This work studies the typical behavior of random integer-valued Lipschitz functions on expander graphs with sufficiently good expansion. We consider two families of functions: M -Lipschitz functions (functions which change by at most M along edges) and integer-homomorphisms (functions which change by exactly 1 along edges). We prove that such functions typically exhibit very small fluctuations....

We present sufficient conditions for the cohomology of a closed aspherical manifold to be proper Lipschitz in sense of Connes-Gromov-Moscovici [CGM]. The conditions are stated in terms of the Stone-Čech compactification of the universal cover of a manifold. We show that these conditions are formally weaker than the sufficient conditions for the Novikov conjecture given in [CP]. Also we show tha...

Foreword These notes are a lightly edited revision of notes written for the course " Graph Partitioning and Expanders " offered at Stanford in Winter 2011 and Winter 2013. I wish to thank the students who attended this course for their enthusiasm and hard work. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily re...

Expanders are graphs that are sparse yet highly connected. The connectivity of the expander is measured by its vertex expansion, which asks that every set S of vertices that is not too large has significantly more than |S| neighbors. Definition. A bipartite (multi)graph with N left-vertices, M right-vertices, and left degree D is specified by a function Γ : [N ]× [D]→ [M ], where Γ(x, y) denote...

Let now φ be an instance of 3SAT with n variables x1, . . . , xn and m clauses. For each variable xi, let occi be the number of occurrences of xi, that is, the number of clauses that involve the literal xi or the literal x̄i. We write xi ∈ Cj if the variable xi occurs in clause Cj . Notice that ∑n i=1 occi = 3m. For each i, construct a 1-expander graph Gi = (Vi, Ei) where Vi has occi vertices, o...

The tissue expander has found widespread use in reconstructive surgery. The donor area is needed, the recipient site with similar characteristics (color, thickness, hair follicle) features, minimal scarring and leads to donor site morbidity. In our study, presented face, scalp and neck reconstruction results obtained with the use of expander. The face, scalp and neck areas expander was used to ...

This paper will introduce expander graphs. Kolmogorov and Barzdin’s proof on the three dimensional realization of networks will be discussed as one of the first examples of expander graphs. The last section will discuss error correcting code as an application of expander graphs to computer science.

of 14 cases who underwent this operation only in one of them because of infection in operation site, we had to extract the expander. also, in the first day of the operation that we injected serum inside the expander of two other patients, the wound opened up and the expander appeared and came out automatically. the remaining ten patients have tolerated the operation after treatment period witho...

This study investigated the effect of the built-in volume ratio of an expander on the performance of a dual-loop Rankine cycle system for the engine waste heat recovery of a vehicle. Varying vehicle operating conditions can cause a positive displacement expander to operate in both underand over-expansion states. Therefore, analysis of the off-design performance of the expander is very important...

Carbon dioxide (CO2) as a natural refrigerant is one candidate alternative to HFC refrigerants. But CO2 system has lower efficiency and is not yet competitive concerning the COP. The most effective method to improve the efficiency of the transcritical CO2 system is to replace the throttling device with an expander. In this paper, a two-cylinder rolling piston expander without inlet valve has be...