Singular Operators Research Articles

Vector-Valued Maximal Inequalities and Multiparameter Oscillation Inequalities for the Polynomial Ergodic Averages Along Multi-dimensional Subsets of Primes

We prove uniform ℓ2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell ^2$$\\end{document}-valued maximal inequalities for polynomial ergodic averages and truncated singular operators of Cotlar type modeled over multidimensional subsets of primes. In the averages case, we combine this with earlier one-parameter oscillation estimates (Mehlhop and Słomian in Math Ann, 2023, https://doi.org/10.1007/s00208-023-02597-8) to prove corresponding multiparameter oscillation estimates. This provides a fuller quantitative description of the pointwise convergence of the mentioned averages and is a generalization of the polynomial Dunford–Zygmund ergodic theorem attributed to Bourgain (Mirek et al. in Rev Mat Iberoam 38:2249–2284, 2022).

Liouville's type results for singular anisotropic operators

Abstract We present two Liouville-type results for solutions to anisotropic elliptic equations that have a growth of power 2 along the first s s coordinate directions and of power p p , with 1 < p < 2 1\lt p\lt 2 along the other ( N − s ) \left(N-s) ones. First, we begin our investigation by assuming that the solution is bounded only from below, deriving a rigidity result for the range p + ( N − s ) ( p − 2 ) > 0 p+\left(N-s)\left(p-2)\gt 0 of non-degeneration, which is a purely parabolic shade. Then we break free from this constraint at the price of assuming the solution to be bounded also from above.

Equivalence of block sequences in Schreier spaces and their duals

We prove that any normalized block sequence in a Schreier space Xξ, of arbitrary order ξ<ω1, admits a subsequence equivalent to a subsequence of the canonical basis of some Schreier space. The analogous result is proved for dual spaces to Schreier spaces. Using these results, we examine the structure of strictly singular operators on Schreier spaces and show that there are 2c many closed operator ideals on a Schreier space of any order, its dual and bidual space.

From area metric backgrounds to the cosmological constant and corrections to the Polyakov action

Area metrics and area metric backgrounds provide a unified framework for quantum gravity. They encode physical degrees of freedom beyond those of a metric. These nonmetric degrees of freedom must be suppressed by a potential at sufficiently high energy scales to ensure that in the infrared regime classical gravity is recovered. On this basis, we first study necessary and sufficient algebraic conditions for an area metric to be induced by a metric. Second, we consider candidate potentials for the area metric and point out a possible connection between the reduction of area metric geometry to metric geometry on the one hand, and the smallness of the cosmological constant on the other. Finally, we consider modifications of the Nambu-Goto action for a string from a metric background to an area metric background. We demonstrate that area metric perturbations introduce an interaction corresponding to a singular vertex operator in the classically equivalent Polyakov action. The implications of these types of vertex operators for the quantum theory remain to be understood. Published by the American Physical Society 2024

Incomplete label distribution learning via label correlation decomposition

Label distribution learning (LDL) has garnered increased attention in recent studies on label ambiguity. However, collecting complete annotations for LDL tasks is often time-consuming and labor-intensive compared to traditional learning paradigms. Therefore, designing effective incomplete LDL algorithms is crucial to broaden LDL’s application scope. In this paper, we propose a novel LDL algorithm, called Incomplete Label Distribution Learning via Label Correlation Decomposition (IncomLDL-LCD), which simultaneously learns label distributions and recovers missing description degrees of labels through label correlations. Specifically, we decompose the label correlation into sparse local label correlation and low-rank global label correlation using a soft-thresholding operator and a singular value thresholding operator, respectively. The former is utilized to capture the related label subsets necessary for reconstructing each possible label, while the latter focuses on extracting the coarse-grained semantic concepts from all labels and exploring the groupings of labels. Additionally, we develop an alternating solution with the accelerated proximal gradient descent method for optimization. Extensive experiments on 16 real-world data sets with varying degrees of missing annotations validate that our algorithm effectively handles incomplete LDL tasks and outperforms state-of-the-art algorithms.

Quantitative Uniform Approximation by Activated Singular Operators

In this article, we study the approximation properties of activated singular integral operators over the real line. We establish their convergence to the unit operator with rates. The kernels here derive from neural network activation functions and their corresponding density functions. The estimates are mostly sharp, and they are pointwise and uniform. The derived inequalities involve the higher order modulus of smoothness.

A class of singular bilinear maximal functions

Lebesgue space bounds Lp1(R1)×Lp2(R1)→Lq(R1) are established for certain singular maximal bilinear operators. The proof combines a single scale trilinear smoothing inequality with Calderón-Zygmund theory.

Taylor dispersion and phase mixing in the non‐cutoff Boltzmann equation on the whole space

AbstractIn this paper we describe the long‐time behavior of the non‐cutoff Boltzmann equation with soft potentials near a global Maxwellian background on the whole space in the weakly collisional limit (that is, infinite Knudsen number ). Specifically, we prove that for initial data sufficiently small (independent of the Knudsen number), the solution displays several dynamics caused by the phase mixing/dispersive effects of the transport operator and its interplay with the singular collision operator. For ‐wavenumbers with , one sees an enhanced dissipation effect wherein the characteristic decay time‐scale is accelerated to , where is the singularity of the kernel ( being the Landau collision operator, which is also included in our analysis); for , one sees Taylor dispersion, wherein the decay time‐scale is accelerated to . Additionally, we prove almost uniform phase mixing estimates. For macroscopic quantities such as the density , these bounds imply almost uniform‐in‐ decay of in due to phase mixing and dispersive decay.

On a fractional derivative operator with a singular kernel: definition, properties and numerical simulation

This paper is concerned with proposing a novel nonlocal fractional derivative operator with a singular kernel. We considered a fractional integral operator as a single integral of convolution type combined with a Mittag-Leffler kernel of Prabhakar type. The proposed singular fractional derivative operator is formulated as a proper inverse of the considered integral operator. We provided some useful features and relationships of the proposed derivative and introduced comparisons with the Caputo derivative which can be utilized for potential applications. Next, we presented numerical solutions for some nonlinear fractional order models incorporating the proposed derivative using a numerical algorithm developed in this paper. As a case study, we discussed the dynamic behavior of a fractional logistic model with the proposed derivative.

Weighted estimates of commutators of singular operators in generalized Morrey spaces beyond Muckenhoupt range and applications

For a certain class of radial weights, we prove weighted norm estimates for commutators with BMO coefficients of singular operators in local generalized Morrey spaces. As a consequence of these estimates, we obtain norm inequalities for such commutators in the generalized Stummel-Morrey spaces. We also discuss a.e. well-posedness of singular operators and their commutators on weighted generalized Morrey spaces. The obtained estimates are applied to prove interior regularity for solutions of elliptic PDEs in the frameworks of the corresponding weighted Sobolev spaces based on the local generalized Morrey spaces or Stummel-Morrey spaces. To this end also conditions for the applicability of the representation formula, for the second-order derivatives of solutions to elliptic PDEs, are found for the case of such weighted spaces. In both results, for commutators and applications, we admit weights beyond the Muckenhoupt range.

On a new singular and degenerate extension of the [formula omitted]-Laplace operator

We study a novel degenerate and singular elliptic operator Δ˜(τ,χ) defined by Δ˜(τ,χ)u=τ(x,Du)(|Du|Δ1u+χ(x,Du)Δ∞u), where the singular weights τ(x,s)>0 and χ(x,s)≥0 are continuous functions on Ω×Rn∖{0}. The operator Δ˜(τ,χ) is an extension of Δ(p,q)u=|Du|qΔ1u+(p−1)|Du|p−2Δ∞u,p≥1,q≥0, introduced by the authors in Baravdishet al. (2020), which in turn is an extension of the p-Laplace operator Δp. We establish the well-posedness of the Neumann boundary value problem for the parabolic equation ut=Δ˜(τ,χ)u in the framework of viscosity solutions. For the solution u, the weight χ controls the evolution along the tangential and the normal directions, respectively, on the level surface of u. The weight τ controls the total speed of the evolution of u. We also prove the consistency and the convergence of the numerical scheme for the finite differences method of the parabolic equation above. Numerical simulations show that our novel nonlinear operator Δ˜(τ,χ) gives better results than both the Perona–Malik (Perona and Malik, 1990) and total variation (TV) methods (Chan and Shen, 2005) when applied to image enhancement.

Quasilinear Schrödinger Equations with a Singular Operator and Critical or Supercritical Growth

Quasilinear Schrödinger Equations with a Singular Operator and Critical or Supercritical Growth

A log-based non-convex relaxation regularized regression for robust face recognition

Regression methods are widely employed in face recognition applications. The vector-based norm regression model only represents the pixel by pixel correlation and exhibits poor performance with contiguous occlusion. In contrast, methods based on the nuclear norm describe the low-rank structural information, but it is difficult to achieve satisfactory performance when dealing with complex noise. To address this, a log-based non-convex relaxation regularized regression (log-NCRR) model for robust face recognition is proposed in this paper. We adopt the log-based matrix loss without additional parameters to characterize the low-rank part of error image. Considering sparsity, we design a log-based vector loss to describe the sparse part of the error matrix, which achieves a balance between the l0-norm and l1-norm. Additionally, a weighted non-convex relaxation of l2,1-norm is proposed to ensure the group sparsity of the regression coefficient. The optimization problem is optimized by ADMM, with the corresponding sub-problems easily solved by the generalized singular value shrinkage operator. Finally, experimental results on four public databases for various types of noise validate the effectiveness and robustness of log-NCRR in face recognition.

INVERSE NODAL PROBLEMS FOR PENCILS OF SINGULAR STURM-LIOUVILLE OPERATORS

In this study, some properties of the pencils of singular Sturm-Liouville operators are investigated. Firstly, the behaviors of eigenvalues and eigenfunctions is learned, then for each discontinuity point

On the incompatibility of object fronting and progressive aspect in Yucatec Maya

In this paper, we present data from an elicitation study and a corpus study that support the observation that the Yucatec Maya progressive aspect auxiliary táan is replaced by the habitual auxiliary k in sentences with contrastively focused fronted objects. Focus has been extensively studied in Yucatec, yet the incompatibility of object fronting and progressive aspect in Yucatec Maya remains understudied. Both our experimental results and our corpus study point in the direction that this incompatibility may very well be categorical. Theoretically, we take a progressive reading to be derived from an imperfectivity operator in combination with a singular operator, and we propose that this singular operator implicates the negation of event plurality, leading to an exhaustive interpretation which ranks below corrective focus on a contrastive focus scale. This means that, in a sentence with object focus fronting, the use of the marked auxiliary táan (as opposed to the more general k) would trigger two contrastive foci, which would be an unlikely and probably dispreferred speech act.

Quantum solution of the relationship between the 19-vertex model and the Jones polynomial

The challenge is to create an efficient quantum algorithm for the bosonic model capable of calculating the Jones polynomials for a knot resulting from interweaving or interlacing n-vertices. This weave is the construction of braid group representations from nineteen-vertex model. We present eigenbases and eigenvalues for lattice generators and their usefulness for the direct computation of Jones polynomials. The calculation shows that the Temperley-Lieb operators can be used for any braid word. Therefore, we propose a quantum sequence using these singular operators as quantum gates operating on the state of n qubits. We show that quantum calculations give the Jones polynomial for achiral knots and links.

SPECTRAL PROPERTIES AND INVERSE NODAL PROBLEMS FOR SINGULAR DIFFUSION EQUATION

In this study, some properties for the pencils of singular Sturm-Liouville operators are investigated. Firstly, the behaviors of eigenvalues were learned, then the solutions of the inverse problem were given to determine the potential function and parameters of the boundary condition with the help of a dense set of nodal points and lastly we obtain a constructive solution to the inverse problems of this class.

Boundedness for Commutators of Singular Integral Operator With Weaker Smooth Kernel on Weighted Hardy and Herz Spaces

In this job, let T be a variant of Hörmander’s singular operator and b ∈ Lipβ,ω(ℝn). We adopt the classic Harmonic analysis method and obtain that commutators are bounded on weighted Hardy space and weighted Herz space.

FEATURES OF THE CONSTRUCTION OF UNIFORM ASYMPTOTIC SOLUTION OF A SYSTEM OF DIFFERENTIAL EQUATIONS WITH A TURNING POINT WITH POSITIVE COEFFICIENTS OF THE MATRIX

The work is devoted to the analysis of the coefficients of the singular operator of the Orr-Sommerfeld type in vector form including the turning point. The system of singularly perturbed differential equations with a small parameter at the highest derivative is investigated. We consider the case when the spectrum of the limit operator contains multiple and identically equal zero elements. Using the method of essentially singular functions, the uniform asymptotic solution of the system is constructed. For the case of a stable turning point, the asymptotics of the solutions of the system are constructed in the sector that contains the turning point. The asymptotics of the first two solutions for the homogeneous problem are constructed using the Airy functions and their derivatives. The third formal solution of a homogeneous system for this case presents certain difficulties. Therefore, taking into account the specified conditions, in order to construct the uniform asymptotics of the solution for the given system, we used the partial solution of the heterogeneous system as the third solution of the homogeneous system.

Perturbation of singular integral operators with piecewise continuous coefficients

In the paper it is shown that the property of singular integral operators with piecewise continuous coefficients to be Noetherian is stable with respect to their perturbation with certain non-compact operators. An example is constructed showing that the corner points of the integration contour significantly affect the Noetherian property of singular operators with translations. These results are obtained using the symbol of the singular operators on contours with corner points, symbol, which is also determined.