Lebesgue Measure Research Articles

Linear and nonlinear transport equations with coordinate-wise increasing velocity fields

We consider linear and nonlinear transport equations with irregular velocity fields, motivated by models coming from mean field games. The velocity fields are assumed to increase in each coordinate, and the divergence therefore fails to be absolutely continuous with respect to the Lebesgue measure in general. For such velocity fields, the well-posedness of first- and second-order linear transport equations in Lebesgue spaces is established, as well as the existence and uniqueness of regular ODE and SDE Lagrangian flows. These results are then applied to the study of certain nonconservative, nonlinear systems of transport type, which are used to model mean field games in a finite state space. A notion of weak solution is identified for which unique minimal and maximal solutions exist, which do not coincide in general. A selection-by-noise result is established for a relevant example to demonstrate that different types of noise can select any of the admissible solutions in the vanishing noise limit.

Topology Meets Number Theory

Liouville proved the existence of a set L of transcendental real numbers now known as Liouville numbers. Erdős proved that while L is a small set in that its Lebesgue measure is zero, and even its s-dimensional Hausdorff measure, for each s > 0, equals zero, it has the Erdős property, that is, every real number is the sum of two numbers in L . He proved L is a dense G δ -subset of R and every dense G δ -subset of R has the Erdős property. While being a dense G δ -subset of R is a purely topological property, all such sets contain c Liouville numbers. Each dense G δ -subset of R , including L , is homeomorphic to the product N ℵ 0 of copies of the discrete space N of all natural numbers. Also this product space is homeomorphic to the space P of all irrational real numbers and the space T of all transcendental real numbers. Hence every dense G δ -subset of R has cardinality c . Indeed, any dense G δ -subset of R has a chain Xm , m ∈ ( 0 , ∞ ) of homeomorphic dense G δ -subsets such that X m ⊂ X n , for n < m, and X n ∖ X m has cardinality c . Finally, every real number r ≠ 1 is equal to ab , for some a , b ∈ L .

Diffeomorphisms with infinitely many Smale horseshoes

ABSTRACT A class of planar diffeomorphims is formulated, with infinitely many coexisting Smale horseshoes, where the Lebesgue measure of the parameters with such strange dynamics is infinite. On each horseshoe, there exists a uniformly hyperbolic invariant set, on which the map is topologically conjugate to the two-sided full-shift on two symbols. Moreover, the topological entropy is infinite in certain parameter regions.

Computation of Choquet integrals: Analytical approach for continuous functions

In the continuous case, analytical computations of the Choquet integral are limited, despite being commonly used in various applications. One can either use the definition, which is computationally demanding and impractical, or apply already existing formulas restricted only to monotone nonnegative functions on a real interval starting at zero. This article aims to present more convenient computational formulas for continuous functions without imposing restrictions on their monotonicity given any real interval. First, a more general approach to monotone functions is provided for both positive and negative functions. Then, reordering techniques are introduced to compute the Choquet integral of an arbitrary continuous function, and with these, a monotone equivalent to every function can be constructed. This equivalent function preserves the final Choquet integral value, implying that only formulas for monotone functions are required. In addition to general fuzzy measures, the article assumes particular cases of distorted Lebesgue measures and distorted probabilities as the most commonly used fuzzy measures.

Parameterized family of annular homeomorphisms with pseudo-circle attractors

In this paper we construct a paramaterized family of annular homeomorphisms with Birkhoff-like rotational attractors that vary continuously with the parameter, are all homeomorphic to a unique topological object, called the R.H. Bing's pseudo-circle, yet display an interesting boundary dynamics. Namely, in the constructed family of homeomorphisms the outer prime ends rotation number vary continuously with the parameter through the interval [0,1/2]. This, in particular, answers a question from Boroński et al. (2020) [15]. Furthermore, these attractors preserve the induced Lebesgue measure from the circle and have strong measure-theoretic and statistical properties. To show main results of the paper we first prove a result of an independent interest, that Lebesgue-measure preserving circle maps generically satisfy the crookedness condition which implies that generically the inverse limits of Lebesgue measure-preserving circle maps are the pseudo-solenoids. For degree one circle maps, this implies that the generic inverse limit in this context is the pseudo-circle.

On the Local Fourier Uniformity Problem for Small Sets

Abstract We consider vanishing properties of exponential sums of the Liouville function $\boldsymbol{\lambda }$ of the form $$ \begin{align*} &amp; \lim_{H\to\infty}\limsup_{X\to\infty}\frac{1}{\log X}\sum_{m\leq X}\frac{1}{m}\sup_{\alpha\in C}\bigg|\frac{1}{H}\sum_{h\leq H}\boldsymbol{\lambda}(m+h)e^{2\pi ih\alpha}\bigg|=0, \end{align*} $$ where $C\subset{{\mathbb{T}}}$. The case $C={{\mathbb{T}}}$ corresponds to the local $1$-Fourier uniformity conjecture of Tao, a central open problem in the study of multiplicative functions with far-reaching number-theoretic applications. We show that the above holds for any closed set $C\subset{{\mathbb{T}}}$ of zero Lebesgue measure. Moreover, we prove that extending this to any set $C$ with non-empty interior is equivalent to the $C={{\mathbb{T}}}$ case, which shows that our results are essentially optimal without resolving the full conjecture. We also consider higher-order variants. We prove that if the linear phase $e^{2\pi ih\alpha }$ is replaced by a polynomial phase $e^{2\pi ih^{t}\alpha }$ for $t\geq 2$ then the statement remains true for any set $C$ of upper box-counting dimension $&amp;lt; 1/t$. The statement also remains true if the supremum over linear phases is replaced with a supremum over all nilsequences coming form a compact countable ergodic subsets of any $t$-step nilpotent Lie group. Furthermore, we discuss the unweighted version of the local $1$-Fourier uniformity problem, showing its validity for a class of “rigid” sets (of full Hausdorff dimension) and proving a density result for all closed subsets of zero Lebesgue measure.

Sumset Estimates in Convex Geometry

Abstract Sumset estimates, which provide bounds on the cardinality of sumsets of finite sets in a group, form an essential part of the toolkit of additive combinatorics. In recent years, probabilistic or entropic forms of many of these inequalities were introduced. We study analogues of these sumset estimates in the context of convex geometry and the Lebesgue measure on ${\mathbb{R}}^{n}$. First, we observe that, with respect to Minkowski summation, volume is supermodular to arbitrary order on the space of convex bodies. Second, we explore sharp constants in the convex geometry analogues of variants of the Plünnecke-Ruzsa inequalities. In the last section of the paper, we provide connections of these inequalities to the classical Rogers-Shephard inequality.

Apportionable matrices and gracefully labelled graphs

To apportion a complex matrix means to apply a similarity so that all entries of the resulting matrix have the same magnitude. We initiate the study of apportionment, both by unitary matrix similarity and general matrix similarity. There are connections between apportionment and classical graph decomposition problems, including graceful labellings of graphs, Hadamard matrices, and equiangular lines, and potential applications to instantaneous uniform mixing in quantum walks. The connection between apportionment and graceful labellings allows the construction of apportionable matrices from trees. A generalization of the well-known Eigenvalue Interlacing Inequalities using graceful labellings is also presented. It is shown that every rank one matrix can be apportioned by a unitary similarity, but there are 2×2 matrices that cannot be apportioned. A necessary condition for a matrix to be apportioned by unitary matrix is established. This condition is used to construct a set of matrices with nonzero Lebesgue measure that are not apportionable by a unitary matrix.

Semiclassical estimates for measure potentials on the real line

We prove an explicit weighted estimate for the semiclassical Schrödinger operator P = - h^{2} \partial^{2}_{x} + V(x;h) on L^{2}(\R) , with V(x;h) a finite signed measure, and where h &gt;0 is the semiclassical parameter. The proof is a one-dimensional instance of the spherical energy method, which has been used to prove Carleman estimates in higher dimensions and in more complicated geometries. The novelty of our result is that the potential need not be absolutely continuous with respect to Lebesgue measure. Two consequences of the weighted estimate are the absence of positive eigenvalues for P , and a limiting absorption resolvent estimate with sharp h -dependence. The resolvent estimate implies exponential time-decay of the local energy for solutions to the corresponding wave equation with a compactly supported measure potential, provided there are no negative eigenvalues and no zero resonance, and provided the initial data have compact support.

Generalization of PINNs for elliptic interface problems

In this letter, we investigate the statistical limits of deep learning for learning solutions of elliptic interface problems from randomly generated data by employing Physics-informed Neural Networks (PINNs). We prove a lower bound and an upper bound for the generalization error of PINNs for solving elliptic interface equations with the Dirichlet boundary condition. In particular, the upper bound on the generalization error of PINNs is obtained by computing the fixed points of the local Rademacher complexity. We show that variations in volume across distinct regions exert influence on the requisite sample complexity. This unveils an underlying principle within the sampling strategy: the sample size of the data is contingent upon the Lebesgue measure of the domain, coupled with the smoothness and dimensionality characteristics of the interface problem. This letter sheds light on the network architecture design for solving interface problems using PINNs.

Critical Gaussian multiplicative chaos for singular measures

Given d≥1, we provide a construction of the random measure – the critical Gaussian Multiplicative Chaos – formally defined as e2dXdμ where X is a log-correlated Gaussian field and μ is a locally finite measure on Rd. Our construction generalizes the one performed in the case where μ is the Lebesgue measure. It requires that the measure μ is sufficiently spread out, namely that for μ almost every x we have ∫B(x,1)μ(dy)|x−y|deρlog1|x−y|<∞, where ρ:R+→R+ can be chosen to be any lower envelope function for the 3-Bessel process (this includes ρ(x)=xα with α∈(0,1/2)). We prove that three distinct random objects converge to a common limit which defines the critical GMC: the derivative martingale, the critical martingale, and the exponential of the mollified field. We also show that the above criterion for the measure μ is in a sense optimal.

A Classification Algorithm Based on Improved Locally Linear Embedding

The current classification is difficult to overcome the high-dimension classification problems. So, we will design the decreasing dimension method. Locally linear embedding is that the local optimum gradually approaches the global optimum, especially the complicated manifold learning problem used in big data dimensionality reduction needs to find an optimization method to adjust k-nearest neighbors and extract dimensionality. Therefore, we intend to use orthogonal mapping to find the optimization closest neighbors k, and the design is based on the Lebesgue measure constraint processing technology particle swarm locally linear embedding to improve the calculation accuracy of popular learning algorithms. So, we propose classification algorithm based on improved locally linear embedding. The experiment results show that the performance of proposed classification algorithm is best compared with the other algorithm.

Physical measures for partially hyperbolic diffeomorphisms with mixed hyperbolicity * *Y Cao is partially supported by National Key R&D Program of China(2022YFA1005802) and NSFC 12371194. Z Mi is partially supported by National Key R&D Program of China(2022YFA1007800) and NSFC 12271260. J Yang was partially supported by CNPq, FAPERJ, PRONEX and NNSFC Grants 12271538, 12071202 and 12161141002.

We study the partially hyperbolic diffeomorphisms whose centre direction admits the u-definite property in the sense that all the central Lyapunov exponents of each ergodic Gibbs u-state are either all positive or all negative. We prove that for this kind of partially hyperbolic diffeomorphisms, there are finitely many physical measures, whose basins cover a full Lebesgue measure subset of the ambient space. We also provide examples of diffeomorphisms whose centres are u-definite, where the supports of different ergodic Gibbs u-states have nontrivial intersections.

On the asymptotic spectral distribution of increasing size matrices: test functions, spectral clustering, and asymptotic estimates of outliers

Let {An}n be a sequence of square matrices such that size(An)=dn→∞ as n→∞. We say that {An}n has an asymptotic spectral distribution described by a Lebesgue measurable function f:D⊂Rk→C if, for every continuous function F:C→C with bounded support,limn→∞⁡1dn∑i=1dnF(λi(An))=1μk(D)∫DF(f(x))dx, where μk is the Lebesgue measure in Rk and λ1(An),…,λdn(An) are the eigenvalues of An. In the last decades, the asymptotic spectral distribution of increasing size matrices has become a subject of investigation by several authors. Special attention has been devoted to matrices An arising from the discretization of differential equations, whose size dn diverges to ∞ along with the mesh-fineness parameter n. However, despite the popularity that the topic has reached nowadays, the role of the so-called test functions F appearing in the above limit relation has been inexplicably neglected so far. In particular, a natural question such as “Which is the largest set of test functions F for which the above limit relation is satisfied?” is still unanswered. In the present paper, we provide a definitive answer to this question by identifying the largest set of test functions F for which the above limit relation is satisfied. We also present some applications of this result to the analysis of spectral clustering, including new asymptotic estimates on the number of outliers. A special attention is devoted to the so-called “inner outliers” of Hermitian block Toeplitz matrices. We conclude the paper with an interpretation of the main result in the context of the vague convergence of probability measures.

Proofs of ergodicity of piecewise Möbius interval maps using planar extensions

We give two results for deducing dynamical properties of piecewise Möbius interval maps from their related planar extensions. First, eventual expansivity and the existence of an ergodic invariant probability measure equivalent to Lebesgue measure both follow from mild finiteness conditions on the planar extension along with a new property “bounded non-full range” used to relax traditional Markov conditions. Second, the “quilting” operation to appropriately nearby planar systems, introduced by Kraaikamp and co-authors, can be used to prove several key dynamical properties of a piecewise Möbius interval map. As a proof of concept, we apply these results to recover known results on the well-studied Nakada α-continued fractions; we obtain similar results for interval maps derived from an infinite family of non-commensurable Fuchsian groups.

On the convergence of discrete dynamic unbalanced transport models

A generalized unbalanced optimal transport distance WBΛ on matrix-valued measures M(Ω, 𝕊n+) was defined in Li and Zou (arXiv:2011.05845) à la Benamou–Brenier, which extends the Kantorovich–Bures and the Wasserstein–Fisher–Rao distances. In this work, we investigate the convergence properties of the discrete transport problems associated with WBΛ. We first present a convergence framework for abstract discretization. Then, we propose a specific discretization scheme that aligns with this framework, whose convergence relies on the assumption that the initial and final distributions are absolutely continuous with respect to the Lebesgue measure. Further, in the case of the Wasserstein–Fisher–Rao distance, thanks to the static formulation, we show that such an assumption can be removed.

An ‘elementary’ perspective on reasoning about probability spaces

Abstract This paper is concerned with a two-sorted probabilistic language, denoted by $\textsf{QPL}$, which contains quantifiers over events and over reals, and can be viewed as an elementary language for reasoning about probability spaces. The fragment of $\textsf{QPL}$ containing only quantifiers over reals is a variant of the well-known ‘polynomial’ language from Fagin et al. (1990, Inform. Comput., 87, 78–128). We shall prove that the $\textsf{QPL}$-theory of the Lebesgue measure on $\left [ 0, 1 \right ]$ is decidable, and moreover, all atomless spaces have the same $\textsf{QPL}$-theory. Also, we shall introduce the notion of elementary invariant for $\textsf{QPL}$ and use it to translate the semantics for $\textsf{QPL}$ into the setting of elementary analysis. This will allow us to obtain further decidability results as well as to provide exact complexity upper bounds for a range of interesting undecidable theories.

Density analysis for coupled forward–backward SDEs with non-Lipschitz drifts and applications

We explore the existence of a continuous marginal law with respect to the Lebesgue measure for each component (X,Y,Z) of the solution to coupled quadratic forward–backward stochastic differential equations (QFBSDEs) for which the drift coefficient of the forward component is either bounded and measurable or Hölder continuous. Our approach relies on a combination of the existence of a weak decoupling field (see Delarue and Guatteri, 2006), the integration with respect to space time local time (see Eisenbaum, 2006), the analysis of the backward Kolmogorov equation associated to the forward component along with an Itô-Tanaka trick (see Flandoli et al., 2009). The framework of this paper is beyond all existing papers on density analysis for Markovian BSDEs and constitutes a major refinement of the existing results. We also derive a comonotonicity theorem for the control variable in this frame and thus extending the works (Chen et al., 2005; Dos Rei and Dos Rei 2013).As applications of our results, we first analyse the regularity of densities of solution to coupled FBSDEs. In the second example, we consider a regime switching term structure interest rate models (see for e.g., Ma et al. (2017)) for which the corresponding FBSDE has discontinuous drift. Our results enables us to: firstly study classical and Malliavin differentiability of the solutions for such models, secondly the existence of density of such solutions. Lastly we consider a pricing and hedging problem of contingent claims on non-tradable underlying, when the dynamic of the latter is given by a regime switching SDE (i.e., the drift coefficient is allowed to be discontinuous). We obtain a representation of the derivative hedge as the weak derivative of the indifference price function, thus extending the result in Ankirchner et al. (2010).

Length and Area of an Apollonian Packing

Summary We prove two theorems about the circular Apollonian packing. The first states that the circles used in the packing have an infinite total length. The second states that the largest circle is covered by the other circles used in the packing, up to an area of Lebesgue measure zero. Our proofs are elementary and only use basic coordinate geometry.

A dimensional mass transference principle from ball to rectangles for projections of Gibbs measures and applications

Mass transference principle are tools designed to provide estimates for the Hausdorff dimension of points approximable at a certain rate by a specific sequence of points (for instance rationals). Usually, such results are established provided that ambiant space is equipped with an Ahlfors regular measure (see [7,28] for instance). In the case of balls, Barral and Seuret established that mass transference principles still holds provided that the ambiant measure is a Gibbs measure or a self-similar measure satisfying the open set condition ([3]). In this article, we establish a mass transference principle “from ball to rectangle” assuming that the ambiant measure is the projection of Gibbs measure on the dyadic grid. Such measures are not Ahlfors regular in general and this result extends in particular the mass transference principle from ball to rectangle in the case of the Lebesgue measure, established in [28], and partially the result of Barral and Seuret [3]. As an application, given b∈N, we estimate the dimension of weighted approximation of points of [0,1]2 approximable by points with rational coordinates whose expansion in base b has prescribed frequencies. These results are new and surprising in the sens that, although the Lebesgue measure “scales” naturally rectangles, this needs not be the case for projection of Gibbs measures.