In this paper, we investigate positive solutions of the critical Choquard–Kirchhoff equation: [Formula: see text] where [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text] is the Hardy–Littlewood–Sobolev critical exponent. We first classify positive solutions by parameter dependence: (i) unique solution for [Formula: see text] and all [Formula: see text]; (ii) unique solution for [Formula: see text] if and only if [Formula: see text] (with [Formula: see text] the ground state of the non-Kirchhoff critical Choquard equation); (iii) two distinct solutions for [Formula: see text] (single solution at critical [Formula: see text]). We then analyze asymptotic behavior of solutions as [Formula: see text]. Moreover, for [Formula: see text] and [Formula: see text], we prove nondegeneracy of positive solutions via spherical harmonics expansion on [Formula: see text], decomposing the linearized operator into radial differential operators. Combining Perron–Frobenius property and spherical harmonic orthogonality, we show the linearized operator’s kernel is spanned by scaling ([Formula: see text]) and translation ([Formula: see text]) perturbations of positive solutions.
Compressed sensing shows that a sparse signal can be stably recovered from incomplete linear measurements. But, in practical applications, some signals have additional structure, where the nonzero elements arise in some blocks. We call such signals as block-sparse signals. In this paper, the [Formula: see text] minimization method for the stable recovery of block-sparse signals is investigated. Sufficient conditions based on block mutual coherence property and associating upper bound estimations of error are established to ensure that block-sparse signals can be stably recovered in the presence of noise via the [Formula: see text] minimization method. To the best of our knowledge, it is the first block mutual coherence property condition of stably reconstructing block-sparse signals by the [Formula: see text] minimization method. To solve the model, we propose an efficient algorithm based on the alternating direction method of multipliers and evidence that the obtained algorithm is globally convergence under moderate assumptions. Additionally, the numerical experiments implemented verify the performance of the [Formula: see text] minimization.
In this paper, we study the following critical Hartree equation with electromagnetic field [Formula: see text] where [Formula: see text] is the imaginary unit, [Formula: see text] is the upper critical exponent in the sense of the Hardy–Littlewood–Sobolev inequality, [Formula: see text] [Formula: see text], [Formula: see text] and [Formula: see text] is a small parameter. The magnetic potential [Formula: see text] is a bounded real-valued vector field on [Formula: see text] and [Formula: see text] denotes an electric potential. Under some suitable assumptions on [Formula: see text] and [Formula: see text], we establish the existence of single-peak solutions by applying the finite-dimensional reduction argument within the framework of critical point theory.
In this paper, we investigate a discrete integrable hierarchy of three-component Volterra-type lattice equations arising from a [Formula: see text] matrix spectral problem. The hierarchy is generated by employing the zero-curvature condition together with Lenard recursion relations. To explore its algebro-geometric aspects, we analyze the characteristic polynomial of the associated Lax matrix and construct a tetragonal spectral curve. The geometry of the corresponding four-sheeted compact Riemann surface, including its genus, holomorphic differentials, and Riemann theta functions, is then examined. Within the framework of algebro-geometric integration, we further study the analytic structure of the Baker–Akhiezer function and related meromorphic functions. As a result, explicit quasi-periodic solutions of the entire hierarchy are derived in terms of Riemann theta functions. This work provides a systematic algebro-geometric characterization of the hierarchy and enriches the theory of integrable lattice equations.
We prove existence of solutions for the system {-div(M(x)del u) + u = -A div (u(M(x)del psi 1 )/1+ psi )+ f(x), -div(M(x)del psi) + psi = u/ B + psi, in the cases B = 1 and B = 0 which leads to a singular second equation.
In this paper, we study the asymptotics of the Fredholm determinant [Formula: see text] of the finite-temperature deformation of the Bessel kernel. This determinant represents the probability that no particles lie in an interval in the edge scaling limit of the finite-temperature fermion system. The logarithmic derivative of the Fredholm determinant, which we denote by [Formula: see text], satisfies an integrable PDE. We derive the large gap asymptotics of [Formula: see text] at finite temperature. Moreover, we obtain the asymptotics of [Formula: see text] in several different regimes in the [Formula: see text]-plane. In the transition regime, we observe a phase transition which is characterized by solutions of the Jimbo–Miwa–Okamoto [Formula: see text]-form of the second Painlevé equation.
In this paper, we introduce a posteriori analysis framework for the weak Chebyshev greedy algorithm (WCGA) and establish new error bounds in terms of the K-functional, which use posteriori information {u(k)} and improve those prior results. The novelty of this framework lies in its ability to not only unify and generalize the classical results of Temlyakov and Jiang, as well as the entropy-based bounds of Li and Siegel, but also establish a quantitative connection between dictionary entropy and the posteriori information for the first time. Furthermore, we prove the stability of the WCGA by extending our analysis to the Approximate WCGA, under certain mild perturbations on the norming functional and approximant. We also establish improved error estimates for the Approximate WCGA. Then, we apply these results to analyze the approximation quality of shallow neural networks with activation functions ReLUs, revealing that higher order activations may yield improved convergence rates. Moreover, the WCGA is nearly optimal for training ReLUs shallow neural networks in L-2 space. These improved error bounds also provide a finer analysis for the performance of the WCGA in the least squares regression problem.
In this paper, we investigate the behavior of semi-discrete sampling operators of the Durrmeyer type within the framework of Sobolev-Orlicz spaces. We begin by addressing the problem of modular convergence through a density approach, using the density of compactly supported smooth functions in Sobolev-Orlicz spaces with respect to the modular topology. To this end, we also employ the boundedness of the celebrated Hardy-Littlewood maximal operator in the Orlicz setting. For completeness, we also provide a quantitative analysis to establish sharper estimates based on suitable Orlicz moduli of continuity, in their strong (with respect to the Luxemburg norm) and weak (modular) version. This is obtained through a direct approach based on the Minkowski inequality within the Orlicz framework, involving phi-functions with weaker growth condition at infinity. Consequently, Luxemburg norm and modular convergence theorems in broader functional spaces are derived. Throughout the paper, we discuss several examples of functional spaces, including the Sobolev versions of Zygmund and exponential-type spaces.
In this paper, we investigate a parabolic equation with [Formula: see text]-bilaplacian operator, where [Formula: see text]. Through an appropriate change of variable, the original problem is transformed into a system of two differential equations. We discretize this system with respect to the time variable, prove the existence and uniqueness of the semi-discrete solution, and prove some a priori estimates. We then use Rothe’s method to prove the existence of a unique weak solution to the original problem. Finally, we investigate the order of convergence and prove some error estimates.
We study the following class of Steklov eigenvalue problems: [Formula: see text] where [Formula: see text] and [Formula: see text] are prescribed positive radial functions, [Formula: see text] is a Lipschitz domain in [Formula: see text] with [Formula: see text] and [Formula: see text] denotes its outward unit normal. Extending classical results in the unweighted case due to Weinstock, the first author, and others, we establish isoperimetric inequalities for low-order eigenvalues under suitable symmetry assumptions on the domain. In the first part, we consider the case [Formula: see text] and [Formula: see text], where the parameters [Formula: see text] satisfy appropriate constraints. Our analysis relies on an explicit computation of the spectrum in the radial case, variational principles, and a family of weighted isoperimetric inequalities with “double density”. In the second part, we address the case [Formula: see text] and [Formula: see text], where [Formula: see text] is a non-decreasing, log-convex function. In this setting, the proof relies, among other tools, on a new weighted isoperimetric inequality, which may be of independent interest.
The oblique dual frames overcome the limitations of traditional dual frames through direct sum decomposition of spaces, and provide a more flexible approach to signal reconstruction. This paper addresses oblique dual Gabor frame theory. We give a characterization of Gabor system-based direct sum decomposition of L 2 (ℝ). Based on this, using our Zak transform matrix techniques we characterize the oblique dual Gabor frame pairs, and present a parametric expression and a uniqueness characterization of the oblique dual Gabor frames for a given subspace Gabor frame. Some examples are also provided to illustrate the generality of our theory.
The coefficients [Formula: see text] in the Maclaurin expansion [Formula: see text] are studied, where [Formula: see text] with [Formula: see text], and [Formula: see text]. In 1973, D. A. Brannan conjectured that [Formula: see text] for each positive odd integer [Formula: see text], and showed it is true for [Formula: see text]. This has recently been proven for all odd integers [Formula: see text] by a number of authors in aggregate for the special case [Formula: see text]. In this paper hypergeometric integral representations and Watson-type approximations are utilized, from which the general problem is reduced to numerically evaluating the minima of certain simple, explicit, slowly-varying functions over compact domains. From the positivity of these constants, it is shown that the conjecture holds for [Formula: see text], [Formula: see text] and [Formula: see text] where [Formula: see text].
This work is devoted to the nonexistence of positive weak solutions for a weighted quasilinear elliptic differential inequality involving nonlocal source and gradient absorption terms in bounded domain, where positive weight functions may be singular or degenerate at the boundary. Under the condition that the weight function of the absorption term is either a sufficiently small positive constant or a more general assumption, we establish new nonexistence results containing the critical cases. The key ingredient in the proof is the rescaled test function method developed by Mitidieri and Pohozaev, which is essentially based on nonlinear capacity estimates adapted to the nonlocal setting of our bounded domain problem.
In this paper, we present pointwise error bounds for Legendre approximations of singular functions in fractional spaces. We begin by recalling the relevant fractional space definitions needed for the pointwise estimates. We then derive explicit and pointwise error bounds for Legendre approximations of functions with interior or endpoint singularities. New analytical techniques are developed to handle the case of endpoint singularities, and all coefficients in our estimates are given by explicit and computable formulas, allowing for a precise quantification of the singularity’s influence and clearly demonstrating the sharpness of the results. Numerical experiments are presented to support the theoretical results.
In this paper, we consider a viscous fluid obeying the 3D Stokes system in a thin layer Omega(& varepsilon;) subset of & Ropf;(3) of thickness 0 < h(& varepsilon;) << 1, which is perforated by & varepsilon;-periodically distributed cylinder shaped obstacles of size 0 < & varepsilon; << 1. On the boundary of the obstacles, we prescribe non-homogeneous Fourier boundary conditions with a parameter alpha(& varepsilon;) is an element of (0, +infinity). Depending on the relation between & varepsilon; and h & varepsilon;, we prove the convergence of the homogenization process when & varepsilon; goes to zero for different values of alpha(& varepsilon;). As a result, we derive different 2D Darcy's laws taking into account the microstructure of the domain by means of local problems. The dimensions of the local problems are 2D or 3D depending on the relation between & varepsilon; and h & varepsilon;, i.e. if h(& varepsilon;) >> & varepsilon; or h(& varepsilon; )approximate to & varepsilon;, respectively. Moreover, depending on the relation between alpha & varepsilon; and & varepsilon;(-1), the boundary conditions of the local problems on the reference obstacle are of three types: homogeneous Dirichlet if alpha(& varepsilon;) >> & varepsilon;(-1), non-homogeneous Fourier if alpha(& varepsilon;) approximate to & varepsilon;(-1), or homogeneous Fourier boundary conditions if alpha(& varepsilon;) << & varepsilon;(-1).
In this paper, we study the existence of nontrivial solutions for the problem div(phi'(vertical bar del u vertical bar/vertical bar del u vertical bar del u) = phi'(vertical bar u vertical bar/vertical bar u vertical bar u, in a bounded smooth domain Omega Omega R-N, subject to the nonlinear boundary condition (phi'(vertical bar del u vertical bar/vertical bar del u vertical bar partial derivative u/partial derivative v = lambda V(x) psi'vertical bar u vertical bar/vertical bar u vertical bar u,, on partial derivative Omega, where phi and psi are given N-functions belonging to the Lieberman class and V is a suitable weight function. We do not assume t bar right arrow phi (root t) is convex on [0, infinity) and the weight V need not be bounded away from zero and may have singular points.
Analyzing large-scale functional data poses significant computational challenges due to high costs and substantial data storage needs. Additionally, traditional batch learning algorithms are not well-equipped to manage streaming data effectively. To address these issues, we propose a fully online learning algorithm designed for functional linear regression, which models the linear relationship between a scalar response and a functional predictor. Our approach employs Tikhonov regularization schemes within the framework of reproducing kernel Hilbert spaces (RKHS). A key feature of this fully online algorithm is its polynomially decaying regularization parameter, which adapts dynamically at each learning step, distinguishing it from the partially online algorithm that uses a fixed parameter. Within the functional linear model framework, we establish sufficient conditions for the convergence of the fully online algorithm in the RKHS norm. Additionally, we employ a capacity-independent approach to derive error bounds and almost sure convergence rates for both prediction and estimation, achieved through careful selection of step sizes and regularization parameters.
A nonlinear Vlasov–Fokker–Planck equation is considered in a bounded domain, with specular reflection imposed on the boundary. It is proven that if the initial datum is sufficiently close, in a weighted L^∞-space, to a radial and spatially homogeneous function that is regular enough, then a corresponding global-in-time weak solution exists, and this one decays with exponential rate to the global Maxwellian. The proof is split into two steps. In the first, the stability of the manifold of spatially homogeneous solutions to this equation is established. In the second part, the close-to-Maxwellian regime is considered. A careful regularization scheme is designed so that this mollified equation can be well-approximated by a linear system. Inspired by the recent preprint [Carrapatoso and Mischler, arxiv:2407.09031], for this linear system, L^2-hypocoercivity along with ultracontractivity properties are derived in order to change the space of functional decay to the L^∞-framework. The properties of the original model are then recovered by means of fixed point arguments and carefully passing to the limit of the regularization parameter.
In this paper, we re-examine a 2D Smoluchowski equation employed for modeling nematic liquid crystalline polymers. Specifically, we provide a novel proof concerning the investigation of steady-state solutions to the 2D Smoluchowski equation. We establish that when the intensity constant is less than or equal to 4, a unique (trivial) solution exists. Conversely, when the intensity constant exceeds 4, precisely two solutions emerge, corresponding to the isotropic and nematic phases, respectively. The proof relies solely on calculus, which is transparent and accessible.
We study a nonlinear Dirichlet problem eigenvalue driven by a differential operator with unbalanced growth (double phase problem) and a reaction that has the competing effects of a singular term and of a superlinear perturbation. We prove an existence and multiplicity theorem which is global in the parameter lambda > 0.
