Among graphs, groups, and latin squares

A latin square of order n is an n × n array in which each row and each column contains each of the numbers {1, 2, . . . , n}. A k-plex in a latin square is a collection of entries which intersects each row and column k times and contains k copies of each symbol. This thesis studies the existence of k-plexes and approximations of k-plexes in latin squares, paying particular attention to latin squares which correspond to multiplication tables of groups. The most commonly studied class of k-plex is the 1-plex, better known as a transversal. Although many latin squares do not have transversals, Brualdi conjectured that every latin square has a near transversal—i.e. a collection of entries with distinct symbols which in- tersects all but one row and all but one column. Our first main result confirms Brualdi’s conjecture in the special case of group-based latin squares. Then, using a well-known equivalence between edge-colorings of complete bipartite graphs and latin squares, we introduce Hamilton 2-plexes. We conjecture that every latin square of order n ≥ 5 has a Hamilton 2-plex and provide a range of evidence for this conjecture. In particular, we confirm our conjecture computationally for n ≤ 8 and show that a suitable analogue of Hamilton 2-plexes always occur in n × n arrays with no symbol appearing more than n/√96 times. To study Hamilton 2-plexes in group-based latin squares, we generalize the notion of harmonious groups to what we call H2-harmonious groups. Our second main result classifies all H2-harmonious abelian groups. The last part of the thesis formalizes an idea which first appeared in a paper of Cameron and Wanless: a (k,l)-plex is a collection of entries which intersects each row and column k times and contains at most l copies of each symbol. We demonstrate the existence of (k, 4k)-plexes in all latin squares and (k, k + 1)-plexes in sufficiently large latin squares. We also find analogues of these theorems for Hamilton 2-plexes, including our third main result: every sufficiently large latin square has a Hamilton (2,3)-plex

A Foundation for Arithmetic

This paper contains a proof of Frege\u27s Theorem: the statement, first discovered by George Boolos, that Gottlob Frege\u27s failed proof of the analyticity of arithmetic could be slightly altered so as to provide an axiomitization of arithmetic with just one proposition. After an expository treatment of the mathematical work in Frege\u27s \u27Foundations of Arithmetic,\u27 the work in which Frege presented his failed proof, a novel, and particularly succinct, proof of the Theorem is provided

Random Embeddings of Graphs: The Expected Number of Faces in Most Graphs is Logarithmic

A random 2-cell embedding of a connected graph $G$ in some orientable surface is obtained by choosing a random local rotation around each vertex. Under this setup, the number of faces or the genus of the corresponding 2-cell embedding becomes a random variable. Random embeddings of two particular graph classes -- those of a bouquet of $n$ loops and those of $n$ parallel edges connecting two vertices -- have been extensively studied and are well-understood. However, little is known about more general graphs despite their important connections with central problems in mainstream mathematics and in theoretical physics (see [Lando & Zvonkin, Springer 2004]). There are also tight connections with problems in computing (random generation, approximation algorithms). The results of this paper, in particular, explain why Monte Carlo methods (see, e.g., [Gross & Tucker, Ann. NY Acad. Sci 1979] and [Gross & Rieper, JGT 1991]) cannot work for approximating the minimum genus of graphs. In his breakthrough work ([Stahl, JCTB 1991] and a series of other papers), Stahl developed the foundation of "random topological graph theory". Most of his results have been unsurpassed until today. In our work, we analyze the expected number of faces of random embeddings (equivalently, the average genus) of a graph $G$. It was very recently shown [Campion Loth & Mohar, arXiv 2022] that for any graph $G$, the expected number of faces is at most linear. We show that the actual expected number of faces is usually much smaller. In particular, we prove the following results: 1) $\frac{1}{2}\ln n - 2 < \mathbb{E}[F(K_n)] \le 3.65\ln n$, for $n$ sufficiently large. This greatly improves Stahl's $n+\ln n$ upper bound for this case. 2) For random models $B(n,\Delta)$ containing only graphs, whose maximum degree is at most $\Delta$, we show that the expected number of faces is $\Theta(\ln n)$.Comment: 44 pages, 6 figure

Detecting transcriptionally active regions using genomic tiling arrays

We have developed a method for interpreting genomic tiling array data, implemented as the program TranscriptionDetector. Probed loci expressed above background are identified by combining replicates in a way that makes minimal assumptions about the data. We performed medium-resolution Anopheles gambiae tiling array experiments and found extensive transcription of both coding and non-coding regions. Our method also showed improved detection of transcriptional units when applied to high-density tiling array data for ten human chromosomes

Cartilage oligomeric matrix protein in idiopathic pulmonary fibrosis

Idiopathic pulmonary fibrosis (IPF) is a progressive and life threatening disease with median survival of 2.5-3 years. The IPF lung is characterized by abnormal lung remodeling, epithelial cell hyperplasia, myofibroblast foci formation, and extracellular matrix deposition. Analysis of gene expression microarray data revealed that cartilage oligomeric matrix protein (COMP), a non-collagenous extracellular matrix protein is among the most significantly up-regulated genes (Fold change 13, p-value <0.05) in IPF lungs. This finding was confirmed at the mRNA level by nCounter® expression analysis in additional 115 IPF lungs and 154 control lungs as well as at the protein level by western blot analysis. Immunohistochemical analysis revealed that COMP was expressed in dense fibrotic regions of IPF lungs and co-localized with vimentin and around pSMAD3 expressing cells. Stimulation of normal human lung fibroblasts with TGF-β1 induced an increase in COMP mRNA and protein expression. Silencing COMP in normal human lung fibroblasts significantly inhibited cell proliferation and negatively impacted the effects of TGF-β1 on COL1A1 and PAI1. COMP protein concentration measured by ELISA assay was significantly increased in serum of IPF patients compared to controls. Analysis of serum COMP concentrations in 23 patients who had prospective blood draws revealed that COMP levels increased in a time dependent fashion and correlated with declines in force vital capacity (FVC). Taken together, our results should encourage more research into the potential use of COMP as a biomarker for disease activity and TGF-β1 activity in patients with IPF. Hence, studies that explore modalities that affect COMP expression, alleviate extracellular matrix rigidity and lung restriction in IPF and interfere with the amplification of TGF-β1 signaling should be persuaded. © 2013 Vuga et al

Sleilwaut’s south shore

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Coloring cayley tables of finite groups

The chromatic number of a latin square L, denoted χ(L), is defined as the minimum number of partial transversals needed to cover all of its cells. It has been conjectured that every latin square L satisfies χ(L) ≤ |L| + 2. If true, this would resolve a longstanding conjecture, commonly attributed to Brualdi, that every latin square has a partial transversal of length |L|−1. Restricting our attention to Cayley tables of finite groups, we prove two results. First, we constructively show that all finite Abelian groups G have Cayley tables with chromatic number |G|+2. Second, we give an upper bound for the chromatic number of Cayley tables of arbitrary finite groups. For |G| ≥ 3, this improves the best-known general upper bound from 2|G| to 3 |G|, while yielding an even stronger result in infinitely many cases