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 hypothesis
aceiaşi	The numbers +8 and +3 have like signs. No two polynomials can have all their roots alike (or the same) unless they differ by a numerical factor.
acelaşi	The preceding discussion may be summed up by saying that the arithmetic for complex numbers is much the same as that for real numbers.
acum	We now want to let j®Ą. 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 y(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(p/17).
admite
la fel admite	The function h(t) also admits decomposition of the form (4).
adresa
ne adresăm la	We now turn to the task of considering the continuous-parameter supermarlingales.
aduce
demonstraţia este adusă	The proof of the theorem is given in [8].
aduna
adunăm	Now write down these inequalities for n, n+1, ..., n+m and add them to obtain the estimation (6).
afirma
afirmă	The fundamental theorem of arithmetic asserts that every integer greater than 1 can be factored in one and only one way, apart from order, as the product of positive prime numbers. Bohm [2] claims that there is such a thing as the primacy of metaphysics in connection with biology.
afirmaţie
afirmaţie mai generală	A more general statement can be found in [15].
afirmaţie necesară	Approaching the limit as N®Ą 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 W. 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 S_r^-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 h may have a different distribution.
altfel	This definition also makes use of the abstraction principle, but in another way.
altfel zis	Otherwise speaking [f_n(x)] = [g_n(x)] if and only if {f_n(x)}~{g_n(x)}.
amănunţit
mai amănunţit	These operations are discussed in more detail in next section.
ambele	Both estimates are quite rough. But the functions f and g are both continuous, and thus the two functions coincide. Multiplying the two sides of (4) by cosmx and integrating, we obtain ...
ambele părţi	The two sides of Young's inequality are equal if and only if b=j(a).
ambiguu	This ambiguity does not lead to misunderstanding.
aminti
amintim că	We recall that the compensator gⁿ of the measure of jumps mⁿ 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
să aplicăm	Let us apply this result to the case of an increasing process A_t.
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 q 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, ..., t_n are arranged in ascending [descending] order.
arăta
arată	Fig.12 depicts the slope of the line Dx versus Dt.
vom arăta	We shall show that in this case p must divide b.
este arătat	The slope of the line Dx versus Dt 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 x(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 1^o ensures the feasibility of the addition. The MIS furnishes (or supplies) data for management.
asociat
asociată cu (referitoare la)	The error is associated with (or is due to) the inaccuracy in computations. The study is concerned with linear functionals of supercritical processes. The first problem is directly related to partial differential equations. Further extension of the notion of a stochastic integral is connected with the extension of the notion of convergence.
asociat la	The polynomial R(t) corresponding to this function by (6) is almost everywhere nonzero. The space, S_2k-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 m related to k by the relation m=4pk.
astfel	Thus the following theorem has been proved. The factors m and k are related as follows: m=4pk.
astfel încât	Let X be a space such that from every open cover we can select a finite cover. We break up the component a into several components in such a way that A and B now belong to different components. We select a basis in the underlying space such that ... Specifically, the parameter can be chosen so that the distribution is unimodal. The model is so constructed as to allow further modification and improvement.
astfel ... ca	We take a point on the line such that ...
astfel ... încât	We take a point on the line such that ... Applying Lemma 20.3 k times, we infer that there exists a periodic function G₀(x) with G₀^(k)=g(x).
aşa numit	In applications we often come across the so-called Heaviside function.
aşa ... pentru care	We take a point on the line such that ...
aşadar	Thus, the integral on the right-hand side of (5) can be analytically continued to the whole domain. The last formula becomes p(la+mq)=b, so that p\|b. Every element in T has norm at least equal to 1, so there must be one of them, g , which is of least positive norm. Since p is of the form 1+4m, Corollary 2.5 is applicable and hence p=x²+y². Therefore P(x,s,t,A) saitsfies the Chapman-Kolmogorov equation. Consequently,
ataşa la
poate fi ataşat la	The process X(t) may be placed (or put) into one of the following three classes depending on its asymptotic behaviour.
atinge
atinge maximum [minimum]	The function y(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 F_m(x,u_m) 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 m more accurately.
se atribuie sens	Analogous meaning is attached to the notation D[0,Ą).

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 R_m-semisimple rings are strongly R_m-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
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 y(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.
avea un caracter
are un caracter	Noise w(t) is stochastic in character (or in nature). Noise w(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 M_2,Ą.
bineînţeles	It is clear that the definitions and properties given below can be extended to s-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 evaluate
calcula integrală dea-lungul curbei	to take the integral along a curve
calcula integrală pe domeniu	to take the integral over a domain
care	In classical analysis there have been considered several sequences of functions which are distributionally convergent to d(x). Choose in D₂ any function that tends to infinity as x®0.
cărui	A variable is a quantity to which an unlimited number of values can be assigned in an investigation.
al cărui	Let A be a strongly positive operator whose domain of definition does not depend on t.
care este	Consider the function l(y), which is a periodical distribution coinciding with d(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 a(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 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 x.
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 M_t, satisfies the conditions of (1) with the stronger requirement that be an L²-martingale.
chiar în cazul când	This example shows that the ordinary derivative does not always coincide with the distributional derivative, even when the ordinary derivative wrists everywhere except at one point.
ci	Thus we need not 8, but 9 parameters to describe the system.
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 d(x).
clar
este clar că	Clearly, the sum, difference and product of such integers are of the same form. It is clear that a and -a are both in the set A.
intuitiv este clar	It is clear intuitively that every collection, finite or infinite, of non-negative integers contains a smallest one.
coeficientul pe lângă	the coefficient of smth. the coefficient multiplying smth.
coeficientul pe lângă termenul superior	A polinomial is monic if its leading coefficent is 1.
coincide
coincide cu	For an arbitrary function F(x) satisfying conditions 1) - 6) there exists a unique probability measure whose distribution function coincides with F(x). The characteristic through A is the same as the one through B.
complet satisfăcător	Completely satisfactory solution of the functional problem is given in [6].
completa
este completată	It is assumed that s-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(j,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 w, other things being equal.
în condiţia că	Thus, the function f(x) is uniquely determined, provided that it exists everywhere. Any term may be transposed from one member of equation to the other, providing its sign is changed. The abstraction principle can be applied to sets of arbitrary elements provided there exists an equivalence relation ~. These functions are locally Riemann-integrable under (or on) the condition that their values are equal at all points of continuity.
în condiţiile teoremei	Under the conditions of Theorem 6.1 the limit exists and is differentiable.
conduce
conduce la ecuaţia	This reasoning yields a simple equation.
confirma ipoteza	to verify a hypothesis
conform
conform algoritmului	The classes can be formed by the FLAMENCO algorithm.
conform criteriului	By the Eisenstein's criterion, the polynomial xⁿ-2 is irreducible over the field of rational numbers.
conform definiţiei	Then, by definition, ...
conform ipotezei inducţiei	By the induction hypothesis, the theorem is true for f(x).
conform lemei	By Lemma 2.3, the sequence {g_n(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, g, 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 f_i(x)≤0, i=l, 2, ..., n.
consecutiv	When an expression contains parentheses within parentheses, they may be removed in succession, beginning with the innermost.
considera
considerăm	Consider the rational number a/b.
poate fi considerată	The problem can be approached in many different ways.
să considerăm	Let us now consider the case a>1. Consider now the case a>1.

sunt considerate	Several other functions, very useful from the practical point of view, are considered in the following section. Some properties of the operator L are dealt with in the next chapter.
considerată	An integral part of the creative approach to the problem under consideration (or under study) is the analysis of its similarity to and difference from the analogous problems.
poate fi considerat ca	C (R₊) can be regarded as a complete separable metric space under the topology of uniform convergence.
în lucrare se consideră	The paper deals with the theory of distributions of finite order.
considera acum
considerăm acum	We now consider the process h(t). Consider now the process h(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.
continuu	In the case under consideration the process a(t) is continuous at every point of the interval (c,d).
continuă din dreapta după t	F_t is continuous from the right (or right-continuous) in t if for all t ≥ 0 .
contrazice
ce contrazice	So p 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®Ą and that (6) holds for all l<k_m.
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 P_n 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 a=eb, where e is a unit.
va fi convenabil	It will be convenient to define the convergence of f_n(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)ÎS if A(x,t) is a process with independent increments.
converge
converge absolut	The series (1) converges absolutely.
converge aproape peste tot	to converge almost everywhere
converge în medie	to converge in (the) mean
converge în probabilitate	to converge in probability
converge uniform	Thus the series (4) converges uniformly.
convinge
ne convingem că	Make (or be) sure that the coefficients have no common factors other than +1. To complete the proof, check that ... Before applying Theorem 10.7, check to see that there is an a in A such that ...
a se convinge în	To prove the convergence X ⁿ®M, it suffices to make sure that ...
trebuie să se convingă că	The investigator must assure himself that the errors are within the admissible limits.
pentru a se convinge în ...	To see that if q satisfies an equation a_nxⁿ + a_n-1xⁿ-1 + ... + a₀ = 0, then there is a rational r such that rq is an algebraic rational, observe that a_nq 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 F_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 j(x) is convex on [c,Ą).
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 {p_n^(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 k_m such that ...
strict crescător	Introduce a strictly increasing funstion g(x) defined on [0,Ą).
creşte
creşte monoton după t	F_t monotonically increases in t, that is F_sĚ F_t, for s<t.
cu ... ce tinde ...	Each summand on the right of (12) becomes vanishingty small with (or as) n®Ą.
cub al numărului	When a is used three times as a factor, the product is called the third power of a, or the cube of a.
cunoaşte
cum noi deja cunoaştem	Since, as we already know, all these trigonometrical integrals vanish, we obtain formula (5).
cunoscut cu	The reader acquainted with the Lebesgue integrals can interpret formula (1) in a still wider sense, viz. as the equality almost everywhere.
este cunoscut că	It is well known that for an arbitrary function F(x) satisfying conditions 1) - 6) there exists a unique probability measure whose distribution function coincides with F(x).
cum este cunoscut	As is well known, the nth partial sum of the series is nothing else than the Dirichlet kernel .
este bine cunoscut	It is well known that there exist sequences {T_n} and {S_n} such that h_n=T_n-S_n.
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=W_st+W_e+W_p.

D
da
dă	The repeated application of relation (2.1) yields ... The following theorem provides the description of the everywhere dense set E.
dăm	We give (or present) below a synthesis of various approaches available.
da posibilitate
dă posibilitate	The martingale approach enables us (or permits) to suggest a new method for constructing solutions of stochastic differential equations. Knowledge of the numerical values of a and b offers a means of (or makes possible) determining ... This method makes it possible to tackle the boundary-value problem under discussion. The main advantage of this approach is that it enables the results of the previous chapter to be extended to the case under consideration.
da restul de la împărţire
dă restul de la împărţire	Every integer a leaves a remainder r, 0 ≤ r < m, on division by m.
dacă	In the course of the proof of the theorem we have described the method for determining the functions F_n (x₀, ..., x_n) 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 z, we write max f = Ą. 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 h(t) if and only if ...
dacă şi numai dacă există	The martingale x(t) is uniformly integrable if and only if there exists a random variable h such that E \|h\| < Ą and x(t) = E {h \| F_t}.
dacă ..., atunci	Evidently, if the inf in (1.5) is attained at a certain point , then the inf F_m(x,u_m) is also attained at that point.
dacă ... nu	Unless u is a power of 2, this can always be done.
dar	Element x belongs to V, whereas (or and) z belongs to W. This value is negative — not positive.
dar atunci	But then we could proceed from the pair (x, y) to a still smaller in the same way as from (x₁, y₁) to (x₂, y₂).
dar nu	With time the number of particles increases, rather than remains constant.
date iniţiale	the initial (or the input) data
datorită	Delacherie and Meyer are well known for their book 'Stochastic Processes and Potentials'. Mathematics, due to (or owing to / because of) its abstract nature, finds wide application in most spheres of human activity.
de aceea	For this reason the points belonging to W are called elementary events. Conversely, if N=1, we have a+b=1, so that either a=0 or b=0. The last formula becomes p(la+mq)=b, so that p\|b. Every element in T has norm at least equal to 1, so there must be one of them, g, which is of least positive norm.
de asemenea	Observe also that \|D_n(s)\| ≤ C logn. From equation (3) we find, as well, . If a and b belong to F, so do a+b, a-b, ab, and a/b if b≠0.
de fapt	In fact, this follows from the consideration that the sequence g_n(x) is bounded for x>0. Since these are continuous processes, we actually have ...
de fiecare dată când	In stating rules, letters are used to represent not only the numbers to be found, but also the numbers that must be given whenever the rule is applied.
de k ori	Applying Lemma 20.3 k times, we infer that there exists a periodic function G₀(x) with G₀^(k)(x)=g(x). Differentiating formula (8) k times, we obtain the equality ...
de la	The Arab notation has been in use since the XVI century. Prof. Romanov is giving a course of lectures on chaos and bifurcations in room 406, commencing October 1. In order to start from the very beginning, we introduce first the basic notions and concepts of the theory of measurements. From this time on, the density function decreases. We shall start with the basic notions and concepts of the theory of measurements.
de unde	Setting () we obtain (), whence, in view of inequality (1), ... The author does not explain from whence* the estimate (7) emerges.
deduce
deducem	Applying Lemma 20.3 k times, we infer (or deduce) that there exists a periodic function G₀(x) with G₀^(k)=g(x).
deducem ecuaţia	We derive (or deduce) the equation for the distribution of cell size at division and the distribution of generation time.
defini
definim	Define unit and prime just as we did for the Gaussian integer.
definită	The function h(x) is defined at every point of the interval [c,d]. The variable, whose value is determined once the value of the independent variable is given, is called the dependent variable.
bine definită	The integral is well defined in accordance with the above definition.
definit anterior	Using the Haar family {g_n} defined above, we introduce functions {h_n}.
pozitiv definit	A complex-valued function f(x) is called positive definite if for any n and complex numbers z_j
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 p_n(x) by functions f_n(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_r<N_a, 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.
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 a_mn can be arranged in a sequence {b_k} convergent to b₀, for otherwise it would suffice to neglect a finite number of initial terms a_1n, a_2n, ..., a_pn 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 y(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.
deriva
deriva ... după x	To differentiate v(t,x) with respect to x.
deriva ... în raport cu x	To take the derivative of v(t,x,u) with respect to x.
derivând	In fact, by differentiating (3) we obtain ... Hence, by successive differentiation we obtain ...
derivând de k ori	Differentiating formula (8) k times, we obtain the equality ...
derivând succesiv	Hence by successive differentiation we obtain ...
derivata după x	A derivative of u(t,x,u) with respect to x.
deschide
deschide posibilităţi	The interpretative approach opens perspectives in that direction.
deschiderea parantezelor	the removal of parentheses
descompune în factori	to expand the number into factors to factor the number to present smth in a factor form to factor smth into components
descompune în serie
descompunem în serie	Expand function j(x) in a Taylor series.
descompunem în serie în raport cu x	Expand function u(t,x,u) into a Taylor series about x.
poate fi descompusă în serie	Under certain conditions a continuous function can be expanded into a Fourier-Bessel series. Function f(x) can be expanded as an infinite power series.
descompunere	Any supermartingale z(t) has exactly one decomposition of the form z(t)=m(t)-a(t).
descompunere în	6 = 23 = 32 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 {a_n} be a decreasing sequence and {b_n} 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 m.
descrisă de ecuaţie	We will consider only those systems whose evolution is given by the equation of form (8).
desemna
desemnează	The symbol F_n(x)® will denote that the sequence {F_n(x)} converges almost uniformly to a function.
desen
pe desen	In Fig.11 the planes p₁ and p₂ 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 {f_n(x)} and {g_n(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 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 x(t) has no discontinuities of the first [second] kind.
discuta
discutăm	In Chapter 8 we discuss the filtering problem.
nu va fi discutată	The validity of this assumption will not be debated here.
distinge
pentru a distinge	The one-to-one correspondence established above makes it unnecessary to distinguish between functions f(x) and distributions [f(x)].
distingere	In the theory of distributions a distinction between the trigonometrical series and the Fourier series is unnecessary since each convergent trigonometrical series is an expansion of a periodic distribution.
distribuţie comună	the joint (or simultaneous) distribution
divide
se divid	Then the numbers 1, 2, ..., p-1 fall into pairs such that the product of any pair is congruent to 1 modulo p.
se divide	The procedure is then broken down (or divided / subdivided) into the three steps: ...
se divide prin	Nine is a multiple of three.
divizibil
sunt divizibile prin	All the terms on the right are divisible by p.
domeniu de definiţie	Let A be a strongly positive operator whose domain of definition does not depend on G.
dori
dorit	Passing to the limit in (6) we obtain the desired result. This is exactly the wanted decomposition.
este de dorit	It is desirable in many applications that the function X_t 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)*m.
enunţul problemei	The most general statement of the problem can be formulated as follows.
evident	A more exact answer to the question requires apparently (or evidently) a refinement of the definition of a system.
este evident	It is obvious (or evident) that X_t 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 W is called an event.
eveniment sigur	the sure event
evident	The proof of this property is obvious since b=0. Applying Theorem 12.3 to the obvious equality (*) we get ...
evita
pentru a evita confuzii	To avoid possible confusion, the continuity condition will be restated in each case.
pentru a evita neînţelegeri	To avoid misunderstanding, we may call the first kind of intervals (including end-points) the closed intervals and the second kind the open intervals.
pentru a evita repetări în viitor	To avoid repetitions in future, we present here basic concepts and definitions.
exact
exact unul	Any prime P 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 n possesses the property (1) for all w and t except for some sets of measure 0. To prove this, interchange the order of integration in the double integral in (6), after which all the integrals except (8) can be evaluated exactly by appropriate change of variables.
exclude
se exclude	The second of these possibilities is excluded, for r(x) is in T.
exclude posibilitate	This approach excludes the possibility of considering the equations having solutions which belong to a more general class of numbers.
exemplu	Another example of the abstraction principle is the Cantor definition of a real number. Let us first look at some simple examples.
de exemplu	For instance (or for example), the coefficient of variation, CV, of size at division is typically 10% and the CV of age at division is typically 20% or more. As an example, let a=5 and b=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 B_n(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.
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®Ą along the subsequence n_k, we have ...
să facem câteva observaţii referitor	Before presenting the main results, we want to make a few comments on the notion of entropy.
face de prisos	The one-to-one correspondence established above makes it unnecessary to distinguish between function f(x) and distribution [f(x)].
face referinţă
de a face referinţă la	To complete the proof, it is sufficient to refer to the Cantor theorem on uniform continuity.
facem referinţă la	For the proof of this theorem, we refer the reader to Theorem 9.
poate să facă referinţă la	For further information the reader is referred to Chapter VI.
fals	That such an impression is erroneous, we demonstrate by exhibiting another simple class of "integers" for which Theorem 8 is false.
familie (set)	M is a collection (or family / set) of processes m_t , tÎT.
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 fie	Now set .
fie că este dat	Let there be given a continuous increasing process x(t).
fie dat	Given point A,construct point B symmetrical to the given point A in relation to plane p. Given any e>0, we fix an index n₀ such that \| f_n(x) - f(x) \|<e for all n>n₀.
fiecare
fiecare dintre care	Let x₁, x₂, ..., x_n be a sequence of random variables each of which is the sum of certain number of mutually independent random variables.
pentru fiecare din	M is the set of processes m_t 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 tÎS. Fix k points -1 ≤ x₁ < x₂ < ... < x_n ≤ 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 equation
formalizare
în formalizarea	When formalizing the notions of probability theory, the first fundamental assumption is that the results of the experiments can be described by means of a certain set W.
formă
de formă	The function h(t) also admits decomposition of the form (4.3). An integral of this form can be evaluated by substitution.
în formă	Write equation (3.2) as (or in the form) ...
în formă explicită	To represent the solution of equation (12) explicitly, it is necessary to solve equation (8) for u(t).
într-o formă concisă	An equation that states a rule in a brief form is called a formula.
într-o formă mai simplă	More recently some of results were obtained in a simplified form.
formula
să formulăm	Let us formulate the conditions under which the statement is valid.
formulare	The new statement of the Fundamental Theorem is this.
formulă
formula schimbului de variabilă	Apply now the change of variables formula to (1).
formulă iniţială	the assumption formula
funcţie
funcţii absolut continue	The class of indefinite Lebesgue integrals of locally integrable functions is the class of absolutely continuous functions.
funcţie caracteristică	E_W(x) is the characteristic function of set W, equal to unity inside W and to zero outside W.
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 n runs into severe (or great) difficulties.
genera
generează	Here w(t) denotes the process which generates the current {F_t}.
generată	F_s is a s-algebra generated by the events {x(s)ÎB}.
general
mai general	More generally, if a sequence {f_n(x)} converges almost everywhere, then it converges also in a distributional sense. If the function f(x) is sectionally continuous, or, more generally, if it is Riemann-integrable, equation (1) is also true except of the points of discontinuities of f(x).
generaliza
pentru a generaliza	To generalize the theorem, it is necessary to modify the definition of convolution.
poate fi generalizat	We now show how the method described above can be extended to prove the existence of m.
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	p_n(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 {h_n} in L₀¹[0,Ą).
important
foarte important	In applications the following theorem is of great importance.
una din cele mai importante	The concept of hierarchy is among the most important in the theory of general systems.
importanţă	The importance of the primes is due to the fact that, together with 0 and ±1, all the other integers can be constructed from them.
impresie falsă	That such an impression is erroneous, we demonstrate by exhibiting another simple class of "integers" for which Theorem 8 is false.
impune
să impunem condiţii asupra	In order that the solutions converge, we must impose the following conditions on the coefficients of equations (2).
include
include (ind. prez. 3 sg.)	It is obvious that the set F includes the set of rational numbers. Equation (8) incorporates normal losses. The dictionary contains (or includes) some information about the appropriate use of collocations.
sunt incluse	Some elements of artificial intelligence are now incorporated in system.
inclusiv	The function f(x,y) possesses bounded partial derivatives up to the third order inclusive. Let F_t be a s-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 q be a random variable independent of the s-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 e^x remains unaltered regardless of the number of times it is differentiated.
mutual independente	Let x₁, x₂, ..., x_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 {F_t} or {F'_t}.
imposibil
este imposibil	But this is impossible for any pair of integers a and b is in J. It is impossible to represent 12 as m² + n². Many of these questions are difficult if not impossible to answer.
inducţie
după inducţie	We shall prove the theorem by induction. Define the random variables M_n 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(Ą), is the limit of this function as x ®Ą.
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®Ą.
iniţial	The distribution of the initial vector converges weakly to a limit as u®0. The reader should be reminded that the initial value of n was chosen to be n₀=7.18.
instructiv	This example is instructive because it displays the structure of the process m(P).
integra prin părţi	to integrate by parts
integrală
integrală definită	the definite integral
integrală improprie	If g(x) is a locally integrable function and the improper integral exists, then g(Ą)=0 provided g(Ą) exists.
integrală nedefinită	By an indefinite integral of a function f(x) we understood each function y(x) such that y'(x) = f(x).
integrant	the integrand (or the expression under the integral sign)
integrare
după integrare	Replacing x by x+t in (1), we have, after integration, ...
integrare prin părţi	the integration by parts
interesant
este interesant de observat că	It is interesting to observe that the above formula remains true if the summation is understood according to the method of arithmetic means.
intra
intră în	This term appears in (or enters into) equation (9) in parentheses.
care intră în ...	The coefficient a^(x,s) entering into* (or involved in) the expression for I _t can be assumed to be constant.
introduce
introducem	Introduce the sequence {a_n}_{n Î N} of positive numbers. Allow sinu within the differential sign.
introducând	Introducing 2n new variables x_i and z_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,e_k).
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_W(x) is the characteristic function of set W, equal to unity inside W and to zero outside W.
în aplicaţii	In applications we often come across the so-called Heaviside function.
în aşa mod încât	For the new distributions one can define addition, subtraction, multiplication by a number or by a function in such a manner that the fundamental properties are preserved. A variable x is said to vary continuously through an interval [a,b] when x increases from a to b in such a manner as to assume all the values between a and b in the order of their magnitudes. It follows from (1), (2) and (3) that the whole set is decomposed into abstraction classes so that two elements are in the same abstraction class if and only if they are equivalent.
în concordanţă cu	In accordance with the law of large numbers, a » mx = x.
în conformitate cu	In accordance with (or by) the law of large numbers, a » mx = x.
în continuare	Next, define M_Ą as the limit of M_t , as t®Ą.
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_W(x) is the characteristic function of set W, equal to unity inside W and to zero outside W. All points of W lie in the interior of y²<d.
în întregime	We will not attempt here to solve the problem in its entirety. Though the macroscopic properties of the cell population as a whole follow a deterministic growth law, the descriptors of individual ceils (such as size at division and age at division) are highly variable. The contents of this book may not be reproduced either in whole or in part.
în loc de această	Instead, we will assume that ...
în mare măsură	The theory is based, to a large extent, on heuristic considerations.
în realitate	Actually, we shall prove somewhat more that we need for the present.
în sfârşit	Finally, to prove (5), suppose p is prime.
în spiritul	The paper is written in the spirit of the Wilkinson's error analysis.
în totalitate	The graphs of these sequences do not intersect and in totality there sequences absorb all jumps of the process Z.
în totalitate	The graphs of these sequences do not intersect and in totality there sequences absorb all jumps of the process Z.
în vigoare	Unless otherwise stated, condition (2) will always be assumed to be in force.
în virtutea	In view of Theorem V, the sequences from Example 2 converge. By virtue of the basic assumptions, ... Right-angle projection is most widely used because of its comparative simplicity. On account of the foregoing remark, the distributions in the earlier sense can be identified with distributions of finite order. By 2.4, the limit of a_n(x) - g_n(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 N_p=0, for otherwise p≠0 so that P=a, 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 a=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 m 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 a(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 n(t) becomes a constant.
în cazul mai sumplu	In the simpler case where a=1 this relation takes the form p=he^2/3.
în acest caz particular	In this particular case only one side of the triangle, namely BC, has traces within the limits of the drawing.
în cel mai bun caz	This problem has received at best only fragmentary treatment.
în multe cazuri	For many purposes (or in many cases / on numerous occasions) we shall need the following assumption.
în mod
în mod analogic	Analogously (or similarly / likewise), it is sometimes useful consider only positive solutions to Eqn (17). This theorem can be proved in much the same way (or manner) as Theorem 1. A similar argument yields ... The plane passing through the vertex A and the vertex C can likewise be constructed.
în mod arbitrar	The first term can be made arbitrarily small by an appropriate choice of e. The variable, to which the values may be assigned at pleasure, is called the independent variable. The initial values of x and z 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 {f_n(x)} converges almost everywhere, then it converges also in the distributional sense. This process is unique in the sense that if {B_t} is any process with properties (1) through (4), then A_t=B_t for every t.
în sensul că	This process is unique in the sense that if {B_t} is any process with properties (1) through (4), then A_t=B_t 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 a 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 a 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 x(t) itself. Here the solution is a non-decreasing function of the drift coefficients and, in a sense, of the jump functions.
înainta
a fost înaintată	A new theory has been put forward this year. We were informed that a new idea had been advanced at the closing session.
înainta o ipoteză	to advance a hypothesis
încă	Many roadblocks must still be overcome before all the subgroups of G are described.
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_Ą is replaced by F_t if tÎ[0,T ].
poate fi înlocuit prin	The integer k can be, if necessary, replaced by any greater integer.
înlocuit ... prin	Replacing in (2) a by b and afterwards x by ax we obtain ...
înlocui a prin b	to replace a with (or by) b to substitute b for a to change a for b
se poate de înlocuit	In case of right-continuity the condition (2) may be replaced by the requirement (3).
putem înlocui prin	In the definition of the mth derivative we can replace the polynomials p_n(x) by functions f_n(x).
însemna
înseamnă	A(C) denotes the completion of B(C) with respect to m_n.
va înseamna	The letter M, with subscripts, will denote positive 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, ..., a_n.
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 d_2p(x) is the derivative of the function E(x/2p), 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) = [f_n(x)] by a number l we understand the distribution [lf_n(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 s-algebras such that ...
înţelegere profundă	This information is needed for a fundamental understanding of (or a deep insight into) the properties of ...

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

L
la fel
la fel ca în cazul	This is proved in the same manner as in the case of discrete-argument submartingales.
la fel ca şi	But then we could proceed from the pair (x,y) to a still smaller pair in the same way as from (x₁, y₁) to (x₂, y₂).
lăsa
lăsăm	The rest we leave to the reader.
va fi lăsată cititorului	The slight change in the original proof which is needed here will be left to the reader.
lega
este legat de	The estimation of n 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 m(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 l=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®Ą, the last term in (8) vanishes.
limită inferioară de integrare	the lower limit of an integral
limită superioară de integrare	the upper limit of an integral
literal	In the term 4x²y the numerical multiplier of the literal part is the coefficient.
lua în consideraţie
luăm în consideraţie	If we consider (or take into consideration) that the series is a series of positive terms, we have ...
fără a lua în consideraţie ...	The analysis has been made without reference (or without regard) to heat losses. Regardless of the last term which is sufficiently small, equation (9) may be solved in radicals for x.
dacă se ia în consideraţie că	The statement becomes apparent when it is considered (or if it is remembered) that ... The statement becomes apparent when it is considered (or taken into account) that ...
trebuie de luat în consideraţie că	One should keep in mind (or it should be kept in mind / it should be remembered) that the partial sums of the series (2.2) are the Fourier transforms of functions B_n(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(p).
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.
margine
margine inferioară exactă	the greatest lower bound
margine superioară exactă	the least upper bound
matematică
matematică aplicată	applied mathematics
matemitică finită	Finite mathematics is considered as superior tool to use, over differential equations.
matematică pură	pure mathematics
material	In order to achieve this, we shall need some material from elementary number theory.
material bogat despre	This volume presents a wealth of material on stochastic integrals.
matrice inversă	the inverse matrix
maxim	We denote by x_max 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 g_n(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 W is called an event. Observe that (x_t ,F_t) is a potential.
vom menţiona că	It should be observed here that this definition of equality is usually adopted in the theory of the Lebesgue integral.
metoda aproximărilor succesive	the method of successive approximations
minuţios	The results reported by Dr. Evans need thoroughly be checked.
modifica
puţin modificând	We shall modify slightly the definition of ...
modificând puţin metoda	By a slight change in the method of (3) the reader can easily prove the theorem by himself.
modificare a metodei	This is a modification (or an adaption) of the method of steepest descent.
moment
din acest moment	From now on, we assume B to be a complete separable metric space. From this time on, the density function decreases.
până la momentul	Define the s-field of events prior to a moment (or a time / an instant) t as Let F_t be a s-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 y(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 s-algebra A is generated by the P-null sets.
mulţime dintr-un singur element	We will assume in the future that the s-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, f_e and f_* need not be identical. Let a(t) be an increasing (but not necessarily adapted) integrable process.
necesar	Hence one of the factors of f(x) necessarily has the same degree as g(x).
necesară	We are going to show that the function F(x) has the required property.
este necesar	In some problems of the filtering theory, it is necessary to work with Wiener processes on R.
este necesar ca	In order that two triangles be congruent, it is necessary that two angles of the first triangle be equal, correspondingly, to those of the second triangle.
necesar pentru ca	G(x) is the time needed for a cell to grow from size a to size x.
necesitate	By Theorem 2.1 it suffices to prove the necessity.
necunoscut	the unknown the unknown quantity the sought-for (or the desired) quantity A number whose value is to be found is called an unknown number.
cu două necunoscute	Thus we obtained an equation in two unknowns.
nedescrescător	The comparison theorem for diffusion equations asserts that the solution of these equations is a non-decreasing function of the drift coefficient.
nici
nici o ... a cărei	Let be p-dimensional cycles, no linear combination of which is homologous to zero.
nici un	No sÎD is an element of G.
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 L 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, ..., P_n.
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 y'(x) = f(x). The Arab notation has been in use since the XVI century. The following designations are used: a=1/a, b=c/a, and g=b/a.
notaţie standardă	the standard notation
noţiune	When formalizing the notions of probability theory, the first fundamental assumption is that the results of experiments can be described by means of a certain set W.
noţiuni şi definiţii de bază	basic concepts and definitions
nu	The line L does not intersect A or B. Last time we were not appealing to Theorem 7. But nothing in this definition indicates from whence the class [x] is obtained. This shows that this method cannot give all solutions of the equation. It is not a mere academic exercise to pick out the cases of equality in such an inequality as Gronwall's. Never did mathematicians worry much about the foundations in algebra. If there is no such x, we write max f = Ą.
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 m(t) and w(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 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 a such that M(t) ≤ C_a for all t. To prove 2^o, 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 » mx = 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=...=a_n=0. Thus we arrive at a factorization m=p₁p₂...p_n, where each p_i is positive and prime.
se obţine	This function is obtained by application to expression (4) a nonlinear differential operator P.
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 R^m.
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 z_N>t.
oricât ar dori	Let e be an arbitrary however small value.
orice
pentru orice	For each wÎW, the function t®X_t(w) is called the sample path. For every number e>0 there exists an index n₀ such that \| a_m - a_n \|<e for all m, n>n₀.

P
pagină
la pagina	Powell's carrier distribution, C(t) on page 245, is our distribution f_m(T).
paranteză
în paranteze	When numbers are collected by the sign of aggregation ( ), they are commonly said to be in parentheses. The expression between (or in) the brackets is precisely the Laplace transform of the function j^*(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®Ą.
în partea stângă	The term on the left-hand side (or on the left) of Eqn (12) vanishes as k®Ą.
pe de altă parte	On the other hand, if [f_n(x)] = f(x), then the first two conditions of Theorem 1 are satisfied.
partiţie	Let t_k, k =1, ..., m, be a partition of the interval (s,t) into subintervals of equal length.
patrat al numărului	When a is used twice as a factor, the product is the second power of a, or the square of a.
păstra
păstrează	Absolute constants retain the same value in all problems.
vom păstra	Since distributions are a generalization of the notion of functions, we shall retain for them the usual notation of functions, i.e., f(x), g(x), etc.
sunt păstrate	For the new distributions one can define addition, subtraction, multiplication by a number or by a function in such a manner that the fundamental properties are preserved.
primi
fi primit	It should be observed that this definition of equality is usually adopted in the theory of the Lebesgue integrals. The height of the triangle is taken to be 12m.
pentru
pentru aceasta	For this purpose we select first the Gaussian integers. To this end it suffices for the first term in (1) to have the distribution F(x) and for the others to be zero with probability 1.
pentru ca	In order that the last property be valid, it is sufficient that ... In order to achieve this, we shall need some material from elementary number theory. To formulate the corresponding result, we consider the following conditions which are the strengthened versions of conditions of Theorem 8.
pentru care	It is therefore reasonable to ask for which classes of "integers" the fundamental theorem holds, and for which it does not.
pentru comoditate	For convenience we define a continuous locally increasing process to be an adapted process which has properties (1) through (4).
pentru nici un	The function cannot be constructed for any value of x.
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 a, if f(x+a) = 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 I_T (F_t), it is sufficient to describe the everywhere dense set I_T²(F_t).
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, ..., x_n 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 n possesses the property (1) for all w and t. We are going to prove that the limits exist and the function F(x) has the required property.
posibil
posibilă	To avoid possible confusion, we shall use this notation ... The error, if any, will be allowed for by visual estimation.
este posibil ca	It is possible that the ordinary derivative does not exist, although the distributional derivative has a value at this point.
este întotdeauna posibil de făcut	This we can always do by choosing s and t as rational integers nearest to A and B respectively.
practic	Practically it suffices to understand integrability in the sense of Riemann or of Lebesgue.
precedent
precedentă	On account of the foregoing remark, the distributions in the earlier sense can be identified with distributions of finite order. 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 s-algebra F₀ is completed by all P-null sets. All stochastic processes considered in this book will be assumed to be measurable. {F_t} is assumed to be right-continuous.
vom presupune că	We assume that FG=H, the operation being defined by FG=RS, where R=TV. Suppose* X_t is a continuous process. Suppose that the theorem is true for k-1. Suppose the conclusion of the theorem to be true for all f(x) of degree between 0 and n-1 inclusive.
presupunem contrariul	Suppose the contrary.
presupunere
presupunerea că	In defining K(x) we have made use of our assumption that b(x)=0 for x<a. This equation contradicts the assumption that G contains points not belonging to D.
presupunerea conform cărei	The assumption under which formula (10) is valid can be somewhat weakened.
presupuneri sunt făcute	The following assumptions are made concerning the random process x(t): 1) the process is F_t-adapted and 2) it satisfies condition (1). Up till now no assumptions have been imposed concerning the convergence of j_n(t).
prezenta în forma
poate fi prezentat în forma	Any potential z_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 x(t), t ≥ 0, is said to be adapted to a current of s-algebras {F_t}_t≥0 if for every t the random variable x(t) is F_t-measurable.
provine
provine de la	The word 'school' has its origin in the Greek word 'schole' meaning 'leisure'.
provin de la	The arithmetic symbols were derived from the Arabs and the Hindus.
punct
în punct	The function has no a derivative at the point 0.
în puncte	Such an identification can be extended to a wider class of functions which have the poles at some points and are locally integrable elsewhere.
punct ce se mişcă dea-lungul	In the equation of straight line x and y are the variable coordinates of a point moving along the line.
punct de vedere
din punct de vedere logic	From the logical point of view, this generalization is exactly of the same type as the generalization of rational numbers in the Cantor theory.
din punct de vedere practic	Several other functions, very useful from the practical point of view, are considered in the following section.
punct de referinţă	Here the starting point is the notion of fundamental sequences of rational numbers. Approximation is our starting point for the definition of distributions.
pune în corespondenţă
operatorul pune în corespondenţă	The operator q, assigns to (or associates with) any function from D a function from D_(-Ą,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_a.
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 A_t 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 a algebraic over F are the roots of the minimal polynomial of a 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_g such that U_gm=U_gU_m.
reduce ecuaţia la formă	to reduce the equation to the form ...
reduce termenii asemenea
reducem termenii asemenea	To solve the equation 2x+x = 12, unite (or group / collect) similar terms in the first member to obtain 3x and then divide both members by 3 to find the value of x to be 4.
referi
se referă la	The Henry's Law refers only to the effect of pressure.
nu se referă la	Evidently the Einstein's restriction does not apply to the case under consideration.
care se referă	We now state a number of results concerning ordinary differential equations.
ceea ce se referă	As regards (or in regard to / in respect to) the solution of equation (3), it may not be argued that it satisfies Condition 2^o. 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 x(t-) is a supermartingale with respect to the family of s-algebras (F_t-) _t≥0. Given point A, construct point B symmetrical to the given point A in relation to plane P. For the present we content ourselves with demonstrating the assertions just made concerning the sets G and H.
relativ la	In relation to W, 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 z_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 [p_n(x)], where p_n(x) are polynomials.
de a reprezenta ... ca ...	To prove this statement, it suffices to present the integral as a limit of .
respectiv	The largest P-factors are n₁ and n₂ respectively.
respinge o ipoteză	to reject the hypothesis
restrânge generalitate
fără a restrânge generalitate	We shall also assume, without restricting the generality, that z₀=s₀=0. We can assume, without any restriction of generality, that all numbers a_mn can be arranged in a sequence {b_k} convergent to b₀. In our discussion we may assume, without loss of generality, that M is a subset of the set of natural numbers.
restricţie	With this additional restriction the problem just posed received an exhaustive solution in the papers of Bawty and Khinchin.
a rezolva ecuaţia în raport cu x explicit [implicit]	to solve the equation for x explicitly [implicitly]
rezolva în raport cu x	to solve for x
rezolvare
la rezolvare problemei	It is convenient, in solving problems, to use letters for the numbers which are sought.
rezulta
rezultă din	The result follows from the induction hypothesis. The term R/n results from the Bohr theory. Much of the utility of the theorem stems from the properties of the binomial coefficients.
rezultă imediat	The proof follows immediately if we observe that ...
rezultă nemijlocit	Since u(t) is continuous, it follows at once that x(u(t),t) satisfies condition (ii).
această rezultă din	This results from the following lemma. This follows from the fact that e ^pi/2 = i.
din cele spuse rezultă că	From what has been said it follows that ... It follows from what has been said that T>0 almost surely on the set B.
din ... rezultă că	It follows from (1), (2), and (3) that the whole set is decomposed into abstraction classes without common elements so that two elements are in the same abstraction class if and only if they are equivalent.
de aici rezultă că	From this it follows that A is consistently defined except on a P-null set. This implies that ... It follows from this that ... It follows that R(s) is a commutative ring.
nemijlocit rezultă din	It follows directly from the definition that ... This assertion follows immediately from relation (2).
ridica
ridica la puterea n	to raise to the nth power
ridica un număr la cub	to cube a number
ridica un număr la patrat	to square a number
ridicarea la putere	The process of finding any required power of a number is called involution.

S
satisface
se satisfac condiţiile	Condition (4) is equivalent to the following conditions being satisfied.
sau ..., sau	Since a satisfies (1), we have that either a=1 or a=p-1.
schimb
schimb de variabilă	Relation (8) follows from the general rule of the change (or the substitution) of variables in integral calculus.
schimb de variabilă la integrare	the change of a variable in integration
schimba
schimbăm ordinea de integrare	To prove this, interchange the order of integration in the double integral in (6).
fără a schimba	The introduction of locally integrable allows us to strengthen Lemma 6.8 without changing its proof.
este schimbat semnul	Any term may be transposed from one member of equation to the other, provided its sign is changed.
schimbare
schimbare mică	A slight change in the original proof will be left to the reader.
schimbarea ordinii de integrare	The main point of the proof is to justify the interchange of the order of integration in (8.4).
fără schimbări	With this transform the first term in (4) vanishes, whereas the last one remains unaltered (or unchanged).
scoate în afară parantezelor
scoatem în afară parantezelor	Now factor out of the expression in parentheses the largest positive factor common to all the b_i. Now take the largest positive factor out of the brackets.
scop (obiectiv)	The purpose (or objective) of this monograph is to make available elementary parts of classical algebraic number theory. The aim of this section is to define the stochastic integral for a wider class of predictable integrands 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 S are the products of continuous functions.
sub semnul integralei	under the integral sign
sens	In the last formula the symbol 0 has two different meanings: on the left-hand side it denotes the number zero and on the right-hand side the zero distribution.
serie absolut [condiţional] convergentă	the absolutely [conditionally] convergent series
sfat	I wish to express gratitude to O. A. Oleinik for her valuable advice.
similar	Progressive measurability of an S-valued process is defined similarly. Reasoning similarly, we convince ourselves that the solution to Egn (17) is also unique.
similar ca	The definition is the same as the one given above, but with R instead of [0,T ].
simplifica
simplificăm	Simplify 4a-(2a-2b).
simplifică	Thus the introduction of distributions makes the 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 j 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 x:W®S is called an S-valued stochastic process and S is called the state space of x.
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 x(t) are sometimes called the states of a stochastic system S and the space S, the phase space of the system S.
specifica
cum a fost specificat mai sus	As has been stated (or specified) above, the functions p_m(x) defined by (1.6) are bounded.
dacă nu este specificat contrariul	We agree to assume, unless otherwise specified, that all processes are adapted. Unless otherwise stated, condition (2) will always be assumed to be in force.
spune
spune (ind. prez. 3 sg.)	The principle of mathematical induction says: "If P is true ..." The second law states (or reads): "If the same current flows ..." The theory holds (or states) that ... Stated in a simple form, the hypothesis runs as follows.
se spune	If conditions (i) and (ii) are also fulfilled, then the current is said to satisfy the 'ordinary' conditions.
vom spune	If the members of a pair are equal, say to a, then a²=1 (mod p).
vom spune că ...	We shall say that two integers a and b are relatively prime if they share no factors except ± 1.
spunând	The function F is, so to say, the optimal cost of the continuation of control.
spunând mai exact	To put it more exactly, ...
din cele spuse mai sus	From what has been outlined (or said) above the following conclusions may be drawn.
stabili
stabileşte	This requirement places (or sets) the upper limit on the admissible error.
stabilim	We first establish the necessity of the condition.
a fost stabilit	This fact was ascertained (or established) from experience.
a fost stabilită	The average value was found to be 16 ± 3.
de a stabili condiţii	Theorem 1.6 allows to lay down the conditions under which the solutions to equations (6) are continuous in parameter z.
strict	Approximations, strictly defined, is our starting point for the definition of distributions.
studia
studiat	The experiments studied in probability theory are carried out when a certain set of conditions Y is satisfied.
substitui
substituim	For this we make the substitution t-x=s.
substituind ... în	Substituting these numbers into (or in) (1), we find that ...
substitui a cu b	to make the substitution of b for a to carry out the change of a for b
substituţia b cu a	the substitution of b for a
suficient	This equation determines a sufficient condition.
suficient de mare	Whatever t may be, one can always find a sufficiently large N such that ... Now EC_t < e/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 a is a Gaussian integer, so is b and hence a+b.
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 f_n(x) = g₁(x) + ... + g_n(x) converges.
sumar	A survey of the general theory of stochastic processes can be found in [4].
supune
se supune legii	The distribution of errors adheres to the law of probability. The kinetics of the system obey (or follow) the ideal gas law.
se supune ecuaţiei	It the coefficients on the right of (2) satisfy the Lipschitz condition, then the function w(t) obeys the equation (3.1). The equilibrium distribution of particles in a gravitational field adheres to (or is governed by) the Boltzmann equation.
sus
mai sus	It is easy to see that the relation of equivalence defined above has the following properties.
de mai sus	The integral is well-defined in accordance with the above definition. The definition is the same as the one given above, but 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	{U_t} is the 'dual predictable projection' in the terminology of Dellacherie and Meyer [42].
timp
în timp ce	Thus 5 and 9 are relatively prime, whereas 6 and 9 are not. As the point x tends to a, the function g(x) approaches the value g(a).
tinde la
când ... tinde la	Each summand on the right of (12) becomes vanishingty small with (or as) n®Ą.
când ... tinde la infinit	to let smth approach (or to let smth go to) infinity
tinde la infinit	to tend to (or to approach) infinity
tinde la zero (ind. prez. 3 sg.)	The limit distribution functions are determined by the fact that they are the limits for sums of an increasing number of independent random variables such that a single summand becomes vanishingly small with n®Ą.
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 H_n^(-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 a_n must be less than a_N.
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 n possesses the property (1) for all w and t.
transforma
a transforma ecuaţia	to rearrange the equation
să transformăm la formă	Let us transform (1) to the form (2).
a se transforma în infinit	to become infinite
transformare inversă	the inverse transform
trebui
trebuie	The problem must be solved as soon as possible. To ensure the existence of a solution to (14), additional assumptions need be made. Much needs to be done to complete the theory. It should be particularly emphasized that condition () is not essential and was introduced solely for the sake of convenience. To comprehend the main idea of my talk, it is essential* (or necessary) that the students be familiar with the generalized Galois theory.
care trebuie	In stating rules, letters are used to represent not only the numbers to be found, but also the numbers that must be given whenever the rule is applied.
ceea ce trebuia	But p is of the form 1+4n, so that a ^p=l+im, as required.
ceea ce trebuia de demonstrat	It follows that the points C₁ and C₂ lie on the line of intersection of the planes a and b, 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®Ą 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 ...

U
unul şi numai unul	Hence there is one and only one x which satisfies the congruence.
unde	We assume that FG=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 s-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=acosj and j=wt+q 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 ...

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^<x_n* from its values on a finite set of points x₁, x₂, ..., x_n.
cu valori în	Let b(j,t) be a random function with values in R^m.

valoare în punct	The Dirac function d(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 q_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 {g_n} is complete in L¹[0,1]. It will readily be seen that that for any d>0 the sequence Z(nd,w) forms a Galton-Watson process.
cum este uşor de văzut	As is easily seen, there exists a Borel function q_s such that ...
verifica
se verifică	In the case of distributions the following identities are valid. Since inequality (4.8) holds true, the function j(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.
verifică condiţia	Assume that the function B(j,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 a_i denote positive constants. Henceforth we shall adhere to the following model of the probability theory. In the future s_n will always be assumed to be bounded. In the future, when continuous local martingales are discussed, the s _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.