181,877 research outputs found
Personal Papers (MS 80-0002)
No full textHandwritten letter from Mrs. R. C. Potts to Mr. and Mrs. Kempner thanking them for the invitation to the annual Kempner family barbecue and promising to attend
Bounds on the complex zeros of (Di)Chromatic polynomials and Potts-model partition functions
No full textWe show that there exist universal constants C(r) such that, for all loopless graphs G of maximum degree less than or equal to r, the zeros (real or complex) of the chromatic polynomial P-G(q) lie in the disc \q\ 7.963907r. This result is a corollary of a more general result on the zeros of the Potts-model partition function Z(G)(q. {v(e)}) in the complex antiferromagnetic regime \1 + v(e)\ less than or equal to 1. The proof is based on a transformation of the Whitney-Tutte-Fortuin-Kasteleyn representation of Z(G)(q,:{v(e)}) to a polymer gas. followed by verification of the Dobrushin-Kotecky-Preiss condition for nonvanishing of a polymer-model partition function. We also show that, for all loopless graphs G of second-largest degree less than or equal to r, the zeros of P-G(q) lie in the disc \q\ < C(r)+ 1. Along the way, I give a simple proof of a generalized (multivariate) Brown-Colbourn conjecture on the zeros of the reliability polynomial for the special case of series-parallel graphs
Bulk and boundary scattering in the q-state potts mode
No full textThis thesis is concerned with the properties of 1 + 1 dimensional massive field theories in both infinite and semi-infinite geometries. Chapters 1, 2 and 3 develop the necessary theoretical framework and review existing work by Chim and Zamolodchikov [1] on integrable perturbations of the (bulk) q-state Potts model, the particular model under consideration in this thesis. Chapter 4 consists of a detailed analysis of the bootstrap for this model, during the course of which unexpected behaviour arises. The treatment of 1] has consequently been revised, but further investigation will be necessary before complete understanding of this behaviour can be reached. In the final chapter, attention turns to the imposition of boundary conditions on two dimensional systems. After looking at this from a statistical mechanical point of view, a brief review of boundary conformal held theory and its integrable perturbations is given. This leads once more to a consideration of the q-state Potts model. After summarising [2], where fixed and free boundary conditions are considered, a third and previously untreated boundary condition is discussed
Computing the Cramer-Rao bound of Markov random field parameters: Application to the Ising and the Potts models
No full textThis letter considers the problem of computing the Cramer–Rao bound for the parameters of a Markov random field. Computation of the exact bound is not feasible for most fields of interest because their likelihoods are intractable and have intractable derivatives. We show here how it is possible to formulate the computation of the bound as a statistical inference problem that can be solve approximately, but with arbitrarily high accuracy, by using a Monte Carlo method. The proposed methodology is successfully applied on the Ising and the Potts models
Distinguishing between long-transient and asymptotic states in a biological aggregation model
No full textAggregations are emergent features common to many biological systems. Mathematical models to understand their emergence are consequently widespread, with the aggregation–diffusion equation being a prime example. Here we study the aggregation–diffusion equation with linear diffusion in one spatial dimension. This equation is known to support solutions that involve both single and multiple aggregations. However, numerical evidence suggests that the latter, which we term ‘multi-peaked solutions’ may often be long-transient solutions rather than asymptotic steady states. We develop a novel technique for distinguishing between long transients and asymptotic steady states via an energy minimisation approach. The technique involves first approximating our study equation using a limiting process and a moment closure procedure. We then analyse local minimum energy states of this approximate system, hypothesising that these will correspond to asymptotic patterns in the aggregation–diffusion equation. Finally, we verify our hypotheses through numerical investigation, showing that our approximate analytic technique gives good predictions as to whether a state is asymptotic or transient. Overall, we find that almost all twin-peaked, and by extension multi-peaked, solutions are transient, except for some very special cases. We demonstrate numerically that these transients can be arbitrarily long-lived, depending on the parameters of the system
Chromatic roots are dense in the whole complex plane
No full textI show that the zeros of the chromatic polynomials P-G(q) for the generalized theta graphs Theta((s.p)) are taken together, dense in the whole complex plane with the possible exception of the disc \q - l\ < l. The same holds for their dichromatic polynomials (alias Tutte polynomials, alias Potts-model partition functions) Z(G)(q,upsilon) outside the disc \q + upsilon\ < \upsilon\. An immediate corollary is that the chromatic roots of not-necessarily-planar graphs are dense in the whole complex plane. The main technical tool in the proof of these results is the Beraha-Kahane-Weiss theorem oil the limit sets of zeros for certain sequences of analytic functions, for which I give a new and simpler proof
Architecture in Ancient Central Italy: connections in Etruscan and early Roman Building
No full textArchitecture in Ancient Central Italy takes studies of individual elements and sites as a starting point to reconstruct a much larger picture of architecture in western central Italy as an industry, and to position the result in space (in the Mediterranean world and beyond) and time (from the second millennium BC to Late Antiquity). This volume demonstrates that buildings in pre-Roman Italy have close connections with Bronze Age and Roman architecture, with practices in local and distant societies, and with the natural world and the cosmos. It also argues that buildings serve as windows into the minds and lives of those who made and used them, revealing the concerns and character of communities in early Etruria, Rome, and Latium. Architecture consequently emerges as a valuable historical source, and moreover a part of life that shaped society as much as reflected it
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