Some Problems in Algebraic and Extremal Graph Theory.

In this dissertation, we consider a wide range of problems in algebraic and extremal graph theory. In extremal graph theory, we will prove that the Tree Packing Conjecture is true for all sequences of trees that are \u27almost stars\u27; and we prove that the Erdos-Sos conjecture is true for all graphs G with girth at least 5. We also conjecture that every graph G with minimal degree k and girth at least $2t+1$ contains every tree T of order $kt+1$ such that $\Delta(T)\leq k.$ This conjecture is trivially true for t = 1. We Prove the conjecture is true for t = 2 and that, for this value of t, the conjecture is best possible. We also provide supporting evidence for the conjecture for all other values of t. In algebraic graph theory, we are primarily concerned with isomorphism problems for vertex-transitive graphs, and with calculating automorphism groups of vertex-transitive graphs. We extend Babai\u27s characterization of the Cayley Isomorphism property for Cayley hypergraphs to non-Cayley hypergraphs, and then use this characterization to solve the isomorphism problem for every vertex-transitive graph of order pq, where p and q distinct primes. We also determine the automorphism groups of metacirculant graphs of order pq that are not circulant, allowing us to determine the nonabelian groups of order pq that are Burnside groups. Additionally, we generalize a classical result of Burnside stating that every transitive group G of prime degree p, is doubly transitive or contains a normal Sylow p-subgroup to all p\sp k, provided that the Sylow p-subgroup of G is one of a specified family. We believe that this result is the most significant contained in this dissertation. As a corollary of this result, one easily gives a new proof of Klin and Poschel\u27s result characterizing the automorphism groups of circulant graphs of order p\sp k, where p is an odd prime

Multicoloured Random Graphs: Constructions and Symmetry

This is a research monograph on constructions of and group actions on countable homogeneous graphs, concentrating particularly on the simple random graph and its edge-coloured variants. We study various aspects of the graphs, but the emphasis is on understanding those groups that are supported by these graphs together with links with other structures such as lattices, topologies and filters, rings and algebras, metric spaces, sets and models, Moufang loops and monoids. The large amount of background material included serves as an introduction to the theories that are used to produce the new results. The large number of references should help in making this a resource for anyone interested in beginning research in this or allied fields.Comment: Index added in v2. This is the first of 3 documents; the other 2 will appear in physic

English for students of mathematics: учебное пособие для студентов Института математики и механики им. Н.И. Лобачевского

Данное учебное пособие представляет собой практические задания по ESP (английский язык для специальных целей)для студентов уровня Pre-Intermediate Института математики и механики I и II курсов. Пособие состоит из 3 разделов, приложения и словаря. Цель данного пособия развить у студентов математической специальности навыки работы со специализированными текстами, включая навыки просмотрового и поискового чтения, навыки монологической речи и навыки ведения дискуссии по актуальным математическим проблемам, расширить словарный запас за счет специальной лексики, а также развить навыки технического перевода с английского на русский и с русского на английский языки01.03.01 Математика01.03.03 Механика и математическое моделирование01.03.04 Прикладная математика02.03.01 Математика и компьютерные науки15.03.03 Прикладная механикаАнглийский языкбакалавриа

Information-theoretic causal inference of lexical flow

This volume seeks to infer large phylogenetic networks from phonetically encoded lexical data and contribute in this way to the historical study of language varieties. The technical step that enables progress in this case is the use of causal inference algorithms. Sample sets of words from language varieties are preprocessed into automatically inferred cognate sets, and then modeled as information-theoretic variables based on an intuitive measure of cognate overlap. Causal inference is then applied to these variables in order to determine the existence and direction of influence among the varieties. The directed arcs in the resulting graph structures can be interpreted as reflecting the existence and directionality of lexical flow, a unified model which subsumes inheritance and borrowing as the two main ways of transmission that shape the basic lexicon of languages. A flow-based separation criterion and domain-specific directionality detection criteria are developed to make existing causal inference algorithms more robust against imperfect cognacy data, giving rise to two new algorithms. The Phylogenetic Lexical Flow Inference (PLFI) algorithm requires lexical features of proto-languages to be reconstructed in advance, but yields fully general phylogenetic networks, whereas the more complex Contact Lexical Flow Inference (CLFI) algorithm treats proto-languages as hidden common causes, and only returns hypotheses of historical contact situations between attested languages. The algorithms are evaluated both against a large lexical database of Northern Eurasia spanning many language families, and against simulated data generated by a new model of language contact that builds on the opening and closing of directional contact channels as primary evolutionary events. The algorithms are found to infer the existence of contacts very reliably, whereas the inference of directionality remains difficult. This currently limits the new algorithms to a role as exploratory tools for quickly detecting salient patterns in large lexical datasets, but it should soon be possible for the framework to be enhanced e.g. by confidence values for each directionality decision

Excursions into Algebra and Combinatorics at q=0q=0

We explore combinatorics associated with the degenerate Hecke algebra at $q=0$, obtaining a formula for a system of orthogonal idempotents, and also exploring various pattern avoidance results. Generalizing constructions for the 0-Hecke algebra, we explore the representation theory of \JJ-trivial monoids. We then discuss two-tensors of crystal bases for $U_q(\tilde{\mathfrak{sl}_2})$, establishing a complementary result to one of Bandlow, Schilling, and Thi\'ery on affine crystals arising from promotion operators. Finally, we give a computer implementation of Stembridge's local axioms for simply-laced crystal bases.Comment: 92 pages, 13 figures. PHd Dissertation accepted at the University of California on July 15th, 2011. arXiv admin note: text overlap with arXiv:1012.136

Mathematical linguistics

but in fact this is still an early draft, version 0.56, August 1 2001. Please d