Lebesgue integration
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High-dimensional Bayesian inference via the unadjusted Langevin algorithm Open
We consider in this paper the problem of sampling a high-dimensional probability distribution $\\pi$ having a density w.r.t. the Lebesgue measure on $\\mathbb{R}^{d}$, known up to a normalization constant $x\\mapsto\\pi(x)=\\mathrm{e}^{-U(…
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Lineability: The Search For Linearity In Mathematics Open
Preliminary Notions and Tools Cardinal numbers Cardinal arithmetic Basic concepts and results of abstract and linear algebra Residual subsets Lineability, spaceability, algebrability, and their variants Real Analysis What one needs to know…
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ResNet with one-neuron hidden layers is a Universal Approximator Open
We demonstrate that a very deep ResNet with stacked modules with one neuron per hidden layer and ReLU activation functions can uniformly approximate any Lebesgue integrable function in $d$ dimensions, i.e. $\ell_1(\mathbb{R}^d)$. Because o…
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ResNet with one-neuron hidden layers is a Universal Approximator Open
We demonstrate that a very deep ResNet with stacked modules that have one neuron per hidden layer and ReLU activation functions can uniformly approximate any Lebesgue integrable function in d dimensions, i.e. ℓ1(Rd). Due to the identity ma…
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On Function Spaces with Mixed Norms — A Survey Open
The targets of this article are threefold. The first one is to give a survey on the recent developments of function spaces with mixed norms, including mixed Lebesgue spaces, iterated weak Lebesgue spaces, weak mixed-norm Lebesgue spaces an…
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Smoothing properties of bilinear operators and Leibniz-type rules in Lebesgue and mixed Lebesgue spaces Open
We prove that bilinear fractional integral operators and similar multipliers are smoothing in the sense that they improve the regularity of functions. We also treat bilinear singular multiplier operators which preserve regularity and obtai…
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Lebesgue functions and Lebesgue constants in polynomial interpolation Open
The Lebesgue constant is a valuable numerical instrument for linear interpolation because it provides a measure of how close the interpolant of a function is to the best polynomial approximant of the function. Moreover, if the interpolant …
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Gradient Descent Only Converges to Minimizers: Non-Isolated Critical Points and Invariant Regions Open
Given a twice continuously differentiable cost function f, we prove that the set of initial conditions so that gradient descent converges to saddle points where \nabla^2 f has at least one strictly negative eigenvalue, has (Lebesgue) measu…
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POINTWISE CONVERGENCE OF SCHRÖDINGER SOLUTIONS AND MULTILINEAR REFINED STRICHARTZ ESTIMATES Open
We obtain partial improvement toward the pointwise convergence problem of Schrödinger solutions, in the general setting of fractal measure. In particular, we show that, for $n\geqslant 3$ , $\lim _{t\rightarrow 0}e^{it\unicode[STIX]{x1D6E5…
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Regularity of the optimal sets for some spectral functionals Open
In this paper we study the regularity of the optimal sets for the shape optimization problem min{λ1(Ω)+⋯+λk(Ω) : Ω⊂Rd open, |Ω|=1}, where λ1(·) , ... , λk(·) denote the eigenvalues of the Dirichlet Laplacian and | · | the d-dimensional Leb…
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Gradient Descent Only Converges to Minimizers: Non-Isolated Critical Points and Invariant Regions Open
Given a non-convex twice differentiable cost function f, we prove that the set of initial conditions so that gradient descent converges to saddle points where \nabla^2 f has at least one strictly negative eigenvalue has (Lebesgue) measure …
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Characterization of Lipschitz spaces via commutators of the Hardy–Littlewood maximal function Open
Let M be the Hardy–Littlewood maximal function and b be a locally integrable function. Denote by and the maximal commutator and the (nonlinear) commutator of M with b . In this paper, the author considers the boundedness of and on Lebesgue…
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On grand and small Lebesgue and Sobolev spaces and some applications to PDE's Open
This paper is essentially a survey on grand and small Lebesgue spaces, which are rearrangement-invariant Banach function spaces of interest not only from the point of view of Function Spaces theory, but also from the point of view of their…
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Rigidity of critical circle maps Open
We prove that any two $C^4$ critical circle maps with the same irrational\nrotation number and the same odd criticality are conjugate to each other by a\n$C^1$ circle diffeomorphism. The conjugacy is $C^{1+\\alpha}$ for Lebesgue\nalmost ev…
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Microwave Imaging by Means of Lebesgue-Space Inversion: An Overview Open
An overview of the recent advancements in the development of microwave imaging procedures based on the exploitation of the regularization theory in Lebesgue spaces is reported in this paper. Such inversion schemes have been found to provid…
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Some Liouville theorems for stationary Navier-Stokes equations in Lebesgue and Morrey spaces Open
Uniqueness of Leray solutions of the 3D Navier-Stokes equations is a challenging open problem. In this article we will study this problem for the 3D stationary Navier-Stokes equations in the whole space \mathbb{R}^{3} . Under some addition…
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On Takens’ last problem: tangencies and time averages near heteroclinic networks Open
We obtain a structurally stable family of smooth ordinary differential\nequations exhibiting heteroclinic tangencies for a dense subset of parameters.\nWe use this to find vector fields $C^2$-close to an element of the family\nexhibiting a…
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THE p-WEAK GRADIENT DEPENDS ON p Open
Given α > 0, we construct a weighted Lebesgue measure on \mathbb{R}^n for which the family of nonconstant curves has p-modulus zero for p ≤ 1 + α but the weight is a Muckenhoupt A_p weight for p > 1 + α. In particular, the p-weak gradient …
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Coifman–Meyer multipliers: Leibniz-type rules and applications to scattering of solutions to PDEs Open
Leibniz-type rules for Coifman–Meyer multiplier operators are studied in the settings of Triebel–Lizorkin and Besov spaces associated with weights in the Muckenhoupt classes. Even in the unweighted case, improvements on the currently known…
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Modeling infinitely many agents Open
This paper offers a resolution to an extensively studied question in theoretical economics: which measure spaces are suitable for modeling many economic agents? We propose the condition of 'nowhere equivalence' to characterize those measur…
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Multiscale Analysis of 1-rectifiable Measures II: Characterizations Open
A measure is 1-rectifiable if there is a countable union of finite length curves whose complement has zero measure. We characterize 1-rectifiable Radon measures μ in n-dimensional Euclidean space for all n ≥ 2 in terms of positivity of the…
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Regular measurable dynamics for reaction-diffusion equations on narrow domains with rough noise Open
This paper is concerned with the regular random dynamics for the reaction-diffusion equation defined on a thin domain and perturbed by rough noise, where the usual Winner process is replaced by a general stochastic process satisfied the ba…
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Simultaneous approximation in Lebesgue and Sobolev norms via eigenspaces Open
We approximate functions defined on smooth bounded domains by elements of the eigenspaces of the Laplacian or the Stokes operator in such a way that the approximations are bounded and converge in both Sobolev and Lebesgue spaces. We prove …
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On well-posedness of semilinear Rayleigh-Stokes problem with fractional derivative on ℝ <sup> <i>N</i> </sup> Open
We are devoted to the study of a semilinear time fractional Rayleigh-Stokes problem on ℝ N , which is derived from a non-Newtonain fluid for a generalized second grade fluid with Riemann-Liouville fractional derivative. We show that a solu…
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Lineability in sequence and function spaces Open
It is proved the existence of large algebraic structures –including large vector subspaces or infinitely generated free algebras– inside, among others, the family of Lebesgue measurable functions that are surjective in a strong sense, the …
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Mean-field sparse Jurdjevic–Quinn control Open
We consider nonlinear transport equations with non-local velocity describing the time-evolution of a measure. Such equations often appear when considering the mean-field limit of finite-dimensional systems modeling collective dynamics. We …
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A Neighborhood Rough Sets-Based Attribute Reduction Method Using Lebesgue and Entropy Measures Open
For continuous numerical data sets, neighborhood rough sets-based attribute reduction is an important step for improving classification performance. However, most of the traditional reduction algorithms can only handle finite sets, and yie…
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Lebesgue‐<i>p</i>NORM Convergence OF Fractional‐Order PID‐Type Iterative Learning Control for Linear Systems Open
This paper discusses first‐ and second‐order fractional‐order PID‐type iterative learning control strategies for a class of Caputo‐type fractional‐order linear time‐invariant system. First, the additivity of the fractional‐order derivative…
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On the Lebesgue constant of weighted Leja points for Lagrange interpolation on unbounded domains Open
Here, this work focuses on weighted Lagrange interpolation on an unbounded domain and analyzes the Lebesgue constant for a sequence of weighted Leja points. The standard Leja points are a nested sequence of points defined on a compact subs…
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Fractional Langevin Equation Involving Two Fractional Orders: Existence and Uniqueness Revisited Open
We consider the nonlinear fractional Langevin equation involving two fractional orders with initial conditions. Using some basic properties of Prabhakar integral operator, we find an equivalent Volterra integral equation with two parameter…