scholarly journals On the Balayage for Logarithmic Potentials

1967 ◽  
Vol 29 ◽  
pp. 229-241 ◽  
Author(s):  
Nobuyuki Ninomiya
Keyword(s):  
Continuous Function ◽  
Compact Support ◽  
Positive Measure ◽  
Logarithmic Potential ◽  
Compact Set ◽  
Logarithmic Potentials ◽  
Capacity Zero

In this paper, we shall consider the logarithmic potential where μ is a positive measure in the plane, P and Q are any points and PQ denotes the distance from P to Q. In general, consider the potential of a positive measure μ taken with respect to a kernel K(P, Q) which is a continuous function in P and Q and may be + ∞ for P = Q. A kernel K (P, Q) is said to satisfy the balayage principle if, given any compact set F and any positive measure μ with compact support, there exists a positive measure μ′ supported by F such that K(P, μ′) = K(P, μ) on F with a possible exception of a set of k-capacity zero and K(P, μ′)≦K(P, μ) everywhere. A kernel K(P, Q) is said to satisfy the equilibrium principle if, given any compact set F there exists a positive measure λ supported by F such that K(P, λ) = V (a constant) on F with a possible exception of a set of K-capacity zero and K(p, λ)≦V everywhere.

scholarly journals Positively Infinite Singularities of a Superharmonic Function

1968 ◽  
Vol 31 ◽  
pp. 89-96
Author(s):  
Kikuji Matsumoto
Keyword(s):  
Complex Plane ◽  
Unit Disc ◽  
Measure Zero ◽  
Logarithmic Potential ◽  
Logarithmic Capacity ◽  
Superharmonic Function ◽  
Compact Set ◽  
Linear Measure ◽  
Capacity Zero ◽  
Positive Capacity

Let E be a compact set of logarithmic capacity zero in the complex plane. Then the following is well-known as Evans-Selberg’s theorem [1] [8]: there is a measure with support contained in E such that its logarithmic potential is positively infinite at each point of E. But such a potential does not exist for E of logarithmic positive capacity. Now suppose that E is contained in the circumference of the unit disc |z| < 1 and is of linear measure zero.

A generalization of the Pompeiu mean-value theorem to compact sets

2019 ◽  
Vol 35 (2) ◽  
pp. 147-152
Author(s):  
LARISA CHEREGI ◽  
VICUTA NEAGOS ◽  
◽  
Keyword(s):  
Continuous Function ◽  
Mean Value ◽  
Mean Value Theorem ◽  
Compact Set ◽  
Compact Sets

We generalize the Pompeiu mean-value theorem by replacing the graph of a continuous function with a compact set.

scholarly journals Two modifications of extension of piyavskii’s global optimization algorithm to a function continuous on a compact interval and its convergence

2021 ◽  
pp. 70-85
Author(s):  
Владислав Иванович Заботин ◽  
Павел Андреевич Чернышевский
Keyword(s):  
Global Optimization ◽  
Continuous Function ◽  
Optimization Algorithm ◽  
Lipschitz Continuity ◽  
Compact Interval ◽  
Compact Set ◽  
Test Functions ◽  
Global Optimization Algorithm ◽  
Lipschitz Property ◽  
Set Function

В работах R.J. Vanderbei доказано, что непрерывная на выпуклом компактном множестве функция обладает свойством $\varepsilon $-липшицевости, обобщающим классическое понятие липшицевости. На основе этого свойства R.J. Vanderbei предложено одно обобщение метода Пиявского поиска глобального минимума непрерывной на отрезке функции. В данной работе предлагаются одна модификация этого метода для положительной $\varepsilon $-константы и одна модификация для положительной $\varepsilon $-константы и условия останова, не зависящего от выбора $\varepsilon $. Доказана сходимость предлагаемых алгоритмов, приведены результаты численных экспериментов на основе применения разработанной программы. Данные методы могут быть применены для оптимизации любых непрерывных на отрезке функций, например, при решении некоторых обратных задачах баллистики и в экономике в прямых задачах потребительского выбора маршаллианского типа с переменными ценами благ и с непрерывной функцией полезности. R.J. Vanderbei in his works proves that any continuous on a compact set function has the $\varepsilon $-Lipschitz property which extends conventional Lipschitz continuity. Based on this feature Vanderbei proposed one extension of Piyavskii’s global optimization algorithm to the continuous function case. In this paper we propose one modification of the Vanderbei’s algorithm for a positive $\varepsilon $-constant and another modification for a positive $\varepsilon $-constant and $\varepsilon $ value independent termination condition. We prove proposed methods convergence and perform several computational experiments with designed software for known test functions.

EFFICIENT HIGHER-ORDER NEURAL NETWORKS FOR CLASSIFICATION AND FUNCTION APPROXIMATION

1992 ◽  
Vol 03 (04) ◽  
pp. 323-350 ◽  
Author(s):  
JOYDEEP GHOSH ◽  
YOAN SHIN
Keyword(s):  
Continuous Function ◽  
Function Approximation ◽  
Order Approximation ◽  
Higher Order ◽  
Compact Set ◽  
Fast Learning ◽  
Learning Rates ◽  
Hidden Layer ◽  
And Function ◽  
Higher Order Networks

This paper introduces a class of higher-order networks called pi-sigma networks (PSNs). PSNs are feedforward networks with a single “hidden” layer of linear summing units and with product units in the output layer. A PSN uses these product units to indirectly incorporate the capabilities of higher-order networks while greatly reducing network complexity. PSNs have only one layer of adjustable weights and exhibit fast learning. A PSN with K summing units provides a constrained Kth order approximation of a continuous function. A generalization of the PSN is presented that can uniformly approximate any continuous function defined on a compact set. The use of linear hidden units makes it possible to mathematically study the convergence properties of various LMS type learning algorithms for PSNs. We show that it is desirable to update only a partial set of weights at a time rather than synchronously updating all the weights. Bounds for learning rates which guarantee convergence are derived. Several simulation results on pattern classification and function approximation problems highlight the capabilities of the PSN. Extensive comparisons are made with other higher order networks and with multilayered perceptrons. The neurobiological plausibility of PSN type networks is also discussed.

scholarly journals On Exceptional Values of Meromorphic Functions with the Set of Singularities of Capacity Zero

1961 ◽  
Vol 18 ◽  
pp. 171-191 ◽  
Author(s):  
Kikuji Matsumoto
Keyword(s):  
Meromorphic Function ◽  
Meromorphic Functions ◽  
Essential Singularity ◽  
Compact Set ◽  
Capacity Zero ◽  
A Value

LetEbe a compact set in thez-plane and letΩbe its complement with respect to the extendedz-plane. Suppose thatEis of capacity zero. ThenΩis a domain and we shall consider a single-valued meromorphic functionw=f(z) onΩwhich has an essential singularity at each point ofE. We shall say that a valuewis exceptional forf(z)at a point ζ ∈Eif there exists a neighborhood of C where the functionf(z)does not take this valuew.

scholarly journals AN ALGORITHM FOR FINDING ALL ZEROS OF VECTOR FUNCTIONS

2008 ◽  
Vol 77 (3) ◽  
pp. 353-363 ◽  
Author(s):  
IBRAHEEM ALOLYAN
Keyword(s):  
Continuous Function ◽  
Theoretical Analysis ◽  
Numerical Simulations ◽  
Numerical Computation ◽  
Compact Set ◽  
Numerical Examples ◽  
Real Roots ◽  
Multiple Variables ◽  
Subdivision Algorithm

AbstractComputing a zero of a continuous function is an old and extensively researched problem in numerical computation. In this paper, we present an efficient subdivision algorithm for finding all real roots of a function in multiple variables. This algorithm is based on a simple computationally verifiable necessity test for the existence of a root in any compact set. Both theoretical analysis and numerical simulations demonstrate that the algorithm is very efficient and reliable. Convergence is shown and numerical examples are presented.

scholarly journals Maximum Principles in the Potential Theory

1963 ◽  
Vol 23 ◽  
pp. 165-187 ◽  
Author(s):  
Masanori Kishi
Keyword(s):  
Maximum Principle ◽  
Potential Theory ◽  
Positive Measure ◽  
Hausdorff Space ◽  
Finite Energy ◽  
Maximum Principles ◽  
Compact Set ◽  
Locally Compact ◽  
Almost Everywhere ◽  
Locally Compact Hausdorff Space

Ninomiya, in his thesis [13] on the potential theory with respect to a positive symmetric continuous kernel G on a locally compact Hausdorff space Ω, proves that G satisfies the balayage (resp. equilibrium) principle if and only if G satisfies the domination (resp. maximum) principle. He starts from the Gauss-Ninomiya variation and shows that for any given compact set K in Ω and any positive upper semi-continuous function u on K, there exists a positive measure μ on K such that its potential Gμ is ≥ u on the support of μ and Gμ≥u on K almost everywhere with respect to any positive measure with finite energy.

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