Lebesgue measure of the universal singular set for the simplest problems in the calculus of variations

1994 ◽  
Vol 35 (6) ◽  
pp. 1220-1233 ◽  
Author(s):  
M. A. Sychëv
Keyword(s):  
Lebesgue Measure ◽  
Calculus Of Variations ◽  
Singular Set

scholarly journals SRB measures for almost Axiom A diffeomorphisms

2015 ◽  
Vol 36 (7) ◽  
pp. 2015-2043 ◽  
Author(s):  
JOSÉ F. ALVES ◽  
RENAUD LEPLAIDEUR
Keyword(s):  
Probability Measure ◽  
Lebesgue Measure ◽  
Compact Manifold ◽  
Invariant Set ◽  
Singular Set ◽  
Axiom A ◽  
Srb Measures ◽  
Hyperbolic Points

We consider a diffeomorphism $f$ of a compact manifold $M$ which is almost Axiom A, i.e. $f$ is hyperbolic in a neighborhood of some compact $f$-invariant set, except in some singular set of neutral points. We prove that if there exists some $f$-invariant set of hyperbolic points with positive unstable Lebesgue measure such that for every point in this set the stable and unstable leaves are ‘long enough’, then $f$ admits an SRB (probability) measure.

On Asymmetric Distances

2013 ◽  
Vol 1 ◽  
pp. 200-231 ◽  
Author(s):  
Andrea C.G. Mennucci
Keyword(s):  
Total Variation ◽  
Calculus Of Variations ◽  
Distance Function ◽  
Metric Space ◽  
Metric Spaces ◽  
Geodesic Segment ◽  
Intrinsic Metric ◽  
Typical Application ◽  
Variation Formula

Abstract In this paper we discuss asymmetric length structures and asymmetric metric spaces. A length structure induces a (semi)distance function; by using the total variation formula, a (semi)distance function induces a length. In the first part we identify a topology in the set of paths that best describes when the above operations are idempotent. As a typical application, we consider the length of paths defined by a Finslerian functional in Calculus of Variations. In the second part we generalize the setting of General metric spaces of Busemann, and discuss the newly found aspects of the theory: we identify three interesting classes of paths, and compare them; we note that a geodesic segment (as defined by Busemann) is not necessarily continuous in our setting; hence we present three different notions of intrinsic metric space.

LEBESGUE MEASURE AND GAMBLING

1993 ◽  
Vol 19 (1) ◽  
pp. 40
Author(s):  
Kanovei ◽  
Linton
Keyword(s):  
Lebesgue Measure

On a Choquet-Stieltjes type integral on intervals

2019 ◽  
Vol 69 (4) ◽  
pp. 801-814 ◽  
Author(s):  
Sorin G. Gal
Keyword(s):  
Lebesgue Measure ◽  
Choquet Integral ◽  
Stieltjes Integral ◽  
Line Integral ◽  
Type Integral ◽  
Choquet Integrals ◽  
Lebesgue Measures ◽  
Stieltjes Type

Abstract In this paper we introduce a new concept of Choquet-Stieltjes integral of f with respect to g on intervals, as a limit of Choquet integrals with respect to a capacity μ. For g(t) = t, one reduces to the usual Choquet integral and unlike the old known concept of Choquet-Stieltjes integral, for μ the Lebesgue measure, one reduces to the usual Riemann-Stieltjes integral. In the case of distorted Lebesgue measures, several properties of this new integral are obtained. As an application, the concept of Choquet line integral of second kind is introduced and some of its properties are obtained.

scholarly journals A Generalization of the Hausdorff Dimension Theorem for Deterministic Fractals

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1546
Author(s):  
Mohsen Soltanifar
Keyword(s):  
Hausdorff Dimension ◽  
Lebesgue Measure ◽  
Virtual World ◽  
Fractal Dimensions ◽  
Definition Of ◽  
The Given

How many fractals exist in nature or the virtual world? In this paper, we partially answer the second question using Mandelbrot’s fundamental definition of fractals and their quantities of the Hausdorff dimension and Lebesgue measure. We prove the existence of aleph-two of virtual fractals with a Hausdorff dimension of a bi-variate function of them and the given Lebesgue measure. The question remains unanswered for other fractal dimensions.

scholarly journals Partial regularity of stable solutions to the fractional Geľfand-Liouville equation

2021 ◽  
Vol 10 (1) ◽  
pp. 1316-1327
Author(s):  
Ali Hyder ◽  
Wen Yang
Keyword(s):  
Liouville Equation ◽  
Weak Solutions ◽  
Partial Regularity ◽  
Singular Set ◽  
Stable Solutions ◽  
Gelfand Problem

Abstract We analyze stable weak solutions to the fractional Geľfand problem ( − Δ ) s u = e u i n Ω ⊂ R n . $$\begin{array}{} \displaystyle (-{\it\Delta})^su = e^u\quad\mathrm{in}\quad {\it\Omega}\subset\mathbb{R}^n. \end{array}$$ We prove that the dimension of the singular set is at most n − 10s.

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