Authors:Patrick L. Combettes, Saverio Salzo, Silvia Villa Pages: 1 - 54 Abstract: Analysis and Applications, Volume 16, Issue 01, Page 1-54, January 2018. This paper proposes a unified framework for the investigation of constrained learning theory in reflexive Banach spaces of features via regularized empirical risk minimization. The focus is placed on Tikhonov-like regularization with totally convex functions. This broad class of regularizers provides a flexible model for various priors on the features, including, in particular, hard constraints and powers of Banach norms. In such context, the main results establish a new general form of the representer theorem and the consistency of the corresponding learning schemes under general conditions on the loss function, the geometry of the feature space, and the modulus of total convexity of the regularizer. In addition, the proposed analysis gives new insight into basic tools such as reproducing Banach spaces, feature maps, and universality. Even when specialized to Hilbert spaces, this framework yields new results that extend the state of the art. Citation: Analysis and Applications PubDate: 2017-10-26T08:57:11Z DOI: 10.1142/S0219530516500202

Authors:Xiaofeng Hou, Hongyun Peng, Changjiang Zhu Pages: 55 - 84 Abstract: Analysis and Applications, Volume 16, Issue 01, Page 55-84, January 2018. In this paper, we investigate the global well-posedness of classical solutions to three-dimensional Cauchy problem of the compressible Navier–Stokes type system with a Korteweg stress tensor under the condition that the initial energy is small. This result improves previous results obtained by Hattori–Li in [H. Hattori and D. Li, Solutions for two dimensional system for materials of Korteweg type, SIAM J. Math. Anal. 25 (1994) 85–98; H. Hattori and D. Li. Global solutions of a high-dimensional system for Korteweg materials. J. Math. Anal. Appl. 198 (1996) 84–97.], where the existence of the classical solution is established for initial data close to an equilibrium in some Sobolev space [math]. Citation: Analysis and Applications PubDate: 2017-10-26T08:57:14Z DOI: 10.1142/S0219530516500123

Authors:Feng Xie, Christian Klingenberg Pages: 85 - 102 Abstract: Analysis and Applications, Volume 16, Issue 01, Page 85-102, January 2018. General radiation magnetic hydrodynamics models include two main parts that are coupled: one part is the macroscopic magnetic fluid part, which is governed by the ideal compressible magnetohydrodynamic (MHD) equations with additional radiation terms; another part is the radiation field, which is described by a transfer equation. It is well known that in radiation hydrodynamics without a magnetic field there are two physical approximations: one is the so-called P1 approximation and the other is the so-called gray approximation. Starting out with a general radiation MHD model one can derive the so-called MHD-P1 approximation model. In this paper, we study the non-relativistic type limit for this MHD-P1 approximation model since the speed of light is much larger than the speed of the macroscopic fluid. This way we achieve a rigorous derivation of a widely used macroscopic model in radiation magnetohydrodynamics. Citation: Analysis and Applications PubDate: 2017-10-26T08:57:08Z DOI: 10.1142/S0219530516500238

Authors:Xiaoyi Chen, Zilong Song, Hui-Hui Dai Pages: 103 - 132 Abstract: Analysis and Applications, Volume 16, Issue 01, Page 103-132, January 2018. This paper studies the planar deformations of a beam composed of a linearly elastic material. Starting from the field equations for the plane-stress problem and adopting a series expansion for the displacement vector about the bottom surface, we deduce the beam equations with two unknowns in a consistent manner. The success relies on using the field equations together with the bottom traction conditions to establish the exact recursion relations, such that all quantities can be represented in terms of the two leading expansion coefficients of the displacements. Another feature is that the remainders of the series can be carried over to the beam equations. Then, based on the general solutions and the error terms of the beam equations, pointwise error estimates for displacement and stress fields are rigorously established. Three benchmark problems are considered, for which the two-dimensional exact solutions are available. It is shown that this new beam theory recovers the exact solutions for these problems. Two cases with boundary layer effects are also discussed in the appendix. Citation: Analysis and Applications PubDate: 2017-10-26T08:57:10Z DOI: 10.1142/S0219530516500135

Authors:Youming Liu, Xiaochen Zeng Pages: 1 - 26 Abstract: Analysis and Applications, Ahead of Print. Using compactly supported wavelets, Giné and Nickl [Uniform limit theorems for wavelet density estimators, Ann. Probab. 37(4) (2009) 1605–1646] obtain the optimal strong [math] convergence rates of wavelet estimators for a fixed noise-free density function. They also study the same problem by spline wavelets [Adaptive estimation of a distribution function and its density in sup-norm loss by wavelet and spline projections, Bernoulli 16(4) (2010) 1137–1163]. This paper considers the strong [math] convergence of wavelet deconvolution density estimators. We first show the strong [math] consistency of our wavelet estimator, when the Fourier transform of the noise density has no zeros. Then strong [math] convergence rates are provided, when the noises are severely and moderately ill-posed. In particular, for moderately ill-posed noises and [math], our convergence rate is close to Giné and Nickl’s. Citation: Analysis and Applications PubDate: 2017-10-19T09:01:10Z DOI: 10.1142/S0219530517500154

Authors:Sibei Yang, Der-Chen Chang, Dachun Yang, Zunwei Fu Pages: 1 - 23 Abstract: Analysis and Applications, Ahead of Print. In this paper, by applying the well-known method for dealing with [math]-Laplace type elliptic boundary value problems, the authors establish a sharp estimate for the decreasing rearrangement of the gradient of solutions to the Dirichlet and the Neumann boundary value problems of a class of Schrödinger equations, under the weak regularity assumption on the boundary of domains. As applications, the gradient estimates of these solutions in Lebesgue spaces and Lorentz spaces are obtained. Citation: Analysis and Applications PubDate: 2017-08-15T02:51:20Z DOI: 10.1142/S0219530517500142

Authors:Seung-Yeal Ha, Hwa Kil Kim, Jinyeong Park Pages: 1 - 39 Abstract: Analysis and Applications, Ahead of Print. The synchronous dynamics of many limit-cycle oscillators can be described by phase models. The Kuramoto model serves as a prototype model for phase synchronization and has been extensively studied in the last 40 years. In this paper, we deal with the complete synchronization problem of the Kuramoto model with frustrations on a complete graph. We study the robustness of complete synchronization with respect to the network structure and the interaction frustrations, and provide sufficient frameworks leading to the complete synchronization, in which all frequency differences of oscillators tend to zero asymptotically. For a uniform frustration and unit capacity, we extend the applicable range of initial configurations for the complete synchronization to be distributed on larger arcs than a half circle by analyzing the detailed dynamics of the order parameters. This improves the earlier results [S.-Y. Ha, H. Kim and J. Park, Remarks on the complete frequency synchronization of Kuramoto oscillators, Nonlinearity, 28 (2015) 1441–1462; Z. Li and S.-Y. Ha, Uniqueness and well-ordering of emergent phase-locked states for the Kuramoto model with frustration and inertia, Math. Models Methods Appl. Sci. 26 (2016) 357–382.] which can be applicable only for initial configurations confined in a half circle. Citation: Analysis and Applications PubDate: 2017-07-05T07:46:04Z DOI: 10.1142/S0219530517500130

Authors:Mourad E. H. Ismail, Ruiming Zhang Pages: 1 - 73 Abstract: Analysis and Applications, Ahead of Print. By applying an integral representation for [math], we systematically derive a large number of new Fourier and Mellin transform pairs and establish new integral representations for a variety of [math]-functions and polynomials that naturally arise from combinatorics, analysis, and orthogonal polynomials corresponding to indeterminate moment problems. These functions include [math]-Bessel functions, the Ramanujan function, Stieltjes–Wigert polynomials, [math]-Hermite and [math]-Hermite polynomials, and the [math]-exponential functions [math], [math] and [math]. Their representations are in turn used to derive many new identities involving [math]-functions and polynomials. In this paper, we also present contour integral representations for the above mentioned functions and polynomials. Citation: Analysis and Applications PubDate: 2017-07-05T07:46:03Z DOI: 10.1142/S0219530517500129

Authors:Hai-Yang Jin, Zhi-An Wang Pages: 1 - 32 Abstract: Analysis and Applications, Ahead of Print. In this paper, we consider the following dual-gradient chemotaxis model ut = Δu −∇⋅ (χu∇v) + ∇⋅ (ξf(u)∇w),x ∈ Ω,t > 0, τ1vt = Δv + αu − βv, x ∈ Ω,t > 0, τ2wt = Δw + γu − δw, x ∈ Ω,t > 0, with [math] for [math] and [math] for [math], where [math] is a bounded domain in [math] with smooth boundary, [math] and [math]. The model was proposed to interpret the spontaneous aggregation of microglia in Alzheimer’s disease due to the interaction of attractive and repulsive chemicals released by the microglia. It has been shown in the literature that, when [math], the solution of the model with homogeneous Neumann boundary conditions either blows up or asymptotically decays to a constant in multi-dimensions depending on the sign of [math], which means there is no pattern formation. In this paper, we shall show as [math], the uniformly-in-time bounded global classical solutions exist in multi-dimensions and hence pattern formation can develop. This is significantly different from the results for the case [math]. We perform the numerical simulations to illustrate the various patterns generated by the model, verify our analytical results and predict some unsolved questions. Biological applications of our results are discussed and open problems are presented. Citation: Analysis and Applications PubDate: 2017-04-11T11:48:28Z DOI: 10.1142/S0219530517500087

Authors:Blanca Bujanda, José L. López, Pedro J. Pagola Pages: 1 - 14 Abstract: Analysis and Applications, Ahead of Print. We consider the incomplete gamma function [math] for [math] and [math]. We derive several convergent expansions of [math] in terms of exponentials and rational functions of [math] that hold uniformly in [math] with [math] bounded from below. These expansions, multiplied by [math], are expansions of [math] uniformly convergent in [math] with [math] bounded from above. The expansions are accompanied by realistic error bounds. Citation: Analysis and Applications PubDate: 2017-04-11T11:48:27Z DOI: 10.1142/S0219530517500099

Authors:Philippe G. Ciarlet, Cristinel Mardare Pages: 1 - 20 Abstract: Analysis and Applications, Ahead of Print. We recast the displacement-traction problem of the Kirchhoff–Love theory of linearly elastic plates as a boundary value problem with the bending moments and stress resultants inside the middle section of the plate as the sole unknowns, instead of the displacement field in the classical formulation. To this end, we show in particular how to recast the Dirichlet boundary conditions satisfied by the displacement field of the middle surface of a plate as boundary conditions satisfied by the bending moments and stress resultants. Citation: Analysis and Applications PubDate: 2017-04-11T11:48:27Z DOI: 10.1142/S0219530517500105

Authors:Chundi Liu, Boyi Wang Pages: 1 - 23 Abstract: Analysis and Applications, Ahead of Print. Quasineutral limit for a model of three-dimensional Euler–Poisson system in half space with a boundary layer is studied. Based on the matched asymptotic expansion method of singular perturbation problem and the elaborate energy method, we prove that the quasineutral regime is the incompressible Euler equation. Citation: Analysis and Applications PubDate: 2017-01-23T07:19:36Z DOI: 10.1142/S0219530517500051

Authors:Alberto Bressan, Yilun Jiang Pages: 1 - 25 Abstract: Analysis and Applications, Ahead of Print. The paper studies optimal strategies for a borrower who needs to repay his debt, in an infinite time horizon. An instantaneous bankruptcy risk is present, which increases with the size of the debt. This induces a pool of risk-neutral lenders to charge a higher interest rate, to compensate for the possible loss of part of their investment. Solutions are interpreted as Stackelberg equilibria, where the borrower announces his repayment strategy [math] at all future times, and lenders adjust the interest rate accordingly. This yields a highly non-standard problem of optimal control, where the instantaneous dynamics depend on the entire future evolution of the system. Our analysis shows the existence of optimal open-loop controls, deriving necessary conditions for optimality and characterizing possible asymptotic limits as [math]. Citation: Analysis and Applications PubDate: 2017-01-23T07:19:36Z DOI: 10.1142/S0219530517500038

Authors:Xiaoping Zhai, Yongsheng Li, Wei Yan Pages: 1 - 43 Abstract: Analysis and Applications, Ahead of Print. In this paper, we investigate the Cauchy problem for the 3D viscous nonhomogeneous incompressible magnetohydrodynamic equations in critical Besov spaces. We aim at proving the local and global well-posedness for respectively large and small initial data having critical Besov regularity, without assumptions of small density variation. Citation: Analysis and Applications PubDate: 2017-01-23T07:19:36Z DOI: 10.1142/S0219530517500014

Authors:Peter Greiner, Yutian Li Pages: 1 - 27 Abstract: Analysis and Applications, Ahead of Print. Let [math] denote the holomorphic tangential vector field to the generalized upper-half plane [math]. In our terminology, [math]. Consider the [math] operator on the boundary of [math], [math]; note that [math] is nowhere elliptic, but it is subelliptic with step three. The principal result of this paper is the derivation of an explicit fundamental solution [math] to [math]. Our approach is based on special functions and their properties. Citation: Analysis and Applications PubDate: 2016-10-19T10:07:24Z DOI: 10.1142/S0219530516500196

Authors:Chunxia Guan, Kai Yan, Xuemei Wei Pages: 1 - 24 Abstract: Analysis and Applications, Ahead of Print. This paper is devoted to the existence and Lipschitz continuity of global conservative weak solutions in time for the modified two-component Camassa–Holm system on the real line. We obtain the global weak solutions via a coordinate transformation into the Lagrangian coordinates. The key ingredients in our analysis are the energy density given by the positive Radon measure and the proposed new distance functions as well. Citation: Analysis and Applications PubDate: 2016-10-19T10:07:15Z DOI: 10.1142/S0219530516500226