Null set
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Automatic differentiation in PCF Open
We study the correctness of automatic differentiation (AD) in the context of a higher-order, Turing-complete language (PCF with real numbers), both in forward and reverse mode. Our main result is that, under mild hypotheses on the primitiv…
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Quantitative stability for sumsets in $\mathbb R^n$ Open
Given a measurable set A\subset \mathbb R^n of positive measure, it is not difficult to show that |A+A|=|2A| if and only if A is equal to its convex hull minus a set of measure zero. We investigate the stability of this statement: If (|A+A…
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Parametrized measure models Open
We develop a new and general notion of parametric measure models and statistical models on an arbitrary sample space $\\Omega$ which does not assume that all measures of the model have the same null sets. This is given by a differentiable …
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Lebesgue measure of escaping sets of entire functions Open
For a transcendental entire function $f$ of finite order in the Eremenko–Lyubich class ${\mathcal{B}}$ , we give conditions under which the Lebesgue measure of the escaping set ${\mathcal{I}}(f)$ of $f$ is zero. This complements the recent…
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Integral Menger curvature and rectifiability of $n$-dimensional Borel sets in Euclidean $N$-space Open
In this paper we show that an $n$-dimensional Borel set in Euclidean $N$-space with finite integral Menger curvature is $n$-rectifiable, meaning that it can be covered by countably many images of Lipschitz continuous functions up to a null…
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A characterization of 1-rectifiable doubling measures with connected supports Open
Garnett, Killip, and Schul have exhibited a doubling measure [math] with support equal to [math] that is [math] -rectifiable, meaning there are countably many curves [math] of finite length for which [math] . In this note, we characterize …
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The Constant Width Measure Set, the Spherical Measure Set and isoperimetric equalities for planar ovals Open
In this paper we introduce the Constant Width Measure Set, which measures the constant width property of an oval, i.e. the planar simple closed strictly convex curve. We study its geometrical properties. We find the exact relation between …
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Structure of porous sets in Carnot groups Open
We show that any Carnot group contains a closed nowhere dense set which has measure zero but is not $\\sigma $-porous with respect to the Carnot–Carathéodory (CC) distance. In the first Heisenberg group, we observe that there exist sets wh…
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Characterization of the Local Growth of Two Cantor-Type Functions Open
The Cantor set and its homonymous function have been frequently utilized as examples for various physical phenomena occurring on discontinuous sets. This article characterizes the local growth of the Cantor’s singular function by means of …
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On cohesive almost zero-dimensional spaces Open
We investigate C-sets in almost zero-dimensional spaces, showing that closed $\sigma $ C-sets are C-sets. As corollaries, we prove that every rim- $\sigma $ -compact almost zero-dimensional space is zero-dimensional and that each cohesive …
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Limit-periodic continuum Schrödinger operators with zero measure Cantor spectrum Open
We consider Schrödinger operators on the real line with limit-periodic potentials and show that, generically, the spectrum is a Cantor set of zero Lebesgue measure and all spectral measures are purely singular continuous. Moreover, we show…
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On sets of zero stationary harmonic measure Open
In this paper, we prove that any subset with an appropriate sub-linear horizontal growth has a non-zero stationary harmonic measure. On the other hand, we also show any subset with super-linear horizontal growth will have a $0$ stationary …
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Local Dimensions of Overlapping Self-Similar Measures Open
We show that any equicontractive, self-similar measure arising from the IFS of contractions (S j ), with self-similar set [0, 1], admits an isolated point in its set of local dimensions provided the images of S j (0, 1) (suitably) overlap …
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Star versions of the Hurewicz basis covering property and strong measure zero spaces Open
In this paper,we introduced the star version of the Hurewicz basis covering property,studied by Babinkostova, Ko?cinac, and Scheepers in 2004. The authors obtained the relationship between star-selection principles and the star version of …
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Universal differentiability sets in Carnot groups of arbitrarily high step Open
We show that any Carnot group G with sufficiently many deformable directions contains a measure zero set N such that every Lipschitz map f: G→ R is differentiable at some point of N. We also prove that model filiform groups satisfy this co…
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The interplay of classes of algorithmically random objects Open
We study algorithmically random closed subsets of $\cs$\!, algorithmically random continuous functions from $\cs$ to $\cs$\!, and algorithmically random Borel probability measures on $\cs$, especially the interplay between these three clas…
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Existence and Non-existence of Solutions to the Coboundary Equation for Measure Preserving Systems Open
Let $(X,\mathcal{B},μ)$ be a standard probability space. We give new fundamental results determining solutions to the coboundary equation: \begin{eqnarray*} f = g - g \circ T \end{eqnarray*} where $f \in L^p$ and $T$ is ergodic invertible …
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Measurable circle squaring Open
Laczkovich proved that if bounded subsets $A$ and $B$ of $R^k$ have the same non-zero Lebesgue measure and the box dimension of the boundary of each set is less than $k$, then there is a partition of $A$ into finitely many parts that can b…
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The Morse-Sard theorem revisited Open
Let $n, m, k$ be positive integers with $k=n-m+1$. We establish an abstract Morse-Sard-type theorem which allows us to deduce, on the one hand, a previous result of De Pascale's for Sobolev $W^{k,p}_{\textrm{loc}}(\mathbb{R}^n, \mathbb{R}^…
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A Full Study of the Dynamics on One-Holed Dilation Tori Open
An open question in the study of dilation surfaces is to determine the typical dynamical behavior of the directional flow on a fixed dilation surface. We show that on any one-holed dilation torus, in all but a measure zero Cantor set of di…
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Harmonic measure on sets of codimension larger than one Open
We introduce a new notion of a harmonic measure for a d -dimensional set in with , that is, when the codimension is strictly bigger than 1. Our measure is associated with a degenerate elliptic PDE, it gives rise to a comprehensive ellipti…
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A genericity property of Fréchet sample means on Riemannian manifolds Open
Let $(M,g)$ be a Riemannian manifold. If $μ$ is a probability measure on $M$ given by a continuous density function, one would expect the Fréchet means of data-samples $Q=(q_1,q_2,\dots, q_N)\in M^N$, with respect to $μ$, to behave ``gener…
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Zero measure spectrum for multi-frequency Schrödinger operators Open
Building on works of Berthé–Steiner–Thuswaldner and Fogg–Nous, we show that on the two-dimensional torus, Lebesgue almost every translation admits a natural coding such that the associated subshift satisfies the Boshernitzan criterion. As …
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Description of Stability for Two and Three-Dimensional Linear Time-Invariant Systems Based on Curvature and Torsion Open
This paper focuses on using curvature and torsion to describe the stability of linear time-invariant system. We prove that for a two-dimensional system $\dot{r}(t)= Ar(t)$, (i) if there exists an initial value, such that zero is not the li…
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Conjugacy in the Cantor set automorphism group Open
We survey, and extend, results on the adjoint action of the homeomorphism group $H(X)$ on the space of surjective continuous maps, $C_s(X)$, where $X$ is a Cantor set. We look also at the restriction of the action to various dynamically de…
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The Family of Central Cantor Sets with Packing Dimension Zero Open
As in the recent article of M. Balcerzak, T. Filipczak and P. Nowakowski, we identify the family CS of central Cantor subsets of [0, 1] with the Polish space X : = (0, 1) ℕ equipped with the probability product measure µ . We investigate t…
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Structure of Porous Sets in Carnot Groups Open
We show that any Carnot group contains a closed nowhere dense set which has measure zero but is not $σ$-porous with respect to the Carnot-Carathéodory (CC) distance. In the first Heisenberg group we observe that there exist sets which are …
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Approximate Drygas mappings on a set of measure zero Open
Let R be the set of real numbers, Y be a Banach space and f : R →Y. We prove the Hyers-Ulam stability for the Drygas functional equationf (x + y) + f (x - y) = 2f (x) + f (y) + f (-y) for all (x, y) ∈ Ω, where Ω⊂ R2 is of Lebesgue measure …
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Almost all non-archimedean Kakeya sets have measure zero Open
We study Kakeya sets over local non-archimedean fields with a probabilistic point of view: we define a probability measure on the set of Kakeya sets as above and prove that, according to this measure, almost all non-archimedean Kakeya sets…
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Large sets avoiding infinite arithmetic / geometric progressions Open
We study some variants of the Erdős similarity problem. We pose the question if every measurable subset of the real line with positive measure contains a similar copy of an infinite geometric progression. We construct a compact subset $E$ …