...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>
2 KB (319 words) - 19:08, 5 February 2016
...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>
674 bytes (100 words) - 17:27, 7 February 2011
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
2 KB (312 words) - 19:38, 15 March 2023
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
2 KB (243 words) - 21:41, 14 April 2014
...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 $(a+b)^{n}$.
3 KB (379 words) - 07:44, 4 November 2023
...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>
1 KB (158 words) - 18:18, 19 October 2014
...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 $c$.
1 KB (172 words) - 13:07, 19 March 2023
...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 $
2 KB (351 words) - 20:40, 16 November 2023
...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
8 KB (1,143 words) - 08:03, 6 June 2020
...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.
3 KB (422 words) - 18:10, 12 December 2014
...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 $10^7$).
3 KB (570 words) - 20:06, 11 April 2014
...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
2 KB (275 words) - 19:39, 29 April 2014
...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 $
5 KB (696 words) - 17:31, 5 June 2020
$#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.
3 KB (471 words) - 19:38, 5 June 2020
"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>
15 KB (2,329 words) - 14:32, 19 March 2023
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
5 KB (787 words) - 16:55, 1 July 2020
...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
3 KB (445 words) - 18:09, 1 October 2017
...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,
10 KB (1,512 words) - 07:24, 25 March 2023
...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,
7 KB (1,099 words) - 19:39, 8 March 2024
...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
11 KB (1,738 words) - 18:15, 24 March 2018
...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,
10 KB (1,550 words) - 13:04, 18 March 2023
...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
11 KB (1,710 words) - 19:08, 6 March 2024
...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
23 KB (3,482 words) - 08:03, 6 June 2020
...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,
9 KB (1,457 words) - 19:16, 24 March 2023
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
4 KB (627 words) - 09:28, 19 March 2023
...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
5 KB (853 words) - 12:16, 22 August 2014
...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).
13 KB (2,014 words) - 20:35, 31 December 2018
...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,
6 KB (895 words) - 12:27, 9 March 2024
...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
10 KB (1,504 words) - 07:34, 10 March 2024
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 $.
9 KB (1,417 words) - 16:09, 1 April 2020
...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
10 KB (1,551 words) - 07:42, 9 March 2024
...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.
10 KB (1,459 words) - 19:56, 5 March 2024
...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
8 KB (1,160 words) - 20:38, 22 September 2016
...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
7 KB (1,015 words) - 19:42, 1 November 2023
...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
7 KB (1,004 words) - 19:40, 8 March 2024
...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.
22 KB (3,256 words) - 21:54, 30 March 2012
...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 $
20 KB (2,829 words) - 18:41, 11 December 2020
...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
11 KB (1,665 words) - 15:52, 25 March 2023
...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
8 KB (1,114 words) - 06:48, 8 October 2023
...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).
11 KB (1,629 words) - 11:46, 23 November 2023
...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
10 KB (1,517 words) - 07:18, 8 March 2024
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
4 KB (631 words) - 19:39, 5 June 2020
...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.
16 KB (2,633 words) - 12:03, 1 November 2023
...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
12 KB (1,727 words) - 10:18, 10 March 2024
...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
14 KB (2,104 words) - 16:21, 17 April 2012
$#C+1 = 13 : ~/encyclopedia/old_files/data/M063/M.0603700 Metric theory of functions
...functions are studied on the basis of the idea of the [[Measure|measure]] of a set.
11 KB (1,633 words) - 08:00, 6 June 2020
...{\pi}$, 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==
18 KB (2,697 words) - 13:11, 13 December 2013
...that many theorems are directly related to the geometry and the topology of Banach spaces.
== History ==
25 KB (4,253 words) - 23:54, 13 April 2019
...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
11 KB (1,612 words) - 18:41, 22 March 2023
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.
17 KB (2,478 words) - 16:09, 1 April 2020