scholarly journals Поточечное условие абсолютной непрерывности функции одной переменной и его применения

2021 ◽  
pp. 41-49
Author(s):  
S.K. Vodopyanov
Keyword(s):  
Continuous Function ◽  
Absolute Continuity ◽  
Sufficient Conditions ◽  
Function Spaces ◽  
Sufficient Condition ◽  
Sobolev Classes ◽  
Absolutely Continuous ◽  
Equivalent Definition ◽  
The Relationship ◽  
Lebesgue Integration

An absolutely continuous function in calculus is precisely such a function that, within the framework of Lebesgue integration, can be restored from its derivative, that is, the Newton--Leibniz theorem on the relationship between integration and differentiation is fulfilled for it. An equivalent definition is that the the sum of the moduli of the increments of the function with respect to arbitrary pair-wise disjoint intervals is less than any positive number if the sum of the lengths of the intervals is small enough. Certain sufficient conditions for absolute continuity are known, for example, the Banach--Zaretsky theorem. In this paper we prove a new sufficient condition for the absolute continuity of a function of one variable and give some of its applications to problems in the theory of function spaces. The proved condition makes it possible to significantly simplify the proof of the theorems on the pointwise description of functions of the Sobolev classes defined on Euclidean spaces and Сarnot groups.

The Maxslope Taxometric Procedure: Mathematical Derivation, Parameter Estimation, Consistency Tests

2004 ◽  
Vol 95 (2) ◽  
pp. 517-550 ◽  
Author(s):  
William M. Grove
Keyword(s):  
Parameter Estimation ◽  
Nonlinear Regression ◽  
Classification Accuracy ◽  
Sufficient Conditions ◽  
Sufficient Condition ◽  
Indicator Variable ◽  
Regression Slope ◽  
The Relationship ◽  
Mathematical Derivation ◽  
Made In

This article first explains concepts in taxometrics, including the meaning of “taxon” in relation to taxometric procedures. It then mathematically develops the MAXSLOPE procedure of Grove and Meehl which relies on nonlinear regression of one taxometric indicator variable on another. Sufficient conditions for MAXSLOPE's validity are set forth. The relationship between the point of maximum regression slope (MAXSLOPE point) and the HITMAX cut, i.e., the point on a variable which, if used as a diagnostic cut-off score, yields maximum classification accuracy, is analyzed. A sufficient condition is given for the MAXSLOPE point to equal the HITMAX cut; however, most distributions have different MAXSLOPE and HITMAX points. Equations and an algorithm are spelled out for making a graphical test for the existence of a taxon, estimating taxometric parameters, and conducting consistency tests; the latter serve as stringent checks on the validity of a taxonic conjecture. The plausibility of assumptions made, in deriving MAXSLOPE equations, is discussed, and the qualitative effects of violations of these assumptions are explained.

scholarly journals A Remark on Isometries of Absolutely Continuous Spaces

2020 ◽  
Vol 2020 ◽  
pp. 1-3
Author(s):  
Alireza Ranjbar-Motlagh
Keyword(s):  
Continuous Function ◽  
Composition Operator ◽  
Real Line ◽  
Weighted Composition Operator ◽  
Function Spaces ◽  
Absolutely Continuous ◽  
The Real ◽  
Absolutely Continuous Function ◽  
Weighted Composition ◽  
Vector Valued

The purpose of this article is to study the isometries between vector-valued absolutely continuous function spaces, over compact subsets of the real line. Indeed, under certain conditions, it is shown that such isometries can be represented as a weighted composition operator.

scholarly journals A NECESSARY AND SUFFICIENT CONDITION FOR CERTAIN MARTINGALE INEQUALITIES IN BANACH FUNCTION SPACES

2007 ◽  
Vol 49 (3) ◽  
pp. 431-447 ◽  
Author(s):  
MASATO KIKUCHI
Keyword(s):  
Function Space ◽  
Probability Space ◽  
Banach Function Space ◽  
Sufficient Conditions ◽  
Function Spaces ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Banach Function Spaces ◽  
Martingale Inequalities ◽  
Necessary And Sufficient

AbstractLet X be a Banach function space over a nonatomic probability space. We investigate certain martingale inequalities in X that generalize those studied by A. M. Garsia. We give necessary and sufficient conditions on X for the inequalities to be valid.

scholarly journals On the convergence of Cesàro means of negative order of Vilenkin-Fourier series

2019 ◽  
Vol 56 (1) ◽  
pp. 22-44
Author(s):  
Gvantsa Shavardenidze
Keyword(s):  
Fourier Series ◽  
Continuous Function ◽  
Uniform Convergence ◽  
Sufficient Conditions ◽  
Sufficient Condition ◽  
Cesàro Means ◽  
Cesaro Means ◽  
Negative Order ◽  
Cesáro Means

Abstract In 1971 Onnewer and Waterman establish a sufficient condition which guarantees uniform convergence of Vilenkin-Fourier series of continuous function. In this paper we consider different classes of functions of generalized bounded oscillation and in the terms of these classes there are established sufficient conditions for uniform convergence of Cesàro means of negative order.

scholarly journals Positive Liapunov exponents and absolute continuity for maps of the interval

1983 ◽  
Vol 3 (1) ◽  
pp. 13-46 ◽  
Author(s):  
P. Collet ◽  
J.-P. Eckmann
Keyword(s):  
Invariant Measure ◽  
Critical Point ◽  
Lebesgue Measure ◽  
Absolute Continuity ◽  
Sufficient Condition ◽  
Liapunov Exponent ◽  
Absolutely Continuous ◽  
Liapunov Exponents ◽  
Unimodal Map ◽  
Weak Regularity

AbstractWe give a sufficient condition for a unimodal map of the interval to have an invariant measure absolutely continuous with respect to the Lebesgue measure. Apart from some weak regularity assumptions, the condition requires positivity of the forward and backward Liapunov exponent of the critical point.

scholarly journals Strong insertion of a $\gamma-$ continuous function

2015 ◽  
Vol 34 (2) ◽  
pp. 43-52
Author(s):  
Majid Mirmiran ◽  
Binesh Naderi
Keyword(s):  
Continuous Function ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Sufficient Condition ◽  
Necessary And Sufficient

Necessary and sufficient conditions in terms of lower cut sets are given for the strong insertion of a $\gamma-$continuous function between two comparable real-valued functions.In this paper, is given a sufficient condition for the weak $\gamma-$insertion property. Also for a  space with the weak $\gamma-$insertion property, we give necessary and sufficient conditions for the space to have the strong $\gamma-$insertion property. Several insertion theorems are obtained as corollaries of these results.

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