Taiwanese Journal of Mathematics Articles (Project Euclid)
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The latest articles from Taiwanese Journal of Mathematics on Project Euclid, a site for mathematics and statistics resources.en-usCopyright 2017 Cornell University LibraryEuclid-L@cornell.edu (Project Euclid Team)Thu, 29 Jun 2017 11:42 EDTThu, 29 Jun 2017 11:42 EDThttp://projecteuclid.org/collection/euclid/images/logo_linking_100.gifProject Euclid
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On Adjacent Vertex-distinguishing Total Chromatic Number of Generalized Mycielski Graphs
http://projecteuclid.org/euclid.twjm/1498750951
<strong>Enqiang Zhu</strong>, <strong>Chanjuan Liu</strong>, <strong>Jin Xu</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Volume 21, Number 2, 253--266.</p><p><strong>Abstract:</strong><br/>
The adjacent vertex-distinguishing total chromatic number of a graph $G$, denoted by $\chi_{at}(G)$, is the smallest $k$ for which $G$ has a proper total $k$-coloring such that any two adjacent vertices have distinct sets of colors appearing on the vertex and its incident edges. In regard of this number, there is a famous conjecture (AVDTCC) which states that for any simple graph $G$, $\chi_{at}(G) \leq \Delta(G)+3$. In this paper, we study this number for the generalized Mycielski graph $\mu_m(G)$ of a graph $G$. We prove that the satisfiability of the conjecture AVDTCC in $G$ implies its satisfiability in $\mu_m(G)$. Particularly we give the exact values of $\chi_{at}(\mu_m(G))$ when $G$ is a graph with maximum degree less than $3$ or a complete graph. Moreover, we investigate $\chi_{at}(G)$ for any graph $G$ with only one maximum degree vertex by showing that $\chi_{at}(G) \leq \Delta(G)+2$ when $\Delta(G) \leq 4$.
</p>projecteuclid.org/euclid.twjm/1498750951_20170629114244Thu, 29 Jun 2017 11:42 EDTMultiplicity of Solutions for a Sublinear Quasilinear Schrödinger Equationhttps://projecteuclid.org/euclid.twjm/1572944419<strong>Gui Bao</strong>, <strong>Tingzhi Cheng</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Advance publication, 10 pages.</p><p><strong>Abstract:</strong><br/>
In this paper, we are concerned with the multiplicity of solutions for a class of quasilinear elliptic equation arising from plasma physics. By using a dual approach, the existence of infinitely many small solutions are obtained. As a main novelty with respect to some previous results, we assume the potential $V$ may changes sign and do not require any condition at infinity on the nonlinear term.
</p>projecteuclid.org/euclid.twjm/1572944419_20200112220114Sun, 12 Jan 2020 22:01 ESTAveraging Operators Along a Certain Type of Surfaces with Hypersingularityhttps://projecteuclid.org/euclid.twjm/1572836421<strong>Jin Bong Lee</strong>, <strong>Jongho Lee</strong>, <strong>Chan Woo Yang</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Advance publication, 13 pages.</p><p><strong>Abstract:</strong><br/>
In this paper we obtain almost sharp decay estimates for $L^2$ operator norm of strongly singular oscillatory integral operators in $\mathbb{R}^{n+1}$ for $n \geq 2$; we prove some necessary condition for $L^2$ estimates. Also, we prove that the operators are bounded on $L^p$ for some $p \neq 2$ and the range of $p$ depends on the hypersingularity of the operators.
</p>projecteuclid.org/euclid.twjm/1572836421_20200112220114Sun, 12 Jan 2020 22:01 ESTMulti-objective Optimization Problems with SOS-convex Polynomials over an LMI Constrainthttps://projecteuclid.org/euclid.twjm/1572422530<strong>Liguo Jiao</strong>, <strong>Jae Hyoung Lee</strong>, <strong>Yuto Ogata</strong>, <strong>Tamaki Tanaka</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Advance publication, 23 pages.</p><p><strong>Abstract:</strong><br/>
In this paper, we aim to find efficient solutions of a multi-objective optimization problem over a linear matrix inequality (LMI in short), in which the objective functions are SOS-convex polynomials. We do this by using two scalarization approaches, that is, the $\epsilon$-constraint method and the hybrid method. More precisely, we first transform the considered multi-objective optimization problem into their scalar forms by the $\epsilon$-constraint method and the hybrid method, respectively. Then, strong duality results, between each formulated scalar problem and its associated semidefinite programming dual problem, are given, respectively. Moreover, for each proposed scalar problem, we show that its optimal solution can be found by solving an associated single semidefinite programming problem, under a suitable regularity condition. As a consequence, we prove that finding efficient solutions to the considered problem can be done by employing any of the two scalarization approaches. Besides, we illustrate our methods through some nontrivial numerical examples.
</p>projecteuclid.org/euclid.twjm/1572422530_20200112220114Sun, 12 Jan 2020 22:01 ESTA Survey on the Lace Expansion for the Nearest-neighbor Models on the BCC Latticehttps://projecteuclid.org/euclid.twjm/1571364135<strong>Satoshi Handa</strong>, <strong>Yoshinori Kamijima</strong>, <strong>Akira Sakai</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Advance publication, 62 pages.</p><p><strong>Abstract:</strong><br/>
The aim of this survey is to explain, in a self-contained and relatively beginner-friendly manner, the lace expansion for the nearest-neighbor models of self-avoiding walk and percolation that converges in all dimensions above 6 and 9, respectively. To achieve this, we consider a $d$-dimensional version of the body-centered cubic (BCC) lattice, on which it is extremely easy to enumerate various random-walk quantities. Also, we choose a particular set of bootstrapping functions, by which a notoriously complicated part of the lace-expansion analysis becomes rather transparent.
</p>projecteuclid.org/euclid.twjm/1571364135_20200112220114Sun, 12 Jan 2020 22:01 ESTExceptional Set of Waring-Goldbach Problem with Unequal Powers of Primeshttps://projecteuclid.org/euclid.twjm/1571191582<strong>Xiaodong Zhao</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Advance publication, 8 pages.</p><p><strong>Abstract:</strong><br/>
In this paper, it is proved that with at most $O(N^{17/42+\varepsilon})$ exceptions, all even positive integer $n$, $n \in [N/2,N]$, can be represented in the form $p_{1}^{2} + p_{2}^{2} + p_{3}^{3} + p_{4}^{3} + p_{5}^{4} + p_{6}^{4}$, where $p_{1}$, $p_{2}$, $p_{3}$, $p_{4}$, $p_{5}$, $p_{6}$ are prime numbers. This improves a recent result $O(N^{13/16+\varepsilon})$ due to Zhang and Li [13].
</p>projecteuclid.org/euclid.twjm/1571191582_20200112220114Sun, 12 Jan 2020 22:01 ESTInvariant Subsets and Homological Properties of Orlicz Modules over Group Algebrashttps://projecteuclid.org/euclid.twjm/1568858572<strong>Rüya Üster</strong>, <strong>Serap Öztop</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Advance publication, 15 pages.</p><p><strong>Abstract:</strong><br/>
Let $G$ be a locally compact group with left Haar measure. We study the closed convex left invariant subsets of $L^{\Phi}(G)$ and characterize affine mappings from the space of nonnegative functions in $L^{1}(G)$ of norm $1$ into $L^{\Phi}(G)$ spaces. We apply the results to the study of the multipliers of $L^{\Phi}(G)$. We also investigate the homological properties of $L^{\Phi}(G)$ as a Banach left $L^{1}(G)$-module such as projectivity, injectivity and flatness.
</p>projecteuclid.org/euclid.twjm/1568858572_20200112220114Sun, 12 Jan 2020 22:01 ESTCofiniteness with Respect to the Class of Modules in Dimension less than a Fixed Integerhttps://projecteuclid.org/euclid.twjm/1568016021<strong>Alireza Vahidi</strong>, <strong>Saeid Morsali</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Advance publication, 16 pages.</p><p><strong>Abstract:</strong><br/>
Let $R$ be a commutative Noetherian ring with non-zero identity, $n$ a non-negative integer, $\mathfrak{a}$ an ideal of $R$ with $\dim(R/\mathfrak{a}) \leq n+1$, and $X$ an arbitrary $R$-module. In this paper, we prove the following results:
(i) If $X$ is an $\mathfrak{a}$-torsion $R$-module such that $\operatorname{Hom}_{R}(R/\mathfrak{a},X)$ and $\operatorname{Ext}_{R}^{1}(R/\mathfrak{a},X)$ are $\operatorname{FD}_{\lt n}$ $R$-modules, then $X$ is an $(\operatorname{FD}_{\lt n},\mathfrak{a})$-cofinite $R$-module;
(ii) The category of $(\operatorname{FD}_{\lt n},\mathfrak{a})$-cofinite $R$-modules is an Abelian category;
(iii) $\operatorname{H}^{i}_{\mathfrak{a}}(X)$ is an $(\operatorname{FD}_{\lt n},\mathfrak{a})$-cofinite $R$-module and $\{ \mathfrak{p} \in \operatorname{Ass}_R(\operatorname{H}^{i}_{\mathfrak{a}}(X)) : \dim(R/\mathfrak{p}) \geq n \}$ is a finite set for all $i$ when $\operatorname{Ext}^{i}_{R}(R/\mathfrak{a},X)$ is an $\operatorname{FD}_{\lt n}$ $R$-module for all $i$.
We observe that, among other things, $\operatorname{Ass}_R(\operatorname{H}^{i}_{\mathfrak{a}}(X))$ is a finite set for all $i$ whenever $R$ is a semi-local ring with $\dim(R/\mathfrak{a}) \leq 2$ and $\operatorname{Ext}^{i}_{R}(R/\mathfrak{a},X)$ is an $\operatorname{FD}_{\lt 1}$ $R$-module for all $i$.
</p>projecteuclid.org/euclid.twjm/1568016021_20200112220114Sun, 12 Jan 2020 22:01 ESTGlobal Stability of Non-monotone Noncritical Traveling Waves for a Discrete Diffusion Equation with a Convolution Type Nonlinearityhttps://projecteuclid.org/euclid.twjm/1568016022<strong>Tao Su</strong>, <strong>Guo-Bao Zhang</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Advance publication, 21 pages.</p><p><strong>Abstract:</strong><br/>
This paper is concerned with the global stability of non-monotone traveling waves for a discrete diffusion equation with a monostable convolution type nonlinearity. It has been proved by Yang and Zhang (Sci. China Math. 61 (2018), 1789--1806) that all noncritical traveling waves (waves with speeds $c \gt c_*$, $c_*$ is minimal speed) are time-exponentially stable, when the initial perturbations around the waves are small. In this paper, we further prove that all traveling waves with large speed are globally stable, when the initial perturbations around the waves in a weighted Sobolev space can be arbitrarily large. The approaches adopted are the nonlinear Halanay's inequality, the technical weighted energy method and Fourier's transform.
</p>projecteuclid.org/euclid.twjm/1568016022_20200112220114Sun, 12 Jan 2020 22:01 ESTHigh Spatial Accuracy Analysis of Linear Triangular Finite Element for Distributed Order Diffusion Equationshttps://projecteuclid.org/euclid.twjm/1566352830<strong>Lin He</strong>, <strong>Jincheng Ren</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Advance publication, 14 pages.</p><p><strong>Abstract:</strong><br/>
In this paper, an effective numerical fully discrete finite element scheme for the distributed order time fractional diffusion equations is developed. By use of the composite trapezoid formula and the well-known $L1$ formula approximation to the distributed order derivative and linear triangular finite element approach for the spatial discretization, we construct a fully discrete finite element scheme. Based on the superclose estimate between the interpolation operator and the Ritz projection operator and the interpolation post-processing technique, the superclose approximation of the finite element numerical solution and the global superconvergence are proved rigorously, respectively. Finally, a numerical example is presented to support the theoretical results.
</p>projecteuclid.org/euclid.twjm/1566352830_20200112220114Sun, 12 Jan 2020 22:01 ESTOn the Bogomolov-Miyaoka-Yau Inequality for Stacky Surfaceshttps://projecteuclid.org/euclid.twjm/1565920823<strong>Jiun-Cheng Chen</strong>, <strong>Hsian-Hua Tseng</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Advance publication, 13 pages.</p><p><strong>Abstract:</strong><br/>
We discuss a generalization of the Bogomolov-Miyaoka-Yau inequality to Deligne-Mumford surfaces of general type.
</p>projecteuclid.org/euclid.twjm/1565920823_20200112220114Sun, 12 Jan 2020 22:01 ESTA Class of Fourth-order Parabolic Equations with Logarithmic Nonlinearityhttps://projecteuclid.org/euclid.twjm/1565683225<strong>Menglan Liao</strong>, <strong>Qingwei Li</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Advance publication, 29 pages.</p><p><strong>Abstract:</strong><br/>
In this paper, we apply the modified potential well method and the logarithmic Sobolev inequality to study the fourth-order parabolic equation with $p$-Laplacian and logarithmic nonlinearity. Some results are obtained under the different initial data conditions. More precisely, we give the global existence of weak solution by combining the classical Galerkin's method with the modified potential well method, decay estimates, and blow-up in finite time when the initial energy is subcritical and critical, respectively. In addition, sufficient conditions for the global existence and blow-up of the weak solution are also provided for supercritical initial energy. These results extend and improve many results in the literature.
</p>projecteuclid.org/euclid.twjm/1565683225_20200112220114Sun, 12 Jan 2020 22:01 ESTAn Application of Liaison Theory to Zero-dimensional Schemeshttps://projecteuclid.org/euclid.twjm/1565337619<strong>Martin Kreuzer</strong>, <strong>Tran N. K. Linh</strong>, <strong>Le Ngoc Long</strong>, <strong>Tu Chanh Nguyen</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Advance publication, 21 pages.</p><p><strong>Abstract:</strong><br/>
Given a $0$-dimensional scheme $\mathbb{X}$ in an $n$-dimensional projective space $\mathbb{P}^n_K$ over an arbitrary field $K$, we use liaison theory to characterize the Cayley-Bacharach property of $\mathbb{X}$. Our result extends the result for sets of $K$-rational points given in [8]. In addition, we examine and bound the Hilbert function and regularity index of the Dedekind different of $\mathbb{X}$ when $\mathbb{X}$ has the Cayley-Bacharach property.
</p>projecteuclid.org/euclid.twjm/1565337619_20200112220114Sun, 12 Jan 2020 22:01 ESTUpper Semicontinuity of Random Attractor for a Kirchhoff Type Suspension Bridge Equation with Strong Damping and White Noisehttps://projecteuclid.org/euclid.twjm/1565337620<strong>Ling Xu</strong>, <strong>Qiaozhen Ma</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Advance publication, 25 pages.</p><p><strong>Abstract:</strong><br/>
This paper is devoted to the well-posedness and long-time behavior of a stochastic Kirchhoff type suspension bridge equation with strong damping. The existence of the random attractor for a Kirchhoff type suspension bridge equation with white noise is established. Moreover, the upper semicontinuity of random attractors is also provided when the coefficient of random term approaches zero.
</p>projecteuclid.org/euclid.twjm/1565337620_20200112220114Sun, 12 Jan 2020 22:01 ESTTopological Sensitivity Analysis and Kohn-Vogelius Formulation for Detecting a Rigid Inclusion in an Elastic Bodyhttps://projecteuclid.org/euclid.twjm/1565337621<strong>Mourad Hrizi</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Advance publication, 44 pages.</p><p><strong>Abstract:</strong><br/>
Our main interest in this work is to detect a rigid inclusion immersed in an isotropic elastic body $\Omega$ from a single pair of Cauchy data on $\partial \Omega$ in two dimensions. We want to completely characterize the unknown rigid inclusion, namely, the shape and the location of inclusion. The idea is to rewrite the inverse problem as an optimization problem, where an energy like functional is minimized with respect to the presence of a small inclusion. A topological sensitivity analysis is derived for an energy like functional. We proposed a non-iterative reconstruction algorithm based on the topological gradient concept. The unknown rigid inclusion is defined by a level curve of a scalar function. The proposed numerical approach is very robust with respect to noisy data. Finally, in order to show the efficiency and accuracy of the proposed algorithm, we present some numerical results.
</p>projecteuclid.org/euclid.twjm/1565337621_20200112220114Sun, 12 Jan 2020 22:01 ESTStrict Monotonicity and Unique Continuation for General Non-local Eigenvalue Problemshttps://projecteuclid.org/euclid.twjm/1565229625<strong>Silvia Frassu</strong>, <strong>Antonio Iannizzotto</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Advance publication, 14 pages.</p><p><strong>Abstract:</strong><br/>
We consider the weighted eigenvalue problem for a general non-local pseudo-differential operator, depending on a bounded weight function. For such problem, we prove that strict (decreasing) monotonicity of the eigenvalues with respect to the weight function is equivalent to the unique continuation property of eigenfunctions. In addition, we discuss some unique continuation results for the special case of the fractional Laplacian.
</p>projecteuclid.org/euclid.twjm/1565229625_20200112220114Sun, 12 Jan 2020 22:01 ESTQuasi-periodic Solutions for Nonlinear Schrödinger Equations with Legendre Potentialhttps://projecteuclid.org/euclid.twjm/1564970424<strong>Guanghua Shi</strong>, <strong>Dongfeng Yan</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Advance publication, 17 pages.</p><p><strong>Abstract:</strong><br/>
In this paper, the nonlinear Schrödinger equations with Legendre potential $\mathbf{i} u_{t} - u_{xx} + V_L(x)u + mu + \sec x \cdot |u|^2 u = 0$ subject to certain boundary conditions is considered, where $V_L(x) = -\frac{1}{2} - \frac{1}{4} \tan^2 x$, $x \in (-\pi/2,\pi/2)$. It is proved that for each given positive constant $m \gt 0$, the above equation admits lots of quasi-periodic solutions with two frequencies. The proof is based on a partial Birkhoff normal form technique and an infinite-dimensional Kolmogorov-Arnold-Moser theory.
</p>projecteuclid.org/euclid.twjm/1564970424_20200112220114Sun, 12 Jan 2020 22:01 ESTThe Explicit Formulae and Evaluations of Ramanujan's Remarkable Product of Theta-functionshttps://projecteuclid.org/euclid.twjm/1564970425<strong>Devasirvatham John Prabhakaran</strong>, <strong>Kumar Ranjith Kumar</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Advance publication, 7 pages.</p><p><strong>Abstract:</strong><br/>
On pages 338 and 339 in his first notebook, Ramanujan defined remarkable product of theta-functions $a_{m,n}$ and also recorded eighteen explicit values depending on two parameters $m$ and $n$. All these values have been established by Berndt et al. In this paper, we establish a new general formulae for the explicit evaluations of $a_{3m,3}$ and $a_{m,9}$ by using $P$-$Q$ mixed modular equation and values for certain class invariant of Ramanujan. Using these formulae, we calculate some new explicit values of $a_{3m,3}$ for $m = 2,7,13,17,25,37$ and $a_{m,9}$ for $m = 17,37$.
</p>projecteuclid.org/euclid.twjm/1564970425_20200112220114Sun, 12 Jan 2020 22:01 ESTA Note on Special Fibers of Shimura Curves and Special Representationshttps://projecteuclid.org/euclid.twjm/1563264243<strong>Yih-Jeng Yu</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Advance publication, 22 pages.</p><p><strong>Abstract:</strong><br/>
We study the geometry of the special fibers of certain Shimura curves and give a direct proof of global-to-local Jacquet-Langlands compatibility by Čerednik-Drinfel'd uniformizations theorem.
</p>projecteuclid.org/euclid.twjm/1563264243_20200112220114Sun, 12 Jan 2020 22:01 ESTQuasi-periodic Solutions of Wave Equations with the Nonlinear Term Depending on the Time and Space Variableshttps://projecteuclid.org/euclid.twjm/1562896821<strong>Yi Wang</strong>, <strong>Jie Rui</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Advance publication, 33 pages.</p><p><strong>Abstract:</strong><br/>
This article is devoted to the study of a wave equation with a constant potential and an $x$-periodic and $t$-quasi-periodic nonlinear term subject to periodic boundary conditions. It is proved that the equation admits small amplitude, linear stable and $t$-quasi-periodic solutions for any constant potential and most frequency vectors.
</p>projecteuclid.org/euclid.twjm/1562896821_20200112220114Sun, 12 Jan 2020 22:01 ESTA Note on Number Knots and the Splitting of the Hilbert Class Fieldhttps://projecteuclid.org/euclid.twjm/1562832028<strong>Yih-Jeng Yu</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Advance publication, 18 pages.</p><p><strong>Abstract:</strong><br/>
Several number knots are defined including the five knots introduced by W. Jehne. The question of the splitting of the group extension of the Hilbert class field can be read off in terms of the triviality of these knots.
</p>projecteuclid.org/euclid.twjm/1562832028_20200112220114Sun, 12 Jan 2020 22:01 ESTOptimality Conditions in Set-valued Optimization Problem with Respect to a Partial Order Relation via Directional Derivativehttps://projecteuclid.org/euclid.twjm/1562724021<strong>Emrah Karaman</strong>, <strong>Mustafa Soyertem</strong>, <strong>İlknur Atasever Güvenç</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Advance publication, 14 pages.</p><p><strong>Abstract:</strong><br/>
In this study, a new directional derivative is defined by using Minkowski difference. Some properties and existence theorems of this directional derivative are given. Moreover, necessary and sufficient optimality conditions are presented for set-valued optimization problems with respect to $m_1$ order relation via directional derivative.
</p>projecteuclid.org/euclid.twjm/1562724021_20200112220114Sun, 12 Jan 2020 22:01 ESTSingularity Formation of the Non-baratropic Compressible Magnetohydrodynamic Equations Without Heat Conductivityhttps://projecteuclid.org/euclid.twjm/1562659215<strong>Xin Zhong</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Advance publication, 26 pages.</p><p><strong>Abstract:</strong><br/>
We study the singularity formation of strong solutions to the three-dimensional full compressible magnetohydrodynamic equations with zero heat conduction in a bounded domain. We show that for the initial density allowing vacuum, the strong solution exists globally if the density $\rho$, the magnetic field $\mathbf{b}$, and the pressure $P$ satisfy $\|\rho\|_{L^{\infty}(0,T;L^{\infty})} + \|\mathbf{b}\|_{L^{\infty}(0,T;L^6)} + \|P\|_{L^{\infty}(0,T;L^{\infty})} \lt \infty$ and the coefficients of viscosity verity $3\mu \gt \lambda$. This extends the corresponding results in Duan (2017), Fan et al. (2018) [1,2] where a blow-up criterion in terms of the upper bounds of the density, the magnetic field and the temperature was obtained under the condition $2\mu \gt \lambda$. Our proof relies on some delicate energy estimates.
</p>projecteuclid.org/euclid.twjm/1562659215_20200112220114Sun, 12 Jan 2020 22:01 ESTTime-asymptotic Dynamics of Hermitian Riccati Differential Equationshttps://projecteuclid.org/euclid.twjm/1562313624<strong>Yueh-Cheng Kuo</strong>, <strong>Huey-Er Lin</strong>, <strong>Shih-Feng Shieh</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Advance publication, 28 pages.</p><p><strong>Abstract:</strong><br/>
The matrix Riccati differential equation (RDE) raises in a wide variety of applications for science and applied mathematics. We are particularly interested in the Hermitian Riccati Differential Equation (HRDE). Radon's lemma gives a solution representation to HRDE. Although solutions of HRDE may show the finite escape time phenomenon, we can investigate the time asymptotic dynamical behavior of HRDE by its extended solutions. In this paper, we adapt the Hamiltonian Jordan canonical form to characterize the time asymptotic phenomena of the extended solutions for HRDE in four elementary cases. The extended solutions of HRDE exhibit the dynamics of heteroclinic, homoclinic and periodic orbits in the elementary cases under some conditions.
</p>projecteuclid.org/euclid.twjm/1562313624_20200112220114Sun, 12 Jan 2020 22:01 ESTSome Properties of the Signless Laplacian and Normalized Laplacian Tensors of General Hypergraphshttps://projecteuclid.org/euclid.twjm/1562054420<strong>Cunxiang Duan</strong>, <strong>Ligong Wang</strong>, <strong>Xihe Li</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Advance publication, 17 pages.</p><p><strong>Abstract:</strong><br/>
In this paper, we obtain some properties of signless Laplacian eigenvalues of general hypergraphs. We give the upper and the lower bound of edge connectivity of general hypergraphs in terms of average degree, minimum degree, the rank and the number of vertices, or analytic connectivity $\alpha(G)$, respectively. We also give the upper bound of analytic connectivity $\alpha(G)$ of general hypergraphs in terms of the degrees of vertices. Finally, we obtain the bounds of the smallest $H^{+}$-eigenvalue of the normalized Laplacian sub-tensors of general hypergraphs.
</p>projecteuclid.org/euclid.twjm/1562054420_20200112220114Sun, 12 Jan 2020 22:01 ESTBackward Stability and Divided Invariance of an Attractor for the Delayed Navier-Stokes Equationhttps://projecteuclid.org/euclid.twjm/1561341624<strong>Yangrong Li</strong>, <strong>Qiangheng Zhang</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Advance publication, 27 pages.</p><p><strong>Abstract:</strong><br/>
We study backward stability of a pullback attractor especially for a delay equation. We introduce a new concept of a backward attractor, which is defined by a compact, pullback attracting and dividedly invariant family. We then show the equivalence between existence of a backward attractor and backward stability of the pullback attractor, and present some criteria by using the backward limit-set compactness of the system. In the application part, we consider the Navier-Stokes equation with a nonuniform Lipschitz delay term and a backward tempered force. Based on the fact that the delay does not change the backward bounds of the velocity field and external forces, we establish the backward-uniform estimates and obtain a backward attractor, which leads to backward stability of the pullback attractor. Some special cases of variable delay and distributed delay are discussed.
</p>projecteuclid.org/euclid.twjm/1561341624_20200112220114Sun, 12 Jan 2020 22:01 ESTInverse Anti-$k$-centrum Problem on Networks with Variable Edge Lengthshttps://projecteuclid.org/euclid.twjm/1561082413<strong>Van Huy Pham</strong>, <strong>Kien Trung Nguyen</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Advance publication, 22 pages.</p><p><strong>Abstract:</strong><br/>
This paper concerns the problem of modifying edge lengths of a network at minimum total costs so as to make a prespecified vertex become an optimal location in the modified environment. Here, we focus on the ordered median objective function with respect to the vector of multipliers $\lambda = (1,\ldots,1,0,\ldots,0)$ with $k$ 1's. This problem is called the inverse anti-$k$-centrum problem. We first show that the inverse anti-$k$-centrum problem is NP-hard even on tree networks. However, for the inverse anti-$k$-centrum problem on cycles, we formulate it as one or two linear programs, depending on odd or even integer $k$. Concerning the special cases with $k = 2,3,M$, we develop combinatorial algorithms that efficiently solve the problem, where $M$ is the number of vertices of the cycle.
</p>projecteuclid.org/euclid.twjm/1561082413_20200112220114Sun, 12 Jan 2020 22:01 ESTOn Hardy's Inequality for Hermite Expansionshttps://projecteuclid.org/euclid.twjm/1559700015<strong>Paweł Plewa</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Advance publication, 15 pages.</p><p><strong>Abstract:</strong><br/>
Sharp multi-dimensional Hardy's inequality for the Laguerre functions of Hermite type is proved for the type parameter $\alpha \in [-1/2,\infty)^d$. As a consequence we obtain the corresponding result for the generalized Hermite expansions. In particular, it validates that the known version of Hardy's inequality for the Hermite functions is sharp.
</p>projecteuclid.org/euclid.twjm/1559700015_20200112220114Sun, 12 Jan 2020 22:01 ESTAlmost Periodicity of All $L^2$-bounded Solutions of a Functional Heat Equationhttps://projecteuclid.org/euclid.twjm/1559095225<strong>Qi-Ru Wang</strong>, <strong>Zhi-Qiang Zhu</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Advance publication, 7 pages.</p><p><strong>Abstract:</strong><br/>
In this paper, we continue the investigations done in the literature about the so called Bohr-Neugebauer property for almost periodic differential equations. More specifically, for a class of functional heat equations, we prove that each $L^2$-bounded solution is almost periodic. This extends a result in [5] to the delay case.
</p>projecteuclid.org/euclid.twjm/1559095225_20200112220114Sun, 12 Jan 2020 22:01 ESTReducibility, Lyapunov Exponent, Pure Point Spectra Property for Quasi-periodic Wave Operatorhttps://projecteuclid.org/euclid.twjm/1558404261<strong>Jing Li</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Advance publication, 35 pages.</p><p><strong>Abstract:</strong><br/>
In the present paper, it is shown that the linear wave equation subject to Dirichlet boundary condition \[ u_{tt} - u_{xx} + \varepsilon V(\omega t, x) u = 0, \quad u(t,-\pi) = u(t,\pi) = 0 \] can be changed by a symplectic transformation into \[ v_{tt} - v_{xx} + \varepsilon M_{\xi} v = 0, \quad v(t,-\pi) = v(t,\pi) = 0, \] where $V$ is finitely smooth and time-quasi-periodic potential with frequency $\omega \in \mathbb{R}^n$ in some Cantor set of positive Lebeague measure and where $M_{\xi}$ is a Fourier multiplier. Moreover, it is proved that the corresponding wave operator $\partial_t^2 - \partial_x^2 + \varepsilon V(\omega t, x)$ possesses the property of pure point spectra and zero Lyapunov exponent.
</p>projecteuclid.org/euclid.twjm/1558404261_20200112220114Sun, 12 Jan 2020 22:01 ESTBoundary Continuity of Nonparametric Prescribed Mean Curvature Surfaceshttps://projecteuclid.org/euclid.twjm/1557972013<strong>Mozhgan Nora Entekhabi</strong>, <strong>Kirk E. Lancaster</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Advance publication, 17 pages.</p><p><strong>Abstract:</strong><br/>
We investigate the boundary behavior of variational solutions of Dirichlet problems for prescribed mean curvature equations at smooth boundary points where certain boundary curvature conditions are satisfied (which preclude the existence of local barrier functions). We prove that if the Dirichlet boundary data $\phi$ is continuous at such a point (and possibly nowhere else), then the solution of the variational problem is continuous at this point.
</p>projecteuclid.org/euclid.twjm/1557972013_20200112220114Sun, 12 Jan 2020 22:01 ESTA Poisson Problem of Transmission-type for the Stokes and Generalized Brinkman Systems in Complementary Lipschitz Domains in $\mathbb{R}^3$https://projecteuclid.org/euclid.twjm/1557799218<strong>Andrei-Florin Albişoru</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Advance publication, 24 pages.</p><p><strong>Abstract:</strong><br/>
The purpose of this paper is to give a well-posedness result for a boundary value problem of transmission type for the Stokes and generalized Brinkman systems in two complementary Lipschitz domains in $\mathbb{R}^3$. In the first part of the paper, we have introduced the classical and weighted $L^2$-based Sobolev spaces on Lipschitz domains in $\mathbb{R}^3$. Afterwards, the trace and conormal derivative operators are defined in the case of both Stokes and generalized Brinkman systems. Also, a summary of the main properties of the layer potential operators for the Stokes system, is provided. In the second part of the work, we exploit the well-posedness of another transmission problem concerning the Stokes system on two complementary Lipschitz domains in $\mathbb{R}^3$ which is based on the Potential Theory for the Stokes system. Then, certain properties of Fredholm operators will allow us to show our main well-posedness result in $L^2$-based Sobolev spaces.
</p>projecteuclid.org/euclid.twjm/1557799218_20200112220114Sun, 12 Jan 2020 22:01 ESTGodunov-type Numerical Scheme for the Shallow Water Equations with Horizontal Temperature Gradienthttps://projecteuclid.org/euclid.twjm/1557475461<strong>Nguyen Xuan Thanh</strong>, <strong>Mai Duc Thanh</strong>, <strong>Dao Huy Cuong</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Advance publication, 45 pages.</p><p><strong>Abstract:</strong><br/>
We present a Godunov-type scheme for the shallow water equations with horizontal temperature gradient and variable topography. First, the exact solutions of the Riemann problem in a computational form are given, where algorithms for computing these solutions are described. Second, a Godunov-type scheme is constructed relying on exact solutions of the local Riemann problems. Computing algorithms for the scheme are given. The scheme is shown to be well-balanced and preserve the positivity of the water height. Numerical tests show that the scheme is convergent with a good accuracy, even for the resonant phenomenon, where the exact solutions contain several distinct waves propagating with the same shock speed. Furthermore, the scheme also provides us with good results for the solution of the wave interaction problem.
</p>projecteuclid.org/euclid.twjm/1557475461_20200112220114Sun, 12 Jan 2020 22:01 ESTComplete Cotorsion Pairs in Exact Categorieshttps://projecteuclid.org/euclid.twjm/1557280827<strong>Zhi-Wei Li</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Advance publication, 12 pages.</p><p><strong>Abstract:</strong><br/>
We generalize a theorem of Saorín-Štovíček on complete cotorsion pairs in exact categories. Our proof is based on a generalized small object argument due to Chorny. As a consequence, we cover some examples which are not covered by the result of Saorín-Štovíček.
</p>projecteuclid.org/euclid.twjm/1557280827_20200112220114Sun, 12 Jan 2020 22:01 ESTOn Orthogonality of Elementary Operators in Norm-attainable Classeshttps://projecteuclid.org/euclid.twjm/1557194426<strong>Nyaare Benard Okelo</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Advance publication, 12 pages.</p><p><strong>Abstract:</strong><br/>
Various notions of orthogonality of elementary operators have been characterized by many mathematicians in different classes. In this paper, we characterize orthogonality of these operators in norm-attainable classes. We first give necessary and sufficient conditions for norm-attainability of Hilbert space operators then we give results on orthogonality of the range and the kernel of elementary operators when they are implemented by norm-attainable operators in norm-attainable classes.
</p>projecteuclid.org/euclid.twjm/1557194426_20200112220114Sun, 12 Jan 2020 22:01 ESTExtension Operators Preserving Janowski Classes of Univalent Functionshttps://projecteuclid.org/euclid.twjm/1556848821<strong>Andra Manu</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Advance publication, 21 pages.</p><p><strong>Abstract:</strong><br/>
In this paper, our main interest is devoted to study the extension operator $\Phi_{n,\alpha,\beta} \colon \mathcal{L}S \to \mathcal{L}S_n$ given by $\Phi_{n,\alpha,\beta}(f)(z) = \big( f(z_1), \widetilde{z}(f(z_1)/z_1)^{\alpha} (f'(z_1))^{\beta} \big)$, $z = (z_1,\widetilde{z}) \in \mathbf{B}^n$, where $\alpha,\beta \geq 0$. We shall prove that if $f \in S$ can be embedded as the first element of a $g$-Loewner chain with $g \colon U \to \mathbb{C}$ given by $g(\zeta) = (1+A\zeta)/(1+B\zeta)$, $|\zeta| \lt 1$, and $-1 \leq B \lt A \leq 1$, then $F = \Phi_{n,\alpha,\beta}(f)$ can be embedded as the first element of a $g$-Loewner chain on the unit ball $\mathbf{B}^n$ for $\alpha \in [0,1]$, $\beta \in [0,1/2]$ and $\alpha + \beta \leq 1$. As a consequence, the operator $\Phi_{n,\alpha,\beta}$ preserves the notions of Janowski starlikeness on $\mathbf{B}^n$ and Janowski almost starlikeness on $\mathbf{B}^n$. Particular cases will be also mentioned.
On the other hand, we are also concerned about some radius problems related to the operator $\Phi_{n,\alpha,\beta}$ and the Janowski class $S^*(a,b)$. We compute the radius $S^*(a,b)$ of the class $S$ (respectively $S^*$).
</p>projecteuclid.org/euclid.twjm/1556848821_20200112220114Sun, 12 Jan 2020 22:01 ESTOn the Existence for an Integral System Including $m$ Equationshttps://projecteuclid.org/euclid.twjm/1556784112<strong>Xiaoqian Liu</strong>, <strong>Yutian Lei</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Advance publication, 17 pages.</p><p><strong>Abstract:</strong><br/>
In this paper, we study an integral system \[ \begin{cases} u_{i}(x) = K_{i}(x) (|x|^{\alpha-n} \ast u^{p_{i+1}}_{i+1})(x), &i = 1,2,\ldots,m-1, \\ u_{m}(x) = K_{m}(x) (|x|^{\alpha-n} \ast u^{p_{1}}_{1})(x). \end{cases} \] When $\alpha \in (0,n)$, $p_{i} \gt 0$ ($i = 1,2,\ldots,m$), the Serrin-type condition is critical for existence of positive solutions for some double bounded functions $K_{i}(x)$ ($i = 1,2,\ldots,m$). When $\alpha \in (0,n)$, $p_{i} \lt 0$ ($i = 1,2,\ldots,m$), the system has no positive solution for any double bounded $K_{i}(x)$ ($i = 1,2,\ldots,m$). When $\alpha \gt n$, $p_{i} \lt 0$ ($i = 1,2,\ldots,m$), and $\max_{i} \{-p_{i}\} \gt \alpha/(\alpha-n)$, then the system exists positive solutions increasing with the rate $\alpha-n$.
</p>projecteuclid.org/euclid.twjm/1556784112_20200112220114Sun, 12 Jan 2020 22:01 ESTProducts of Composition, Multiplication and Iterated Differentiation Operators Between Banach Spaces of Holomorphic Functionshttps://projecteuclid.org/euclid.twjm/1555639218<strong>Shuming Wang</strong>, <strong>Maofa Wang</strong>, <strong>Xin Guo</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Advance publication, 22 pages.</p><p><strong>Abstract:</strong><br/>
Let $H(\mathbb{D})$ denote the space of holomorphic functions on the unit disk $\mathbb{D}$ of $\mathbb{C}$, $\psi,\varphi \in H(\mathbb{D})$, $\varphi(\mathbb{D}) \subset \mathbb{D}$ and $n \in \mathbb{N} \cup \{0\}$. Let $C_{\varphi}$, $M_{\psi}$ and $D^n$ denote the composition, multiplication and iterated differentiation operators, respectively. To treat the operators induced by products of these operators in a unified manner, we introduce a sum operator $\sum_{j=0}^n M_{\psi_j} C_{\varphi} D^j$. We characterize the boundedness and compactness of this sum operator mapping from a large class of Banach spaces of holomorphic functions into the $k$th weighted-type space $\mathcal{W}_{\mu}^{(k)}$ (or $\mathcal{W}_{\mu,0}^{(k)}$), $k \in \mathbb{N} \cup \{0\}$, and give its estimates of norm and essential norm. Our results show that the boundedness and compactness of the sum operator depend only on the symbols and the norm of the point-evaluation functionals on the domain space. Our results cover many known results in the literature. Moreover, we introduce the order boundedness of the sum operator and turn its study into that of the boundedness and compactness.
</p>projecteuclid.org/euclid.twjm/1555639218_20200112220114Sun, 12 Jan 2020 22:01 ESTSecond-order Necessary Optimality Conditions for an Optimal Control Problemhttps://projecteuclid.org/euclid.twjm/1555552827<strong>Toan Nguyen</strong>, <strong>Thuy Le</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Advance publication, 40 pages.</p><p><strong>Abstract:</strong><br/>
Second-order necessary optimality conditions for an optimal control problem with a nonconvex cost function and state-control constraints are studied in this paper. By establishing an abstract result on second-order necessary optimality conditions for a mathematical programming problem, we obtain second-order necessary optimality conditions for an optimal control problem.
</p>projecteuclid.org/euclid.twjm/1555552827_20200112220114Sun, 12 Jan 2020 22:01 ESTGround State Solutions for Kirchhoff-type Problems with Critical Nonlinearityhttps://projecteuclid.org/euclid.twjm/1555315213<strong>Yiwei Ye</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Advance publication, 17 pages.</p><p><strong>Abstract:</strong><br/>
In this paper, we study the Kirchhoff-type equation with critical exponent \[ -\left( a + b \int_{\mathbb{R}^3} |\nabla u|^2 \, dx \right) \Delta u + V(x)u = a(x) f(u) + u^5 \quad \textrm{in $\mathbb{R}^3$}, \] where $a,b \gt 0$ are constants, $V \in C(\mathbb{R}^3,\mathbb{R})$, $\lim_{|x| \to \infty} V(x) = V_{\infty} \gt 0$ and $V(x) \leq V_{\infty} + C_1 e^{-b |x|}$ for some $C_1 \gt 0$ and $|x|$ large enough. Via variational methods, we prove the existence of ground state solution.
</p>projecteuclid.org/euclid.twjm/1555315213_20200112220114Sun, 12 Jan 2020 22:01 ESTIntrinsic Square Function Characterizations of Variable Weak Hardy Spaceshttps://projecteuclid.org/euclid.twjm/1555315214<strong>Xianjie Yan</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Advance publication, 20 pages.</p><p><strong>Abstract:</strong><br/>
Let $p(\cdot) \colon \mathbb{R}^n \to (0,\infty)$ be a variable exponent function satisfying the globally log-Hölder continuous condition. In this article, via using the atomic and Littlewood-Paley function characterizations of variable weak Hardy space $W\!H^{p(\cdot)}(\mathbb{R}^n)$, the author establishes its intrinsic square function characterizations including the intrinsic Littlewood-Paley $g$-function, the intrinsic Lusin area function and the intrinsic $g_{\lambda}^{\ast}$-function.
</p>projecteuclid.org/euclid.twjm/1555315214_20200112220114Sun, 12 Jan 2020 22:01 ESTNonseparating Independent Sets of Cartesian Product Graphshttps://projecteuclid.org/euclid.twjm/1554170479<strong>Fayun Cao</strong>, <strong>Han Ren</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Advance publication, 17 pages.</p><p><strong>Abstract:</strong><br/>
A set of vertices $S$ of a connected graph $G$ is a nonseparating independent set if $S$ is independent and $G-S$ is connected. The nsis number $\mathcal{Z}(G)$ is the maximum cardinality of a nonseparating independent set of $G$. It is well known that computing the nsis number of graphs is NP-hard even when restricted to $4$-regular graphs. In this paper, we first present a new sufficient and necessary condition to describe the nsis number. Then, we completely solve the problem of counting the nsis number of hypercubes $Q_{n}$ and Cartesian product of two cycles $C_{m} \square C_{n}$, respectively. We show that $\mathcal{Z}(Q_{n}) = 2^{n-2}$ for $n \geq 2$, and $\mathcal{Z}(C_{m} \square C_{n}) = n + \lfloor (n+2)/4 \rfloor$ if $m = 4$, $m + \lfloor (m+2)/4 \rfloor$ if $n = 4$ and $\lfloor mn/3 \rfloor$ otherwise. Moreover, we find a maximum nonseparating independent set of $Q_{n}$ and $C_{m} \square C_{n}$, respectively.
</p>projecteuclid.org/euclid.twjm/1554170479_20200112220114Sun, 12 Jan 2020 22:01 ESTExistence and Multiplicity of Solutions for a Class of $(p,q)$-Laplacian Equations in $\mathbb{R}^N$ with Sign-changing Potentialhttps://projecteuclid.org/euclid.twjm/1554105651<strong>Nian Zhang</strong>, <strong>Gao Jia</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Advance publication, 20 pages.</p><p><strong>Abstract:</strong><br/>
In this paper, we use variational approaches to establish the existence of weak solutions for a class of $(p,q)$-Laplacian equations on $\mathbb{R}^N$, for $1 \lt q \lt p \lt q^{*} := Nq/(N-q)$, $p \lt N$, with a sign-changing potential function and a Carathéodory reaction term which do not satisfy the Ambrosetti-Rabinowitz type growth condition. By linking theorem with Cerami condition, the fountain theorem and dual fountain theorem with Cerami condition, we obtain some existence of weak solutions for the above equations under our considerations which are different from those used in related papers.
</p>projecteuclid.org/euclid.twjm/1554105651_20200112220114Sun, 12 Jan 2020 22:01 ESTThe Hilbert Curve of a $4$-dimensional Scroll with a Divisorial Fiberhttps://projecteuclid.org/euclid.twjm/1552377618<strong>Antonio Lanteri</strong>, <strong>Andrea Luigi Tironi</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Advance publication, 11 pages.</p><p><strong>Abstract:</strong><br/>
In dimension $n = 2m-2 \geq 4$ adjunction theoretic scrolls over a smooth $m$-fold may not be classical scrolls, due to the existence of divisorial fibers. A $4$-dimensional scroll $(X,L)$ over $\mathbb{P}^3$ of this type is considered, and the equation of its Hilbert curve $\Gamma$ is determined in two ways, one of which relies on the fact that $(X,L)$ is at the same time a classical scroll over a threefold $Y \neq \mathbb{P}^3$. It turns out that $\Gamma$ does not perceive divisorial fibers. The equation we obtain also shows that a question raised in [2] has negative answer in general for non-classical scrolls over a $3$-fold. More precisely, the answer for $(X,L)$ is negative or positive according to whether $(X,L)$ is regarded as an adjunction theoretic scroll or as a classical scroll; in other words, it is the answer to this question to distinguish between the existence of jumping fibers or not.
</p>projecteuclid.org/euclid.twjm/1552377618_20200112220114Sun, 12 Jan 2020 22:01 ESTExistence of Nonnegative Solutions for Fourth Order Elliptic Equations of Kirchhoff Type with General Subcritical Growthhttps://projecteuclid.org/euclid.twjm/1551150032<strong>Jianping Huang</strong>, <strong>Qi Zhang</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Advance publication, 16 pages.</p><p><strong>Abstract:</strong><br/>
This paper is dedicated to investigating the following fourth-order elliptic equation with Kirchhoff-type \[ \begin{cases} \displaystyle \Delta^{2} u - \left( a + b \int_{\mathbb{R}^{N}} |\nabla u|^{2} \, dx \right) \Delta u + cu = f(u) &\textrm{in $\mathbb{R}^{N}$}, \\ u \in H^{2}(\mathbb{R}^{N}), \end{cases} \] where $a \gt 0$, $b \geq 0$ and $c \gt 0$ are constants. By using cut-off functional and monotonicity tricks, we prove that the above problem has a positive solution. Our result cover the case where the nonlinearity satisfies asymptotically linear and superlinear condition at the infinity, which extend the results of related literatures.
</p>projecteuclid.org/euclid.twjm/1551150032_20200112220114Sun, 12 Jan 2020 22:01 ESTIterated Commutators of Multilinear Maximal Square Functions on Some Function Spaceshttps://projecteuclid.org/euclid.twjm/1578970817<strong>Zengyan Si</strong>, <strong>Qingying Xue</strong>, <strong>Pu Zhang</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Advance publication, 22 pages.</p><p><strong>Abstract:</strong><br/>
In this paper, the iterated commutators of multilinear maximal square function and pointwise multiplication with functions in Lipschitz spaces are studied. Some new estimates for the iterated commutators with kernels satisfying some Dini type conditions on Lebesgue spaces, homogenous Lipschitz spaces and homogenous Triebel-Lizorkin spaces will be given, respectively.
</p>projecteuclid.org/euclid.twjm/1578970817_20200113220122Mon, 13 Jan 2020 22:01 ESTThe Monochromatic Connectivity of Graphshttps://projecteuclid.org/euclid.twjm/1578884422<strong>Zemin Jin</strong>, <strong>Xueliang Li</strong>, <strong>Kaijun Wang</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Advance publication, 31 pages.</p><p><strong>Abstract:</strong><br/>
In 2011, Caro et al. introduced the monochromatic connection of graphs. An edge-coloring of a connected graph $G$ is called a monochromatically connecting (MC-coloring, for short) if there is a monochromatic path joining any two vertices. The monochromatic connection number $\operatorname{mc}(G)$ of a graph $G$ is the maximum integer $k$ such that there is a $k$-edge-coloring, which is an MC-coloring of $G$. Clearly, a monochromatic spanning tree can monochromatically connect any two vertices. So for a graph $G$ of order $n$ and size $m$, $\operatorname{mc}(G) \geq m-n+2$. Caro et al. proved that both triangle-free graphs and graphs of diameter at least three meet the lower bound.
In this paper, we consider the monochromatic connectivity of graphs containing triangles which meet the lower bound too. Also, in order to study the graphs of diameter two, we present the formula for the monochromatic connectivity of join graphs. This will be helpful to solve the problem for graphs of diameter two.
</p>projecteuclid.org/euclid.twjm/1578884422_20200113220122Mon, 13 Jan 2020 22:01 ESTSingular Limit Solutions for a 4-dimensional Semilinear Elliptic System of Liouville Typehttps://projecteuclid.org/euclid.twjm/1576724419<strong>Sami Baraket</strong>, <strong>Imen Bazarbacha</strong>, <strong>Rima Chetouane</strong>, <strong>Abdellatif Messaoudi</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Advance publication, 55 pages.</p><p><strong>Abstract:</strong><br/>
We consider the existence of singular limit solutions for a nonlinear elliptic system of Liouville type with Navier boundary conditions. We use the nonlinear domain decomposition method and a Pohozaev type identity.
</p>projecteuclid.org/euclid.twjm/1576724419_20200113220122Mon, 13 Jan 2020 22:01 ESTA Parabolic Flow of Almost Balanced Metricshttps://projecteuclid.org/euclid.twjm/1576227617<strong>Masaya Kawamura</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Advance publication, 14 pages.</p><p><strong>Abstract:</strong><br/>
We define a parabolic flow of almost balanced metrics. We show that the flow has a unique solution on compact almost Hermitian manifolds.
</p>projecteuclid.org/euclid.twjm/1576227617_20200113220122Mon, 13 Jan 2020 22:01 ESTA Note on Special Fibers of Shimura Curves and Special Representationshttps://projecteuclid.org/euclid.twjm/1589875217<strong>Yih-Jeng Yu</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Volume 24, Number 3, 523--544.</p><p><strong>Abstract:</strong><br/>
We study the geometry of the special fibers of certain Shimura curves and give a direct proof of global-to-local Jacquet-Langlands compatibility by Čerednik-Drinfel'd uniformizations theorem.
</p>projecteuclid.org/euclid.twjm/1589875217_20200519040036Tue, 19 May 2020 04:00 EDTThe Explicit Formulae and Evaluations of Ramanujan's Remarkable Product of Theta-functionshttps://projecteuclid.org/euclid.twjm/1589875218<strong>Devasirvatham John Prabhakaran</strong>, <strong>Kumar Ranjith Kumar</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Volume 24, Number 3, 545--551.</p><p><strong>Abstract:</strong><br/>
On pages 338 and 339 in his first notebook, Ramanujan defined remarkable product of theta-functions $a_{m,n}$ and also recorded eighteen explicit values depending on two parameters $m$ and $n$. All these values have been established by Berndt et al. In this paper, we establish a new general formulae for the explicit evaluations of $a_{3m,3}$ and $a_{m,9}$ by using $P$-$Q$ mixed modular equation and values for certain class invariant of Ramanujan. Using these formulae, we calculate some new explicit values of $a_{3m,3}$ for $m = 2,7,13,17,25,37$ and $a_{m,9}$ for $m = 17,37$.
</p>projecteuclid.org/euclid.twjm/1589875218_20200519040036Tue, 19 May 2020 04:00 EDTAn Application of Liaison Theory to Zero-dimensional Schemeshttps://projecteuclid.org/euclid.twjm/1589875219<strong>Martin Kreuzer</strong>, <strong>Tran N. K. Linh</strong>, <strong>Le Ngoc Long</strong>, <strong>Tu Chanh Nguyen</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Volume 24, Number 3, 553--573.</p><p><strong>Abstract:</strong><br/>
Given a $0$-dimensional scheme $\mathbb{X}$ in an $n$-dimensional projective space $\mathbb{P}^n_K$ over an arbitrary field $K$, we use liaison theory to characterize the Cayley-Bacharach property of $\mathbb{X}$. Our result extends the result for sets of $K$-rational points given in [8]. In addition, we examine and bound the Hilbert function and regularity index of the Dedekind different of $\mathbb{X}$ when $\mathbb{X}$ has the Cayley-Bacharach property.
</p>projecteuclid.org/euclid.twjm/1589875219_20200519040036Tue, 19 May 2020 04:00 EDTBackward Stability and Divided Invariance of an Attractor for the Delayed Navier-Stokes Equationhttps://projecteuclid.org/euclid.twjm/1589875220<strong>Yangrong Li</strong>, <strong>Qiangheng Zhang</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Volume 24, Number 3, 575--601.</p><p><strong>Abstract:</strong><br/>
We study backward stability of a pullback attractor especially for a delay equation. We introduce a new concept of a backward attractor, which is defined by a compact, pullback attracting and dividedly invariant family. We then show the equivalence between existence of a backward attractor and backward stability of the pullback attractor, and present some criteria by using the backward limit-set compactness of the system. In the application part, we consider the Navier-Stokes equation with a nonuniform Lipschitz delay term and a backward tempered force. Based on the fact that the delay does not change the backward bounds of the velocity field and external forces, we establish the backward-uniform estimates and obtain a backward attractor, which leads to backward stability of the pullback attractor. Some special cases of variable delay and distributed delay are discussed.
</p>projecteuclid.org/euclid.twjm/1589875220_20200519040036Tue, 19 May 2020 04:00 EDTSingularity Formation of the Non-barotropic Compressible Magnetohydrodynamic Equations Without Heat Conductivityhttps://projecteuclid.org/euclid.twjm/1589875221<strong>Xin Zhong</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Volume 24, Number 3, 603--628.</p><p><strong>Abstract:</strong><br/>
We study the singularity formation of strong solutions to the three-dimensional full compressible magnetohydrodynamic equations with zero heat conduction in a bounded domain. We show that for the initial density allowing vacuum, the strong solution exists globally if the density $\rho$, the magnetic field $\mathbf{b}$, and the pressure $P$ satisfy $\|\rho\|_{L^{\infty}(0,T;L^{\infty})} + \|\mathbf{b}\|_{L^{\infty}(0,T;L^6)} + \|P\|_{L^{\infty}(0,T;L^{\infty})} \lt \infty$ and the coefficients of viscosity verity $3\mu \gt \lambda$. This extends the corresponding results in Duan (2017), Fan et al. (2018) [1,2] where a blow-up criterion in terms of the upper bounds of the density, the magnetic field and the temperature was obtained under the condition $2\mu \gt \lambda$. Our proof relies on some delicate energy estimates.
</p>projecteuclid.org/euclid.twjm/1589875221_20200519040036Tue, 19 May 2020 04:00 EDTQuasi-periodic Solutions of Wave Equations with the Nonlinear Term Depending on the Time and Space Variableshttps://projecteuclid.org/euclid.twjm/1589875222<strong>Yi Wang</strong>, <strong>Jie Rui</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Volume 24, Number 3, 629--661.</p><p><strong>Abstract:</strong><br/>
This article is devoted to the study of a wave equation with a constant potential and an $x$-periodic and $t$-quasi-periodic nonlinear term subject to periodic boundary conditions. It is proved that the equation admits small amplitude, linear stable and $t$-quasi-periodic solutions for any constant potential and most frequency vectors.
</p>projecteuclid.org/euclid.twjm/1589875222_20200519040036Tue, 19 May 2020 04:00 EDTQuasi-periodic Solutions for Nonlinear Schrödinger Equations with Legendre Potentialhttps://projecteuclid.org/euclid.twjm/1589875223<strong>Guanghua Shi</strong>, <strong>Dongfeng Yan</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Volume 24, Number 3, 663--679.</p><p><strong>Abstract:</strong><br/>
In this paper, the nonlinear Schrödinger equations with Legendre potential $\mathbf{i} u_{t} - u_{xx} + V_L(x)u + mu + \sec x \cdot |u|^2 u = 0$ subject to certain boundary conditions is considered, where $V_L(x) = -\frac{1}{2} - \frac{1}{4} \tan^2 x$, $x \in (-\pi/2,\pi/2)$. It is proved that for each given positive constant $m \gt 0$, the above equation admits lots of quasi-periodic solutions with two frequencies. The proof is based on a partial Birkhoff normal form technique and an infinite-dimensional Kolmogorov-Arnold-Moser theory.
</p>projecteuclid.org/euclid.twjm/1589875223_20200519040036Tue, 19 May 2020 04:00 EDTStrict Monotonicity and Unique Continuation for General Non-local Eigenvalue Problemshttps://projecteuclid.org/euclid.twjm/1589875224<strong>Silvia Frassu</strong>, <strong>Antonio Iannizzotto</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Volume 24, Number 3, 681--694.</p><p><strong>Abstract:</strong><br/>
We consider the weighted eigenvalue problem for a general non-local pseudo-differential operator, depending on a bounded weight function. For such problem, we prove that strict (decreasing) monotonicity of the eigenvalues with respect to the weight function is equivalent to the unique continuation property of eigenfunctions. In addition, we discuss some unique continuation results for the special case of the fractional Laplacian.
</p>projecteuclid.org/euclid.twjm/1589875224_20200519040036Tue, 19 May 2020 04:00 EDTHigh Spatial Accuracy Analysis of Linear Triangular Finite Element for Distributed Order Diffusion Equationshttps://projecteuclid.org/euclid.twjm/1589875225<strong>Lin He</strong>, <strong>Jincheng Ren</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Volume 24, Number 3, 695--708.</p><p><strong>Abstract:</strong><br/>
In this paper, an effective numerical fully discrete finite element scheme for the distributed order time fractional diffusion equations is developed. By use of the composite trapezoid formula and the well-known $L1$ formula approximation to the distributed order derivative and linear triangular finite element approach for the spatial discretization, we construct a fully discrete finite element scheme. Based on the superclose estimate between the interpolation operator and the Ritz projection operator and the interpolation post-processing technique, the superclose approximation of the finite element numerical solution and the global superconvergence are proved rigorously, respectively. Finally, a numerical example is presented to support the theoretical results.
</p>projecteuclid.org/euclid.twjm/1589875225_20200519040036Tue, 19 May 2020 04:00 EDTOptimality Conditions in Set-valued Optimization Problem with Respect to a Partial Order Relation via Directional Derivativehttps://projecteuclid.org/euclid.twjm/1589875226<strong>Emrah Karaman</strong>, <strong>Mustafa Soyertem</strong>, <strong>İlknur Atasever Güvenç</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Volume 24, Number 3, 709--722.</p><p><strong>Abstract:</strong><br/>
In this study, a new directional derivative is defined by using Minkowski difference. Some properties and existence theorems of this directional derivative are given. Moreover, necessary and sufficient optimality conditions are presented for set-valued optimization problems with respect to $m_1$ order relation via directional derivative.
</p>projecteuclid.org/euclid.twjm/1589875226_20200519040036Tue, 19 May 2020 04:00 EDTA Survey on the Lace Expansion for the Nearest-neighbor Models on the BCC Latticehttps://projecteuclid.org/euclid.twjm/1589875227<strong>Satoshi Handa</strong>, <strong>Yoshinori Kamijima</strong>, <strong>Akira Sakai</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Volume 24, Number 3, 723--784.</p><p><strong>Abstract:</strong><br/>
The aim of this survey is to explain, in a self-contained and relatively beginner-friendly manner, the lace expansion for the nearest-neighbor models of self-avoiding walk and percolation that converges in all dimensions above 6 and 9, respectively. To achieve this, we consider a $d$-dimensional version of the body-centered cubic (BCC) lattice, on which it is extremely easy to enumerate various random-walk quantities. Also, we choose a particular set of bootstrapping functions, by which a notoriously complicated part of the lace-expansion analysis becomes rather transparent.
</p>projecteuclid.org/euclid.twjm/1589875227_20200519040036Tue, 19 May 2020 04:00 EDT