scholarly journals A necessary and sufficient condition for global existence for a quasilinear reaction-diffusion system

2005 ◽  
Vol 2005 (11) ◽  
pp. 1809-1818 ◽  
Author(s):  
Alan V. Lair
Keyword(s):  
Global Solution ◽  
Smooth Boundary ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient ◽  
Special Case

We show that the reaction-diffusion systemut=Δφ(u)+f(v),vt=Δψ(v)+g(u), with homogeneous Neumann boundary conditions, has a positive global solution onΩ×[0,∞)if and only if∫∞ds/f(F−1(G(s)))=∞(or, equivalently,∫∞ds/g(G−1(F(s)))=∞), whereF(s)=∫0sf(r)drandG(s)=∫0sg(r)dr. The domainΩ⊆ℝN(N≥1)is bounded with smooth boundary. The functionsφ,ψ,f, andgare nondecreasing, nonnegativeC([0,∞))functions satisfyingφ(s)ψ(s)f(s)g(s)>0fors>0andφ(0)=ψ(0)=0. Applied to the special casef(s)=spandg(s)=sq,p>0,q>0, our result proves that the system has a global solution if and only ifpq≤1.

A Classification and Closure Properties of Languages for Describing Concurrent System Behaviours

1981 ◽  
Vol 4 (3) ◽  
pp. 531-549 ◽  
Author(s):  
Miklós Szijártó
Keyword(s):  
Formal Languages ◽  
Parallel Program ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Concurrent System ◽  
Closure Properties ◽  
Sequential Program ◽  
Necessary And Sufficient ◽  
Special Case ◽  
Independence Relations

The correspondence between sequential program schemes and formal languages is well known (Blikle and Mazurkiewicz (1972), Engelfriet (1974)). The situation is more complicated in the case of parallel program schemes, and trace languages (Mazurkiewicz (1977)) have been introduced to describe them. We introduce the concept of the closure of a language on a so called independence relation on the alphabet of the language, and formulate several theorems about them and the trace languages. We investigate the closedness properties of Chomsky classes under closure on independence relations, and as a special case we derive a new necessary and sufficient condition for the regularity of the commutative closure of a language.

scholarly journals On composition of formal power series

2002 ◽  
Vol 30 (12) ◽  
pp. 761-770 ◽  
Author(s):  
Xiao-Xiong Gan ◽  
Nathaniel Knox
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Open Problem ◽  
Constant Term ◽  
Radius Of Convergence ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient ◽  
Formal Power ◽  
Special Case

Given a formal power seriesg(x)=b0+b1x+b2x2+⋯and a nonunitf(x)=a1x+a2x2+⋯, it is well known that the composition ofgwithf,g(f(x)), is a formal power series. If the formal power seriesfabove is not a nonunit, that is, the constant term offis not zero, the existence of the compositiong(f(x))has been an open problem for many years. The recent development investigated the radius of convergence of a composed formal power series likefabove and obtained some very good results. This note gives a necessary and sufficient condition for the existence of the composition of some formal power series. By means of the theorems established in this note, the existence of the composition of a nonunit formal power series is a special case.

On the general bilinear time series model

1988 ◽  
Vol 25 (3) ◽  
pp. 553-564 ◽  
Author(s):  
Jian Liu ◽  
Peter J. Brockwell
Keyword(s):  
Time Series ◽  
Stationary Solution ◽  
Time Series Model ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Special Cases ◽  
Non Linear ◽  
Necessary And Sufficient ◽  
Special Case

A sufficient condition is derived for the existence of a strictly stationary solution of the general bilinear time series equations. The condition is shown to reduce to the conditions of Pham and Tran (1981) and Bhaskara Rao et al. (1983) in the special cases which they consider. Under the condition specified, a solution is constructed which is shown to be causal, stationary and ergodic. It is moreover the unique causal solution and the unique stationary solution of the defining equations. In the special case when the defining equations contain no non-linear terms, our condition reduces to the well-known necessary and sufficient condition for existence of a causal stationary solution.

scholarly journals Characterization of $[1,k]$-Bar Visibility Trees

10.37236/1116 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Guantao Chen ◽  
Joan P. Hutchinson ◽  
Ken Keating ◽  
Jian Shen
Keyword(s):  
Knapsack Problem ◽  
Unit Length ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Visibility Representation ◽  
Visibility Graphs ◽  
Necessary And Sufficient ◽  
Positive Integers ◽  
Special Case

A unit bar-visibility graph is a graph whose vertices can be represented in the plane by disjoint horizontal unit-length bars such that two vertices are adjacent if and only if there is a unobstructed, non-degenerate, vertical band of visibility between the corresponding bars. We generalize unit bar-visibility graphs to $[1,k]$-bar-visibility graphs by allowing the lengths of the bars to be between $1/k$ and $1$. We completely characterize these graphs for trees. We establish an algorithm with complexity $O(kn)$ to determine whether a tree with $n$ vertices has a $[1,k]$-bar-visibility representation. In the course of developing the algorithm, we study a special case of the knapsack problem: Partitioning a set of positive integers into two sets with sums as equal as possible. We give a necessary and sufficient condition for the existence of such a partition.

scholarly journals Companion matrices and their relations to Toeplitz and Hankel matrices

2015 ◽  
Vol 3 (1) ◽  
Author(s):  
Yousong Luo ◽  
Robin Hill
Keyword(s):  
Hankel Matrix ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Hankel Matrices ◽  
Hankel Structure ◽  
Companion Matrices ◽  
Necessary And Sufficient ◽  
Special Case

AbstractIn this paper we describe some properties of companion matrices and demonstrate some special patterns that arisewhen a Toeplitz or a Hankel matrix is multiplied by a related companion matrix.We present a necessary and sufficient condition, generalizing known results, for a matrix to be the transforming matrix for a similarity between a pair of companion matrices. A special case of our main result shows that a Toeplitz or a Hankel matrix can be extended using associated companion matrices, preserving the Toeplitz or Hankel structure respectively.

INDEPENDENCE OF DOUBLE WIENER INTEGRALS

2001 ◽  
Vol 17 (6) ◽  
pp. 1143-1155
Author(s):  
Seiji Nabeya
Keyword(s):  
Quadratic Forms ◽  
Continuity Condition ◽  
Mathematical Statistics ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Mutual Independence ◽  
Pairwise Independence ◽  
Necessary And Sufficient ◽  
Discontinuous Kernels ◽  
Special Case

In this paper a necessary and sufficient condition is obtained for two double Wiener integrals to be statistically independent, first in the case of symmetric and continuous kernels. It is also shown that, for more than two double Wiener integrals, pairwise independence implies mutual independence. After that, the continuity condition on the kernels is somewhat relaxed, and it is shown that Craig's (1943, Annals of Mathematical Statistics 14, 195–197) theorem on the independence of quadratic forms in normal variables is a special case of the result obtained for the case of discontinuous kernels.

EXPONENTIALLY DECAYING COMPONENT OF A GLOBAL SOLUTION TO A REACTION–DIFFUSION SYSTEM

1998 ◽  
Vol 08 (05) ◽  
pp. 897-904 ◽  
Author(s):  
HIROKI HOSHINO ◽  
SHUICHI KAWASHIMA
Keyword(s):  
Global Solution ◽  
Large Time ◽  
Reaction Diffusion ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Spatial Average ◽  
Exponential Rate ◽  
Chemical Substances ◽  
Irreversible Chemical Reaction ◽  
The Difference

A reaction–diffusion system which is related to a simple irreversible chemical reaction between two chemical substances is considered. When a non-negative global solution for the system converges uniformly to zero with polynomial rate as time goes to infinity, large-time approximation of the solution is studied. It is shown that the difference of the solution and its spatial average tends to zero with exponential rate via a global solution for the corresponding system of ordinary differential equations.

scholarly journals Existence of global solution and nontrivial steady states for a system modeling chemotaxis

2006 ◽  
Vol 2006 ◽  
pp. 1-23 ◽  
Author(s):  
Zhenbu Zhang
Keyword(s):  
Steady State ◽  
Global Existence ◽  
Global Solution ◽  
System Modeling ◽  
Reaction Diffusion ◽  
Steady States ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Steady State Solutions

We consider a reaction-diffusion system modeling chemotaxis, which describes the situation of two species of bacteria competing for the same nutrient. We use Moser-Alikakos iteration to prove the global existence of the solution. We also study the existence of nontrivial steady state solutions and their stability.

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