A theorem on average Liapunov functions

A theorem on average Liapunov functions for dynamical systems is generalized. As an illustration the result is used to establish a rather strong coexistence criterion for an ecological system.  

Asymptotic bounds for some bipartite graph: complete graph Ramsey numbers

The Ramsey number r(H,K_n) is the smallest integer N so that each graph on N vertices that fails to contain H as a subgraph has independence number at least n. It is shown that r(K_2,m,K_n)(m−1+o(1))(n/log n)² and r(C_2m,K_n)c(n/log n)^m/(m−1) for m fixed and n→∞. Also r(K_2,n,K_n)=Θ(n³/log² n) and .  

Relative perturbation bounds for the unitary polar factor

LetB be anm×n (m≥n) complex (or real) matrix. It is known that there is a uniquepolar decomposition B=QH, whereQ*Q=I, then×n identity matrix, andH is positive definite, providedB has full column rank. Existing perturbation bounds suggest that in the worst case, for complex matrices the change inQ be proportional to the reciprocal ofB's least singular value, or the reciprocal of the sum ofB's least and second least singular values if matrices are real. However, there are situations where this unitary polar factor is much more accurately determined by the data than the existing perturbation bounds would indicate. In this paper the following question is addressed: how much mayQ change ifB is perturbed to $\tilde B = D_1^* BD_2 $ , whereD ₁ andD ₂ are nonsingular and close to the identity matrices of suitable dimensions? It is shown that for a such kind of perturbation, the change inQ is bounded only by the distances fromD ₁ andD ₂ to identity matrices and thus is independent ofB's singular values. Such perturbation is restrictive, but not unrealistic. We show how a frequently used scaling technique yields such a perturbation and thus scaling may result in better-conditioned polar decompositions.  

Numerical analysis of oscillations in multiple well problems

Summary. Variational problems which fail to be convex occur often in the study of ordered materials such as crystals. In these problems, the energy density for the material has multiple potential wells. In this paper, we study multiple well problems by first, determining the analytic properties of energy minimizing sequences and then, by estimating the continuous problem by an approximation using piecewise linear finite elements. We show that even when there is no minimizer of the energy, the approximations still take on a predictable structure. Received May 18, 1994  

Asymptotic Upper Bounds for Ramsey Functions

 We show that for any graph G with N vertices and average degree d, if the average degree of any neighborhood induced subgraph is at most a, then the independence number of G is at least Nf _{a +1}(d), where f _{a +1}(d)=∫₀ ¹(((1−t)^{1/( a +1)})/(a+1+(d−a−1)t))dt. Based on this result, we prove that for any fixed k and l, there holds r(K _k + _l ,K _n )≤ (l+o(1))n ^k /(logn) ^{k −1}. In particular, r(K _k , K _n )≤(1+o(1))n ^{k −1}/(log n) ^{k −2}. Received: May 11, 1998 Final version received: March 24, 1999  

The Baum-Connes conjecture for hyperbolic groups

We prove the Baum-Connes conjecture for hyperbolic groups and their subgroups. Oblatum 20-VI-2001 & 24-VIII-2001?Published online: 15 April 2002 RID="*" ID="*"The second author is partially supported by NSF and MSRI.  

Edge Coloring of Embedded Graphs with Large Girth

Let G be a simple graph embedded in the surface of Euler characteristic ()0. Denote _e(G), and g the edge chromatic number, the maximum degree and the girth of the graph G, respectively. The paper shows that _e(G)= if 5 and g4, or 4 and g5, or 3 and g9. In addition, if ()>0, then _e(G)= if 3 and g8. Acknowledgments.The authors would like to thank Dr. C.Q. Zhang for carefully reading several versions of this paper during its preparation and for suggesting several stylistic changes that have improved the overall presentation.  

Optimal Control of the Obstacle for an Elliptic Variational Inequality

An optimal control problem for an elliptic obstacle variational inequality is considered. The obstacle is taken to be the control and the solution to the obstacle problem is taken to be the state. The goal is to find the optimal obstacle from H ¹ ₀ (Ω) so that the state is close to the desired profile while the H ¹ (Ω) norm of the obstacle is not too large. Existence, uniqueness, and regularity as well as some characterizations of the optimal pairs are established. Accepted 11 September 1996  

Topological design of telecommunication networks-local access design methods

Computer communication networks and telecommunication systems are growing at an explosive rate. Some of the major factors influencing this phenomenal growth rate have been technology driven, deregulation of the telecommunication industry and the breakup of AT&T, product and service introductions and competition, new application areas, price reductions and improved services. Corporations have discovered how to use telecommunication-based systems and computer networks as a strategic competitive weapon. Modern computer networks consist of backbone networks which serve as major highways to transfer large volumes of communication traffic, and local access networks which feed traffic between the backbone network and end user nodes. The design of the local access network is a complex process which builds on many difficult combinatorial optimization problems. This paper surveys many of the problems, presents the state of the art in solving them, and demonstrates a variety of solution procedures. The paper concludes with a list of open problems and areas open for further investigation.This research was partially supported by a Dean's grant for faculty research at the Owen Graduate School of Management, Vanderbilt University, Nashville, TN 37204, USA.