Dictionar Matematic Roman-Englez de Fraze Uzuale

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 hypothesisaceiaşi 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 (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.This value was found to be cos(/17).

admite        la fel admite The function (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

1

        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 set .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 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.

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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=(a).ambiguu This ambiguity does not lead to misunderstanding.aminti        amintim că We recall that the compensator n of the measure of jumps

n 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).        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

3

        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 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 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 (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

4

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 la The polynomial R(t) corresponding to this function by (6) is almost everywhere nonzero.The space, S2k-1, and the projective metric, L(r,s), associated with it will play an important part in the future.It is convenient to replace k by a new factor related to k by the relation =4k.

astfelThus the following theorem has been proved.The factors and k are related as follows: =4k.

        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 numitIn 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, , 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

5

following three classes depending on its asymptotic behaviour.

atinge        atinge maximum [minimum] The function (x) reaches its maximum [minimum] at

x*=3h/2.Let b1(s,w) coincide with the z for which the 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 m(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 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 approach

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

avea

6

        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 (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        dacă are sens

The 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.

7

avea un caracter        are un caracter Noise (t) is stochastic in character (or in nature).

Noise (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 -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, 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 evaluatecalcula integrală dea-lungul curbei

to take the integral along a curve

calcula integrală pe domeniu to take the integral over a domaincare In classical analysis there have been considered several

8

sequences of functions which are distributionally convergent to (x).Choose in D2 any function that tends to infinity as x0.

        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 (y), which is a periodical distribution coinciding with (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 (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        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

9

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 .        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.

ciThus we need not 8, but 9 parameters to describe the system.

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 (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 lângăthe coefficient of smth.the coefficient multiplying smth.

        coeficientul pe lângă termenul

A polinomial is monic if its leading coefficent is 1.

10

        superiorcoincide        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 -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.

        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(,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 , 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

11

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 hypothesisconform        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 presurunerii By assumption, E(1)=0.

By the assumption of the lemma, the solution to (1) is positive.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, , 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.       

12

        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 (t).

Consider now the process (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        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.

continuuIn the case under consideration the process (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)

13

in t  if for all t ≥ 0 .contrazice        ce contrazice So 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.        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 =, where 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

14

common factors other than +1.To complete the proof, check that ...Before applying Theorem 10.7, check to see that there is an in A such that ...

        a se convinge în To prove the convergence X nM, 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 satisfies an equation anxn + an-1xn-1 + ... + a0 = 0, then there is a rational r such that r is an algebraic rational, observe that an 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 b(x).

        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 (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).

15

        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

than the Dirichlet kernel .        este bine cunoscut It is well known that there exist sequences {Tn} and

{Sn} such that n=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

16

        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 n (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.

        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 , 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 (t) if and only if ...        dacă şi numai dacă există The martingale  (t)  is uniformly integrable if and only

if there exists a random variable    such that  E || <   and  (t) = E { | Ft}.

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

dacă ... nu Unless u is a power of 2, this can always be done.dar Element belongs to V, whereas (or and) 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) datadatorită 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 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.

17

de asemenea Observe also that |Dn(s)| ≤ C logn.From equation (3) we find, as well,

.If and belong to F, so do and if ≠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

18

dependent variable.        bine definită

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 N<N, 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        fără demonstraţie We shall assume without proof the following theorem.

19

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 (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

monotonically dependent on t and right-continuous in t.

20

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 parenthesesdescompune în factori to expand the number into factors

to factor the numberto present smth in a factor formto factor smth into components

descompune în serie        descompunem în serie Expand function (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.

descompunereAny supermartingale (t) has exactly one decomposition of the form (t)=(t)-(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        descris anterior We now show how the method described above can be

extended to move the existence of .        descrisă de ecuaţie We will consider only those systems whose evolution is

given by the equation of form (8).desemna

21

        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 1 and 2 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        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

22

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 (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) distributiondivide        se divid Then the numbers 1, 2, ..., p-1 fall into pairs such that

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.

23

        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.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)*.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

24

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 is called an event.        eveniment sigur the sure eventevident 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 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.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 possesses the property (1) for all 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

25

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 =5 and =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        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.

26

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.

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.

27

        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 ,  tT.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

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 fieNow set .

        fie că este dat Let there be given a continuous increasing process (t).        fie dat Given point A,construct point B symmetrical to the

given point A in relation to plane .Given any >0, we fix an index n0 such that | fn(x) - f(x) |< for all n>n0.

fiecare        fiecare dintre care Let 1, 2, ..., n be a sequence of random variables each

of which is the sum of certain number of mutually independent random variables.

28

        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 tS.

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".

        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

equationformalizare        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 .

formă        de formă The function (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

29

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 formulafuncţie        funcţii absolut continue The class of indefinite Lebesgue integrals of locally

integrable functions is the class of absolutely continuous functions.

        funcţie caracteristică E(x) is the characteristic function of set , equal to unity inside and to zero outside .

        funcţie inversă the inverse function

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 runs into severe (or great)

difficulties.genera        generează Here w(t) denotes the process which generates the

current {Ft}.        generată Fs is a -algebra generated by the events {(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

30

definition of convolution.         poate fi generalizat We now show how the method described above can be

extended to prove the existence of .        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 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

31

        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 -algebra of the events observed up to and 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 be a random variable independent of the -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 1, 2, ..., n 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.

32

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 infinitesimal) with n.

iniţial The distribution of the initial vector converges weakly to a limit as u0.The reader should be reminded that the initial value of was chosen to be 0=7.18.

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

integra prin părţi to integrate by partsintegrală        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 (x) such that '(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 partsinteresant        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

33

        introducem Introduce the sequence {an}n N of positive numbers.Allow sinu within the differential sign.

        introducând Introducing 2n new variables i and i 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.

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 E(x) is the characteristic function of set , equal to unity inside and to zero outside .

        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

34

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 m = x.

        în conformitate cu In accordance with (or by) the law of large numbers, a m = x.

        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 E(x) is the characteristic function of set , equal to unity

inside and to zero outside .All points of 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

35

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.

Moreover Np=0, for otherwise p≠0 so that =, 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 =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 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 (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 (t) becomes a constant.

        în cazul mai sumplu In the simpler case where =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).

36

This theorem can be proved in much the same way (or manner) as Theorem 1.A similar argument yields ...The plane passing through the vertex A and the vertex C can likewise be constructed.

        în mod arbitrar The first term can be made arbitrarily small by an appropriate choice of .The variable, to which the values may be assigned at pleasure, is called the independent variable.The initial values of and 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 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 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 (t) itself.Here the solution is a non-decreasing function of the drift coefficients and, in a sense, of the jump functions.

înainta

37

        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 hypothesisîncă Many roadblocks must still be overcome before all the

subgroups of G are described.        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) by and afterwards x by x we

obtain ...         înlocui a prin b to replace a with (or by) b

to substitute b for ato 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 .        va înseamna The letter M, with subscripts, will denote positive

38

constants.'ln' signifies the natural logarithm.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 (x) is the derivative of the function

E(x/2), 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 we understand the distribution [fn(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 -algebras such that ...

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

J

39

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 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 (t) with continuous characteristics.For the present we shall content ourselves with 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

40

=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 integralliteral 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().        mai mult ca atât Moreover, every solution of (9) may be represented in

the form (3).        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.

41

margine        margine inferioară exactă the greatest lower bound        margine superioară exactă the least upper boundmatematică        matematică aplicată applied mathematics        matemitică finită Finite mathematics is considered as superior tool to

use, over differential equations.        matematică pură pure mathematics

materialIn 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 matrixmaxim 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.

        menţionăm We note that any arbitrary subset of is called an event.Observe that (t ,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 approximationsminuţios 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

42

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 -field of events prior to a moment (or a time / an instant) as

Let Ft be a -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 (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 -algebra A is generated by the P-null sets.        mulţime dintr-un singur        element

We will assume in the future that the -algebra, N, contains all single-element sets.

N

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

neapărat        nu neapărat In general, e and * need not be identical.

Let (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

43

size x.necesitate By Theorem 2.1 it suffices to prove the necessity.necunoscut the unknown

the unknown quantitythe sought-for (or the desired) quantityA 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 sD is an element of .        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 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.notaţie Let us introduce the following notation

where '(x) = f(x).The Arab notation has been in use since the XVI century.The following designations are used: =1/a, =c/a, and =b/a.

        notaţie standardă the standard notationnoţiune 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 .

        noţiuni şi definiţii de bază basic concepts and definitionsnu The line L does not intersect A or B.

Last time we were not appealing to Theorem 7.

44

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 , 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 (t) and (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.

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

45

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 such that   M(t) ≤ C  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 ...        trebuie de observat că It is worth noticing that, in accordance with the law of

large numbers, a m = 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 .

        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.

46

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).oricare n-ar fi Whatever (or no matter what) t (may be), one can

always find a sufficiently large N to satisfy the inequality N>t.

oricât ar dori Let be an arbitrary however small value.orice        pentru orice For each , the function is called the sample

path.For every number >0 there exists an index n0 such that | am - an |< for all m, n>n0.

P

pagină        la pagina Powell's carrier distribution, C() on page 245, is our

distribution fm(T).

47

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 *(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.        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.

48

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.        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 ,

if f(x+) = 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 possesses the property (1) for all and

49

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.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 -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.

50

        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 (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 n(t).

prezenta în forma        poate fi prezentat în forma Any potential t 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 interes Another theoretical approach of great interest consists

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 (t), t ≥ 0, is said to be adapted to a

current of -algebras  {Ft}t≥0  if for every t the random variable (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.

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        punct ce se mişcă dea-lungulIn 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        corespondenţă

The operator , assigns to (or associates with) any 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>N.

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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.

        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 algebraic over F are the roots of the minimal polynomial of 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 U such that U=UU.

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

53

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.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 (t-) is a supermartingale with respect to the family of -algebras (Ft-) t≥0.Given point A, construct point B symmetrical to the given point A in relation to plane .For the present we content ourselves with demonstrating the assertions just made concerning the sets G and H.

        relativ la In relation to , 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 t 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

54

.respectiv The largest -factors are 1 and 2 respectively.respinge o ipoteză to reject the hypothesisrestrânge generalitate        fără a restrânge generalitate We shall also assume, without restricting the

generality, that z0=s0=0.We can assume, without any restriction of 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 xrezolvare        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 i/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.

55

        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

ridicarea la putereThe 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

56

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 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   are the products of

continuous functions.        sub semnul integralei under the integral signsens 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.

57

        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

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 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 S  is called an S-valued stochastic

process and S is called the state space of .        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 (t) are sometimes called the states of a stochastic system and the space S, the phase space of the system .

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 We agree to assume, unless otherwise specified, that

58

        contrariul all processes are adapted.Unless otherwise stated, condition (2) will always be 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

59

substituţia b cu a the substitution of b for asuficient 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 < /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 is a Gaussian integer, so is

and hence .        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 parţială

A 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.

60

The definition is the same as the one given above, but 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].timp

61

        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) infinitytinde la infinit to tend to (or to approach) infinitytinde 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 possesses the property (1) for all 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 infinitetransformare inversă the inverse transformtrebui        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 (*) 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.

62

        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 and , 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

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

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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 -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=acos and t+ 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

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 ...

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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 (,t) be a random function with values in Rm.                valoare în punct The Dirac function (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 j ,  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 >0 the sequence Z(n,) forms a Galton-Watson process.

        cum este uşor de văzut As is easily seen, there exists a Borel function s such that ...

verifica        se verifică In the case of distributions the following identities are

valid.Since inequality (4.8) holds true, the function (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.

        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.

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        verifică condiţia Assume that the function B(,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 n will always be assumed to be bounded.In the future, when continuous local martingales are discussed, the 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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