Linear colorings of simplicial complexes and collapsing

A vertex coloring of a simplicial complex $\Delta$ is called a linear coloring if it satisfies the property that for every pair of facets $(F_1, F_2)$ of $\Delta$, there exists no pair of vertices $(v_1, v_2)$ with the same color such that $v_1\in F_1\backslash F_2$ and $v_2\in F_2\backslash F_1$. We show that every simplicial complex $\Delta$ which is linearly colored with $k$ colors includes a subcomplex $\Delta'$ with $k$ vertices such that $\Delta'$ is a strong deformation retract of $\Delta$. We also prove that this deformation is a nonevasive reduction, in particular, a collapsing.Comment: 18 page

Evasiveness of Graph Properties and Topological Fixed-Point Theorems

Many graph properties (e.g., connectedness, containing a complete subgraph) are known to be difficult to check. In a decision-tree model, the cost of an algorithm is measured by the number of edges in the graph that it queries. R. Karp conjectured in the early 1970s that all monotone graph properties are evasive -- that is, any algorithm which computes a monotone graph property must check all edges in the worst case. This conjecture is unproven, but a lot of progress has been made. Starting with the work of Kahn, Saks, and Sturtevant in 1984, topological methods have been applied to prove partial results on the Karp conjecture. This text is a tutorial on these topological methods. I give a fully self-contained account of the central proofs from the paper of Kahn, Saks, and Sturtevant, with no prior knowledge of topology assumed. I also briefly survey some of the more recent results on evasiveness.Comment: Book version, 92 page

An Exposition of Discrete Morse Theory and Applications

The classical Morse theory is a powerful tool to study topological properties of a smooth manifold by examining critical points of some differentiable functions on that manifold. Robin Forman developed a discrete variant of Morse theory by adapting it on abstract simplicial complexes that resulted in a new theory with wide applications in other fields of mathematics, computer science, data science, and others. In this thesis, we present Forman’s construction of discrete Morse theory, as well as its main theorems such as the Collapse theorem, discrete Morse inequalities, the theorem for cancelling critical simplices, and some additional topics. We also discuss some applications of discrete Morse theory with a major focus on the concept of persistence in topological data analysis. While many results exist in the literature, we wrote our own proofs, added more details and explanations, and adapted it to our own settings with a strong topological flavor whenever possible

An Exposition of Discrete Morse Theory and Applications

The classical Morse theory is a powerful tool to study topological properties of a smooth manifold by examining critical points of some differentiable functions on that manifold. Robin Forman developed a discrete variant of Morse theory by adapting it on abstract simplicial complexes that resulted in a new theory with wide applications in other fields of mathematics, computer science, data science, and others. In this thesis, we present Forman’s construction of discrete Morse theory, as well as its main theorems such as the Collapse theorem, discrete Morse inequalities, the theorem for cancelling critical simplices, and some additional topics. We also discuss some applications of discrete Morse theory with a major focus on the concept of persistence in topological data analysis. While many results exist in the literature, we wrote our own proofs, added more details and explanations, and adapted it to our own settings with a strong topological flavor whenever possible

Synchrotron Radiation

In this book, the readers will find chapters exposing them to different, useful applications of synchrotron radiation in various fields of physics, archeology, and biosciences. Different research has been carried out in this field and has conveyed a novel approach to the applications of synchrotron radiation.The chapters are crisp and precise and will motivate students, young researchers, and professionals to carry out research with novel ideas. The authors have successfully tried to convey their ideas in a very simple text. It will be useful for both the novice and those who are currently doing research in this field. I hope it will complete my task of enriching researcher into synchrotron radiation with better applications and also giving students a better platform for understanding the subject

A fish fit for Ozymandias? : the ecology, growth and osteology of Leedsichthys (Pachycormidae, Actinopterygii)

In this thesis, I describe work to resolve issues of bone identifications that have been outstanding since Smith Woodward's initial description in 1889, to assess the taxonomic validity of material assigned to the hypodigm of Leedsichthys and the interrelationships of the members of Family Pachycormidae. In addition I look at the palaeoecology of this animal on the basis of its size and growth and its locomotion capabilities and its likely feeding abilities and behaviour. Chapter 2 includes a review of the history of work on Leedsichthys, with particular reference to the discoveries made in the Peterborough district. In chapter 3, archival photographs and papers are used to establish the distinction between the type material, the tail specimen, and the gill basket specimen. In chapter 4, occurrences of Leedsichthys outwith the Peterborough district are considered, including the announcement of a new locality extending the range of the taxon into the Kimmeridgian. Some identifications of previously misidentified bones are made, specifically the hypobranchial and dorsal fin-rays. Feeding trace fossils are interpreted in the context of Leedsichthys. In chapter 5, a new Callovian pachycormid is described from the Oxford Clay of Peterborough district, and used in a reworking of Lambers' 1992 phylogenetic analysis of the interrelationships of the Pachycormidae. The Pachycormiformes are redefined on the basis of derived characters. In chapter 6, the value of gill rakers as a source of taxonomic characters is considered, with specific reference to their use in Lambers' 1992 character set, and the validity of Leedsichthys notocetes as a distinct species. In chapter 7, specimens are analysed using growth marks and scaling, in order to establish estimates of length-at-age for Leedsichthys. In chapter 8, the bone identifications of Smith Woodward (1889b) are revised, and further bone morphologies identified from within the hypodigm of the genus. In chapter 9, the size estimates derived in chapter 7 are used to inform interpretation of Leedsichthys palaeoecology, focussing primarily on locomotion and feeding. In the conclusions, an up-to-date reconstruction is presented

Aeronautical engineering: A continuing bibliography with indexes (supplement 301)

This bibliography lists 1291 reports, articles, and other documents introduced into the NASA scientific and technical information system in Feb. 1994. Subject coverage includes: design, construction and testing of aircraft and aircraft engines; aircraft components, equipment, and systems; ground support systems; and theoretical and applied aspects of aerodynamics and general fluid dynamics

Note: The Smallest Nonevasive Graph Property

A property of n-vertex graphs is called evasive if every algorithm testing this property by asking questions of the form “is there an edge between vertices u and v” requires, in the worst case, to ask about all pairs of vertices. Most “natural” graph properties are either evasive or conjectured to be such, and of the few examples of nontrivial nonevasive properties scattered in the literature the smallest one has n = 6. We exhibit a nontrivial, nonevasive property of 5-vertex graphs and show that it is essentially the unique such with n ≤ 5