Showing papers in "Taiwanese Journal of Mathematics in 2000"
TL;DR: In this paper, the major directions of research in abstract frame theory and some sample techniques from each of the areas are discussed, and some of the important open questions and limitations of the existing theory are discussed.
Abstract: The theory of frames for a Hilbert space plays a fundamental role in signal processing, image processing, data compression, sampling theory and more, as well as being a fruitful area of research in abstract mathematics. In this “tutorial” on abstract frame theory, we will try to point out the major directions of research in abstract frame theory and give some sample techniques from each of the areas. We will also bring out some of the important open questions, discuss some of the limitations of the existing theory, and point to some new directions for research.
524 citations
TL;DR: Hardy-type inequalities for n-dimensional Hermite and special Hermite expansions were proved in this article, and Paley-type theorems for these expansions were also deduced.
Abstract: Hardy-type inequalities are proved for n-dimensional Hermite and special Hermite expansions. Paley-type theorems for these expansions are also deduced.
269 citations
TL;DR: In this paper, the most important mathematical properties of dispersive limits of non-linear Schr¨odinger type equations are presented, e.g., turbulent diffusion appears naturally in the zero dispersion limit.
Abstract: In this review paper we present the most important mathematical properties of dispersive limits of (non)linear Schr¨odinger type equations. Different formulations are used to study these singular limits, e.g., the kinetic formulation of the linear Schr¨odinger equation based on the Wigner transform is well suited for global-in-time analysis without using WKB-(expansion) techniques, while the modified Madelung transformation reformulating Schr¨odinger equations in terms of a dispersive perturbation of a quasilinear symmetric hyperbolic system usually only gives local-in-time results due to the hyperbolic nature of the limit equations. Deterministic analogues of turbulence are also discussed. There, turbulent diffusion appears naturally in the zero dispersion limit.
87 citations
TL;DR: In this article, the authors considered a semilinear equation arising from the Navier-Stokes equations and related model equations and showed that the non-local terms play an intriguing role for the blow-up and/or global existence of the solutions and that the convection term plays a very decisive role.
Abstract: We consider a semilinear equation arising from the Navier- Stokes equations – the governing equations of viscous fluid motion – and related model equations. The solutions of the semilinear equation represent a certain class of exact solutions of the Navier-Stokes equations. Both the equation and our models have nonlocal terms. We will show that the nonlocality will play an intriguing role for the blow-up and/or global existence of the solutions and that the convection term, which is often neglected in the study of the blow-up problems, plays a very decisive role. In addition to our new contributions, open problems and known facts are surveyed.
58 citations
TL;DR: In this article, the problem of determining the largest number of maximum independent sets of a graph of order n is studied, and solutions to this problem are given for various classes of graphs, including general graphs, trees, forests, (connected) graphs with at most one cycle, connected graphs and triangle-free graphs.
Abstract: In this paper, we study the problem of determining the largest number of maximum independent sets of a graph of order n. Solutions to this problem are given for various classes of graphs, including general graphs, trees, forests, (connected) graphs with at most one cycle, connected graphs and triangle-free graphs. Extremal graphs achieving the maximum values are also given.
51 citations
TL;DR: In this article, it was shown that when the number of items is sufficiently large, and the fraction of defective items is at least $1/log 3/2/3, individual testing is optimal.
Abstract: Adaptive group testing in the presence of a large percentage of defectives is best done by individual testing rather than by pooling. The fraction of items which must be defective to make individual testing optimal remains unknown, and is conjectured to be 1/3. In this paper it is shown that when the number of items is sufficiently large, and the fraction of defective items is at least $1/\log_{3/2}3$, individual testing is optimal.
35 citations
TL;DR: In this article, a renewed algorithm is presented to calculate the mixed volume of the support of a polynomial system, where the key ingredient is a specially tailored application of LP feasibility tests.
Abstract: A renewed algorithm is presented to calculate the mixed volume of the support ${\cal A}=({\cal A}_1,\dots,{\cal A}_n)$ of a polynomial system $P({\bf x})=(p_1({\bf x}),\dots, p_n({\bf x}))$ in $\Bbb C^n$. The key ingredient is a specially tailored application of LP feasibility tests, which allows us to calculate the {\em mixed cells}, their volumes constituting the mixed volume, in a {\em mixed subdivision} of ${\cal A}$ more efficiently. The problem of finding mixed cells plays a crucial role in polyhedral homotopy methods for finding all isolated zeros of $P({\bf x})$. Our new algorithm advances the speed of mixed volume computation by a considerable margin, illustrated by numerical examples.
25 citations
TL;DR: In this paper, a discrete version of the Aleksandrov -Bakelman maximum principle for general meshes in Euclidean space has been established in terms of a discrete analogue of the determinant of the coefficient matrix.
Abstract: In previous works, we have established discrete versions of the Aleksandrov -Bakelman maximum principle for elliptic operators, on general meshes in Euclidean space. In this paper, we prove a variant of these estimates in terms of a discrete analogue of the determinant of the coefficient matrix in the differential operator case. Our treatment depends on an interesting connection between the determinant and volumes of cells in the underlying mesh.
25 citations
TL;DR: An order-theoretic version of the Bishop-Parsons theorem, based on the interplay of the notions of partial order and completeness, was presented in this article.
Abstract: In this report1, we present an order-theoretic version of the Cantor theorem. This result, which is based on the interplay of the notions of partial order and of completeness, permits to give a unified and simplified account to a long list of results related to the Bishop– Phelps theorem. We survey briefly only its simplest applications and refer the reader to [10] for a complete presentation of the results.
22 citations
TL;DR: Many interesting families of rapidly convergent series representations for the Riemann Zeta function were considered recently by various authors as discussed by the authors, and a systematic investigation of these series representations is presented in this survey-cum-expository paper.
Abstract: Many interesting families of rapidly convergent series representations for the Riemann Zeta function $\zeta (2n+1)$ $(n\in {\Bbb N})$ were considered recently by various authors In this survey-cum-expository paper, the author presents a systematic (and historical) investigation of these series representations Relevant connections of the results presented here with several other known series representations for $\zeta (2n+1)$ $(n\in {\Bbb N})$ are also pointed out In one of many computationally useful special cases presented here, it is observed that $\zeta (3)$ can be represented by means of a series which converges much faster than that in Euler's celebrated formula as well as the series used recently by Ap\'{e}ry in his proof of the irrationality of $\zeta (3)$ Symbolic and numerical computations using {\em Mathematica} (Version 40) for Linux show, among other things, that only 50 terms of this series are capable of producing an accuracy of seven decimal places
19 citations
TL;DR: The relation between $\lambda_d $ and $\sigma_d$ as well $\ lambda_d^\prime$ and $\ sigma _d^prime$ is studied and the join of graphs and the multipartite graphs are studied.
Abstract: For positive integers $k$ and $d\geq 2$, a $k$-$S(d, 1)$-$labelling$ of a graph $G$ is a function on the vertex set of $G$, $f:V(G)\to\{0,1,2,\cdots,k-1\}$, such that $$ |f(u)-f(v)|_k\geq\left\{\begin{array}{ll} d &\text{\rm if }\ d_G(u,v)=1;\\ 1 &\text{\rm if }\ d_G(u,v)=2,\end{array}\right. $$ where $|x|_k=\min\{|x|,k-|x|\}$ is the {\em circular difference} modulo $k$. In general, this kind of labelling is called the $S(d,1)$-$labelling$. The $\sigma_d$-number of $G$, $\sigma_d(G)$, is the minimum $k$ of a $k$-$S(d,1)$-labelling of $G$. If the labelling is required to be injective, then we have analogous $k$-$S'(d,1)$-$labelling$, $S'(d,1)$-$labelling$ and $\sigma _d^\prime(G)$. If the circular difference in the definition above is replaced by the absolute difference, then $f$ is an $L(d,1)$-labelling of $G$. The $span$ of an $L(d,1)$-labelling is the difference of the maximum and the minimum labels used. The $\lambda_d$-number of $G$, $\lambda_d(G)$, is defined as the minimum span among all $L(d,1)$-labellings of $G$. In this case, we have the corresponding $L'(d,1)$-labelling and $\lambda_d'(G)$ for the labelling with injective condition. We will first study the relation between $\lambda_d$ and $\sigma_d$ as well $\lambda_d^\prime$ and $\sigma _d^\prime$. Then we consider these parameters on cycles and trees. Finally, we study the join of graphs and the multipartite graphs.
TL;DR: In this article, the authors survey some recent results on the study of viscous and invisid compressible flow with vacuum and show that most of the important questions about vacuum are still open due to the singularities caused by vacuum which need new mathematical tools and techniques to handle.
Abstract: In this paper, we will survey some recent results on the study of the viscous and invisid compressible flow with vacuum. It is wellknown that the study on vacuum has significance in the investigation on some important physical phenomena. However, most of the important questions about vacuum are still open due to the singularities caused by vacuum which need new mathematical tools and techniques to handle.
TL;DR: In this paper, the authors give some inequalities of capacity in Gaussian channel with or without feedback, and show that the non-feedback capacity and the feedback capacity are concave functions of the same type.
Abstract: We give some inequalities of capacity in Gaussian channel with or without feedback. The nonfeedback capacity $C_{n,Z}(P)$ and the feedback capacity $C_{n,FB,Z}(P)$ are both concave functions of $P$. Though it is shown that $C_{n,Z}(P)$ is a convex function of $Z$ in some sense, $C_{n,FB,Z}(P)$ is a convex-like function of $Z$.
TL;DR: A graph labelling problem which has two constraints instead of one is proposed and the question of finding the minimum edge of this labelling is considered.
Abstract: The radio channel assignment problem can be cast as a graph coloring problem. Vertices correspond to transmitter locations and their labels (colors) to radio channels. The assignment of frequencies to each transmitter (vertex) must avoid interference which depends on the seperation each pair of vertices has. Two levels of interference are assumed in the problem we are concerned. Based on this channel assignment problem, we proposed a graph labelling problem which has two constraints instead of one. We consider the question of finding the minimum edge of this labelling. Several classes of graphs including one that is important to a telecommunication problem have been studied.
TL;DR: In this paper, the independence of the sample mean and the standard deviation of the normal population was shown to characterize the gamma distribution, where the random samples are drawn from a population having a continuous density function on the R-dimensional space.
Abstract: In this paper, earlier works of the present authors and a method due to Anosov for solving certain intego-functional equations are combined to show that the independence of the sample mean $\bar{X}_n$ and the $Z_n$-statistic characterizes the normal population, when the random samples are iid from a population having a continuous density function on ${\Bbb R}$, and the sample size $n\geq 3$; obviously the sample standard deviation is a $Z_n$-statistic. Further, an important subclass of $Z_n$-statistic with the form of a linear combination $\sum^n_{i=1} a_iX_{(i)}$ of order statistics is found, where $a_1 \leq \cdots \leq a_n$, not all equal and $\sum^n_{i=1}a_i=0$, which includes Gini$^\prime$s mean difference and the sample range but not the sample standard deviation. Similar approach can be applied to prove that the independence of $\bar{X}_n$ and $Z_n/\bar{X}_n$ characterizes the gamma distribution; obviously the independence of sample mean and sample coefficient of variation characterizes the gamma distribution. The study of identifying $Z_n$ to more known statistics will be the future work.
TL;DR: In this paper, it was shown that for any integer n ≥ 4 and any rational number r ≥ 2, there exists a graph G$ which has circular chromatic number r and which does not contain k_n as a minor.
Abstract: This paper proves that for any integer $n\geq 4$ and any rational number $r$, $2\leq r\leq n-2$, there exists a graph $G$ which has circular chromatic number $r$ and which does not contain $K_n$ as a minor.
TL;DR: The main object of as discussed by the authors is to show how some of the most recent contributions on this subject, involving the Weber equations and their various generalized forms, can be obtained by suitably applying a general theorem on particular solutions of a certain family of fractional differintegral equations.
Abstract: In recent years, several workers demonstrated the usefulness of fractional calculus in the derivation of particular solutions of a number of familiar second-order differential equations associated (for example) with Gauss, Legendre, Jacobi, Chebyshev, Coulomb, Whittaker, Euler, Hermite, and Weber equations. The main object of this paper is to show how some of the most recent contributions on this subject, involving the Weber equations and their various generalized forms, can be obtained by suitably applying a general theorem on particular solutions of a certain family of fractional differintegral equations.
TL;DR: In this article, Supsets and infsets for subsets of a partially ordered topological linear space were introduced. And these notions generalize the usual notions of supremum and infimum in Riesz spaces.
Abstract: We introduce supsets and infsets for subsets of a partially ordered topological linear space. These notions generalize the usual notions of supremum and infimum in Riesz spaces. We shall investigate properties of supsets and infsets in this paper.
TL;DR: In this paper, it was shown that if the involutions are of different kinds, such a map cannot exist (modulo some low-dimensional counterexamples) and if both the second kind is determined by a related associative isomorphism, then such map cannot be found.
Abstract: Let $R$ and $S$ be *-prime GPI rings with involution, with respective skew elements $K$ and $L$, with respective extended centroids $C$ and $D$, and let $\alpha :[K, K]/[K, K]\cap C\to [L, L]\cap D$ be a Lie isomorphism. If both involutions are of the second kind it is shown that $\alpha$ is determined by a related associative isomorphism and if the involutions are of different kinds it is shown that such a map $\alpha$ cannot exist (modulo some low-dimensional counterexamples).
TL;DR: In this article, the authors present two convergence theorems for the H1-integral, which are equivalent to the equi-integrability theorem with the strong Lusin condition.
Abstract: We present two convergence theorems for the H1-integral. The Henstock integral is now relatively well-known. An attempt has been made by Garces, Lee, and Zhao (2) to define the Henstock integral as the Moore-Smith limit of Riemann sums. The resulting integral is the so-called H1-integral. It has the property that a function f is Henstock integrable on (a;b) if and only if there is an H1-integrable function g such that f(x) = g(x) almost everywhere in (a;b). Every integral has a corresponding convergence theorem. For example, the Denjoy integral has the controlled convergence the- orem, whereas the Perron integral has the generalized dominated convergence theorem. Corresponding to the Henstock integral, which is equivalent to both the integrals of Denjoy and Perron, is the equi-integrability theorem with the strong Lusin condition. It is the purpose of the current paper to present two (well-known) convergence theorems that hold for the H1-integral. We assume that the reader is familiar with the definition of the Henstock integral (5). A division D of (a;b) is a finite set of interval-point pairs ((u;v);») such that the intervals (u;v) are non-overlapping, (a;b) = ((u;v), and » 2 (u;v). If -(x) > 0 for x 2 (a;b), then a division D = f((u;v);»)g is said to be --fine if » 2 (u;v) ‰ (» i -(»);» + -(»)) for each ((u;v);») 2 D. A function f is said to be Henstock integrable to a real number A on (a;b) if for every † > 0 there exists a positive function - on (a;b) such that for every --fine division D, we have
TL;DR: In this article, the authors studied the Cauchy problem for a 3 × 3-system of conservation laws describing the phase transition, where a phase boundary is said to be admissible if it satisfies the Abeyaratne-Knowles kinetic condition.
Abstract: We study the Cauchy problem for a $3\times 3$-system of conservation laws describing the phase transition: $u_t-v_x=0$, $v_t-\sigma(u)_x=0$, $(e+\frac{1}{2}v^2)_t-(\sigma v)_x=0$. A phase boundary is said to be admissible if it satisfies the Abeyaratne-Knowles kinetic condition. We give a physical account of the kinetic condition by means of the $Gibbs function$. We also obtain a useful description of the entropy function using the Gibbs function.
TL;DR: An overview of the MGMRES and LAN/MGMRES iterative methods for solving large sparse linear systems is presented.
Abstract: We present an overview of the MGMRES and LAN/MGMRES iterative methods for solving large sparse linear systems.
TL;DR: In this paper, a characterization of unconditional convergent series is given for the case of sequentially complete locally convex spaces, and it is shown that if $E$ is a barrelled space with continuous dual $E', then ($E'$, $\beta (E'-, $E', $E)) contains no copy of ($c_0,~\|\cdot \|_\infty$) if and only if every continuous linear operator $T:E\to l_1$ is both compact and sequentially compact.
Abstract: A characterization of unconditional convergent series is given for the case of sequentially complete locally convex spaces. From it we show that if $E$ is a barrelled space with continuous dual $E'$, then ($E'$, $\beta (E'$, $E$)) contains no copy of ($c_0,~\|\cdot \|_\infty$) if and only if every continuous linear operator $T:E\to l_1$ is both compact and sequentially compact.
TL;DR: In this paper, the authors studied the third-order ODEs which are relevant to the steady state of the Kuramoto-Sivashinsky equation, and/or to a model of dendritic growth of needle crystals.
Abstract: We study the third-order ordinary differential equations (ODEs) which are relevant to the steady state of the Kuramoto-Sivashinsky equation, and/or to a model of dendritic growth of needle crystals. We show that there is no monotonic solution for certain range of parameter.
TL;DR: In this paper, Couette-Taylor problems of the perturbation to Couette flow between two rotating cylinders are considered, and it is shown that the stationary bifurcation occurs when the Taylor number increases.
Abstract: We consider Couette-Taylor problems of the perturbation to Couette flow between two rotating cylinders, and show that the stationary bifurcation or Hopf bifurcation occurs when Taylor number increases. We make precise analysis of the eigenvalue problems
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TL;DR: A survey of qualitative shock wave theory can be found in this paper, which is meant to explain some of the key issues close to the interest of the author and raise open questions.
Abstract: We survey the qualitative shock wave theory. The survey is meant to explain some of the key issues close to the interest of the author and raise open questions.
TL;DR: In this article, two kinds of Palais-Smale conditions for nonsmooth functionals are studied, and it is shown that they are equivalent for convex functionals.
Abstract: Two kinds of Palais-Smale condition, $(PS)_c$ and $(PS)^*_c$, for nondifferentiable functionals are studied. It is shown that $(PS)_c$ implies $(PS)^*_c$ and that they are equivalent for convex functionals. This points out a gap in the proof of Costa and Goncalves [5, Proposition 3]. Some other nonsmooth versions of known smooth results are also obtained.
TL;DR: Special Issue of the Proceedings of 1999 International Conference on Nonlinear Analysis (October 16–20, 1999, Academia Sinica, Taipei, Taiwan) in honor of the 60th birthday of Fon-Che Liu.
Abstract: Special Issue of the Proceedings of 1999 International Conference on Nonlinear Analysis (October 16–20, 1999, Academia Sinica, Taipei, Taiwan) in honor of the 60th birthday of Fon-Che Liu
TL;DR: In this article, the exact boundary controllability for onedimensional linear heat equation with coefficients depending on the space variable and the time variable was studied and it was shown that the functions of Gevrey class 2 are reachable when the initial functions are continuous.
Abstract: We consider the exact boundary controllability for onedimensional linear heat equation with coefficients depending on the space variable and the time variable. We show that the functions of Gevrey class 2 are reachable when the initial functions are continuous.
TL;DR: In this article, an expository article concerning complex analysis, in particular, several complex variables, is presented to demonstrate the development and the diversity of complex variables and to motivate the reader's interest to this subject.
Abstract: This is an expository article concerning complex analysis, in particular, several complex variables. Several subjects are discussed here to demonstrate the development and the diversity of several complex variables. Hopefully, the brief introduction to complex analysis in several variables would motivate the reader’s interests to this subject.