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...with any element $a$ also contains the sum $a+1$, is necessarily the whole of $\mathbf{N}$. See [[Natural sequence]]; [[Arithmetic, formal]]. ...lign="top"> , ''The history of mathematics from Antiquity to the beginning of the XIX-th century'' , '''1''' , Moscow (1970) (In Russian)</TD></TR>

...ich the counters (bones, pebbles, etc.) were moved. The Chinese equivalent of the abacus — swan-pan — was used in the Far East, while the "schetysch ...> <TD valign="top"> W.W. Rouse Ball, "A short account of the history of mathematics" , Dover, reprint (1960) pp. 123–125</TD></TR></table>

An algebraic equation of degree three, ''i.e.'' an equation of the form beginning of that century, S. Ferro solved the equation $x^3+px=q$, where

A nice illustration of some of the simpler properties of (countably) infinite sets. There is also room for an infinity of new guests. Indeed, shift the existing guest in room $1$ to room $2$, the o

...es of an equilateral triangle and each of the remaining numbers is the sum of the two numbers above it to the left and right: there appear the coefficients of the expansion of the binomial $(a+b)^{n}$.

...r the solution of the problem of the [[Quadrature of the circle|quadrature of the circle]]. For example: the [[Dinostratus quadratrix|dinostratus quadrat ...lign="top">[a1]</TD> <TD valign="top"> Th.L. Heath, "A history of Greek mathematics" , '''1–2''' , Dover, reprint (1981)</TD></TR>

...ers|coprime]] $(k + 1)$-tuple together with $n$. This is a generalisation of Euler's [[totient function]], which is $J_1$. ...d) = n^k $. The [[Average order of an arithmetic function|average order]] of $J_k(n)$ is $c n^k$ for some $c$.

...d, in general, two elements of a [[Euclidean ring]]) or the common measure of two intervals. It was described in geometrical form in Euclid's Elements (3 ...tive integers $a \ge b$, the method is as follows. Division with remainder of $a$ by $b$ always leads to the result $a = n b + b_1$, where the quotient $

...30 : ~/encyclopedia/old_files/data/N067/N.0607960 Numbers, representations of ...symbols began to appear, so number systems began to vary in the character of their numerical signs and in the principles governing the form in which the

...s, recursive functions, normal algorithms, and others — one may, by virtue of Church' thesis, restrict oneself to intuitively obvious constructions, and ...cannot be solved in the framework of this or that precision of the notion of algorithm.

...ion of numbers (cf. [[Numbers, representations of|Numbers, representations of]]). ...rom about 3000–2500 B.C.) there were individual signs to denote the powers of ten (up to $10^7$).

...lem can be reduced to finding a function $y(x)$ that constitutes a minimum of the functional where $a$ and $b$ are the abscissas of points $A$ and $B$. The brachistochrone is a [[Cycloid|cycloid]] with a hor

...rrently charged for an individual insurance contract be a weighted average of the form is the observed mean claim amount per unit of risk exposed for the individual and $ \mu $

$#C+1 = 30 : ~/encyclopedia/old_files/data/E036/E.0306840 Exhaustion, method of ...ns of antiquity in order to determine areas and volumes. The name "method of exhaustion" was introduced in the 17th century.

"a history of Hilbert’s program for the foundations of mathematics, initiated by his Problems Address given in Paris, 1900" -- see [[Hilbert p ...ed to” (proved indirectly as a result of proving directly) the consistency of analysis:<ref>Hilbert (1902) §2</ref>

Out of 16 formulas, 16 were replaced by TEX code.--> ...er must provide a reserve, $V _ { t }$, which by statute is the mean value of future discounted benefits less premiums. Splitting into payments before an

...od for obtaining the incomplete fraction and the remainder in the division of a polynomial ...coefficients $a,a_0,\ldots,a_n$ lie in a certain field, e.g. in the field of complex numbers. Any polynomial $f(x)$ can be uniquely represented in the f

...ighted by the author(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribution Share-Al '''Summary.''' A physician and writer, the creator of John Bull,

...ighted by the author(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribution Share-Al The second of three sons of François Reymond, Écuyer,

...eory of functions of a real variable typically involves a wide application of set-theoretic methods, along, naturally, with classical methods. ...rnoulli, J. d'Alembert, J.L. Lagrange, and others), although the formation of this theory took place in the 19th century (J. Fourier, A.L. Cauchy, N.I. L

...ighted by the author(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribution Share-Al '''Summary.''' Jevons pioneered the use of mathematics in economics,

...ighted by the author(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribution Share-Al '''Summary''' Poisson, the apostle of Laplacian science, was the master of

...f mathematics on the other, have determined the development of the concept of a number. ...provided by linguistics and ethnography. Primitive man clearly had no need of counting skills to determine whether or not a given collection was complete

...ighted by the author(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribution Share-Al the presence of statistical dependence. In the design of electronic circuitry,

The Dirichlet convolution of two [[arithmetic function]]s $f(n)$ and $g(n)$ is defined as where the sum is over the positive divisors $d$ of $n$. General background material on the Dirichlet convolution can be found

...y can be identified in a natural way with the ideals (cf. [[Ideal|Ideal]]) of $A$. ...the factorization of the corresponding divisor $\phi(a)$ into the product of prime ideal numbers can be looked at as a substitute for unique factorizati

...ence of mathematics at a speed without parallel during the long history of mathematics. ...ete mathematics — and only the very beginning of the influence of discrete mathematics is witnessed now (1990).

...ighted by the author(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribution Share-Al as ''The Divorce Problem: A Study in Statistics'', (Studies in History,

...ighted by the author(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribution Share-Al '''Summary'''. G. von Mayr was the foremost representative of German

In its basic form, the study of the interplay between an embedding of a projective manifold (cf. [[Projective algebraic set|Projective algebraic is the cotangent bundle of $ X $.

...ighted by the author(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribution Share-Al ...ad of smallpox, which was prevalent at the time, and argued the advantages of

...ighted by the author(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribution Share-Al the probability calculus, by means of his dispersion theory.

...ighted by the author(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribution Share-Al general and President of

...ighted by the author(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribution Share-Al Summary some fifty years before the least sum of squared residuals

...ighted by the author(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribution Share-Al '''Summary'''. Arguably the greatest French mathematician of

...ed in mathematics is continuously expanding; so this general definition of mathematics becomes ever richer in content. ...he first problems of mechanics and physics were already in this collection of fundamental mathematical ideas.

...en functions irrespective of any special properties of these functions and of any real processes described by these functions (see, e.g., [[Metric space| ...ofs of geometric theorems also appeared during that period. The [[Elements-of-Euclid| $ Elements $

...ighted by the author(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribution Share-Al '''Summary.''' Gauss shaped the treatment of observations into

...ighted by the author(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribution Share-Al treatment of outliers in statistics and to the application of probability

...ighted by the author(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribution Share-Al ...y edition of Statisticians of the Centuries and elsewhere is actually that of the politician Louis Legendre (1752--1797).

...ighted by the author(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribution Share-Al '''Summary'''. Shcherbina was the most eminent of the zemstvo

One of the original programming languages for problems of computational mathematics, developed in 1954–1956 for the IBM 704 machine, and in the version Fortr ...eping the name Fortran. Most of them were constructed as extensions of one of the two standard ones (1962–1964): Basic Fortran and Fortran IV, and subs

...ighted by the author(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribution Share-Al political circumstances, Christiaan Huygens turned to science and mathematics.

...ighted by the author(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribution Share-Al '''Summary.''' Mendel was definitively recognized as the originator of

...easurable, and the probability of each subset is the sum of probabilities of its elements. ...nuous]], [[Singular distribution|singular]] or mixed), as well as the set of all continuous functions $[0,\infty)\to\R$ with the [[Wiener measure]]. Th

$#C+1 = 13 : ~/encyclopedia/old_files/data/M063/M.0603700 Metric theory of functions ...functions are studied on the basis of the idea of the [[Measure|measure]] of a set.

...{\pi}$, while the statement "the ratio of the length of the circumference of the circle to its diameter is greater than three and ten seventy-firsts and ==Early history==

...that many theorems are directly related to the geometry and the topology of Banach spaces. == History ==

...ighted by the author(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribution Share-Al <ref> Abridged version of an article

The branch of mathematics in which one studies algebraic operations (cf. [[Algebraic operation|Algebr ...e next three centuries, during which views as to the proper subject matter of this discipline kept radically changing.

...ighted by the author(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribution Share-Al '''Summary''' Author of the monumental ''Histoire Naturelle'', Buffon also

...ighted by the author(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribution Share-Al '''Summary'''. Active in many area of science, notably mathematics,

...hic. In other words, the variety of all heaps is equivalent to the variety of all groups. ...ho_3]=\rho_1\rho_2^{-1}\rho_3$. The set of all invertible partial mappings of $A$ into $B$ is also closed with respect to the triple multiplication and i

...ighted by the author(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribution Share-Al '''Summary.''' Slutsky contributed to the understanding of stochastic

...ion of elementary methods is extended by bringing in the simplest elements of mathematical analysis. Traditionally, proofs are deemed to be non-elementar ...elementary solution of some problem, one also finds an elementary solution of it.

...ighted by the author(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribution Share-Al especially concerning its use in actuarial mathematics. He headed the British

$#C+1 = 37 : ~/encyclopedia/old_files/data/C025/C.0205340 Constructive mathematics, ''constructive trend in mathematics''

...ighted by the author(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribution Share-Al Dalmatia, who developed in 1760 a simple geometrical method of fitting a

...ighted by the author(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribution Share-Al prominent exponent of 19th

...inear functions or polynomials, and it is in fact such functions which are of interest, more than other functions, in applications. be defined in some neighbourhood of a point $ x _ {0} $.

...c propositions of geometry and other branches of mathematics, on the basis of which all the theories are developed in a rigorously logical fashion. ...equal; and 5) the whole is greater than any of its parts (in some editions of the Elements there are four additional axioms).

Out of 93 formulas, 91 were replaced by TEX code.--> ...uch media and more exactly, to the [[Spectral analysis|spectral analysis]] of the [[Schrödinger equation|Schrödinger equation]] with a periodic potenti

...ighted by the author(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribution Share-Al Neyman, made fundamental contributions to the theory of hypothesis testing.

...ighted by the author(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribution Share-Al of a calculus of probabilities to contain a form of local

A [[recursive sequence]] of integers in which each term is the sum of the two preceding. ...onacci numbers were produced for the first time in 1228 in the manuscripts of Leonardo da Pisa (Fibonacci).

...ighted by the author(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribution Share-Al introduction of quantitative methods into psychology. He also developed a

...ighted by the author(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribution Share-Al '''Summary.''' Malthus was a pivotal figure in the development of

...ighted by the author(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribution Share-Al Bernoulli, together with his brother Johann one of the pioneers of the

...ighted by the author(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribution Share-Al original ways ways the application of the probability calculus to statistical

...ighted by the author(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribution Share-Al ...s or robotic satellites. Astrophysics is the study of the intrinsic nature of astronomical bodies and the processes by which they interact and evolve. Th

...erations performed on numbers. The properties of integers form the subject of number theory (cf. [[Elementary number theory|Elementary number theory]]; [ ...rmed on objects of very different kinds: "matrix arithmetic" , "arithmetic of quadratic forms" , etc.

...teristics of (uniformly) continuous functions. The modulus of a continuity of a continuous function $f:\mathbb R^n \supset E \to \mathbb R^k$ is given by ...modulus of continuity was introduced by H. Lebesgue in 1910 for functions of one real variable, although in essence the concept was known earlier. The d

...r the solution of a given infinite series of problems of a given type, and of finding such an algorithm if it exists. ...portant nevertheless, since it makes it possible to speak of unsolvability of given algorithms in a wide sense, which is generally understood by mathemat

...ighted by the author(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribution Share-Al '''Summary.''' A man of many talents, Professor (at the Copenhagen

...ighted by the author(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribution Share-Al '''Summary''' Chebyshev is regarded as the founder of the St. Petersburg School

...ighted by the author(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribution Share-Al and as critic (indeed as polemicist) that he is known to history.

...ighted by the author(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribution Share-Al '''Summary'''. D'Alembert's doubts concerning the calculus of probabilities

of Hermitian operators (cf. also [[Semi-group of operators|Semi-group of operators]]) on a [[Hilbert space|Hilbert space]] $ L _ {2} ( X, \mu ) $ ...nces|[a9]]] to prove that the Hamiltonian operators arising in some models of [[Quantum field theory|quantum field theory]] are bounded below. In the imp

$#C+1 = 33 : ~/encyclopedia/old_files/data/A011/A.0101910 Algorithms, theory of ...he concept of an algorithm more precise by the introduction of the concept of a [[Normal algorithm|normal algorithm]] [[#References|[10]]]. A.N. Kolmogor

...ighted by the author(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribution Share-Al co-founder,together with Gheorghe Mihoc (1906-1981), of the Romanian

...ighted by the author(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribution Share-Al '''Summary.''' Pastor Süssmilch was the German ancestor of demographic

An area of mathematics concerned with geometric figures on a sphere, in the same way as planimetry ...circle as section; if the intersecting plane passes through the centre $O$ of the sphere, then a so-called great circle is obtained as the intersection.

...many concepts in abstract group theory have had their origin in the theory of finite groups. ...fficult task for the best modern computers. In general, the classification of finite $ p $-

...ighted by the author(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribution Share-Al is best known for his important proof of the central limit theorem.

...ighted by the author(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribution Share-Al socialists and slums, Bowley, a pioneer in the development of

...ighted by the author(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribution Share-Al <ref> Abridged version of an article

...Three kinds of such problems were particularly important in the history of mathematics. ...y the method of exhaustion (cf. [[Exhaustion, method of|Exhaustion, method of]]), in which infinitesimal quantities are used merely to prove that two giv

...which are sought for in the algebraic integers of some algebraic extension of the field $ \mathbf Q $ of rational numbers, of the field of $ p $-adic numbers, etc.

...ighted by the author(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribution Share-Al Von Mises was principally known for his work on the foundations of

...ighted by the author(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribution Share-Al and originality in economic theory and policy, made him one of the most

...ular, continuous, piecewise-linear or smooth (diffeomorphisms); imbeddings of one object into another and also immersions (local imbeddings). (cf. [[Home ...the mapping is a homomorphism, an imbedding or an immersion at any moment of time.

...ighted by the author(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribution Share-Al ...d application of random processes. His textbook influenced the development of probability and statistics internationally.

...as a unit" . Rigorous theories of real numbers were constructed at the end of the 19th century by K. Weierstrass, G. Cantor and R. Dedekind. Real numbers form a non-empty totality of elements which contains more than one element and displays the following pr

...ighted by the author(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribution Share-Al '''Summary.''' Arguably the Father of Modern Statistics, Graunt produced

...l equations. A limit cycle corresponds to a periodic non-constant solution of the system. ...er the cycle, which may require some quite detailed knowledge of the shape of the cycle itself.

...ighted by the author(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribution Share-Al of a classical education, Edgeworth made contributions to the moral

...ighted by the author(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribution Share-Al '''Summary.''' Unacknowledged in his own times as founder of mathematical

...: ~/encyclopedia/old_files/data/A013/A.0103000 Approximation of functions of a complex variable ...n conformal mapping, integral representation, potential theory, the theory of function algebras, etc., play an important role in approximation theory.

Out of 120 formulas, 118 were replaced by TEX code.--> ''Zermelo–Fraenkel set theory with the axiom of choice''

Out of 61 formulas, 61 were replaced by TEX code.--> ...statics, that is, without rapid time variations), which arises from a set of physical laws known as the [[Maxwell equations|Maxwell equations]]. Electro

...function $f$ is given on $X$ (and also that the variable $y$ is a function of the variable $x$, or that $y$ depends on $x$) and one writes $f:X\to Y$. ...as a correspondence between two arbitrary sets (not necessarily consisting of numbers) was formulated by R. Dedekind in 1887 [[#References|[3]]].

...ighted by the author(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribution Share-Al '''Summary'''. After his celebrated contributions to pure mathematics, Borel

...multiplicative [[Cobordism|cobordism]] invariants taking values in a ring of modular forms. The following is an attempt to present the simplest case — ...#References|[a7]]]) and view a [[Modular form|modular form]] as a function of elliptic curves with a chosen invariant differential (cf. also [[Elliptic c

...ighted by the author(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribution Share-Al Memorial Medal of the Faculty of Mathematics and Physics, Charles

...n [[Arithmetic|arithmetic]] connected with the multiplication and division of integers. ...Perfect number|Perfect number]]) was studied; and the solution in integers of the equation $ x ^ {2} + y ^ {2} = z ^ {2} $

...ighted by the author(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribution Share-Al '''Summary''' Pascal introduced the concept of mathematical expectation and

...ighted by the author(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribution Share-Al use of probability to describe one's beliefs about scientific ideas, and

Out of 55 formulas, 55 were replaced by TEX code.--> for a set $P : = \{ p _ { 1 } , \dots , p _ { m } \}$ of $m &gt; 3$ given non-collinear points with corresponding positive weights $

of class $ S $, The problem of finding an accurate estimate of the coefficients for the class $ S $

...ighted by the author(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribution Share-Al Department of Mathematics, Indiana University,

...ighted by the author(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribution Share-Al '''Summary.''' From a strictly statistical point of view, Gini's

...of a mathematical system (a structure) has become a fundamental concept in mathematics. ...ents $x$, $y$ in $G$, the left quotient and the right quotient of division of $b$ by $a$, such that $ax=b$, and $ya=b$.

...ighted by the author(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribution Share-Al '''Summary.''' This member of the Bernoulli dynasty

...'the discretization program'".</ref>These efforts took place over a period of about 50 years, with the following results: # the establishment of fundamental concepts related to limits

...ighted by the author(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribution Share-Al '''Summary.''' Founding Director of the Central Bureau of Statistics in

...of some specific (function) or abstract Banach space, that is, an equation of the form is some, generally speaking non-linear, operator transforming elements of a $ B $-

...ighted by the author(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribution Share-Al '''Summary.''' The data of Pastor Caspar Neumann on births and funerals in Breslau

...ighted by the author(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribution Share-Al == 1. HISTORY ==

...ighted by the author(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribution Share-Al '''Summary.''' At the end of the eighteenth century, Eden's concern with

...ighted by the author(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribution Share-Al '''Summary.''' Francis Galton developed the basic statistical concepts of

...d approach to establishing connections between local and global properties of topological spaces (in particular geometric objects) and which is a powerfu ...stalk-wise algebraic operations and the projection $p$ is called the sheaf of Abelian groups (rings, etc.) over $X$ associated with the pre-sheaf $F$.

...ighted by the author(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribution Share-Al plans and contributions to Statistical Systems of other countries as

...remises as [[Euclidean geometry|Euclidean geometry]], except for the axiom of parallelism (see [[Fifth postulate|Fifth postulate]]). In Euclidean geometr ...nt to require that there is at most one straight line, since the existence of a non-intersecting line can be proved by successively drawing lines $ PQ

...ighted by the author(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribution Share-Al '''Summary.''' Shewhart was the father of statistical methods of

...ighted by the author(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribution Share-Al '''Summary.''' Long recognized as a leading proponent of

$#C+1 = 161 : ~/encyclopedia/old_files/data/S091/S.0901970 System of subvarieties, ''system of submanifolds''

...ighted by the author(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribution Share-Al popularize British mathematical statistics. He was a pioneer in the theory of

''calculus of variations'' ...(cf. [[Variation|Variation]]), i.e. by the method of small perturbations of the arguments and functionals; such problems, in the wider sense, are oppos

...ighted by the author(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribution Share-Al '''Summary.''' Bienaymé was a Civil Servant. A disciple of Laplace, he

Out of 225 formulas, 223 were replaced by TEX code.--> The notion of a Buchsbaum ring (and module) is a generalization of that of a [[Cohen–Macaulay ring|Cohen–Macaulay ring]] (respectively, module). L

...ighted by the author(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribution Share-Al together of statistics and probability into mathematical statistics in the

...ighted by the author(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribution Share-Al '''Summary.''' Founding father of

''of curves and surfaces'' ...ns properties of sufficiently small pieces of them. Properties of families of curves and surfaces are also studied (see, for example, [[Congruence|Congru

...ighted by the author(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribution Share-Al associated with the discovery of the ''t''-distribution and its use, and he

...atics. In conjunction with algebra, topology forms a general foundation of mathematics, and promotes its unity. ...th of the class of geometric objects that fall within the sphere of action of its laws.

...e (1768), who considered the following variational problem: Find a surface of least area stretched across a given closed contour. Assuming that the requi Later G. Monge (1776) discovered that the condition for minimality of a surface leads to the condition $ H = 0 $,

...alytic functions forms the main subject of the general theory of functions of a complex variable. ==Analytic functions of one complex variable==

...ighted by the author(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribution Share-Al population or a part of it, such as the residents of a country or the

...tatement of his 2nd problem, initiating his program for the foundations of mathematics'' -- see [[Hilbert problems]] By about 1820, mathematicians had developed deductively a large part of analysis using the real numbers and their properties as a starting point.

The branch of mathematics dealing with geometric objects connected with commutative rings: algebraic ...he conversion of a tentative reasoning into a proof as for the formulation of such proofs in their most obvious and most general form.

$#C+1 = 83 : ~/encyclopedia/old_files/data/F040/F.0400960 Foundations of geometry A branch of geometry in which the basic concepts of geometry, the relations between them and related questions are studied.

...8]]] contains 23 problems, though at the meeting Hilbert discussed but ten of them (problems 1, 2, 6, 7, 8, 13, 16, 19, 21, 22). For a translation, see [ Cantor's problem on the [[Continuum, cardinality of the|cardinal number of the continuum]].

...were built in England for the purpose of cryptanalysis. Consult the books of D. Kahn, in particular [[#References|[a13]]], for a detailed historical per The purpose of a cryptosystem is to encipher an intelligible cleartext (also called plaint