A study of saturation number

Thesis (M.S.) University of Alaska Fairbanks, 2017This paper seeks to provide complete proofs in modern notation of (early) key saturation number results and give some new results concerning the semi-saturation number. We highlight relevant results from extremal theory and present the saturation number for the complete graph Kk; and the star K₁,t, elaborating on the proofs provided in the 1964 paper A Problem in Graph Theory by Erdos, Hajnal and Moon and the 1986 paper Saturated Graphs with Minimal Number of Edges by Kászonyi and Tuza. We discuss the proof of a general bound on the saturation number for a family of target graphs provided by Kászonyi and Tuza. A discussion of related results showing that the complete graph has the maximum saturation number among target graphs of the same order and that the star has the maximum saturation number among target trees of the same order is included. Before presenting our result concerning the semi-saturation number for the path Pk; we discuss the structure of some Pk-saturated trees of large order as well as the saturation number of Pk with respect to host graphs of large order.Chapter 1: Introduction -- 1.1 Basic definitions -- 1.2 Saturation number -- 1.3 Chapter overview. Chapter 2: A brief history of saturation number -- 2.1 Extremal theory -- 2.2 The minimal Kk-saturated Graph: Ak(n). Chapter 3: General saturation number results -- 3.1 General bounds for sat(n,F) -- 3.2 Stars. Chapter 4: Saturation numbers for paths and other families of trees -- 4.1 Isolated edges -- 4.2 Paths -- 4.3 Trees of minimum saturation number -- 4.4 Other tree saturation number results -- 4.4.1 Properties of subtrees and saturation number bounds -- 4.4.2 Trees T for which there exists a minimal T-saturated forest. Chapter 5: Semi-saturation number -- 5.1 Motivation -- 5.2 The semi-saturation number for Pk. Chapter 6: Further questions -- References

Error, bias, and long-branch attraction in data for two chloroplast photosystem genes in seed plants

Sequences of two chloroplast photosystem genes, psaA and psbB, together comprising about 3,500 bp, were obtained for all five major groups of extant seed plants and several outgroups among other vascular plants. Strongly supported, but significantly conflicting, phylogenetic signals were obtained in parsimony analyses from partitions of the data into first and second codon positions versus third positions. In the former, both genes agreed on a monophyletic gymnosperms, with Gnetales closely related to certain conifers. In the latter, Gnetales are inferred to be the sister group of all other seed plants, with gymnosperms paraphyletic. None of the data supported the modern ‘‘anthophyte hypothesis,’’ which places Gnetales as the sister group of flowering plants. A series of simulation studies were undertaken to examine the error rate for parsimony inference. Three kinds of errors were examined: random error, systematic bias (both properties of finite data sets), and statistical inconsistency owing to long-branch attraction (an asymptotic property). Parsimony reconstructions were extremely biased for third-position data for psbB. Regardless of the true underlying tree, a tree in which Gnetales are sister to all other seed plants was likely to be reconstructed for these data. None of the combinations of genes or partitions permits the anthophyte tree to be reconstructed with high probability. Simulations of progressively larger data sets indicate the existence of long-branch attraction (statistical inconsistency) for third-position psbB data if either the anthophyte tree or the gymnosperm tree is correct. This is also true for the anthophyte tree using either psaA third positions or psbB first and second positions. A factor contributing to bias and inconsistency is extremely short branches at the base of the seed plant radiation, coupled with extremely high rates in Gnetales and nonseed plant outgroups. M. J. Sanderson,* M. F. Wojciechowski,*† J.-M. Hu,* T. Sher Khan,* and S. G. Brad

Journey to the West: Trans-Pacific Historical Biogeography of Fringehead Blennies in the Genus Neoclinus (Teleostei: Blenniiformes).

Several temperate marine taxa of the northern hemisphere follow a trans-Pacific biogeographic track with representatives on either side of the intervening boreal waters. Shelter-dwelling blenniiform fishes of the genus Neoclinus exhibit this trans-Pacific distribution pattern with three species in the eastern North Pacific and eight species in the western North Pacific. We reconstructed the phylogeny of the Neocliniini (Neoclinus and the monotypic Mccoskerichthys) using six genetic markers: four mitochondrial genes (COI, cytochrome b, 12S and 16S), and two nuclear genes (RAG-1, TMO-4C4). Ancestral state reconstruction and molecular clock dating were used to explore hypothetical ancestral distributions and area relationships, and to estimate divergent times within this group. The monophyly of the genus Neoclinus, and the reciprocal monophyly of the eastern Pacific and western Pacific lineages were supported. Available evidence, including the eastern Pacific and western Atlantic occurrence of a New World clade of blennioid fishes that includes this lineage, supports the origin of the Neocliniini in the eastern Pacific with a single divergence event to the west across the North Pacific by the ancestor of the western Pacific clade. Estimated divergence time of the eastern and western Pacific clades of Neoclinus was 24.14 million year ago, which falls during the Oligocene epoch. Estimated times of divergence in other trans-Pacific lineages of marine fishes vary widely, from recent Pleistocene events to as early as 34 mya

On the universal structure of human lexical semantics

How universal is human conceptual structure? The way concepts are organized in the human brain may reflect distinct features of cultural, historical, and environmental background in addition to properties universal to human cognition. Semantics, or meaning expressed through language, provides direct access to the underlying conceptual structure, but meaning is notoriously difficult to measure, let alone parameterize. Here we provide an empirical measure of semantic proximity between concepts using cross-linguistic dictionaries. Across languages carefully selected from a phylogenetically and geographically stratified sample of genera, translations of words reveal cases where a particular language uses a single polysemous word to express concepts represented by distinct words in another. We use the frequency of polysemies linking two concepts as a measure of their semantic proximity, and represent the pattern of such linkages by a weighted network. This network is highly uneven and fragmented: certain concepts are far more prone to polysemy than others, and there emerge naturally interpretable clusters loosely connected to each other. Statistical analysis shows such structural properties are consistent across different language groups, largely independent of geography, environment, and literacy. It is therefore possible to conclude the conceptual structure connecting basic vocabulary studied is primarily due to universal features of human cognition and language use.Comment: Press embargo in place until publicatio

Deciding definability in FO2(<h,<v) on trees

We provide a decidable characterization of regular forest languages definable in FO2(<h,<v). By FO2(<h,<v) we refer to the two variable fragment of first order logic built from the descendant relation and the following sibling relation. In terms of expressive power it corresponds to a fragment of the navigational core of XPath that contains modalities for going up to some ancestor, down to some descendant, left to some preceding sibling, and right to some following sibling. We also show that our techniques can be applied to other two variable first-order logics having exactly the same vertical modalities as FO2(<h,<v) but having different horizontal modalities