Square-integrable function ≈ Square-integrable function
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Improved boundary regularity for a Stokes-Lam\\'e system Open
This paper recalls a partial differential equations system, which is the\nlinearization of a recognized fluid-elasticity interaction three-dimensional\nmodel. A collection of regularity results for the traces of the fluid variable\non the …
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Consistent nonparametric estimation for heavy-tailed sparse graphs Open
We study graphons as a non-parametric generalization of stochastic block models, and show how to obtain compactly represented estimators for sparse networks in this framework. Our algorithms and analysis go beyond previous work in several …
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Nonlocal gravity with a Weyl-square term Open
Recent work has shown that modifications of General Relativity based on the addition of a non-local term $R\\,\\Box^{-2}R$ produce a dynamical model of dark energy, which is cosmologically viable both at the background level and at the lev…
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Derivatives Pricing in Energy Markets: An Infinite-Dimensional Approach Open
Based on forward curves modelled as Hilbert-space valued processes, we analyze the pricing of various options relevant in energy markets. In particular, we connect empirical evidence about energy forward prices known from the literature to…
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Optimal quadrature formulas for non-periodic functions in Sobolev space and its application to CT image reconstruction Open
In the present paper, optimal quadrature formulas in the sense of Sard are constructed for numerical integration of the integral ?ba e2?i?x?(x)dx with ? ? R in the Sobolev space L(m)2 [a,b] of complexvalued functions which are square integ…
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Mean-field stochastic differential equations and associated PDEs Open
In this paper we consider a mean-field stochastic differential equation, also called Mc Kean-Vlasov equation, with initial data $(t,x)\in[0,T]\times R^d,$ which coefficients depend on both the solution $X^{t,x}_s$ but also its law. By cons…
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A Monte Carlo Method for Integration of Multivariate Smooth Functions Open
We study a Monte Carlo algorithm that is based on a specific (randomly shifted and dilated) lattice point set. The main result of this paper is that the mean squared error for a given compactly supported, square-integrable function is boun…
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Hoeffding's lemma for Markov Chains and its applications to statistical learning Open
We extend Hoeffding's lemma to general-state-space and not necessarily reversible Markov chains. Let $\{X_i\}_{i \ge 1}$ be a stationary Markov chain with invariant measure $π$ and absolute spectral gap $1-λ$, where $λ$ is defined as the o…
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Evolution of superoscillations for Schrödinger equation in a uniform magnetic field Open
Aharonov-Berry superoscillations are band-limited functions that oscillate faster than their fastest Fourier component. Superoscillations appear in several fields of science and technology, such as Aharonov’s weak measurement in quantum me…
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Series solutions of Heun-type equation in terms of orthogonal polynomials Open
We introduce a nine-parameter Heun-type differential equation and obtain three classes of its solution as series of square integrable functions written in terms of the Jacobi polynomial. The expansion coefficients of the series satisfy thr…
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Survey Article: Self-adjoint ordinary differential operators and their spectrum Open
We survey the theory of ordinary self-adjoint differential operators in\nHilbert space and their spectrum. Such an operator is generated by a\nsymmetric differential expression and a boundary condition. We discuss the\nvery general modern …
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Spectral instability of the peaked periodic wave in the reduced Ostrovsky equations Open
We show that the peaked periodic traveling wave of the reduced Ostrovsky equations with quadratic and cubic nonlinearity is spectrally unstable in the space of square integrable periodic functions with zero mean and the same period. We dis…
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Global Jacquet-Langlands Correspondence for Division Algebras in Characteristic $p$ Open
International audience
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Quantum harmonic analysis on locally compact groups Open
On a locally compact group we introduce covariant quantization schemes and analogs of phase space representations as well as mixed-state localization operators. These generalize corresponding notions for the affine group and the Heisenberg…
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Control problem on space of random variables and master equation Open
In this article, we study a control problem in an appropriate space of random variables; in fact, in our set up, we can consider an arbitrary Hilbert space, yet we specialize only to a Hilbert space of square-integrable random variables. W…
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Second order Stein: SURE for SURE and other applications in high-dimensional inference Open
Stein's formula states that a random variable of the form $z^\top f(z) - \text{div} f(z)$ is mean-zero for functions $f$ with integrable gradient. Here, $\text{div} f$ is the divergence of the function $f$ and $z$ is a standard normal vect…
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On solutions of some delay Volterra integral problems on a half-line Open
In this paper, we study the existence of a.e. monotonic solutions of some general delay integral problems for both fractional and integer orders in the space of Lebesgue integrable functions on the interval R+ = [0;1) and in the space of l…
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Global well-posedness and scattering for the quantum Zakharov system in 𝐿² Open
We study the Cauchy problem for the quantum Zakharov system in the class of square-integrable functions on the Euclidean space of general dimensions. The local well-posedness is proven for dimensions up to eight, together with global exist…
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Fractional Diffusion in Gaussian Noisy Environment Open
We study the fractional diffusion in a Gaussian noisy environment as described by the fractional order stochastic heat equations of the following form: \(D_t^{(\alpha)} u(t, x)=\textit{B}u+u\cdot \dot W^H\), where \(D_t^{(\alpha)}\) is the…
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3. Weck’s selection theorem: The Maxwell compactness property for bounded weak Lipschitz domains with mixed boundary conditions in arbitrary dimensions Open
It is proved that the space of differential forms with weak exterior and co-derivative, is compactly embedded into the space of square integrable differential forms. Mixed boundary conditions on weak Lipschitz domains are considered. Furth…
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DIFFERENCE EQUATIONS IN A MULTIDIMENSIONAL SPACE Open
One considers a general difference equation in a multidimensional space with continuous coefficients in the space of integrable functions. Necessary and sufficient conditions for a Fredholm property are obtained with a help of the Fourier …
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Global existence and non-uniqueness for the Cauchy problem associated to 3D Navier–Stokes equations perturbed by transport noise Open
We show global existence and non-uniqueness of probabilistically strong, analytically weak solutions of the three-dimensional Navier–Stokes equations perturbed by Stratonovich transport noise. We can prescribe either: (i) any divergence-fr…
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A Second Regularized Trace Formula for a Fourth Order Differential Operator Open
In applications, many states given for a system can be expressed by orthonormal elements, called “state elements”, taken in a separable Hilbert space (called “state space”). The exact nature of the Hilbert space depends on the system; for …
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The sup-norm problem on the Siegel modular space of rank two Open
Let F be a square integrable Maass form on the Siegel upper half space of rank 2 for the Siegel modular group Sp(4, Z) with Laplace eigenvalue lambda. If, in addition, F is a joint eigenfunction of the Hecke algebra, we show a power-saving…
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Pseudo-bosons and bi-coherent states out of ℒ2(ℝ) Open
In this paper we continue our analysis on deformed canonical commutation relations and on their related pseudo-bosons and bi-coherent states. In particular, we show how to extend the original approach outside the Hilbert space , le…
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Line with attached segment as a model of Helmholtz resonator: Resonant states completeness Open
Quantum graph consisting of a line with attached segment is considered as a simple model of the Helmholtz resonator. Completeness of resonant states in the space of square integrable functions on the segment is proved. Relation between the…
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New Function Spaces Associated to Representations of Nilpotent Lie Groups and Generalized Time-Frequency Analysis Open
We study function spaces that are related to square-integrable, irreducible, unitary representations of several low-dimensional nilpotent Lie groups. These are new examples of coorbit theory and yield new families of function spaces on $\m…
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Spectral properties of non-selfadjoint Sturm-Liouville operator equation on the real axis Open
In this paper, we analyze the non-selfadjoint Sturm-Liouville operator $L$ defined in the Hilbert space $L_{2}(\mathbb{R},H)$ of vector-valued functions which are strongly-measurable and square-integrable in $ \mathbb{R} $. $L$ is defined …
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Asymptotic properties of solutions to third order neutral differential equations with delay Open
This paper concerns the asymptotic properties of solutions of a class of third-order neutral differential equations with delay. We give sufficient conditions for every solution to be converges to zero, bounded and square integrable. An exa…
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Existence of density for the stochastic wave equation with space-time homogeneous Gaussian noise Open
In this article, we consider the stochastic wave equation on $\\mathbb{R} _{+} \\times \\mathbb{R} $, driven by a linear multiplicative space-time homogeneous Gaussian noise whose temporal and spatial covariance structures are given by loc…