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We consider the numerical evaluation of a class of double integrals with respect to a pair of self-similar measures over a self-similar fractal set (the attractor of an iterated function system), with a weakly singular integrand of logarithmic or algebraic type. In a recent paper (Gibbs et al. Numer. Algorithms 92, 2071–2124 2023), it was shown that when the fractal set is "disjoint" in a certain sense (an example being the Cantor set), the self-similarity of the measures, combined with the homogeneity properties of the integrand, can be exploited to express the singular integral exactly in terms of regular integrals, which can be readily approximated numerically. In this paper, we present a methodology for extending these results to cases where the fractal is non-disjoint but non-overlapping (in the sense that the open set condition holds). Our approach applies to many well-known examples including the Sierpinski triangle, the Vicsek fractal, the Sierpinski carpet, and the Koch snowflake. [ABSTRACT FROM AUTHOR]

Let X = { X (t) , t ∈ R N } be a centered space-time anisotropic Gaussian random field in R d with stationary increments, where the components X i (i = 1 , ... , d) are independent but distributed differently. Under certain conditions, we not only give the Hausdorff dimension of the graph sets of X in the asymmetric metric in the recurrent case, but also determine the exact Hausdorff measure functions of the graph sets of X in the transient and recurrent cases, respectively. Moreover, we establish a uniform Hausdorff dimension result for the image sets of X. Our results extend the corresponding results on fractional Brownian motion and space or time anisotropic Gaussian random fields. [ABSTRACT FROM AUTHOR]

We characterize measure spaces such that the canonical map L ∞ → L 1 ∗ is surjective. In case of d -dimensional Hausdorff measure on a complete separable metric space X , we give two equivalent conditions. We give examples of X and d so that whether these conditions are met is undecidable in ZFC, including one with d equals the Hausdorff dimension of X. [ABSTRACT FROM AUTHOR]

This article, titled "Sub-Riemannian Coarea Formula for Classes of Noncontact Mappings of Carnot Groups," discusses the metric properties of level sets of noncontact mappings on Carnot groups. The author explores the possibility of extracting "sub-Riemannian" properties for such mappings and presents the sub-Riemannian analogs of differentiability for classes of mappings. The article also introduces definitions and theorems related to the sub-Riemannian metric properties of noncontact mappings. The results provide insights into the structure of the kernel of the differential and the Hausdorff dimension of level sets. The article concludes with the coarea formula for open sets not containing characteristic points. This research was supported by the Mathematical Center in Akademgorodok. [Extracted from the article]

Sets of infinite multifractal measures are awkward to work with, and reducing them to sets of positive finite multifractal measures is a very useful simplification. The aim of this paper is to show that the multifractal Hausdorff measures satisfy the "subset of positive and finite measure" property. We apply our main result to prove that the multifractal function dimension is defined as the supremum over the multifractal dimension of all Borel probability measures. [ABSTRACT FROM AUTHOR]

We prove a sub-Lorentzian analog of the area formula for intrinsically Lipschitz mappings of open subsets of Carnot groups of arbitrary depth with a sub-Lorentzian structure introduced on the image space. [ABSTRACT FROM AUTHOR]

We prove that the strong convexity supporting condition is typical for an arbitrary convex compact set in . It is shown that, in a certain sense for almost all points, the metric projection onto a convex compact set satisfies the Lipschitz condition with Lipschitz constant strictly less than 1. This condition characterizes the strong convexity supporting condition. The linear convergence of the alternating projection method for a convex compact set with the strong convexity supporting condition and for a proximally smooth set is proved under a certain relation between the constant in the strong convexity supporting condition and the proximal smoothness constant. [ABSTRACT FROM AUTHOR]

We show a new method of estimating the Hausdorff measure of a set from below. The method requires computing the subsequent closest return times of a point to itself. [ABSTRACT FROM AUTHOR]

In this article, we prove the various minimality of the product of a 1-codimensional calibrated manifold and a paired calibrated set. This is motivated by the attempt to classify all possible singularities for Almgren minimal sets – Plateau's problem in the setting of sets. The Almgren minimality was introduced by Almgren to modernize Plateau's problem. It gives a very good description of local behavior for soap films. The natural question of whether the product of any two Almgren minimal sets is still minimal is still open, although it seems obvious in intuition. We prove the Almgren minimality for the product of two large classes of Almgren minimal sets – the class of 1-codimensional calibrated manifolds and the class of paired calibrated sets. The general idea is to properly combine different topological conditions (separation and spanning) under different homology groups, to set up a reasonable topological condition and prove the minimality for the product under this condition, which will imply the Almgren minimality. A main difficulty comes from the codimension – algebraic coherences such as multiplicity, separation and orientation do not exist anymore for codimensions larger than 1. An unexpectedly useful thing in the present paper is the flow of the calibrations. Its most important role among all is helping us to do the decomposition of a competitor with the help of the first projections along the flows. [ABSTRACT FROM AUTHOR]

Our aim in this note is to establish exponential integrability and vanishing exponential integrability for Riesz potentials of functions in Morrey–Orlicz spaces. Our work generalizes results by Adams and Hurri‐Syrjänen (2003) and the authors (2007) from Orlicz case. We also discuss the size of the exceptional sets by using a capacity and Hausdorff measure. [ABSTRACT FROM AUTHOR]

We study the families of measures on Carnot groups that have vanishing pp-module, which we call Mp{M}_{p}-exceptional families. We found necessary and sufficient Conditions for the family of intrinsic Lipschitz surfaces passing through a common point to be Mp{M}_{p}-exceptional for p≥1p\ge 1. We describe a wide class of Mp{M}_{p}-exceptional intrinsic Lipschitz surfaces for p∈(0,∞)p\in \left(0,\infty ).

For contact mappings of Carnot groups of depth two whose image is endowed with a sub-Lorentzian structure, we prove local properties of the surfaces-images and explicitly deduce a sub-Lorentzian analog of the area formula. The result in particular also holds for Lipschitz mappings in the sub-Riemannian sense. [ABSTRACT FROM AUTHOR]

Given positive integers ℓ < n and a real d ∈ (ℓ , n) , we construct sets K ⊂ R n with positive and finite Hausdorff d–measure such that the Radon–Nikodym derivative associated to all projections on ℓ -dimensional planes, when it exists, is not an L p function for any p > 1 . [ABSTRACT FROM AUTHOR]

In this paper, we investigate the Hewitt-Stromberg h-measures H h and P h which lie between the Hausdorff h-measure H h and packing h-measure P h where h is a dimension function. We formulate a new version of density theorem given by Raymond and Tricot (1988) and then set up a necessary and sufficient condition for which these measures are equivalent. Also we study the notion of weak and strong regularity of sets. As an application we study the Hewitt-Stromberg measures of cartesian product sets by means of the measure of their components. We prove also that if, 0 < H t (E) = P t (E) < + ∞ , then t ∈ N . [ABSTRACT FROM AUTHOR]

We present and analyse numerical quadrature rules for evaluating regular and singular integrals on self-similar fractal sets. The integration domain Γ ⊂ R n is assumed to be the compact attractor of an iterated function system of contracting similarities satisfying the open set condition. Integration is with respect to any "invariant" (also known as "balanced" or "self-similar") measure supported on Γ , including in particular the Hausdorff measure H d restricted to Γ , where d is the Hausdorff dimension of Γ . Both single and double integrals are considered. Our focus is on composite quadrature rules in which integrals over Γ are decomposed into sums of integrals over suitable partitions of Γ into self-similar subsets. For certain singular integrands of logarithmic or algebraic type, we show how in the context of such a partitioning the invariance property of the measure can be exploited to express the singular integral exactly in terms of regular integrals. For the evaluation of these regular integrals, we adopt a composite barycentre rule, which for sufficiently regular integrands exhibits second-order convergence with respect to the maximum diameter of the subsets. As an application we show how this approach, combined with a singularity-subtraction technique, can be used to accurately evaluate the singular double integrals that arise in Hausdorff-measure Galerkin boundary element methods for acoustic wave scattering by fractal screens. [ABSTRACT FROM AUTHOR]

This work sets the matrix variate Birnbaum–Saunders theory in the context of singular distributions and elliptical models. The termed singular matrix variate generalized Birnbaum–Saunders distribution is obtained with respect the Hausdorff measure. Several basic properties and particular cases of this distribution are also derived. [ABSTRACT FROM AUTHOR]

In this work, we focus on the centered Hausdorff measure, the packing measure, and the Hewitt–Stromberg measure that determines the modified lower box dimension Moran fractal sets. The equivalence of these measures for a class of Moran is shown by having a strong separation condition. We give a sufficient condition for the equality of the Hewitt–Stromberg dimension, Hausdorff dimension, and packing dimensions. As an application, we obtain some relevant conclusions about the Hewitt–Stromberg measures and dimensions of the image measure of a τ -invariant ergodic Borel probability measures. Moreover, we give some statistical interpretation to dimensions and corresponding geometrical measures. [ABSTRACT FROM AUTHOR]

We give an estimate for the volume of an analytic variety (or more generally the mass of a positive closed current) close to a real submanifold M. Applications are given to the Hausdorff measure of the intersection of the variety with M and the exponential integrability of plurisubharmonic functions on M. [ABSTRACT FROM AUTHOR]

We prove that every finite Borel measure \(\mu \) in \(\mathbb R^N \) that is bounded from above by the Hausdorff measure \(\mathcal{H}^s \) can be split in countable many parts \(\mu{\lfloor_{_{{E_k}}}} \) that are bounded from above by the Hausdorff content \(\mathcal{H}^s_\infty \). Such a result generalises a theorem due to R. Delaware that says that any Borel set with finite Hausdorff measure can be decomposed as a countable disjoint union of straight sets. We apply this decomposition to give a simpler proof for the existence of solutions of a Dirichlet problem involving an exponential nonlinearity. [ABSTRACT FROM AUTHOR]

In this paper, we answer a question raised by David H. Fremlin about the Hausdorff measure of $\mathbb R^2$ with respect to a distance inducing the Euclidean topology. In particular we prove that the Hausdorff $n$-dimensional measure of $\mathbb R^n$ is never $0$ when considering a distance inducing the Euclidean topology. Finally, we show via counterexamples that the previous result does not hold in general if we remove the assumption on the topology. [ABSTRACT FROM AUTHOR]

We consider the construction of a family {KN} of 3-dimensional Koch-type surfaces, with a corresponding family of 3-dimensional Koch-type “snowflake analogues” {CN}, where N > 1 are integers with N ≢ 0 (mod 3). We first establish that the Koch surfaces KN are sN -sets with respect to the sN -dimensional Hausdorff measure, for sN = log(N² +2)/log(N) the Hausdorff dimension of each Koch-type surface KN. Using self-similarity, one deduces that the same result holds for each Koch-type crystal CN. We then develop lower and upper approximation monotonic sequences converging to the sN -dimensional Hausdorff measure on each Koch-type surface KN, and consequently, one obtains upper and lower bounds for the Hausdorff measure for each set CN. As an application, we consider the realization of Robin boundary value problems over the Koch-type crystals CN, for N > 2. [ABSTRACT FROM AUTHOR]

A thin set is defined to be an uncountable dense zero-dimensional subset of measure zero and Hausdorff measure zero of an Euclidean space. A fat set is defined to be an uncountable dense path-connected subset of an Euclidean space which has full measure, that is, its complement has measure zero. While there are well-known pathological maps of a set of measure zero, such as the Cantor set, onto an interval, we show that the standard addition on $\mathbb {R}$ maps a thin set onto a fat set; in fact the fat set is all of $\mathbb {R}$. Our argument depends on the theorem of Paul Erdős that every real number is a sum of two Liouville numbers. Our thin set is the set $\mathcal {L}^{2}$ , where $\mathcal {L}$ is the set of all Liouville numbers, and the fat set is $\mathbb {R}$ itself. Finally, it is shown that $\mathcal {L}$ and $\mathcal {L}^{2}$ are both homeomorphic to $\mathbb {P}$ , the space of all irrational numbers. [ABSTRACT FROM AUTHOR]

We extend the results of Hasselblatt and Schmeling [Dimension product structure of hyperbolic sets. Modern Dynamical Systems and Applications. Eds. B. Hasselblatt, M. Brin and Y. Pesin. Cambridge University Press, New York, 2004, pp. 331–345] and of Rams and Simon [Hausdorff and packing measure for solenoids. Ergod. Th. & Dynam. Sys.23 (2003), 273–292] for $C^{1+\varepsilon }$ hyperbolic, (partially) linear solenoids $\Lambda $ over the circle embedded in $\mathbb {R}^3$ non-conformally attracting in the stable discs $W^s$ direction, to nonlinear solenoids. Under the assumptions of transversality and on the Lyapunov exponents for an appropriate Gibbs measure imposing thinness, as well as the assumption that there is an invariant $C^{1+\varepsilon }$ strong stable foliation, we prove that Hausdorff dimension $\operatorname {\mathrm {HD}}(\Lambda \cap W^s)$ is the same quantity $t_0$ for all $W^s$ and else $\mathrm {HD}(\Lambda)=t_0+1$. We prove also that for the packing measure, $0 , but for Hausdorff measure, $\mathrm {HM}_{t_0}(\Lambda \cap W^s)=0$ for all $W^s$. Also $0 and $\mathrm {HM}_{1+t_0}(\Lambda)=0$. A technical part says that the holonomy along unstable foliation is locally Lipschitz, except for a set of unstable leaves whose intersection with every $W^s$ has measure $\mathrm {HM}_{t_0}$ equal to 0 and even Hausdorff dimension less than $t_0$. The latter holds due to a large deviations phenomenon. [ABSTRACT FROM AUTHOR]

Let Q = { q k } k ≥ 1 be a sequence of positive integers with q k ≥ 2 for every k ≥ 1 and T Q n : [ 0 , 1 ] → [ 0 , 1) be the transformation with T Q n (x) = q 1 ⋯ q n x − ⌊ q 1 ⋯ q n x ⌋ which induces the Cantor series expansion. In the paper, we show that the Lebesgue measure and generalized Hausdorff measure of the following shrinking target set { (x , y) ∈ [ 0 , 1 ] 2 : | T Q n (x) − y | < φ (n)   for i.m.   n ∈ ℕ } satisfy a dichotomy law, which depends on the convergence or divergence of a certain series, where φ : ℕ → ℝ + is a positive function such that φ (n) → 0 as n → ∞ and i.m. stands for infinitely many. In addition, for every integer d ≥ 1 and integer i with 1 ≤ i ≤ d , let f i : [ 0 , 1 ] → [ 0 , 1 ] be a Lipschitz function with Lipschitz constant L i . Assume that { k i } 1 ≤ i ≤ d ⊂ ℝ + with k 1 ≤ k 2 ≤ ⋯ ≤ k d , we obtain the Hausdorff dimension of the following set: { x = (x 1 , ... , x d) ∈ [ 0 , 1 ] d : | T Q n (x 1) − f 1 (x 1) | < (q 1 ⋯ q n) − k 1 , ... , | T Q n (x d) − f d (x d) | < (q 1 ⋯ q n) − k d   for i.m.   n ∈ ℕ }. [ABSTRACT FROM AUTHOR]

By the Galvin–Mycielski–Solovay theorem, a subset X of the line has Borel's strong measure zero if and only if $M+X\neq \mathbb {R}$ for each meager set M. A set $X\subseteq \mathbb {R}$ is meager-additive if $M+X$ is meager for each meager set M. Recently a theorem on meager-additive sets that perfectly parallels the Galvin–Mycielski–Solovay theorem was proven: A set $X\subseteq \mathbb {R}$ is meager-additive if and only if it has sharp measure zero, a notion akin to strong measure zero. We investigate the validity of this result in Polish groups. We prove, e.g., that a set in a locally compact Polish group admitting an invariant metric is meager-additive if and only if it has sharp measure zero. We derive some consequences and calculate some cardinal invariants. [ABSTRACT FROM AUTHOR]

In this work we completely describe the superdifferential of the Takagi-Van der Waerden functions and, as a consequence, the local maxima of these functions are characterized. Regarding the set of points where the superdifferential is not empty, we calculate its Hausdorff dimension as well as its corresponding Hausdorff measure. To do so, for any even integer greater than or equal to two we determine the 1/2-dimensional Hausdorff measure of the set of points where Takagi-Van der Waerden functions attain their global maximum. [ABSTRACT FROM AUTHOR]

We establish a refined version of parametric transversality theorems such as Thom’s parametric transversality theorem and its improvement which was given by John Mather in 1973. [ABSTRACT FROM AUTHOR]

We define rectifiability in Rn × R with a parabolic metric in terms of C¹ graphs and Lipschitz graphs with small Lipschitz constants and we characterize it in terms of approximate tangent planes and tangent measures. We also discuss relations between the parabolic rectifiability and other notions of rectifiability. [ABSTRACT FROM AUTHOR]

We present a complete study of measure-theoretic area formulas in metric spaces, providing different measurability conditions. [ABSTRACT FROM AUTHOR]

We continue the study of the fractional variation following the distributional approach developed in the previous works Bruè et al. (2021), Comi and Stefani (2019), Comi and Stefani (2019). We provide a general analysis of the distributional space B V α , p (R n) of L p functions, with p ∈ [ 1 , + ∞ ] , possessing finite fractional variation of order α ∈ (0 , 1) . Our two main results deal with the absolute continuity property of the fractional variation with respect to the Hausdorff measure and the existence of the precise representative of a B V α , p function. [ABSTRACT FROM AUTHOR]

In this work, we focus on the centered Hausdorff measure, the packing measure, and the Hewitt–Stromberg measure that determines the modified lower box dimension Moran fractal sets. The equivalence of these measures for a class of Moran is shown by having a strong separation condition. We give a sufficient condition for the equality of the Hewitt–Stromberg dimension, Hausdorff dimension, and packing dimensions. As an application, we obtain some relevant conclusions about the Hewitt–Stromberg measures and dimensions of the image measure of a τ-invariant ergodic Borel probability measures. Moreover, we give some statistical interpretation to dimensions and corresponding geometrical measures.

We construct a dynamical model of the deformation of a classical diffusion process into superdiffusion implementing the interaction between the diffusion background medium and the external medium. We show how the transformation law of energy characteristics of this deformation is formed gradually. [ABSTRACT FROM AUTHOR]

We prove that a pseudoconvex complex space X of pure dimension n + 1 {n+1} is holomorphically convex provided that its singular set has only compact connected components (e.g., X has isolated singularities) and rank x ⁡ 풪 X = n {\operatorname{rank}_{x}\mathscr{O}_{X}=n} for all smooth points x outside a compact set of X. This extends a known result due to Ohsawa asserting that a weakly 1-complete, smooth complex surface is holomorphically convex if it admits a globally defined non-constant holomorphic function. We also revise an example of Markoe concerning non-invariance of holomorphic convexity under normalization. [ABSTRACT FROM AUTHOR]

Let be a class of extended numerical functions on a domain of -dimensional Euclidean space , . Given and , we write if there is a function such that on . We consider this special preorder for a pair of subharmonic functions and on in cases when is the space of all harmonic functions on or is the convex cone of all subharmonic functions on . Main results are dual equivalent forms for this preorder in terms of balayage processes for Riesz measures of subharmonic functions and , for Jensen and Arens–Singer (representing) measures, for potentials of these measures, and for special test functions generated by subharmonic functions on complements of non-empty precompact subsets . Applications to holomorphic functions on a domain relate to the distribution of zero sets of functions under upper restrictions on . If a domain is a finitely connected domain with non-empty exterior or a simply connected domain with two different points on the boundary of , then our conditions for the distribution of zeros of with on are both necessary and sufficient. [ABSTRACT FROM AUTHOR]

The mean density estimation of a random closed set in R d , based on a single observation, is a crucial problem in several application areas. In the case of stationary random sets, a common practice to estimate the mean density is to take the n-dimensional volume fraction with observation window as large as possible. In the present paper, we provide large and moderate deviation results for these estimators when the random closed set Θ n belongs to the quite general class of stationary Boolean models with Hausdorff dimension n < d . Moreover, we establish a central limit theorem and a Berry–Esseen bound for the family of estimators under study. Our findings allow to recover some well-known results in the literature on Boolean models. Finally, we also provide a guideline for the estimation of the mean density of non-stationary Boolean models characterized by high intensity of the underlying Poisson point process. [ABSTRACT FROM AUTHOR]

Geometrical measurements of biological objects form the basis of many quantitative analyses. Hausdorff measures such as the volume and the area of objects are simple and popular descriptors of individual objects; however, for most biological processes, the interaction between objects cannot be ignored, and the shape and function of neighboring objects are mutually influential. In this paper, we present a theory on the geometrical interaction between objects inspired by K-functions for spatial point-processes. Our theory describes the relation between two objects: a reference and an observed object. We generate the r-parallel sets of the reference object, calculate the intersection between the r-parallel sets and the observed object, and define measures on these intersections. The measures are simple, like the volume or surface area, but describe further details about the shape of individual objects and their pairwise geometrical relation. Finally, we propose a summary-statistics. To evaluate these measures, we present a new segmentation of cell membrane, mitochondria, synapses, vesicles, and endoplasmic reticulum in a publicly available FIB-SEM 3D brain tissue data set and use our proposed method to analyze key biological structures herein. [ABSTRACT FROM AUTHOR]

Substituting in the definition of outer measure the addition with the maximum (or the supremum, or the join) operation we obtain a new set function called retuo measure. It is proved that every retuo measure is an outer measure. We give necessary and sufficient conditions for a set function to be a retuo measure. Similarly as in the case of outer measure, we propose a way to construct retuo measures. We consider some theoretical applications for constructed pairs of outer and retuo measures in the image of the Hausdorff measure and dimension.

We show that affine subspaces of Euclidean space are of Khintchine type for divergence under certain multiplicative Diophantine conditions on the parameterizing matrix. This provides evidence towards the conjecture that all affine subspaces of Euclidean space are of Khintchine type for divergence, or that Khintchine's theorem still holds when restricted to the subspace. This result is proved as a special case of a more general Hausdorff measure result from which the Hausdorff dimension of W (τ) intersected with an appropriate subspace is also obtained. [ABSTRACT FROM AUTHOR]

We consider finitely generated free semigroup actions on (X, d) and generalize Boshernitzan's quantitative recurrence theorem to general free semigroup actions. Let G be a finitely generated free semigroup endowed with a Bernoulli probability measure ℙ a ̲ and S be the corresponding continuous semigroup continuous semigroup action. Assume that, for some α > 0, the Hausdorff measure ν = Hα(X) as invariant by every generator in G. ν in X invariant by every generator in G. Then, for ℙ a -almost every ω and ν-almost x ∈ X, one has the following: lim inf n → ∞ n 1 α d (x , f ω n (x)) ≤ 1. [ABSTRACT FROM AUTHOR]

In this paper, we discuss a problem of multiplicative Diophantine approximation of the expansions under different bases restricted on a line. More specifically, let φ : ℕ → [ 0 , ∞) be a positive function. A dichotomy law of the Hausdorff measure for the following set: M (φ) = { x ∈ [ 0 , 1 ] : ∥ 2 n x ∥ ⋅ ∥ 3 n x ∥ < φ (n) for i.m. n ∈ ℕ } , is obtained, which depends the convergence or divergence of certain series. [ABSTRACT FROM AUTHOR]