Lebesgue Measure Research Articles

Wasserstein asymptotics for the empirical measure of fractional Brownian motion on a flat torus

We establish asymptotic upper and lower bounds for the Wasserstein distance of any order p≥1 between the empirical measure of a fractional Brownian motion on a flat torus and the uniform Lebesgue measure. Our inequalities reveal an interesting interaction between the Hurst index H and the dimension d of the state space, with a “phase-transition” in the rates when d=2+1/H, akin to the Ajtai–Komlós–Tusnády theorem for the optimal matching of i.i.d. points in two-dimensions. Our proof couples PDE’s and probabilistic techniques, and also yields a similar result for discrete-time approximations of the process, as well as a lower bound for the same problem on Rd.

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.

Smoothness of Class $$C^2$$ of Nonautonomous Linearization Without Spectral Conditions

We prove that smoothness of nonautonomous linearization is of class \(C^2.\) Our approach admits the existence of stable and unstable manifolds determined by a family of nonautonomous hyperbolicities, including the non uniform exponential case, while for the classic exponential dichotomy we obtain the same class of differentiability except for a zero Lebesgue measure set. Moreover, our goal is reached without spectral conditions.

Limit-periodic Dirac operators with thin spectra

We prove that limit-periodic Dirac operators generically have spectra of zero Lebesgue measure and that a dense set of them have spectra of zero Hausdorff dimension. The proof combines ideas of Avila from a Schrödinger setting with a new commutation argument for generating open spectral gaps. This overcomes an obstacle previously observed in the literature; namely, in Schrödinger-type settings, translation of the spectral measure corresponds to small L∞-perturbations of the operator data, but this is not true for Dirac or CMV operators. The new argument is much more model-independent. To demonstrate this, we also apply the argument to prove generic zero-measure spectrum for CMV matrices with limit-periodic Verblunsky coefficients.

The missing (A,D,r) diagram

In this paper we are interested in “optimal” universal geometric inequalities involving the area, diameter and inradius of convex bodies. The term “optimal” is to be understood in the following sense: we tackle the issue of minimizing/maximizing the Lebesgue measure of a convex body among all convex sets of given diameter and inradius. The minimization problem in the two-dimensional case has been solved in a previous work, by M. Hernandez-Cifre and G. Salinas. In this article, we provide a generalization to the n-dimensional case based on a different approach, as well as the complete solving of the maximization problem in the two-dimensional case. This allows us to completely determine the so-called 2-dimensional Blaschke–Santaló diagram for planar convex bodies with respect to the three magnitudes area, diameter and inradius in euclidean spaces, denoted (A,D,r). Such a diagram is used to determine the range of possible values of the area of convex sets depending on their diameter and inradius. Although this question of convex geometry appears to be quite elementary, it had not been answered until now. This is probably related to the fact that the diagram description uses unexpected particular convex sets, such as a kind of smoothed nonagon inscribed in an equilateral triangle.

Entire Symmetric Functions on the Space of Essentially Bounded Integrable Functions on the Union of Lebesgue-Rohlin Spaces

The class of measure spaces which can be represented as unions of Lebesgue-Rohlin spaces with continuous measures contains a lot of important examples, such as Rn for any n∈N with the Lebesgue measure. In this work we consider symmetric functions on Banach spaces of all complex-valued integrable essentially bounded functions on such unions. We construct countable algebraic bases of algebras of continuous symmetric polynomials on these Banach spaces. The completions of such algebras of polynomials are Fréchet algebras of all complex-valued entire symmetric functions of bounded type on the abovementioned Banach spaces. We show that each such Fréchet algebra is isomorphic to the Fréchet algebra of all complex-valued entire symmetric functions of bounded type on the complex Banach space of all complex-valued essentially bounded functions on [0,1].

ON SOLID CORES AND HULLS OF WEIGHTED BERGMAN SPACES A_{mu }^1

We consider weighted Bergman spaces A_mu ^1 on the unit disc as well as the corresponding spaces of entire functions, defined using non-atomic Borel measures with radial symmetry. By extending the techniques from the case of reflexive Bergman spaces, we characterize the solid core of A_mu ^1. Also, as a consequence of a characterization of solid A_mu ^1-spaces, we show that, in the case of entire functions, there indeed exist solid A_mu ^1-spaces. The second part of the article is restricted to the case of the unit disc and it contains a characterization of the solid hull of A_mu ^1, when mu equals the weighted Lebesgue measure with the weight v. The results are based on the duality relation of the weighted A^1- and H^infty-spaces, the validity of which requires the assumption that -log v belongs to the class mathcal {W}_0, studied in a number of publications; moreover, v has to satisfy the condition (b), introduced by the authors. The exponentially decreasing weight v(z) = exp ( -1 /(1-|z|) provides an example satisfying both assumptions.

The structure of (local) ordinal Bayesian incentive compatible random rules

We explore the structure of locally ordinal Bayesian incentive compatible (LOBIC) random Bayesian rules (RBRs). We show that under lower contour monotonicity, for almost all prior profiles (with full Lebesgue measure), a LOBIC RBR is locally dominant strategy incentive compatible (LDSIC). We further show that for almost all prior profiles, a unanimous and LOBIC RBR on the unrestricted domain is random dictatorial, and thereby extend the result in Gibbard (Econometrica 45:665–681, 1977) for Bayesian rules. Next, we provide a sufficient condition on a domain so that for almost all prior profiles, unanimous RBRs on it are tops-only. Finally, we provide a wide range of applications of our results on single-peaked (on arbitrary graphs), hybrid, multiple single-peaked, single-dipped, single-crossing, multi-dimensional separable domains, and domains under partitioning. Since OBIC implies LOBIC by definition, all our results hold for OBIC RBRs.

Sample distribution theory using Coarea Formula

Let ( Ω , Σ , p ) be a probability measure space and let X : Ω → R k be a (vector valued) random variable. We suppose that the probability pX induced by X is absolutely continuous with respect to the Lebesgue measure on R k and set fX as its density function. Let ϕ : R k → R n be a C 1-map and let us consider the new random variable Y = ϕ ( X ) : Ω → R n . Setting m : = max { rank ( J ϕ ( x ) ) : x ∈ R k } , we prove that the probability pY induced by Y has a density function fY with respect to the Hausdorff measure H m on ϕ ( R k ) which satisfies f Y ( y ) = ∫ ϕ − 1 ( y ) f X ( x ) 1 J m ϕ ( x ) d H k − m ( x ) , for H m − a . e . y ∈ ϕ ( R k ) . Here J m ϕ is the m-dimensional Jacobian of ϕ . When J ϕ has maximum rank we allow the map ϕ to be only locally Lipschitz. We also consider the case of X having probability concentrated on some m-dimensional sub-manifold E ⊆ R k and provide, besides, several examples including algebra of random variables, order statistics, degenerate normal distributions, Chi-squared and “Student's t” distributions.

Convergence analysis for gradient flows in the training of artificial neural networks with ReLU activation

Gradient descent (GD) type optimization schemes are the standard methods to train artificial neural networks (ANNs) with rectified linear unit (ReLU) activation. Such schemes can be considered as discretizations of the corresponding gradient flows (GFs). In this work we analyze GF processes in the training of ANNs with ReLU activation and three layers. In particular, in this article we prove in the case where the distribution of the input data is absolutely continuous with respect to the Lebesgue measure that the risk of every bounded GF trajectory converges to the risk of a critical point. In addition, we show in the case of a 1-dimensional affine target function and a uniform input distribution that the risk of every bounded GF trajectory converges to zero if the initial risk is sufficiently small. Finally, we show that the boundedness assumption can be removed if the hidden layer consists of only one neuron.

On discrete L_p Brunn–Minkowski type inequalities

L_p Brunn–Minkowski type inequalities for the lattice point enumerator mathrm {G}_n(cdot ) are shown, pge 1, both in a geometrical and in a functional setting. In particular, we prove that Gn((1-λ)·K+pλ·L+(-1,1)n)p/n≥(1-λ)Gn(K)p/n+λGn(L)p/n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned}\\mathrm {G}_n\\bigl ((1-\\lambda )\\cdot K +_p \\lambda \\cdot L + (-1,1)^n\\bigr )^{p/n}\\ge (1-\\lambda )\\mathrm {G}_n(K)^{p/n}+\\lambda \\mathrm {G}_n(L)^{p/n} \\end{aligned}$$\\end{document}for any K, Lsubset mathbb {R}^n bounded sets with integer points and all lambda in (0,1). We also show that these new discrete analogues (for mathrm {G}_n(cdot )) imply the corresponding results concerning the Lebesgue measure.

Existence and uniqueness of solutions of differential equations with respect to non‐additive measures

By taking Sugeno‐derivative into account, first, we investigate the existence of solutions to the initial value problems (IVP) of first‐order differential equations with respect to non‐additive measure (more precisely, distorted Lebesgue measure). It particularly occurs in the mathematical modeling of biology. We begin by expressing the differential equation in terms of ordinary derivative and the derivative with respect to the distorted Lebesgue measure. Then, by using the fixed point theorem on cones, we construct an operator and prove the existence of positive non‐decreasing solutions on cones in semi‐order Banach spaces. In addition, we also use Picard–Lindelöf theorem to prove the existence and uniqueness of the solution of the equation.Second, we investigate the existence of a solution to the boundary value problem (BVP) on cones with integral boundary conditions of a mix‐order differential equation with respect to non‐additive measures. Moreover, the Krasnoselskii fixed point theorem is also applied to both BVP and IVP and obtains at least one positive non‐decreasing solution. Examples with graphs are provided to validate the results.

Real zeros of random trigonometric polynomials with dependent coefficients

We further investigate the relations between the large degree asymptotics of the number of real zeros of random trigonometric polynomials with dependent coefficients and the underlying correlation function. We consider trigonometric polynomials of the form \[ f n ( t ) ≔ 1 n ∑ k = 1 n a k cos ⁡ ( k t ) + b k sin ⁡ ( k t ) , x ∈ [ 0 , 2 π ] , f_n(t)≔\frac {1}{\sqrt {n}}\sum _{k=1}^{n}a_k \cos (kt)+b_k\sin (kt), ~x\in [0,2\pi ], \] where the sequences ( a k ) k ≥ 1 (a_k)_{k\geq 1} and ( b k ) k ≥ 1 (b_k)_{k\geq 1} are two independent copies of a stationary Gaussian process centered with variance one and correlation function ρ \rho with associated spectral measure μ ρ \mu _{\rho } . We focus here on the case where μ ρ \mu _{\rho } is not purely singular and we denote by ψ ρ \psi _{\rho } its density component with respect to the Lebesgue measure λ \lambda . Quite surprisingly, we show that the asymptotics of the number of real zeros N ( f n , [ 0 , 2 π ] ) \mathcal {N}(f_n,[0,2\pi ]) of f n f_n in [ 0 , 2 π ] [0,2\pi ] is not related to the decay of the correlation function ρ \rho but instead to the Lebesgue measure of the vanishing locus of ψ ρ \psi _{\rho } . Namely, assuming that ψ ρ \psi _{\rho } is C 1 \mathcal {C}^1 with Hölder derivative on an open set of full measure, one establishes that \[ lim n → + ∞ E [ N ( f n , [ 0 , 2 π ] ) ] n = λ ( { ψ ρ = 0 } ) π 2 + 2 π − λ ( { ψ ρ = 0 } ) π 3 . \lim _{n \to +\infty } \frac {\mathbb {E}\left [\mathcal {N}(f_n,[0,2\pi ])\right ]}{n}= \frac {\lambda (\{\psi _{\rho }=0\})}{\pi \sqrt {2}} + \frac {2\pi - \lambda (\{\psi _{\rho }=0\})}{\pi \sqrt {3}}. \] On the other hand, assuming a sole log-integrability condition on ψ ρ \psi _{\rho } , which implies that it is positive almost everywhere, we recover the asymptotics of the independent case: \[ lim n → + ∞ E [ N ( f n , [ 0 , 2 π ] ) ] n = 2 3 . \lim _{n \to +\infty } \frac {\mathbb {E}\left [\mathcal {N}(f_n,[0,2\pi ])\right ]}{n}= \frac {2}{\sqrt {3}}. \] The latter asymptotics thus broadly generalizes the main result of Angst, Dalmao, and Poly [Proc. Amer. Math. Soc. 147 (2019), pp. 205–214] where the spectral density was assumed to be continuous and bounded from below. Besides, with further assumptions of regularity and existence of negative moment for ψ ρ \psi _{\rho } , which encompass e.g. the case of random coefficients being increments of fractional Brownian motion with any Hurst parameter, we moreover show that the above convergence in expectation can be strengthened to an almost sure convergence.

Shape of Eigenvectors for the Decaying Potential Model

We consider the 1d Schrödinger operator with decaying random potential and study the joint scaling limit of the eigenvalues and the measures associated with the corresponding eigenfunctions, which is based on the formulation by Rifkind and Virág (Geom Funct Anal 28:1394–1419, 2018). As a result, we have completely different behavior depending on the decaying rate $$\alpha > 0$$ of the potential: The limiting measure is equal to (1) Lebesgue measure for the supercritical case ( $$\alpha > 1/2$$ ), (2) a measure of which the density has power-law decay with Brownian fluctuation for critical case ( $$\alpha =1/2$$ ), and (3) the delta measure with its atom being uniformly distributed for the subcritical case ( $$\alpha <1/2$$ ). This result is consistent with the previous study on spectral and statistical properties.

Quasi-periodic breathers in Newton’s cradle

We consider the parameterized Newton’s cradle lattice with Hertzian interactions in this paper. The positive parameters are {βn : |n| ≤ b} with a fixed integer b ≥ 0, and the Hertzian potential is V(x)=11+α|x|1+α for a fixed real number α &amp;gt; α* ≔ 12b + 25. Corresponding to a large Lebesgue measure set of (βj)|j|≤b∈R+2b+1, we show the existence of a family of small amplitude, linearly stable, quasi-periodic breathers for Newton’s cradle lattice, which are quasi-periodic in time with 2b + 1 frequencies and localized in space with rate 1|n|1+α as |n| ≫ 1. To overcome obstacles in applying the Kolmogorov–Arnold–Moser (KAM) method due to the finite smoothness of V, especially when α is not an integer and to obtain a sharp estimate of the localization rate of the quasi-periodic breathers, the proof of our result uses the Jackson–Moser–Zehnder analytic approximation technique but with refined estimates on error bounds, depending on the smoothness and dimension, which provide crucial controls on the convergence of KAM iterations.

Multivariate ρ-quantiles: A spatial approach

By substituting an Lp loss function for the L1 loss function in the optimization problem defining quantiles, one obtains Lp-quantiles that, as shown recently, dominate their classical L1-counterparts in financial risk assessment. In this work, we propose a concept of multivariate Lp-quantiles generalizing the spatial (L1-)quantiles introduced by Probal Chaudhuri (J. Amer. Statist. Assoc. 91 (1996) 862–872). Rather than restricting to power loss functions, we actually allow for a large class of convex loss functions ρ. We carefully study existence and uniqueness of the resulting ρ-quantiles, both for a general probability measure over Rd and for a spherically symmetric one. Interestingly, the results crucially depend on ρ and on the nature of the underlying probability measure. Building on an investigation of the differentiability properties of the objective function defining ρ-quantiles, we introduce a companion concept of spatial ρ-depth, that generalizes the classical spatial depth. We study extreme ρ-quantiles and show in particular that extreme Lp-quantiles behave in fundamentally different ways for p≤2 and p>2. Finally, we establish Bahadur representation results for sample ρ-quantiles and derive their asymptotic distributions. Throughout, we impose only very mild assumptions on the underlying probability measure, and in particular we never assume absolute continuity with respect to the Lebesgue measure.

On the measure of products from the middle-third Cantor set

We prove upper and lower bounds for the Lebesgue measure of the set of products xy with x and y in the middle-third Cantor set. Our method is inspired by Athreya, Reznick and Tyson, but a different subdivision of the Cantor set provides a more rapidly converging approximation formula.

Disguised toric dynamical systems

We study families of polynomial dynamical systems inspired by biochemical reaction networks. We focus on complex balanced mass-action systems, which have also been called toric. They are known or conjectured to enjoy very strong dynamical properties, such as existence and uniqueness of positive steady states, local and global stability, persistence, and permanence. We consider the class of disguised toric dynamical systems, which contains toric dynamical systems, and to which all dynamical properties mentioned above extend naturally. By means of (real) algebraic geometry we show that some reaction networks have an empty toric locus or a toric locus of Lebesgue measure zero in parameter space, while their disguised toric locus is of positive measure. We also propose some algorithms one can use to detect the disguised toric locus.

Semi-discrete optimal transport: hardness, regularization and numerical solution

Semi-discrete optimal transport problems, which evaluate the Wasserstein distance between a discrete and a generic (possibly non-discrete) probability measure, are believed to be computationally hard. Even though such problems are ubiquitous in statistics, machine learning and computer vision, however, this perception has not yet received a theoretical justification. To fill this gap, we prove that computing the Wasserstein distance between a discrete probability measure supported on two points and the Lebesgue measure on the standard hypercube is already #P-hard. This insight prompts us to seek approximate solutions for semi-discrete optimal transport problems. We thus perturb the underlying transportation cost with an additive disturbance governed by an ambiguous probability distribution, and we introduce a distributionally robust dual optimal transport problem whose objective function is smoothed with the most adverse disturbance distributions from within a given ambiguity set. We further show that smoothing the dual objective function is equivalent to regularizing the primal objective function, and we identify several ambiguity sets that give rise to several known and new regularization schemes. As a byproduct, we discover an intimate relation between semi-discrete optimal transport problems and discrete choice models traditionally studied in psychology and economics. To solve the regularized optimal transport problems efficiently, we use a stochastic gradient descent algorithm with imprecise stochastic gradient oracles. A new convergence analysis reveals that this algorithm improves the best known convergence guarantee for semi-discrete optimal transport problems with entropic regularizers.

Poincaré inequality meets Brezis–Van Schaftingen–Yung formula on metric measure spaces

Let (X,ρ,μ) be a metric measure space of homogeneous type, and let Cc⁎(X) denote the set of all Lipschitz functions f on X such that limr→0+⁡supy∈B(⋅,r)⁡|f(⋅)−f(y)|/r converges uniformly to a compactly supported continuous function Lip f on X. Let p∈[1,∞) and let f∈Cc⁎(X) be such that the pair (f,Lipf) satisfies a certain Poincaré inequality. Assume that α∈R∖{0} if p∈(1,∞), and α∈(0,∞) if p=1. Under these assumptions, the authors prove the following result which extends a recent formula of H. Brezis, A. Seeger, J. Van Schaftingen, and P.-L. Yung on finite dimensional Euclidean spaces:supλ∈(0,∞)⁡λp∬Dλ[U(x,y)]α−1dμ(x)dμ(y)∼∫X[Lipf(x)]pdμ(x), whereDλ:={(x,y)∈X×X:|f(x)−f(y)|>λρ(x,y)[U(x,y)]αp} for any λ∈(0,∞), V(x,y):=μ(B(x,ρ(x,y))) and U(x,y):=min⁡{V(x,y),V(y,x)} for any x,y∈X, and the positive constants of equivalence are independent of f. A similar result remains true if p∈(1,∞) and f lies in a wider class of functions in the Hajłasz–Sobolev space. As an application, the authors establish new fractional Sobolev and Gagliardo–Nirenberg inequalities on X. These results are applicable to many classical examples, such as Euclidean spaces, with weighted Lebesgue measure, and complete finite dimensional Riemannian manifolds, with non-negative Ricci curvature.