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Authors:Shengbing Deng, Fang Yu Pages: 1 - 49 Abstract: Analysis and Applications, Ahead of Print. In this paper, we study the following biharmonic elliptic problem involving slightly subcritical non-power nonlinearity {Δ2u = u 2∗−2u [ln(e+ u )]ðœ–inΩ,Δu = u = 0 on∂Ω, where [math] is a bounded smooth domain, [math], [math], [math] is a small parameter. By Lyapunov–Schmidt procedure, under suitable assumptions on the Robin function related to [math], we construct the multi-peak solutions which blow-up and concentrate in different points of [math] as [math] goes to [math]. Citation: Analysis and Applications PubDate: 2024-07-05T07:00:00Z DOI: 10.1142/S0219530524500295

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Authors:Yun Cai, Qian Zhang, Ruifang Hu Pages: 1 - 20 Abstract: Analysis and Applications, Ahead of Print. In this paper, we study an unconstrained [math] minimization and its associated iteratively reweighted least squares algorithm (UBIRLS) for recovering block sparse signals. Wang et al. [Y. Wang, J. Wang and Z. Xu, On recovery of block-sparse signals via mixed [math] [math] norm minimization, EURASIP J. Adv. Signal Process. 2013(76) (2013) 76] have used numerical experiments to show the remarkable performance of UBIRLS algorithm for recovering a block sparse signal, but no theoretical analysis such as convergence and convergence rate analysis of UBIRLS algorithm was given. We focus on providing convergence and convergence rate analysis of UBIRLS algorithm for block sparse recovery problem. First, the convergence of UBIRLS is proved strictly. Second, based on the block restricted isometry property (block RIP) of linear measurement matrix [math], we give the error bound analysis of the UBIRLS algorithm. Lastly, we also characterize the local convergence behavior of the UBIRLS algorithm. The simplicity of UBIRLS algorithm, along with the theoretical guarantees provided in this paper, will make a compelling case for its adoption as a standard tool for block sparse recovery. Citation: Analysis and Applications PubDate: 2024-07-03T07:00:00Z DOI: 10.1142/S0219530524500283

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Authors:H. R. Clark, L. Friz, M. Rojas-Medar Pages: 1 - 24 Abstract: Analysis and Applications, Ahead of Print. This paper deals with a study on an initial-boundary value problem for incompressible non-Newtonian fluids of degree two, in a bounded and simply connected open set [math] of [math]. It will be shown that the global-in-time weak solution is uniformly stable (i.e. the behavior of the solution changes continuously with the data), and consequently this solution is unique. Furthermore, an exponential decay rate for the energy of the weak solution will also be established. Citation: Analysis and Applications PubDate: 2024-06-29T07:00:00Z DOI: 10.1142/S0219530524300011

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Authors:Zhiguang Zhang, Yuxiang Li Pages: 1 - 28 Abstract: Analysis and Applications, Ahead of Print. In this work, we study the no-flux initial-boundary value problem for the migration-consumption taxis system involving singular density-suppressed motility (⋆) ut = Δ(ulϕ(v)),vt = Δv − uvm in a bounded domain [math] [math], where [math] generalizes the singular prototype given by [math] [math] with [math]. We prove that if [math] and [math], then the model (⋆) possesses a global weak-strong solution. Citation: Analysis and Applications PubDate: 2024-06-26T07:00:00Z DOI: 10.1142/S0219530524500258

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Authors:Si-Ying Long, An-Ping Liu, Teng-Fei Zhang Pages: 1 - 45 Abstract: Analysis and Applications, Ahead of Print. In this paper, we study the Cauchy problem of the two-species incompressible viscoelastic fluid of Oldroyd-B system, which involving a reaction effect between two species of polymers. We prove the local existence with initial data in [math] in a classical solution framework, and then provide a blow-up criteria. We concentrate on the a priori estimate, by using the energy method. In particular, the variant system in a general formulation is also studied, and the corresponding local well-posedness is established. Citation: Analysis and Applications PubDate: 2024-06-15T07:00:00Z DOI: 10.1142/S021953052450026X

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Authors:Bojing Shi Pages: 1 - 33 Abstract: Analysis and Applications, Ahead of Print. For a family of second-order parabolic systems with periodic, oscillating and time-dependent coefficients, we establish the uniform boundary Hölder and Lipschitz estimates for Neumann problems in [math] and [math] cylinders, respectively, by using the convergence rate method. Moreover, we establish the uniform [math] estimates for initial-Neumann problems in [math] cylinders for [math] by using the real-variable method. As a by-product, we also obtain the Gaussian estimates for Neumann function and its derivatives. Citation: Analysis and Applications PubDate: 2024-06-08T07:00:00Z DOI: 10.1142/S0219530524500246

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Authors:Zhenhai Liu, Nikolaos S. Papageorgiou Pages: 1 - 18 Abstract: Analysis and Applications, Ahead of Print. We consider a nonlinear Dirichlet problem driven by the double phase operator. The reaction has the combined effects of parametric concave term and of an indefinite convex one (“concave-convex” problem with indefinite weight). Using the Nehari method we prove the existence of two bounded, positive ground state solutions. Citation: Analysis and Applications PubDate: 2024-06-01T07:00:00Z DOI: 10.1142/S0219530524500222

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Authors:José Francisco de Oliveira, Jeferson Silva Pages: 1 - 30 Abstract: Analysis and Applications, Ahead of Print. Critical Sobolev-type inequality for a class of weighted Sobolev spaces on the entire space is established. We also investigate the existence of extremal function for the associated variational problem. As an application, we prove the existence of a weak solution for a general class of critical semilinear elliptic equations related to the polyharmonic operator. Citation: Analysis and Applications PubDate: 2024-05-31T07:00:00Z DOI: 10.1142/S0219530524500210

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Authors:Ahmed Abdeljawad, Philipp Grohs Pages: 1 - 30 Abstract: Analysis and Applications, Ahead of Print. While it is well-known that neural networks enjoy excellent approximation capabilities, it remains a big challenge to compute such approximations from point samples. Based on tools from Information-based complexity, recent work by Grohs and Voigtlaender [Proof of the theory-to-practice gap in deep learning via sampling complexity bounds for neural network approximation spaces, preprint (2021), arXiv:2104.02746] developed a rigorous framework for assessing this so-called ”theory-to-practice gap”. More precisely, in that work it is shown that there exist functions that can be approximated by neural networks with ReLU activation function at an arbitrary rate while requiring an exponentially growing (in the input dimension) number of samples for their numerical computation. This study extends these findings by showing analogous results for the ReQU activation function. Citation: Analysis and Applications PubDate: 2024-05-31T07:00:00Z DOI: 10.1142/S0219530524500234

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Authors:Dan Dai, Luming Yao, Yu Zhai Pages: 1 - 35 Abstract: Analysis and Applications, Ahead of Print. In this paper, we investigate a determinantal point process on the interval [math], associated with the confluent hypergeometric kernel. Let [math] denote the trace class integral operator acting on [math] with the confluent hypergeometric kernel. Our focus is on deriving the asymptotics of the Fredholm determinant [math] as [math], while simultaneously [math] in a super-exponential region. In this regime of double scaling limit, our asymptotic result also gives us asymptotics of the eigenvalues [math] of the integral operator [math] as [math]. Based on the integrable structure of the confluent hypergeometric kernel, we derive our asymptotic results by applying the Deift–Zhou nonlinear steepest descent method to analyze the related Riemann–Hilbert problem. Citation: Analysis and Applications PubDate: 2024-05-14T07:00:00Z DOI: 10.1142/S0219530524500192

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Authors:Zhiwei Hao, Xinru Ding, Libo Li, Ferenc Weisz Pages: 1 - 28 Abstract: Analysis and Applications, Ahead of Print. In this paper, we study the real interpolation theory for variable Lorentz–Karamata spaces as well as for the corresponding martingale Hardy spaces. As a consequence, the generalization of Doob’s maximal inequality will be proved. Moreover, the atomic decompositions of the variable martingale Hardy–Lorentz–Karamata spaces are also presented. The results obtained here are new even for martingale Hardy–Lorentz–Karamata spaces. Citation: Analysis and Applications PubDate: 2024-05-10T07:00:00Z DOI: 10.1142/S0219530524500209

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Authors:Sibei Yang, Zhenyu Yang Pages: 1 - 50 Abstract: Analysis and Applications, Ahead of Print. In this paper, the authors establish a general (two-weight) boundedness criterion for a pair of functions, [math], on [math] in the scale of weighted Lebesgue spaces, weighted Lorentz spaces, (Lorentz–)Morrey spaces, and variable Lebesgue spaces. As applications, the authors give a unified approach to prove the (two-weight) boundedness of Calderón–Zygmund operators, Littlewood–Paley [math]-functions, Lusin area functions, Littlewood–Paley [math]-functions, and fractional integral operators, in the aforementioned function spaces. Moreover, via applying the above (two-weight) boundedness criterion, the authors further obtain the (two-weight) boundedness of Riesz transforms, Littlewood–Paley [math]-functions, and fractional integral operators associated with second-order divergence elliptic operators with complex bounded measurable coefficients on [math] in the aforementioned function spaces. Citation: Analysis and Applications PubDate: 2024-05-08T07:00:00Z DOI: 10.1142/S0219530524500180

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Authors:Rui Li, Qingyue Zhang Pages: 1 - 13 Abstract: Analysis and Applications, Ahead of Print. In this paper, we study the problem of phase retrieval with short-time linear canonical transform (STLCT). The relation between signal, window and their STLCT is provided through Fourier transform. Based on this theorem, a uniqueness result is established for all square integrable functions. For nonseparable real continuous signal, we prove the uniqueness theorems under some weaker conditions. In complex bandlimited and cardinal [math]-spline spaces, uniqueness results are provided with magnitude-only STLCT. Citation: Analysis and Applications PubDate: 2024-04-27T07:00:00Z DOI: 10.1142/S0219530524500155

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Authors:Lorenza D’Elia, Michela Eleuteri, Elvira Zappale Pages: 1 - 48 Abstract: Analysis and Applications, Ahead of Print. We propose a homogenized supremal functional rigorously derived via [math]-approximation by functionals of the type ess-sup[math], when [math] is a bounded open set of [math] and [math]. The homogenized functional is also deduced directly in the case where the sublevel sets of [math] satisfy suitable convexity properties, as a corollary of homogenization results dealing with pointwise gradient constrained integral functionals. Citation: Analysis and Applications PubDate: 2024-04-24T07:00:00Z DOI: 10.1142/S0219530524500179

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Authors:Lei Shi, Zihan Zhang Pages: 1 - 37 Abstract: Analysis and Applications, Ahead of Print. Kernel methods are popular in nonlinear and nonparametric regression due to their solid mathematical foundations and optimal statistical properties. However, scalability remains the primary bottleneck in applying kernel methods to large-scale data regression analysis. This paper aims to improve the scalability of kernel methods. We combine Nyström subsampling and the preconditioned conjugate gradient method to solve regularized kernel regression. Our theoretical analysis indicates that achieving optimal convergence rates requires only [math] memory and [math] time (up to logarithmic factors). Numerical experiments show that our algorithm outperforms existing methods in time efficiency and prediction accuracy on large-scale datasets. Notably, compared to the FALKON algorithm [A. Rudi, L. Carratino and L. Rosasco, Falkon: An optimal large scale kernel method, in Advances in Neural Information Processing Systems (Curran Associates, 2017), pp. 3891–3901], which is known as the optimal large-scale kernel method, our method is more flexible (applicable to non-positive definite kernel functions) and has a lower algorithmic complexity. Additionally, our established theoretical analysis further relaxes the restrictive conditions on hyperparameters previously imposed in convergence analyses. Citation: Analysis and Applications PubDate: 2024-04-22T07:00:00Z DOI: 10.1142/S0219530524500131

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Authors:Rongrong Lin, Shimin Li, Yulan Liu Pages: 1 - 22 Abstract: Analysis and Applications, Ahead of Print. Computing the proximal operator of the sparsity-promoting piece-wise exponential (PiE) penalty [math] with a given shape parameter [math], which is treated as a popular nonconvex surrogate of [math]-norm, is fundamental in feature selection via support vector machines, image reconstruction, zero-one programming problems, compressed sensing, neural networks, etc. Due to the nonconvexity of PiE, for a long time, its proximal operator is frequently evaluated via an iteratively reweighted [math] algorithm, which substitutes PiE with its first-order approximation, however, the obtained solutions only are the critical point. Based on the exact characterization of the proximal operator of PiE, we explore how the iteratively reweighted [math] solution deviates from the true proximal operator in certain regions, which can be explicitly identified in terms of [math], the initial value and the regularization parameter in the definition of the proximal operator. Moreover, the initial value can be adaptively and simply chosen to ensure that the iteratively reweighted [math] solution belongs to the proximal operator of PiE. Citation: Analysis and Applications PubDate: 2024-04-22T07:00:00Z DOI: 10.1142/S0219530524500143

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Authors:Xinghong Pan, Chao-Jiang Xu Pages: 1 - 59 Abstract: Analysis and Applications, Ahead of Print. In this paper, we prove the global existence of small Gevrey-2 solutions to the 3D axially symmetric Prandtl equations. The index 2 is the optimal index for well-posedness result in smooth Gevrey function spaces for data without monotonic assumptions. The novelty of our paper lies in two aspects: one is the tangentially weighted energy construction to match the [math] weight in the incompressibility and the other is introducing of the new linearly good unknowns to obtain the fast decay of the lower order Gevrey-2 norms of the solutions and auxiliary functions. Citation: Analysis and Applications PubDate: 2024-04-22T07:00:00Z DOI: 10.1142/S0219530524500167

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Authors:Shivam Bajpeyi, Dhiraj Patel, S. Sivananthan Pages: 1 - 20 Abstract: Analysis and Applications, Ahead of Print. In this paper, we aim to provide a general paradigm for dealing with the sampling and random sampling problem in a reproducing kernel subspace of Orlicz space [math]. We consider the function space [math] as the image of an idempotent integral operator on [math], where the integral kernel satisfies certain off-diagonal decay and regularity conditions. The model example of such reproducing kernel subspace of [math] includes the finitely generated shift-invariant space and signal space with a finite rate of innovation. We show that a signal in [math] can be stably reconstructed from its samples at distinct points separated by a sufficiently small gap. Next, we deduce that the random sampling inequality holds with a high probability for the class of functions in [math] concentrated on a cube [math], when the samples collected at i.i.d. random points are drawn on [math] of order [math]. Citation: Analysis and Applications PubDate: 2024-04-10T07:00:00Z DOI: 10.1142/S021953052450012X

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Authors:Philipp Trunschke Pages: 1 - 29 Abstract: Analysis and Applications, Ahead of Print. We consider the problem of approximating a function in a general nonlinear subset of [math], when only a weighted Monte Carlo estimate of the [math]-norm is accessible. The concept of sample complexity, i.e. the number of sample points necessary to achieve a prescribed error with high probability, is of particular interest in this setting. Reasonable worst-case bounds for this quantity exist only for particular model classes, like linear spaces or sets of sparse vectors. However, the existing bounds are very pessimistic for more general sets, like tensor networks or neural networks. Restricting the model class to a neighborhood of the best approximation allows us to derive improved worst-case bounds for the sample complexity. When the considered neighborhood is a manifold with positive local reach, its sample complexity can be estimated through the sample complexities of the tangent and normal spaces and the manifold’s curvature. Citation: Analysis and Applications PubDate: 2024-04-05T07:00:00Z DOI: 10.1142/S0219530524500271

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Authors:Mirela Kohr, Radu Precup Pages: 1 - 21 Abstract: Analysis and Applications, Ahead of Print. We analyze a general class of coupled systems of stationary Navier–Stokes type equations with variable coefficients and non-homogeneous terms of reaction type in the incompressible case. Existence of solutions satisfying the homogeneous Dirichlet condition in a bounded domain in [math], [math], and localization results for the corresponding kinetic energy and enstrophy are obtained by using a variational approach and the fixed point index theory. Citation: Analysis and Applications PubDate: 2024-03-25T07:00:00Z DOI: 10.1142/S0219530524500118

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Authors:Bao-Huai Sheng, Jian-Li Wang Pages: 1 - 42 Abstract: Analysis and Applications, Ahead of Print. We give investigations on a kernel function approximation problem arising from learning theory and show the convergence rate from the view of classical Fourier analysis. First, we provide the general definition for a modulus of smoothness and a [math]-functional, and show that they are equivalent. In particular, we give explicit representation for some moduli of smoothness. Second, we establish some Jackson-type inequalities for the approximation error associated with some non-radial kernels. Also we apply these results to some concrete classical kernel function spaces and give Jackson-type inequalities for some concrete RKHS approximation problems. Finally, we apply these discussions to learning theory and describe the learning rates with the moduli of smoothness. The tools we used are Fourier analysis and the semigroup operator. The results show that the Jackson-type inequalities of approximation by some radial kernel functions on compact set with nonempty interiors cannot be expressed with the classical moduli of smoothness. Citation: Analysis and Applications PubDate: 2024-03-13T07:00:00Z DOI: 10.1142/S021953052450009X

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Authors:Hyung Jun Choi, Woocheol Choi, Jinmyoung Seok Pages: 1 - 29 Abstract: Analysis and Applications, Ahead of Print. In this work, we establish the linear convergence estimate for the gradient descent involving the delay [math] when the cost function is [math]-strongly convex and [math]-smooth. This result improves upon the well-known estimates in [Y. Arjevani, O. Shamir and N. Srebro, A tight convergence analysis for stochastic gradient descent with delayed updates, Proc. Mach. Learn. Res. 117 (2020) 111–132; S. U. Stich and S. P. Karimireddy, The error-feedback framework: Better rates for SGD with delayed gradients and compressed updates, J. Mach. Learn. Res. 21(1) (2020) 9613–9648] in the sense that it is non-ergodic and is still established in spite of weaker constraint of cost function. Also, the range of learning rate [math] can be extended from [math] to [math] for [math] and [math] for [math], where [math] is the Lipschitz continuity constant of the gradient of cost function. In a further research, we show the linear convergence of cost function under the Polyak–Łojasiewicz[math](PL) condition, for which the available choice of learning rate is further improved as [math] for the large delay [math]. The framework of the proof for this result is also extended to the stochastic gradient descent with time-varying delay under the PL condition. Finally, some numerical experiments are provided in order to confirm the reliability of the analyzed results. Citation: Analysis and Applications PubDate: 2024-03-11T07:00:00Z DOI: 10.1142/S0219530524500106

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Authors:Hans-Peter Beise, Steve Dias Da Cruz Pages: 1 - 16 Abstract: Analysis and Applications, Ahead of Print. In [A. Radhakrishnan, M. Belkin and C. Uhler, Overparameterized neural networks implement associative memory, Proc. Natl. Acad. Sci. USA 117(44) (2020) 27162–27170], the authors empirically show that autoencoders trained with standard SGD methods form basins of attraction around their training data. We consider network functions of width not exceeding the input dimension and prove that in this situation, such basins of attraction are bounded and their complement cannot have bounded components. Our conditions in these results are met in several experiments reported in [A. Radhakrishnan, M. Belkin and C. Uhler, Overparameterized neural networks implement associative memory, Proc. Natl. Acad. Sci. USA 117(44) (2020) 27162–27170] and we thus address a question posed therein. We also show that under some more restrictive conditions, the basins of attraction are path-connected. The necessity of the conditions in our results is demonstrated by means of examples. Citation: Analysis and Applications PubDate: 2024-02-29T08:00:00Z DOI: 10.1142/S0219530524500076

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Authors:David Cruz-Uribe, Michael Penrod Pages: 1 - 25 Abstract: Analysis and Applications, Ahead of Print. We extend the theory of matrix weights to the variable Lebesgue spaces. The theory of matrix [math] weights has attracted considerable attention beginning with the work of Nazarov, Treil, and Volberg in the 1990s. We extend this theory by generalizing the matrix [math] condition to the variable exponent setting. We prove boundedness of the convolution operator [math] for [math], and show that the approximate identity defined using [math] converges in matrix weighted, variable Lebesgue spaces [math] for [math] in matrix [math]. Our approach to this problem is through averaging operators; these results are of interest in their own right. As an application of our work, we prove a version of the classical [math] theorem for matrix weighted, variable exponent Sobolev spaces. Citation: Analysis and Applications PubDate: 2024-02-09T08:00:00Z DOI: 10.1142/S0219530524500027