Measure Zero Research Articles

Measurable Tilings by Abelian Group Actions

Abstract Let $X$ be a measure space with a measure-preserving action $(g,x) \mapsto g \cdot x$ of an abelian group $G$. We consider the problem of understanding the structure of measurable tilings $F \odot A = X$ of $X$ by a measurable tile $A \subset X$ translated by a finite set $F \subset G$ of shifts, thus the translates $f \cdot A$, $f \in F$ partition $X$ up to null sets. Adapting arguments from previous literature, we establish a “dilation lemma” that asserts, roughly speaking, that $F \odot A = X$ implies $F^{r} \odot A = X$ for a large family of integer dilations $r$, and use this to establish a structure theorem for such tilings analogous to that established recently by the second and fourth authors. As applications of this theorem, we completely classify those random tilings of finitely generated abelian groups that are “factors of iid”, and show that measurable tilings of a torus ${\mathbb{T}}^{d}$ can always be continuously (in fact linearly) deformed into a tiling with rational shifts, with particularly strong results in the low-dimensional cases $d=1,2$ (in particular resolving a conjecture of Conley, the first author, and Pikhurko in the $d=1$ case).

Topological Analysis of Vector-Field Guided Path Following on Manifolds

A path-following control algorithm enables a system’s trajectories under its guidance to converge to and evolve along a given geometric desired path. There exist various such algorithms, but many of them can only guarantee local convergence to the desired path in its neighborhood. In contrast, the control algorithms using a well-designed guiding vector field can ensure almost global convergence of trajectories to the desired path; here, “almost” means that in some cases, a measure-zero set of trajectories converge to the singular set where the vector field becomes zero (with all other trajectories converging to the desired path). In this article, we first generalize the guiding vector field from the Euclidean space to a general smooth Riemannian manifold. This generalization can deal with path-following in some abstract configuration space (such as robot arm joint space). Then, we show several theoretical results from a topological viewpoint. Specifically, we are motivated by the observation that singular points of the guiding vector field exist in many examples where the desired path is homeomorphic to the unit circle, but it is unknown whether the existence of singular points always holds in general (i.e., is inherent in the topology of the desired path). In the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$n$</tex-math></inline-formula> -dimensional Euclidean space, we provide an affirmative answer, and conclude that it is not possible to guarantee global convergence to desired paths that are homeomorphic to the unit circle. Furthermore, we show that there always exist nonpath-converging trajectories (i.e., trajectories that do not converge to the desired path) starting from the boundary of a ball containing the desired path in an <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$n$</tex-math></inline-formula> -dimensional Euclidean space where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$n \geq 3$</tex-math></inline-formula> . Examples are provided to illustrate the theoretical results.

ERDŐS PROPERTIES OF SUBSETS OF THE MAHLER SETS

AbstractErdős proved that every real number is the sum of two Liouville numbers. A setWof complex numbers is said to have the Erdős property if every real number is the sum of two members ofW. Mahler divided the set of all transcendental numbers into three disjoint classesS,TandUsuch that, in particular, any two complex numbers which are algebraically dependent lie in the same class. The set of Liouville numbers is a proper subset of the setUand has Lebesgue measure zero. It is proved here, using a theorem of Weil on locally compact groups, that if$m\in [0,\infty )$, then there exist$2^{\mathfrak {c}}$dense subsetsWofSeach of Lebesgue measuremsuch thatWhas the Erdős property and no two of theseWare homeomorphic. It is also proved that there are$2^{\mathfrak {c}}$dense subsetsWofSeach of full Lebesgue measure, which have the Erdős property. Finally, it is proved that there are$2^{\mathfrak {c}}$dense subsetsWofSsuch that every complex number is the sum of two members ofWand such that no two of theseWare homeomorphic.

Subspaces of Hilbert-generated Banach spaces and the quantification of super weak compactness

We introduce a measure of super weak noncompactness Γ defined for bounded subsets and bounded linear operators in Banach spaces that allows to state and prove a characterization of the Banach spaces which are subspaces of a Hilbert-generated space. The use of super weak compactness and Γ casts light on the structure of these Banach spaces and complements the work of Argyros, Fabian, Farmaki, Godefroy, Hájek, Montesinos, Troyanski and Zizler on this subject. A particular kind of relatively super weakly compact sets, namely uniformly weakly null sets, plays an important role and exhibits connections with Banach-Saks type properties.

SIGEST

The SIGEST article in this issue, “Improbability of Collisions of Point-Vortices in Bounded Planar Domains,” by Martin Donati, is based on the 2022 SIAM Journal on Mathematics Analysis article “Two-Dimensional Point Vortex Dynamics in Bounded Domains: Global Existence for Almost Every Initial Data.” This work concerns point-vortex dynamics in a very general bounded domain in the plane. The main result is that the set of initial configurations which lead to finite-time collision, although nonempty, has Lebesgue measure zero. This is an elegant and highly nontrivial extension to general bounded domains with $C^{2, \alpha}$ boundary of a result previously known only for three domains: the full plane, a disk, and the complement of a disk. It is notable that the result also extends to point-vortex dynamics in multiply connected domains. In preparing this SIGEST version, the author has added a new introductory section that explains the context of the new results. Moreover, some of the technical details from the original article have been replaced by more accessible high-level explanations. Recent references on the topics of vortex collisions and desingularization problems are also included. This SIGEST article will be of particular interest to researchers in fluid mechanics, partial differential equations, and applied analysis.

Uniformly convergent Fourier series with universal power parts on closed subsets of measure zero

Given a closed subset E of Lebesgue measure zero on the unit circle T there is a function f on T with uniformly convergent symmetric Fourier series Sn(f,ζ)=∑k=−nnfˆ(k)ζk⇉Tf(ζ),such that for every continuous function g on E, there is a subsequence of partial power sums Sn+(f,ζ)=∑k=0nfˆ(k)ζkof f, which converges to g uniformly on E. Here fˆ(k)=∫Tζ̄kf(ζ)dm(ζ),and m is the normalized Lebesgue measure on T.

О взаимоотношении движений динамических систем в сепарабельном локально компактном метрическом пространствес инвариантной мерой

In this paper, we study the interrelation between recurrent and outgoing motions of dynamical systems. An outgoing motion is a motion whose α- and ω-limit sets are either empty or non-compact. It is shown that in a separable locally compact metric space Σ with invariant Caratheodory measure, almost all points lie on trajectories of motions that are either recurrent or outgoing, i. e. in the space Σ, the set of points Γ lying on the trajectories of nonoutgoing and non-recurrent motions has measure zero. Moreover, any motion located in Γ is both positively and negatively asymptotic with respect to the corresponding compact minimal sets. The proof of this assertion essentially relies on the classical Poincare-Caratheodory and Hopf recurrence theorems. From this proof and Hopf’s theorem, it follows that in a separable locally compact metric space, there can exist non-recurrent Poisson-stable motions, but all these motions must necessarily be outgoing. At the same time, in the compact space Σ any Poisson-stable motion is recurrent.

On the generalized Bessel potential and perfect functional completion

he class of generalized Bessel potentials is the main object of study in this article. The generalized Bessel potential is a negative real power of the operator (I − Δγ), where Δγ = (...)∂xk - Laplace - Bessel operator, γ = (γ1,...,γn) - multi-index consisting of positive fixed real numbers. When solving various problems for differential equations, proving embedding theorems for some classes of functions, and inverting integral operators, there is a need to consider functions up to some small, from the point of view of the problem under consideration, set. Often a set of Lebesgue measure zero is taken as such a small set. However, often the sets of Lebesgue measure zero turn out to be too large to be neglected. For example, when solving a boundary problem, the behavior of the solution on the boundary is essential. In this regard, there was a need to construct complete classes of admissible functions suitable for solving specific problems. N. Aronshine and K. Smith presented two stages of constructing a functional completion. The first of these is finding a suitable class of exceptional sets. The second is to find the functions defined modulo these exceptional classes that need to be attached to get a complete functional class. It turns out that there can be infinitely many suitable exceptional classes in a particular problem, but each of them corresponds essentially to one functional completion. It is clear that the most suitable functional completion is the one whose exceptional class is the smallest, since then the functions will be defined with the best possible accuracy. Whenever such a minimal exceptional class exists, the corresponding functional completion is called a perfect completion. In this paper, perfect completions are constructed from the norm associated with the kernel of the generalized Bessel potential.

An inclusive study on the fundamentals of interval-valued fuzzy hypersoft set

When compared to its extension, hypersoft set, a soft set only deals with a single set of attributes, while a hypersoft set deals with several attribute-valued disjoint sets that correspond to various attributes. Several researchers have developed models based on soft sets, but the majority of these models suffer from limitations since they are inappropriate for interval-type data or uncertain data. In order to address these issues, a novel model interval-valued fuzzy hypersoft set (IV F HS -set) is presented in this research article. This model not only resolves the inadequacy of soft set for distinct attributes for non-overlapping attribute-valued sets, but also addresses the limitations of soft set-like models with having data in interval environment. This work modifies the current fuzzy hypersoft set concept and introduces certain fundamental ideas, such as subset, not set, whole set, and absolute relative null set, relative absolute set and aggregation operations e.g. intersection, union, extended intersection, restricted union, complement, OR, AND, difference, restricted difference are discussed under IV F HS -set environment with illustrated examples. Some new hybrids of fuzzy hypersoft set under interval-valued settings are also discussed. Moreover, some extensions of IV F HS -set are presented along with different operations.

Near fixed point theorems in near Banach spaces

&lt;abstract&gt;&lt;p&gt;The near vector space in which the additive inverse element does not necessarily exist is introduced in this paper. The reason is that an element in a near vector space which subtracts itself may not be a zero element. Therefore, the concept of a null set is introduced in this paper to play the role of a zero element. A near vector space can also be endowed with a norm to define a so-called near normed space. Based on this norm, the concept of a Cauchy sequence can be similarly defined. A near Banach space can also be defined according to the concept of completeness using the Cauchy sequences. The main aim of this paper is to establish the so-called near fixed point theorems and Meir-Keeler type of near fixed point theorems in near Banach spaces.&lt;/p&gt;&lt;/abstract&gt;

An Identity Theorem for the Fourier–Laplace Transform of Polytopes on Nonzero Complex Multiples of Rationally Parameterizable Hypersurfaces

A set mathcal {S} of points in mathbb {R}^n is called a rationally parameterizable hypersurface if there is vector function varvec{sigma }:mathbb {R}^{n-1}rightarrow mathbb {R}^n having as components rational functions defined on some common domain D such that mathcal {S}={varvec{sigma }(textbf{t}):textbf{t}in D}. A generalized n-dimensional polytope in mathbb {R}^n is a union of a finite number of convex n-dimensional polytopes in mathbb {R}^n. The Fourier–Laplace transform of such a generalized polytope mathcal {P} in mathbb {R}^n is defined by F_{mathcal {P}}(textbf{z})=int _{mathcal {P}}e^{textbf{z}cdot textbf{x}},textbf{dx}. Let gamma be a fixed nonzero complex number. We prove that F_{mathcal {P}_1}(gamma varvec{sigma }(textbf{t}))=F_{mathcal {P}_2}(gamma varvec{sigma }(textbf{t})) for all textbf{t} in O implies mathcal {P}_1=mathcal {P}_2 if O is an open subset of D satisfying some well-defined conditions and we present similar results for the null set of the Fourier–Laplace transform of mathcal {P}. Moreover we show that this theorem can be applied to quadric hypersurfaces that do not contain a line, but at least two points, i.e., in particular to spheres.

Some properties of Vitali sets

Abstract If V 1 , … , V n {V_{1},\dots,V_{n}} are translations of a Vitali set by rational numbers, then we prove that ⋃ i = 1 n V i {\bigcup_{i=1}^{n}V_{i}} contains no measurable subset of positive measure. This provides a decomposition of ℝ {{\mathbb{R}}} as a countable union of disjoint sets, any finite union of which has Lebesgue inner measure zero. As a consequence, we present a function δ : ℝ → ( 0 , + ∞ ) {\delta:{\mathbb{R}}\rightarrow(0,+\infty)} for which there is no measurable function f : ℝ → ℝ {f:{\mathbb{R}}\rightarrow{\mathbb{R}}} satisfying 0 &lt; f ≤ δ {0&lt;f\leq\delta} , on any measurable set of positive Lebesgue measure.

Bipolar Spherical Fuzzy Soft Topology with Applications to Multi-Criteria Group Decision-Making in Buildings Risk Assessment

A generalized soft set model that is more accurate, useful, and realistic is the bipolar spherical fuzzy soft set (BSFSs). It is a more developed variant of current fuzzy soft set models that may be applied to characterize erroneous data in practical applications. Bipolar spherical fuzzy soft sets and bipolar spherical fuzzy soft topology are novel ideas that are intended to be introduced in this work. Bipolar spherical fuzzy soft intersection, bipolar spherical fuzzy soft null set, spherical fuzzy soft absolute set, and other operations on bipolar spherical fuzzy soft sets are some of the fundamental ideas defined in this work. The bipolar spherical fuzzy soft open set, the bipolar spherical fuzzy soft close set, the bipolar spherical fuzzy soft closure, and the spherical fuzzy soft interior are also defined. Additionally, the characteristics of this specified set are covered and described using pertinent instances. The innovative notion of BSFSs makes it easier to describe the symmetry of two or more objects. Moreover, a group decision-making algorithm based on the TOPSIS (Technique of Order Preference by Similarity to an Ideal Solution) approach to problem-solving is described. We analyze the symmetry of the optimal decision and ranking of feasible alternatives. A numerical example is used to show how the suggested approach may be used. The extensive benefits of the proposed work over the existing techniques have been listed.

Localization Properties of High-Energy Eigenfunctions on Flat Tori

Abstract We consider the question of when the Laplace eigenfunctions on an arbitrary flat torus ${\mathbb {T}}_\Gamma :={\mathbb {R}}^d/\Gamma $ are flexible enough to approximate, over the natural length scale of order $1/\sqrt \lambda $ where $\lambda \gg 1$ is the eigenvalue, an arbitary solution of the Helmholtz equation $\Delta h + h=0$ on ${\mathbb {R}}^d$. This problem is motivated by the fact that, by the asymptotics for the local Weyl law, “approximate Laplace eigenfunctions” do have this approximation property on any compact Riemannian manifold. What we find is that the answer depends solely on the arithmetic properties of the spectrum. Specifically, recall that the eigenvalues of ${\mathbb {T}}_\Gamma $ are of the form $\lambda _k=Q_\Gamma (k)$, where $Q_\Gamma $ is a quadratic form and $k\in {\mathbb {Z}}^d$. Our main result is that the eigenfunctions of ${\mathbb {T}}_\Gamma $ have the desired approximation property if and only if $Q_\Gamma $ is a multiple of a quadratic form with integer coefficients. In particular, the set of lattices $\Gamma $ for which this approximation property holds has measure zero but includes all rational lattices. A consequence of this fact is that when $Q_\Gamma $ is a multiple of a quadratic form with integer coefficients, Laplace eigenfunctions exhibit an extremely flexible behavior over scales of order $1/\sqrt \lambda $. In particular, there are eigenfunctions of arbitrarily high energy that exhibit nodal components diffeomorphic to any compact hypersurface of diameter $O(1/\sqrt \lambda )$.

Chemical Probe-Based Nanopore Sequencing to Selectively Assess the RNA Modifications.

Nanopore direct RNA sequencing (dRNA-Seq) reads reveal RNA modifications through consistent error profiles specific to a modified nucleobase. However, a null data set is required to identify actual RNA modification-associated errors for distinguishing it from confounding highly intrinsic sequencing errors. Here, we reveal that inosine creates a signature mismatch error in dRNA-Seq reads and obviates the need for a null data set by harnessing the selective reactivity of acrylonitrile for validating the presence of actual inosine modifications. Selective reactivity of acrylonitrile toward inosine altered multiple dRNA-Seq parameters like signal intensity and trace value. We also deduced the stoichiometry of inosine modification through deviation in signal intensity and trace value using this chemical biology approach. Furthermore, we devised Nano ICE-Seq, a protocol to overcome the low coverage issue associated with direct RNA sequencing. Taken together, our chemical probe-based approach may facilitate the knockout-free detection of disease-associated RNA modifications in clinical scenarios.

Radner equilibrium and systems of quadratic BSDEs with discontinuous generators

Motivated by an equilibrium problem, we establish the existence of a solution for a family of Markovian backward stochastic differential equations with quadratic nonlinearity and discontinuity in Z. Using unique continuation and backward uniqueness, we show that the set of discontinuity has measure zero. In a continuous-time stochastic model of an endowment economy, we prove the existence of an incomplete Radner equilibrium with nondegenerate endogenous volatility.

Tightness of supercritical Liouville first passage percolation

Liouville first passage percolation (LFPP) with parameter \xi &gt;0 is the family of random distance functions \{D_h^\epsilon\}_{\epsilon &gt;0} on the plane obtained by integrating e^{\xi h_\epsilon} along paths, where h_\epsilon for \epsilon &gt;0 is a smooth mollification of the planar Gaussian free field. Previous work by Ding–Dubédat–Dunlap–Falconet and Gwynne–Miller has shown that there is a critical value \xi_{\mathrm{crit}} &gt; 0 such that for \xi &lt; \xi_{\mathrm{crit}} , LFPP converges under appropriate re-scaling to a random metric on the plane which induces the same topology as the Euclidean metric (the so-called \gamma - Liouville quantum gravity metric for \gamma = \gamma(\xi)\in (0,2) ). We show that for all \xi &gt; 0 , the LFPP metrics are tight with respect to the topology on lower semicontinuous functions. For \xi &gt; \xi_{\mathrm{crit}} , every possible subsequential limit D_h is a metric on the plane which does not induce the Euclidean topology: rather, there is an uncountable, dense, Lebesgue measure-zero set of points z\in\mathbb C such that D_h(z,w) = \infty for every w\in\mathbb C\setminus \{z\} . We expect that these subsequential limiting metrics are related to Liouville quantum gravity with matter central charge in (1,25) .

Generic Identifiability of the DINA Model and Blessing of Latent Dependence.

Cognitive diagnostic models are a powerful family of fine-grained discrete latent variable models in psychometrics. Within this family, the DINA model is a fundamental and parsimonious one that has received significant attention. Similar to other complex latent variable models, identifiability is an important issue for CDMs, including the DINA model. Gu and Xu (Psychometrika 84(2):468-483, 2019) established the necessary and sufficient conditions for strict identifiability of the DINA model. Despite being the strongest possible notion of identifiability, strict identifiability may impose overly stringent requirements on designing the cognitive diagnostic tests. This work studies a slightly weaker yet very useful notion, generic identifiability, which means parameters are identifiable almost everywhere in the parameter space, excluding only a negligible subset of measure zero. We propose transparent generic identifiability conditions for the DINA model, relaxing existing conditions in nontrivial ways. Under generic identifiability, we also explicitly characterize the forms of the measure-zero sets where identifiability breaks down. In addition, we reveal an interesting blessing-of-latent-dependence phenomenon under DINA-that is, dependence between the latent attributes can restore identifiability under some otherwise unidentifiable [Formula: see text]-matrix designs. The blessing of latent dependence provides useful practical implications and reassurance for real-world designs of cognitive diagnostic assessments.

Zero measure spectrum for multi-frequency Schrödinger operators

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 a consequence, we show that for these torus translations, every quasi-periodic potential can be approximated uniformly by one for which the associated Schrödinger operator has Cantor spectrum of zero Lebesgue measure. We also describe a framework that can allow this to be extended to higher-dimensional tori.

Singular vectors on manifolds over totally real number fields

We extend the notion of singular vectors to the context of Diophantine approximation of real numbers with elements of a totally real number field K. For $$m\ge 1$$ , we establish a version of Dani’s correspondence in number fields and prove that under a class of ‘friendly measures’ in $$K_S^m$$ , the set of singular vectors has measure zero. Here S is the set of Archimedean valuations of K and $$K_S$$ is the product of the completions of $$\sigma (K)$$ , $$\sigma \in S$$ . On the other hand, we show the existence of uncountably many non-trivial singular vectors on suitable submanifolds of $$K^m_S$$ under the action of a certain one parameter subgroup of $${\mathrm {SL}}_{m+1}(K_S)$$ .