Among their many uses, growth processes (probabilistic amplification), were used for constructing reliable networks from unreliable components, and deriving complexity bounds of various classes of functions. Hence, determining the initial conditions for such processes is an important and challenging problem. In this paper we characterize growth processes by their initial conditions and derive conditions under which results such as Valiant's[Val84] hold. First, we completely characterize growth processes that use linear connectives. Second, by extending Savicky's [Sav90] analysis, via "Restriction Lemmas", we characterize growth processes that use monotone connectives, and show that our technique is applicable to growth processes that use other connectives as well. Additionally, we obtain explicit bounds on the convergence rates of several growth processes, including the growth process studied by Savicky (1990).

Let f be a de Morgan read-once function of n variables. Let f " be the random restriction obtained by independently assigning to each variable of f , the value 0 with probability (1 \Gamma ")=2, the value 1 with the same probability, and leaving it unassigned with probability ". We show that f " depends, on the average, on only O(" ff n + "n 1=ff ) variables, where ff = log p 5\Gamma1 2 ' 3:27. This result is asymptotically the tightest possible. It improves a similar result obtained recently by Hastad, Razborov and Yao. 1 Introduction Obtaining non-trivial lower bounds on the complexity of Boolean functions is currently a very difficult task. Only a handful of methods yielding such lower bounds are currently known and even they work only in suitably restricted models. The current state of affairs in this respect is summerized in the books of Dunne [5] and Wegener [15] and the survey paper of Boppana and Sipser [2]. Many of the currently known methods for obtaining complexity l...

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In this note we consider the problem of computing threshold functions using directed monotone contact networks. We give constructions of monotone contact networks of size (k − 1)(n − k + 2) ⌈log(n − k + 2) ⌉ computing T n k, for 2 ≤ k ≤ n − 1. Our upper bound is close to the Ω(kn log(n/(k − 1))) lower bound for small thresholds and the k(n − k + 1) lower bound for large thresholds. Our networks are described explicitly; we do not use probabilistic existence arguments.

Moore and Shannon have shown that relays with arbitrarily high reliability can be built from relays with arbitrarily poor reliability. Valiant used similar methods to construct monotone readonce formulae of size O(n ff+2 ) (where ff = log p 5\Gamma1 2 ' 3:27) that amplify (/ \Gamma 1 n ; / + 1 n ) (where / = p 5\Gamma1 2 ' 0:62) to (2 \Gamman ; 1 \Gamma 2 \Gamman ) and deduced as a consequence the existence of monotone formulae of the same size that compute the majority of n bits. Boppana has shown that any monotone read-once formula that amplifies (p \Gamma 1 n ; p + 1 n ) to ( 1 4 ; 3 4 ) (where 0 ! p ! 1 is constant) has size\Omega\Gamma n ff ) and that any monotone, not necessarily read-once, contact network (and in particular any monotone formula) that amplifies ( 1 4 ; 3 4 ) to (2 \Gamman ; 1 \Gamma 2 \Gamman ) has size\Omega\Gamma n 2 ). We extend Boppana's results in two ways. We first show that his two lower bounds hold for general read-onc...

Abstract: In this Chapter, we address fault tolerance in wireless sensor networks. In order to make the presentation self-contained, we start by providing a short summary of sensor networks and classical fault tolerance techniques. After that, we discuss the three phases of fault tolerance (fault models, fault detection and identification and resiliency mechanisms) at four levels of abstractions (hardware, system software, middleware, and applications) and four scopes (components of individual node, individual node, network, and the distributed system). The technical cores of the chapter are two case-studies on heterogeneous fault tolerance and discrepancy minimization-based fault detection and correction. We conclude the chapter with a brief survey of the future directions for fault tolerance research in wireless sensor networks.

this article, we will use the term network reliability in a broad sense and cover several subtopics. We will start with network availability and performability, and then discuss survivable network design, followed by fault detection, isolation, and restoration as well as preplanning. We will conclude with a short discussion on recent issues and literature.

This paper develops and tests a model of how organizational structure influences organizational performance. Organizational structure, conceptualized as the decision-making structure among a group of individuals, is shown to affect the number of initiatives pursued by organizations, and the omission and commission errors (Type I and II errors, respectively) made by organizations. The empirical setting are over 150,000 stock-picking decisions made by 609 mutual funds. Mutual funds offer an ideal and rare setting to test the theory, as detailed records exist on the projects they face, the decisions they make, and the outcomes of these decisions. The independent variable of the study, organizational structure, is coded from fund management descriptions made by Morningstar, and the estimates of the omission and commission errors are computed by a novel technique that uses bootstrapping to create measures which are comparable across funds. The findings suggest that organizational structure has relevant and predictable effects on a wide range of

A certain type of integer grid, called here an echelon grid, is an object found both in coherent systems whose components have a finite or countable number of levels and in algebraic geometry. If α = (α1,...,αd) is an integer vector representing the state of a system, then the corresponding algebraic object is a monomial x α1 1 · · ·xα d d in the indeterminates x1,...,xd. The idea is to relate a coherent system to monomial ideals, so that the so-called Scarf complex of the monomial ideal yields an inclusion–exclusion identity for the probability of failure, which uses many fewer terms than the classical identity. Moreover in the “general position ” case we obtain via the Scarf complex the tube bounds given by Naiman and Wynn [J. Inequal. Pure Appl. Math. (2001) 2 1–16]. Examples are given for the binary case but the full utility is for general multistate coherent systems and a comprehensive example is given. 1. Introduction. The

this paper is that it is easily possible to analyze substantially different distributions for choosing the formula with little additional difficulty.