1,720,959 research outputs found
Exotic number bases with application to combinatorics
No full textWhile it is well known that our traditional base 10 number system can be generalized to other bases such as binary or hexadecimal, such generalizations can be taken farther and be far more useful than many realize. For example, they can give new insights into Fibonacci numbers, solve problems from combinatorics, solve and generalize the game of Nim, compute digits of pi, and even create fractals.
In this talk we\u27ll give a smattering of results we\u27ve happened across over the last couple of years as we\u27ve pursued one question: how far can you push the idea of a number base and still have the fundamental property where every number has exactly one representation
Informal Calculus Subtitle: With Applications to Biology and Environmental Science
No full text307 p.This book is an approachable introduction to calculus with applications to biology and environmental science. For example, one application in the book is determining the volume of earth moved in the 1959 earthquake that created Quake Lake. Another application uses differential equations to model various biological examples, including moose and wolf populations at Isle Royale National Park, ranavirus in amphibians, and competing species of protozoa. The text focuses on intuitive understanding of concepts, but still covers most of the algebra and calculations common in a survey of calculus course
Packings and Realizations of Degree Sequences with Specified Substructures
Get PDFThis dissertation focuses on the intersection of two classical and fundamental areas in graph theory: graph packing and degree sequences. The question of packing degree sequences lies naturally in this intersection, asking when degree sequences have edge-disjoint realizations on the same vertex set. The most significant result in this area is Kundu\u27s k-Factor Theorem, which characterizes when a degree sequence packs with a constant sequence. We prove a series of results in this spirit, and we particularly search for realizations of degree sequences with edge-disjoint 1-factors. Perhaps the most fundamental result in degree sequence theory is the Erdos-Gallai Theorem, characterizing when a degree sequence has a realization. After exploring degree sequence packing, we develop several proofs of this famous theorem, connecting it to many other important graph theory concepts.We are also interested in locating edge-disjoint 1-factors in dense graphs. Before tackling this question, we build on the work of Katerinis to find the largest k such that a graph has a k-factor, where the value of k depends on the minimum degree of the graph. This gives an upper bound on the number of edge-disjoint 1-factors.The question of finding edge-disjoint 1-factors leads us to a conjecture of Bollobas and Scott about finding spanning balanced bipartite subgraphs with vertices of high degree. We first prove a degree-sequence version of the Bollobas--Scott Conjecture which we apply to the question of edge-disjoint 1-factors. We then generalize and prove an approximate version of the conjecture, yielding balanced partitions with many edges going to each part. This version has many applications, including finding edge-disjoint 1-factors and edge-disjoint Hamiltonian cycles.
Adviser: Stephen G. Hartk
The Traveling Philosopher and Avoiding Szemerédi\u27s Regularity Lemma
No full textSuppose a wandering philosopher regularly visits every major city in an area, but she gets bored if she travels on the same path, and therefore never wants to use a road twice. How many times can she travel to every city before she is forced to reuse a road? In graph theory, this problem is finding as many edge-disjoint Hamiltonian cycles as possible.
Recently, in the case where each vertex has large degree (each city has many roads leading out of it), Christofides, Kühn, and Osthus proved you could find a surprisingly large number of Hamiltonian cycles. They used one of the most powerful tools in graph theory, originated by Szemerédi, known as the Regularity Lemma. However, there are some drawbacks to using the Regularity Lemma, and recently there has been a push to develop tools and proof methods that replace it.
We developed a partition theorem similar in flavor to the Regularity Lemma that is easier to use. Using our partition theorem, we were able to give a shorter proof of the Christofides, Kühn, Osthus result that applies in more cases. Proving our partition theorem will allow us to talk about another of the most important ideas in graph theory: the probabilistic method
Mathematical Models of Image Processing
Get PDFThe purpose of this thesis is to develop various advanced linear algebra techniques that apply to image processing. With the increasing use of computers and digital photography, being able to manipulate digital images efficiently and with greater freedom is extremely important. By applying the tools of linear algebra, we hope to improve the ability to process such images. We are especially interested in developing techniques that allow computers to manipulate images with the least amount of human guidance. In Chapter 2 and Chapter 3, we develop the basic definitions and linear algebra concepts that lay the foundation for later chapters. Then, in Chapter 4, we demonstrate techniques that allow a computer to rotate an image to the correct orientation automatically, and similarly, for the computer to correct a certain class of color distortion automatically. In both cases, we use certain properties of the eigenvalues and eigenvectors of covariance matrices. We then model color clashing and color variation in Chapter 5 using a powerful tool from linear algebra known as the Perron-Frobenius theorem. Finally, we explore ways to determine whether an image is a blur of another image using invariant functions. The inspiration behind these functions are recent applications of Lie Groups and Lie algebra to image processing
A Prisoner Coordination Puzzle and Some Generalizations
No full textOne hundred prisoners are playing a game for their freedom. They start by developing a collective strategy. Then the warden places a hat and a hat number (range 1 to 100, repetition allowed) on each prisoner’s head such that they can see everyone’s number except their own. They must then, simultaneously, shout a number. If everyone shouts the same number and that one number appears on at least one hat, the prisoners win. Otherwise, they lose.
We discuss the elegant solution to this puzzle and how it relates to Sperner’s lemma from combinatorial topology and a hardness-of-approximation result regarding hypergraph labelings. We then discuss generalizations including when the shouted number must appear more than once, and what happens when the underlying graph dictating hat visibility is changed
Going Beyond Counting First Authors in Author Co-citation Analysis
Get PDFThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
Variations on the Author
Get PDF“Variations on the Author” discusses two of Eduardo Coutinho’s recent films (Um Dia na Vida, from 2010, and Últimas Conversas, posthumously released in 2015) and their contribution to the general question of documentary authorship. The director’s filmography is characterized by a consistent yet self-effacing form of authorial self-inscription: Coutinho often features as an interviewer that rather than express opinions propels discourses; an interviewer that is good at listening. This mode of self-inscription characterizes him as an author who is not expressive but who is nonetheless markedly present on the screen. In Um Dia na Vida, however, Coutinho is completely absent form the image, while Últimas Conversas, on the contrary, includes a confessional prologue that moves the director from the margins to the center of his films. This article examines the ways in which these works stand out in the filmography of a director who offers new insights into the notion of cinematic authorship
Appropriate Similarity Measures for Author Cocitation Analysis
Get PDFWe provide a number of new insights into the methodological discussion about author cocitation analysis. We first argue that the use of the Pearson correlation for measuring the similarity between authors’ cocitation profiles is not very satisfactory. We then discuss what kind of similarity measures may be used as an alternative to the Pearson correlation. We consider three similarity measures in particular. One is the well-known cosine. The other two similarity measures have not been used before in the bibliometric literature. Finally, we show by means of an example that our findings have a high practical relevance.information science;Pearson correlation;cosine;similarity measure;author cocitation analysis
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