Combinatorial designs represent an important area of contemporary discrete mathematics closely related to such fields as finite geometries, regular graphs and multigraphs, factorizations of graphs, linear algebra, number theory, finite fields, group and quasigroup theory, Latin squares, and matroids. It has a history of more than 150 years when it started as a collection of unrelated problems. Nowadays the field is a well developed theory with deep mathematical results and a wide range of applications in coding theory, cryptography, computer science, and other areas. In the most general setting, a combinatorial design consists of a ground set of elements and a collection of subsets of these elements satisfying some specific restrictions; the latter are often expressed in the language of graphs.
On the other side, hypergraph theory is a relatively new field which started in early 60s of the last century as a generalization of graph theory. A hypergraph consists of a ground set of elements and a collection of subsets of these elements without any specific restrictions. In this sense the concept of hypergraph is more general than the concept of combinatorial design. While it started as a generalization of graph theory, hypergraph theory soon became a separate subject because many new properties have been discovered that miss or degenerate in graphs. Compared to graph theory, the language of hypergraphs not only allows us to formulate and solve more general problems, it also helps us to understand and solve several graph theory problems by simplifying and unifying many previously unrelated concepts.
The main feature of this book is applying the hypergraph approach to the theory of combinatorial designs. An alternative title of it could be "Combinatorial designs as hypergraphs". There is no analogue to this book on the market. Its primary audience is researchers and graduate students taking courses in design theory, combinatorial geometry, finite geometry, discrete mathematics, graph theory, combinatorics, cryptography, information and coding theory, and similar areas. The aim of this book is to show the connection and mutual benefit between hypergraph theory and design theory. It does not intend to give a survey of all important results or methods in any of these subjects. (Imprint: Nova)
Preface
Chapter 1: Graphs
Chapter 2: Hypergraphs
Chapter 3: Hypergraph Coloring
Chapter 4: Berge’s Conjecture for Linear Hypergraphs
Chapter 5: Quasigroups and Latin Squares
Chapter 6: STS(v): Steiner Triple Systems
Chapter 7: Steiner Quadruple Systems
Chapter 8: Steiner Systems
Chapter 9: Constructions of Steiner Systems
Chapter 10: Blocking Sets in Steiner Systems
Chapter 11: Balanced Incomplete Block Designs
Chapter 12: G-Designs
References
Index
Series:
Mathematics Research Developments
Binding: Hardcover
Pub. Date: 2014 - 4th Quarter
Pages: 7x10 - (NBC-C)
ISBN: 978-1-63321-911-3
This book presents original studies on the leading edge of linear algebra. Each chapter has been carefully selected in an attempt to present substantial research results across a broad spectrum. The main goal of Chapter One is to define and investigate the restricted generalized inverses corresponding to minimization of constrained quadratic form. As stated in Chapter Two, in systems and control theory, Linear Time Invariant (LTI) descriptor (Differential-Algebraic) systems are intimately related to the matrix pencil theory. A review of the most interesting properties of the Projective Equivalence and the Extended Hermite Equivalence classes is presented in the chapter. New determinantal representations of generalized inverse matrices based on their limit representations are introduced in Chapter Three. Using the obtained analogues of the adjoint matrix, Cramer’s rules for the least squares solution with the minimum norm and for the Drazin inverse solution of singular linear systems have been obtained in the chapter. In Chapter Four, a very interesting application of linear algebra of commutative rings to systems theory, is explored. Chapter Five gives a comprehensive investigation to behaviors of a general Hermitian quadratic matrix-valued function by using ranks and inertias of matrices. In Chapter Six, the theory of triangular matrices (tables) is introduced. The main "characters" of the chapter are special triangular tables (which will be called triangular matrices) and their functions paradeterminants and parapermanents. The aim of Chapter Seven is to present the latest developments in iterative methods for solving linear matrix equations. The problems of existence of common eigenvectors and simultaneous triangularization of a pair of matrices over a principal ideal domain with quadratic minimal polynomials are investigated in Chapter Eight. Two approaches to define a noncommutative determinant (a determinant of a matrix with noncommutative elements) are considered in Chapter Nine. The last, Chapter 10, is an example of how the methods of linear algebra are used in natural sciences, particularly in chemistry. In this chapter, it is shown that in a First Order Chemical Kinetics Mechanisms matrix, all columns add to zero, all the diagonal elements are non-positive and all the other matrix entries are non-negative. As a result of this particular structure, the Gershgorin Circles Theorem can be applied to show that all the eigenvalues are negative or zero. (Imprint: Nova)
Preface
Chapter 1. Minimization of Quadratic Forms and Generalized Inverses
(Predrag S. Stanimirovi?, Dimitrios Pappas and Vasilios N. Katsikis, University of Ni?, Faculty of Sciences and Mathematics, Ni?, Serbia and others)
Chapter 2. The Study of the Invariants of Homogeneous Matrix Polynomials Using the Extended Hermite Equivalence εrh
(Grigoris I. Kalogeropoulos Athanasios D. Karageorgos and Athanasios A. Pantelous, Department of Mathematics, University of Athens, Greece and others)
Chapter 3. Cramer's Rule for Generalized Inverse Solutions
(Ivan I. Kyrchei, Pidstryhach Institute for Applied Problems of Mechanics and Mathematics, Lviv, Ukraine)
Chapter 4. Feedback Actions on Linear Systems over Von Neumann Regular Rings
(Andres Saez-Schwedt, Departamento de Matematicas, Universidad de Leon, Campus de Vegazana, Leon, Spain)
Chapter 5. How to Characterize Properties of General Hermitian Quadratic Matrix-Valued Functions by Rank and Inertia
(Yongge Tian, CEMA, Central University of Finance and Economics, Beijing, China)
Chapter 6. Introduction to the Theory of Triangular Matrices (Tables)
(Roman Zatorsky, Precarpathian Vasyl Stefanyk National University, Ivano-Frankivsk, Ukraine)
Chapter 7. Recent Developments in Iterative Algorithms for Solving Linear Matrix Equations
(Masoud Hajarian, Department of Mathematics, Faculty of Mathematical Sciences, Shahid Beheshti University, General Campus, Evin, Tehran, Iran)
Chapter 8. Simultaneous Triangularization of a Pair of Matrices over a Principal Ideal Domain with Quadratic Minimal Polynomials
(Volodymyr M. Prokip, Pidstryhach Institute for Applied Problems of Mechanics and Mathematics, Lviv, Ukraine)
Chapter 9. Relation of Row-Column Determinants with Quasideterminants of Matrices over a Quaternion Algebra
(Aleks Kleyn, American Mathematical Society and Ivan I. Kyrchei, Pidstryhach Institute for Applied Problems of Mechanics and Mathematics, NAS of Ukraine)
Chapter 10. First Order Chemical Kinetics Matrices and Stability of O.D.E. Systems
(Victor Martinez-Luaces, Electrochemistry Engineering Multidisciplinary Research Group. UdelaR. Uruguay)
Index
Series:
Mathematics Research Developments
Binding: Hardcover
Pub. Date: 2015 - 1st Quarter
Pages: 7x10 - (NBC-C)
ISBN: 978-1-63463-565-3
Status: FP
December 2014 232 pages SAGE Publications, Inc
Series: Quantitative Applications in the Social Sciences, Volume 174
Paperback ISBN: 9781452282015
Introducing time series methods and their application in social science research, this practical guide to time series models is the first in the field written for a non-econometrics audience. Giving readers the tools they need to apply models to their own research, this unique book demonstrates the use of?and the assumptions underlying?common models of time series data, including finite distributed lag; autoregressive distributed lag; moving average; differenced data; and GARCH, ARMA, ARIMA, and error correction models.
PREFACE
ACKNOWLEDGEMENTS
ABOUT THE AUTHOR
SERIES EDITOR'S INTRODUCTION
1. Thinking Time-serially
2. Fundamental Concepts in Time Series Analysis
3. Static Time Series Models and Ordinary Least Squares Estimation
4. Introducing Dynamic Time Series Models
5. Autoregressive Moving Average (ARMA) Models
6. Models for Integrated and Cointegrated Data
Conclusion
Series: Advanced Courses in Mathematics - CRM Barcelona
2014, VIII, 112 p.
Information
ISBN 978-3-0348-0873-6
Offers a modern and self-contained introduction to integral geometry
Includes both the classical theory and some very recent progress in integral geometry
Presents in a unified and accessible way a collection of results that have revolutionized the theory of convex valuations over the last decade
Valuations are finitely additive functionals on the space of convex bodies. Their study has become a central subject in convexity theory, with fundamental applications to integral geometry. In the last years there has been significant progress in the theory of valuations, which in turn has led to important achievements in integral geometry. This book originated from two courses delivered by the authors at the CRM and provides a self-contained introduction to these topics, covering most of the recent advances.
The first part, by Semyon Alesker, is devoted to the theory of convex valuations, with emphasis on the latest developments. A special focus is put on the new fundamental structures of the space of valuations discovered after Alesker's irreducibility theorem. Moreover, the author describes the newly developed theory of valuations on manifolds.
In the second part, Joseph H. G. Fu gives a modern introduction to integral geometry in the sense of Blaschke and Santalo, based on the notions and tools presented in the first part. At the core of this approach lies the close relationship between kinematic formulas and Alesker's product of valuations. This original viewpoint not only enlightens the classical integral geometry of Euclidean space, it has also produced previously unreachable results, such as the explicit computation of kinematic formulas in Hermitian spaces.
Part I: New Structures on Valuations and Applications.- Translation invariant valuations on convex sets.- Valuations on manifolds.- Part II: Algebraic Integral Geometry.- Classical integral geometry.- Curvature measures and the normal cycle.- Integral geometry of euclidean spaces via Alesker theory.- Valuations and integral geometry on isotropic manifolds.- Hermitian integral geometry.?
Popular Content within this publication
Series: Advanced Courses in Mathematics - CRM Barcelona
2014, IX, 198 p. 20 illus., 10 illus. in color.
Softcover
ISBN 978-3-0348-0870-5
Due: November 14, 2014
Provides an introduction to Foliation Theory with a comprehensive overview of some recent developments of the theory
Includes results that so far were only available in original research articles
The different topics are presented by the best experts with a detailed discussion of many examples, making the text accessible to a wide audience
This book is an introduction to several active research topics in Foliation Theory and its connections with other areas. It contains expository lectures showing the diversity of ideas and methods arising and used in the study of foliations.
The lectures by A. El Kacimi Alaoui offer an introduction to Foliation Theory, with emphasis on examples and transverse structures. S. Hurder's lectures apply ideas from smooth dynamical systems to develop useful concepts in the study of foliations, like limit sets and cycles for leaves, leafwise geodesic flow, transverse exponents, stable manifolds, Pesin Theory, and hyperbolic, parabolic, and elliptic types of foliations, all of them illustrated with examples. The lectures by M. Asaoka are devoted to the computation of the leafwise cohomology of orbit foliations given by locally free actions of certain Lie groups, and its application to the description of the deformation of those actions. In the lectures by K. Richardson, he studies the geometric and analytic properties of transverse Dirac operators for Riemannian foliations and compact Lie group actions, and explains a recently proved index formula.
Fundamentals of Foliation Theory.- Foliation Dynamics.- Deformation of Locally Free Actions and Leafwise Cohomology.- Transversal Dirac Operators on Distributions, Foliations, and G-Manifolds.?
Popular Content within this publication
Transversal Dirac Operators on Distributions, Foliations, and G-ManifoldsRichardson, Ken
Fundaments of Foliation TheoryEl Kacimi Alaoui, Aziz
Deformation of Locally Free Actions and Leafwise CohomologyAsaoka, Masayuki
Lectures on Foliation DynamicsHurder, Steven