Related papers: Some notes on variational principle for metric mea…
This paper defines and discusses the dimension notion of topological slow entropy of any subset for Z^d actions. Also, the notion of measure-theoretic slow entropy for Z^d actions is presented, which is modified from Brin and Katok [2].…
We study the pointwise dimension for a new class of projection measures on arbitrary fractal limit sets without separation conditions. We prove that the pointwise dimension exists a.e. for this class of measures associated to equilibrium…
The paper deals with the variational principles for evaluation of the spectral radii of transfer and weighted shift operators associated with a dynamical system. These variational principles have been the matter of numerous investigations…
A variational principle is introduced to provide a new formulation and resolution for several boundary value problems with a variational structure. This principle allows one to deal with problems well beyond the weakly compact structure. As…
We show that the naive application of the maximum entropy principle can yield answers which depend on the level of description, i.e. the result is not invariant under coarse-graining. We demonstrate that the correct approach, even for…
We consider the case with boundary of the classical Kazdan-Warner problem in dimension greater or equal than three, i.e. the prescription of scalar and boundary mean curvatures via conformal deformations of the metric. We deal in particular…
The maximum principle forms an important qualitative property of second order elliptic equations, therefore its discrete analogues, the so-called discrete maximum principles (DMPs) have drawn much attention. In this paper DMPs are…
Entropy and information can be considered dual: entropy is a measure of the subspace defined by the information constraining the given ambient space. Negative entropies, arising in na\"ive extensions of the definition of entropy from…
The discrete data encoded in the power moments of a positive measure, fast decaying at infinity on euclidean space, is incomplete for recovery, leading to the concept of moment indeterminateness. On the other hand, classical integral…
Entropy is useful in statistical problems as a measure of irreversibility, randomness, mixing, dispersion, and number of microstates. However, there remains ambiguity over the precise mathematical formulation of entropy, generalized beyond…
This article extends to the vector setting the results of our previous work Kruger et al. (2015) which refined and slightly strengthened the metric space version of the Borwein-Preiss variational principle due to Li and Shi, J. Math. Anal.…
It has been a long history in testing whether a mean vector with a fixed dimension has a specified value. Some well-known tests include the Hotelling $T^2$-test and the empirical likelihood ratio test proposed by Owen [Biometrika 75 (1988)…
Inspired by the recent work [MRT21], we prove a non-universal non-central Moderate Deviation principle for the nodal length of arithmetic random waves (Gaussian Laplace eigenfunctions on the standard flat torus) both on the whole manifold…
The maximum entropy approach operating with quite general entropy measure and constraint is considered. It is demonstrated that for a conditional or parametrized probability distribution $f(x|\mu)$ there is a "universal" relation among the…
Peres and Solomyak proved that on $\mathbb R^n$, the limits defining the $L^q$-dimension for any $q\in(0,\infty)\setminus\{1\}$, and the entropy dimension of a self-conformal measure exist, without assuming any separation condition. By…
The problem of how to properly quantify redundant information is an open question that has been the subject of much recent research. Redundant information refers to information about a target variable S that is common to two or more…
Metric mean dimension is a metric invariant of dynamical systems. It is a dynamical analogue of Minkowski dimension of metric spaces. We explain that old ideas of Bowen (1972) can be used for clarifying the local nature of metric mean…
The variational principle and the corresponding differential equation for geodesic circles in two dimensional (pseudo)-Riemannian space are being discovered. The relationship with the physical notion of uniformly accelerated relativistic…
In this paper, we introduce the notions of upper metric mean dimension, $u$-upper metric mean dimension, $l$-upper metric mean dimension of free semigroup actions for non-compact sets via Carath\'{e}odory-Pesin structure. Firstly, the lower…
Since its introduction by Vapnik and Chervonenkis in the 1960s, the VC dimension and its variants have played a central role in numerous fields. In this paper, we investigate several variants of the VC dimension and their applications to…