Robert Kilmoyer

Bio: Robert Kilmoyer is an academic researcher from University of Wisconsin-Madison. The author has contributed to research in topics: Structure (category theory) & Semisimple algebra. The author has an hindex of 1, co-authored 1 publications receiving 28 citations. Previous affiliations of Robert Kilmoyer include Clark University.

Analysis of systematic scan Metropolis algorithms using Iwahori-Hecke algebra techniques

Abstract: When faced with a complex task, is it better to be systematic or to proceed by making random adjustments? We study aspects of this problem in the context of generating random elements of a finite group. For example, suppose we want to fill n empty spaces with zeros and ones such that the probability of configuration x = (x1, . . . , xn) is θ n−|x|(1− θ)|x|, with |x| the number of ones in x. A systematic scan approach works left to right, filling each successive place with a θ coin toss. A random scan approach picks places at random, and a given site may be hit many times before all sites are hit. The systematic approach takes order n steps and the random approach takes order 1 4n log n steps. Realistic versions of this toy problem arise in image analysis and Ising-like simulations, where one must generate a random array by a Monte Carlo Markov chain. Systematic updating and random updating are competing algorithms that are discussed in detail in Section 2. There are some successful analyses for random scan algorithms, but the intuitively appealing systematic scan algorithms have resisted analysis. Our main results show that the binary problem just described is exceptional; for the examples analyzed in this paper, systematic and random scans converge in about the same number of steps. Let W be a finite Coxeter group generated by simple reflections s1, s2, . . . , sn, where s2 i = id. For example, W may be the permutation group Sn+1 with si = (i, i + 1). The length function `(w) is the smallest k such that w = si1si2 · · · sik . Fix 0 < θ ≤ 1 and define a probability distribution on W by π(w) = θ −`(w) PW (θ−1) , where PW(θ −1) = ∑

Generalized polygons, SCABs and GABs

Abstract: E 8 root lattices. Miscellaneous problems and examples. Connections with finite group theory. References. Introduction Recently there has been a great deal of activity in the geometric and group theoretic study of "building-like geometries" One of the directions of this activity has concerned GABs ("geometries that are almost buildings") and SCABs ("chamber systems that are almost buildings"). (Other names for these are "chamber systems of type M" and "geometries of type M" in [Ti 7], and "Tits geometries of type M" or "Tits chamber systems of type M" in [AS; Tim 1,2].) The main goal of this paper is to survey these developments with special emphasis on their relationships with finite geometries. The study of SCABs is primarily based upon the work of Tits, and especially on the seminal paper [Ti 7]. While our Chapter B briefly describes his results, few proofs are given. (In fact, relatively few proofs will be provided throughout this survey.) On the other hand, our point of view will be somewhat elementary (especially when compared with [Ti 7] and Tits' paper in these Proceedings), in the sense that a great deal of space will be devoted to ways in which familiar geometric objects (e.g. hyper-ovals, difference sets or root systems) can be easily used in order to construct new or less familiar ones. Consequently, it will be clear both that the subject is still in its infancy and that there are many opportunities to enter into this area. 81 Generalized polygons are the building bricks from which SCABs and GABs are obtained. A great deal has been written about these: they are rank 2 buildings, they include projective planes, and they are very natural from a variety of geometric, group theoretic and combinatorial viewpoints. In Chapter A we will construct almost all of the known finite examples that are not projective planes, and indicate their group theoretic origins when appropriate. However, we will not discuss the large number of characterizations presently known-especially of finite generalized quadrangles; that is outside the scope (and size) of this paper. Instead, Chapter A is intended to function as the source for generalized polygons arising later. Moreover, Chapter A highlights one basic question concerning the constructions in Chapter C: what nonclassical finite generalized polygons can appear in finite SCABs? (In fact, the only such SCABs that seem to be known are similar to those in (C.6.11).) While Chapter B contains …

Analysis of systematic scan Metropolis algorithms using Iwahori-Hecke algebra techniques

Abstract: When faced with a complex task, is it better to be systematic or to proceed by making random adjustments? We study aspects of this problem in the context of generating random elements of a finite group. For example, suppose we want to fill n empty spaces with zeros and ones such that the probability of configuration x = (x1, . . . , xn) is θ n−|x|(1− θ)|x|, with |x| the number of ones in x. A systematic scan approach works left to right, filling each successive place with a θ coin toss. A random scan approach picks places at random, and a given site may be hit many times before all sites are hit. The systematic approach takes order n steps and the random approach takes order 1 4n log n steps. Realistic versions of this toy problem arise in image analysis and Ising-like simulations, where one must generate a random array by a Monte Carlo Markov chain. Systematic updating and random updating are competing algorithms that are discussed in detail in Section 2. There are some successful analyses for random scan algorithms, but the intuitively appealing systematic scan algorithms have resisted analysis. Our main results show that the binary problem just described is exceptional; for the examples analyzed in this paper, systematic and random scans converge in about the same number of steps. Let W be a finite Coxeter group generated by simple reflections s1, s2, . . . , sn, where s2 i = id. For example, W may be the permutation group Sn+1 with si = (i, i + 1). The length function `(w) is the smallest k such that w = si1si2 · · · sik . Fix 0 < θ ≤ 1 and define a probability distribution on W by π(w) = θ −`(w) PW (θ−1) , where PW(θ −1) = ∑

Invariant Relations, Coherent Configurations and Generalized Polygons

Abstract: A high point in the combinatorial approach to the theory of finite permutation groups is Wielandt’s theory of invariant relations, culminating in his theorem on groups of degree p2 [16]. In section 1 we give a few rudiments of Wielandt’s theory in the context of the theory of G-spaces, illustrating the concepts by a proof, which seems first to have been made explicit by R. Liebler [12], of a theorem of Alperin [1].