...heory of probabilistic primality tests (cf. [[Probabilistic primality test|Probabilistic primality test]]), as they show that Fermat's theorem, to wit $ a^p \equiv
...only if $\lambda(n) \mid n-1$. From this it follows that every Carmichael number is odd, square-free, and has at least $3$ distinct prime factors.
3 KB (485 words) - 17:34, 18 October 2014
''(on the normal number of prime factors of an integer)''
For any integer $n \ge 2$, let $\omega(n)$ denote the number of distinct prime factors of $n$. The Hardy–Ramanujan theorem [[#Referenc
4 KB (647 words) - 07:30, 18 March 2023
A result in [[probabilistic number theory]] characterising those [[additive function]]s that possess a limiting distr
...ef; Mitrinović, Dragoslav S.; Crstici, Borislav, eds. ''Handbook of number theory I''. Dordrecht: Springer-Verlag (2006). pp. 564–566. {{ISBN|1-4020-4215-9
2 KB (230 words) - 12:04, 23 November 2023
...unction $\omega(n)$, which is the number of distinct prime divisors of the number $n$, is strongly additive; and the function $\log m$ is completely additive
An arithmetic function is also called a number-theoretic function.
1 KB (206 words) - 05:52, 15 April 2023
''number-theoretic function''
...er than $x$ — describes the distribution of primes; $\pi(x,q,l)$ gives the number of primes not larger than $x$ in the arithmetic progression $p\equiv l\pmod
4 KB (608 words) - 08:18, 4 November 2023
Let $f$ be a function on the [[natural number]]s. We say that the ''normal order'' of $f$ is $g$ if for every $\epsilon
* The [[Hardy–Ramanujan theorem]]: the normal order of $\omega(n)$, the number of distinct [[prime factor]]s of $n$, is $\log\log n$;
1 KB (235 words) - 08:12, 4 November 2023
Let $f$, $g$ be functions on the [[natural number]]s. We say that $f$ has average order $g$ if the [[asymptotic equality]]
* The average order of $d(n)$, the [[number of divisors]] of $n$, is $\log n$;
1 KB (231 words) - 19:41, 17 November 2023
''of a natural number $n$''
The sum of the positive integers divisors of a natural number $n$, including $1$ and $n$:
2 KB (268 words) - 19:41, 17 November 2023
[[Number theory|number theory]] in which ideas and methods from
[[Probability theory|probability theory]] are used.
10 KB (1,718 words) - 04:55, 8 August 2018
...mber generation is possible by performing a measurement process in quantum theory.
...ision) may be potentially influenced by cosmic rays making the calculation probabilistic.
12 KB (1,954 words) - 17:47, 26 December 2013
A central technique in the probabilistic method. It is used to prove the existence of a "good" object even when th
Here, the number of events, $|I|$, may be arbitrarily large, giving the Lovász local lemma
4 KB (586 words) - 19:47, 18 December 2014
The branch of [[mathematical programming]] in which one studies the theory and methods for the solution of conditional extremal problems, given incomp
...all (or nearly all) realizations are called rigid. Restrictions are called probabilistic if discrepancies are admissible in the conditions of the problem with a pro
5 KB (768 words) - 13:11, 3 May 2016
$#C+1 = 180 : ~/encyclopedia/old_files/data/P110/P.1100230 Probabilistic metric space
...tion was introduced by K. Menger in 1942 and has since been developed by a number of authors. A treatment, comprehensive up to 1983, may be found in {{Cite|S
12 KB (1,757 words) - 08:07, 6 June 2020
The union of a finite number of sets thin at the point $ y _ {0} \in \mathbf R ^ {n} $
...]</TD> <TD valign="top"> N.S. Landkof, "Foundations of modern potential theory" , Springer (1972) (Translated from Russian)</TD></TR></table>
4 KB (696 words) - 08:25, 6 June 2020
For a given set $B\subseteq\mathbf N$ one writes $r_{B,h}(x)$ for the number of representations of the non-negative integer $x$ as a sum of $h$ terms fr
...y many positive integers $x$. See also [[#References|[a5]]] for a modified probabilistic construction which can be derandomized to yield a polynomial-time algorithm
4 KB (658 words) - 19:37, 29 March 2024
* E.D. Cashwell, C.J. Everett,. "The ring of number-theoretic functions", ''Pacific J. Math.'' '''9''' (1959) 975-985 {{ZBL|009
* Henri Cohen, "Number Theory: Volume II: Analytic and Modern Tools", Graduate Texts in Mathematics '''24
2 KB (358 words) - 17:25, 11 November 2023
...error of $a$. To characterize the error one usually states bounds on it. A number $\Delta(a)$ such that
is called a bound on the absolute error. A number $\delta(a)$ such that
3 KB (519 words) - 22:31, 1 November 2014
$#C+1 = 22 : ~/encyclopedia/old_files/data/N067/N.0607940 Number theory
...figures, were the first and the most ancient mathematical concepts. Number theory arose from problems in [[Arithmetic|arithmetic]] connected with the multipl
10 KB (1,503 words) - 08:03, 6 June 2020
The collective name for systems consisting of a large number of interconnected elements. It must be emphasized that this is an informal
...ir names, such as systems analysis, system-technology, and general systems theory, among others.
4 KB (532 words) - 17:22, 7 February 2011
Traditionally, a composite natural number $n$ is called a pseudo-prime if $2^{n-1} \equiv 1$ modulo $n$, for it has l
...ude any composite number that acts like a prime in some realization of a [[probabilistic primality test]]. That is, it satisfies some easily computable necessary, b
3 KB (516 words) - 17:58, 8 November 2014
''in the theory of random walk''
...f the different factorizations of the type indicated which are obtained by probabilistic consideration and are written in terms of characteristic functions of the j
2 KB (347 words) - 11:27, 1 August 2014
...s, account must be taken of the size of the lot to be controlled (i.e. the number of items it contains), the results of the control of previous lots and othe
...ry of statistical acceptance control, methods are devised to calculate the probabilistic characteristics of the control plans and to estimate the efficiency of the
9 KB (1,409 words) - 08:23, 6 June 2020
''in probability theory''
...from the very beginnings of mathematical [[Probability theory|probability theory]].
15 KB (2,228 words) - 22:12, 5 June 2020
...schemes where the (random) results of an experiment can be described by a number or a finite collection of numbers to schemes where the results of an experi
The basic concepts of probability theory, such as the [[Characteristic function|characteristic function]], the [[Mat
5 KB (804 words) - 08:09, 6 June 2020
...he algebraic and geometric analogues of such problems concerning algebraic number fields and sets of lattice points. These problems are said to be additive,
...s branches of number theory — analytic, algebraic and probabilistic number theory.
10 KB (1,609 words) - 13:03, 8 February 2020
A probabilistic model for the study of frequency characteristics of various graph parameter
...ses the concept of a random graph makes it possible to utilize probability theory as a tool in order to obtain asymptotic solutions of enumeration problems.
6 KB (999 words) - 20:05, 15 March 2023
...also, quite apart from his many and striking contributions to probability theory.
probabilistic work of the Petersburg School and wrote a splendid commentary on
10 KB (1,435 words) - 08:56, 25 March 2023
E.V. Novoselov's theory of integration for arithmetic functions (see [[#References|[a3]]]) also lea
...<TD valign="top"> E.V. Novoselov, "A new method in probabilistic number theory" ''Transl. Amer. Math. Soc.'' , '''52''' (1966) pp. 217–275 ''Izv. Ak
3 KB (410 words) - 11:03, 21 October 2017
...hms (cf. [[Algorithms, theory of|Algorithms, theory of]]), there emerged a number of modifications of the original definition of Turing. The version given he
...sented as an automatically-functioning system capable of being in a finite number of internal states and endowed with an infinite external memory, called a t
14 KB (2,355 words) - 12:40, 28 December 2013
...The mathematical ingredients of quantum probability theory derive from the theory of operator algebras, as founded by J. von Neumann and developed by M.A. Na
The fundamental object of Kolmogorov's probability theory is a triple $ ( \Omega , {\mathcal F} , {\mathsf P} ) $,
11 KB (1,494 words) - 08:08, 6 June 2020
century probability theory. His major probabilistic work, the ''
probability theory, which contained, according to Laplace's
16 KB (2,329 words) - 12:09, 28 October 2023
...practical conclusions. Here, by statistical data is meant information on a number of objects in some, more or less extensive, collection, which have some spe
...a very preliminary orientation on the situation from the statistics on the number of excellent, good, adequate, and inadequate appraisals made by his or her
21 KB (3,113 words) - 12:48, 13 January 2024
...ndent functions and has applications both in probability theory and in the theory of orthogonal series.
In probabilistic terminology the theorem above means that if $ \sum c _ {n} ^ {2} < + \in
5 KB (763 words) - 08:09, 6 June 2020
[[Category:Probabilistic number theory]]
...alpha$, $0 < \alpha < 1$, having the following property: For every natural number $s$, any given $s$-tuple $\delta = (\delta_1,\ldots,\delta_s)$ consisting o
6 KB (960 words) - 08:11, 4 November 2023
a) $t ( M ; 1,1 )$ is the number of bases of $M$;
b) $t ( M ; 2,1 )$ is the number of independent subsets of $M$;
11 KB (1,736 words) - 06:27, 15 February 2024
...utational complexity studies continuous problems and tends to use the real number model; see [[#References|[a1]]], [[#References|[a3]]] as well as [[Optimiza
...se (see also [[Bayesian numerical analysis|Bayesian numerical analysis]]), probabilistic, randomized (see also [[Monte-Carlo method|Monte-Carlo method]]; [[Stochast
6 KB (871 words) - 17:45, 1 July 2020
...plitudes of disjoint events add, the amplitude of an event being a complex number whose square is the probability of the event. A more precise way of saying
...ssical probabilistic model. Since this axiom cannot be true in the quantum probabilistic model, the following problems naturally arise:
22 KB (3,198 words) - 08:08, 6 June 2020
argued fashion to humanistic and social problems. His probabilistic work is full
in terms of a finite number of explicit functions, 40 years before Liouville.
8 KB (1,269 words) - 18:56, 7 March 2024
...l computational complexity theory (cf. also [[Complexity theory|Complexity theory]]; [[Computational complexity classes|Computational complexity classes]]).
...machines (RAMs) is to enhance the RAM by giving it access to an unbounded number of quantum bits which can be controlled by applying quantum gates (cf. [[Qu
15 KB (2,154 words) - 17:45, 1 July 2020
[[Category:Distribution theory]]
the number of trials) is large and $ p $(
7 KB (971 words) - 17:34, 6 January 2024
...heorem on multiplicative functions" , ''Proc. Amalfi Conf. Analytic Number Theory'' , '''1989''' (1992) pp. 17–34</TD></TR>
...op">[a5]</TD> <TD valign="top"> P.D.T.A. Elliott, "Probabilistic number theory" , '''I–II''' , Springer (1979–1980)</TD></TR>
5 KB (715 words) - 10:00, 9 January 2021
R.P. Stanley [[#References|[a11]]] proved that the number of acyclic orientations of a graph $ G $
of the [[chromatic polynomial]] $ \chi ( G;k ) $ (computing this number is $ \# P $-complete [[#References|[a13]]]). More generally, suppose that
10 KB (1,497 words) - 06:32, 26 March 2023
...y theory stating conditions under which sums or other functions of a large number of independent or weakly-dependent random variables have a probability dist
...his purpose there are inequalities and asymptotic expansions (and also the theory of [[Probability of large deviations|probability of large deviations]]; see
20 KB (2,992 words) - 07:15, 21 March 2023
The mathematical discipline devoted to the theory and methods of solving problems about extrema of linear functions on sets o
The basis for studying properties of linear programming problems is the theory of duality. The dual problem to the problem \eqref{eq:1}–\eqref{eq:3} is
12 KB (1,882 words) - 22:25, 1 September 2016
$#C+1 = 23 : ~/encyclopedia/old_files/data/D110/D.1100030 De Haan theory
...riation, initiated in 1930 by the Yugoslav mathematician J. Karamata. This theory studies asymptotic relations of the form
5 KB (688 words) - 17:32, 5 June 2020
The theory of Walsh series is distinguished from that of other orthonormal series (cf.
...any Walsh series forms a [[Martingale|martingale]]. This opens the way for probabilistic techniques to be used on Walsh series, e.g., martingale convergence theorem
4 KB (668 words) - 08:28, 6 June 2020
...uters and of programming them relies, to a large extent, on ideas from the theory of algorithms and mathematical cybernetics. In turn, practical work with co
...input. A probabilistic machine is a machine that is provided with a random-number generator or one that has a program in which the transition from one comman
18 KB (2,656 words) - 04:11, 6 June 2020
...olution of problems of enumeration, in particular the determination of the number of configurations of a given class. The simplest examples of combinatorial
The number of combinations of size $ m $
28 KB (4,167 words) - 17:45, 4 June 2020
In the case of a finite number of states, equations (3) and (4) hold, provided that the limits in (5) exis
|valign="top"|{{Ref|GS}}|| I.I. Gihman, A.V. Skorohod, "The theory of stochastic processes" , '''2''' , Springer (1979) (Translated from Russi
5 KB (778 words) - 22:14, 5 June 2020
L.E. Dickson was the first to seriously study various algebraic and number-theoretic properties of these polynomials. In particular, he studied these
...dditional details. See [[#References|[a10]]] for a survey of algebraic and number-theoretic properties as well as applications of Dickson polynomials; see al
15 KB (2,207 words) - 16:45, 1 July 2020
French mathematics from 1815 to 1840. His contribution to probability theory is
importance of Laplace's probabilistic opus. It was Poisson who gave the first
11 KB (1,710 words) - 19:08, 6 March 2024
...nt which must be estimated from the sampling data). In the latter case the theory is usually employed to study the properties of random objects (for example,
...] and are often employed in sociological studies. According to probability theory, the sample will correctly reproduce the properties of the population as a
11 KB (1,705 words) - 08:12, 6 June 2020
...r, this is not achievable anymore in a perfect way. The effects of quantum theory may begin to have an influence on the outcome of the calculation.
Quantum theory would not be probabilistical without measurements. The probabilistic nature could be explained as a result of an interaction between the measure
27 KB (4,213 words) - 18:53, 26 April 2014
The [[Capacity|capacity]] theory for Riesz potentials can be constructed, for example, on the basis of the c
...tential is called after M. Riesz (see [[#References|[2]]]), who obtained a number of important properties of Riesz potentials; for the first time such potent
6 KB (860 words) - 06:02, 12 July 2022
...mata, reversible automata, etc. (cf. [[Automaton, probabilistic|Automaton, probabilistic]]). Structural automata also have several generalizations, mainly consistin
...of a non-deterministic automaton is reflected in the fact that a different number of states may be necessary to represent the same event using a non-determin
18 KB (2,794 words) - 10:01, 25 April 2020
[[Category:Distribution theory]]
...ion on the straight line, on a finite-dimensional Euclidean space and to a number of other, even more general, cases. The one-dimensional case will be consid
11 KB (1,489 words) - 09:00, 13 January 2024
...ge behaviour and, furthermore, the sheer size of the required rule set and number of lexical items has up to now (2000) prevented linguists from providing a
...} { 2 n } \\ { n - 1 } \end{array} \right)$, where $\mathcal{C}_n$ is the number of ways to parenthesize a formula of length $n$ [[#References|[a5]]].
18 KB (2,627 words) - 17:03, 1 July 2020
physics. His published works are mainly in German, although a number,
usually remembered is his famous example on the number of men
13 KB (1,926 words) - 18:33, 17 April 2024
is the condition number of the matrix $ A $,
...hen one is solving problems in which the solution is the result of a large number of consecutively performed arithmetic operations.
18 KB (2,636 words) - 08:00, 13 May 2022
...languages with infinitely many sentences. The main task of formal language theory is the study of finitary specifications of infinite languages.
...such [[Cryptography|cryptography]], is inseparably connected with language theory. The input and output sets of a computational device can be viewed as langu
28 KB (4,202 words) - 08:53, 1 February 2020
| This article ''Queueing Theory'' was adapted from an original article by U. Narayan Bhat, which appeared i
<center>'''Queueing Theory'''</center>
20 KB (3,169 words) - 11:35, 3 April 2016
[[Category:Distribution theory]]
...hin canonical representation|Lévy–Khinchin canonical representation]]. The probabilistic meaning of the functions $ N $
8 KB (1,006 words) - 01:14, 19 January 2022
if there exists a number $ R > 0 $
belongs to an infinite number of shadows $ {\mathcal S} _ {o} ( go,R ) $,
8 KB (1,145 words) - 22:11, 5 June 2020
.... More importantly, they can be of no use in the modern context of a large number of users. Codes (cf. [[Code]]) are instances of restricted cryptosystems. T
...ey has actually been used. General cryptosystems should allow a very large number of possible keys, so as to discourage exhaustive search (trying systematica
35 KB (5,528 words) - 19:33, 19 January 2024
the theory of Brownian movement concided with those of Einstein. The
during the period in L'vov, and is heavily probabilistic and
11 KB (1,571 words) - 07:00, 15 March 2024
...ce of a "true" water level which is identified with a [[Real number|real number]] times the measurement unit. But this is not realistic. The formal descrip
...t )$. Specializing the membership functions from fuzzy set theory, a fuzzy number is characterized by its so-called characterizing function $\xi ( . )$, whic
11 KB (1,572 words) - 17:45, 1 July 2020
20 papers in analytic number theory over the next seven years.
subsequent fundamental probabilistic contributions.
13 KB (2,000 words) - 18:58, 5 March 2024
...timators used; and the investigation of the probability distributions of a number of statistics that are used to construct statistical tests for the verifica
...hese models and the analysis of their properties (more precisely, and in a probabilistic formulation, of their distribution laws, confidence regions, etc.). Thus, l
27 KB (3,850 words) - 14:44, 7 June 2020
...ecision problems accepted by Turing machines using at most some polynomial number of steps, $P = \cup _ { k = 1 } ^ { \infty } \operatorname { DTIME } [ n ^
...that any computation may take on this input, not the exponentially greater number of steps required to simulate all possible computations on that input.
22 KB (3,250 words) - 17:43, 1 July 2020
...\Delta ( G ) \leq \chi ^ { \prime } ( G )$. The origins of chromatic graph theory may be traced back to 1852, with the birth of the [[Four-colour problem|fou
...l N P$-complete (cf. [[#References|[a12]]]; [[Complexity theory|Complexity theory]]), and that its solution would imply the four-colour theorem. Very little
12 KB (1,873 words) - 09:33, 19 January 2021
An extremum problem having several, or an unknown number of, local extrema.
...lations of the non-local search process can be optimized with respect to a number of indices taking account of limitations on the calculating means, a priori
9 KB (1,335 words) - 08:01, 6 June 2020
...: ~/encyclopedia/old_files/data/S110/S.1100290 Stochastic limit of quantum theory
Here, for each real number $ t $(
8 KB (1,210 words) - 08:23, 6 June 2020
as $t \rightarrow 0$, where $A$ is the area, $L$ the perimeter and $r$ the number of holes of $\Omega$, so at least these features of the domain can be recov
...( \Omega _ { 2 } )$ for all $t \in ( 0,1 )$. Another example of the use of probabilistic methods is the proof of (a10) by R. Bañuelos and T. Carroll [[#References|
12 KB (1,753 words) - 10:12, 16 March 2023
...concept of an arithmetical semi-group $G$ (cf. [[Abstract analytic number theory]]; [[Semi-group]]).
...ion. Theorems of the latter kind may then be referred to as abstract prime number theorems within the context considered.
14 KB (2,037 words) - 09:45, 11 November 2023
...re $\mu$, the cost of optimal methods does not depend exponentially on the number of variables, see [[#References|[a8]]]. On the other hand, often a [[Curse
...al analysis" S.S. Gupta (ed.) J.O. Berger (ed.) , ''Statistical Decision Theory and Related Topics IV'' , '''1''' , Springer (1988) pp. 163–175</td></t
6 KB (908 words) - 18:44, 21 March 2024
popularize British mathematical statistics. He was a pioneer in the theory of
problems arising in the general theory of relativity. Fortet wrote
11 KB (1,766 words) - 18:41, 6 March 2024
queueing theory.
with a thesis on number theory under supervision of I. Shur.
12 KB (1,866 words) - 14:20, 10 March 2024
A set function arising in [[Potential theory|potential theory]] as the analogue of the physical concept of the electrostatic capacity.
is the number
17 KB (2,487 words) - 06:29, 30 May 2020
...te evolution of the process is observed by the controller. A corresponding
theory has also been developed in the case of partial observations (incomplete dat
...opediaofmath.org/legacyimages/c/c026/c026010/c026010102.png" /> is a fixed
number. By defining suitable functions <img align="absmiddle" border="0" src="http
107 KB (14,649 words) - 21:43, 15 December 2020
The logarithm to the base 2 of the smallest number of points in an $ \epsilon $-
...nd definitions of information theory (cf. [[Information theory|Information theory]]). $ {\mathcal H} _ \epsilon ( C , X ) $
12 KB (1,685 words) - 20:48, 2 January 2021
of queueing theory for application to telephone traffic and was a precursor
to much modern theory of stochastic processes.
7 KB (990 words) - 13:25, 18 March 2023
of mathematics, which encompassed path-breaking work in [[probability theory]]. The
capabilities" which was in part a stimulus for both the probabilistic works:
12 KB (1,774 words) - 20:00, 7 March 2024
...ry of random fields of a general form is almost identical with the general theory of random functions. One can only obtain more interesting concrete results
The theory of Markov random fields and $ L $-
9 KB (1,319 words) - 08:09, 6 June 2020
He worked simultaneously at a number of other institutions: his first
statistics relates to the theory of correlation and regression, the bivariate
10 KB (1,403 words) - 16:21, 10 March 2024
'''Summary.''' En'ko was a Russian physician whose probabilistic
In 1760, [[Bernoulli, Daniel|Daniel Bernoulli]] produced a model for the number of susceptibles in
9 KB (1,380 words) - 08:31, 9 March 2024
of the divinity of number"; this grew into a deep love of mathematics,
theory of (linear) sets which are now called after him, to break the confines
14 KB (2,139 words) - 18:37, 8 March 2024
Fréchet in particular that we owe the first theory
Kolmogorov was to make use of this to axiomatize probability theory,
11 KB (1,737 words) - 12:11, 23 November 2023
...er of bit operations. This is a tiny fraction: the binary logarithm of the number of atoms in the known Universe is under $300$.
..., only essential errors (i.e., not correctable from the context) have this probabilistic detection guarantee. There is a proofreader procedure, also running in poly
11 KB (1,701 words) - 10:26, 24 August 2014
...1 = 79 : ~/encyclopedia/old_files/data/S087/S.0807320 Statistical decision theory
...n a broader interpretation of the term, statistical decision theory is the theory of choosing an optimal non-deterministic behaviour in incompletely known si
12 KB (1,679 words) - 08:23, 6 June 2020
$#C+1 = 56 : ~/encyclopedia/old_files/data/Q076/Q.0706850 Queueing theory
The branch of probability theory in which one studies mathematical models of various kinds of real queues (c
12 KB (1,824 words) - 08:09, 6 June 2020
...ong-running 1960's TV show "Let's make a Deal". Selvin's letter provoked a number of people to write to the author, and he published a second letter in respo
Vos Savant proceded to give a number of simple arguments for the good answer: switch, it doubles your chance of
22 KB (3,718 words) - 20:19, 12 March 2016
A [[Markov process|Markov process]] with finite or countable state space. The theory of Markov chains was created by [[Markov, Andrei Andreevich|A.A. Markov]] w
In the probabilistic description of a Markov chain $ \xi ( t) $
12 KB (1,726 words) - 08:04, 14 January 2024
...|Gauss–Bonnet theorem]] and its multi-dimensional variants. Subsequently a number of generalizations of index formulas were obtained for objects of a more co
Formula (2) admits a number of modifications. The rational cohomology class $ \mathop{\rm ch} [ \si
41 KB (5,908 words) - 22:12, 5 June 2020
...fundamental work of C. Shannon [[#References|[1]]] published in 1948. The theory of alphabetical coding is a branch of applied mathematics that consists in
...uage of the messages. In the simplest case, the source of information is a probabilistic scheme at the output of which one of the letters of $ B $
9 KB (1,453 words) - 18:06, 15 November 2023
$#C+1 = 55 : ~/encyclopedia/old_files/data/S083/S.0803330 Scheduling theory
...tive) sets of operations. The area of application of results in scheduling theory include management, production, transportation, computer systems, construct
21 KB (3,117 words) - 10:04, 18 February 2021
...f $G$. Here, $2$ is the reciprocal of the contraction ratio and $3$ is the number of parts of $G$ similar to the whole in that ratio (see, for example, [[#Re
...g" /> (see [[#References|[a19]]], Chapts. 2–6 and 10, where a mathematical theory of complex dimensions is developed in the one-dimensional case).
26 KB (3,837 words) - 19:21, 17 March 2024
Complètes'', Volume VIII). In this work, he applied the theory of
of Cournot's earlier work, in the context of game theory. Nash was awarded
13 KB (1,947 words) - 18:43, 4 March 2024
architect of the statistical theory of extreme values.
statistics. The hostility of an increasing number of his colleagues
12 KB (1,919 words) - 18:58, 7 March 2024
One of the basic facts of the theory of martingales is that the structure of a martingale (submartingale) $ X
Among the basic results of the theory of martingales is Doob's inequality: If $ X = ( X _ {n} , {\mathcal F} _
12 KB (1,716 words) - 08:14, 9 January 2024
problem" for which his name persists in probability theory.
the theory and empirical data on the other. It
13 KB (2,017 words) - 09:45, 15 August 2023