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We prove that the vector play operator with a uniformly prox-regular characteristic set of constraints is continuous with respect to the BV-norm and to the BV-strict metric in the space of rectifiable curves, i.e., in the space of continuous functions of bounded variation. We do not assume any further regularity of the characteristic set. We also prove that the non-convex play operator is rate independent. [ABSTRACT FROM AUTHOR]

This paper focuses on characterizing the optimality of a kind of partial set order robust solutions, which are defined by Minkowski set difference, for an uncertain multi-objective optimization problem via oriented distance function and image space analysis. Firstly, the relationships between partial set order robust efficiency and upper (lower) set order robust efficiency are illustrated. Secondly, the optimality conditions to partial set order robust solutions are presented by utilizing image space analysis. Furthermore, characterizations are also established for partial set order robust solutions under the assumption of generalized monotonicity, which is determined by an oriented distance function. Finally, an application, namely a shortest path problem, is discussed to verify the effectiveness for the obtained results. [ABSTRACT FROM AUTHOR]

This paper defines local weighted Hardy spaces with variable exponents. Local Hardy spaces permit atomic decomposition, which is one of the main themes in this paper. A consequence is that the atomic decomposition is obtained for the functions in the Lebesgue spaces with exponentially decaying exponent. As applications, we obtain the boundedness of singular integral operators, the Littlewood–Paley characterization, and wavelet decomposition. [ABSTRACT FROM AUTHOR]

This paper is concerned with the estimation of the partial derivatives of a probability density function of directional data on the d-dimensional torus within the local thresholding framework. The estimators here introduced are built by means of the toroidal needlets, a class of wavelets characterised by excellent concentration properties in both the real and the harmonic domains. In particular, we discuss the convergence rates of the Lp-risks for these estimators, investigating their minimax properties and proving their optimality over a scale of Besov spaces, here taken as nonparametric regularity function spaces. [ABSTRACT FROM AUTHOR]

In this paper, using the method of Hicks, we prove some new related fixed point theorems for pair of mappings on two d-complete topological spaces under the suitable conditions. We also derive some related fixed point theorems from our main results. Finally, by using w-continuity, we establish a related fixed point theorem in d-complete Hausdorff topological spaces. [ABSTRACT FROM AUTHOR]

We consider general nonlinear function‐on‐scalar (FOS) regression models, where the functional response depends on multiple scalar predictors in a general unknown nonlinear form. Existing methods either assume specific model forms (e.g., additive models) or directly estimate the nonlinear function in a space with dimension equal to the number of scalar predictors, which can only be applied to models with a few scalar predictors. To overcome these shortcomings, motivated by the classic universal approximation theorem used in neural networks, we develop a functional universal approximation theorem which can be used to approximate general nonlinear FOS maps and can be easily adopted into the framework of functional data analysis. With this theorem and utilizing smoothness regularity, we develop a novel method to fit the general nonlinear FOS regression model and make predictions. Our new method does not make any specific assumption on the model forms, and it avoids the direct estimation of nonlinear functions in a space with dimension equal to the number of scalar predictors. By estimating a sequence of bivariate functions, our method can be applied to models with a relatively large number of scalar predictors. The good performance of the proposed method is demonstrated by empirical studies on various simulated and real datasets. [ABSTRACT FROM AUTHOR]

We provide a complete description of zero sets of some new analytic area Nevanlinna type spaces in the unit disk. Our theorems extend recent sharp results of E. Rodikova to larger analytic function spaces of area Nevanlinna type. [ABSTRACT FROM AUTHOR]

In the countably normed space of smooth functions on the unit circle, we consider a Toeplitz operator with a smooth symbol. Questions about the boundedness, Fredholm property, and invertibility of such operators are studied. The concepts of smooth canonical degenerate factorization of minus type of smooth functions and the associated local degenerate canonical factorization of minus type are introduced. Criteria are obtained in terms of the symbol for the existence of a canonical degenerate factorization of minus type. Just as in the classical case of the Toeplitz operator in spaces of integrable functions with Wiener symbols, the Fredholm property of the Toeplitz operator turned out to be equivalent to the existence of a smooth degenerate canonical factorization of the minus type of its symbol. The equivalence of degenerate canonical factorizability and similar local factorizability is established, which permits one to use the localization of the symbol on certain characteristic arcs of a circle when studying invertibility issues. Relations between the spectra of some Toeplitz operators in spaces of smooth and integrable functions are obtained. A description of the resolvent set of the Toeplitz operator with a smooth symbol is given. [ABSTRACT FROM AUTHOR]

This article continues the study of the Hadamard–Bergman operators in the unit disk of the complex plane. These operators arose as a natural generalization of orthogonal projections and represent an integral realization of multiplier operators. However, the study of operators in integral form offers a number of advantages in the context of the application of the theory of integral operators as well as in the study of certain function spaces such as holomorphic Hölder functions to which the multiplier theory does not apply. As a main result, we prove boundedness theorems for the Hadamard–Bergman operators and variable Hadamard–Bergman operators using the technique of operators with homogeneous kernels earlier developed in real analysis. [ABSTRACT FROM AUTHOR]

For any set X we let F (X) denote the complex vector space of functions f : X → C . Let X = S be a magma, and let V be a subspace of F (S) , which is invariant under left or right translations. It is known for an abelian group S that if p 1 χ 1 , ⋯ , p n χ n ∈ F (S) are nonzero exponential polynomials with distinct exponentials χ 1 , ⋯ , χ n then p 1 χ 1 + ⋯ + p n χ n ∈ V ⇒ p 1 χ 1 , ⋯ , p n χ n ∈ V and χ 1 , ⋯ , χ n ∈ V . We extend this to magmas. Our results imply that any exponential polynomial solution f ∈ F (S) of f (x y) = f (x) χ (y) + χ (x) f (y) where χ ∈ F (S) is an exponential, has the form f = a χ where a ∈ F (S) is additive, even when χ has zeros. [ABSTRACT FROM AUTHOR]

We prove new inequalities and equalities for the generalized and the joint spectral radius (and their essential versions) of Hadamard (Schur) geometric means of bounded sets of positive kernel operators on Banach function spaces. In the case of non-negative matrices that define operators on Banach sequences, we obtain additional results. Our results extend the results of several authors that appeared relatively recently. [ABSTRACT FROM AUTHOR]

It is known, by Gelfand theory, that every commutative JB $ ^* $ ∗ -triple admits a representation as a space of continuous functions of the form \[ C_0^{\mathbb{T}}(L) = \{ a\in C_0(L) : a(\lambda t) = \lambda a(t), \ \forall \lambda\in \mathbb{T}, t\in L\}, \] C 0 T (L) = { a ∈ C 0 (L) : a (λ t) = λ a (t) , ∀ λ ∈ T , t ∈ L } , where L is a principal $ \mathbb {T} $ T -bundle and $ \mathbb {T} $ T denotes the unit circle in $ \mathbb {C}. $ C. We provide a full technical description of all orthogonality preserving (non-necessarily continuous nor bijective) linear maps between commutative JB $ ^* $ ∗ -triples. Among the consequences of this representation, we obtain that every linear bijection preserving orthogonality between commutative JB $ ^* $ ∗ -triples is automatically continuous and bi-orthogonality preserving. [ABSTRACT FROM AUTHOR]

New families of direct serendipity and direct mixed finite elements on general planar, strictly convex polygons were recently defined by the authors. The finite elements of index r are H 1 and H (div) conforming, respectively, and approximate optimally to order r + 1 while using the minimal number of degrees of freedom. The shape function space consists of the full set of polynomials defined directly on the element and augmented with a space of supplemental functions. The supplemental functions were constructed as rational functions, which can be difficult to integrate accurately using numerical quadrature rules when the index is high. This can result in a loss of accuracy in certain cases. In this work, we propose alternative ways to construct the supplemental functions on the element as continuous piecewise polynomials. One approach results in supplemental functions that are in H p for any p ≥ 1 . We prove the optimal approximation property for these new finite elements. We also perform numerical tests on them, comparing results for the original supplemental functions and the various alternatives. The new piecewise polynomial supplements can be integrated accurately, and therefore show better robustness with respect to the underlying meshes used. [ABSTRACT FROM AUTHOR]

We introduce a method based on Lipschitz pointwise transformations to define a distance on a Banach function space from its norm. We show how some specific lattice geometric properties (p-convexity, p-concavity, p-regularity) or, equivalently, some types of summability conditions (for example, when the terms of the terms in the sums in the range of the operator are restricted to the interval [ − 1 , 1 ] ) can be studied by adapting the classical analytical techniques of the summability of operators on Banach lattices, which recalls the work of Maurey. We show a technique to prove new integral dominations (equivalently, operator factorizations), which involve non-homogeneous expressions constructed by pointwise composition with Lipschitz maps. As an example, we prove a new family of integral bounds for certain operators on Lorentz spaces. [ABSTRACT FROM AUTHOR]

In this research article we focus on the study of existence of global solution for a three-dimensional fractional Porous medium equation. The main objectives of studying the fractional porous medium equation in the corresponding critical function spaces are to show the existence of unique global mild solution under the condition of small initial data. Applying Fourier transform methods gives an equivalent integral equation of the model equation. The linear and nonlinear terms are then estimated in the corresponding Lei and Lin spaces. Further, the analyticity of solution to the fractional Porous medium equation is also obtained. [ABSTRACT FROM AUTHOR]

In this paper we study the initial boundary value problem for a class of integro-differential time fractional order parabolic equations with a small positive parameter ε. Using the Laplace transform, Mittag-Leffler operator family, C 0 -semigroup, resolvent operator, and weighted function space, we get the existence of a mild solution. For suitable indices p ∈ [ 1 , + ∞) and s ∈ (1 , + ∞) , we first prove that the mild solution of the approximating problem converges to that of the corresponding limit problem in L p ((0 , T) , L s (Ω)) as ε → 0 + . Then for the linear approximating problem with ε and the corresponding limit problem, we give the continuous dependence of the solutions. Finally, for a class of semilinear approximating problems and the corresponding limit problems with initial data in L s (Ω) , we prove the local existence and uniqueness of the mild solution and then give the continuous dependence on the initial data. [ABSTRACT FROM AUTHOR]

Semiflows on some spaces of continuous functions generated by a semilinear partial differential equation are studied. Sufficient conditions for the existence of invariant measures with strong ergodic properties are given. These properties imply chaotic behaviour of the semiflows, that is, the existence of dense trajectories and that each trajectory is unstable. In the proofs we use the Lévy d -parameter Brownian motion and we show that these semiflows are isomorphic with translation along the radii. [ABSTRACT FROM AUTHOR]

This article treats finite element simulations of controlled wrinkle formation experiments of a soft bulk material with a thin, stiff layer on top. The wrinkling process is triggered by a stress mismatch between the bulk material and the thin layer. For the finite element simulations, we model the bulk material using a three‐dimensional hyperelastic material and the thin layer with a geometrically nonlinear elastic Cosserat shell. For the finite element simulations, we model the bulk material using a three‐dimensional hyperelastic material and the thin layer with a geometrically nonlinear elastic Cosserat shell. We use Lagrange finite elements for the bulk material and geodesic finite elements for the shell. The resulting minimization problem is nonlinear and nonconvex. We prove existence of minimizers in the continuous and the discrete function space. Finally, we solve the resulting nonconvex minimization problem numerically using a Riemannian trust‐region algorithm and compare our simulations to real experiments. [ABSTRACT FROM AUTHOR]

We study the machine learning task for models with operators mapping between the Wasserstein space of probability measures and a space of functions, like e.g. in mean-field games/control problems. Two classes of neural networks based on bin density and on cylindrical approximation, are proposed to learn these so-called mean-field functions, and are theoretically supported by universal approximation theorems. We perform several numerical experiments for training these two mean-field neural networks, and show their accuracy and efficiency in the generalization error with various test distributions. Finally, we present different algorithms relying on mean-field neural networks for solving time-dependent mean-field problems, and illustrate our results with numerical tests for the example of a semi-linear partial differential equation in the Wasserstein space of probability measures. [ABSTRACT FROM AUTHOR]

Let Φ be a locally convex space and let Ψ be a quasi-complete bornological nuclear space (like spaces of smooth functions and distributions) with dual spaces Φ ′ and Ψ ′. In this work we introduce sufficient conditions for time regularity properties of the Ψ ′ -valued stochastic convolution ∫ 0 t ∫ U S (t − r) ′ R (r , u) M (d r , d u) , t ∈ [ 0 , T ] , where (S (t) ′ : t ≥ 0) is the dual semigroup to a C0-semigroup (S (t) : t ≥ 0) on Ψ , R (r , ω , u) is a suitable operator form Φ ′ into Ψ ′ , and M is a cylindrical-martingale valued measure on Φ ′. Our result is latter applied to study time regularity of solutions to Ψ ′ -valued stochastic evolutions equations. [ABSTRACT FROM AUTHOR]

This work contributes to nonlocal vector calculus as an indispensable mathematical tool for the study of nonlocal models that arises in a variety of applications. We define the nonlocal half-ball gradient, divergence and curl operators with general kernel functions (integrable or fractional type with finite or infinite supports) and study the associated nonlocal vector identities. We study the nonlocal function space on bounded domains associated with zero Dirichlet boundary conditions and the half-ball gradient operator and show it is a separable Hilbert space with smooth functions dense in it. A major result is the nonlocal Poincaré inequality, based on which a few applications are discussed, and these include applications to nonlocal convection–diffusion, nonlocal correspondence model of linear elasticity and nonlocal Helmholtz decomposition on bounded domains. [ABSTRACT FROM AUTHOR]

Beam vibrations with allowance for deformation effects in the transverse direction are modeled using a nonlinear Sobolev-type differential equation, for which the Cauchy problem is investigated in the space of continuous functions. Conditions for solution blow-up on a finite time interval are considered. [ABSTRACT FROM AUTHOR]

We consider optimal control problems with integer-valued controls and a total variation regularization penalty in the objective on domains of dimension two or higher. The penalty yields that the feasible set is sequentially closed in the weak-\ast topology and closed in the strict topology in the space of functions of bounded variation. In turn, we derive first-order optimality conditions of the optimal control problem as well as trust-region subproblems with partially linearized model functions using local variations of the level sets of the feasible control functions. We also prove that a recently proposed function space trust-region algorithm---sequential linear integer programming---produces sequences of iterates whose limits are first-order optimal points. [ABSTRACT FROM AUTHOR]

Vast amounts of monitoring data can be obtained through various optical sensors, and mask detection based on deep learning integrates neural science into a variety of applications in everyday life. However, mask detection poses technical challenges such as small targets, complex scenes, and occlusions, which necessitate high accuracy and robustness in multi-scene target detection networks. Considering that multi-scale features can increase the receptive field and attention mechanism can improve the detection effect of small targets, we propose the YOLO-MSM network based on the multi-scale residual (MSR) block, multi-scale residual cascaded channel-spatial attention (MSR-CCSA) block, enhanced residual CCSA (ER-CCSA) block, and enhanced residual PCSA (ER-PCSA) block. Considering the performance and parameters, we use YOLOv5 as the baseline network. Firstly, for the MSR block, we construct hierarchical residual connections in the residual blocks to extract multi-scale features and obtain finer features. Secondly, to realize the joint attention function of channel and space, both the CCSA block and PCSA block are adopted. In addition, we construct a new dataset named Multi-Scene-Mask, which contains various scenes, crowd densities, and mask types. Experiments on the dataset show that YOLO-MSM achieves an average precision of 97.51%, showing better performance than other detection networks. Compared with the baseline network, the mAP value of YOLO-MSM is increased by 3.46%. Moreover, we propose a module generalization improvement strategy (GIS) by training YOLO-MSM on the dataset augmented with white Gaussian addition noise to improve the generalization ability of the network. The test results verify that GIS can greatly improve the generalization of the network and YOLO-MSM has stronger generalization ability than the baseline. [ABSTRACT FROM AUTHOR]

It is known that the nonnegativity of Li coefficients is a necessary and sufficient condition for the Riemann hypothesis. We show that it is a necessary and sufficient condition for the Riemann hypothesis that all Li coefficients are norms of certain concrete functions on the real line. Such conditional formulas for Li coefficients are understood as a kind of Weil's criterion for the Riemann hypothesis. [ABSTRACT FROM AUTHOR]

In a normed space, we study the perturbed minimal time function determined by a bounded closed convex set and a proper lower semicontinuous function. It covers the perturbed distance function as a special case. In particular, we show that the ε-subdifferential and the limiting subdifferential of the perturbed minimal time function are representable by virtue of corresponding subdifferential of the associated function and the support function of the constraint set. [ABSTRACT FROM AUTHOR]

It is shown that the uniform radius of spatial analyticity σ (t) of solutions at time t to the Ostrovsky equation with positive dispersion cannot decay faster than | t | − 4 3 as | t | → ∞ given initial data that is analytic with fixed radius σ 0 . The main ingredients in the proof are almost conservation law for the solution to the Ostrovsky equation in space of analytic functions and space-time dyadic bilinear estimates associated with the Ostrovsky equation. [ABSTRACT FROM AUTHOR]

In this paper, we introduced a reproducing kernel space which is a particular class of Hilbert space. We discuss various properties of the reproducing kernel. In particular, our aim is to construct kernel in reproducing kernel Hilbert space of the specific function space (Sobolev space) with the improved inner product and norm. Also, we derive the reproducing kernel for periodic boundary conditions. [ABSTRACT FROM AUTHOR]

Louveau showed that if a Borel set in a Polish space happens to be in a Borel Wadge class Γ , then its Γ -code can be obtained from its Borel code in a hyperarithmetical manner. We extend Louveau's theorem to Borel functions: If a Borel function on a Polish space happens to be a Σ ~ t -function, then one can find its Σ ~ t -code hyperarithmetically relative to its Borel code. More generally, we prove extension-type, domination-type, and decomposition-type variants of Louveau's theorem for Borel functions. [ABSTRACT FROM AUTHOR]

Introduction: Head-down bed rest (HDBR) has long been used as an analog to microgravity, and it also enables studying the changes occurring with aging. Exercise is the most effective countermeasure for the deleterious effects of inactivity. The aim of this study was to investigate the efficacy of an exercise countermeasure in healthy older participants on attenuating musculoskeletal deconditioning, cardiovascular fitness level, and muscle strength during 14 days of HDBR as part of the standard measures of the Canadian Space Agency. Methods: Twenty-three participants (12 males and 11 females), aged 55–65 years, were admitted for a 26-day inpatient stay at the McGill University Health Centre. After 5 days of baseline assessment tests, they underwent 14 days of continuous HDBR followed by 7 days of recovery with repeated tests. Participants were randomized to passive physiotherapy or an exercise countermeasure during the HDBR period consisting of 3 sessions per day of either high-intensity interval training (HIIT) or low-intensity cycling or strength exercises for the lower and upper body. Peak aerobic power (V̇O2peak) was determined using indirect calorimetry. Body composition was assessed by dual-energy X-ray absorptiometry, and several muscle group strengths were evaluated using an adjustable chair dynamometer. A vertical jump was used to assess whole-body power output, and a tilt test was used to measure cardiovascular and orthostatic challenges. Additionally, changes in various blood parameters were measured as well as the effects of exercise countermeasure on these measurements. Results: There were no differences at baseline in main characteristics between the control and exercise groups. The exercise group maintained V̇O2peak levels similar to baseline, whereas it decreased in the control group following 14 days of HDBR. Body weight significantly decreased in both groups. Total and leg lean masses decreased in both groups. However, total body fat mass decreased only in the exercise group. Isometric and isokinetic knee extension muscle strength were significantly reduced in both groups. Peak velocity, flight height, and flight time were significantly reduced in both groups with HDBR. Conclusion: In this first Canadian HDBR study in older adults, an exercise countermeasure helped maintain aerobic fitness and lean body mass without affecting the reduction of knee extension strength. However, it was ineffective in protecting against orthostatic intolerance. These results support HIIT as a promising approach to preserve astronaut health and functioning during space missions, and to prevent deconditioning as a result of hospitalization in older adults. [ABSTRACT FROM AUTHOR]

A function that is analytic on a domain of C n is holonomic if it is the solution to a holonomic system of linear homogeneous differential equations with polynomial coefficients. We define and study the Bernstein–Sato polynomial of a holonomic function on a smooth algebraic variety. We analyze the structure of certain sheaves of holonomic functions, such as the algebraic functions along a hypersurface, determining their direct sum decompositions into indecomposables, that further respect decompositions of Bernstein–Sato polynomials. When the space is endowed with the action of a linear algebraic group G, we study the class of G-finite analytic functions, i.e. functions that under the action of the Lie algebra of G generate a finite dimensional rational G-module. These are automatically algebraic functions on a variety with a dense orbit. When G is reductive, we give several representation-theoretic techniques toward the determination of Bernstein–Sato polynomials of G-finite functions. We classify the G-finite functions on all but one of the irreducible reduced prehomogeneous vector spaces, and compute the Bernstein–Sato polynomials for distinguished G-finite functions. The results can be used to construct explicitly equivariant D -modules. [ABSTRACT FROM AUTHOR]

We prove a functional limit theorem in a space of analytic functions for the random Dirichlet series D (α ; z) = ∑ n ≥ 2 (log n) α (η n + i θ n) / n z , properly scaled and normalized, where (η n , θ n) n ∈ N is a sequence of independent copies of a centered R 2 -valued random vector (η , θ) with a finite second moment and α > − 1 / 2 is a fixed real parameter. As a consequence, we show that the point processes of complex and real zeros of D (α ; z) converge vaguely, thereby obtaining a universality result. In the real case, that is, when P { θ = 0 } = 1 , we also prove a law of the iterated logarithm for D (α ; z) , properly normalized, as z → (1 / 2) +. [ABSTRACT FROM AUTHOR]

We study in this paper a few remarkable properties of twisted sums Z(κ, X) of c0(κ) and a Banach space X. We first prove a representation theorem for such twisted sums from which we will obtain, among others, the following: (a) twisted sums of c0(κ) and c0(I) are either subspaces of ℓ∞(κ) or contain a complemented copy of c0(κ+); (b) under the hypothesis [p = c], when K is either a suitable Corson compact, a separable Rosenthal compact or a scattered compact of finite height, there is a twisted sum of c0 and C(K) that is not isomorphic to a space of continuous functions; (c) all twisted sums Z(κ, X) are isomorphically Lindenstrauss spaces when X is a Lindenstrauss space; (d) all twisted sums Z(κ, X) are isomorphically polyhedral when X is a polyhedral space with a σ-discrete boundary, which solves a problem of Castillo and Papini. [ABSTRACT FROM AUTHOR]

We consider an analog of the problem of the existence of the summable distributional density of a random variable in the form of power series on the dyadic half-line which was originally proposed and partially solved by Erdös on the standard real line. Given a random variable as a series of the powers of , we address the question of such that the density of belongs to the space of the function whose modulus is summable on the dyadic half-line. We answer the question for some values of , and consider the so-called dual problem when is fixed, but the coefficients of the formula for have more degrees of freedom. Also we obtain some criteria for the existence of density in terms of the solution of the refinement equation tied directly to as well as in terms of the coefficients defining . [ABSTRACT FROM AUTHOR]

We prove the modular convexity of the mixed norm L p (ℓ 2) on the Sobolev space W 1 , p (Ω) in a domain Ω ⊂ R n under the sole assumption that the exponent p (x) is bounded away from 1, i.e., we include the case sup x ∈ Ω p (x) = ∞ . In particular, the mixed Sobolev norm is uniformly convex if 1 < inf x ∈ Ω p (x) ≤ sup x ∈ Ω p (x) < ∞ and W 0 1 , p (Ω) is uniformly convex. [ABSTRACT FROM AUTHOR]

The study presented in this paper follows a line of research familiar for Geometric Function Theory, which consists in defining new integral operators and conducting studies for revealing certain geometric properties of those integral operators such as univalence, starlikness, or convexity. The present research focuses on the Bessel function of the first kind and order ν unveiling the conditions for this function to be univalent and further using its univalent form in order to define a new integral operator on the space of holomorphic functions. For particular values of the parameters implicated in the definition of the new integral operator involving the Bessel function of the first kind, the well-known Alexander, Libera, and Bernardi integral operators can be obtained. In the first part of the study, necessary and sufficient conditions are obtained for the Bessel function of the first kind and order ν to be a starlike function or starlike of order α ∈ [ 0 , 1) . The renowned prolific method of differential subordination due to Sanford S. Miller and Petru T. Mocanu is employed in the reasoning. In the second part of the study, the outcome of the first part is applied in order to introduce the new integral operator involving the form of the Bessel function of the first kind, which is starlike. Further investigations disclose the necessary and sufficient conditions for this new integral operator to be starlike or starlike of order 1 2 . [ABSTRACT FROM AUTHOR]

The flanged parallel-plate waveguide is analysed based on the method of Kobayashi potential (KP) using Fourier function space. The presentation of the method is free from intricate mathematics. Standard integral identities are used for problem formulation, without direct use of Weber-Schafheitlin (WS) integrals. The Fourier function space is exploited for the construction of the governing linear equations instead of Jacobi polynomials. A simple strategy is suggested for the evaluation of the required improper integrals. Near-field results are validated through convergence analysis. Far-field patterns are compared with predictions of the surface equivalence theorem (SET). [ABSTRACT FROM AUTHOR]

This paper aims to explore the optimal nonlinear approximation on the class of functions in Sobolev space in the probabilistic and average case settings. Specifically, we investigate the optimal approximation through the utilization of a set with finite pseudo-dimension, measured by the Kolmogorov probabilistic nonlinear (n , δ) -width (or say, the Kolmogorov probabilistic pseudo- (n , δ) -width). Furthermore, we provide an estimation of the exact asymptotic order of the Kolmogorov probabilistic nonlinear (n , δ) -width and p -average nonlinear n -width on the class of functions in Sobolev space. [ABSTRACT FROM AUTHOR]

This paper concerns the existence of multiple solutions for a Schrödinger logarithmic equation of the form − 2 Δ u + V (x) u = u log u 2 , in ℝ N , u ∈ H 1 (ℝ N) , (P ) where V : ℝ N → ℝ is a continuous function that satisfies some technical conditions and is a positive parameter. We will establish the multiplicity of solution for (P ) by using the notion of Lusternik–Schnirelmann category, by introducing a new function space where the energy functional is C 1 . [ABSTRACT FROM AUTHOR]

Let X be a ball Banach function space on R n . In this article, under some mild extra assumptions about both X and the boundedness of the Hardy–Littlewood maximal operator on both X and the associate space of its convexification, the authors successfully recover the homogeneous ball Banach Sobolev semi-norm ‖ | ∇ f | ‖ X via the functional sup λ ∈ (0 , ∞) λ ∫ { y ∈ R n : | f (·) - f (y) | > λ | · - y | 1 + γ q } · - y γ - n d y 1 q X for any distributions f with | ∇ f | ∈ X , as well as the corresponding limiting identities with the limit for λ → ∞ when γ ∈ (0 , ∞) or the limit for λ → 0 + when γ ∈ (- ∞ , 0) , where γ ∈ R \ { 0 } and where q ∈ (0 , ∞) is related to X. In particular, some of these results are still new even when X : = L p (R n) with p ∈ [ 1 , ∞) . As applications, the authors obtain some fractional Sobolev-type and some fractional Gagliardo–Nirenberg-type inequalities in the setting of X. All these results are of quite wide generality and are applied to various specific function spaces, including Morrey, mixed-norm (or variable or weighted) Lebesgue, Lorentz, and Orlicz (or Orlicz-slice) spaces, some of which are new even in all these special cases. The novelty of this article is to use both the method of the extrapolation and the boundedness of the Hardy–Littlewood maximal operator on both X and the associate space of its convexification to overcome the essential difficulties caused by the deficiency of both the translation and the rotation invariance and an explicit expression of the norm of X. [ABSTRACT FROM AUTHOR]

In this paper, we introduce new 2‐microlocal Besov and Triebel–Lizorkin spaces of Jaffard type, which contain many classical functions. We establish local theory of these function spaces by the φ‐transform and the wavelet decomposition coefficients. Moreover, we give a characterization of these local spaces by a local polynomial approximation. [ABSTRACT FROM AUTHOR]

We are interested in the singular behavior at the origin of solutions to the equation H\rho=e on a half-axis, where H is the one-sided Hilbert transform, p an unknown solution, and e a known function. This is a simpler model problem on the path to understanding wave field singularities caused by curve-shaped scatterers in a planar domain. We prove that \rho has a singularity of the form M [e](1/2), where is the Mellin transform. To do this, we use specially built function spaces.. A. 6) by Zemanian, and these allow us to precisely investigate the relationship between the Mellin and Hilbert transforms. Fourier comes into play in the sense that the Mellin transform is simpy the Fourier transform on the locally compact Abelian multiplicative group of the half-line, and as a more familiar operator, it guides our investigation. [ABSTRACT FROM AUTHOR]

We study nonstationary frames of matrix-valued Gabor systems and wavelet systems in the matrix-valued function space L 2 (ℝ , ℂ l × l). First, we show that a diagonal matrix-valued window function constitutes a frame for L 2 (ℝ , ℂ l × l) whenever each diagonal entry constitutes a frame for the space L 2 (ℝ). This is not true for arbitrary nonzero matrix-valued function. Using this, we prove the existence of nonstationary matrix-valued Gabor frames associated with the Weyl–Heisenberg group in terms of density of real numbers. We give a representation of the frame operator of matrix-valued nonstationary Gabor system. A necessary condition with explicit frame bounds for nonstationary matrix-valued Gabor frames associated with the Weyl–Heisenberg group is given. We discuss matrix-valued frame preserving maps in terms of adjointablity, with respect to the matrix-valued inner product, of bounded linear operators acting on L 2 (ℝ , ℂ l × l). It is shown that the image of a matrix-valued Gabor frame under bounded, linear and invertible operator on L 2 (ℝ , ℂ l × l) may not be a frame for L 2 (ℝ , ℂ l × l). In this direction, we give sufficient conditions on bounded linear operators which can preserve frame conditions. Finally, we give necessary and sufficient condition for the existence of nonstationary matrix-valued wavelet frames associated with the extended affine group. [ABSTRACT FROM AUTHOR]

For positive C_0-semigroups S and T on a Banach lattice such that S(t) \le T(t) for all times t, we prove that analyticity of T implies analyticity of S. This answers an open problem posed by Arendt in 2004. Our proof is based on a spectral theoretic argument: we apply spectral theory of positive operators to multiplication operators that are induced by S and T on a vector-valued function space. [ABSTRACT FROM AUTHOR]

Embedding data in hyperbolic spaces has proven beneficial for many advanced machine learning applications. However, working in hyperbolic spaces is not without difficulties as a result of its curved geometry (e.g., computing the Fréchet mean of a set of points requires an iterative algorithm). In Euclidean spaces, one can resort to kernel machines that not only enjoy rich theoretical properties but that can also lead to superior representational power (e.g., infinite-width neural networks). In this paper, we introduce valid kernel functions for hyperbolic representations. This brings in two major advantages, 1. kernelization will pave the way to seamlessly benefit the representational power from kernel machines in conjunction with hyperbolic embeddings, and 2. the rich structure of the Hilbert spaces associated with kernel machines enables us to simplify various operations involving hyperbolic data. That said, identifying valid kernel functions on curved spaces is not straightforward and is indeed considered an open problem in the learning community. Our work addresses this gap and develops several positive definite kernels in hyperbolic spaces (modeled by a Poincaré ball), the proposed kernels include the rich universal ones (e.g., Poincaré RBF kernel), or realize the multiple kernel learning scheme (e.g., Poincaré radial kernel). We comprehensively study the proposed kernels on a variety of challenging tasks including few-shot learning, zero-shot learning, person re-identification, deep metric learning, knowledge distillation and self-supervised learning. The consistent performance gain over different tasks shows the benefits of the kernelization for hyperbolic representations. [ABSTRACT FROM AUTHOR]

It is known that a tempered exponential dichotomy (which is the version of nonuniform exponential dichotomy for random dynamical systems) can be described by admissibility of a pair of function classes defined in terms of Lyapunov norms. Moreover, for MET-systems (i.e. random dynamical systems which satisfy the assumptions of the Multiplicative Ergodic Theorem) over an ergodic base system, tempered exponential dichotomy can be described by admissibility of a pair of function spaces which are defined in terms of the original norm. In present paper, we find an admissibility description of tempered exponential dichotomies for MET-systems in terms of new output and input classes. These classes can be regarded as a critical case of admissibility classes in the known result. Finally, we use our new pair of admissible spaces to prove the roughness of tempered exponential dichotomies for parametric MET-systems and give a Hölder continuous dependence of the associated projections on the parameter. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the iteration of the Laplace transform over certain spaces of generalized functions. As a consequence we prove a new Post-Widder-type inversion formula for this transform over distributions of compact support. [ABSTRACT FROM AUTHOR]

The direction estimation of the coherent source in a uniform circular array is an essential part of the signal processing area of the array, but the traditional uniform circular array algorithm has a low localization accuracy and a poor localization effect on the coherent source. To solve this problem, this paper proposes a two-dimensional direction of arrival (DOA) estimation for the coherent source in broadband. Firstly, the central frequency of the coherent sound source is estimated using the frequency estimation method of the delayed data, and a real-valued beamformer is constructed using the concept of the multiloop phase mode. Then, the cost function in the beam space is obtained. Finally, the cost function is searched in two dimensions to locate the sound source. In this paper, we simulate the DOA of the sound source at different frequencies and signal-to-noise ratios and analyze the resolution of the circular array. The simulation results show that the proposed algorithm can estimate the direction of arrival with high precision and achieve the desired results. [ABSTRACT FROM AUTHOR]

We study the estimation of a seasonality perturbed by a continuous time process admitting a C [ 0 , δ ] -valued autoregressive representation where C [ 0 , δ ] is the Banach space of continuous functions on [ 0 , δ ] , δ > 0. We provide the almost sure convergence, asymptotic normality and compact iterated logarithm law. Following Antoniadis A. (1982) we construct in the framework of functional autoregressive processes (non i.i.d. case), confidence balls for the seasonality in the space C [ 0 , δ ] from compact iterated logarithm law. Then when seasonality belongs to a finite dimensional space (dimensional reduction), we study the seasonality estimation giving its asymptotic properties. Finally, we examine an estimator of the dimension of this space when it is unknown. Numerical simulations illustrate the asymptotic results of the estimators. [ABSTRACT FROM AUTHOR]

We study multiple notions of Hilbert spaces of functions which, via the respective inner products, reproduce function values, or differences of function values. We do this by extending results from the more familiar settings of reproducing kernel Hilbert spaces, RKHSs. Our main results deal with operations on infinite graphs G = (V , E) of vertices and edges, and associated Hilbert spaces. For electrical network models, the differences f (x) − f (y) represent voltage differences for pairs of vertices x, y. In these cases, relative RKHSs will depend on choices of conductance functions c, where an appropriate function c is specified as a positive function defined on the edge-set E from G. Our present study of higher order differences, using choices of relative RKHSs, is motivated in part by existing numerical algorithms for discretization of PDEs. Our approach to higher order differences uses both combinatorial operations on graphs, and operator theory for the respective RKHSs. Starting with a graph G = (V , E) , we introduce an induced graph G ′ such that the vertices in G ′ are the edges in E from G, while the edges in G ′ are pairs of neighboring edges from G. [ABSTRACT FROM AUTHOR]