Enumerating spanning trees

In 1889 Arthur Cayley stated his well-known and widely used theorem that there are n° - 2 trees on n labeled vertices [6, p. 70]. Since he originally stated it, the theorem has received much attention: people have proved it in many different ways. In this paper we consider three of these proofs. The first is an algebraic . result using Kirchhoff s Matrix Tree theorem. The second proof shows a one-to-one correspondence between trees on labeled vertices and sequences known as Prufer codes. The final proof involves degree sequences and multinomial coefficients. In addition, we extend each of these three proofs to find a result for the number·of spanning trees on the complete bipartite graph, and extend the first two results to count the number of spanning trees on the complete tripartite graph. We conclude with a brief generalization to the number of spanning trees on the complete k-partite graph

An Introduction to Combinatorics via Cayley\u27s Theorem

In this paper, we explore some of the methods that are often used to solve combinatorial problems by proving Cayley’s theorem on trees in multiple ways. The intended audience of this paper is undergraduate and graduate mathematics students with little to no experience in combinatorics. This paper could also be used as a supplementary text for an undergraduate combinatorics course

Constructive analysis of two dimensional Fermi systems at finite temperature

We consider a dilute Fermion system in continuum two spatial dimensions with short-range interaction. We prove nonperturbatively that at low temperature the renormalized perturbation expansion has non-zero radius of convergence. The convergence radius shrinks when the energy scale goes to the infrared cutoff. The shrinking rate of the convergence radius is established to be dependent of the sign of the coupling constant by a detailed analysis of the so-called ladder contributions. We prove further that the self-energy of the model is uniformly of C1, in the analytic domain of the theory. The proofs are based on renormalization of the Fermi surface and multiscale analysis employing mathematical renormalization group technique. Tree expansion is introduced to reorganize perturbation expansion nicely. Finally we apply these techniques to construct a half-filled Hubbard model on honeycomb bilayer lattice with local interaction

Graph Theory and Universal Grammar

Tese arquivada ao abrigo da Portaria nº 227/2017 de 25 de Julho-Registo de Grau EstrangeiroIn the last few years, Noam Chomsky (1994; 1995; 2000; 2001) has gone quite far in the direction of simplifying syntax, including eliminating X-bar theory and the levels of D-structure and S-structure entirely, as well as reducing movement rules to a combination of the more primitive operations of Copy and Merge. What remain in the Minimalist Program are the operations Merge and Agree and the levels of LF (Logical Form) and PF (Phonological form). My doctoral thesis attempts to offer an economical theory of syntactic structure from a graph-theoretic point of view (cf. Diestel, 2005), with special emphases on the elimination of category and projection labels and the Inclusiveness Condition (Chomsky 1994). The major influences for the development of such a theory have been Chris Collins’ (2002) seminal paper “Eliminating labels”, John Bowers (2001) unpublished manuscript “Syntactic Relations” and the Cartographic Paradigm (see Belletti, Cinque and Rizzi’s volumes on OUP for a starting point regarding this paradigm). A syntactic structure will be regarded here as a graph consisting of the set of lexical items, the set of relations among them and nothing more

Modeling and inference of changing dependence among multiple time-series

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2009.Cataloged from PDF version of thesis.Includes bibliographical references (p. 183-190).In this dissertation we investigate the problem of reasoning over evolving structures which describe the dependence among multiple, possibly vector-valued, time-series. Such problems arise naturally in variety of settings. Consider the problem of object interaction analysis. Given tracks of multiple moving objects one may wish to describe if and how these objects are interacting over time. Alternatively, consider a scenario in which one observes multiple video streams representing participants in a conversation. Given a single audio stream, one may wish to determine with which video stream the audio stream is associated as a means of indicating who is speaking at any point in time. Both of these problems can be cast as inference over dependence structures. In the absence of training data, such reasoning is challenging for several reasons. If one is solely interested in the structure of dependence as described by a graphical model, there is the question of how to account for unknown parameters. Additionally, the set of possible structures is generally super-exponential in the number of time series. Furthermore, if one wishes to reason about structure which varies over time, the number of structural sequences grows exponentially with the length of time being analyzed. We present tractable methods for reasoning in such scenarios. We consider two approaches for reasoning over structure while treating the unknown parameters as nuisance variables. First, we develop a generalized likelihood approach in which point estimates of parameters are used in place of the unknown quantities. We explore this approach in scenarios in which one considers a small enumerated set of specified structures.(cont.) Second, we develop a Bayesian approach and present a conjugate prior on the parameters and structure of a model describing the dependence among time-series. This allows for Bayesian reasoning over structure while integrating over parameters. The modular nature of the prior we define allows one to reason over a super-exponential number of structures in exponential-time in general. Furthermore, by imposing simple local or global structural constraints we show that one can reduce the exponential-time complexity to polynomial-time complexity while still reasoning over a super-exponential number of candidate structures. We cast the problem of reasoning over temporally evolving structures as inference over a latent state sequence which indexes structure over time in a dynamic Bayesian network. This model allows one to utilize standard algorithms such as Expectation Maximization, Viterbi decoding, forward-backward messaging and Gibbs sampling in order to efficiently reasoning over an exponential number of structural sequences. We demonstrate the utility of our methodology on two tasks: audio-visual association and moving object interaction analysis. We achieve state-of-the-art performance on a standard audio-visual dataset and show how our model allows one to tractably make exact probabilistic statements about interactions among multiple moving objects.by Michael Richard Siracusa.Ph.D

2018-2019 academic catalog

This annual catalog from the University of South Carolina Upstate includes the following: academic calendar, general information, admissions information, financial information, student affairs, and more

2004-2005 Ursinus College Course Catalogue

A digitized copy of the 2004-2005 Ursinus College Catalog. It contains details of the curriculum, departmental requirements and courses of instruction as well as lists of students, faculty and administrators. Student life, terms of admission, expenses and financial aid are also included as well as descriptions of the buildings and equipment available to students.https://digitalcommons.ursinus.edu/uccatalog/1057/thumbnail.jp