Taiwanese Journal of Mathematics Articles (Project Euclid)
http://projecteuclid.org/euclid.twjm
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
http://projecteuclid.org/
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 EDTLDG Methods for Reaction-diffusion Systems with Application of Krylov Implicit Integration Factor Methodshttps://projecteuclid.org/euclid.twjm/1537927428<strong>Na An</strong>, <strong>Chaobao Huang</strong>, <strong>Xijun Yu</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Advance publication, 23 pages.</p><p><strong>Abstract:</strong><br/>
In this paper, we present an efficient fully-discrete local discontinuous Galerkin (LDG) method for nonlinear reaction-diffusion systems, which are often used as mathematical models for many physical and biological applications. We can derive numerical approximations not only for solutions but also for their gradients at the same time, while most of methods derive numerical solutions only. And due to the strict time-step restriction ($\Delta t = O(h^2_{\min})$) of explicit schemes for stability, we introduce the implicit integration factor (IIF) method based on Krylov subspace approximation, in which the time step can be taken as $\Delta t = O(h_{\min})$. Moreover, the method allows us to compute element by element and avoid solving a global system of nonlinear algebraic equations as the standard implicit schemes do, which can reduce the computational cost greatly. Numerical experiments about the reaction-diffusion equations with exact solutions and the well-studied Schnakenberg system are conducted to illustrate the accuracy, capability and advantages of the method.
</p>projecteuclid.org/euclid.twjm/1537927428_20190114220130Mon, 14 Jan 2019 22:01 ESTSpectral Approximations for Nonlinear Fractional Delay Diffusion Equations with Smooth and Nonsmooth Solutionshttps://projecteuclid.org/euclid.twjm/1537927429<strong>Haiyu Liu</strong>, <strong>Shujuan Lü</strong>, <strong>Hu Chen</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Advance publication, 20 pages.</p><p><strong>Abstract:</strong><br/>
A fully discrete scheme is proposed for the nonlinear fractional delay diffusion equations with smooth solutions, where the fractional derivative is described in Caputo sense with the order $\alpha$ ($0 \lt \alpha \lt 1$). The scheme is constructed by combining finite difference method in time and Legendre spectral approximation in space. Stability and convergence are proved rigorously. Moreover, a modified scheme is proposed for the equation with nonsmooth solutions by adding correction terms to the approximations of fractional derivative operator and nonlinear term. Numerical examples are carried out to support the theoretical analysis.
</p>projecteuclid.org/euclid.twjm/1537927429_20190114220130Mon, 14 Jan 2019 22:01 ESTRestricted Arc Connectivity of Unidirectional Hypercubes and Unidirectional Folded Hypercubeshttps://projecteuclid.org/euclid.twjm/1535680829<strong>Shang-wei Lin</strong>, <strong>Na-qi Fan</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Advance publication, 15 pages.</p><p><strong>Abstract:</strong><br/>
Unidirectional hypercubes and unidirectional folded hypercubes are generalizations of hypercubes and folded hypercubes to digraphs. The super-$\lambda$ property of a digraph is a index for network reliability, which can be measured by the restricted arc-connectivity quantitatively. In this paper, we first show that the restricted arc-connectivity of the $n$-dimensional unidirectional hypercube is $n-1$ when $n$ is even and is $n-2$ when $n$ is odd, and then we show that the restricted arc-connectivity of the $n$-dimensional unidirectional folded hypercube is $n-1$ when $n$ is even and is $n$ when $n$ is odd. As a consequence, we prove that both unidirectional hypercube and unidirectional folded hypercube are super-$\lambda$.
</p>projecteuclid.org/euclid.twjm/1535680829_20190114220130Mon, 14 Jan 2019 22:01 ESTSemi-classical Limit for the Quantum Zakharov Systemhttps://projecteuclid.org/euclid.twjm/1534730421<strong>Yung-Fu Fang</strong>, <strong>Hung-Wen Kuo</strong>, <strong>Hsi-Wei Shih</strong>, <strong>Kuan-Hsiang Wang</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Advance publication, 25 pages.</p><p><strong>Abstract:</strong><br/>
In this paper, we prove the semi-classical limit for the quantum Zakharov system, that is, the quantum Zakharov system converges to the classical Zakharov system as the quantum parameter goes to zero, including a convergence rate. We improve the results of Guo-Zhang-Guo [11].
</p>projecteuclid.org/euclid.twjm/1534730421_20190114220130Mon, 14 Jan 2019 22:01 ESTSkew Generalized Power Series Rings and the McCoy Propertyhttps://projecteuclid.org/euclid.twjm/1534125615<strong>Masoome Zahiri</strong>, <strong>Rasul Mohammadi</strong>, <strong>Abdollah Alhevaz</strong>, <strong>Ebrahim Hashemi</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Advance publication, 23 pages.</p><p><strong>Abstract:</strong><br/>
Given a ring $R$, a strictly totally ordered monoid $(S,\preceq)$ and a monoid homomorphism $\omega \colon S \to \operatorname{End}(R)$, one can construct the skew generalized power series ring $R[[S,\omega,\preceq]]$, consisting all of the functions from a monoid $S$ to a coefficient ring $R$ whose support is artinian and narrow, where the addition is pointwise, and the multiplication is given by convolution twisted by an action $\omega$ of the monoid $S$ on the ring $R$. In this paper, we consider the problem of determining some annihilator and zero-divisor properties of the skew generalized power series ring $R[[S,\omega,\preceq]]$ over an associative non-commutative ring $R$. Providing many examples, we investigate relations between McCoy property of skew generalized power series ring, namely $(S,\omega)$-McCoy property, and other standard ring-theoretic properties. We show that if $R$ is a local ring such that its Jacobson radical $J(R)$ is nilpotent, then $R$ is $(S,\omega)$-McCoy. Also if $R$ is a semicommutative semiregular ring such that $J(R)$ is nilpotent, then $R$ is $(S,\omega)$-McCoy ring.
</p>projecteuclid.org/euclid.twjm/1534125615_20190114220130Mon, 14 Jan 2019 22:01 ESTBreathers of Discrete One-dimensional Nonlinear Schrödinger Equations in Inhomogeneous Mediahttps://projecteuclid.org/euclid.twjm/1533866418<strong>Shuguan Ji</strong>, <strong>Zhenhua Wang</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Advance publication, 16 pages.</p><p><strong>Abstract:</strong><br/>
This paper is concerned with the breathers of discrete one-dimensional nonlinear Schrödinger equations in inhomogeneous media. By using a constrained minimization approach known as the Nehari variational principle or the Nehari manifold approach, we obtain the existence of nontrivial breathers.
</p>projecteuclid.org/euclid.twjm/1533866418_20190114220130Mon, 14 Jan 2019 22:01 ESTArbitrary High-order EQUIP Methods for Stochastic Canonical Hamiltonian Systemshttps://projecteuclid.org/euclid.twjm/1533866419<strong>Xiuyan Li</strong>, <strong>Chiping Zhang</strong>, <strong>Qiang Ma</strong>, <strong>Xiaohua Ding</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Advance publication, 23 pages.</p><p><strong>Abstract:</strong><br/>
This paper is concerned with arbitrary high-order energy-preserving numerical methods for stochastic canonical Hamiltonian systems. Energy and quadratic invariants-preserving (EQUIP) methods for deterministic Hamiltonian systems are applied to stochastic canonical Hamiltonian systems and analyzed accordingly. A class of stochastic parametric Runge-Kutta methods with a truncation technique of random variables are obtained. Increments of Wiener processes are replaced by some truncated random variables. We prove the replacement doesn't change the convergence order under some conditions. The methods turn out to be symplectic for any given parameter. It is shown that there exists a parameter $\alpha_{n}^{*}$ at each step such that the energy-preserving property holds, and the energy-preserving methods retain the order of the underlying stochastic Gauss Runge-Kutta methods. Numerical results illustrate the effectiveness of EQUIP methods when applied to stochastic canonical Hamiltonian systems.
</p>projecteuclid.org/euclid.twjm/1533866419_20190114220130Mon, 14 Jan 2019 22:01 ESTAn Expectation Formula Based on a Maclaurin Expansionhttps://projecteuclid.org/euclid.twjm/1533866420<strong>Mingjin Wang</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Advance publication, 12 pages.</p><p><strong>Abstract:</strong><br/>
In this paper, we obtain an expectation formula with respect to the $q$-probability distribution $W(x,y;q)$ based on a Maclaurin expansion. The formula has many applications in mathematics. Some of the applications are also given, which include a probability version of the Al-Salam and Verma $q$-integral.
</p>projecteuclid.org/euclid.twjm/1533866420_20190114220130Mon, 14 Jan 2019 22:01 ESTWell-posedness and Stability of Two Classes of Plate Equations with Memory and Strong Time-dependent Delayhttps://projecteuclid.org/euclid.twjm/1533866421<strong>Baowei Feng</strong>, <strong>Gongwei Liu</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Advance publication, 34 pages.</p><p><strong>Abstract:</strong><br/>
Two classes of plate equations with past history and strong time-dependent delay in the internal feedback are considered. Our results contain the global well-posedness and exponential stability of the two systems. We prove the global well-posedness of a system with rotational inertia without any restrictions on $\mu_1$, $\mu_2$, and the system without rotational inertia under the assumption $|\mu_2| \leq \mu_1$. For the system with rotational inertia, we establish exponential stability to the plate equation with the memory term only to control the delay term if the amplitude of the time delay term is small, and the stability result also holds for the plate equation with strong anti-damping. For the system without rotational inertia, we obtain the exponential stability under the assumption $|\mu_2| \lt \sqrt{1-d} \mu_1$.
</p>projecteuclid.org/euclid.twjm/1533866421_20190114220130Mon, 14 Jan 2019 22:01 ESTTwo Positive Solutions for Kirchhoff Type Problems with Hardy-Sobolev Critical Exponent and Singular Nonlinearitieshttps://projecteuclid.org/euclid.twjm/1533110480<strong>Yu-Ting Tang</strong>, <strong>Jia-Feng Liao</strong>, <strong>Chun-Lei Tang</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Advance publication, 23 pages.</p><p><strong>Abstract:</strong><br/>
We consider the following singular Kirchhoff type equation with Hardy-Sobolev critical exponent \[ \begin{cases} \displaystyle -\left( a + b \int_{\Omega} |\nabla u|^2 \, dx \right) \Delta u = \frac{u^{3}}{|x|} + \frac{\lambda}{|x|^{\beta} u^{\gamma}}, & x \in \Omega, \\ u > 0, & x \in \Omega, \\ u = 0, & x \in \partial \Omega, \end{cases} \] where $\Omega \subset \mathbb{R}^{3}$ is a bounded domain with smooth boundary $\partial \Omega$, $0 \in \Omega$, $a,b,\lambda \gt 0$, $0 \lt \gamma \lt 1$, and $0 \leq \beta \lt (5+\gamma)/2$. Combining with the variational method and perturbation method, two positive solutions of the equation are obtained.
</p>projecteuclid.org/euclid.twjm/1533110480_20190114220130Mon, 14 Jan 2019 22:01 ESTGramian Schauder Basic Measures and Gramian Uniformly Bounded Linearly Stationary Processeshttps://projecteuclid.org/euclid.twjm/1532397614<strong>Yûichirô Kakihara</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Advance publication, 23 pages.</p><p><strong>Abstract:</strong><br/>
For a normal Hilbert $B(H)$-module $X$ gramian Schauder basic $X$-valued measures are considered. Some equivalence conditions are given for an $X$-valued measure to be gramian Schauder basic. As an application gramian uniformly bounded linearly stationary $X$-valued processes are characterized.
</p>projecteuclid.org/euclid.twjm/1532397614_20190114220130Mon, 14 Jan 2019 22:01 ESTA Calculation Approach to Scalarization for Polyhedral Sets by Means of Set Relationshttps://projecteuclid.org/euclid.twjm/1532333185<strong>Hui Yu</strong>, <strong>Koichiro Ike</strong>, <strong>Yuto Ogata</strong>, <strong>Tamaki Tanaka</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Advance publication, 13 pages.</p><p><strong>Abstract:</strong><br/>
In this paper, we focus on certain functions as scalarization for six types of set relations and discuss calculation algorithms for them between polyhedral sets, while those between polytopes have been already investigated. A major difference between polyhedral sets and polytopes is in boundedness. Polyhedral sets are no longer necessarily bounded. Methods for calculating types (1), (2), (4), (6) are easily available by a similar way to existing ideas. However, those for types (3) and (5), which are actually the most famous and long-standing types, require some technical ways approaching to the value of them by using the fact that finitely generatedness and polyhedrality coincide and can be algorithmically switched in finite-dimensional spaces. As a result, we show all types are reduced to a finite number of linear programming problems. Also, we demonstrate our methods through an example and give detailed calculation process.
</p>projecteuclid.org/euclid.twjm/1532333185_20190114220130Mon, 14 Jan 2019 22:01 ESTA Menon-type Identity with Multiplicative and Additive Charactershttps://projecteuclid.org/euclid.twjm/1531382426<strong>Yan Li</strong>, <strong>Xiaoyu Hu</strong>, <strong>Daeyeoul Kim</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Advance publication, 11 pages.</p><p><strong>Abstract:</strong><br/>
This paper studies Menon-type identities involving both multiplicative characters and additive characters. In the paper, we shall give the explicit formula of the following sum \[ \sum_{\substack{a \in \mathbb{Z}_n^{\ast} \\ b_1, \ldots, b_k \in \mathbb{Z}_n}} \gcd(a-1, b_1, \ldots, b_k, n) \chi(a) \lambda_1(b_1) \cdots \lambda_k(b_k), \] where for a positive integer $n$, $\mathbb{Z}_n^{\ast}$ is the group of units of the ring $\mathbb{Z}_n = \mathbb{Z}/n\mathbb{Z}$, $\gcd$ represents the greatest common divisor, $\chi$ is a Dirichlet character modulo $n$, and for a nonnegative integer $k$, $\lambda_1, \ldots, \lambda_k$ are additive characters of $\mathbb{Z}_n$. Our formula further extends the previous results by Sury [13], Zhao-Cao [17] and Li-Hu-Kim [4].
</p>projecteuclid.org/euclid.twjm/1531382426_20190114220130Mon, 14 Jan 2019 22:01 ESTBounds for the Lifespan of Solutions to Fourth-order Hyperbolic Equations with Initial Data at Arbitrary Energy Levelhttps://projecteuclid.org/euclid.twjm/1547715693<strong>Bin Guo</strong>, <strong>Xiaolei Li</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Advance publication, 17 pages.</p><p><strong>Abstract:</strong><br/>
This paper deals with lower and upper bounds for the lifespan of solutions to a fourth-order nonlinear hyperbolic equation with strong damping: \[ u_{tt} + \Delta^{2} u - \Delta u - \omega \Delta u_t + \alpha(t) u_t = |u|^{p-2} u. \] First of all, the authors construct a new control function and apply the Sobolev embedding inequality to establish some qualitative relationships between initial energy value and the norm of the gradient of the solution for supercritical case ($2(N-2)/(N-4) \lt p \lt 2N/(N-4)$, $N \geq 5$). And then, the concavity argument is used to prove that the solution blows up in finite time for initial data at low energy level, at the same time, an estimate of the upper bound of blow-up time is also obtained.
Subsequently, for initial data at high energy level, the authors prove the monotonicity of the $L^{2}$ norm of the solution under suitable assumption of initial data, furthermore, we utilize the concavity argument and energy methods to prove that the solution also blows up in finite time for initial data at high energy level.
At last, for the supercritical case, a new control functional with a small dissipative term and an inverse Hölder inequality with correction constants are employed to overcome the difficulties caused by the failure of the embedding inequality ($H^{2}(\Omega) \cap H^{1}_{0}(\Omega) \hookrightarrow L^{2p-2}$ for $2(N-2)/(N-4) \lt p \lt 2N/(N-4)$) and then an explicit lower bound for blow-up time is obtained. Such results extend and improve those of [S. T. Wu, J. Dyn. Control Syst. 24 (2018), no. 2, 287--295].
</p>projecteuclid.org/euclid.twjm/1547715693_20190117040240Thu, 17 Jan 2019 04:02 ESTGeneralized Fractional Integral Operators and Their Commutators with Functions in Generalized Campanato Spaces on Orlicz Spaceshttps://projecteuclid.org/euclid.twjm/1546506192<strong>Minglei Shi</strong>, <strong>Ryutaro Arai</strong>, <strong>Eiichi Nakai</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Advance publication, 26 pages.</p><p><strong>Abstract:</strong><br/>
We investigate the commutators $[b,I_{\rho}]$ of generalized fractional integral operators $I_{\rho}$ with functions $b$ in generalized Campanato spaces and give a necessary and sufficient condition for the boundedness of the commutators on Orlicz spaces. To do this we define Orlicz spaces with generalized Young functions and prove the boundedness of generalized fractional maximal operators on the Orlicz spaces.
</p>projecteuclid.org/euclid.twjm/1546506192_20190117040240Thu, 17 Jan 2019 04:02 ESTA Numerical Method Based on the Jacobi Polynomials to Reconstruct an Unknown Source Term in a Time Fractional Diffusion-wave Equationhttps://projecteuclid.org/euclid.twjm/1546419620<strong>Somayeh Nemati</strong>, <strong>Afshin Babaei</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Advance publication, 19 pages.</p><p><strong>Abstract:</strong><br/>
In this paper, we consider an inverse problem of identifying an unknown time dependent source function in a time-fractional diffusion-wave equation. First, some basic properties of the shifted Jacobi polynomials (SJPs) are presented. Then, the analytical solution of the direct problem is given and used to obtain an approximation of the unknown source function in a series of SJPs. Due to ill-posedness of this inverse problem, the Tikhonov regularization method with Morozov's discrepancy principle criterion is applied to find a stable solution. After that, an error bound is obtained for the approximation of the unknown source function. Finally, some numerical examples are provided to show effectiveness and robustness of the proposed algorithm.
</p>projecteuclid.org/euclid.twjm/1546419620_20190117040240Thu, 17 Jan 2019 04:02 ESTAdmissibly Stable Manifolds for a Class of Partial Neutral Functional Differential Equations on a Half-linehttps://projecteuclid.org/euclid.twjm/1545966022<strong>Thieu Huy Nguyen</strong>, <strong>Xuan Yen Trinh</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Advance publication, 27 pages.</p><p><strong>Abstract:</strong><br/>
For the following class of partial neutral functional differential equations \[ \begin{cases} \frac{\partial}{\partial t} Fu_t = B(t) u(t) + \Phi(t,u_t) &t \in (0,\infty), \\ u_0 = \phi \in \mathcal{C} := C([-r,0],X) \end{cases} \] we prove the existence of a new type of invariant stable and center-stable manifolds, called admissibly invariant manifolds of $\mathcal{E}$-class for the solutions. The existence of such manifolds is obtained under the conditions that the family of linear partial differential operators $(B(t))_{t \geq 0}$ generates the evolution family $\{U(t,s)\}_{t \geq s \geq 0}$ (on Banach space $X$) having an exponential dichotomy or trichotomy on the half-line and the nonlinear delay operator $\Phi$ satisfies the $\varphi$-Lipschitz condition, i.e., $\|\Phi(t,\phi)-\Phi(t,\psi)\| \leq \varphi(t) \|\phi-\psi\|_{\mathcal{C}}$ for $\phi,\psi \in \mathcal{C}$, where $\varphi(t)$ belongs to some admissible function space on the half-line. Our main method is based on Lyapunov-Perrons equations combined with the admissibility of function spaces and fixed point arguments.
</p>projecteuclid.org/euclid.twjm/1545966022_20190117040240Thu, 17 Jan 2019 04:02 ESTPower-free Values of Strongly $Q$-additive Functionshttps://projecteuclid.org/euclid.twjm/1545966023<strong>Karam Aloui</strong>, <strong>Mohamed Mkaouar</strong>, <strong>Walid Wannes</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Advance publication, 22 pages.</p><p><strong>Abstract:</strong><br/>
Let $f$ be a strongly $q$-additive function with integer values. Given an integer $k \geq 2$, we try to estimate the number of positive integers $n \leq N$ (resp. primes $p \leq N$) for which $f(n)$ is $k$-free (resp. $f(p)$ is $k$-free).
</p>projecteuclid.org/euclid.twjm/1545966023_20190117040240Thu, 17 Jan 2019 04:02 ESTAn Analytic Version of Wiener-Itô Decomposition on Abstract Wiener Spaceshttps://projecteuclid.org/euclid.twjm/1545966024<strong>Yuh-Jia Lee</strong>, <strong>Hsin-Hung Shih</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Advance publication, 19 pages.</p><p><strong>Abstract:</strong><br/>
In this paper, we first establish an analogue of Wiener-Itô theorem on finite-dimensional Gaussian spaces through the inverse $S$-transform, that is, the Gauss transform on Segal-Bargmann spaces. Based on this point of view, on infinite-dimensional abstract Wiener space $(H,B)$, we apply the analyticity of the $S$-transform, which is an isometry from the $L^2$-space onto the Bargmann-Segal-Dwyer space, to study the regularity. Then, by defining the Gauss transform on Bargmann-Segal-Dwyer space and showing the relationship with the $S$-transform, an analytic version of Wiener-Itô decomposition will be obtained.
</p>projecteuclid.org/euclid.twjm/1545966024_20190117040240Thu, 17 Jan 2019 04:02 ESTTraveling Waves for a Spatial SIRI Epidemic Modelhttps://projecteuclid.org/euclid.twjm/1545361214<strong>Zhiting Xu</strong>, <strong>Yixin Xu</strong>, <strong>Yehui Huang</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Advance publication, 26 pages.</p><p><strong>Abstract:</strong><br/>
The aim of this paper is to study the traveling waves in a spatial SIRI epidemic model arising from herpes viral infection. We obtain the complete information about the existence and non-existence of traveling waves in the model. Namely, we prove that when the basic reproduction number $\mathcal{R}_0 \gt 1$, there exists a critical wave speed $c^* \gt 0$ such that for each $c \gt c^*$, the model admits positive traveling waves; and for $c \lt c^*$, the model has no non-negative and bounded traveling wave. We also give some numerical simulations to illustrate our analytic results.
</p>projecteuclid.org/euclid.twjm/1545361214_20190117040240Thu, 17 Jan 2019 04:02 ESTCounting Permutations by Simsun Successionshttps://projecteuclid.org/euclid.twjm/1531382427<strong>Yen-Chi Roger Lin</strong>, <strong>Shi-Mei Ma</strong>, <strong>Yeong-Nan Yeh</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Advance publication, 13 pages.</p><p><strong>Abstract:</strong><br/>
In this paper, we introduce the definitions of simsun succession statistics and simsun patterns. In addition to its original definition by Brenti, we give two more combinatorial interpretations of the $q$-Eulerian polynomials using simsun successions. We also present a bijection between permutations avoiding the simsun pattern $132$ and set partitions.
</p>projecteuclid.org/euclid.twjm/1531382427_20190117040240Thu, 17 Jan 2019 04:02 ESTDynamics of Riemannian $1$-foliations on $3$-manifoldshttps://projecteuclid.org/euclid.twjm/1531382429<strong>Jaeyoo Choy</strong>, <strong>Hahng-Yun Chu</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Advance publication, 13 pages.</p><p><strong>Abstract:</strong><br/>
In this paper we study several dynamical properties of a Riemannian $1$-dimensional foliation $\mathcal{L}$ on an oriented closed $3$-manifold $M$. Carrière [6] classified such pairs $(M,\mathcal{L})$. Using the classification we describe in detail recurrence points, $\omega$-limit sets and attractors. Finally, using the fact that the Poincaré map on a transversal surface for a Riemannian $1$-dimensional foliation is an isometry, we show the nonhyperbolicity of $(M,\mathcal{L})$.
</p>projecteuclid.org/euclid.twjm/1531382429_20190117040240Thu, 17 Jan 2019 04:02 ESTThe Diameter of Unit Graphs of Ringshttps://projecteuclid.org/euclid.twjm/1529028016<strong>Huadong Su</strong>, <strong>Yangjiang Wei</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Advance publication, 10 pages.</p><p><strong>Abstract:</strong><br/>
Let $R$ be a ring. The unit graph of $R$, denoted by $G(R)$, is the simple graph defined on all elements of $R$, and where two distinct vertices $x$ and $y$ are linked by an edge if and only if $x+y$ is a unit of $R$. The diameter of a simple graph $G$, denoted by $\operatorname{diam}(G)$, is the longest distance between all pairs of vertices of the graph $G$. In the present paper, we prove that for each integer $n \geq 1$, there exists a ring $R$ such that $n \leq \operatorname{diam}(G(R)) \leq 2n$. We also show that $\operatorname{diam}(G(R)) \in \{ 1,2,3,\infty \}$ for a ring $R$ with $R/J(R)$ self-injective and classify all those rings with $\operatorname{diam}(G(R)) = 1,2,3$ and $\infty$, respectively. This extends [12, Theorem 2 and Corollary 1].
</p>projecteuclid.org/euclid.twjm/1529028016_20190117040240Thu, 17 Jan 2019 04:02 ESTFactors of Sums and Alternating Sums of Products of $q$-binomial Coefficients and Powers of $q$-integershttps://projecteuclid.org/euclid.twjm/1528509849<strong>Victor J. W. Guo</strong>, <strong>Su-Dan Wang</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Advance publication, 17 pages.</p><p><strong>Abstract:</strong><br/>
We prove that, for all positive integers $n_1, \ldots, n_m$, $n_{m+1} = n_1$, and non-negative integers $j$ and $r$ with $j \leq m$, the following two expressions \begin{gather*} \frac{1}{[n_1+n_m+1]} {n_1+n_{m} \brack n_1}^{-1} \sum_{k=0}^{n_1} q^{j(k^2+k) - (2r+1)k} [2k+1]^{2r+1} \prod_{i=1}^m {n_i+n_{i+1}+1 \brack n_i-k}, \\ \frac{1}{[n_1+n_m+1]} {n_1+n_{m} \brack n_1}^{-1} \sum_{k=0}^{n_1} (-1)^k q^{\binom{k}{2} + j(k^2+k) - 2rk} [2k+1]^{2r+1} \prod_{i=1}^m {n_i+n_{i+1}+1 \brack n_i-k} \end{gather*} are Laurent polynomials in $q$ with integer coefficients, where $[n] = 1+q+\cdots+q^{n-1}$ and ${n \brack k} = \prod_{i=1}^k (1-q^{n-i+1})/(1-q^i)$. This gives a $q$-analogue of some divisibility results of sums and alternating sums involving binomial coefficients and powers of integers obtained by Guo and Zeng. We also confirm some related conjectures of Guo and Zeng by establishing their $q$-analogues. Several conjectural congruences for sums involving products of $q$-ballot numbers $\left( {2n \brack n-k} - {2n \brack n-k-1} \right)$ are proposed in the last section of this paper.
</p>projecteuclid.org/euclid.twjm/1528509849_20190117040240Thu, 17 Jan 2019 04:02 ESTNorm-attaining Composition Operators on Lipschitz Spaceshttps://projecteuclid.org/euclid.twjm/1528509850<strong>Antonio Jiménez-Vargas</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Advance publication, 16 pages.</p><p><strong>Abstract:</strong><br/>
Every composition operator $C_{\varphi}$ on the Lipschitz space $\operatorname{Lip}_0(X)$ attains its norm. This fact is essentially known and we give in this paper a sequential characterization of the extremal functions for the norm of $C_{\varphi}$ on $\operatorname{Lip}_0(X)$. We also characterize the norm-attaining composition operators $C_{\varphi}$ on the little Lipschitz space $\operatorname{lip}_0(X)$ which separates points uniformly and identify the extremal functions for the norm of $C_{\varphi}$ on $\operatorname{lip}_0(X)$. We deduce that compact composition operators on $\operatorname{lip}_0(X)$ are norm-attaining whenever the sphere unit of $\operatorname{lip}_0(X)$ separates points uniformly. In particular, this condition is satisfied by spaces of little Lipschitz functions on Hölder compact metric spaces $(X,d^{\alpha})$ with $0 \lt \alpha \lt 1$.
</p>projecteuclid.org/euclid.twjm/1528509850_20190117040240Thu, 17 Jan 2019 04:02 ESTParametrized Multilinear Littlewood-Paley Operators on Hardy Spaceshttps://projecteuclid.org/euclid.twjm/1528509851<strong>Sha He</strong>, <strong>Qingying Xue</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Advance publication, 15 pages.</p><p><strong>Abstract:</strong><br/>
In this paper, we study the parametrized multilinear Marcinkiewicz integral $\mu^{\rho}$ and the multilinear Littlewood-Paley $g_{\lambda}^{*}$-function. We proved that if the kernel $\Omega$ associated to parametrized multilinear Marcinkiewicz integral $\mu^{\rho}$ is homogeneous of degree zero and satisfies the Lipschitz continuous condition, or the kernel $K$ associated to the multilinear Littlewood-Paley $g_{\lambda}^{*}$-function satisfies the Hörmander condition, then they are bounded from $H^{p_1} \times \cdots \times H^{p_m}$ to $L^p$ with $mn/(mn+\gamma) \lt p_1, \ldots, p_m \leq 1$ and $1/p = 1/p_1 + \cdots + 1/p_m$.
</p>projecteuclid.org/euclid.twjm/1528509851_20190117040240Thu, 17 Jan 2019 04:02 ESTPhantom Ideals and Cotorsion Pairs in Extriangulated Categorieshttps://projecteuclid.org/euclid.twjm/1528509854<strong>Tiwei Zhao</strong>, <strong>Zhaoyong Huang</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Advance publication, 33 pages.</p><p><strong>Abstract:</strong><br/>
In this paper, we introduce and study relative phantom morphisms in extriangulated categories defined by Nakaoka and Palu. Then using their properties, we show that if $(\mathscr{C},\mathbb{E},\mathfrak{s})$ is an extriangulated category with enough injective objects and projective objects, then there exists a bijective correspondence between any two of the following classes: (1) special precovering ideals of $\mathscr{C}$; (2) special preenveloping ideals of $\mathscr{C}$; (3) additive subfunctors of $\mathbb{E}$ having enough special injective morphisms; and (4) additive subfunctors of $\mathbb{E}$ having enough special projective morphisms. Moreover, we show that if $(\mathscr{C},\mathbb{E}, \mathfrak{s})$ is an extriangulated category with enough injective objects and projective morphisms, then there exists a bijective correspondence between the following two classes: (1) all object-special precovering ideals of $\mathscr{C}$; (2) all additive subfunctors of $\mathbb{E}$ having enough special injective objects.
</p>projecteuclid.org/euclid.twjm/1528509854_20190117040240Thu, 17 Jan 2019 04:02 ESTClassification and Evolution of Bifurcation Curves for a Dirichlet-Neumann Boundary Value Problem and its Applicationhttps://projecteuclid.org/euclid.twjm/1527127365<strong>Da-Chang Kuo</strong>, <strong>Shin-Hwa Wang</strong>, <strong>Yu-Hao Liang</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Advance publication, 25 pages.</p><p><strong>Abstract:</strong><br/>
We study the classification and evolution of bifurcation curves of positive solutions for the one-dimensional Dirichlet-Neumann boundary value problem \[ \begin{cases} u''(x) + \lambda f(u) = 0, \quad 0 \lt x \lt 1, \\ u(0) = 0, \quad u'(1) = -c \lt 0, \end{cases} \] where $\lambda \gt 0$ is a bifurcation parameter and $c \gt 0$ is an evolution parameter. We mainly prove that, under some suitable assumptions on $f$, there exists $c_{1} \gt 0$, such that, on the $(\lambda,\|u\|_{\infty})$-plane, (i) when $0 \lt c \lt c_{1}$, the bifurcation curve is $S$-shaped; (ii) when $c \geq c_{1}$, the bifurcation curve is $\subset$-shaped. Our results can be applied to the one-dimensional perturbed Gelfand equation with $f(u) = \exp \big( \frac{au}{a+u} \big)$ for $a \geq 4.37$.
</p>projecteuclid.org/euclid.twjm/1527127365_20190117040240Thu, 17 Jan 2019 04:02 ESTA Characterization of Weighted Carleson Measure Spaceshttps://projecteuclid.org/euclid.twjm/1526889718<strong>Hsun-Wu Liu</strong>, <strong>Kunchuan Wang</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Advance publication, 25 pages.</p><p><strong>Abstract:</strong><br/>
Using Frazier and Jawerth's $\varphi$-transform, we characterize weighted generalized Carleson measure spaces $\dot{C}MO^{\alpha,q}_{p,w}$ for a weight $w$ and show that the definition of this space is well-defined by a Plancherel-Pôlya inequality. Note that $\dot{C}MO^{0,2}_{1,w}$ is the weighted $BMO$ space.
</p>projecteuclid.org/euclid.twjm/1526889718_20190117040240Thu, 17 Jan 2019 04:02 ESTA Survey of Threshold Regression for Time-to-event Analysis and Applicationshttps://projecteuclid.org/euclid.twjm/1548406820<strong>Mei-Ling Ting Lee</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Volume 23, Number 2, 293--305.</p><p><strong>Abstract:</strong><br/>
In analyzing time-to-event data, proportional hazards (PH) regression is an ubiquitous model used in many fields. PH regression, however, requires a strong assumption that is not always appropriate. Threshold regression (TR) is one of the alternative models. A first-hitting-time (FHT) survival model postulates a health status process for a patient that gradually declines until the patient dies when the health level first reaches a critical threshold. In this article, we review the development of threshold regression models and their applications.
</p>projecteuclid.org/euclid.twjm/1548406820_20190320040039Wed, 20 Mar 2019 04:00 EDTLocalized Front Structures in FitzHugh-Nagumo Equationshttps://projecteuclid.org/euclid.twjm/1544086877<strong>Chao-Nien Chen</strong>, <strong>Che-Hao Lin</strong>, <strong>Shyuh-yaur Tzeng</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Volume 23, Number 2, 333--349.</p><p><strong>Abstract:</strong><br/>
We are interested in various types of localized waves in FitzHugh-Nagumo equations. Variational methods have been successfully worked out to establish the existence of traveling and standing waves. Starting with a simple planar traveling front, an ordered method is employed to demonstrate different front propagation between two stable equilibria. If these two stable equilibria are in the same energy level, a saddle-focus condition ensures that there are infinite number of standing waves with multiple fronts.
</p>projecteuclid.org/euclid.twjm/1544086877_20190320040039Wed, 20 Mar 2019 04:00 EDTWeighted Endpoint Estimates for Singular Integral Operators Associated with Zygmund Dilationshttps://projecteuclid.org/euclid.twjm/1545361216<strong>Yongsheng Han</strong>, <strong>Ji Li</strong>, <strong>Chin-Cheng Lin</strong>, <strong>Chaoqiang Tan</strong>, <strong>Xinfeng Wu</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Volume 23, Number 2, 375--408.</p><p><strong>Abstract:</strong><br/>
The main purpose of this paper is to study multi-parameter singular integral operators which commute with Zygmund dilations. We develop the theory of the weighted multi-parameter Hardy space $H^p_{\mathfrak{z},w}$ and prove the boundedness for these operators on $H^p_{\mathfrak{z},w}$ for certain $p \leq 1$, which provide endpoint estimates for those singular integral operators studied by Ricci-Stein [31] and Fefferman-Pipher [15]. We also establish the Calderón-Zygmund decomposition and interpolation theorem in this setting.
</p>projecteuclid.org/euclid.twjm/1545361216_20190320040039Wed, 20 Mar 2019 04:00 EDTExact Bounds and Approximating Solutions to the Fredholm Integral Equations of Chandrasekhar Typehttps://projecteuclid.org/euclid.twjm/1542855640<strong>Sheng-Ya Feng</strong>, <strong>Der-Chen Chang</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Volume 23, Number 2, 409--425.</p><p><strong>Abstract:</strong><br/>
In this paper, we study the $L^p$ solutions of the Fredholm integral equations with Chandrasekhar kernels. The Hilbert type inequality is resorted to establish an existence and uniqueness result for the Fredholm integral equation associated with Chandrasekhar kernel. A couple of examples well support the condition and extend the classical results in the literature with one generalizing the classical Chandrasekhar kernel. In order to approximate the original solution, a truncated operator is introduced to overcome the non-compactness of the integral operator. An error estimate of the convergence is made in terms of the truncated parameter, the upper bounds of the symbolic function constituting the integral kernel and initial data to the equation.
</p>projecteuclid.org/euclid.twjm/1542855640_20190320040039Wed, 20 Mar 2019 04:00 EDTBall Average Characterizations of Variable Besov-type Spaceshttps://projecteuclid.org/euclid.twjm/1545361215<strong>Ciqiang Zhuo</strong>, <strong>Der-Chen Chang</strong>, <strong>Dachun Yang</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Volume 23, Number 2, 427--452.</p><p><strong>Abstract:</strong><br/>
In this article, the authors characterize the variable Besov-type spaces $B_{p(\cdot),q(\cdot)}^{s(\cdot),\phi}(\mathbb{R}^n)$, with $1/p(\cdot)$ and $1/q(\cdot)$ satisfying the globally log-Hölder continuous conditions, via Peetre maximal functions and averages on balls. The latter characterization, via averages on balls, gives one way to introduce these spaces on metric measure spaces.
</p>projecteuclid.org/euclid.twjm/1545361215_20190320040039Wed, 20 Mar 2019 04:00 EDTCompressed Hierarchical Schur Algorithm for Frequency-domain Analysis of Photonic Structureshttps://projecteuclid.org/euclid.twjm/1544432420<strong>Cheng-Han Du</strong>, <strong>Yih-Peng Chiou</strong>, <strong>Weichung Wang</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Volume 23, Number 2, 473--501.</p><p><strong>Abstract:</strong><br/>
Three-dimensional finite-difference frequency-domain analyses of partially periodic photonic structures result in large-scale ill-conditioned linear systems. Due to the lack of efficient preconditioner and reordering scheme, existed general-purpose iterative and direct solvers are inadequate to solve these linear systems in time or memory. We propose an efficient direct solver to tackle this problem. By exploring the physical properties, the coefficient matrix structure, and hardware computing efficiency, we extend the concepts of grid geometry manipulation and multi-level Schur method to propose the Compressed Hierarchical Schur algorithm (CHiS). The proposed CHiS algorithm can use less memory and remove redundant computational workloads due to the homogeneity and periodicity of photonic structures. Moreover, CHiS relies on dense BLAS3 operations of sub-matrices that can be computed efficiently with strong scalability by the latest multicore processors or accelerators. The implementation and benchmarks of CHiS demonstrate promising memory usage, timing, and scalability results. The feasibility of future hardware acceleration for CHiS is also addressed using computational data. This high-performance analysis tool can improve the design and modeling capability for various photonic structures.
</p>projecteuclid.org/euclid.twjm/1544432420_20190320040039Wed, 20 Mar 2019 04:00 EDTDecay Solutions and Decay Rate for a Class of Retarded Abtract Semilinear Fractional Evolution Inclusionshttps://projecteuclid.org/euclid.twjm/1542013227<strong>Do Lan</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Volume 23, Number 3, 625--651.</p><p><strong>Abstract:</strong><br/>
In this paper, we prove the existence of decay integral solutions to a class of fractional differential inclusions with finite delays and estimate their decay rate. For these purposes, we have to construct a suitable regular measure of noncompactness on the space of solutions and then deploy the fixed point theory for condensing multivalued maps. An application to a class of fractional PDE with almost sectorial operator is also given.
</p>projecteuclid.org/euclid.twjm/1542013227_20190520040400Mon, 20 May 2019 04:04 EDTChromatic Number and Orientations of Graphs and Signed Graphshttps://projecteuclid.org/euclid.twjm/1563436867<strong>Hao Qi</strong>, <strong>Tsai-Lien Wong</strong>, <strong>Xuding Zhu</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Volume 23, Number 4, 767--776.</p><p><strong>Abstract:</strong><br/>
Assume $D$ is a digraph, and $D'$ is a spanning sub-digraph of $D$. We say $D'$ is a modulo-$k$ Eulerian sub-digraph of $D$ if for each vertex $v$ of $D'$, $d_{D'}^+(v) \equiv d_{D'}^-(v) \pmod{k}$. A modulo-$k$ Eulerian sub-digraph $D'$ of $D$ is special if for every vertex $v$, $d_D^+(v) = 0$ implies $d_{D'}^-(v) = 0$ and $d_{D'}^+(v) = d_D^+(v) > 0$ implies $d_{D'}^-(v) > 0$. We denote by $\operatorname{OE}_k(D)$ or $\operatorname{EE}_k(D)$ (respectively, $\operatorname{OE}_k^s(D)$ or $\operatorname{EE}_k^s(D)$) the sets of spanning modulo-$k$ Eulerian sub-digraphs (respectively, the sets of spanning special modulo-$k$ Eulerian sub-digraphs) of $D$ with an odd number or even number of edges. Matiyasevich [A criterion for vertex colorability of a graph stated in terms of edge orientations, (in Russia), Diskretnyi Analiz, issue 26, 65--71 (1974)] proved that a graph $G$ is $k$-colourable if and only if $G$ has an orientation $D$ such that $|\operatorname{OE}_k(D)| \neq |\operatorname{EE}_k(D)|$. In this paper, we give another characterization of $k$-colourable graphs: a graph $G$ is $k$-colourable if and only if $G$ has an orientation $D$ such that $|\operatorname{OE}_{k-1}^s(D)| \neq |\operatorname{EE}_{k-1}^s(D)|$. We extend the characterizations of $k$-colourable graphs to $k$-colourable signed graphs: If $k$ is an even integer, then a signed graph $(G,\sigma)$ is $k$-colourable if and only if $G$ has an orientation $D$ such that $|\operatorname{OE}_k(D)| \neq |\operatorname{EE}_k(D)|$; if $k$ is an odd integer, then $(G,\sigma)$ is $k$-colourable if and only if $G$ has an orientation $D$ such that $|\operatorname{OE}_{k-1}^s(D)| \neq |\operatorname{EE}_{k-1}^s(D)|$, where a (special) modulo-$k$ Eulerian sub-digraph is even or odd if it has an even or odd number of positive edges. The characterization of $k$-colourable signed graphs for even $k$ (respectively, for odd $k$) fails for odd $k$ (respectively, for even $k$).
</p>projecteuclid.org/euclid.twjm/1563436867_20190718040119Thu, 18 Jul 2019 04:01 EDTPower-free Values of Strongly $Q$-additive Functionshttps://projecteuclid.org/euclid.twjm/1563436868<strong>Karam Aloui</strong>, <strong>Mohamed Mkaouar</strong>, <strong>Walid Wannes</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Volume 23, Number 4, 777--798.</p><p><strong>Abstract:</strong><br/>
Let $f$ be a strongly $q$-additive function with integer values. Given an integer $k \geq 2$, we try to estimate the number of positive integers $n \leq N$ (resp. primes $p \leq N$) for which $f(n)$ is $k$-free (resp. $f(p)$ is $k$-free).
</p>projecteuclid.org/euclid.twjm/1563436868_20190718040119Thu, 18 Jul 2019 04:01 EDTMinimal Ideals and Primitivity in Near-ringshttps://projecteuclid.org/euclid.twjm/1563436869<strong>Gerhard Wendt</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Volume 23, Number 4, 799--820.</p><p><strong>Abstract:</strong><br/>
We address and answer the question when a minimal ideal of a zero symmetric near-ring is a primitive near-ring. This implies that a minimal ideal of a zero symmetric near-ring is a simple near-ring in many natural situations.
</p>projecteuclid.org/euclid.twjm/1563436869_20190718040119Thu, 18 Jul 2019 04:01 EDTWeighted $L^p$ Boundary Value Problems for Laplace's Equation on (Semi-)Convex Domainshttps://projecteuclid.org/euclid.twjm/1563436870<strong>Sibei Yang</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Volume 23, Number 4, 821--840.</p><p><strong>Abstract:</strong><br/>
Let $n \geq 2$ and $\Omega$ be a bounded (semi-)convex domain in $\mathbb{R}^n$. Assume that $p \in (1,\infty)$ and $\omega \in A_p(\partial \Omega)$, where $A_p(\partial \Omega)$ denotes the Muckenhoupt weight class on $\partial \Omega$, the boundary of $\Omega$. In this article, the author proves that the Dirichlet and Neumann problems for Laplace's equation on $\Omega$ with boundary data in the weighted space $L^p_{\omega}(\partial \Omega)$ are uniquely solvable. Moreover, the unique solvability of the Regularity problem for Laplace's equation on $\Omega$ with boundary data in the weighted Sobolev space $\dot{W}^p_{1,\omega}(\partial \Omega)$ is also obtained. Furthermore, the weighted $L^p_{\omega}(\partial \Omega)$-estimates for the Dirichlet, Regularity and Neumann problems are established.
</p>projecteuclid.org/euclid.twjm/1563436870_20190718040119Thu, 18 Jul 2019 04:01 EDTGap Theorems on Critical Point Equation of the Total Scalar Curvature with Divergence-free Bach Tensorhttps://projecteuclid.org/euclid.twjm/1563436871<strong>Gabjin Yun</strong>, <strong>Seungsu Hwang</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Volume 23, Number 4, 841--855.</p><p><strong>Abstract:</strong><br/>
On a compact $n$-dimensional manifold, it is well known that a critical metric of the total scalar curvature, restricted to the space of metrics with unit volume is Einstein. It has been conjectured that a critical metric of the total scalar curvature, restricted to the space of metrics with constant scalar curvature of unit volume, will be Einstein. This conjecture, proposed in 1987 by Besse, has not been resolved except when $M$ has harmonic curvature or the metric is Bach flat. In this paper, we prove some gap properties under divergence-free Bach tensor condition for $n \geq 5$, and a similar condition for $n = 4$.
</p>projecteuclid.org/euclid.twjm/1563436871_20190718040119Thu, 18 Jul 2019 04:01 EDTInfinitely Many Solutions for Sublinear Modified Nonlinear Schrödinger Equations Perturbed from Symmetryhttps://projecteuclid.org/euclid.twjm/1563436872<strong>Liang Zhang</strong>, <strong>Xianhua Tang</strong>, <strong>Yi Chen</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Volume 23, Number 4, 857--882.</p><p><strong>Abstract:</strong><br/>
In this paper, we consider the existence of infinitely many solutions for the following perturbed modified nonlinear Schrödinger equations \[ \begin{cases} -\Delta u - \Delta(|u|^{\alpha}) |u|^{\alpha-2}u = g(x,u) + h(x,u) &x \in \Omega, \\ u = 0 &x \in \partial \Omega, \end{cases} \] where $\Omega$ is a bounded smooth domain in $\mathbb{R}^N$ ($N \geq 1$) and $\alpha \geq 2$. Under the condition that $g(x,u)$ is sublinear near origin with respect to $u$, we study the effect of non-odd perturbation term $h(x,u)$ which breaks the symmetry of the associated energy functional. With the help of modified Rabinowitz's perturbation method and the truncation method, we prove that this equation possesses a sequence of small negative energy solutions approaching to zero.
</p>projecteuclid.org/euclid.twjm/1563436872_20190718040119Thu, 18 Jul 2019 04:01 EDTAttractors for a Class of Kirchhoff Models with $p$-Laplacian and Time Delayhttps://projecteuclid.org/euclid.twjm/1563436873<strong>Sun-Hye Park</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Volume 23, Number 4, 883--896.</p><p><strong>Abstract:</strong><br/>
This paper is concerned with a class of Kirchhoff models with time delay and perturbation of $p$-Laplacian type \[ u_{tt}(x,t) + \Delta^2 u(x,t) - \Delta_p u(x,t) - a_0 \Delta u_t(x,t) + a_1 u_t(x,t-\tau) + f(u(x,t)) = g(x), \] where $\Delta_p u = \operatorname{div}(|\nabla u|^{p-2} \nabla u)$ is the usual $p$-Laplacian operator. Many researchers have studied well-posedness and decay rates of energy for these equations without delay effects. But, there are not many studies on attractors for other delayed systems. Thus we establish the existence of global attractors and the finite dimensionality of the attractors by establishing some functionals which are related to the norm of the phase space to our problem.
</p>projecteuclid.org/euclid.twjm/1563436873_20190718040119Thu, 18 Jul 2019 04:01 EDTAdmissibly Stable Manifolds for a Class of Partial Neutral Functional Differential Equations on a Half-linehttps://projecteuclid.org/euclid.twjm/1563436874<strong>Thieu Huy Nguyen</strong>, <strong>Xuan Yen Trinh</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Volume 23, Number 4, 897--923.</p><p><strong>Abstract:</strong><br/>
For the following class of partial neutral functional differential equations \[ \begin{cases} \frac{\partial}{\partial t} Fu_t = B(t) u(t) + \Phi(t,u_t) &t \in (0,\infty), \\ u_0 = \phi \in \mathcal{C} := C([-r,0],X) \end{cases} \] we prove the existence of a new type of invariant stable and center-stable manifolds, called admissibly invariant manifolds of $\mathcal{E}$-class for the solutions. The existence of such manifolds is obtained under the conditions that the family of linear partial differential operators $(B(t))_{t \geq 0}$ generates the evolution family $\{U(t,s)\}_{t \geq s \geq 0}$ (on Banach space $X$) having an exponential dichotomy or trichotomy on the half-line and the nonlinear delay operator $\Phi$ satisfies the $\varphi$-Lipschitz condition, i.e., $\|\Phi(t,\phi)-\Phi(t,\psi)\| \leq \varphi(t) \|\phi-\psi\|_{\mathcal{C}}$ for $\phi,\psi \in \mathcal{C}$, where $\varphi(t)$ belongs to some admissible function space on the half-line. Our main method is based on Lyapunov-Perrons equations combined with the admissibility of function spaces and fixed point arguments.
</p>projecteuclid.org/euclid.twjm/1563436874_20190718040119Thu, 18 Jul 2019 04:01 EDTSemi-classical Limit for the Quantum Zakharov Systemhttps://projecteuclid.org/euclid.twjm/1563436875<strong>Yung-Fu Fang</strong>, <strong>Hung-Wen Kuo</strong>, <strong>Hsi-Wei Shih</strong>, <strong>Kuan-Hsiang Wang</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Volume 23, Number 4, 925--949.</p><p><strong>Abstract:</strong><br/>
In this paper, we prove the semi-classical limit for the quantum Zakharov system, that is, the quantum Zakharov system converges to the classical Zakharov system as the quantum parameter goes to zero, including a convergence rate. We improve the results of Guo-Zhang-Guo [11].
</p>projecteuclid.org/euclid.twjm/1563436875_20190718040119Thu, 18 Jul 2019 04:01 EDTTraveling Wave Solutions of a Diffusive SEIR Epidemic Model with Nonlinear Incidence Ratehttps://projecteuclid.org/euclid.twjm/1563436876<strong>Lin Zhao</strong>, <strong>Liang Zhang</strong>, <strong>Haifeng Huo</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Volume 23, Number 4, 951--980.</p><p><strong>Abstract:</strong><br/>
This paper is concerned with the existence and nonexistence of traveling wave solutions of a diffusive SEIR epidemic model with nonlinear incidence rate, which are determined by the basic reproduction number $R_0$ and the minimal wave speed $c^*$. Namely, the system admits a nontrivial traveling wave solution if $R_0 \gt 1$ and $c \geq c^*$ and then the non-existence of traveling wave solutions of the system is established if $R_0 \gt 1$ and $0 \lt c \lt c^*$. Especially, using numerical simulation, we give the basic framework of traveling wave solutions of the system.
</p>projecteuclid.org/euclid.twjm/1563436876_20190718040119Thu, 18 Jul 2019 04:01 EDTSpectral Approximations for Nonlinear Fractional Delay Diffusion Equations with Smooth and Nonsmooth Solutionshttps://projecteuclid.org/euclid.twjm/1563436877<strong>Haiyu Liu</strong>, <strong>Shujuan Lü</strong>, <strong>Hu Chen</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Volume 23, Number 4, 981--1000.</p><p><strong>Abstract:</strong><br/>
A fully discrete scheme is proposed for the nonlinear fractional delay diffusion equations with smooth solutions, where the fractional derivative is described in Caputo sense with the order $\alpha$ ($0 \lt \alpha \lt 1$). The scheme is constructed by combining finite difference method in time and Legendre spectral approximation in space. Stability and convergence are proved rigorously. Moreover, a modified scheme is proposed for the equation with nonsmooth solutions by adding correction terms to the approximations of fractional derivative operator and nonlinear term. Numerical examples are carried out to support the theoretical analysis.
</p>projecteuclid.org/euclid.twjm/1563436877_20190718040119Thu, 18 Jul 2019 04:01 EDTStable Conical Regularization by Constructible Dilating Cones with an Application to $L^{p}$-constrained Optimization Problemshttps://projecteuclid.org/euclid.twjm/1563436878<strong>Baasansuren Jadamba</strong>, <strong>Akhtar A. Khan</strong>, <strong>Miguel Sama</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Volume 23, Number 4, 1001--1023.</p><p><strong>Abstract:</strong><br/>
We study a convex constrained optimization problem that suffers from the lack of Slater-type constraint qualification. By employing a constructible representation of the constraint cone, we devise a new family of dilating cones and use it to introduce a family of regularized problems. We establish novel stability estimates for the regularized problems in terms of the regularization parameter. To show the feasibility and efficiency of the proposed framework, we present applications to some $L^{p}$-constrained least-squares problems.
</p>projecteuclid.org/euclid.twjm/1563436878_20190718040119Thu, 18 Jul 2019 04:01 EDTOn the Average Size of an $(\overline{s},\overline{t})$-Core Partitionhttps://projecteuclid.org/euclid.twjm/1540195382<strong>Joseph L. P. Wang</strong>, <strong>Jane Y. X. Yang</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Volume 23, Number 5, 1025--1040.</p><p><strong>Abstract:</strong><br/>
Let $s$ and $t$ be two coprime integers. Bessenrodt and Olsson obtained the number of $(\overline{s},\overline{t})$-cores for odd $s$ and odd $t$ by establishing a bijection between the lattice paths in $(s,t)$ Yin-Yang diagram and $(\overline{s},\overline{t})$-cores. In this paper, motivated by their results, we extend the definition of Yin-Yang diagram and the bijection to all possible coprime pairs $(s,t)$, then obtain that the number of $(\overline{s},\overline{t})$-cores is $\binom{\lfloor s/2 \rfloor + \lfloor t/2 \rfloor}{\lfloor s/2 \rfloor}$. Furthermore, based on the identities of Chen-Huang-Wang, we determine the average size of an $(\overline{s},\overline{t})$-core depending on the parity of $s$, which is $(s-1) (t-1) (s+t-2)/48$ if $s$ and $t$ are both odd, or $(t-1) (s^2+st-3s+2t+2)/48$ if $s$ is even and $t$ is odd.
</p>projecteuclid.org/euclid.twjm/1540195382_20190919220037Thu, 19 Sep 2019 22:00 EDTOpen Problem on $\sigma$-invarianthttps://projecteuclid.org/euclid.twjm/1542790915<strong>Kinkar Ch. Das</strong>, <strong>Seyed Ahmad Mojallal</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Volume 23, Number 5, 1041--1059.</p><p><strong>Abstract:</strong><br/>
Let $G$ be a graph of order $n$ with $m$ edges. Also let $\mu_1 \geq \mu_2 \geq \cdots \geq \mu_{n-1} \geq \mu_n = 0$ be the Laplacian eigenvalues of graph $G$ and let $\sigma = \sigma(G)$ ($1 \leq \sigma \leq n$) be the largest positive integer such that $\mu_{\sigma} \geq 2m/n$. In this paper, we prove that $\mu_2(G) \geq 2m/n$ for almost all graphs. Moreover, we characterize the extremal graphs for any graphs. Finally, we provide the answer to Problem 3 in [8], that is, the characterization of all graphs with $\sigma = 1$.
</p>projecteuclid.org/euclid.twjm/1542790915_20190919220037Thu, 19 Sep 2019 22:00 EDTWaring-Goldbach Problem: Two Squares and Three Biquadrateshttps://projecteuclid.org/euclid.twjm/1542790912<strong>Yingchun Cai</strong>, <strong>Li Zhu</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Volume 23, Number 5, 1061--1071.</p><p><strong>Abstract:</strong><br/>
Assume that $\psi$ is a function of positive variable $t$, monotonically increasing to infinity and $0 \lt \psi(t) \ll \log t/(\log \log t)$. Let $\mathcal{R}_{3}(n)$ denote the number of representations of the integer $n$ as sums of two squares and three biquadrates of primes and we write $\mathcal{E}_{3}(N)$ for the number of integers $n$ satisfying $n \leq N$, $n \equiv 5, 53, 101 \pmod{120}$ and \[ \left| \mathcal{R}_{3}(n) - \frac{\Gamma^{2}(1/2) \Gamma^{3}(1/4)}{\Gamma(7/4)} \frac{\mathfrak{S}_{3}(n) n^{3/4}}{\log^{5}n} \right| \geq \frac{n^{3/4}}{\psi(n) \log^{5}n}, \] where $0 \lt \mathfrak{S}_{3}(n) \ll 1$ is the singular series. In this paper, we prove \[ \mathcal{E}_{3}(N) \ll N^{23/48+\varepsilon} \psi^{2}(N) \] for any $\varepsilon \gt 0$. This result constitutes a refinement upon that of Friedlander and Wooley [2].
</p>projecteuclid.org/euclid.twjm/1542790912_20190919220037Thu, 19 Sep 2019 22:00 EDTDiophantine Approximation with Mixed Powers of Primeshttps://projecteuclid.org/euclid.twjm/1550890836<strong>Huafeng Liu</strong>, <strong>Jing Huang</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Volume 23, Number 5, 1073--1090.</p><p><strong>Abstract:</strong><br/>
Let $k$ be an integer with $k \geq 3$. Let $\lambda_1$, $\lambda_2$, $\lambda_3$ be non-zero real numbers, not all negative. Assume that $\lambda_1/\lambda_2$ is irrational and algebraic. Let $\mathcal{V}$ be a well-spaced sequence, and $\delta \gt 0$. In this paper, we prove that, for any $\varepsilon \gt 0$, the number of $\upsilon \in \mathcal{V}$ with $\upsilon \leq X$ such that the inequality \[ |\lambda_1 p_1^2 + \lambda_2 p_2^2 + \lambda_3 p_3^k - \upsilon| \lt \upsilon^{-\delta} \] has no solution in primes $p_1$, $p_2$, $p_3$ does not exceed $O(X^{1-2/(7m_2(k))+2\delta+\varepsilon})$, where $m_2(k)$ relies on $k$. This refines a recent result. Furthermore, we briefly describe how a similar method can refine a previous result on a Diophantine problem with two squares of primes, one cube of primes and one $k$-th power of primes.
</p>projecteuclid.org/euclid.twjm/1550890836_20190919220037Thu, 19 Sep 2019 22:00 EDTCharacter Formulas for Simple Modules of Hamiltonian Lie Superalgebras of Odd Typehttps://projecteuclid.org/euclid.twjm/1543546838<strong>Wende Liu</strong>, <strong>Jixia Yuan</strong>, <strong>Shujuan Wang</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Volume 23, Number 5, 1091--1113.</p><p><strong>Abstract:</strong><br/>
In this paper, character formulas are explicitly characterized for all simple restricted modules of Hamiltonian Lie superalgebras of odd type over an algebraically closed field of characteristic $p \gt 3$. In the process we use the lengths and highest weights of simple quotients of restricted Kac modules of atypical weights with respect to a series of Borel subalgebras to determine the composition factors, composition series and the character formulas for the restricted Kac modules of atypical weights for the Lie superalgebras under consideration.
</p>projecteuclid.org/euclid.twjm/1543546838_20190919220037Thu, 19 Sep 2019 22:00 EDTCastelnuovo-Mumford Regularity and Hilbert Coefficients of Parameter Idealshttps://projecteuclid.org/euclid.twjm/1548817227<strong>Cao Huy Linh</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Volume 23, Number 5, 1115--1131.</p><p><strong>Abstract:</strong><br/>
Let $A$ be a noetherian local ring of dimension $d \geq 1$ and $\operatorname{depth}(A) \geq d-1$. In this paper, we study the non-positivity for the Hilbert coefficients of parameter ideals in the ring $A$. Moreover, we establish a bound for the Castelnuovo-Mumford regularity of associated graded ring of $A$ with respect to parameter ideal in terms of the first Hilbert coefficient and the dimension.
</p>projecteuclid.org/euclid.twjm/1548817227_20190919220037Thu, 19 Sep 2019 22:00 EDTCharacterization of Temperatures Associated to Schrödinger Operators with Initial Data in Morrey Spaceshttps://projecteuclid.org/euclid.twjm/1542790913<strong>Qiang Huang</strong>, <strong>Chao Zhang</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Volume 23, Number 5, 1133--1151.</p><p><strong>Abstract:</strong><br/>
Let $\mathcal{L}$ be a Schrödinger operator of the form $\mathcal{L} = -\Delta + V$ acting on $L^2(\mathbb{R}^n)$ where the nonnegative potential $V$ belongs to the reverse Hölder class $B_q$ for some $q \geq n$. Let $L^{p,\lambda}(\mathbb{R}^{n})$, $0 \leq \lambda \lt n$ denote the Morrey space on $\mathbb{R}^{n}$. In this paper, we will show that a function $f \in L^{2,\lambda}(\mathbb{R}^{n})$ is the trace of the solution of $\mathbb{L}u := u_{t} + \mathcal{L}u = 0$, $u(x,0) = f(x)$, where $u$ satisfies a Carleson-type condition \[ \sup_{x_B,r_B} r_B^{-\lambda} \int_0^{r_B^2} \!\! \int_{B(x_B,r_B)} |\nabla u(x,t)|^2 \, dx dt \leq C \lt \infty. \] Conversely, this Carleson-type condition characterizes all the $\mathbb{L}$-carolic functions whose traces belong to the Morrey space $L^{2,\lambda}(\mathbb{R}^{n})$ for all $0 \leq \lambda \lt n$. This result extends the analogous characterization found by Fabes and Neri in [8] for the classical BMO space of John and Nirenberg.
</p>projecteuclid.org/euclid.twjm/1542790913_20190919220037Thu, 19 Sep 2019 22:00 EDTEigenvalue Problems for Fractional $p(x,y)$-Laplacian Equations with Indefinite Weighthttps://projecteuclid.org/euclid.twjm/1555984822<strong>Nguyen Thanh Chung</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Volume 23, Number 5, 1153--1173.</p><p><strong>Abstract:</strong><br/>
In this paper, we consider a class of eigenvalue problems for fractional $p(x,y)$-Laplacian equations with indefinite weight in fractional Sobolev space with variable exponent. Under some suitable conditions on the growth rates involved in the problem, we establish some results on the existence of a continuous family of eigenvalues using variational techniques and Ekeland's variational principle.
</p>projecteuclid.org/euclid.twjm/1555984822_20190919220037Thu, 19 Sep 2019 22:00 EDTSchur Product with Operator-valued Entrieshttps://projecteuclid.org/euclid.twjm/1543546839<strong>Oscar Blasco</strong>, <strong>Ismael García-Bayona</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Volume 23, Number 5, 1175--1199.</p><p><strong>Abstract:</strong><br/>
In this paper we characterize Toeplitz matrices with entries in the space of bounded operators on Hilbert spaces $\mathcal{B}(H)$ which define bounded operators acting on $\ell^2(H)$ and use it to get the description of the right Schur multipliers acting on $\ell^2(H)$ in terms of certain operator-valued measures.
</p>projecteuclid.org/euclid.twjm/1543546839_20190919220037Thu, 19 Sep 2019 22:00 EDTOptimal Energy Decay for a Transmission Problem of Waves Under a Nonlocal Boundary Controlhttps://projecteuclid.org/euclid.twjm/1550631631<strong>Halim Atoui</strong>, <strong>Abbes Benaissa</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Volume 23, Number 5, 1201--1225.</p><p><strong>Abstract:</strong><br/>
In this paper, we consider a transmission problem in the presence of a boundary control condition of nonlocal type. We prove well-posedness by using the semigroup theory. Also we establish an optimal decay result by frequency domain method and Borichev-Tomilov theorem.
</p>projecteuclid.org/euclid.twjm/1550631631_20190919220037Thu, 19 Sep 2019 22:00 EDTGeneral Decay Rates for a Laminated Beam with Memoryhttps://projecteuclid.org/euclid.twjm/1542855636<strong>Zhijing Chen</strong>, <strong>Wenjun Liu</strong>, <strong>Dongqin Chen</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Volume 23, Number 5, 1227--1252.</p><p><strong>Abstract:</strong><br/>
In previous work [23], Mustafa considered a viscoelastic laminated beam system with structural damping in the case of equal-speed wave propagations, and established explicit energy decay formula which gives the best decay rates. In this paper, we continue to consider the similar problems and establish the general decay result for the energy, to system with structural damping in the case of non-equal wave speeds and to system without structural damping in the case of equal wave speeds, respectively. For the first case, we use the second-order energy method to overcome the difficulty of estimating the non-equal speeds term. For the second case, we construct an appropriated perturbed functional to estimate $\|w_{t}\|^{2}_{2}$ so as to overcome the absence of structural damping.
</p>projecteuclid.org/euclid.twjm/1542855636_20190919220037Thu, 19 Sep 2019 22:00 EDTA Method-of-lines Approach for Solving American Option Problemshttps://projecteuclid.org/euclid.twjm/1541667765<strong>Min-Sun Horng</strong>, <strong>Tzyy-Leng Horng</strong>, <strong>Chih-Yuan Tien</strong>. <p><strong>Source: </strong>Taiwanese Journal of Mathematics, Volume 23, Number 5, 1253--1270.</p><p><strong>Abstract:</strong><br/>
The early exercise property of American option changes the original Black-Scholes equation to an inequality that cannot be solved via traditional finite difference method. Therefore, finding the early exercise boundary prior to spatial discretization is a must in each time step. This overhead slows down the computation and the accuracy of solution relies on if the early exercise boundary can be accurately located. A simple numerical method based on finite difference and method of lines is proposed here to overcome this difficulty in American option valuation. Our method averts the otherwise necessary procedure of locating the optimal exercise boundary before applying finite difference discretization. The method is efficient and flexible to all kinds of pay-off. Computations of American put, American call with dividend, American strangle, two-factor American basket put option, and two-factor convertible bond with embedded call and put options are demonstrated to show the efficiency of the current method.
</p>projecteuclid.org/euclid.twjm/1541667765_20190919220037Thu, 19 Sep 2019 22:00 EDT