A universal sequence of integers generating balanced Steinhaus figures modulo an odd number

In this paper, we partially solve an open problem, due to J.C. Molluzzo in 1976, on the existence of balanced Steinhaus triangles modulo a positive integer n, that are Steinhaus triangles containing all the elements of Z/nZ with the same multiplicity. For every odd number n, we build an orbit in Z/nZ, by the linear cellular automaton generating the Pascal triangle modulo n, which contains infinitely many balanced Steinhaus triangles. This orbit, in Z/nZ, is obtained from an integer sequence called the universal sequence. We show that there exist balanced Steinhaus triangles for at least 2/3 of the admissible sizes, in the case where n is an odd prime power. Other balanced Steinhaus figures, such as Steinhaus trapezoids, generalized Pascal triangles, Pascal trapezoids or lozenges, also appear in the orbit of the universal sequence modulo n odd. We prove the existence of balanced generalized Pascal triangles for at least 2/3 of the admissible sizes, in the case where n is an odd prime power, and the existence of balanced lozenges for all admissible sizes, in the case where n is a square-free odd number.     

Infinite horizon asymptotic average optimality for large-scale parallel server networks

We study infinite-horizon asymptotic average optimality for parallel server networks with multiple classes of jobs and multiple server pools in the Halfin–Whitt regime. Three control formulations are considered: (1) minimizing the queueing and idleness cost, (2) minimizing the queueing cost under constraints on idleness at each server pool, and (3) fairly allocating the idle servers among different server pools. For the third problem, we consider a class of bounded-queue, bounded-state (BQBS) stable networks, in which any moment of the state is bounded by that of the queue only (for both the limiting diffusion and diffusion-scaled state processes). We show that the optimal values for the diffusion-scaled state processes converge to the corresponding values of the ergodic control problems for the limiting diffusion. We present a family of state-dependent Markov balanced saturation policies (BSPs) that stabilize the controlled diffusion-scaled state processes. It is shown that under these policies, the diffusion-scaled state process is exponentially ergodic, provided that at least one class of jobs has a positive abandonment rate. We also establish useful moment bounds, and study the ergodic properties of the diffusion-scaled state processes, which play a crucial role in proving the asymptotic optimality.     

Average consensus in networks of multi-agents with both switching topology and coupling time-delay

This paper is devoted to the study of the average-consensus problem in directed networks of agents with both switching topology and time-delay. The stability analysis is performed based on a proposed Lyapunov-Krasovskii function. Sufficient conditions in terms of linear matrix inequalities (LMIs) are given to guarantee the average consensus under arbitrary switching of the network topology even if the time-delay is time-varying. Numerical simulations show the effectiveness of our theoretical results.     

Orthogonal schedules in single round robin tournaments

A measure for the flexibility of a Home-Away Pattern set (HAP-set) is the width. The width of a HAP-set equals the size of the largest set of schedules compatible with the HAP-set, for which no match is scheduled in the same round in any two schedules. We prove lower and upper bounds on the width, and identify HAP-sets with largest possible width when the number of teams is a power of 2.     

有界平衡拟凸域上星形映射的一个新特征

假定Ω是Cn中具有C2定义函数的有界平衡拟凸域.本文证明Ω上的每一星形映射f能表示成Schwarz映射的极限形式.作为应用,得到有界平衡拟凸域上星形映射的增长定理.最后给出星形映射的一个的等价刻画.