scholarly journals Continuity properties of weakly monotone Orlicz–Sobolev functions

2018 ◽  
Vol 0 (0) ◽  
Author(s):  
Menita Carozza ◽  
Andrea Cianchi
Keyword(s):  
Sobolev Spaces ◽  
Hausdorff Measure ◽  
Additional Assumption ◽  
Monotone Function ◽  
General Setting ◽  
Monotone Functions ◽  
Finite Distortion ◽  
Continuity Properties ◽  
Definition Of ◽  
Generalized Zero

Abstract The notion of weakly monotone functions extends the classical definition of monotone function, that can be traced back to Lebesgue. It was introduced, in the framework of Sobolev spaces, by Manfredi, in connection with the analysis of the regularity of maps of finite distortion appearing in the theory of nonlinear elasticity. Diverse authors, including Iwaniecz, Kauhanen, Koskela, Maly, Onninen, Zhong, thoroughly investigated continuity properties of monotone functions in the more general setting of Orlicz–Sobolev spaces, in view of the analysis of continuity, openness and discreteness properties of maps under minimal integrability assumptions on their distortion. The present paper complements and augments the available Orlicz–Sobolev theory of weakly monotone functions. In particular, a variant is proposed in a customary condition ensuring the continuity of functions from this class, which avoids a technical additional assumption, and applies in certain situations when the latter is not fulfilled. The continuity outside sets of zero Orlicz capacity, and outside sets of (generalized) zero Hausdorff measure are also established when everywhere continuity fails.

scholarly journals Functions of Bounded Variation on Complete and Connected One-Dimensional Metric Spaces

2020 ◽  
Author(s):  
Panu Lahti ◽  
Xiaodan Zhou
Keyword(s):  
Bounded Variation ◽  
Hausdorff Measure ◽  
Metric Spaces ◽  
General Setting ◽  
Functions Of Bounded Variation ◽  
One Dimensional ◽  
Underlying Space ◽  
Finite Perimeter ◽  
Bv Functions ◽  
Definition Of

Abstract In this paper, we study functions of bounded variation on a complete and connected metric space with finite one-dimensional Hausdorff measure. The definition of BV functions on a compact interval based on pointwise variation is extended to this general setting. We show this definition of BV functions is equivalent to the BV functions introduced by Miranda [18]. Furthermore, we study the necessity of conditions on the underlying space in Federer’s characterization of sets of finite perimeter on metric measure spaces. In particular, our examples show that the doubling and Poincaré inequality conditions are essential in showing that a set has finite perimeter if the codimension one Hausdorff measure of the measure-theoretic boundary is finite.

scholarly journals Stability and memory-loss go hand-in-hand: three results in dynamics and computation

2020 ◽  
Vol 476 (2242) ◽  
pp. 20200563
Author(s):  
G. Manjunath
Keyword(s):  
State Of The Art ◽  
Memory Loss ◽  
General Setting ◽  
Biologically Inspired ◽  
Driven Systems ◽  
Hardware Implementations ◽  
Internal States ◽  
Definition Of ◽  
Necessary And Sufficient ◽  
Dedicated Hardware

The search for universal laws that help establish a relationship between dynamics and computation is driven by recent expansionist initiatives in biologically inspired computing. A general setting to understand both such dynamics and computation is a driven dynamical system that responds to a temporal input. Surprisingly, we find memory-loss a feature of driven systems to forget their internal states helps provide unambiguous answers to the following fundamental stability questions that have been unanswered for decades: what is necessary and sufficient so that slightly different inputs still lead to mostly similar responses? How does changing the driven system’s parameters affect stability? What is the mathematical definition of the edge-of-criticality? We anticipate our results to be timely in understanding and designing biologically inspired computers that are entering an era of dedicated hardware implementations for neuromorphic computing and state-of-the-art reservoir computing applications.

The Generalized Baker–Schmidt Problem on Hypersurfaces

2019 ◽  
Author(s):  
Mumtaz Hussain ◽  
Johannes Schleischitz ◽  
David Simmons
Keyword(s):  
Hausdorff Measure ◽  
General Setting ◽  
Dimensional Hausdorff Measure ◽  
Planar Curves ◽  
Open Question

Abstract The generalized Baker–Schmidt problem (1970) concerns the $f$-dimensional Hausdorff measure of the set of $\psi $-approximable points on a nondegenerate manifold. There are two variants of this problem concerning simultaneous and dual approximation. Beresnevich–Dickinson–Velani (in 2006, for the homogeneous setting) and Badziahin–Beresnevich–Velani (in 2013, for the inhomogeneous setting) proved the divergence part of this problem for dual approximation on arbitrary nondegenerate manifolds. The corresponding convergence counterpart represents a major challenging open question and the progress thus far has only been attained over planar curves. In this paper, we settle this problem for hypersurfaces in a more general setting, that is, for inhomogeneous approximations and with a non-monotonic multivariable approximating function.

A characterisation of tangential exceptional sets for αp = n

1996 ◽  
Vol 126 (3) ◽  
pp. 625-641
Author(s):  
Carme Cascante ◽  
Joaquín Ortega
Keyword(s):  
Sobolev Spaces ◽  
Hausdorff Measure ◽  
Exceptional Sets

In this paper we characterise some closed tangential exceptional sets for Hardy–Sobolev spaces , αp = n, 1 < p ≦ 2, in terms of the annihilation of a nonisotropic Hausdorff measure.

scholarly journals A nonlinear complementarity problem for monotone functions

1979 ◽  
Vol 20 (2) ◽  
pp. 227-231 ◽  
Author(s):  
Sribatsa Nanda ◽  
Ujagar Patel
Keyword(s):  
Unique Solution ◽  
Complementarity Problem ◽  
Monotone Function ◽  
Nonlinear Complementarity Problem ◽  
Monotone Functions ◽  
Nonlinear Complementarity

In this note we prove that for a monotone function that fixes the origin, the complementarity problem for Cn always admits a solution. If, moreover, the function is strictly monotone, then zero is the unique solution. These results are stronger than known results in this direction for two reasons: firstly, there is no condition on the nature of the cone and secondly, no feasibility assumptions are made.

scholarly journals From Matrix to Operator Inequalities

2012 ◽  
Vol 55 (2) ◽  
pp. 339-350 ◽  
Author(s):  
Terry A. Loring
Keyword(s):  
Technical Condition ◽  
Monotone Functions ◽  
Bounded Operators ◽  
Operator Inequalities ◽  
Residually Finite ◽  
Finite Dimensional ◽  
Operator Monotone ◽  
Noncommutative Polynomials ◽  
Definition Of

AbstractWe generalize Löwner's method for proving that matrix monotone functions are operator monotone. The relation x ≤ y on bounded operators is our model for a definition of C*-relations being residually finite dimensional.Our main result is a meta-theorem about theorems involving relations on bounded operators. If we can show there are residually finite dimensional relations involved and verify a technical condition, then such a theorem will follow from its restriction to matrices.Applications are shown regarding norms of exponentials, the norms of commutators, and “positive” noncommutative ∗-polynomials.

scholarly journals On rectifiable measures in Carnot groups: representation

2021 ◽  
Vol 61 (1) ◽  
Author(s):  
Gioacchino Antonelli ◽  
Andrea Merlo
Keyword(s):  
Large Class ◽  
Hausdorff Measure ◽  
Carnot Group ◽  
Lipschitz Functions ◽  
Carnot Groups ◽  
Geodesic Sphere ◽  
Area Formula ◽  
Definition Of ◽  
Lipschitz Graphs ◽  
Almost All

AbstractThis paper deals with the theory of rectifiability in arbitrary Carnot groups, and in particular with the study of the notion of $$\mathscr {P}$$ P -rectifiable measure. First, we show that in arbitrary Carnot groups the natural infinitesimal definition of rectifiabile measure, i.e., the definition given in terms of the existence of flat tangent measures, is equivalent to the global definition given in terms of coverings with intrinsically differentiable graphs, i.e., graphs with flat Hausdorff tangents. In general we do not have the latter equivalence if we ask the covering to be made of intrinsically Lipschitz graphs. Second, we show a geometric area formula for the centered Hausdorff measure restricted to intrinsically differentiable graphs in arbitrary Carnot groups. The latter formula extends and strengthens other area formulae obtained in the literature in the context of Carnot groups. As an application, our analysis allows us to prove the intrinsic $$C^1$$ C 1 -rectifiability of almost all the preimages of a large class of Lipschitz functions between Carnot groups. In particular, from the latter result, we obtain that any geodesic sphere in a Carnot group equipped with an arbitrary left-invariant homogeneous distance is intrinsic $$C^1$$ C 1 -rectifiable.

EXISTENCE AND BOX DIMENSION OF GENERAL RECURRENT FRACTAL INTERPOLATION FUNCTIONS

2020 ◽  
pp. 1-13
Author(s):  
HUO-JUN RUAN ◽  
JIAN-CI XIAO ◽  
BING YANG
Keyword(s):  
Iterated Function System ◽  
General Setting ◽  
Iterated Function Systems ◽  
Box Dimension ◽  
Fractal Interpolation ◽  
Fractal Interpolation Functions ◽  
Iterated Function ◽  
Recurrent System ◽  
Definition Of ◽  
Interpolation Functions

Abstract The notion of recurrent fractal interpolation functions (RFIFs) was introduced by Barnsley et al. [‘Recurrent iterated function systems’, Constr. Approx.5 (1989), 362–378]. Roughly speaking, the graph of an RFIF is the invariant set of a recurrent iterated function system on $\mathbb {R}^2$ . We generalise the definition of RFIFs so that iterated functions in the recurrent system need not be contractive with respect to the first variable. We obtain the box dimensions of all self-affine RFIFs in this general setting.

scholarly journals On semi-infinite cohomology of finite-dimensional graded algebras

2010 ◽  
Vol 146 (2) ◽  
pp. 480-496 ◽  
Author(s):  
Roman Bezrukavnikov ◽  
Leonid Positselski
Keyword(s):  
Quantum Group ◽  
General Setting ◽  
Derived Categories ◽  
Graded Algebras ◽  
Root Of Unity ◽  
Finite Dimensional ◽  
Small Quantum ◽  
Definition Of

AbstractWe describe a general setting for the definition of semi-infinite cohomology of finite-dimensional graded algebras, and provide an interpretation of such cohomology in terms of derived categories. We apply this interpretation to compute semi-infinite cohomology of some modules over the small quantum group at a root of unity, generalizing an earlier result of Arkhipov (posed as a conjecture by B. Feigin).

Monotone Functions on Linear Lattices

1963 ◽  
Vol 15 ◽  
pp. 226-236 ◽  
Author(s):  
H. W. Ellis ◽  
Hidegorô Nakano
Keyword(s):  
Monotone Function ◽  
Continuous Linear ◽  
Monotone Functions ◽  
Linear Lattice ◽  
Image Position ◽  
Linear Lattices

If R is a sequentially continuous linear lattice, a function f(x), denned on R+ = {x:0 ≤ x ∊ R} with 0 ≤ f(x) ≤ + ∞, will be called a monotone function if it satisfies

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