A concept in the theory of harmonic functions (cf. [[Harmonic function|Harmonic function]]) connect
As regards applications to the theory of functions of a complex variable, the dependence of a harmonic measure on
8 KB (1,149 words) - 19:43, 5 June 2020
...]] defined on the Wiener space (cf. [[Wiener space, abstract|Wiener space, abstract]]). Let $C$ be the space of continuous functions from $\mathbf R_+$ to $\ma
b) they arise in connection with classical [[Potential theory|potential theory]], cf. [[#References|[a1]]];
7 KB (1,053 words) - 22:49, 7 March 2019
...ctions are also found in the work of F. Hartogs [[#References|[1]]] on the theory of analytic functions of several complex variables; the systematic study of
...ewton potential|Newton potential]] and [[Logarithmic potential|logarithmic potential]] of non-negative masses, when written with a minus sign, are subharmonic f
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As an example of this abstract theory, consider a free vibration of a membrane occupying a region $\Omega$. It is
...orms of vibration, of the ratio of the average kinetic energy over average potential energy, computed over a single cycle of vibration.
9 KB (1,326 words) - 07:37, 24 November 2023
...ysis|Harmonic analysis]]; [[Harmonic analysis, abstract|Harmonic analysis, abstract]]; for the notion of a hypergroup, see also [[Generalized displacement oper
...fundamentals and examples, as well as results oriented towards probability theory, see [[#References|[a2]]]. It contains an almost complete and up-to-date (1
9 KB (1,358 words) - 22:11, 5 June 2020
|valign="top"|{{Ref|He}}|| L. Helms, "Potential theory". Universitext. Springer-Verlag London, Ltd., London, 2009. {{MR|2526019}}
|valign="top"|{{Ref|HS}}|| E. Hewitt, K.R. Stromberg, "Real and abstract analysis" , Springer (1965) {{MR|0188387}} {{ZBL|0137.03202}}
8 KB (1,260 words) - 15:29, 5 January 2017
to which Fredholm's theory is applicable (cf. also [[Fredholm theorems|Fredholm theorems]]). For this
to which Fredholm's theory (cf. [[#References|[5]]]) is applicable (cf. [[Fredholm theorems|Fredholm t
33 KB (4,788 words) - 08:53, 13 May 2022
...th the success of the intuitionists there appeared in [[Proof theory|proof theory]] (created by D. Hilbert with the aim of justifying set-theoretic mathemati
...e framework of the [[Abstraction of potential realizability|abstraction of potential realizability]] with total exclusion of the idea of an actual infinity; 3)
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...simplest case the Lagrangian is the difference between the kinetic and the potential energy of the system, and the motions of the system coincide with the extre
...n the presence of "non-classical" constraints of inequality type, in the theory of optimal control, a necessary condition for a strong extremum of the func
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...f. [[Optimal control, mathematical theory of|Optimal control, mathematical theory of]]), but this name is rarely used in English: the most common name is (co
...lities in martingale theory, Chernov's inequality in classical probability theory, Kullback's inequality in statistics, etc.), via the easy to prove but fund
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...]], [[#References|[a2]]]), allowing one to prove deep results in potential theory. Let $ \Omega $
...top">[a4]</TD> <TD valign="top"> E. Hewitt, K.R. Stromberg, "Real and abstract analysis" , Springer (1965) {{MR|0188387}} {{ZBL|0137.03202}} </TD></TR><
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...golyubov in the justification of the dispersion relations in quantum field theory ([[#References|[1]]], Appendix A). The modern formulation is as follows. Le
...m field theory, in the theory of partial differential equations and in the theory of boundary values of holomorphic functions (especially of functions of sev
11 KB (1,586 words) - 07:17, 13 June 2022
...integral|Cauchy integral]] formula plays a dominant and unique role in the theory of functions. In contrast, in higher dimensions there are numerous generali
...lace operator|Laplace operator]]) and $N$ is the [[Newton potential|Newton potential]] on $\mathbf{R} ^ { 2 n }$ ($= {\bf C}^ { n }$). As in dimension one, ther
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...erturbation theory (a branch of general [[Perturbation theory|perturbation theory]]). Thus, the spectrum is an upper semi-continuous function of the operator
...he theory of wave operators, closely connected with the quantum-mechanical theory of scattering (see [[#References|[2]]]). The wave operator $ W ( A , B )
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$#C+1 = 37 : ~/encyclopedia/old_files/data/P075/P.0705430 Proof theory
...ented his formalization method, which is one of the basic methods in proof theory.
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...t the same time classes of locally convex spaces that are important in the theory and applications) are normed spaces, countably-normed spaces and Fréchet s
...e with its dual or [[Adjoint space|adjoint space]]. The foundation of this theory of [[Duality|duality]] for locally convex spaces is the [[Hahn–Banach the
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...p">[3a]</TD> <TD valign="top"> N. Wiener, "Certain notions in potential theory" ''J. Math. Phys.'' , '''3''' (1924) pp. 24–51 {{MR|}} {{ZBL|50.0646
...ldysh theorem]]). All these results have their counterpart in the abstract theory of harmonic spaces (cf. [[Harmonic space|Harmonic space]]), cf. [[#Referenc
10 KB (1,417 words) - 08:05, 6 June 2020
...se existence can be verified by a potentially realizable construction, and abstract set-theoretic objects of study. In classical mathematics methods of establi
...cism of classical mathematics is not directly related to antinomies in set theory (cf. [[Antinomy|Antinomy]]). The appearance of antinomies can be regarded a
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...is called in mechanics, a material point) moving in a potential field with potential $ U ( x _ {1} , x _ {2} , x _ {3} ) $
...this manifold reflects the fact that the energy of a particle moving in a potential field is conserved, i.e. $ p ^ {2} / 2m + U ( x) $
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...theory, Riemann surfaces, subharmonic functions and function algebras. The theory of boundary properties of analytic functions is closely connected with vari
three main approaches can be distinguished in the theory of boundary properties of analytic functions.
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