Algorithms for projecting a point onto a level surface of a continuous function on a compact set

2014 ◽  
Vol 54 (9) ◽  
pp. 1395-1401 ◽  
Author(s):  
N. K. Arutyunova ◽  
A. M. Dulliev ◽  
V. I. Zabotin
Keyword(s):  
Continuous Function ◽  
Level Surface ◽  
Compact Set

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 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 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 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 On compactness and connectedness of the paratingent

2016 ◽  
Vol 70 (2) ◽  
pp. 91
Author(s):  
Wojciech Zygmunt
Keyword(s):  
Continuous Function ◽  
Lipschitz Condition ◽  
Compact Set ◽  
Connected Set

In this note we shall prove that for a continuous function \(\varphi : \Delta\to\mathbb{R}^n\), where \(\Delta\subset\mathbb{R}\),  the paratingent of \(\varphi\) at \(a\in\Delta\) is a non-empty and compact set in \(\mathbb{R}^n\) if and only if \(\varphi\) satisfies Lipschitz condition in a neighbourhood of \(a\). Moreover, in this case the paratingent is a connected set.

Generalized ʎ-Closed Sets and (ʎ-ʏ)*-Continuous Function

2017 ◽  
Vol 4 (ICBS Conference) ◽  
pp. 1-17 ◽  
Author(s):  
Alias Khalaf ◽  
Sarhad Nami
Keyword(s):  
Continuous Function ◽  
Closed Sets

Almost Somewhat Near Continuity and Near Regularity

2021 ◽  
Vol 7 (1) ◽  
pp. 88-99
Author(s):  
Zanyar A. Ameen
Keyword(s):  
Continuous Function ◽  
Continuous Functions ◽  
One To One ◽  
Nearly Continuous

AbstractThe notions of almost somewhat near continuity of functions and near regularity of spaces are introduced. Some properties of almost somewhat nearly continuous functions and their connections are studied. At the end, it is shown that a one-to-one almost somewhat nearly continuous function f from a space X onto a space Y is somewhat nearly continuous if and only if the range of f is nearly regular.

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