Showing papers in "Taiwanese Journal of Mathematics in 2017"
TL;DR: In this paper, the authors introduced the variable anisotropic Hardy space and established its radial or non-tangential maximal function characterizations, respectively, by means of atoms, finite atoms, the Lusin area function, the Littlewood-Paley $g$-function or $g
Abstract: Let $p(\cdot) \colon \mathbb{R}^n \to (0,\infty]$ be a variable exponent function satisfying the globally log-Holder continuous condition and $A$ a general expansive matrix on $\mathbb{R}^n$. In this article, the authors first introduce the variable anisotropic Hardy space $H_A^{p(\cdot)}(\mathbb{R}^n)$ associated with $A$, via the non-tangential grand maximal function, and then establish its radial or non-tangential maximal function characterizations. Moreover, the authors also obtain various equivalent characterizations of $H_A^{p(\cdot)}(\mathbb{R}^n)$, respectively, by means of atoms, finite atoms, the Lusin area function, the Littlewood-Paley $g$-function or $g_{\lambda}^{\ast}$-function. As applications, the authors first establish a criterion on the boundedness of sublinear operators from $H^{p(\cdot)}_A(\mathbb{R}^n)$ into a quasi-Banach space. Then, applying this criterion, the authors show that the maximal operators of the Bochner-Riesz and the Weierstrass means are bounded from $H^{p(\cdot)}_A(\mathbb{R}^n)$ to $L^{p(\cdot)}(\mathbb{R}^n)$ and, as consequences, some almost everywhere and norm convergences of these Bochner-Riesz and Weierstrass means are also obtained. These results on the Bochner-Riesz and the Weierstrass means are new even in the isotropic case.
43 citations
TL;DR: In this article, the induced Toeplitz operator of the positive Borel measure on the complex plane was characterized for a weighted Fock space on which the induced TOEplitz is bounded.
Abstract: Let $F^{p}(\phi)$ be the weighted Fock space on the complex plane $\mathbb{C}$, where $\phi$ is subharmonic with $\Delta \phi \, dA$ a doubling measure. In this paper, we characterize the positive Borel measure $\mu$ on $\mathbb{C}$ for which the induced Toeplitz operator $T_\mu$ is bounded (or compact) from one weighted Fock space $F^{p}(\phi)$ to another $F^{q}(\phi)$ for $0 < p, q < \infty$.
21 citations
TL;DR: In this article, the following Schrodinger-Poisson system was studied and the following conditions were established: Δ u + V(x,u + K(x) \phi(x), x t > 0, \; \tau
eq 0\] with constant $\theta_0 \in (0,1)$ instead of the usual Nehari-type monotonic condition on $f(x,t)/|t|^3$.
Abstract: This paper is dedicated to studying the following Schrodinger-Poisson system\[ \begin{cases} -\Delta u + V(x)u + K(x) \phi(x)u = f(x,u), x t > 0, \; \tau
eq 0\]with constant $\theta_0 \in (0,1)$, instead of $\lim_{|t| \to \infty} \left( \int_0^t f(x,s) \, \mathrm{d}s \right)/|t|^4 = \infty$ uniformly in $x \in \mathbb{R}^3$ and the usual Nehari-type monotonic condition on $f(x,t)/|t|^3$.
18 citations
TL;DR: In this paper, a fully discrete spectral scheme was proposed on the basis of finite difference discretization in time and Legendre spectral approximation in space, and the stability and convergence of the fully discrete scheme were rigorously established.
Abstract: The numerical approximation of the nonlinear time fractional Klein-Gordon equation in a bounded domain is considered. The time fractional derivative is described in the Caputo sense with the order $\gamma$ ($1 < \gamma < 2$). A fully discrete spectral scheme is proposed on the basis of finite difference discretization in time and Legendre spectral approximation in space. The stability and convergence of the fully discrete scheme are rigorously established. The convergence rate of the fully discrete scheme in $H^1$ norm is $\mathrm{O}(\tau^{3-\gamma} + N^{1-m})$, where $\tau$, $N$ and $m$ are the time-step size, polynomial degree and regularity in the space variable of the exact solution, respectively. Numerical examples are presented to support the theoretical results.
17 citations
TL;DR: In this paper, a method for solving some well-known classes of fractional Lane-Emden type equations which are nonlinear ordinary differential equations on the semi-infinite domain is proposed by utilizing Haar wavelets in conjunction with Adomian's decomposition method.
Abstract: In this paper, we propose a method for solving some well-known classes of fractional Lane-Emden type equations which are nonlinear ordinary differential equations on the semi-infinite domain. The method is proposed by utilizing Haar wavelets in conjunction with Adomian's decomposition method. The operational matrices for the Haar wavelets are derived and constructed. Procedure of implementation and convergence analysis of the method are presented. The method is tested on the fractional standard Lane-Emden equation and the fractional isothermal gas spheres equation. We compare the results produce by present method with some well-known results to show the accuracy and applicability of the method.
17 citations
Abstract: By using variational approaches, we study a class of quasilinear Schrodinger equations involving critical Sobolev exponents \[ -\Delta u + V(x)u + \frac{1}{2} \kappa [\Delta(u^2)]u = |u|^{p-2}u + |u|^{2^*-2}u, \quad x \in \mathbb{R}^N, \] where $V(x)$ is the potential function, $\kappa \gt 0$, $\max \{ (N+3)/(N-2),2 \} \lt p \lt 2^* := 2N/(N-2)$, $N \geq 4$. If $\kappa \in [0,\overline{\kappa})$ for some $\overline{\kappa} \gt 0$, we prove the existence of a positive solution $u(x)$ satisfying $\max_{x \in \mathbb{R}^N} |u(x)| \leq \sqrt{1/(2\kappa)}$.
16 citations
TL;DR: In this article, the generalized normalized Casorati curvatures for real hypersurfaces of complex two-plane Grassmannians and complex hyperbolic two-planar Grassmannian were obtained.
Abstract: In this paper we obtain two types of optimal inequalities consisting of the normalized scalar curvature and the generalized normalized $\delta$-Casorati curvatures for real hypersurfaces of complex two-plane Grassmannians and complex hyperbolic two-plane Grassmannians. We also find the conditions on which the equalities hold.
16 citations
TL;DR: In this article, the existence and uniqueness theorem of mild solutions for the associated linear evolution equation with non-instantaneous impulses is established by using perturbation technique and iterative method.
Abstract: In this paper, we are concerned with the existence of mild solutions for the initial value problem to a new class of abstract evolution equations with non-instantaneous impulses on ordered Banach spaces. The existence and uniqueness theorem of mild solution for the associated linear evolution equation with non-instantaneous impulses is established. With the aid of this theorem, the existence of mild solutions for nonlinear evolution equation with non-instantaneous impulses is obtained by using perturbation technique and iterative method under the situation that the corresponding solution semigroup $T(\cdot)$ and non-instantaneous impulsive function $g_k$ are compact, $T(\cdot)$ is not compact and $g_k$ is compact, $T(\cdot)$ and $g_k$ are not compact, respectively. The results obtained in this paper essentially improve and extend some related conclusions on this topic. Two concrete examples to parabolic partial differential equations with non-instantaneous impulses are given to illustrate that our results are valuable.
15 citations
TL;DR: In this article, it was shown that Okounkov bodies of a pseudoeffective divisor which admits the birational good Zariski decomposition is a rational polytope with respect to some admissible flag.
Abstract: We first prove some basic properties of Okounkov bodies and give a characterization of Nakayama and positive volume subvarieties of a pseudoeffective divisor in terms of Okounkov bodies. Next, we show that each valuative and limiting Okounkov bodies of a pseudoeffective divisor which admits the birational good Zariski decomposition is a rational polytope with respect to some admissible flag. This is an extension of the result of Anderson-Kuronya-Lozovanu about the rational polyhedrality of Okounkov bodies of big divisors with finitely generated section rings.
14 citations
TL;DR: In this paper, the Calabi-Yau type structure of normal projective surfaces and Mori dream spaces admitting non-trivial polarized endomorphisms is discussed and discussed.
Abstract: We discuss the Calabi-Yau type structure of normal projective surfaces and Mori dream spaces admitting non-trivial polarized endomorphisms.
14 citations
TL;DR: In this paper, the authors considered the wave equation with variable coefficients and Balakrishnan-Taylor damping and derived the uniform decay rates of the energy without imposing any restrictive growth near zero assumption on the damping term.
Abstract: In this paper, we consider the wave equation with variable coefficients and Balakrishnan-Taylor damping and source terms. This work is devoted to prove, under suitable conditions on the initial data, the uniform decay rates of the energy without imposing any restrictive growth near zero assumption on the damping term.
TL;DR: In this paper, a family of new finite difference (NFD) methods for solving the convection-diffusion equation with singularly perturbed parameters are considered, which can achieve the predicted convergence orders on uniform mesh regardless of the perturbed parameter.
Abstract: In this paper, a family of new finite difference (NFD) methods for solving the convection-diffusion equation with singularly perturbed parameters are considered. By taking account of infinite terms in the Taylor's expansions and using the triangle function theorem, we construct a series of NFD schemes for the one-dimensional problems firstly and derive the error estimates as well. Then, applying the ADI technique, the idea is extended to two dimensional equations. Besides no numerical oscillation, there are mainly three advantages for the proposed methods: one is that the schemes can achieve the predicted convergence orders on uniform mesh regardless of the perturbed parameter for 1D equations; Secondly, no matter which convergence order the scheme is, the generated linear systems have diagonal structures; Thirdly, the methods are easily expanded to the special mesh technique such as Shishkin mesh. Some numerical experiments are shown to verify the prediction.
TL;DR: In this article, the authors considered the quantum Zakharov system in one spatial dimension and proved the global well-posedness of the system with Schrodinger data and some wave data.
Abstract: In this paper, we consider the quantum Zakharov system in one spatial dimension. We prove the global well-posedness of the system with $L^2$-Schrodinger data and some wave data. The regularity of the wave data is in the largest set. We give counterexamples for the boundary of the set. As the quantum parameter tends to zero, we formally recover the result of Colliander-Holmer-Tzirakis for the classical Zakharov system.
TL;DR: In this article, the authors considered the blow-up problem for a nonlocal reaction-diffusion equation with null Robin boundary conditions and established sufficient conditions to guarantee the solution exists globally or blows up at finite time under appropriate measure sense.
Abstract: This work is concerned with the blow-up phenomena for a nonlocal reaction-diffusion equation with null Robin boundary conditions. We establish sufficient conditions to guarantee the solution exists globally or blows up at finite time under appropriate measure sense. Moreover, upper and lower bounds for the blow-up time are derived in higher dimensional spaces. Finally, some application examples are presented.
TL;DR: In this article, all the solutions of the Diophantine equation in nonnegative integer variables (n,m,\ell,a) were found in the nonnegative setting.
Abstract: In this paper, we find all the solutions of the Diophantine equation $P_\ell + P_m + P_n = 2^a$, in nonnegative integer variables $(n,m,\ell,a)$ where $P_k$ is the $k$-th term of the Pell sequence $\{P_n\}_{n \geq 0}$ given by $P_0 = 0$, $P_1 = 1$ and $P_{n+1} = 2P_{n} + P_{n-1}$ for all $n \geq 1$.
TL;DR: In this paper, a linear viscoelastic wave equation with density and time delay in the whole space was considered, and a general decay result of solution for the initial value problem by using energy perturbation method was established.
Abstract: A linear viscoelastic wave equation with density and time delay in the whole space $\mathbb{R}^n$ ($n \geq 3$) is considered. In order to overcome the difficulties in the non-compactness of some operators, we introduce some weighted spaces. Under suitable assumptions on the relaxation function, we establish a general decay result of solution for the initial value problem by using energy perturbation method. Our result extends earlier results.
TL;DR: In this paper, it was shown that for any integer r ≥ 1, there exists an r-AP graph of order 4r− 1, where r is the number of vertices in the graph.
Abstract: The center and the periphery of a graph is the set of vertices with minimum resp. maximum eccentricity in it. A graph is almost self-centered (ASC) if it contains exactly two non-central vertices and is almost peripheral (AP) if all its vertices but one lie in the periphery. Answering a question from (Taiwanese J. Math. 18 (2014), 463–471) it is proved that for any integer r ≥ 1 there exists an r-AP graph of order 4r− 1. Using this result it is proved that any graph G can be embedded into an r-AP graph by adding at most 4r − 2 vertices to G. A construction of ASC graphs from (Acta Math. Sin. (Engl. Ser.) 27 (2011), 2343–2350) is corrected and refined. Two new constructions of ASC graphs are also presented. Strong product graphs that are AP graphs are also characterized and it is shown that there are no strong product graphs that are ASC graphs. We conclude with some related open problems.
TL;DR: In this paper, it was shown that there is a pair of smooth complex quartic K3 surfaces (S_1$ and S_2$ ) that are isomorphic as abstract varieties but not Cremona isomorphic.
Abstract: We show that there is a pair of smooth complex quartic K3 surfaces $S_1$ and $S_2$ in $\mathbb{P}^3$ such that $S_1$ and $S_2$ are isomorphic as abstract varieties but not Cremona isomorphic. We also show, in a geometrically explicit way, that there is a pair of smooth complex quartic K3 surfaces $S_1$ and $S_2$ in $\mathbb{P}^3$ such that $S_1$ and $S_2$ are Cremona isomorphic, but not projectively isomorphic. This work is much motivated by several e-mails from Professors Tuyen Truong and Janos Kollar.
TL;DR: In this paper, the authors complete Mori's program for Kontsevich's moduli space of degree $2$ stable maps to the Grassmannian of lines, and describe all birational models in terms of moduli spaces (of curves and sheaves), incidence varieties and Kirwan's partial desingularization.
Abstract: We complete Mori's program for Kontsevich's moduli space of degree $2$ stable maps to the Grassmannian of lines. We describe all birational models in terms of moduli spaces (of curves and sheaves), incidence varieties, and Kirwan's partial desingularization.
TL;DR: In this paper, the authors studied the optimal control problem of second order stochastic evolution hemivariational inequalities with Poisson jumps by virtue of cosine operator theory in the Hilbert space.
Abstract: The purpose of this article is to study the optimal control problem of second order stochastic evolution hemivariational inequalities with Poisson jumps by virtue of cosine operator theory in the Hilbert space. Initially, the sufficient conditions for existence of mild solution of the proposed system are verified by applying properties of Clarke's subdifferential operator and fixed point theorem in multivalued maps. Further, we formulated and proved the existence results for optimal control of the proposed system with corresponding cost function by using Balder theorem. Finally an example is provided to illustrate the main results.
TL;DR: In this paper, the authors consider the viscoelastic equation with variable coefficients and Balakrishnan-Taylor damping and source terms and prove the asymptotic stability without imposing any restrictive growth assumption on the damping term and weakening of the usual assumptions on the relaxation function.
Abstract: In this paper, we consider the viscoelastic equation with variable coefficients and Balakrishnan-Taylor damping and source terms. This work is devoted to prove, under suitable conditions on the initial data, the asymptotic stability without imposing any restrictive growth assumption on the damping term and weakening of the usual assumptions on the relaxation function.
TL;DR: In this paper, the authors show how approximation involving continued fractions combined with recent results of Kuronya and Lozovanu on Okounkov bodies of line bundles on surfaces lead to effective statements considerably restricting possible values of Seshadri constants.
Abstract: Starting with the pioneering work of Ein and Lazarsfeld [9] restrictions on values of Seshadri constants on algebraic surfaces have been studied by many authors [2,5,10,12,18,20,22,24]. In the present note we show how approximation involving continued fractions combined with recent results of Kuronya and Lozovanu on Okounkov bodies of line bundles on surfaces [13,14] lead to effective statements considerably restricting possible values of Seshadri constants. These results in turn provide strong additional evidence to a conjecture governing the Seshadri constants on algebraic surfaces with Picard number $1$.
TL;DR: In this article, the authors obtained new explicit criteria for uniform asymptotic stability of non-convolutional non-convex integro-differential equations of the Volterra type.
Abstract: Using a novel approach, we get some new explicit criteria for uniform asymptotic stability and exponential asymptotic stability of linear Volterra time-varying integro-differential equations of non-convolution type. Some examples are given to illustrate the obtained results.
TL;DR: In this paper, the authors give a characterization of AU-rigid modules over Auslander algebras in terms of projective dimension of modules, and show that for any AU algebra, there is a finite number of non-isomorphic basic tilting modules and tilting AU modules.
Abstract: We give a characterization of $\tau$-rigid modules over Auslander algebras in terms of projective dimension of modules. Moreover, we show that for an Auslander algebra $\Lambda$ admitting finite number of non-isomorphic basic tilting $\Lambda$-modules and tilting $\Lambda^{\operatorname{op}}$-modules, if all indecomposable $\tau$-rigid $\Lambda$-modules of projective dimension $2$ are of grade $2$, then $\Lambda$ is $\tau$-tilting finite.
TL;DR: In this article, the authors investigated the relation between the number of rational points over a finite field and their periods in terms of hypergeometric functions and provided an algorithm to determine the coefficients involved.
Abstract: In this paper we investigate the relation between the number of rational points over a finite field $\mathbb{F}_{p^n}$ on a family of higher genus curves and their periods in terms of hypergeometric functions. For the case $y^\ell = x(x-1)(x-\lambda)$ we find a closed form in terms of hypergeometric functions associated with the periods of the curve. For the general situation $y^\ell = x^{a_1}(x-1)^{a_2}(x-\lambda)^{a_3}$ we show that the number of rational points is a linear combination of hypergeometric series, and we provide an algorithm to determine the coefficients involved.
TL;DR: In this article, the distance spectral radii of several classes of circulant graphs are given and the exact values of the vertex-forwarding indices and some bounds of the edge forwarding indices for these kinds of graphs are presented.
Abstract: In this paper, we give the distance spectral radii of several classes of circulant graphs. We also list the elements in the first rows of their corresponding distance matrices, with which all other distance eigenvalues can be obtained. In addition, we get the relationships between the distance spectral radii and forwarding indices of circulant graphs. Finally, the exact values of the vertex-forwarding indices and some bounds of the edge-forwarding indices for these kinds of graphs are presented.
TL;DR: In this article, it was shown that for graphs of order n, f(K_m,K_{n,n}) = 2n^2+2n-1, where n is the number of vertices in the graph.
Abstract: For graphs $G$, $G_1$ and $G_2$, let $G \to (G_1,G_2)$ signify that any red/blue edge-coloring of $G$ contains a red $G_1$ or a blue $G_2$, and let $f(G_1,G_2)$ be the minimum $N$ such that there is a graph $G$ of order $N$ with $\omega(G) = \max \{\omega(G_1),\omega(G_2)\}$ and $G \to (G_1,G_2)$. It is shown that $c_1(n/\!\log n)^{(m+1)/2} \leq f(K_m,K_{n,n}) \leq c_2 n^{m-1}$, where $c_i = c_i(m) > 0$ are constants. In particular, $cn^2/\log n \leq f(K_3,K_{n,n}) \leq 2n^2+2n-1$. Moreover, $f(K_m,T_n) \leq m^2(n-1)$ for all $n \geq m \geq 2$, where $T_n$ is a tree on $n$ vertices.
TL;DR: In this article, a conditional variational principle for the topological entropy of level sets of Birkhoff averages for maps with the gluing orbit property was obtained. But this result can be easily extended to flows.
Abstract: In this paper, we obtain a conditional variational principle for the topological entropy of level sets of Birkhoff averages for maps with the gluing orbit property. Our result can be easily extended to flows.
TL;DR: In this paper, the Lipschitzian stability of parametric eTRS was investigated using the Frechet and Mordukhovich coderivative tool, in which the trust region intersects a ball with a single linear inequality constraint.
Abstract: This paper deals with the Frechet and Mordukhovich coderivatives of the normal cone mapping related to the parametric extended trust region subproblems (eTRS), in which the trust region intersects a ball with a single linear inequality constraint. We use the obtained results to investigate the Lipschitzian stability of parametric eTRS. We also propose a necessary condition for the local (or global) solution of the eTRS by using the coderivative tool.
TL;DR: In this article, the existence of infinitely many weak solutions for the fractional Laplace equation was proved under appropriate assumptions on $V$ and $f$ under appropriate conditions on the Laplacian.
Abstract: We use two variant fountain theorems to prove the existence of infinitely many weak solutions for the following fractional $p$-Laplace equation\[ (-\Delta)^\alpha_p u + V(x) |u|^{p-2}u = f(x,u), \quad x \in \mathbb{R}^N,\]where $N \geq 2$, $p \geq 2$, $\alpha \in (0,1)$, $(-\Delta)^\alpha_p$ is the fractional $p$-Laplacian and $f$ is either asymptotically linear or subcritical $p$-superlinear growth. Under appropriate assumptions on $V$ and $f$, we prove the existence of infinitely many nontrivial high or small energy solutions. Our results generalize and extend some existing results.