Related papers: Regularity versus smoothness of measures
The paper treats density measures as typical examples of finitely additive measures in $\mathbb{R}^n$. We study their structure and derive basic properties. In addition, estimates for related integrals are provided. The results are applied…
Depth is a complexity measure for natural systems of the kind studied in statistical physics and is defined in terms of computational complexity. Depth quantifies the length of the shortest parallel computation required to construct a…
The Sobolev regularity of invariant measures for diffusion processes is proved on non-smooth metric measure spaces with synthetic lower Ricci curvature bounds. As an application, the symmetrizability of semigroups is characterized, and the…
We develop a systematic study about the spectrality of measures supported on piecewise smooth curves by studying the support of the tempered distributions arising from the tiling equation of some singular spectral measures. In doing so, we…
We study the relationship between the smoothness of a plane curve and that of its evolute, especially in the cases where the parent curve is no more two or three times continuously differentiable, and exhibit the same kind of apparent…
In this paper, we discuss various basic properties of moduli of smoothness of functions from $L_p(\mathbb{R}^d)$, $0<p\le \infty$. In particular, complete versions of Jackson-, Marchaud-, and Ulyanov-type inequalities are given for the…
We prove two-sided inequalities between the integral moduli of smoothness of a function on $\mathbb{R}^d/\mathbb{T}^d$ and the weighted tail-type integrals of its Fourier transform/series. Sharpness of obtained results in particular is…
This paper studies the relationship between volume and surface uniform measures on n-dimensional p-balls under the p-norm. It is proved that for p=1, p=2 and p=infinity, and only for these values of p, radial projection maps a…
Measuring interdisciplinarity is a pertinent but challenging issue in quantitative studies of science. There seems to be a consensus in the literature that the concept of interdisciplinarity is multifaceted and ambiguous. Unsurprisingly,…
We develop a test to determine whether a function lying in a fixed $L_2$-Sobolev-type ball of smoothness $t$, and generating a noisy signal, is in fact of a given smoothness $s\geq t$ or not. While it is impossible to construct a uniformly…
In this work we investigate and characterize linear functionals $L:\mathbb{R}[x_1,\dots,x_n]\to\mathbb{R}$ with absolutely continuous representing measures $\mu$, i.e., $\mathrm{d}\mu(x) = g(x)\,\mathrm{d} x$ for some density $g$. We focus…
We are going to classify sets by a given mean in two ways. Firstly we study small and big sets regarding a given mean. Secondly we study sets that have the same weight according to a mean. We also generalize the notion of roundness and get…
We study various generalizations of concentration of measure on the unit sphere, in particular by means of log-Sobolev inequalities. First, we show Sudakov-type concentration results and local semicircular laws for weighted random matrices.…
We develop a theory of generically stable and smooth Keisler measures in NIP metric theories, generalizing the case of classical logic. Using smooth extensions, we verify that fundamental properties of (Borel)-definable measures and the…
Interpreting experimental data in high school experiments can be a difficult task for students, especially when there is large variation in the data. At the same time, calculating the standard deviation poses a challenge for students. In…
In this paper, we study the summability properties of double sequences of real constants which map sequences of random variables to sequences of random variables that are defined on the same probability sample space. We show that a regular…
We classify quasiconformal Anosov flows whose strong stable and unstable distributions are at least two dimensional and the sum of these two distributions is smooth. We deduce from this classification result the complete classification of…
A measure of complexity based on a probabilistic description of physical systems is proposed. This measure incorporates the main features of the intuitive notion of such a magnitude. It can be applied to many physical situations and to…
We introduce a suitable notion of asymptotic smoothness on infinite dimensional Banach spaces, and we prove that, under some structural restrictions on the space, the convex envelope of an asymptotically smooth function is asymptotically…
Piecewise smooth dynamical systems make use of discontinuities to model switching between regions of smooth evolution. This introduces an ambiguity in prescribing dynamics at the discontinuity: should it be given by a limiting value on one…