Exact conditions for countable inclusion-exclusion identity and extensions

Simplification of inclusion-exclusion on intersections of unions with application to network systems reliability

Reliability of safety-critical systems is an important issue in system engineering and in most practical situations the reliability of a non series-parallel network system has to be calculated. Some methods for calculating reliability use the probability principle of inclusion-exclusion. When dealing with complex networks, this leads to very long mathematical expressions which are usually computationally very expensive to calculate. In this paper, we provide a new expression to simplify the probability principle of inclusion-exclusion's formula for intersections of unions, which appear when ca…

Inclusion-exclusion principles for convex hulls and the Euler relation

Consider $n$ points $X_1,\ldots,X_n$ in $\mathbb R^d$ and denote their convex hull by $Π$. We prove a number of inclusion-exclusion identities for the system of convex hulls $Π_I:=conv(X_i\colon i\in I)$, where $I$ ranges over all subsets of $\{1,\ldots,n\}$. For instance, denoting by $c_k(X)$ the number of $k$-element subcollections of $(X_1,\ldots,X_n)$ whose convex hull contains a point $X\in\mathbb R^d$, we prove that $$ c_1(X)-c_2(X)+c_3(X)-\ldots + (-1)^{n-1} c_n(X) = (-1)^{\dim Π} $$ for all $X$ in the relative interior of $Π$. This confirms a conjecture of R. Cowan [Adv. Appl. Probab.,…

The inclusion process: duality and correlation inequalities

We prove a comparison inequality between a system of independent random walkers and a system of random walkers which either interact by attracting each other -- a process which we call here the symmetric inclusion process (SIP) -- or repel each other -- a generalized version of the well-known symmetric exclusion process. As an application, new correlation inequalities are obtained for the SIP, as well as for some interacting diffusions which are used as models of heat conduction, -- the so-called Brownian momentum process, and the Brownian energy process. These inequalities are counterparts of…

可数等价关系的共谱半径

Co-spectral radius for countable equivalence relations

本文定义可数等价关系包含的共谱半径，推广2-3方法证明有界度单模随机有根图的游走增长存在，并研究其在超有限关系与渗流临界指数中的应用。

简单排他过程的图形构造及其在随机环境中的应用

Graphical constructions of simple exclusion processes with applications to random environments

本文利用图形构造方法在弱化条件下建立了可数集上SSEP和SEP的Feller过程及其生成元核性质，并将结果应用于随机环境，推广了淬火流体力学极限。