Abstract: We investigate the relation between broken time-reversal symmetry and localization of the electronic states, in the explicitly tractable case of the Landau model. We first review, for the reader’s convenience, the symmetries of the Landau Hamiltonian and the relation of the latter with the Segal-Bargmann representation of Quantum Mechanics. We then study the localization properties of the Landau eigenstates by applying an abstract version of the Balian-Low Theorem to the operators corresponding to the coordinates of the centre of the cyclotron orbit in the classical theory. Our proof of the Balian-Low Theorem, although based on Battle’s main argument, has the advantage of being representation-independent. PubDate: 2019-03-18

Abstract: In this paper, we introduce iterative regularization methods for solving the multiple-sets split feasibility problem, that is to find a point closest to a family of closed convex subsets in one space such that its image under a bounded linear mapping will be closest to another family of closed convex subsets in the image space. We consider the cases, when the families are either finite or infinite. We also give two numerical examples for illustrating our main method. PubDate: 2019-03-07

Abstract: In this paper, we show that bounded weak solutions of the Cauchy problem for general degenerate parabolic equations of the form $$ u_{t} + \operatorname{div}f(x,t,u) = \operatorname{div}\bigl( u ^{\alpha } \nabla u\bigr), \quad x \in \mathbb{R}^{n} , \ t > 0, $$ where \(\alpha > 0 \) is constant, decrease to zero, under fairly broad conditions on the advection flux \(f\) . Besides that, we derive a time decay rate for these solutions. PubDate: 2019-03-05

Abstract: This paper is devoted to identify a space-dependent source term in a multi-dimensional time fractional diffusion-wave equation from a part of noisy boundary data. Based on the series expression of solution for the direct problem, we improve the regularity of the weak solution for the direct problem under strong conditions. And we obtain the uniqueness of inverse space-dependent source term problem by the Titchmarsh convolution theorem and the Duhamel principle. Further, we use a non-stationary iterative Tikhonov regularization method combined with a finite dimensional approximation to find a stable source term. Numerical examples are provided to show the effectiveness of the proposed method. PubDate: 2019-03-05

Abstract: The Kaczmarz algorithm is an iterative method for solving a system of linear equations. It can be extended so as to reconstruct a vector \(x\) in a (separable) Hilbert space from the inner-products \(\{\langle x, \phi _{n} \rangle \}\) . The Kaczmarz algorithm defines a sequence of approximations from the sequence \(\{\langle x, \phi _{n} \rangle \}\) ; these approximations only converge to \(x\) when \(\{\phi _{n}\}\) is effective. We dualize the Kaczmarz algorithm so that \(x\) can be obtained from \(\{\langle x, \phi _{n} \rangle \}\) by using a second sequence \(\{\psi _{n}\}\) in the reconstruction. This allows for the recovery of \(x\) even when the sequence \(\{\phi _{n}\}\) is not effective; in particular, our dualization yields a reconstruction when the sequence \(\{\phi _{n}\}\) is almost effective. We also obtain some partial results characterizing when the sequence of approximations from \(\{\langle x, \phi _{n} \rangle \}\) using \(\{\psi _{n}\}\) converges to \(x\) , in which case \(\{(\phi _{n}, \psi _{n})\}\) is called an effective pair. PubDate: 2019-02-14

Abstract: We discuss a parametric eigenvalue problem, where the differential operator is of \((p,2)\) -Laplacian type. We show that, when \(p\neq 2\) , the spectrum of the operator is a half line, with the end point formulated in terms of the parameter and the principal eigenvalue of the Laplacian with zero Dirichlet boundary conditions. Two cases are considered corresponding to \(p>2\) and \(p<2\) , and the methods that are applied are variational. In the former case, the direct method is applied, whereas in the latter case, the fibering method of Pohozaev is used. We will also discuss a priori bounds and regularity of the eigenfunctions. In particular, we will show that, when the eigenvalue tends towards the end point of the half line, the supremum norm of the corresponding eigenfunction tends to zero in the case of \(p>2\) , and to infinity in the case of \(p < 2\) . PubDate: 2019-02-07

Abstract: In this paper we first establish a decomposition theorem for size-biased Poisson random measures. As consequences of this decomposition theorem, we get a spine decomposition theorem and a 2-spine decomposition theorem for some critical superprocesses. Then we use these spine decomposition theorems to give probabilistic proofs of the asymptotic behavior of the survival probability and Yaglom’s exponential limit law for critical superprocesses. PubDate: 2019-02-07

Abstract: In order to solve global minimization problems involving best proximity points, we introduce general Mann algorithm for nonself nonexpansive mappings and then prove weak and strong convergence of the proposed algorithm under some suitable conditions in real Hilbert spaces. Furthermore, we also provide numerical experiment to illustrate the convergence behavior of our proposed algorithm. PubDate: 2019-02-06

Abstract: This paper is devoted to the study of the \(p\) -fractional Schrödinger–Kirchhoff equations with electromagnetic fields and critical nonlinearity. By using the variational methods, we obtain the existence of mountain pass solutions \(u_{\varepsilon }\) which tend to the trivial solutions as \(\varepsilon \rightarrow 0\) . Moreover, we get \(m^{\ast }\) pairs of solutions for the problem in absence of magnetic effects under some extra assumptions. PubDate: 2019-02-04

Abstract: Fractal interpolation and approximation received a lot of attention in the last thirty years. The main aim of the current article is to study a fractal trigonometric approximants which converge to the given continuous function even if the magnitude of the scaling factors does not approach zero. In this paper, we first introduce a new class of fractal approximants, namely, Bernstein \(\alpha \) -fractal functions using the theory of fractal approximation and Bernstein polynomial. Using the proposed class of fractal approximants and imposing no condition on corresponding scaling factors, we establish that the set of Bernstein \(\alpha \) -fractal trigonometric functions is fundamental in the space of continuous periodic functions. Fractal version of Gauss formula of trigonometric interpolation is obtained by means of Bernstein trigonometric fractal polynomials. We study the Bernstein fractal Fourier series of a continuous periodic function \(f\) defined on \([-l,l]\) . The Bernstein fractal Fourier series converges to \(f\) even if the magnitude of the scaling factors does not approach zero. Existence of the \(\mathcal{C}^{r}\) -Bernstein fractal functions is investigated, and Bernstein cubic spline fractal interpolation functions are proposed based on the theory of \(\mathcal{C}^{r}\) -Bernstein fractal functions. PubDate: 2019-02-01

Abstract: In this research we give the existence of solutions to a elliptic problem containing two lower order terms, the first nonlinear term satisfying the growth conditions and without sign conditions and the second is a continuous function on ℝ. Not also that for right hand side, it is assumed that to be merely integrable. This results in formulation of the problem in Musielak-Orlicz-Sobolev spaces. PubDate: 2019-02-01

Abstract: We study a model of three interacting species in a food chain composed by a prey, an specific predator and a generalist predator. The capture of the prey by the specific predator is modelled as a modified Holling-type II non-differentiable functional response. The other predatory interactions are both modelled as Holling-type I. Moreover, our model follows a Leslie-Gower approach, in which the function that models the growth of each predator is of logistic type, and the corresponding carrying capacities depend on the sizes of their associated available preys. The resulting model has the form of a set of nonlinear ordinary differential equations which includes a non-differentiable term. By means of topological equivalences and suitable changes of parameters, we find that there exists an Allee threshold for the survival of the prey population in the food chain, given, effectively, as a critical level for the generalist predator. The dynamics of the model is studied with analytical and computational tools for bifurcation theory. We present two-parameter bifurcation diagrams that contain both local phenomena (Hopf, saddle-node transcritical, cusp, Bogdanov-Takens bifurcations) and global events (homoclinic and heteroclinic connections). In particular, we find that two types of heteroclinic cycles can be formed, both of them containing connections to the origin. One of these cycles is planar involving the absence of the specific predator. In turn, the other heteroclinic cycle is formed by connections in the full three-dimensional phase space. PubDate: 2019-01-24

Abstract: In this paper, we consider a class of non-cooperative elliptic systems of Kirchhoff type involving \(p\) -biharmonic operator and critical growth. With the help of the Limit index theory due to Li (Nonlinear Anal. TMA 30(7):4619–4627, 1997) and the concentration compactness principle, we establish the existence of infinitely many solutions for the problem under the suitable conditions on the nonlinearity. PubDate: 2019-01-23

Authors:Sergey Bezuglyi; Palle E. T. Jorgensen Abstract: In our study of electrical networks we develop two themes: finding explicit formulas for special classes of functions defined on the vertices of a transient network, namely monopoles, dipoles, and harmonic functions. Secondly, our interest is focused on the properties of electrical networks supported on Bratteli diagrams. We show that the structure of Bratteli diagrams allows one to describe algorithmically harmonic functions as well as monopoles and dipoles. We also discuss some special classes of Bratteli diagrams (stationary, Pascal, trees), and we give conditions under which the harmonic functions defined on these diagrams have finite energy. PubDate: 2018-05-31 DOI: 10.1007/s10440-018-0189-7

Authors:Zhen-Qing Chen; Yan-Xia Ren; Ting Yang Abstract: The goal of this paper is twofold. First, we establish skeleton and spine decompositions for superprocesses whose underlying processes are general symmetric Hunt processes. Second, we use these decompositions to obtain weak and strong law of large numbers for supercritical superprocesses where the spatial motion is a symmetric Hunt process on a locally compact metric space \(E\) and the branching mechanism takes the form $$ \psi _{\beta }(x,\lambda )=-\beta (x)\lambda +\alpha (x)\lambda ^{2}+ \int _{(0,{\infty })}\bigl(e^{-\lambda y}-1+\lambda y\bigr)\pi (x,dy) $$ with \(\beta \in \mathcal{B}_{b}(E)\) , \(\alpha \in \mathcal{B}^{+}_{b}(E)\) and \(\pi \) being a kernel from \(E\) to \((0,{\infty })\) satisfying \(\sup_{x\in E}\int _{(0,{\infty })} (y\wedge y^{2}) \pi (x,dy)<{\infty }\) . The limit theorems are established under the assumption that an associated Schrödinger operator has a spectral gap. Our results cover many interesting examples of superprocesses, including super Ornstein-Uhlenbeck process and super stable-like process. The strong law of large numbers for supercritical superprocesses are obtained under the assumption that the strong law of large numbers for an associated supercritical branching Markov process holds along a discrete sequence of times, extending an earlier result of Eckhoff et al. (Ann. Probab. 43(5):2594–2659, 2015) for superdiffusions to a large class of superprocesses. The key for such a result is due to the skeleton decomposition of superprocess, which represents a superprocess as an immigration process along a supercritical branching Markov process. PubDate: 2018-05-29 DOI: 10.1007/s10440-018-0190-1

Authors:Yaling Zhao; Zuhan Liu; Ling Zhou Abstract: In this paper we focus on a nonlocal reaction-diffusion population model. Such a model can be used to describe a single species which is diffusing, aggregating, reproducing and competing for space and resources, with the free boundary representing the expanding front. The main objective is to understand the influence of the nonlocal term in the form of an integral convolution on the dynamics of the species. Precisely, when the species successfully spreads into infinity as \(t\rightarrow \infty \) , it is proved that the species stabilizes at a positive equilibrium state under rather mild conditions. Furthermore, we obtain a upper bound for the spreading of the expanding front. PubDate: 2018-05-22 DOI: 10.1007/s10440-018-0188-8

Authors:Feliz Minhós; Robert de Sousa Abstract: This work considers two types of a second order impulsive coupled system of differential equations with generalized jump conditions in half-line. For both problems it will be presented two localization results with different monotonicity assumptions on the nonlinearities and on the impulsive conditions. The arguments apply lower and upper solutions method combined with Nagumo type condition and truncature techniques. Last section contains an applications of one of this impulsive problems to logging timber by helicopter. PubDate: 2018-05-08 DOI: 10.1007/s10440-018-0187-9

Authors:Kwok-Pun Ho Abstract: We show that when the infimum of the exponent function equals to 1, the fractional integral operator is a bounded operator from the Morrey space with variable exponent to the weak Morrey space with variable exponent. PubDate: 2018-04-26 DOI: 10.1007/s10440-018-0181-2

Authors:X. Blasco; G. Reynoso-Meza; E. A. Sánchez-Pérez; J. V. Sánchez-Pérez Abstract: Given a finite dimensional asymmetric normed lattice, we provide explicit formulae for the optimization of the associated (non-Hausdorff) asymmetric “distance” among a subset and a point. Our analysis has its roots and finds its applications in the current development of effective algorithms for multi-objective optimization programs. We are interested in providing the fundamental theoretical results for the associated convex analysis, fixing in this way the framework for this new optimization tool. The fact that the associated topology is not Hausdorff forces us to define a new setting and to use a new point of view for this analysis. Existence and uniqueness theorems for this optimization are shown. Our main result is the translation of the original abstract optimal distance problem to a clear optimization scheme. Actually, this justifies the algorithms and shows new aspects of the numerical and computational methods that have been already used in visualization of multi-objective optimization problems. PubDate: 2018-04-26 DOI: 10.1007/s10440-018-0184-z

Authors:Rasul Ganikhodzhaev; Farrukh Mukhamedov; Mansoor Saburov Abstract: The paper is devoted to the study of elliptic quadratic operator equations over the finite dimensional Euclidean space. We provide necessary and sufficient conditions for the existence of solutions of elliptic quadratic operator equations. The iterative Newton-Kantorovich method for stable solutions is also presented. PubDate: 2018-04-26 DOI: 10.1007/s10440-018-0183-0