Optimizing the robustness of electrical power systems against cascading failures

Electrical power systems are one of the most important infrastructures that support our society. However, their vulnerabilities have raised great concern recently due to several large-scale blackouts around the world. In this paper, we investigate the robustness of power systems against cascading failures initiated by a random attack. This is done under a simple yet useful model based on global and equal redistribution of load upon failures. We provide a complete understanding of system robustness by i) deriving an expression for the final system size as a function of the size of initial attacks; ii) deriving the critical attack size after which system breaks down completely; iii) showing that complete system breakdown takes place through a first-order (i.e., discontinuous) transition in terms of the attack size; and iv) establishing the optimal load-capacity distribution that maximizes robustness. In particular, we show that robustness is maximized when the difference between the capacity and initial load is the same for all lines; i.e., when all lines have the same redundant space regardless of their initial load. This is in contrast with the intuitive and commonly used setting where capacity of a line is a fixed factor of its initial load.Comment: 18 pages including 2 pages of supplementary file, 5 figure

A note on the Banaś modulus of smoothness in the Bynum space

AbstractRecently, the Banaś modulus of smoothness for the Bynum space b2,∞ was obtained by Zuo and Cui (Z. Zuo, Y. Cui, Some modulus and normal structure in Banach space, J. Inequal. Appl. 2009 (2009) 15. doi:10.1155/2009/676373. Article ID 676373). It is however not true in general. In this note, we will present the exact value for this modulus in the b2,∞ space

Fast Evaluation of Generalized Todd Polynomials: Applications to MacMahon's Partition Analysis and Integer Programming

The Todd polynomials $td_k=td_k(b_1,b_2,\dots,b_m)$ are defined by their generating functions $$\sum_{k\ge 0} td_k s^k = \prod_{i=1}^m \frac{b_i s}{e^{b_i s}-1}.$$ It appears as a basic block in Todd class of a toric variety, which is important in the theory of lattice polytopes and in number theory. We find generalized Todd polynomials arise naturally in MacMahon's partition analysis, especially in Erhart series computation.We give fast evaluation of generalized Todd polynomials for numerical $b_i$'s. In order to do so, we develop fast operations in the quotient ring $\mathbb{Z}_p[[x]]$ modulo $s^d$ for large prime $p$. As applications, i) we recompute the Ehrhart series of magic squares of order 6, which was first solved by the first named author. The running time is reduced from 70 days to about 1 day; ii) we give a polynomial time algorithm for Integer Linear Programming when the dimension is fixed, with a good performance.Comment: 2 table

Two proofs of the q,tq,t-symmetry of the generalized q,tq,t-Catalan number C(k1,k2,k3)(q,t)C_{(k_1,k_2,k_3)}(q,t)

We give two proofs of the $q,t$-symmetry of the generalized $q,t$-Catalan number $C_{\vec{k}}(q,t)$ for $\vec{k}=(k_1,k_2,k_3)$. One is by MacMahon's partition analysis as we proposed; the other is by a direct bijection.Comment: 13 pages, 2 figure

Inverting the General Order Sweep Map

Inspired by Thomas-Williams work on the modular sweep map, Garsia and Xin gave a simple algorithm for inverting the sweep map on rational $(m,n)$-Dyck paths for a coprime pairs $(m,n)$ of positive integers. We find their idea naturally extends for general Dyck paths. Indeed, we define a class of Order sweep maps on general Dyck paths, using different sweep orders on level $0$. We prove that each such Order sweep map is a bijection. This includes sweep map for general Dyck paths and incomplete general Dyck paths as special cases