The two sides of Young's inequality
are equal if and only
if b=ϕ(a).
ambiguu This ambiguity does not lead to misunderstanding.
aminti amintim că 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.
analiza pentru a analiza To analyse the equation, it is convenient to go over to
non-dimensional variables. este analizată To analyse the equation, it is convenient to go over to
non-dimensional variables. The sum S is analysed analogously.
analogic The proof of (11) is completely analogous to that of (10).
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analogic cu ... 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 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.
care nu aparţine This equation contradicts the assumption that G contains points not belonging to D.
apărea apare The problem arises of finding the conditions under
which the distribution p(u) is unimodal.
aplica să aplicăm Let us apply this result to the case of an increasing
process At.
pot fi aplicate Formulas (1) and (2) may be employed on condition that the contour C has no loops.
se poate de aplicat Since p is of the form 1+4m, Corollary 2.5 is applicable and hence p=x2+y2.
aproape The temperature was observed to be near zero. aproape de The angle θ is closed to 45 deg. aproape peste tot The Lebesque-integrable functions are equal if and
only if they have the same values almost everywhere. aproape toate Nearly all scientific problems deal with quantities and
relations among them. aproape toţi It is easy to verify that f(x)<C for almost all x.
aproximativ acelaşi As the numbers of the molecules moving in opposite directions are much the same, it is natural to assume ...
aproximaţie în primă aproximaţie
As (or in) a first approximation, x(t) may be chosen
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to be t2.
aranja în ordine crescătoare [descrescătoare] sunt aranjaţi în ordine crescă- toare [descrescătoare]
The jump moments t1, t2, ..., tn are arranged in ascending [descending] order.
arăta arată Fig.12 depicts the slope of the line ∆x versus ∆t. vom arăta We shall show that in this case p must divide b. este arătat The slope of the line ∆x versus ∆t is shown
schematically in Fig.12. pentru a arăta că 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.
cum este arătat mai sus 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.
argumente There are doubtful points in the above reasoning.
aritmetică Arithmetic is the elementary branch of mathematics dealing with the properties of numbers and their operations.
asigura asigură Property 1o ensures the feasibility of the addition.
The MIS furnishes (or supplies) data for management.
asociat asociată cu (referitoare la) 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.
asociat laThe 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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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 Thus the following theorem has been proved. The factors µ and k are related as follows: µ=4πk.
astfel încât 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.
astfel ... ca We take a point on the line such that ... astfel ... încât 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 Chapman-Kolmogorov equation.
Consequently,
ataşa la poate fi ataşat la The process X(t) may be placed (or put) into one of
the following three classes depending on its asymptotic behaviour.
atinge atinge maximum [minimum]
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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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 In order to attain the purpose, we shall not avail
ourselves of the methods of functional analysis. este atins scopul This aim is attained (or objective is accomplished)
by introducing non-dimensional variables x' = x/x0 and y' = y/y0.
atinge se atinge 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.
atrage atenţia să atragă atenţia 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.
se atrage atenţia Recently much attention (or thought / consideration) has been given to the study of the diffusion processes.
atribui sens pentru a atribui sens To attach a meaning to the solution of this stochastic
equation, we must define the measure µ more accurately.
se atribuie sens Analogous meaning is attached to the notation D[0,∞).
atribui valoarea atribuim valoarea We assign the value 0 to this state. valori pot fi atribuite 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 To appreciate the merits (or the advantages) of this
theory, it suffices to observe that ... avantaj principal
The main (or principal / major) advantage of this
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approach is that it enables the results of the previous chapter to be extended to the case under consideration.
avea avem Since a satisfies (1), we have Y 1=Y 2. nu are For instance, the function
has not a derivative at the point 0. The prime number p possess no factors other than ± 1 and ± p.
poate avea It follows that the solution can have only a finite number of zeros in D.
nu poate avea It follows that the solution cannot have more than a finite number of zeros in D.
avea de-a face avem de-a face In this section we deal only with rational integers.
avea intenţia vom avea intenţia 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.
avea loc această nu poate avea loc This cannot be, for l/p is not a rational number. această are loc în cazul This happens to be the case for those equations in
which S increases.
avea maxim [minim] are maxim [minim] The function ψ(x) has a maximum [minimum] at the
point x*=3h/2.
avea neajunsul au neajunsul These equations possess a fundamental defect (or
demerit / draw back / disadvantage). are neajunsul His hypothesis has limitations of its own.
avea nevoie de vom avea nevoie de In order to achieve this, we shall need some material
from elementary number theory.
avea o formă ... are o formă ... Equation (3.2) is of the following form ...
avea o mare importanţă are o mare importanţă In applications the following theorem is of great
importance.
avea sens
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dacă are sensThe symbol , if it is meaningful, always
represents a number. nu are sens If the common part is empty, the difference has no
sense.
avea soluţie unică are soluţie unică Equation (7) has a unique solution.
avea un caracter are un caracter Noise ω(t) is stochastic in character (or in nature).
Noise ω(t) is of a stochastic character (or nature).
avea valoarea are valoarea 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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the following properties hold true.
ca urmare Such a solution does not exist in consequence (or on account) of (6) and Remark 4.
calcula As is easy to calculate ... sunt calculate The coefficients of expansion (9) are evaluated (or
computed) from formulas (5). cum este uşor de calculat as is easy to calculate
as is easy to evaluate
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 x→0.
cărui A variable is a quantity to which an unlimited number of values can be assigned in an investigation.
al cărui Let A be a strongly positive operator whose domain of definition does not depend on t.
care este Consider the function λ(y), which is a periodical distribution coinciding with δ(y) for | y | < 1/2.
caz caz particular This formula can be considered as a particular case of
formula (14). cazul considerat In the case under consideration the process α(t) is
continuous. Let us generalize equation (6) to the case under discussion.
căuta vom căuta We shall seek a solution to (12) in the form
z(t,u) = x(t) y(u). sunt căutate It is convenient, in solving problems, to use letters for
the numders which are sought. căutată The sought-for quantity M is involved in the equation
reciprocally. The desired value of m can be determined from conition (4).
care va fi căutată 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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în căutarea In looking (or in search) for the optimal solution we discovered that ...
câteva 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.
câteva modalităţi 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 the in-depth study cercetare ulterioară To determine whether these ideals coincide, we must
make a further study of the polynomials.
cere se cere It is required to estimate in the best possible manner
the value of a random variable ξ. se cere întrebarea It is therefore reasonable to ask for which classes of
"integers" the fundamental theorem holds, and for which it does not.
cerinţă mai puternică The process Mt, satisfies the conditions of (1) with the
stronger requirement that be an L2-martingale.
chiar în cazul când 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.
ci Thus we need not 8, but 9 parameters to describe the system.
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ciocni cu ne ciocnim cu 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 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.
intuitiv este clar It is clear intuitively that every collection, finite or infinite, of non-negative integers contains a smallest one.
coeficientul pe lângă the coefficient of smth. the coefficient multiplying smth.
coeficientul pe lângă termenul superior
A polinomial is monic if its leading coefficent is 1.
coincide coincide cu 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.
complet satisfăcător Completely satisfactory solution of the functional problem is given in [6].
completa este completată 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 This result was established in another way under
somewhat weaker conditions on the diffusion coefficients.
condiţiile conform cărora The problem arose of finding the conditions under which the distribution p(t) is unimodal.
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condiţie de convergenţă la The condition for convergence to a Wiener process is stated in the next section.
condiţie de existenţă şi unicitate a soluţiei
Let the coefficients of equation (9) satisfy the condition for the existence and uniqueness of the solution.
condiţii exacte pentru The exact conditions for equilibria to exist and be stable are examined.
condiţie mai fină The tangent bundle of the manifolds provides a sharper necessary condition.
condiţii sunt verificate 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.
alte condiţii fiind aceleaşi The distinction between (14) and (15) is that the variable x entering into (15) depends on ω, other things being equal.
în condiţia că 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.
în condiţiile teoremei 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 The classes can be formed by the FLAMENCO
algorithm. conform criteriului By the Eisenstein's criterion, the polynomial xn-2 is
irreducible over the field of rational numbers. conform definiţiei Then, by definition, ... conform ipotezei inducţiei By the induction hypothesis, the theorem is true for f
(x). conform lemei By Lemma 2.3, the sequence gn(x) is fundamental.
conform presuruneriiBy assumption, E(1)=0. By the assumption of the lemma, the solution to (1) is positive.
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Since ... by hypothesis, (4) is a bounded sequence. conform proprietăţii 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.
conform teoremei By Theorem 1.1, we can find q and r so that b=dq+r, 0 ≤ r < d.
consecinţă consecinţă din To complete the proof of the lemma, we use the
corollary of Lemma 2. este o consecinţă The inequality f(x)≤0 is a direct consequence of the
system fi(x)≤0, i=l, 2, ..., n.
consecutiv When an expression contains parentheses within parentheses, they may be removed in succession, beginning with the innermost.
considera considerăm Consider the rational number a/b. poate fi considerată The problem can be approached in many different
ways. să considerăm Let us now consider the case a>1.
Consider now the case a>1. sunt considerate 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.
considerată 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.
poate fi considerat ca C (R+) can be regarded as a complete separable metric space under the topology of uniform convergence.
în lucrare se consideră The paper deals with the theory of distributions of finite order.
considera acum considerăm acum We now consider the process η(t).
Consider now the process η(t).
considera de la început vom considera de la început Consider first the simplest case where a=0.
We will first confine our attention to discrete-parameter martingales.
considera util
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considerăm util 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 ... The abstraction principle consists (or lies) in the
identification of objects (mathematical entities) which have a common property.
construcţie By a slight change in the method of (7), the construction of such functions is easily carried out.
construi pentru a construi The abstraction principle is often applied in
mathematics to construct new notions.
continuu In the case under consideration the process α(t) is continuous at every point of the interval (c,d).
continuă din dreapta după t 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<km.
contribui pentru a contribui la răspândirea
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.
conţine se conţine The subgroup Pn is contained in the centralizer of the
element p in B.
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conţine proprietăţi The following list contains the fundamental properties of the norm.
convenabil convenabilă In some cases the following complex form of
trigonometrical series is advantageous.
mai convenabil It is more convenient to break up the experiment into three phases.
este convenabil It is convenient to call two Gaussian integers associates if α=εβ, where ε is a unit.
va fi convenabil It will be convenient to define the convergence of fn(x) in the following sense.
deseori este convenabil It is often convenient to establish the position of points by giving their coordinates.
conveni vom conveni We shall agree to write A(x,t)∈S if A(x,t) is a process
with independent increments.
converge converge absolut The series (1) converges absolutely. converge aproape peste tot to converge almost everywhere converge în medie to converge in (the) mean converge în probabilitate to converge in probability converge uniform Thus the series (4) converges uniformly.
convinge ne convingem că 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 ...
a se convinge în To prove the convergence X n→M, it suffices to make sure that ...
trebuie să se convingă că The investigator must assure himself that the errors are within the admissible limits.
pentru a se convinge în ... 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.
pentru a se convinge în aceasta
To see this, notice that Powell's frequency function is identical with our function Φb(x).
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se poate de convins că Then, similarly as has been done in [1], one ascertains that the limit function satisfies the equation (4).
este uşor de convins că It is easy to make sure that ϕ(x) is convex on [c,∞).
coordona se coordonează 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 There is an increasing sequence of integers km such
that ... strict crescător Introduce a strictly increasing funstion g(x) defined
on [0,∞).
creşte creşte monoton după t Ft monotonically increases in t, that is Fs⊂ Ft, for
s<t.
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 Since, as we already know, all these trigonometrical
integrals vanish, we obtain formula (5). cunoscut cu The reader acquainted with the Lebesgue integrals
can interpret formula (1) in a still wider sense, viz. as the equality almost everywhere.
este cunoscut că 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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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 This relation is known as the Biot-Savart law.
cuvânt cu alte cuvinte To put it in another way, the gross weight W can be
represented as W=Wst+We+Wp.
D
da dă The repeated application of relation (2.1) yields ...
The following theorem provides the description of the everywhere dense set E.
dăm We give (or present) below a synthesis of various approaches available.
da posibilitate dă posibilitate 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 boundary-value 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.
da restul de la împărţire dă restul de la împărţire Every integer a leaves a remainder r, 0 ≤ r < m, on
division by m.
dacă 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.
dacă acesta este cazul If this is the case, the deterministic approach is quite suited for this purpose.
dacă acesta există Hence, it follows that the representation of f(x) in the form (4), if it exists, is unique.
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dacă este necesar The integer k can be, if necessary, replaced by any greater integer.
dacă n-ar fi Were is not for the discontinous coefficient,the problem would be much simpler.
dacă nu If there is no such ζ, we write max f = ∞. Unless otherwise stated, condition (2) will always be assumed to be in force.
dacă numai Thus, the function f(x) is uniquely determined, provided (that) it exists everywhere.
dacă şi numai dacă R is a realization of η(t) if and only if ... dacă şi numai dacă există 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.
dar atunci 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 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 asemeneaObserve 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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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 Applying Lemma 20.3 k times, we infer (or deduce)
that there exists a periodic function G0(x) with G0(k)=g
(x). deducem ecuaţia We derive (or deduce) the equation for the
distribution of cell size at division and the distribution of generation time.
defini definim Define unit and prime just as we did for the Gaussian
integer. definită 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 is well defined in
accordance with the above definition. definit anterior Using the Haar family gn defined above, we
introduce functions hn.
pozitiv definit A complex-valued function f(x) is called positive definite if for any n and complex numbers zj
definire în definirea In defining K(x) we have made use of our assumption
that b(x)=0 for x<a.
definiţie In the definition of the mth derivative we can replace the polynomials pn(x) by functions fn(x).
din definiţie From the definition it follows at once that G(u) satisfies condition (2).
definiţie echivalentă It is possible to give an equivalent definition of integral (1) as the limit of sums (2).
definiţie exactă A precise definition is given below.
demonstra demonstrează This proves that d is a factor shared by both a and b. pentru a demonstra To prove the theorem, it suffices to show that ... este demonstrată Since Nρ<Nα, the inequality (3) is established.
pentru a demonstra 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.
nu este ce demonstra If p|a, there is nothing to prove. 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 teorema
Before proving the theorem we illustrate its applications.
trebuie să demonstrăm că We must prove that the two factorizations differ at most in the order in which the primes appear.
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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fără demonstraţie 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 ...
deoarece în caz contrar 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).
deoarece ..., atunci 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.
deosebi se deosebeşte prin The function f(x) differs from F(x)(k-1)by a constant. se deosebesc prin We must prove that the two factorizations differ at
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 To this end, we establish first that there exists a
constant L not depending on s and n such that ... nu depinde de alegere The distribution (1) does not depend on the choice of
the indefinite integral ψ(x). depinde esenţial de In this new definition the fact that the choice of the
integer k depends on the interval [a,b] is essential. depinde monoton de t
Let (2) be a family of finite stopping times which are
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monotonically dependent on t and right-continuous in t.
deriva deriva ... după x To differentiate v(t,x) with respect to x. deriva ... în raport cu x To take the derivative of v(t,x,u) with respect to x. derivând In fact, by differentiating (3) we obtain ...
Hence, by successive differentiation we obtain ... derivând de k ori Differentiating formula (8) k times, we obtain the
equality ... derivând succesiv Hence by successive differentiation we obtain ...
derivata după x A derivative of u(t,x,u) with respect to x.
deschide deschide posibilităţi 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 raport cu x
Expand function u(t,x,u) into a Taylor series about 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 Any supermartingale ζ(t) has exactly one decomposition of the form ζ(t)=µ(t)-α(t).
descompunere în 6 = 2*3 = 3*2 are the only factorizations of 6 into positive prime factors.
descompunere în fracţii elementare
The decomposition of a fraction into partial fractions.
descompunere în serie 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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descris anterior We now show how the method described above can be extended to move the existence of µ.
descrisă de ecuaţie We will consider only those systems whose evolution is given by the equation of form (8).
desemna desemnează The symbol Fn(x)→ will denote that the sequence
Fn(x) converges almost uniformly to a function.
desen pe desen In Fig.11 the planes π1 and π2 are shown as coinciding.
din desen se vede că In Fig.10 it is seen that ... în limitele desenului In this particular case only one side of the triangle,
namely BC, has traces within the limits of the drawing.
deşi It is possible that the ordinary derivative does not exist, although the distributional derivative has a value at this point.
detaliat mai detaliat It is necessary to consider this problem in greater
detail.
detaliile 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.
determina determină Two sequences fn(x) and gn(x) determine the same
distribution if and only if they are equivalent.
determinat cu exactitate de 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.
deveni devine 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.
dezacord This explains the discrepancy in behaviour of the two functions.
diferenţă
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unică diferenţă între The only difference among them is the order of appearance of the factors.
diferi diferă de The function f(x) differs from F(k-1)(x) by a constant.
The proof is not like that of Theorem 5.6. diferite de Consider the numbers which are different from 0 and
±1. The prime numbers p possess no factors other than ±1 and ± p.
dificil este mai dificil It is slightly more difficult to see that
dificultate dificultate poate fi trecută 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 The interpretative approach opens perspectives in that
direction.
discontinuitate de speţa întâia [a doua]
With probability 1 ξ(t) has no discontinuities of the first [second] kind.
discuta discutăm In Chapter 8 we discuss the filtering problem. nu va fi discutată The validity of this assumption will not be debated
here.
distinge pentru a distinge 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
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the product of any pair is congruent to 1 modulo p. se divide The procedure is then broken down (or divided /
subdivided) into the three steps: ... se divide prin Nine is a multiple of three.
divizibil sunt divizibile prin All the terms on the right are divisible by p.
domeniu de definiţie Let A be a strongly positive operator whose domain of definition does not depend on G.
dori dorit Passing to the limit in (6) we obtain the desired result.
This is exactly the wanted decomposition. este de dorit It is desirable in many applications that the function Xt
be also measurable.
după ce 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 From equation (2) we find that ... ecuaţie cu n necunoscute the equation in n unknowns ecuaţie de gradul n the equation of degree n ecuaţie iniţială the input equation ecuaţie în raport cu (necunoscuta) x
the equation in (the unknown) x
ecuaţie rezolvabilă în raport cu x
the equation solvable for x
editarea The publication of this monograph was made possible due to ... He has over 70 published works.
efectua pot fi efectuate The calculations can be carried out (or performed) in
the same way as in Section 15. de a efectua transformarea On many occasions it is useful to take the Laplace
transform of equation (15).
egal sunt egali The vectors and are equal in magnitude.
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elabora era elaborată The statistical theory has been developed quite
recently.
elementar The solution of equation (5) is obtained in an elementary manner by expanding function x(t) in a Fourier series.
elibera să ne eliberăm de 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.
este evident 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+.
eveniment We note that any subset of Ω is called an event. eveniment sigur 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 To avoid possible confusion, the continuity condition
will be restated in each case. pentru a evita neînţelegeri To avoid misunderstanding, we may call the first kind
of intervals (including end-points) the closed intervals and the second kind the open intervals.
pentru a evita repetări în viitor
To avoid repetitions in future, we present here basic concepts and definitions.
exact exact unul Any prime Π divides exactly one positive rational
prime p. mai exact More precisely, we shall assume that the following
condition is satisfied.
exactitate cu exactitate de
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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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 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.
exclude se exclude The second of these possibilities is excluded, for r(x) is
in T. exclude posibilitate This approach excludes the possibility of considering
the equations having solutions which belong to a more general class of numbers.
exemplu Another example of the abstraction principle is the Cantor definition of a real number. Let us first look at some simple examples.
de exemplu 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.
prin exemple The nature of this problem can best be made plain by examples. We shall explain it by means of examples.
exemplu interesant We shall construct an interesting example of such a process in this chapter.
exemplu tipic The polynomials Bn(t) are typical examples of generalized Brownian functionals.
unele exemple Some other examples of fundamental sequences will be given after the following lemma.
în calitate de exemplu As an example, consider the primitive polynomials x2+3 and 3x2+7x-11.
exista există limita With probability 1 there exists the limit
.
existenţă The existence of the Wiener process follows easily from the Kolmogorov's criterion.
explica
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vom explica We shall explain the essence of the problem by giving several examples.
pot fi explicate The variations are greater than can be accounted for by chance.
pentru a explica 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 ... To express the solution of equation (2) in terms of w
(t), it is sufficient to observe that w(t) satisfies equation (3).
să exprim recunoştină I wish to express deep gratitude to O. A. Oleinik for her valuable advice and her interest in my work.
expunere 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.
extinde va fi extinsă The definition of a stochastic integral will be further
extended to cover the alternating functions. extindem ... asupra We now extend the definition of a stochastic integral to
a wider class of random processes. de extins în cazul In §21 we show how to extend the basic definitions
and theorems to the case of distributions of infinite order.
extinde rezultatele de a extinde rezultatele The main purpose of this section is to extend the
results of the preceding section to the processes of a continuous argument.
rezultatele pot fi extinse These results can be extended to all other cases.
extinderile noţiunii de 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 Evolution may be defined as the process of finding a root of a number, or as the inverse of involution.
extremitate cu extremităţi In every closed subinterval with ends a and b there is at
most a finite number of poles.
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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 ... să facem câteva observaţii referitor
Before presenting the main results, we want to make a few comments on the notion of entropy.
face de prisos The one-to-one correspondence established above makes it unnecessary to distinguish between function f(x) and distribution [f(x)].
face referinţă de a face referinţă la To complete the proof, it is sufficient to refer to the
Cantor theorem on uniform continuity. facem referinţă la For the proof of this theorem, we refer the reader to
Theorem 9. poate să facă referinţă la 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 In case of right-continuity the condition (2) may be
replaced by the requirement (i).
fi de prisos este de prisos 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.
fi în relaţie sunt în aşa relaţie încât 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.
fi în stare suntem în stare We are now in a position to prove the fundamental
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theorem stated in the opening paragraph of the chapter.
fi recunoscător ... pentru ... sunt recunoscător ... pentru ...
I am particularly indebted to Miss Leila R. Raines for her painstaking assistance in the revision and preparation of the manuscript for publication.
fie 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.
acum fie Now set .
fie că este dat Let there be given a continuous increasing process ξ(t).
fie dat Given point A,construct point B symmetrical to the given point A in relation to plane π. Given any ε>0, we fix an index n0 such that | fn(x) - f(x) |<ε for all n>n0.
fiecare fiecare dintre care Let ξ1, ξ2, ..., ξn be a sequence of random variables
each of which is the sum of certain number of mutually independent random variables.
pentru fiecare din M is the set of processes mt for each of which there exists a random variable m such that ...
fix acelaşi 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 Indeed, let c>0 be an arbitrary fixed number. este fixată A quantity whose value is fixed in any investigation is
called a constant.
folosi 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].
se foloseşte 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 ... Let us now make use of the fact that the trajectory of process is right-continuous.
font aldin The title should be printed in bold type.
forma formează The numbers 0, 1, 2, ..., p-1 form a complete residue
system modulo p. a forma ecuaţia to set up (or to formulate / to form / to write) the
equation
formalizare în formalizarea 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.
formă de formă The function η(t) also admits decomposition of the
form (4.3). An integral of this form can be evaluated by substitution.
în formă Write equation (3.2) as (or in the form) ... în formă explicită To represent the solution of equation (12) explicitly, it
is necessary to solve equation (8) for u(t). într-o formă concisă An equation that states a rule in a brief form is called a
formula. într-o formă mai simplă More recently some of results were obtained in a
simplified form.
formula să formulăm Let us formulate the conditions under which the
statement is valid.
formulare The new statement of the Fundamental Theorem is this.
formulă formula schimbului de variabilă Apply now the change of variables formula to (1).
formulă iniţială the assumption formula
funcţie funcţii absolut continue The class of indefinite Lebesgue integrals of locally
integrable functions is the class of absolutely continuous functions.
funcţie caracteristică EΩ(x) is the characteristic function of set Ω, equal to unity inside Ω and to zero outside Ω.
funcţie inversă the inverse function
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G
garanta garantează This condition ensures (or assures) the uniqueness of
the solution.
găsi găsim From (5), (6) and (7), we find ... se poate găsi The facts used below can be found in [5]-[7].
greutăţi mari The estimation of ν runs into severe (or great) difficulties.
genera generează Here w(t) denotes the process which generates the
current Ft.
generată Fs is a σ-algebra generated by the events ξ(s)∈B.
general mai general 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).
generaliza pentru a generaliza To generalize the theorem, it is necessary to modify the
definition of convolution. poate fi generalizat We now show how the method described above can be
extended to prove the existence of µ. poate uşor fi generalizat Theorem 6 can be easily generalized.
generaliza ... pentru cazul Let us generalize equation (6) to the case under discussion.
poate fi generalizată pentru cazul
This theorem may be generalized to the case of multiplicative error.
generalizare Distributions are a generalization of functions.
grad de gradul n pn(x) is a polynomial of degree n.
grupa grupăm termenii asemenea To solve the equation 2x+x=12, unite (or group /
collect) similar terms in the first member to obtain 3x
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and then divide both members by 3 to find the value of x to be 4.
I
imediat Using Lemma 2 we can prove at once that G(u) satisfies condition (ii). It follows at once from 12.4 that ...
imita vom imita 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.
implica această implică This implies the completeness of hn in L0
1[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.) 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.
sunt incluse Some elements of artificial intelligence are now incorporated in system.
inclusiv 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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including the moment t.
independent By a slight change in the method of [3] the reader can easily prove the theorem by himself.
independent de Let θ be a random variable independent of the σ-algebra. 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.
mutual independente Let ξ1, ξ2, ..., ξn be a sequence of random variables each of which is the sum of certain number of mutually independent variables.
indica indică că These results indicate (or suggest) that ...
indiferent este indiferent It makes no difference whether we say that t is a
stopping time relative to Ft or F't.
imposibil este imposibil 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.
inducţie după inducţie We shall prove the theorem by induction.
Define the random variables Mn inductively.
inducţie după k We shall prove the relation by induction in k.
infinit la infinit The value of a function f(x) at infinity, written f(∞), is
the limit of this function as x →∞. până la infinit to infinity infinit mic
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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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 the definite integral integrală improprie If g(x) is a locally integrable function and the
improper integral exists, then g(∞)=0
provided g(∞) exists. integrală nedefinită By an indefinite integral of a function f(x) we
understood each function ψ(x) such that ψ'(x) = f(x).
integrant the integrand (or the expression under the integral sign)
integrare după integrare Replacing x by x+t in (1), we have, after integration,
... integrare prin părţi the integration by parts
interesant este interesant de observat că
It is interesting to observe that the above formula remains true if the summation is understood according to the method of arithmetic means.
intra intră în This term appears in (or enters into) equation (9) in
parentheses. care intră în ... The coefficient a*(x,s) entering into (or involved in)
the expression for I t can be assumed to be constant.
introduce introducem Introduce the sequence ann ∈ N of positive numbers.
Allow sinu within the differential sign. introducând Introducing 2n new variables ξi and ζi we deduced 2n
equations. a fost introdusă The notion of distribution was introduced to
mathematics by S. Sobolyev in 1936. sunt introduse Data are entered (or fed) into the computer from the
keyboard.
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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 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.
în afară EΩ(x) is the characteristic function of set Ω, equal to unity inside Ω and to zero outside Ω.
în aplicaţii In applications we often come across the so-called Heaviside function.
în aşa mod încât 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.
în concordanţă cu In accordance with the law of large numbers, a ≈ mξ = x.
în conformitate cu In accordance with (or by) the law of large numbers, a ≈ mξ = x.
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în continuare Next, define M∞ as the limit of Mt , as t→∞.
în fond Basically, the difficulty with establishing inequality (6) is that ...
în grupe The cards may be processed singly or grouped (or in groups).
în interior EΩ(x) is the characteristic function of set Ω, equal to unity inside Ω and to zero outside Ω. All points of Ω lie in the interior of y2<d.
în întregime We will not attempt here to solve the problem in its entirety. Though the macroscopic properties of the cell population as a whole follow a deterministic growth law, the descriptors of individual ceils (such as size at division and age at division) are highly variable. The contents of this book may not be reproduced either in whole or in part.
în loc de această Instead, we will assume that ... în mare măsură The theory is based, to a large extent, on heuristic
considerations. în realitate Actually, we shall prove somewhat more that we need
for the present. în sfârşit Finally, to prove (5), suppose p is prime. în spiritul The paper is written in the spirit of the Wilkinson's
error analysis. în totalitate The graphs of these sequences do not intersect and in
totality there sequences absorb all jumps of the process Z.
în totalitate The graphs of these sequences do not intersect and in totality there sequences absorb all jumps of the process Z.
în vigoare Unless otherwise stated, condition (2) will always be assumed to be in force.
în virtutea In view of Theorem V, the sequences from Example 2 converge. By virtue of the basic assumptions, ... Right-angle projection is most widely used because of its comparative simplicity. On account of the foregoing remark, the distributions in the earlier sense can be identified with distributions of finite order. By 2.4, the limit of an(x) - gn(x) is also a polynomial of degree less than l.
în caz In the case discussed above the left-hand side of (1) can also interpreted as ...
în caz contrar In the opposite case the point x0 is said to be singular.
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Moreover Np=0, for otherwise p≠0 so that Π=α, contrary to assumption.
în caz general This derivative is, in general, not a function, but a distribution.
în caz particular 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.
în caz special In the special case where the L2-martingale is continuous, the variation <M> will also be continuous.
în cazul acesta In this case ... în cazul când 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.
în cazul considerat In the case under consideration the process α(t) is continuous.
în cazul dacă In the case of a negative determinant a quadratic equation has no roots.
în cazul limită In the limiting case the stochastic process ν(t) becomes a constant.
în cazul mai sumplu In the simpler case where α=1 this relation takes the form p=he2/3.
în acest caz particular In this particular case only one side of the triangle, namely BC, has traces within the limits of the drawing.
în cel mai bun caz This problem has received at best only fragmentary treatment.
în multe cazuri For many purposes (or in many cases / on numerous occasions) we shall need the following assumption.
în mod în mod analogic 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 arbitrarThe first term can be made arbitrarily small by an appropriate choice of ε. The variable, to which the values may be assigned at
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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 Distributions may be, in an intuitive sense,
approximated by functions. în sens larg The reader acquainted with the Lebesgue integrals can
interpret formula (1) in a still wider sense, viz. as the equality almost everywhere.
în sens uzual A sequence of constant functions converges in the distributional sense if and only if it converges in the usual sense.
în sensul 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.
în sensul că 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.
în sensul larg in a broad (or wide / loose) sense în sensul nou So α has n conjugate in the new sense, but m in the
old. în sensul următor The converse holds in the following sense. în sensul vechi 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.
într-un anumit sens 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.
înainta a fost înaintată A new theory has been put forward this year.
We were informed that a new idea had been advanced at the closing session.
înainta o ipoteză to advance a hypothesis
încă Many roadblocks must still be overcome before all the subgroups of G are described.
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încă mai mult 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.
încă unul Let us formulate one further assumption. To conclude, consider one more example.
începe vom începe cu To find the sum of the series, we start from the known
formula (1). se începe Each command begins with a forward slash (/).
început de la început We first establish the necessity of the condition.
First let us define ...
încheia demonstrarea pentru a încheia demonstrarea To complete the proof, it suffices to observe that ...
îndoială fără îndoială 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 F∞ is replaced by Ft if t∈[0,T ].
poate fi înlocuit prin The integer k can be, if necessary, replaced by any greater integer.
înlocuit ... prin Replacing in (2) α by β and afterwards x by αx we obtain ...
înlocui a prin b to replace a with (or by) b to substitute b for a to change a for b
se poate de înlocuit In case of right-continuity the condition (2) may be replaced by the requirement (3).
putem înlocui prin In the definition of the mth derivative we can replace the polynomials pn(x) by functions fn(x).
însemna înseamnă A(C) denotes the completion of B(C) with respect to
µν.
va înseamnaThe letter M, with subscripts, will denote positive constants. 'ln' signifies the natural logarithm.
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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.
această înseamnă că This means that a is different from any of a1, a2, ..., an.
întâmpina dificultăţi întâmpinăm dificultăţi 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.
întoarce ne întoarcem We turn to the proof of the theorem.
între 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 ...
întreaba se întreabă It is asked what properties the limit distribution
function must possess.
întrebare considerată These ideas can bring advantages in attacking the problem under consideration, over the usual theories.
întrece care nu întrece 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 Practically it suffices to understand integrability in the sence of Riemann or of Lebesgue.
înţelegem By the integral of the P-measurable function f we mean the Lebesgue-Stiltjes integral.
prin ... vom înţelege ... 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).
prin ... se înţelege A current is understood to be an increasing family of σ-algebras such that ...
înţelegere profundă This information is needed for a fundamental understanding of (or a deep insight into) the properties of ...
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J
jos mai jos The operations introduced below will be identical with
the usual operations on functions. de mai jos The definitions given below are for the processes
defined on R+.
juca un rol va juca un rol decisiv The following fact will play a decisive role in the
proof. joacă un rol important The class of normal functionals plays an important
role from the viewpoint of the causal calculus.
just Let us formulate the conditions under which the statement is valid (or true).
L
la fel la fel ca în cazul This is proved in the same manner as in the case of
discrete-argument submartingales. la fel ca şi 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 The rest we leave to the reader. va fi lăsată cititorului The slight change in the original proof which is
needed here will be left to the reader.
lega este legat de The estimation of ν runs into (or entails) great
difficulties.
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
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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.
este limitată de Accuracy is limited only by that of the original data. ne vom limita la cazul 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 ... fără a lua în consideraţie ... The analysis has been made without reference (or
without regard) to heat losses. Regardless of the last term which is sufficiently small, equation (9) may be solved in radicals for x.
dacă se ia în consideraţie că The statement becomes apparent when it is 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 că
One should keep in mind (or it should be kept in 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 This chapter deals mainly (or chiefly /
predominantly) with semigroups. mai mic decât Now p is in T and has norm less than N(π). mai mult ca atât Moreover, every solution of (9) may be represented in
the form (3).
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mai mult decât ... Over (or more than) two thousand scientists took part in the 1st World Congress.
mai multe ... decât The set B contains much more elements than the set A. The set B contains twice as many elements as the set A.
margine margine inferioară exactă the greatest lower bound margine superioară exactă the least upper bound
matematică matematică aplicată applied mathematics matemitică finită Finite mathematics is considered as superior tool to
use, over differential equations. matematică pură pure mathematics
material In order to achieve this, we shall need some material from elementary number theory.
material bogat despre 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 We can restrict the variable so that it shall take on
only values lying between a and b. mărginit 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
The first thing to notice is a wide variety of concepts of entropy in the scientific literature.
menţionămWe note that any arbitrary subset of Ω is called an event.
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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 We shall modify slightly the definition of ... modificând puţin metoda By a slight change in the method of (3) the reader
can easily prove the theorem by himself.
modificare a metodei This is a modification (or an adaption) of the method of steepest descent.
moment din acest moment From now on, we assume B to be a complete
separable metric space. From this time on, the density function decreases.
până la momentul 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.
până la acest moment Up till now no assumptions have been imposed concerning the convergence of ψ(t).
în momentul t în punctul x Let a(t,x) be a velocity of the particle at time t at point x.
multiplu Twelve is a multiple of three. multiplu întreg 12.6 is a whole (or an integer) multiple of 4.2.
mulţime mulţime de măsură 0 The σ-algebra A is generated by the P-null sets. mulţime dintr-un singur element
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 ...
neapărat nu neapărat
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In general, φe and φ* need not be identical.
Let α(t) be an increasing (but not necessarily adapted) integrable process.
necesar Hence one of the factors of f(x) necessarily has the same degree as g(x).
necesară We are going to show that the function F(x) has the required property.
este necesar In some problems of the filtering theory, it is necessary to work with Wiener processes on R.
este necesar ca 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.
necesar pentru ca 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.
cu două necunoscute Thus we obtained an equation in two unknowns.
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
of which is homologous to zero. nici un No s∈D is an element of Γ. nici ... No linear combination of is homologous to zero.
nici ..., nici ... The number 0 is neither positive nor negative.
nota vom nota We denote by Λ the set of all points of discontinuity of
F(x). să notăm prin 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.
este notat prin The area of the triangle FGH is denoted by S. This volume is designated as V.
sunt notate prin Variables are denoted usually by the later letters of the alphabet.
pentru a nota The symbol f(x) is used to denote a function of x.
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notaţie Let us introduce the following notation
where
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.
notaţie standardă the standard notation
noţiune 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.
noţiuni şi definiţii de bază basic concepts and definitions
nu 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 = ∞.
nu ..., ci ... 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.
nu numai 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.
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număr număr întreg 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 Let 1 and -1 be called units.
We shall call r the radius of the set D. se numeşte 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.
se va numi 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 Notice that U is not unique. se observă că 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<t .
este util de observat 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.
de la început observăm că Observe first that ...
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trebuie de observat că 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.
obţine obţinem 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.
se obţine This function is obtained by application to expression (4) a nonlinear differential operator Π.
obţine forma In this case equation (7) becomes (or takes the form / assumes the form) ...
obţinem în rezultat After k similar steps we finally obtain ... obţine valori Let f(x,t) take on (or assume) values in Rm.
ocupa ocupă poziţie în spaţiu Two lines may occupy the following relative positions
in space: 1) they may be parallel, 2) they may be intersect or, 3) not intersect.
ocupă un loc special The prime numbers occupy a special position among integers.
omite putem omite
Since and n > 200, we may drop the last term on the right of (19).
omitem 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.
este omis The proof is omitted here.
operaţie operaţii aritmetice asupra 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.
operaţie inversă 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.
ordine în ordine aleatoare Numbers may be added in any order. ordinea integrării To prove this, interchange the order of integration in
the double integral in (6).
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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, figurate]
The signs of aggregation are the parentheses (), the brackets [] and the braces .
paranteze pot fi deschise When an expression contains parentheses within parentheses, the parentheses may be removed in succession, beginning with the innermost.
parcurge parcurge valorile 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.
parte parte componentă 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.
parte dreaptă The right-hand side of equation (12). parte întreagă The integer (or whole) part of a real number is the
greatest integer not exceeding the number. parte stângă The left-hand side (or member) of equation (12). partea fracţionară The fractional part of 5.6 is 0.6.
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în partea dreaptă The term on the right-hand side (or on the right) of (12) vanishes as k→∞.
în partea stângă The term on the left-hand side (or on the left) of Eqn (12) vanishes as k→∞.
pe de altă parte 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 Absolute constants retain the same value in all
problems. vom păstra 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.
sunt păstrate 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.
primi fi primit 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.
pentru pentru aceasta 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.
pentru ca 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.
pentru care It is therefore reasonable to ask for which classes of "integers" the fundamental theorem holds, and for which it does not.
pentru comoditate For convenience we define a continuous locally increasing process to be an adapted process which has properties (1) through (4).
pentru nici un The function cannot be constructed for any value of x.
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pentru simplitate 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 The function H(p(x)) is everywhere equal to 0. peste tot densă To determine IT (Ft), it is sufficient to describe the
everywhere dense set IT2(Ft).
principial nou We will describe a radically (or fundamentally) new approach to the problem.
probabilitate cu probabilitatea With probability 1 there exists the limit
.
polinom de By a polinomial in x1, x2, ..., xn we mean a sum of the form (6).
poseda posedă 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.
posedă o proprietate 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 To avoid possible confusion, we shall use this
notation ... The error, if any, will be allowed for by visual estimation.
este posibil ca It is possible that the ordinary derivative does not exist, although the distributional derivative has a value at this point.
este întotdeauna posibil de făcut
This we can always do by choosing s and t as rational integers nearest to A and B respectively.
practic Practically it suffices to understand integrability in the sense of Riemann or of Lebesgue.
precedent precedentă
On account of the foregoing remark, the distributions in the earlier sense can be identified with distributions of finite order.
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This follows from the previous theorem. nivele precedente 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 It is assumed that σ-algebra F0 is completed by all P-
null sets. All stochastic processes considered in this book will be assumed to be measurable. Ft is assumed to be right-continuous.
vom presupune că 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.
presupunem contrariul Suppose the contrary.
presupunere presupunerea că In defining K(x) we have made use of our assumption
that b(x)=0 for x<a. This equation contradicts the assumption that G contains points not belonging to D.
presupunerea conform cărei The assumption under which formula (10) is valid can be somewhat weakened.
presupuneri sunt făcute The following assumptions are made concerning the random process ξ(t): 1) the process is Ft-adapted and 2) it satisfies condition (1). Up till now no assumptions have been imposed concerning the convergence of ϕn(t).
prezenta în forma poate fi prezentat în forma Any potential ζt can be represented in the form ...
prezenta un interes prezintă un interes Of interest also are the induced statistics.
The results have encouraged (or aroused) a more serious interest in precision methods of control.
va prezenta un interes I hope this paper will be of interest to you. prezintă un interes aparte To prove the existence of <a,a>t, the following lemma,
which is of interest in its own right, will be required. care prezintă un mare Another theoretical approach of great interest consists
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interes in looking at the evolution toward equal phenotypic fitness.
prin prin alegere This we can always do by choosing s and t as rational
integers nearest to A and B respectively.
probabil puţin probabil These conditions are unlikely to occur in practice.
probabilist There is presented below a glossary translating set-theoretic 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 Ftt≥0 if for every t the random variable ξ(t) is Ft-measurable.
provine provine de la The word 'school' has its origin in the Greek word
'schole' meaning 'leisure'. provin de la The arithmetic symbols were derived from the Arabs
and the Hindus.
punct în punct The function has no a
derivative at the point 0. în puncte 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 From the logical point of view, this generalization is
exactly of the same type as the generalization of rational numbers in the Cantor theory.
din punct de vedere practic Several other functions, very useful from the practical point of view, are considered in the following section.
punct de referinţă Here the starting point is the notion of fundamental sequences of rational numbers. Approximation is our starting point for the definition of distributions.
pune în corespondenţă operatorul pune în The operator θ, assigns to (or associates with) any
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corespondenţă function from D a function from D(−∞,0].
este pusă în corespondenţă A probability is assigned to each outcome.
purta numele poartă numele It is instructive to study the Ito's method of defining the
stochastic integral that now bears his name.
putea nu putem If n=1, we cannot impose this condition. nu se poate 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 There are doubtful points in the above reasoning. Following the argument of N. N. Bautin, we can now easily show that ...
raţiona raţionând analogic Reasoning similarly we convince ourselves that the
solution to Eqn (7) is also unique. raţionând ca Reasoning as in the proof of Theorem 7, we can
prove, under suitable assumptions, that relation (1) holds for s>Nα.
rădăcină rădăcină cubică din 2 is the third, or cube, root of 8. rădăcină de ordinul n din indicates the nth root of a. rădăcină din număr When the factors of a number are all equal, any of
the factors is called the root of the number. rădăcină patrată din 5 is the square root of 25.
rămânea rămâne 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 de demonstrat Theoretically, it still remains to prove for each particular system considered that an equilibrium actually exists and is stable.
rămâne în vigoare Theorem 4 and Lemma 9.5 remain true. This result holds true if we proceed from the group G to any of its subgroups.
răspuns The answer to the question must be postponed until later.
reaminti reamintim Because of its great importance in this book, we
recall the definition of a normal distribution. reaminteşte că 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.
trebuie de reamintit că It should also be recalled that R(t) is almost everywhere nonzero.
reciproc şi reciproc Equal chords cut off equal arcs, and the converse.
reduce reduce la This fact reduces the construction of the solution to
(2) to that of a process such that
is a martingale.
se reduce la Under these conditions equation (3) reduces to (4). This investigation amounts (or boils down) to finding the operator Uγ such that Uγµ=UγUµ.
reduce ecuaţia la formă to reduce the equation to the form ...
reduce termenii asemenea reducem termenii asemenea 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.
referi se referă la The Henry's Law refers only to the effect of
pressure. nu se referă la Evidently the Einstein's restriction does not apply to
the case under consideration. care se referă We now state a number of results concerning
ordinary differential equations. ceea ce se referă 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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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.
relativ la In relation to Ω, an experiment is completely characterized by a certain class of events.
remarca este cazul de a remarca It is relevant to remark that the diffusion
corresponds to the increase of entropy.
repeta poate fi repetată This procedure can be repeated with q2, q3, ... until
all the prime factors on one side are exhausted. repetată The repeated application of this procedure proves
the assertion.
reprezenta reprezentat în formă 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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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 It is convenient, in solving problems, to use letters
for the numbers which are sought.
rezulta rezultă din 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.
rezultă imediat The proof follows immediately if we observe that ... rezultă nemijlocit Since u(t) is continuous, it follows at once that x(u
(t),t) satisfies condition (ii). această rezultă din This results from the following lemma.
This follows from the fact that e πi/2 = i. din cele spuse rezultă că From what has been said it follows that ...
It follows from what has been said that T>0 almost surely on the set B.
din ... rezultă că 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.
de aici rezultă că 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.
nemijlocit rezultă din It follows directly from the definition that ... This assertion follows immediately from relation (2).
ridica ridica la puterea n to raise to the nth power ridica un număr la cub to cube a number ridica un număr la patrat to square a number
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ridicarea la putere The process of finding any required power of a number is called involution.
S
satisface se satisfac condiţiile Condition (4) is equivalent to the following
conditions being satisfied.
sau ..., sau Since a satisfies (1), we have that either a=1 or a=p-1.
schimb schimb de variabilă Relation (8) follows from the general rule of the
change (or the substitution) of variables in integral calculus.
schimb de variabilă la integrare the change of a variable in integration
schimba schimbăm ordinea de integrare
To prove this, interchange the order of integration in the double integral in (6).
fără a schimba The introduction of locally integrable allows us to strengthen Lemma 6.8 without changing its proof.
este schimbat semnul Any term may be transposed from one member of equation to the other, provided its sign is changed.
schimbare schimbare mică A slight change in the original proof will be left to the
reader. schimbarea ordinii de integrare
The main point of the proof is to justify the interchange of the order of integration in (8.4).
fără schimbări 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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and, at the same time, to relax the restrictions on M. scopul acestei lucrări The purpose of this paper is to present the
elementary exposition of the theory of distributions. scopul constă în ... 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 The function F(x) can be written in the form ... putem scrie Then we can write for the division-size density
function (up to a normalization factor) .
scriind Writing down these inequalities for n, n+1, ..., n+m and adding them we obtain the estimation ...
vom scrie Write down the coordinates of the points A, B and C. If there is no such G, we write max f = 0.
scriem ... în formă Write the equation as (or in the form) (2).
scriere în scrierea In writing numbers, the figures are separated into
groups of three figures each, called periods.
semn cu semn opus Subtracting any number is equivelent to adding it with
its sign changed (or with reversal of its sign). semnul opus The numbers +2 and -2 have unlike signs.
The number -2 is opposite in sign to +2. sub semnul The products under the sign of Σ are the products of
continuous functions. sub semnul integralei under the integral sign
sens 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.
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.
similar ca The definition is the same as the one given above, but with R instead of [0,T ].
simplifica simplificăm Simplify 4a-(2a-2b). simplifică Thus the introduction of distributions makes the
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calculation easier. pentru a simplifica To simplify the equation, make the substitution
u=2x2-1. poate fi simplificată This equation can be simplified by the substitution
u=2x2-1.
simplifica fracţie prin a To reduce the fraction by the factor a.
simplifica prin putem simplifica prin We can divide out (p-1)! from each side to obtain the
conclusion.
sistem sistem de notaţii the system of notation sistem zecimal 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.
situa situate aproape This quantity may take on any values, however closely
spaced.
slăbi putem slăbi Hence, we can weaken condition ϕ in the following
way. pentru a slăbi restricţiile referitor ...
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.
soluţie soluţie unică Equation (7) has a unique solution.
spaţiu spaţiu de stări A function ξ:Ω→S is called an S-valued stochastic
process and S is called the state space of ξ. spaţiu metric separabil complet
C (R+) can be regarded as a complete separable metric space under the topology of uniform convergence.
spaţiu-fază 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.
specifica cum a fost specificat mai sus As has been stated (or specified) above, the
functions pm(x) defined by (1.6) are bounded.
dacă nu este specificat contrariul We agree to assume, unless otherwise specified, that
all processes are adapted. Unless otherwise stated, condition (2) will always be
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assumed to be in force.
spune spune (ind. prez. 3 sg.) 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.
se spune If conditions (i) and (ii) are also fulfilled, then the current is said to satisfy the 'ordinary' conditions.
vom spune If the members of a pair are equal, say to a, then a2=1 (mod p).
vom spune că ... We shall say that two integers a and b are relatively prime if they share no factors except ± 1.
spunând The function F is, so to say, the optimal cost of the continuation of control.
spunând mai exact To put it more exactly, ... din cele spuse mai sus From what has been outlined (or said) above the
following conclusions may be drawn.
stabili stabileşte This requirement places (or sets) the upper limit on
the admissible error. stabilim We first establish the necessity of the condition. a fost stabilit This fact was ascertained (or established) from
experience. a fost stabilită The average value was found to be 16 ± 3. de a stabili condiţii Theorem 1.6 allows to lay down the conditions under
which the solutions to equations (6) are continuous in parameter z.
strict Approximations, strictly defined, is our starting point for the definition of distributions.
studia studiat The experiments studied in probability theory are
carried out when a certain set of conditions Y is satisfied.
substitui substituim For this we make the substitution t-x=s. substituind ... în 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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suficient This equation determines a sufficient condition. suficient de mare Whatever t may be, one can always find a sufficiently
large N such that ... Now ECt < ε/2, if n is taken large enough.
suficient de mic The term can be made sufficiently small by makingthe partition of the interval [0,T ] fine enough.
este suficient By theorem 2.1, it suffices to prove the necessity. este suficient de arătat că ... It suffices to show that if α is a Gaussian integer, so
is β and hence α+β. este suficient de spus că ... Suffice it to say that ...
It suffices to say that ... It is enough to say that ...
este suficient pentru ca In order that the series (1) of independent random variables converge, it is sufficient that the series (2) and (3) converge.
pentru această este suficient ca
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.
suma parţialăA series is said to be convergent if the
sequence of its partial sums fn(x) = g1(x) + ... + gn(x) converges.
sumar A survey of the general theory of stochastic processes can be found in [4].
supune se supune legii The distribution of errors adheres to the law of
probability. The kinetics of the system obey (or follow) the ideal gas law.
se supune ecuaţiei 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
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with R instead of [0,T].
T
tăia taie Equal chords cut off equal arcs, and the converse.
târziu mai târziu We shall show later on that this generalization contains
also class of non-continuous functions.
teoremă teorema despre valoare medie the theorem of the mean
teorema probabilităţii totale the theorem of total probability teoremă preliminară Some preliminary theorems will be established first. teoremă reciprocă The converse theorem in not true.
teoretic Theoretically, it still remains to prove for each particular system considered that an equilibrium actually exists and is stable.
teorie teoria jocurilor the theory of games teoria mulţimilor the theory of numbers (or the set theory) teoria numerilor the theory of numbers (or the number theory) teoria probabilităţilor the theory of probability teoria proceselor stocastice the theory of stochastic processes teorie unitară the unified theory teoria ce va fi dezvoltată Continuous functions on a fixed interval are the starting
point for the theory to be developed here.
termen Any term may be transposed from one member of equation to the other, provided its sign is changed.
termen cu termen Roughly speaking, each convergent series can be differentiated term by term (or termwise).
termina termină This completes (or concludes) the proof of Lemma 4
and also that of Theorem 2. am terminat Then r=s and we are done.
terminologie în terminologia Ut is the 'dual predictable projection' in the
terminology of Dellacherie and Meyer [42].
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timp în timp ce 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).
tinde la când ... tinde la Each summand on the right of (12) becomes
vanishingty small with (or as) n→∞. când ... tinde la infinit 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.
toţi următorii Every succeeding (or subsequent) member an must be less than aN.
nu toţi However, not all distributions can be represented in the form [f(x)], i.e., not all distributions can be identified with continuous functions.
pentru toţi The function ν possesses the property (1) for all ω and t.
transforma a transforma ecuaţia to rearrange the equation să transformăm la formă Let us transform (1) to the form (2). a se transforma în infinit 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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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.
care trebuie 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.
ceea ce trebuia But p is of the form 1+4n, so that a p=l+im, as required.
ceea ce trebuia de demonstrat
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.
trece trecând la limită 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).
de a trece la limita To obtain equation (3), it is necessary to pass to the limit in relation (1).
poate fi trecut dintr-o parte a ecuaţiei în altă
Any term may be transposed from one member of equation to the other, provided its sign is changed.
trecere trecerea de la ... The identification of equivalent elements consists in
the passage from the elements of the set in question to the abstraction classes.
trecerea la limită This equation is obtained from relation (3) by passing to the limit.
la trecerea la limită 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
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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.
univoc determinată Thus, the function f(x) is uniquely determined provided it exists everywhere.
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 Such an element is unique. următoare 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.
uşor este uşor de ... It is not difficult to establish the existence of an
infinite number of rational primes.
util 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.
utiliza putem utiliza metoda In order to construct such a process, we can take such
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an approach. va fi utilizat permanent We shall prove below a very useful result which will be
constantly used in our work. utilizate de obicei 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 The absolute value of b is greater that of a.
după valoarea absolută These numbers are equal in absolute magnitude.
variabile aleatoare cu acelaşi repartiţie
Let θj , j=1, 2, ..., n, be mutually independent identically distributed random variables.
vedea vedeţi See, for example, the deduction of inequality (11) in
the proof of Theorem 1. este uşor de văzut că 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.
cum este uşor de văzut As is easily seen, there exists a Borel function θs such that ...
verifica se verifică In the case of distributions the following identities are
valid. Since inequality (4.8) holds true, the function ϕ(x) is non-negative.
sunt verificate 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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verifică cetinţe The functionals L and M satisfy (or meet) the aforementioned requirements.
pentru a verifica To verify that h(t) is the desired solution, observe that ...
este uşor de verificat că It is easily verified that the polynomial xn-2 is irreducible over the field of rational numbers.
dacă o serie de condiţii este verificată
The experiments studied in probability theory are carried out when a certain set of conditions F is satisfied.
verifică condiţia Assume that the function B(ϕ,t) satisfies the local Lipschitz condition.
verifică ecuaţie We shall prove that the function F(x) satisfies (or fulfils) equation (1).
vom verifica ipoteza despre ... We shall test the hypothesis that ...
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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