181 research outputs found
On the structure of dense graphs, and other extremal problems
Extremal combinatorics is an area of mathematics populated by problems that are easy to state, yet often difficult to resolve. The typical question in this field is the following: What is the maximum or minimum size of a collection of finite objects (e.g., graphs, finite families of sets) subject to some set of constraints? Despite its apparent simplicity, this question has led to a rather rich body of work. This dissertation consists of several new results in this field.The first two chapters concern structural results for dense graphs, thus justifying the first part of my title. In the first chapter, we prove a stability result for edge-maximal graphs without complete subgraphs of fixed size, answering questions of Tyomkyn and Uzzell. The contents of this chapter are based on joint work with Kamil Popielarz and Julian Sahasrabudhe.The second chapter is about the interplay between minimum degree and chromatic number in graphs which forbid a specific set of `small\u27 graphs as subgraphs. We determine the structure of dense graphs which forbid triangles and cycles of length five. A particular consequence of our work is that such graphs are 3-colorable. This answers questions of Messuti and Schacht, and Oberkampf and Schacht. This chapter is based on joint work with Shoham Letzter.Chapter 3 departs from undirected graphs and enters the domain of directed graphs. Specifically, we address the connection between connectivity and linkedness in tournaments with large minimum out-degree. Making progress on a conjecture of Pokrovskiy, we show that, for any positive integer , any -connected tournament with large enough minimum out-degree is -linked. This chapter is based on joint work with Ant{\\u27o}nio Gir{\~a}o.ArrayThe final chapter leaves the world of graphs entirely and examines a problem in finite set systems.More precisely, we examine an extremal problem on a family of finite sets involving constraints on the possible intersectionsizes these sets may have. Such problems have a long history in extremal combinatorics. In this chapter, we are interested in the maximum number of disjoint pairs a family of sets can have under various restrictions on intersection sizes. We obtain several new results in this direction. The contents of this chapter are based on joint work with Ant{\\u27o}nio Gir{\~a}o
Basic Neutrosophic Algebraic Structures and their Application to Fuzzy and Neutrosophic Models
The involvement of uncertainty of varying degrees when the total of the
membership degree exceeds one or less than one, then the newer mathematical
paradigm shift, Fuzzy Theory proves appropriate. For the past two or more
decades, Fuzzy Theory has become the potent tool to study and analyze
uncertainty involved in all problems. But, many real-world problems also abound
with the concept of indeterminacy. In this book, the new, powerful tool of
neutrosophy that deals with indeterminacy is utilized. Innovative neutrosophic
models are described. The theory of neutrosophic graphs is introduced and
applied to fuzzy and neutrosophic models. This book is organized into four
chapters. In Chapter One we introduce some of the basic neutrosophic algebraic
structures essential for the further development of the other chapters. Chapter
Two recalls basic graph theory definitions and results which has interested us
and for which we give the neutrosophic analogues. In this chapter we give the
application of graphs in fuzzy models. An entire section is devoted for this
purpose. Chapter Three introduces many new neutrosophic concepts in graphs and
applies it to the case of neutrosophic cognitive maps and neutrosophic
relational maps. The last section of this chapter clearly illustrates how the
neutrosophic graphs are utilized in the neutrosophic models. The final chapter
gives some problems about neutrosophic graphs which will make one understand
this new subject.Comment: 149 pages, 130 figure
Digraph Coloring Games and Game-Perfectness
In this thesis the game chromatic number of a digraph is introduced as a game-theoretic variant of the dichromatic number. This notion generalizes the well-known game chromatic number of a graph. An extended model also takes into account relaxed colorings and asymmetric move sequences. Game-perfectness is defined as a game-theoretic variant of perfectness of a graph, and is generalized to digraphs. We examine upper and lower bounds for the game chromatic number of several classes of digraphs. In the last part of the thesis, we characterize game-perfect digraphs with small clique number, and prove general results concerning game-perfectness. Some results are verified with the help of a computer program that is discussed in the appendix
Inference and experimental design for percolation and random graph models.
The problem of optimal arrangement of nodes of a random weighted graph is
studied in this thesis. The nodes of graphs under study are fixed, but their edges
are random and established according to the so called edge-probability function.
This function is assumed to depend on the weights attributed to the pairs of graph
nodes (or distances between them) and a statistical parameter. It is the purpose
of experimentation to make inference on the statistical parameter and thus to
extract as much information about it as possible. We also distinguish between two
different experimentation scenarios: progressive and instructive designs.
We adopt a utility-based Bayesian framework to tackle the optimal design
problem for random graphs of this kind. Simulation based optimisation methods,
mainly Monte Carlo and Markov Chain Monte Carlo, are used to obtain
the solution. We study optimal design problem for the inference based on partial
observations of random graphs by employing data augmentation technique.
We prove that the infinitely growing or diminishing node configurations asymptotically
represent the worst node arrangements. We also obtain the exact solution
to the optimal design problem for proximity graphs (geometric graphs) and numerical
solution for graphs with threshold edge-probability functions.
We consider inference and optimal design problems for finite clusters from bond
percolation on the integer lattice Zd and derive a range of both numerical and
analytical results for these graphs. We introduce inner-outer plots by deleting
some of the lattice nodes and show that the ‘mostly populated’ designs are not
necessarily optimal in the case of incomplete observations under both progressive
and instructive design scenarios.
Finally, we formulate a problem of approximating finite point sets with lattice
nodes and describe a solution to this problem
Recommended from our members
Combinatorics and Probability
The main theme of this workshop was the use of probabilistic methods in combinatorics and theoretical computer science. Although these methods have been around for decades, they are being refined all the time: they are getting more and more sophisticated and powerful. Another theme was the study of random combinatorial structures, either for their own sake, or to tackle extremal questions. Both themes were richly represented at the workshop, with many recent exciting results presented by the lecturers
Evolutionary games on graphs
Game theory is one of the key paradigms behind many scientific disciplines
from biology to behavioral sciences to economics. In its evolutionary form and
especially when the interacting agents are linked in a specific social network
the underlying solution concepts and methods are very similar to those applied
in non-equilibrium statistical physics. This review gives a tutorial-type
overview of the field for physicists. The first three sections introduce the
necessary background in classical and evolutionary game theory from the basic
definitions to the most important results. The fourth section surveys the
topological complications implied by non-mean-field-type social network
structures in general. The last three sections discuss in detail the dynamic
behavior of three prominent classes of models: the Prisoner's Dilemma, the
Rock-Scissors-Paper game, and Competing Associations. The major theme of the
review is in what sense and how the graph structure of interactions can modify
and enrich the picture of long term behavioral patterns emerging in
evolutionary games.Comment: Review, final version, 133 pages, 65 figure
A study of the relationship between surface features and the in-flight performance of footballs
Football is widely regarded as the most popular sport in the world involving over 270 million
people from different countries and cultures. It can be argued that the football is one of most
important aspects of the game and hence the flight of the ball, if unexpected, can alter the
outcome of the game. This thesis provides an engineering perspective and contribution to the
continued understanding and improvement of the in-flight performance of FIFA approved
footballs.
Skilful players will impart spin onto a ball to induce a curve in-flight to try and deceive
opponents. This flight is generally smooth, although subtle variations in the orientation and spin
rate may cause conditions that affect the path and final ball position, in a manner considered to be
unpredictable due to aerodynamic effects. Ball designs and manufacturing techniques are
evolving and certain seam configurations are known to induce asymmetric pressure distributions
resulting in lateral movement during flight.
Aerodynamic research of sport balls has primarily focused on drag and the effects of high spin
rates. Studies have shown the introduction of surface roughness affects the boundary layer state
compared to a smooth sphere. Surface roughness on a football takes many forms including seam
configurations and micro surface textures. The influence of changing the density, distribution and
dimensions of the surface roughness with respect to the aerodynamic behaviour has been
researched. The principle focus of this thesis is concerned with the influence on the lateral
component as a result of applying surface roughness to the outer surfaces.
The influence of the surface roughness on the drag and lateral components were determined using
established wind tunnel techniques. Real balls and full size prototypes were tested. A
mathematical flight model was employed to simulate realistic multiple flight trajectories based on
empirical aerodynamic data. Mathematical and statistical techniques, including R.M.S and AutoCorrelation
Functions were used to analyse the data. The results from this research showed how
small variations in surface texture affected the complex nature of the lateral forces. Trajectories
varied significantly depending on initial orientation and slow spin rate sensitivities. In conclusion,
ball characterisation techniques were developed that identified lateral deviation and shape
measures and considered a gradient profiling approach. Application of these novel parameters
through multiple trajectory analysis allowed for an in-flight performance measure of footballs
designs
Dagstuhl Reports : Volume 1, Issue 2, February 2011
Online Privacy: Towards Informational Self-Determination on the Internet (Dagstuhl Perspectives Workshop 11061) : Simone Fischer-Hübner, Chris Hoofnagle, Kai Rannenberg, Michael Waidner, Ioannis Krontiris and Michael Marhöfer Self-Repairing Programs (Dagstuhl Seminar 11062) : Mauro Pezzé, Martin C. Rinard, Westley Weimer and Andreas Zeller Theory and Applications of Graph Searching Problems (Dagstuhl Seminar 11071) : Fedor V. Fomin, Pierre Fraigniaud, Stephan Kreutzer and Dimitrios M. Thilikos Combinatorial and Algorithmic Aspects of Sequence Processing (Dagstuhl Seminar 11081) : Maxime Crochemore, Lila Kari, Mehryar Mohri and Dirk Nowotka Packing and Scheduling Algorithms for Information and Communication Services (Dagstuhl Seminar 11091) Klaus Jansen, Claire Mathieu, Hadas Shachnai and Neal E. Youn
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