Quantitative graph theory mathematical foundations and applications

CERN Document Server

Dehmer, Matthias

2014-01-01

The first book devoted exclusively to quantitative graph theory, Quantitative Graph Theory: Mathematical Foundations and Applications presents and demonstrates existing and novel methods for analyzing graphs quantitatively. Incorporating interdisciplinary knowledge from graph theory, information theory, measurement theory, and statistical techniques, this book covers a wide range of quantitative-graph theoretical concepts and methods, including those pertaining to real and random graphs such as:Comparative approaches (graph similarity or distance)Graph measures to characterize graphs quantitat

Discrete Mathematics and Curriculum Reform.

Science.gov (United States)

Kenney, Margaret J.

1996-01-01

Defines discrete mathematics as the mathematics necessary to effect reasoned decision making in finite situations and explains how its use supports the current view of mathematics education. Discrete mathematics can be used by curriculum developers to improve the curriculum for students of all ages and abilities. (SLD)

Discrete Morse functions for graph configuration spaces

International Nuclear Information System (INIS)

Sawicki, A

2012-01-01

We present an alternative application of discrete Morse theory for two-particle graph configuration spaces. In contrast to previous constructions, which are based on discrete Morse vector fields, our approach is through Morse functions, which have a nice physical interpretation as two-body potentials constructed from one-body potentials. We also give a brief introduction to discrete Morse theory. Our motivation comes from the problem of quantum statistics for particles on networks, for which generalized versions of anyon statistics can appear. (paper)

Finite Mathematics and Discrete Mathematics: Is There a Difference?

Science.gov (United States)

Johnson, Marvin L.

Discrete mathematics and finite mathematics differ in a number of ways. First, finite mathematics has a longer history and is therefore more stable in terms of course content. Finite mathematics courses emphasize certain particular mathematical tools which are useful in solving the problems of business and the social sciences. Discrete mathematics…

Discrete Mathematics Re "Tooled."

Science.gov (United States)

Grassl, Richard M.; Mingus, Tabitha T. Y.

1999-01-01

Indicates the importance of teaching discrete mathematics. Describes how the use of technology can enhance the teaching and learning of discrete mathematics. Explorations using Excel, Derive, and the TI-92 proved how preservice and inservice teachers experienced a new dimension in problem solving and discovery. (ASK)

Theoretical Basics of Teaching Discrete Mathematics

Directory of Open Access Journals (Sweden)

Y. A. Perminov

2012-01-01

Full Text Available  The paper deals with the research findings concerning the process of mastering the theoretical basics of discrete mathematics by the students of vocational pedagogic profile. The methodological analysis is based on the subject and functions of the modern discrete mathematics and its role in mathematical modeling and computing. The modern discrete mathematics (i.e. mathematics of the finite type structures plays the important role in modernization of vocational training. It is especially rele- vant to training students for vocational pedagogic qualifications, as in the future they will be responsible for training the middle and the senior level specialists in engineer- ing and technical spheres. Nowadays in different industries, there arise the problems which require for their solving both continual – based on the classical mathematical methods – and discrete modeling. The teaching course of discrete mathematics for the future vocational teachers should be relevant to the target qualification and aimed at mastering the mathematical modeling, systems of computer mathematics and computer technologies. The author emphasizes the fundamental role of mastering the language of algebraic and serial structures, as well as the logical, algorithmic, combinatory schemes dominating in dis- crete mathematics. The guidelines for selecting the content of the course in discrete mathematics are specified. The theoretical findings of the research can be put into practice whilst developing curricula and working programs for bachelors and masters’ training. 

Co-Roman domination in graphs

Indian Academy of Sciences (India)

1National Centre for Advanced Research in Discrete Mathematics ... 3Department of Computer Science, Ball State University, Muncie, IN, USA .... The corona of two disjoint graphs G1 and G2 is defined to be the graph G = G1 ◦ G2,.

Simplicial complexes of graphs

CERN Document Server

Jonsson, Jakob

2008-01-01

A graph complex is a finite family of graphs closed under deletion of edges. Graph complexes show up naturally in many different areas of mathematics, including commutative algebra, geometry, and knot theory. Identifying each graph with its edge set, one may view a graph complex as a simplicial complex and hence interpret it as a geometric object. This volume examines topological properties of graph complexes, focusing on homotopy type and homology. Many of the proofs are based on Robin Forman's discrete version of Morse theory. As a byproduct, this volume also provides a loosely defined toolbox for attacking problems in topological combinatorics via discrete Morse theory. In terms of simplicity and power, arguably the most efficient tool is Forman's divide and conquer approach via decision trees; it is successfully applied to a large number of graph and digraph complexes.

Expander graphs in pure and applied mathematics

OpenAIRE

Lubotzky, Alexander

2012-01-01

Expander graphs are highly connected sparse finite graphs. They play an important role in computer science as basic building blocks for network constructions, error correcting codes, algorithms and more. In recent years they have started to play an increasing role also in pure mathematics: number theory, group theory, geometry and more. This expository article describes their constructions and various applications in pure and applied mathematics.

Nonlocal discrete regularization on weighted graphs: a framework for image and manifold processing.

Science.gov (United States)

Elmoataz, Abderrahim; Lezoray, Olivier; Bougleux, Sébastien

2008-07-01

We introduce a nonlocal discrete regularization framework on weighted graphs of the arbitrary topologies for image and manifold processing. The approach considers the problem as a variational one, which consists of minimizing a weighted sum of two energy terms: a regularization one that uses a discrete weighted p-Dirichlet energy and an approximation one. This is the discrete analogue of recent continuous Euclidean nonlocal regularization functionals. The proposed formulation leads to a family of simple and fast nonlinear processing methods based on the weighted p-Laplace operator, parameterized by the degree p of regularity, the graph structure and the graph weight function. These discrete processing methods provide a graph-based version of recently proposed semi-local or nonlocal processing methods used in image and mesh processing, such as the bilateral filter, the TV digital filter or the nonlocal means filter. It works with equal ease on regular 2-D and 3-D images, manifolds or any data. We illustrate the abilities of the approach by applying it to various types of images, meshes, manifolds, and data represented as graphs.

Performance analysis of chi models using discrete-time probabilistic reward graphs

NARCIS (Netherlands)

Trcka, N.; Georgievska, S.; Markovski, J.; Andova, S.; Vink, de E.P.

2008-01-01

We propose the model of discrete-time probabilistic reward graphs (DTPRGs) for performance analysis of systems exhibiting discrete deterministic time delays and probabilistic behavior, via their interpretation as discrete-time Markov reward chains, full-fledged platform for qualitative and

Discrete Mathematics and Its Applications

Science.gov (United States)

Oxley, Alan

2010-01-01

The article gives ideas that lecturers of undergraduate Discrete Mathematics courses can use in order to make the subject more interesting for students and encourage them to undertake further studies in the subject. It is possible to teach Discrete Mathematics with little or no reference to computing. However, students are more likely to be…

chromatic number of a complete balanced multipartite graph

African Journals Online (AJOL)

2006-12-29

Dec 29, 2006 ... †Applied Mathematics Division, Department of Mathematical Sciences, University of ..... classes originally comprising discrete entities (graph vertices) are replaced by collections of ...... Journal of Combinatorial Theory, 6, pp.

Discrete Mathematics and the Secondary Mathematics Curriculum.

Science.gov (United States)

Dossey, John

Discrete mathematics, the mathematics of decision making for finite settings, is a topic of great interest in mathematics education at all levels. Attention is being focused on resolving the diversity of opinion concerning the exact nature of the subject, what content the curriculum should contain, who should study that material, and how that…

Ancestral Genres of Mathematical Graphs

Science.gov (United States)

Gerofsky, Susan

2011-01-01

Drawing from sources in gesture studies, cognitive science, the anthropology of religion and art/architecture history, this article explores cultural, bodily and cosmological resonances carried (unintentionally) by mathematical graphs on Cartesian coordinates. Concepts of asymmetric bodily spaces, grids, orthogonality, mapping and sacred spaces…

Discrete mathematics using a computer

CERN Document Server

Hall, Cordelia

2000-01-01

Several areas of mathematics find application throughout computer science, and all students of computer science need a practical working understanding of them. These core subjects are centred on logic, sets, recursion, induction, relations and functions. The material is often called discrete mathematics, to distinguish it from the traditional topics of continuous mathematics such as integration and differential equations. The central theme of this book is the connection between computing and discrete mathematics. This connection is useful in both directions: • Mathematics is used in many branches of computer science, in applica­ tions including program specification, datastructures,design and analysis of algorithms, database systems, hardware design, reasoning about the correctness of implementations, and much more; • Computers can help to make the mathematics easier to learn and use, by making mathematical terms executable, making abstract concepts more concrete, and through the use of software tools su...

Discrete Mathematics in the Schools. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, Volume 36.

Science.gov (United States)

Rosenstein, Joseph G., Ed.; Franzblau, Deborah S., Ed.; Roberts, Fred S., Ed.

This book is a collection of articles by experienced educators and explains why and how discrete mathematics should be taught in K-12 classrooms. It includes evidence for "why" and practical guidance for "how" and also discusses how discrete mathematics can be used as a vehicle for achieving the broader goals of the major…

Isospectral discrete and quantum graphs with the same flip counts and nodal counts

Science.gov (United States)

Juul, Jonas S.; Joyner, Christopher H.

2018-06-01

The existence of non-isomorphic graphs which share the same Laplace spectrum (to be referred to as isospectral graphs) leads naturally to the following question: what additional information is required in order to resolve isospectral graphs? It was suggested by Band, Shapira and Smilansky that this might be achieved by either counting the number of nodal domains or the number of times the eigenfunctions change sign (the so-called flip count) (Band et al 2006 J. Phys. A: Math. Gen. 39 13999–4014 Band and Smilansky 2007 Eur. Phys. J. Spec. Top. 145 171–9). Recent examples of (discrete) isospectral graphs with the same flip count and nodal count have been constructed by Ammann by utilising Godsil–McKay switching (Ammann private communication). Here, we provide a simple alternative mechanism that produces systematic examples of both discrete and quantum isospectral graphs with the same flip and nodal counts.

Some unsolved problems in discrete mathematics and mathematical cybernetics

Science.gov (United States)

Korshunov, Aleksei D.

2009-10-01

There are many unsolved problems in discrete mathematics and mathematical cybernetics. Writing a comprehensive survey of such problems involves great difficulties. First, such problems are rather numerous and varied. Second, they greatly differ from each other in degree of completeness of their solution. Therefore, even a comprehensive survey should not attempt to cover the whole variety of such problems; only the most important and significant problems should be reviewed. An impersonal choice of problems to include is quite hard. This paper includes 13 unsolved problems related to combinatorial mathematics and computational complexity theory. The problems selected give an indication of the author's studies for 50 years; for this reason, the choice of the problems reviewed here is, to some extent, subjective. At the same time, these problems are very difficult and quite important for discrete mathematics and mathematical cybernetics. Bibliography: 74 items.

Some unsolved problems in discrete mathematics and mathematical cybernetics

International Nuclear Information System (INIS)

Korshunov, Aleksei D

2009-01-01

There are many unsolved problems in discrete mathematics and mathematical cybernetics. Writing a comprehensive survey of such problems involves great difficulties. First, such problems are rather numerous and varied. Second, they greatly differ from each other in degree of completeness of their solution. Therefore, even a comprehensive survey should not attempt to cover the whole variety of such problems; only the most important and significant problems should be reviewed. An impersonal choice of problems to include is quite hard. This paper includes 13 unsolved problems related to combinatorial mathematics and computational complexity theory. The problems selected give an indication of the author's studies for 50 years; for this reason, the choice of the problems reviewed here is, to some extent, subjective. At the same time, these problems are very difficult and quite important for discrete mathematics and mathematical cybernetics. Bibliography: 74 items.

What Is Discrete Mathematics?

Science.gov (United States)

Sharp, Karen Tobey

This paper cites information received from a number of sources, e.g., mathematics teachers in two-year colleges, publishers, and convention speakers, about the nature of discrete mathematics and about what topics a course in this subject should contain. Note is taken of the book edited by Ralston and Young which discusses the future of college…

Discrete mathematics in the high school curriculum

NARCIS (Netherlands)

Anderson, I.; Asch, van A.G.; van Lint, J.H.

2004-01-01

In this paper we present some topics from the field of discrete mathematics which might be suitable for the high school curriculum. These topics yield both easy to understand challenging problems and important applications of discrete mathematics. We choose elements from number theory and various

Some unsolved problems in discrete mathematics and mathematical cybernetics

Energy Technology Data Exchange (ETDEWEB)

Korshunov, Aleksei D [S.L. Sobolev Institute for Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk (Russian Federation)

2009-10-31

There are many unsolved problems in discrete mathematics and mathematical cybernetics. Writing a comprehensive survey of such problems involves great difficulties. First, such problems are rather numerous and varied. Second, they greatly differ from each other in degree of completeness of their solution. Therefore, even a comprehensive survey should not attempt to cover the whole variety of such problems; only the most important and significant problems should be reviewed. An impersonal choice of problems to include is quite hard. This paper includes 13 unsolved problems related to combinatorial mathematics and computational complexity theory. The problems selected give an indication of the author's studies for 50 years; for this reason, the choice of the problems reviewed here is, to some extent, subjective. At the same time, these problems are very difficult and quite important for discrete mathematics and mathematical cybernetics. Bibliography: 74 items.

How Bob Barker Would (Probably) Teach Discrete Mathematics

Science.gov (United States)

Urness, Timothy

2010-01-01

This article proposes a discrete mathematics course in which games from "The Price Is Right" are used to engage students in a deeper, practical study of discrete mathematics. The games themselves are not the focus of the course; rather, the mathematical principles of the games give motivation for the concepts being taught. The game examples are…

Is Discrete Mathematics the New Math of the Eighties?

Science.gov (United States)

Hart, Eric W.

1985-01-01

Considered are what discrete mathematics includes, some parallels and differences between new math and discrete mathematics (listed in a table), and lessons to be learned. A list of references is included. (MNS)

DISCRETE MATHEMATICS AS FUNDAMENTAL DISCIPLINE IS IN SYSTEM OF MATHEMATICAL PREPARATION OF FUTURE SOFTWARE ENGINEER

OpenAIRE

D. Shchedrolosev

2010-01-01

Fundamental mathematical background is an important part of training future engineers and programmers. The paper considers existing approaches to teaching the fundamentals of discrete mathematics specialist IT profile, a comparative analysis of modern textbooks on discrete mathematics for IT professionals was conducted

CDM: Teaching Discrete Mathematics to Computer Science Majors

Science.gov (United States)

Sutner, Klaus

2005-01-01

CDM, for computational discrete mathematics, is a course that attempts to teach a number of topics in discrete mathematics to computer science majors. The course abandons the classical definition-theorem-proof model, and instead relies heavily on computation as a source of motivation and also for experimentation and illustration. The emphasis on…

DISCRETE MATHEMATICS AS FUNDAMENTAL DISCIPLINE IS IN SYSTEM OF MATHEMATICAL PREPARATION OF FUTURE SOFTWARE ENGINEER

Directory of Open Access Journals (Sweden)

D. Shchedrolosev

2010-04-01

Full Text Available Fundamental mathematical background is an important part of training future engineers and programmers. The paper considers existing approaches to teaching the fundamentals of discrete mathematics specialist IT profile, a comparative analysis of modern textbooks on discrete mathematics for IT professionals was conducted

GDSCalc: A Web-Based Application for Evaluating Discrete Graph Dynamical Systems.

Science.gov (United States)

Elmeligy Abdelhamid, Sherif H; Kuhlman, Chris J; Marathe, Madhav V; Mortveit, Henning S; Ravi, S S

2015-01-01

Discrete dynamical systems are used to model various realistic systems in network science, from social unrest in human populations to regulation in biological networks. A common approach is to model the agents of a system as vertices of a graph, and the pairwise interactions between agents as edges. Agents are in one of a finite set of states at each discrete time step and are assigned functions that describe how their states change based on neighborhood relations. Full characterization of state transitions of one system can give insights into fundamental behaviors of other dynamical systems. In this paper, we describe a discrete graph dynamical systems (GDSs) application called GDSCalc for computing and characterizing system dynamics. It is an open access system that is used through a web interface. We provide an overview of GDS theory. This theory is the basis of the web application; i.e., an understanding of GDS provides an understanding of the software features, while abstracting away implementation details. We present a set of illustrative examples to demonstrate its use in education and research. Finally, we compare GDSCalc with other discrete dynamical system software tools. Our perspective is that no single software tool will perform all computations that may be required by all users; tools typically have particular features that are more suitable for some tasks. We situate GDSCalc within this space of software tools.

Discrete geometric analysis of message passing algorithm on graphs

Science.gov (United States)

Watanabe, Yusuke

2010-04-01

We often encounter probability distributions given as unnormalized products of non-negative functions. The factorization structures are represented by hypergraphs called factor graphs. Such distributions appear in various fields, including statistics, artificial intelligence, statistical physics, error correcting codes, etc. Given such a distribution, computations of marginal distributions and the normalization constant are often required. However, they are computationally intractable because of their computational costs. One successful approximation method is Loopy Belief Propagation (LBP) algorithm. The focus of this thesis is an analysis of the LBP algorithm. If the factor graph is a tree, i.e. having no cycle, the algorithm gives the exact quantities. If the factor graph has cycles, however, the LBP algorithm does not give exact results and possibly exhibits oscillatory and non-convergent behaviors. The thematic question of this thesis is "How the behaviors of the LBP algorithm are affected by the discrete geometry of the factor graph?" The primary contribution of this thesis is the discovery of a formula that establishes the relation between the LBP, the Bethe free energy and the graph zeta function. This formula provides new techniques for analysis of the LBP algorithm, connecting properties of the graph and of the LBP and the Bethe free energy. We demonstrate applications of the techniques to several problems including (non) convexity of the Bethe free energy, the uniqueness and stability of the LBP fixed point. We also discuss the loop series initiated by Chertkov and Chernyak. The loop series is a subgraph expansion of the normalization constant, or partition function, and reflects the graph geometry. We investigate theoretical natures of the series. Moreover, we show a partial connection between the loop series and the graph zeta function.

Modern approaches to discrete curvature

CERN Document Server

Romon, Pascal

2017-01-01

 This book provides a valuable glimpse into discrete curvature, a rich new field of research which blends discrete mathematics, differential geometry, probability and computer graphics. It includes a vast collection of ideas and tools which will offer something new to all interested readers. Discrete geometry has arisen as much as a theoretical development as in response to unforeseen challenges coming from applications. Discrete and continuous geometries have turned out to be intimately connected. Discrete curvature is the key concept connecting them through many bridges in numerous fields: metric spaces, Riemannian and Euclidean geometries, geometric measure theory, topology, partial differential equations, calculus of variations, gradient flows, asymptotic analysis, probability, harmonic analysis, graph theory, etc. In spite of its crucial importance both in theoretical mathematics and in applications, up to now, almost no books have provided a coherent outlook on this emerging field.

(3, 1)*-Choosability of graphs of nonnegative characteristic without ...

Indian Academy of Sciences (India)

School of Mathematical Science, Huaiyin Normal University, 111 Changjiang ... For the used but undefined terminologies and notations, we refer the reader to the book ..... A (3,1)∗-choosable theorem on toroidal graphs, Discrete Appl. Math.

An Infinite Family of Circulant Graphs with Perfect State Transfer in Discrete Quantum Walks

OpenAIRE

Zhan, Hanmeng

2017-01-01

We study perfect state transfer in a discrete quantum walk. In particular, we show that there are infinitely many $4$-regular circulant graphs that admit perfect state transfer between antipodal vertices. To the best of our knowledge, previously there was no infinite family of $k$-regular graphs with perfect state transfer, for any $k\\ge 3$.

Web-Based Implementation of Discrete Mathematics

Science.gov (United States)

Love, Tanzy; Keinert, Fritz; Shelley, Mack

2006-01-01

The Department of Mathematics at Iowa State University teaches a freshman-level Discrete Mathematics course with total enrollment of about 1,800 students per year. The traditional format includes large lectures, with about 150 students each, taught by faculty and temporary instructors in two class sessions per week and recitation sections, with…

Geometric structure of chemistry-relevant graphs zigzags and central circuits

CERN Document Server

Deza, Michel-Marie; Shtogrin, Mikhail Ivanovitch

2015-01-01

The central theme of the present book is zigzags and central-circuits of three- or four-regular plane graphs, which allow a double covering or covering of the edgeset to be obtained. The book presents zigzag and central circuit structures of geometric fullerenes and several other classes of graph of interest in the fields of chemistry and mathematics. It also discusses the symmetries, parameterization and the Goldberg–Coxeter construction for those graphs. It is the first book on this subject, presenting full structure theory of such graphs. While many previous publications only addressed particular questions about selected graphs, this book is based on numerous computations and presents extensive data (tables and figures), as well as algorithmic and computational information. It will be of interest to researchers and students of discrete geometry, mathematical chemistry and combinatorics, as well as to lay mathematicians.

Parallel algorithms for finding cliques in a graph

International Nuclear Information System (INIS)

Szabo, S

2011-01-01

A clique is a subgraph in a graph that is complete in the sense that each two of its nodes are connected by an edge. Finding cliques in a given graph is an important procedure in discrete mathematical modeling. The paper will show how concepts such as splitting partitions, quasi coloring, node and edge dominance are related to clique search problems. In particular we will discuss the connection with parallel clique search algorithms. These concepts also suggest practical guide lines to inspect a given graph before starting a large scale search.

Logic and discrete mathematics a concise introduction

CERN Document Server

Conradie, Willem

2015-01-01

A concise yet rigorous introduction to logic and discrete mathematics. This book features a unique combination of comprehensive coverage of logic with a solid exposition of the most important fields of discrete mathematics, presenting material that has been tested and refined by the authors in university courses taught over more than a decade.  The chapters on logic - propositional and first-order - provide a robust toolkit for logical reasoning, emphasizing the conceptual understanding of the language and the semantics of classical logic as well as practical applications through the easy

A Note on Discrete Mathematics and Calculus.

Science.gov (United States)

O'Reilly, Thomas J.

1987-01-01

Much of the current literature on the topic of discrete mathematics and calculus during the first two years of an undergraduate mathematics curriculum is cited. A relationship between the recursive integration formulas and recursively defined polynomials is described. A Pascal program is included. (Author/RH)

Discrete mathematics in deaf education: a survey of teachers' knowledge and use.

Science.gov (United States)

Pagliaro, Claudia M; Kritzer, Karen L

The study documents what deaf education teachers know about discrete mathematics topics and determines if these topics are present in the mathematics curriculum. Survey data were collected from 290 mathematics teachers at center and public school programs serving a minimum of 120 students with hearing loss, grades K-8 or K-12, in the United States. Findings indicate that deaf education teachers are familiar with many discrete mathematics topics but do not include them in instruction because they consider the concepts too complicated for their students. Also, regardless of familiarity level, deaf education teachers are not familiar with discrete mathematics terminology; nor is their mathematics teaching structured to provide opportunities to apply the real-world-oriented activities used in discrete mathematics instruction. Findings emphasize the need for higher expectations of students with hearing loss, and for reform in mathematics curriculum and instruction within deaf education.

Discrete Mathematics Course Supported by CAS MATHEMATICA

Science.gov (United States)

Ivanov, O. A.; Ivanova, V. V.; Saltan, A. A.

2017-01-01

In this paper, we discuss examples of assignments for a course in discrete mathematics for undergraduate students majoring in business informatics. We consider several problems with computer-based solutions and discuss general strategies for using computers in teaching mathematics and its applications. In order to evaluate the effectiveness of our…

A short course in discrete mathematics

CERN Document Server

Bender, Edward A

2004-01-01

What sort of mathematics do I need for computer science? In response to this frequently asked question, a pair of professors at the University of California at San Diego created this text. Its sources are two of the university's most basic courses: Discrete Mathematics, and Mathematics for Algorithm and System Analysis. Intended for use by sophomores in the first of a two-quarter sequence, the text assumes some familiarity with calculus. Topics include Boolean functions and computer arithmetic; logic; number theory and cryptography; sets and functions; equivalence and order; and induction, seq

A discrete transition to advanced mathematics

CERN Document Server

Richmond, Bettina

2009-01-01

As the title indicates, this book is intended for courses aimed at bridging the gap between lower-level mathematics and advanced mathematics. The text provides a careful introduction to techniques for writing proofs and a logical development of topics based on intuitive understanding of concepts. The authors utilize a clear writing style and a wealth of examples to develop an understanding of discrete mathematics and critical thinking skills. While including many traditional topics, the text offers innovative material throughout. Surprising results are used to motivate the reader. The last thr

Conference on "Mathematical Technology of Networks"

CERN Document Server

2015-01-01

Bringing together leading researchers in the fields of functional analysis, mathematical physics and graph theory, as well as natural scientists using networks as a tool in their own research fields, such as neuroscience and machine learning, this volume presents recent advances in functional, analytical, probabilistic, and spectral aspects in the study of graphs, quantum graphs, and complex networks. The contributors to this volume explore the interplay between theoretical and applied aspects of discrete and continuous graphs. Their work helps to close the gap between different avenues of research on graphs, including metric graphs and ramified structures. All papers were presented at the conference "Mathematical Technology of Networks," held December 4–7, 2013 at the Zentrum für interdisziplinäre Forschung (ZiF) in Bielefeld, Germany, and are supplemented with detailed figures illustrating both abstract concepts as well as their real-world applications. Dynamical models on graphs or random graphs a...

Gems of combinatorial optimization and graph algorithms

CERN Document Server

Skutella, Martin; Stiller, Sebastian; Wagner, Dorothea

2015-01-01

Are you looking for new lectures for your course on algorithms, combinatorial optimization, or algorithmic game theory?  Maybe you need a convenient source of relevant, current topics for a graduate student or advanced undergraduate student seminar?  Or perhaps you just want an enjoyable look at some beautiful mathematical and algorithmic results, ideas, proofs, concepts, and techniques in discrete mathematics and theoretical computer science?   Gems of Combinatorial Optimization and Graph Algorithms is a handpicked collection of up-to-date articles, carefully prepared by a select group of international experts, who have contributed some of their most mathematically or algorithmically elegant ideas.  Topics include longest tours and Steiner trees in geometric spaces, cartograms, resource buying games, congestion games, selfish routing, revenue equivalence and shortest paths, scheduling, linear structures in graphs, contraction hierarchies, budgeted matching problems, and motifs in networks.   This ...

On the discrete spectrum of the Dirac operator on bent chain quantum graph

Directory of Open Access Journals (Sweden)

Belov Michail

2017-01-01

Full Text Available We study Dirac operators on an infinite quantum graph of a bent chain form which consists of identical rings connected at the touching points by δ-couplings with a parameter α ∈ ℝ. We are interested in the discrete spectrum of the corresponding Hamiltonian. It can be non-empty due to a local (geometrical perturbation of the corresponding infinite chain of rings. The quantum graph of analogous geometry with the Schrodinger operator on the edges was considered by Duclos, Exner and Turek in 2008. They showed that the absence of δ-couplings at vertices (i.e. the Kirchhoff condition at the vertices lead to the absence of eigenvalues. We consider the relativistic particle (the Dirac operator instead of the Schrodinger one but the result is analogous. Quantum graphs of such type are suitable for description of grapheme-based nanostructures. It is established that the negativity of α is the necessary and sufficient condition for the existence of eigenvalues of the Dirac operator (i.e. the discrete spectrum of the Hamiltonian in this case is not empty. The continuous spectrum of the Hamiltonian for bent chain graph coincides with that for the corresponding straight infinite chain. Conditions for appearance of more than one eigenvalue are obtained. It is related to the bending angle. The investigation is based on the transfer-matrix approach. It allows one to reduce the problem to an algebraic task. δ-couplings was introduced by the operator extensions theory method.

Equivalence of massive propagator distance and mathematical distance on graphs

International Nuclear Information System (INIS)

Filk, T.

1992-01-01

It is shown in this paper that the assignment of distance according to the massive propagator method and according to the mathematical definition (length of minimal path) on arbitrary graphs with a bound on the degree leads to equivalent large scale properties of the graph. Especially, the internal scaling dimension is the same for both definitions. This result holds for any fixed, non-vanishing mass, so that a really inequivalent definition of distance requires the limit m → 0

Introduction to graph theory

CERN Document Server

Trudeau, Richard J

1994-01-01

Preface1. Pure Mathematics Introduction; Euclidean Geometry as Pure Mathematics; Games; Why Study Pure Mathematics?; What's Coming; Suggested Reading2. Graphs Introduction; Sets; Paradox; Graphs; Graph diagrams; Cautions; Common Graphs; Discovery; Complements and Subgraphs; Isomorphism; Recognizing Isomorphic Graphs; Semantics The Number of Graphs Having a Given nu; Exercises; Suggested Reading3. Planar Graphs Introduction; UG, K subscript 5, and the Jordan Curve Theorem; Are there More Nonplanar Graphs?; Expansions; Kuratowski's Theorem; Determining Whether a Graph is Planar or

A Design of Computer Aided Instructions (CAI) for Undirected Graphs in the Discrete Math Tutorial (DMT). Part 1.

Science.gov (United States)

1990-06-01

The objective of this thesis research is to create a tutorial for teaching aspects of undirected graphs in discrete math . It is one of the submodules...of the Discrete Math Tutorial (DMT), which is a Computer Aided Instructional (CAI) tool for teaching discrete math to the Naval Academy and the

A Design of Computer Aided Instructions (CAI) for Undirected Graphs in the Discrete Math Tutorial (DMT). Part 2

Science.gov (United States)

1990-06-01

The objective of this thesis research is to create a tutorial for teaching aspects of undirected graphs in discrete math . It is one of the submodules...of the Discrete Math Tutorial (DMT), which is a Computer Aided Instructional (CAI) tool for teaching discrete math to the Naval Academy and the

Gromov hyperbolicity in lexicographic product graphs

Indian Academy of Sciences (India)

41

on the group [17]. The concept of hyperbolicity appears also in discrete mathematics, algorithms and networking. For .... graph (of a presentation with solvable word problem) there is an algorithm which allows to decide if it is ...... of Theorem 3.14, i.e., dG1◦{w}(Vp, [π(x)π(z)] ∪ [π(z)π(y)]) = δ(G1) with π the canonical projection.

Derivatives in discrete mathematics: a novel graph-theoretical invariant for generating new 2/3D molecular descriptors. I. Theory and QSPR application.

Science.gov (United States)

Marrero-Ponce, Yovani; Santiago, Oscar Martínez; López, Yoan Martínez; Barigye, Stephen J; Torrens, Francisco

2012-11-01

In this report, we present a new mathematical approach for describing chemical structures of organic molecules at atomic-molecular level, proposing for the first time the use of the concept of the derivative ([Formula: see text]) of a molecular graph (MG) with respect to a given event (E), to obtain a new family of molecular descriptors (MDs). With this purpose, a new matrix representation of the MG, which generalizes graph's theory's traditional incidence matrix, is introduced. This matrix, denominated the generalized incidence matrix, Q, arises from the Boolean representation of molecular sub-graphs that participate in the formation of the graph molecular skeleton MG and could be complete (representing all possible connected sub-graphs) or constitute sub-graphs of determined orders or types as well as a combination of these. The Q matrix is a non-quadratic and unsymmetrical in nature, its columns (n) and rows (m) are conditions (letters) and collection of conditions (words) with which the event occurs. This non-quadratic and unsymmetrical matrix is transformed, by algebraic manipulation, to a quadratic and symmetric matrix known as relations frequency matrix, F, which characterizes the participation intensity of the conditions (letters) in the events (words). With F, we calculate the derivative over a pair of atomic nuclei. The local index for the atomic nuclei i, Δ(i), can therefore be obtained as a linear combination of all the pair derivatives of the atomic nuclei i with all the rest of the j's atomic nuclei. Here, we also define new strategies that generalize the present form of obtaining global or local (group or atom-type) invariants from atomic contributions (local vertex invariants, LOVIs). In respect to this, metric (norms), means and statistical invariants are introduced. These invariants are applied to a vector whose components are the values Δ(i) for the atomic nuclei of the molecule or its fragments. Moreover, with the purpose of differentiating

Instructional Efficiency of the Integration of Graphing Calculators in Teaching and Learning Mathematics

Science.gov (United States)

Tajuddin, Nor'ain Mohd; Tarmizi, Rohani Ahmad; Konting, Mohd Majid; Ali, Wan Zah Wan

2009-01-01

This quasi-experimental study with non-equivalent control group post-test only design was conducted to investigate the effects of using graphing calculators in mathematics teaching and learning on Form Four Malaysian secondary school students' performance and their meta-cognitive awareness level. Graphing calculator strategy refers to the use of…

Homology groups for particles on one-connected graphs

Science.gov (United States)

MaciÄ Żek, Tomasz; Sawicki, Adam

2017-06-01

We present a mathematical framework for describing the topology of configuration spaces for particles on one-connected graphs. In particular, we compute the homology groups over integers for different classes of one-connected graphs. Our approach is based on some fundamental combinatorial properties of the configuration spaces, Mayer-Vietoris sequences for different parts of configuration spaces, and some limited use of discrete Morse theory. As one of the results, we derive the closed-form formulae for ranks of the homology groups for indistinguishable particles on tree graphs. We also give a detailed discussion of the second homology group of the configuration space of both distinguishable and indistinguishable particles. Our motivation is the search for new kinds of quantum statistics.

Logic and discrete mathematics a concise introduction : solutions manual

CERN Document Server

Conradie, Willem; Robinson, Claudette

2015-01-01

Solutions manual to accompany Logic and Discrete Mathematics: A Concise Introduction This book features a unique combination of comprehensive coverage of logic with a solid exposition of the most important fields of discrete mathematics, presenting material that has been tested and refined by the authors in university courses taught over more than a decade. Written in a clear and reader-friendly style, each section ends with an extensive set of exercises, most of them provided with complete solutions which are available in this accompanying solutions manual.

Application of multivariate splines to discrete mathematics

OpenAIRE

Xu, Zhiqiang

2005-01-01

Using methods developed in multivariate splines, we present an explicit formula for discrete truncated powers, which are defined as the number of non-negative integer solutions of linear Diophantine equations. We further use the formula to study some classical problems in discrete mathematics as follows. First, we extend the partition function of integers in number theory. Second, we exploit the relation between the relative volume of convex polytopes and multivariate truncated powers and giv...

Mathematical aspects of the discrete space-time hypothesis

International Nuclear Information System (INIS)

Sardanashvili, G.A.

1979-01-01

A hypothesis of a microcosm space discreteness is considered from the theoretical-mathematical point of view. The type of topological spaces, which formalizes representations on the discrete space-time, is determined. It is explained, how these spaces arise in physical models. The physical task, in which the discrete space could arise as a version of its solution, is considered. It is shown that the discrete structure of space can arise with a certain interaction type in the system, for example, with its considerable self-shielding, which can take place, in particular, in the particles or in the cosmological and astrophysical singularities

Discrete Mathematics - Special Issue: Graph Theory - dedicated to Carsten Thomassen on his 60th birthday

DEFF Research Database (Denmark)

2011-01-01

Carsten Thomassen belongs to the worlds's absolute top graph theorists, and to the world's top mathematicians in general. The special issue is a rather somewhat random collection of good papers in graph theory, by many different authors, dedicated to Carsten Thomassen on his 60th birthday. Guest ...

Discrete mathematics course supported by CAS MATHEMATICA

Science.gov (United States)

Ivanov, O. A.; Ivanova, V. V.; Saltan, A. A.

2017-08-01

In this paper, we discuss examples of assignments for a course in discrete mathematics for undergraduate students majoring in business informatics. We consider several problems with computer-based solutions and discuss general strategies for using computers in teaching mathematics and its applications. In order to evaluate the effectiveness of our approach, we conducted an anonymous survey. The results of the survey provide evidence that our approach contributes to high outcomes and aligns with the course aims and objectives.

Inevitable randomness in discrete mathematics

CERN Document Server

Beck, Jozsef

2009-01-01

Mathematics has been called the science of order. The subject is remarkably good for generalizing specific cases to create abstract theories. However, mathematics has little to say when faced with highly complex systems, where disorder reigns. This disorder can be found in pure mathematical arenas, such as the distribution of primes, the 3n+1 conjecture, and class field theory. The purpose of this book is to provide examples--and rigorous proofs--of the complexity law: (1) discrete systems are either simple or they exhibit advanced pseudorandomness; (2) a priori probabilities often exist even when there is no intrinsic symmetry. Part of the difficulty in achieving this purpose is in trying to clarify these vague statements. The examples turn out to be fascinating instances of deep or mysterious results in number theory and combinatorics. This book considers randomness and complexity. The traditional approach to complexity--computational complexity theory--is to study very general complexity classes, such as P...

Introduction to quantum graphs

CERN Document Server

Berkolaiko, Gregory

2012-01-01

A "quantum graph" is a graph considered as a one-dimensional complex and equipped with a differential operator ("Hamiltonian"). Quantum graphs arise naturally as simplified models in mathematics, physics, chemistry, and engineering when one considers propagation of waves of various nature through a quasi-one-dimensional (e.g., "meso-" or "nano-scale") system that looks like a thin neighborhood of a graph. Works that currently would be classified as discussing quantum graphs have been appearing since at least the 1930s, and since then, quantum graphs techniques have been applied successfully in various areas of mathematical physics, mathematics in general and its applications. One can mention, for instance, dynamical systems theory, control theory, quantum chaos, Anderson localization, microelectronics, photonic crystals, physical chemistry, nano-sciences, superconductivity theory, etc. Quantum graphs present many non-trivial mathematical challenges, which makes them dear to a mathematician's heart. Work on qu...

Particle transport in breathing quantum graph

International Nuclear Information System (INIS)

Matrasulov, D.U.; Yusupov, J.R.; Sabirov, K.K.; Sobirov, Z.A.

2012-01-01

Full text: Particle transport in nanoscale networks and discrete structures is of fundamental and practical importance. Usually such systems are modeled by so-called quantum graphs, the systems attracting much attention in physics and mathematics during past two decades [1-5]. During last two decades quantum graphs found numerous applications in modeling different discrete structures and networks in nanoscale and mesoscopic physics (e.g., see reviews [1-3]). Despite considerable progress made in the study of particle dynamics most of the problems deal with unperturbed case and the case of time-dependent perturbation has not yet be explored. In this work we treat particle dynamics for quantum star graph with time-dependent bonds. In particular, we consider harmonically breathing quantum star graphs, the cases of monotonically contracting and expanding graphs. The latter can be solved exactly analytically. Edge boundaries are considered to be time-dependent, while branching point is assumed to be fixed. Quantum dynamics of a particle in such graphs is studied by solving Schrodinger equation with time-dependent boundary conditions given on a star graph. Time-dependence of the average kinetic energy is analyzed. Space-time evolution of the Gaussian wave packet is treated for harmonically breathing star graph. It is found that for certain frequencies energy is a periodic function of time, while for others it can be non-monotonically growing function of time. Such a feature can be caused by possible synchronization of the particles motion and the motions of the moving edges of graph bonds. (authors) References: [1] Tsampikos Kottos and Uzy Smilansky, Ann. Phys., 76, 274 (1999). [2] Sven Gnutzmann and Uzy Smilansky, Adv. Phys. 55, 527 (2006). [3] S. GnutzmannJ.P. Keating, F. Piotet, Ann. Phys., 325, 2595 (2010). [4] P.Exner, P.Seba, P.Stovicek, J. Phys. A: Math. Gen. 21, 4009 (1988). [5] J. Boman, P. Kurasov, Adv. Appl. Math., 35, 58 (2005)

Predictions of first passage times in sparse discrete fracture networks using graph-based reductions

Science.gov (United States)

Hyman, J.; Hagberg, A.; Srinivasan, G.; Mohd-Yusof, J.; Viswanathan, H. S.

2017-12-01

We present a graph-based methodology to reduce the computational cost of obtaining first passage times through sparse fracture networks. We derive graph representations of generic three-dimensional discrete fracture networks (DFNs) using the DFN topology and flow boundary conditions. Subgraphs corresponding to the union of the k shortest paths between the inflow and outflow boundaries are identified and transport on their equivalent subnetworks is compared to transport through the full network. The number of paths included in the subgraphs is based on the scaling behavior of the number of edges in the graph with the number of shortest paths. First passage times through the subnetworks are in good agreement with those obtained in the full network, both for individual realizations and in distribution. Accurate estimates of first passage times are obtained with an order of magnitude reduction of CPU time and mesh size using the proposed method.

Introductory graph theory

CERN Document Server

Chartrand, Gary

1984-01-01

Graph theory is used today in the physical sciences, social sciences, computer science, and other areas. Introductory Graph Theory presents a nontechnical introduction to this exciting field in a clear, lively, and informative style. Author Gary Chartrand covers the important elementary topics of graph theory and its applications. In addition, he presents a large variety of proofs designed to strengthen mathematical techniques and offers challenging opportunities to have fun with mathematics. Ten major topics - profusely illustrated - include: Mathematical Models, Elementary Concepts of Grap

Conversations about Curriculum Change: Mathematical Thinking and Team-Based Learning in a Discrete Mathematics Course

Science.gov (United States)

Paterson, Judy; Sneddon, Jamie

2011-01-01

This article reports on the learning conversations between a mathematician and a mathematics educator as they worked together to change the delivery model of a third year discrete mathematics course from a traditional lecture mode to team-based learning (TBL). This change prompted the mathematician to create team tasks which increasingly focused…

The Mathematics of Networks Science: Scale-Free, Power-Law Graphs and Continuum Theoretical Analysis

Science.gov (United States)

Padula, Janice

2012-01-01

When hoping to initiate or sustain students' interest in mathematics teachers should always consider relevance, relevance to students' lives and in the middle and later years of instruction in high school and university, accessibility. A topic such as the mathematics behind networks science, more specifically scale-free graphs, is up-to-date,…

Lecture Note on Discrete Mathematics: Predicates and Quantifiers

DEFF Research Database (Denmark)

Nordbjerg, Finn Ebertsen

2016-01-01

This lecture note supplements the treatment of predicates and quantifiers given in standard textbooks on Discrete Mathematics (e.g.: [1]) and introduces the notation used in this course. We will present central concepts that are important, when predicate logic is used for specification...

An excursion through elementary mathematics, volume iii discrete mathematics and polynomial algebra

CERN Document Server

Caminha Muniz Neto, Antonio

2018-01-01

This book provides a comprehensive, in-depth overview of elementary mathematics as explored in Mathematical Olympiads around the world. It expands on topics usually encountered in high school and could even be used as preparation for a first-semester undergraduate course. This third and last volume covers Counting, Generating Functions, Graph Theory, Number Theory, Complex Numbers, Polynomials, and much more. As part of a collection, the book differs from other publications in this field by not being a mere selection of questions or a set of tips and tricks that applies to specific problems. It starts from the most basic theoretical principles, without being either too general or too axiomatic. Examples and problems are discussed only if they are helpful as applications of the theory. Propositions are proved in detail and subsequently applied to Olympic problems or to other problems at the Olympic level. The book also explores some of the hardest problems presented at National and International Mathematics Ol...

Adventures in graph theory

CERN Document Server

Joyner, W David

2017-01-01

This textbook acts as a pathway to higher mathematics by seeking and illuminating the connections between graph theory and diverse fields of mathematics, such as calculus on manifolds, group theory, algebraic curves, Fourier analysis, cryptography and other areas of combinatorics. An overview of graph theory definitions and polynomial invariants for graphs prepares the reader for the subsequent dive into the applications of graph theory. To pique the reader’s interest in areas of possible exploration, recent results in mathematics appear throughout the book, accompanied with examples of related graphs, how they arise, and what their valuable uses are. The consequences of graph theory covered by the authors are complicated and far-reaching, so topics are always exhibited in a user-friendly manner with copious graphs, exercises, and Sage code for the computation of equations. Samples of the book’s source code can be found at github.com/springer-math/adventures-in-graph-theory. The text is geared towards ad...

A Study of the Use of a Handheld Computer Algebra System in Discrete Mathematics

Science.gov (United States)

Powers, Robert A.; Allison, Dean E.; Grassl, Richard M.

2005-01-01

This study investigated the impact of the TI-92 handheld Computer Algebra System (CAS) on student achievement in a discrete mathematics course. Specifically, the researchers examined the differences between a CAS section and a control section of discrete mathematics on students' in-class examinations. Additionally, they analysed student approaches…

Teaching Discrete Mathematics Entirely from Primary Historical Sources

Science.gov (United States)

Barnett, Janet Heine; Bezhanishvili, Guram; Lodder, Jerry; Pengelley, David

2016-01-01

We describe teaching an introductory discrete mathematics course entirely from student projects based on primary historical sources. We present case studies of four projects that cover the content of a one-semester course, and mention various other courses that we have taught with primary source projects.

Modern graph theory

CERN Document Server

Bollobás, Béla

1998-01-01

The time has now come when graph theory should be part of the education of every serious student of mathematics and computer science, both for its own sake and to enhance the appreciation of mathematics as a whole. This book is an in-depth account of graph theory, written with such a student in mind; it reflects the current state of the subject and emphasizes connections with other branches of pure mathematics. The volume grew out of the author's earlier book, Graph Theory -- An Introductory Course, but its length is well over twice that of its predecessor, allowing it to reveal many exciting new developments in the subject. Recognizing that graph theory is one of several courses competing for the attention of a student, the book contains extensive descriptive passages designed to convey the flavor of the subject and to arouse interest. In addition to a modern treatment of the classical areas of graph theory such as coloring, matching, extremal theory, and algebraic graph theory, the book presents a detailed ...

An Application of Discrete Mathematics to Coding Theory.

Science.gov (United States)

Donohoe, L. Joyce

1992-01-01

Presents a public-key cryptosystem application to introduce students to several topics in discrete mathematics. A computer algorithms using recursive methods is presented to solve a problem in which one person wants to send a coded message to a second person while keeping the message secret from a third person. (MDH)

Mathematical Biology Modules Based on Modern Molecular Biology and Modern Discrete Mathematics

Science.gov (United States)

Davies, Robin; Hodge, Terrell; Enyedi, Alexander

2010-01-01

We describe an ongoing collaborative curriculum materials development project between Sweet Briar College and Western Michigan University, with support from the National Science Foundation. We present a collection of modules under development that can be used in existing mathematics and biology courses, and we address a critical national need to introduce students to mathematical methods beyond the interface of biology with calculus. Based on ongoing research, and designed to use the project-based-learning approach, the modules highlight applications of modern discrete mathematics and algebraic statistics to pressing problems in molecular biology. For the majority of projects, calculus is not a required prerequisite and, due to the modest amount of mathematical background needed for some of the modules, the materials can be used for an early introduction to mathematical modeling. At the same time, most modules are connected with topics in linear and abstract algebra, algebraic geometry, and probability, and they can be used as meaningful applied introductions into the relevant advanced-level mathematics courses. Open-source software is used to facilitate the relevant computations. As a detailed example, we outline a module that focuses on Boolean models of the lac operon network. PMID:20810955

Mathematical biology modules based on modern molecular biology and modern discrete mathematics.

Science.gov (United States)

Robeva, Raina; Davies, Robin; Hodge, Terrell; Enyedi, Alexander

2010-01-01

We describe an ongoing collaborative curriculum materials development project between Sweet Briar College and Western Michigan University, with support from the National Science Foundation. We present a collection of modules under development that can be used in existing mathematics and biology courses, and we address a critical national need to introduce students to mathematical methods beyond the interface of biology with calculus. Based on ongoing research, and designed to use the project-based-learning approach, the modules highlight applications of modern discrete mathematics and algebraic statistics to pressing problems in molecular biology. For the majority of projects, calculus is not a required prerequisite and, due to the modest amount of mathematical background needed for some of the modules, the materials can be used for an early introduction to mathematical modeling. At the same time, most modules are connected with topics in linear and abstract algebra, algebraic geometry, and probability, and they can be used as meaningful applied introductions into the relevant advanced-level mathematics courses. Open-source software is used to facilitate the relevant computations. As a detailed example, we outline a module that focuses on Boolean models of the lac operon network.

Discrete Mathematics in Deaf Education: A Survey of Teachers' Knowledge and Use

Science.gov (United States)

Pagliaro, C.; Kritzer, K. L.

2005-01-01

The study documents what deaf education teachers know about discrete mathematics topics and determines if these topics are present in the mathematics curriculum. Survey data were collected from 290 mathematics teachers at center and public school programs serving a minimum of 120 students with hearing loss, grades K-8 or K-12, in the United…

a Discrete Mathematical Model to Simulate Malware Spreading

Science.gov (United States)

Del Rey, A. Martin; Sánchez, G. Rodriguez

2012-10-01

With the advent and worldwide development of Internet, the study and control of malware spreading has become very important. In this sense, some mathematical models to simulate malware propagation have been proposed in the scientific literature, and usually they are based on differential equations exploiting the similarities with mathematical epidemiology. The great majority of these models study the behavior of a particular type of malware called computer worms; indeed, to the best of our knowledge, no model has been proposed to simulate the spreading of a computer virus (the traditional type of malware which differs from computer worms in several aspects). In this sense, the purpose of this work is to introduce a new mathematical model not based on continuous mathematics tools but on discrete ones, to analyze and study the epidemic behavior of computer virus. Specifically, cellular automata are used in order to design such model.

Asymptote Misconception on Graphing Functions: Does Graphing Software Resolve It?

Directory of Open Access Journals (Sweden)

Mehmet Fatih Öçal

2017-01-01

Full Text Available Graphing function is an important issue in mathematics education due to its use in various areas of mathematics and its potential roles for students to enhance learning mathematics. The use of some graphing software assists students’ learning during graphing functions. However, the display of graphs of functions that students sketched by hand may be relatively different when compared to the correct forms sketched using graphing software. The possible misleading effects of this situation brought a discussion of a misconception (asymptote misconception on graphing functions. The purpose of this study is two- fold. First of all, this study investigated whether using graphing software (GeoGebra in this case helps students to determine and resolve this misconception in calculus classrooms. Second, the reasons for this misconception are sought. The multiple case study was utilized in this study. University students in two calculus classrooms who received instructions with (35 students or without GeoGebra assisted instructions (32 students were compared according to whether they fell into this misconception on graphing basic functions (1/x, lnx, ex. In addition, students were interviewed to reveal the reasons behind this misconception. Data were analyzed by means of descriptive and content analysis methods. The findings indicated that those who received GeoGebra assisted instruction were better in resolving it. In addition, the reasons behind this misconception were found to be teacher-based, exam-based and some other factors.

Packing Degenerate Graphs Greedily

Czech Academy of Sciences Publication Activity Database

Allen, P.; Böttcher, J.; Hladký, J.; Piguet, Diana

2017-01-01

Roč. 61, August (2017), s. 45-51 ISSN 1571-0653 R&D Projects: GA ČR GJ16-07822Y Institutional support: RVO:67985807 Keywords : tree packing conjecture * graph packing * graph processes Subject RIV: BA - General Mathematics OBOR OECD: Pure mathematics

Simulating triangulations. Graphs, manifolds and (quantum) spacetime

International Nuclear Information System (INIS)

Krueger, Benedikt

2016-01-01

Triangulations, which can intuitively be described as a tessellation of space into simplicial building blocks, are structures that arise in various different branches of physics: They can be used for describing complicated and curved objects in a discretized way, e.g., in foams, gels or porous media, or for discretizing curved boundaries for fluid simulations or dissipative systems. Interpreting triangulations as (maximal planar) graphs makes it possible to use them in graph theory or statistical physics, e.g., as small-world networks, as networks of spins or in biological physics as actin networks. Since one can find an analogue of the Einstein-Hilbert action on triangulations, they can even be used for formulating theories of quantum gravity. Triangulations have also important applications in mathematics, especially in discrete topology. Despite their wide occurrence in different branches of physics and mathematics, there are still some fundamental open questions about triangulations in general. It is a prior unknown how many triangulations there are for a given set of points or a given manifold, or even whether there are exponentially many triangulations or more, a question that relates to a well-defined behavior of certain quantum geometry models. Another major unknown question is whether elementary steps transforming triangulations into each other, which are used in computer simulations, are ergodic. Using triangulations as model for spacetime, it is not clear whether there is a meaningful continuum limit that can be identified with the usual and well-tested theory of general relativity. Within this thesis some of these fundamental questions about triangulations are answered by the use of Markov chain Monte Carlo simulations, which are a probabilistic method for calculating statistical expectation values, or more generally a tool for calculating high-dimensional integrals. Additionally, some details about the Wang-Landau algorithm, which is the primary used

Simulating triangulations. Graphs, manifolds and (quantum) spacetime

Energy Technology Data Exchange (ETDEWEB)

Krueger, Benedikt

2016-07-01

Triangulations, which can intuitively be described as a tessellation of space into simplicial building blocks, are structures that arise in various different branches of physics: They can be used for describing complicated and curved objects in a discretized way, e.g., in foams, gels or porous media, or for discretizing curved boundaries for fluid simulations or dissipative systems. Interpreting triangulations as (maximal planar) graphs makes it possible to use them in graph theory or statistical physics, e.g., as small-world networks, as networks of spins or in biological physics as actin networks. Since one can find an analogue of the Einstein-Hilbert action on triangulations, they can even be used for formulating theories of quantum gravity. Triangulations have also important applications in mathematics, especially in discrete topology. Despite their wide occurrence in different branches of physics and mathematics, there are still some fundamental open questions about triangulations in general. It is a prior unknown how many triangulations there are for a given set of points or a given manifold, or even whether there are exponentially many triangulations or more, a question that relates to a well-defined behavior of certain quantum geometry models. Another major unknown question is whether elementary steps transforming triangulations into each other, which are used in computer simulations, are ergodic. Using triangulations as model for spacetime, it is not clear whether there is a meaningful continuum limit that can be identified with the usual and well-tested theory of general relativity. Within this thesis some of these fundamental questions about triangulations are answered by the use of Markov chain Monte Carlo simulations, which are a probabilistic method for calculating statistical expectation values, or more generally a tool for calculating high-dimensional integrals. Additionally, some details about the Wang-Landau algorithm, which is the primary used

The fascinating world of graph theory

CERN Document Server

Benjamin, Arthur; Zhang, Ping

2015-01-01

Graph theory goes back several centuries and revolves around the study of graphs-mathematical structures showing relations between objects. With applications in biology, computer science, transportation science, and other areas, graph theory encompasses some of the most beautiful formulas in mathematics-and some of its most famous problems. The Fascinating World of Graph Theory explores the questions and puzzles that have been studied, and often solved, through graph theory. This book looks at graph theory's development and the vibrant individuals responsible for the field's growth. Introducin

Mathematics of Web science: structure, dynamics and incentives.

Science.gov (United States)

Chayes, Jennifer

2013-03-28

Dr Chayes' talk described how, to a discrete mathematician, 'all the world's a graph, and all the people and domains merely vertices'. A graph is represented as a set of vertices V and a set of edges E, so that, for instance, in the World Wide Web, V is the set of pages and E the directed hyperlinks; in a social network, V is the people and E the set of relationships; and in the autonomous system Internet, V is the set of autonomous systems (such as AOL, Yahoo! and MSN) and E the set of connections. This means that mathematics can be used to study the Web (and other large graphs in the online world) in the following way: first, we can model online networks as large finite graphs; second, we can sample pieces of these graphs; third, we can understand and then control processes on these graphs; and fourth, we can develop algorithms for these graphs and apply them to improve the online experience.

Constructing Contracts: Making Discrete Mathematics Relevant to Beginning Programmers

Science.gov (United States)

Gegg-Harrison, Timothy S.

2005-01-01

Although computer scientists understand the importance of discrete mathematics to the foundations of their field, computer science (CS) students do not always see the relevance. Thus, it is important to find a way to show students its relevance. The concept of program correctness is generally taught as an activity independent of the programming…

Praxis II mathematics content knowledge test (0061)

CERN Document Server

McCune, Ennis Donice

2007-01-01

Your guide to a higher score on the Praxis II?: Mathematics Content Knowledge Test (0061) Why CliffsTestPrep Guides? Go with the name you know and trust Get the information you need--fast! Written by test-prep specialists About the contents: Introduction * Overview of the exam * How to use this book * Proven study strategies and test-taking tips Part I: Subject Review * Focused review of all exam topics: arithmetic and basic algebra, geometry, trigonometry, analytic geometry, functions and their graphs, calculus, probability and statistics, discrete mathematics, linear algebra, compute

Introduction to the discrete Fourier series considering both mathematical and engineering aspects - A linear-algebra approach

Directory of Open Access Journals (Sweden)

Ludwig Kohaupt

2015-12-01

Full Text Available The discrete Fourier series is a valuable tool developed and used by mathematicians and engineers alike. One of the most prominent applications is signal processing. Usually, it is important that the signals be transmitted fast, for example, when transmitting images over large distances such as between the moon and the earth or when generating images in computer tomography. In order to achieve this, appropriate algorithms are necessary. In this context, the fast Fourier transform (FFT plays a key role which is an algorithm for calculating the discrete Fourier transform (DFT; this, in turn, is tightly connected with the discrete Fourier series. The last one itself is the discrete analog of the common (continuous-time Fourier series and is usually learned by mathematics students from a theoretical point of view. The aim of this expository/pedagogical paper is to give an introduction to the discrete Fourier series for both mathematics and engineering students. It is intended to expand the purely mathematical view; the engineering aspect is taken into account by applying the FFT to an example from signal processing that is small enough to be used in class-room teaching and elementary enough to be understood also by mathematics students. The MATLAB program is employed to do the computations.

Isospectral graphs with identical nodal counts

International Nuclear Information System (INIS)

Oren, Idan; Band, Ram

2012-01-01

According to a recent conjecture, isospectral objects have different nodal count sequences (Gnutzmann et al 2005 J. Phys. A: Math. Gen. 38 8921–33). We study generalized Laplacians on discrete graphs, and use them to construct the first non-trivial counterexamples to this conjecture. In addition, these examples demonstrate a surprising connection between isospectral discrete and quantum graphs. (paper)

Levels of line graph question interpretation with intermediate elementary students of varying scientific and mathematical knowledge and ability: A think aloud study

Science.gov (United States)

Keller, Stacy Kathryn

This study examined how intermediate elementary students' mathematics and science background knowledge affected their interpretation of line graphs and how their interpretations were affected by graph question levels. A purposive sample of 14 6th-grade students engaged in think aloud interviews (Ericsson & Simon, 1993) while completing an excerpted Test of Graphing in Science (TOGS) (McKenzie & Padilla, 1986). Hand gestures were video recorded. Student performance on the TOGS was assessed using an assessment rubric created from previously cited factors affecting students' graphing ability. Factors were categorized using Bertin's (1983) three graph question levels. The assessment rubric was validated by Padilla and a veteran mathematics and science teacher. Observational notes were also collected. Data were analyzed using Roth and Bowen's semiotic process of reading graphs (2001). Key findings from this analysis included differences in the use of heuristics, self-generated questions, science knowledge, and self-motivation. Students with higher prior achievement used a greater number and variety of heuristics and more often chose appropriate heuristics. They also monitored their understanding of the question and the adequacy of their strategy and answer by asking themselves questions. Most used their science knowledge spontaneously to check their understanding of the question and the adequacy of their answers. Students with lower and moderate prior achievement favored one heuristic even when it was not useful for answering the question and rarely asked their own questions. In some cases, if students with lower prior achievement had thought about their answers in the context of their science knowledge, they would have been able to recognize their errors. One student with lower prior achievement motivated herself when she thought the questions were too difficult. In addition, students answered the TOGS in one of three ways: as if they were mathematics word problems

Handbook of mathematical methods in imaging

CERN Document Server

2015-01-01

The Handbook of Mathematical Methods in Imaging provides a comprehensive treatment of the mathematical techniques used in imaging science. The material is grouped into two central themes, namely, Inverse Problems (Algorithmic Reconstruction) and Signal and Image Processing. Each section within the themes covers applications (modeling), mathematics, numerical methods (using a case example) and open questions. Written by experts in the area, the presentation is mathematically rigorous. This expanded and revised second edition contains updates to existing chapters and 16 additional entries on important mathematical methods such as graph cuts, morphology, discrete geometry, PDEs, conformal methods, to name a few. The entries are cross-referenced for easy navigation through connected topics. Available in both print and electronic forms, the handbook is enhanced by more than 200 illustrations and an extended bibliography. It will benefit students, scientists and researchers in applied mathematics. Engineers and com...

Mathematical Minute: Rotating a Function Graph

Science.gov (United States)

Bravo, Daniel; Fera, Joseph

2013-01-01

Using calculus only, we find the angles you can rotate the graph of a differentiable function about the origin and still obtain a function graph. We then apply the solution to odd and even degree polynomials.

Application of methods of discrete mathematics at modular synthesis of mechatronic devices

OpenAIRE

Nikiforov, S.; Nikiforov, B.; Mandarov, E.; Rabdanova, N.

2010-01-01

The article is devoted to application of methods of discrete mathematics (the theory of counts, the method of matrix code and others) and synthesis of executive mechanisms of mechatronic handling devices

System for Automatic Generation of Examination Papers in Discrete Mathematics

Science.gov (United States)

Fridenfalk, Mikael

2013-01-01

A system was developed for automatic generation of problems and solutions for examinations in a university distance course in discrete mathematics and tested in a pilot experiment involving 200 students. Considering the success of such systems in the past, particularly including automatic assessment, it should not take long before such systems are…

Graph comprehension in science and mathematics education: Objects and categories

DEFF Research Database (Denmark)

Voetmann Christiansen, Frederik; May, Michael

types of registers. In the second part of the paper, we consider how diagrams in science are often composites of iconic and indexical elements, and how this fact may lead to confusion for students. In the discussion the utility of the Peircian semiotic framework for educational studies......, the typological mistake of considering graphs as images is discussed related to litterature, and two examples from engineering education are given. The educational implications for science and engineering are discussed, with emphasis on the need for students to work explicitly with conversions between different...... of representational forms in science is discussed, and how the objects of mathematics and science relate to their semiotic representations....

The discrete Fourier transform theory, algorithms and applications

CERN Document Server

Sundaraajan, D

2001-01-01

This authoritative book provides comprehensive coverage of practical Fourier analysis. It develops the concepts right from the basics and gradually guides the reader to the advanced topics. It presents the latest and practically efficient DFT algorithms, as well as the computation of discrete cosine and Walsh-Hadamard transforms. The large number of visual aids such as figures, flow graphs and flow charts makes the mathematical topic easy to understand. In addition, the numerous examples and the set of C-language programs (a supplement to the book) help greatly in understanding the theory and

Graph theory favorite conjectures and open problems 1

CERN Document Server

Hedetniemi, Stephen; Larson, Craig

2016-01-01

This is the first in a series of volumes, which provide an extensive overview of conjectures and open problems in graph theory. The readership of each volume is geared toward graduate students who may be searching for research ideas. However, the well-established mathematician will find the overall exposition engaging and enlightening. Each chapter, presented in a story-telling style, includes more than a simple collection of results on a particular topic. Each contribution conveys the history, evolution, and techniques used to solve the authors’ favorite conjectures and open problems, enhancing the reader’s overall comprehension and enthusiasm. The editors were inspired to create these volumes by the popular and well attended special sessions, entitled “My Favorite Graph Theory Conjectures," which were held at the winter AMS/MAA Joint Meeting in Boston (January, 2012), the SIAM Conference on Discrete Mathematics in Halifax (June,2012) and the winter AMS/MAA Joint meeting in Baltimore(January, 2014). In...

Generalized connectivity of graphs

CERN Document Server

Li, Xueliang

2016-01-01

Noteworthy results, proof techniques, open problems and conjectures in generalized (edge-) connectivity are discussed in this book. Both theoretical and practical analyses for generalized (edge-) connectivity of graphs are provided. Topics covered in this book include: generalized (edge-) connectivity of graph classes, algorithms, computational complexity, sharp bounds, Nordhaus-Gaddum-type results, maximum generalized local connectivity, extremal problems, random graphs, multigraphs, relations with the Steiner tree packing problem and generalizations of connectivity. This book enables graduate students to understand and master a segment of graph theory and combinatorial optimization. Researchers in graph theory, combinatorics, combinatorial optimization, probability, computer science, discrete algorithms, complexity analysis, network design, and the information transferring models will find this book useful in their studies.

Cross Coursing in Mathematics: Physical Modelling in Differential Equations Crossing to Discrete Dynamical Systems

Science.gov (United States)

Winkel, Brian

2012-01-01

We give an example of cross coursing in which a subject or approach in one course in undergraduate mathematics is used in a completely different course. This situation crosses falling body modelling in an upper level differential equations course into a modest discrete dynamical systems unit of a first-year mathematics course. (Contains 1 figure.)

Graph-theoretic analysis of discrete-phase-space states for condition change detection and quantification of information

Science.gov (United States)

Hively, Lee M.

2014-09-16

Data collected from devices and human condition may be used to forewarn of critical events such as machine/structural failure or events from brain/heart wave data stroke. By monitoring the data, and determining what values are indicative of a failure forewarning, one can provide adequate notice of the impending failure in order to take preventive measures. This disclosure teaches a computer-based method to convert dynamical numeric data representing physical objects (unstructured data) into discrete-phase-space states, and hence into a graph (structured data) for extraction of condition change.

Differential-discrete mathematical model of two phase flow heat exchanger

International Nuclear Information System (INIS)

Debeljkovic, D.Lj.; Zitek, Pavel; Simeunovic, G.; Inard, Christian

2007-01-01

A dynamic thermal-hydraulic mathematical model of evaporator dynamics of a once - through sub critical steam generator is derived and presented. This model allows the investigation of evaporator dynamics including its transients responses. The evaporator was considered as a part of three-section (economizer, evaporator and super-heater) model with time varying phase boundaries and is described by a set of linearized discrete - difference equations which, with some other algebraic equations, constitutes a closed system of equations possible for exact computer solution. This model has been derived upon the fundamental equations of mass, energy and momentum balance. For the first time, a discrete differential approach has been applied in order to investigate such complex, two phase processes. Namely, this approach allows one to escape from the model of this process usually described by a set of partial differential equations and enables one, using this method, to simulate evaporators dynamics in an extraordinarily simple way. In current literature this approach is sometimes called physical discretization. (author)

Labeled Embedding Of (n, n-2-Graphs In Their Complements

Directory of Open Access Journals (Sweden)

Tahraoui M.-A.

2017-11-01

Full Text Available Graph packing generally deals with unlabeled graphs. In [4], the authors have introduced a new variant of the graph packing problem, called the labeled packing of a graph. This problem has recently been studied on trees [M.A. Tahraoui, E. Duchêne and H. Kheddouci, Labeled 2-packings of trees, Discrete Math. 338 (2015 816-824] and cycles [E. Duchˆene, H. Kheddouci, R.J. Nowakowski and M.A. Tahraoui, Labeled packing of graphs, Australas. J. Combin. 57 (2013 109-126]. In this note, we present a lower bound on the labeled packing number of any (n, n − 2-graph into Kn. This result improves the bound given by Woźniak in [Embedding graphs of small size, Discrete Appl. Math. 51 (1994 233-241].

GENERAL TASKS OF MATHEMATICAL EDUCATION DEVELOPMENT

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V. A. Testov

2014-01-01

Full Text Available The paper discusses basic implementation aspects of the Mathematical Education Development Concept, adopted by the Russian Government in 2013. According to the above document, the main problems of mathematical education include: low motivation of secondary and higher school students for studying the discipline, resulted from underestimation of mathematical knowledge; and outdated educational content, overloaded by technical elements. In the author’s opinion, a number of important new mathematical fields, developed over the last years, - the graph theory, discrete mathematics, encoding theory, fractal geometry, etc – have a large methodological and applied educational potential. However, these new subdisciplines have very little representation both in the secondary and higher school mathematical curricula. As a solution for overcoming the gap between the latest scientific achievements and pedagogical practices, the author recommends integration of the above mentioned mathematical disciplines in educational curricula instead of some outdated technical issues. In conclusion, the paper emphasizes the need for qualified mathematical teachers’ training for solving the problems of students’ motivation development and content updates.

Recognition of fractal graphs

NARCIS (Netherlands)

Perepelitsa, VA; Sergienko, [No Value; Kochkarov, AM

1999-01-01

Definitions of prefractal and fractal graphs are introduced, and they are used to formulate mathematical models in different fields of knowledge. The topicality of fractal-graph recognition from the point of view, of fundamental improvement in the efficiency of the solution of algorithmic problems

RNA Secondary Structure Prediction by Using Discrete Mathematics: An Interdisciplinary Research Experience for Undergraduate Students

Science.gov (United States)

Ellington, Roni; Wachira, James

2010-01-01

The focus of this Research Experience for Undergraduates (REU) project was on RNA secondary structure prediction by using a lattice walk approach. The lattice walk approach is a combinatorial and computational biology method used to enumerate possible secondary structures and predict RNA secondary structure from RNA sequences. The method uses discrete mathematical techniques and identifies specified base pairs as parameters. The goal of the REU was to introduce upper-level undergraduate students to the principles and challenges of interdisciplinary research in molecular biology and discrete mathematics. At the beginning of the project, students from the biology and mathematics departments of a mid-sized university received instruction on the role of secondary structure in the function of eukaryotic RNAs and RNA viruses, RNA related to combinatorics, and the National Center for Biotechnology Information resources. The student research projects focused on RNA secondary structure prediction on a regulatory region of the yellow fever virus RNA genome and on an untranslated region of an mRNA of a gene associated with the neurological disorder epilepsy. At the end of the project, the REU students gave poster and oral presentations, and they submitted written final project reports to the program director. The outcome of the REU was that the students gained transferable knowledge and skills in bioinformatics and an awareness of the applications of discrete mathematics to biological research problems. PMID:20810968

RNA secondary structure prediction by using discrete mathematics: an interdisciplinary research experience for undergraduate students.

Science.gov (United States)

Ellington, Roni; Wachira, James; Nkwanta, Asamoah

2010-01-01

The focus of this Research Experience for Undergraduates (REU) project was on RNA secondary structure prediction by using a lattice walk approach. The lattice walk approach is a combinatorial and computational biology method used to enumerate possible secondary structures and predict RNA secondary structure from RNA sequences. The method uses discrete mathematical techniques and identifies specified base pairs as parameters. The goal of the REU was to introduce upper-level undergraduate students to the principles and challenges of interdisciplinary research in molecular biology and discrete mathematics. At the beginning of the project, students from the biology and mathematics departments of a mid-sized university received instruction on the role of secondary structure in the function of eukaryotic RNAs and RNA viruses, RNA related to combinatorics, and the National Center for Biotechnology Information resources. The student research projects focused on RNA secondary structure prediction on a regulatory region of the yellow fever virus RNA genome and on an untranslated region of an mRNA of a gene associated with the neurological disorder epilepsy. At the end of the project, the REU students gave poster and oral presentations, and they submitted written final project reports to the program director. The outcome of the REU was that the students gained transferable knowledge and skills in bioinformatics and an awareness of the applications of discrete mathematics to biological research problems.

Mathematical methods in biology and neurobiology

CERN Document Server

Jost, Jürgen

2014-01-01

Mathematical models can be used to meet many of the challenges and opportunities offered by modern biology. The description of biological phenomena requires a range of mathematical theories. This is the case particularly for the emerging field of systems biology. Mathematical Methods in Biology and Neurobiology introduces and develops these mathematical structures and methods in a systematic manner. It studies:   • discrete structures and graph theory • stochastic processes • dynamical systems and partial differential equations • optimization and the calculus of variations.   The biological applications range from molecular to evolutionary and ecological levels, for example:   • cellular reaction kinetics and gene regulation • biological pattern formation and chemotaxis • the biophysics and dynamics of neurons • the coding of information in neuronal systems • phylogenetic tree reconstruction • branching processes and population genetics • optimal resource allocation • sexual recombi...

Comparison of university students’ understanding of graphs in different contexts

Directory of Open Access Journals (Sweden)

Maja Planinic

2013-07-01

Full Text Available This study investigates university students’ understanding of graphs in three different domains: mathematics, physics (kinematics, and contexts other than physics. Eight sets of parallel mathematics, physics, and other context questions about graphs were developed. A test consisting of these eight sets of questions (24 questions in all was administered to 385 first year students at University of Zagreb who were either prospective physics or mathematics teachers or prospective physicists or mathematicians. Rasch analysis of data was conducted and linear measures for item difficulties were obtained. Average difficulties of items in three domains (mathematics, physics, and other contexts and over two concepts (graph slope, area under the graph were computed and compared. Analysis suggests that the variation of average difficulty among the three domains is much smaller for the concept of graph slope than for the concept of area under the graph. Most of the slope items are very close in difficulty, suggesting that students who have developed sufficient understanding of graph slope in mathematics are generally able to transfer it almost equally successfully to other contexts. A large difference was found between the difficulty of the concept of area under the graph in physics and other contexts on one side and mathematics on the other side. Comparison of average difficulty of the three domains suggests that mathematics without context is the easiest domain for students. Adding either physics or other context to mathematical items generally seems to increase item difficulty. No significant difference was found between the average item difficulty in physics and contexts other than physics, suggesting that physics (kinematics remains a difficult context for most students despite the received instruction on kinematics in high school.

Mathematical Modeling of Contact Problems of Elasticity Theory with Unilateral Discrete Contact

Directory of Open Access Journals (Sweden)

I. V. Stankevich

2015-01-01

Full Text Available Development and operation of modern machinery and latest technology require reliable estimates of the strength characteristics of the critical elements of structures and technological equipment under the impact of high-intensity thermomechanical loading, accompanied, as a rule, by complex contact interaction. Mathematical modeling of stress-strain state of such parts and components in the contact area, based on adequate mathematical models, modern numerical methods and efficient algorithms that implement the direct determination of displacement fields, strains and stresses, is the main tool that allows fast acquisition of data required for the calculations of strength and durability. The paper considers an algorithm for constructing the numerical solution of the contact problem of elasticity theory in relation to the body, which has an obvious one-sided discrete contact interaction with an elastic half-space. The proposed algorithm is specially designed to have a correction of the tangential forces at discrete contact points, allowing us to achieve sufficiently accurate implementation of the adopted law of friction. The algorithm is embedded in a general finite element technology, with which the application code is generated. Numerical study of discrete unilateral contact interaction of an elastic plate and a rigid half-space showed a high efficiency of the developed algorithm and the application code that implements it.

Tracing the Construction of Mathematical Activity with an Advanced Graphing Calculator to Understand the Roles of Technology Developers, Teachers and Students

Science.gov (United States)

Hillman, Thomas

2014-01-01

This article examines mathematical activity with digital technology by tracing it from its development through its use in classrooms. Drawing on material-semiotic approaches from the field of Science and Technology Studies, it examines the visions of mathematical activity that developers had for an advanced graphing calculator. It then follows the…

A Median-Type Condition for Graph Tiling

Czech Academy of Sciences Publication Activity Database

Piguet, Diana; Saumell, Maria

2017-01-01

Roč. 61, August (2017), s. 979-985 ISSN 1571-0653 R&D Projects: GA ČR GJ16-07822Y Grant - others:GA MŠk(CZ) LO1506 Institutional support: RVO:67985807 Keywords : extremal graph theory * graph tiling * regularity lemma * LP-duality Subject RIV: BA - General Mathematics OBOR OECD: Pure mathematics

ANALYSIS OF SUBJECT DISCRETE MATHEMATICS PARTS AND PROPOSAL OF E-COURSE MODEL FOLLOWING PETRI NETS FOR INFORMATICS EDUCATION

Directory of Open Access Journals (Sweden)

TURČÁNI, Milan

2013-03-01

Full Text Available Nowadays, quality Mathematical basis - Informatics is an inherent part of study. Mathematical basis is provided by Discrete Mathematics that is taught as a compulsory subject in stated study program in the Department of Mathematics. Authors clarify significance and importance of simple thematic units of subject Discrete Mathematics in teaching technical-system subjects in study programme Applied Informatics. Mentioned subject is being taught in first year of University study and knowledge that students acquire during the study of this course are the "cornerstone" for their further development in technical-system study. Justness and importance of individual topics were analysed based on the evaluation of questionnaires, in which pedagogues teaching professional IT subjects alloted weighted coefficients to individual thematic units. Weighted coefficients were alloted based on the significance of the given topic of the subject Discrete Math, with regard to the IT subject they are teaching. Upon designing the e-course, experience with the creation of linear and branch teaching software were used. For the simulation of the transition of students through individual lessons as well as the whole course, authors employed the method of the teaching process simulation using Petri nets.

A seminar on graph theory

CERN Document Server

Harary, Frank

2015-01-01

Presented in 1962-63 by experts at University College, London, these lectures offer a variety of perspectives on graph theory. Although the opening chapters form a coherent body of graph theoretic concepts, this volume is not a text on the subject but rather an introduction to the extensive literature of graph theory. The seminar's topics are geared toward advanced undergraduate students of mathematics.Lectures by this volume's editor, Frank Harary, include ""Some Theorems and Concepts of Graph Theory,"" ""Topological Concepts in Graph Theory,"" ""Graphical Reconstruction,"" and other introduc

Teaching Proofs and Algorithms in Discrete Mathematics with Online Visual Logic Puzzles

Science.gov (United States)

Cigas, John; Hsin, Wen-Jung

2005-01-01

Visual logic puzzles provide a fertile environment for teaching multiple topics in discrete mathematics. Many puzzles can be solved by the repeated application of a small, finite set of strategies. Explicitly reasoning from a strategy to a new puzzle state illustrates theorems, proofs, and logic principles. These provide valuable, concrete…

Investigating a Link between Pre-Calculus Students' Uses of Graphing Calculators and Their Understanding of Mathematical Symbols

Science.gov (United States)

Kenney, Rachael H.

2014-01-01

This study examined ways in which students make use of a graphing calculator and how use relates to comfort and understanding with mathematical symbols. Analysis involved examining students' words and actions in problem solving to identify evidence of algebraic insight. Findings suggest that some symbols and symbolic structures had strong…

Graphing and Percentage Applications Using the Personal Computer.

Science.gov (United States)

Innes, Jay

1985-01-01

The paper describes how "IBM Graphing Assistant" and "Apple Softgraph" can foster a multifaceted approach to application of mathematical concepts and how a survey can be undertaken using the computer as word processor, data bank, and source of visual displays. Mathematical skills reinforced include estimating, rounding, graphing, and solving…

Phase-modified CTQW unable to distinguish strongly regular graphs efficiently

International Nuclear Information System (INIS)

Mahasinghe, A; Wijerathna, J K; Izaac, J A; Wang, J B

2015-01-01

Various quantum walk-based algorithms have been developed, aiming to distinguish non-isomorphic graphs with polynomial scaling, within both the discrete-time quantum walk (DTQW) and continuous-time quantum walk (CTQW) frameworks. Whilst both the single-particle DTQW and CTQW have failed to distinguish non-isomorphic strongly regular graph families (prompting the move to multi-particle graph isomorphism (GI) algorithms), the single-particle DTQW has been successfully modified by the introduction of a phase factor to distinguish a wide range of graphs in polynomial time. In this paper, we prove that an analogous phase modification to the single particle CTQW does not have the same distinguishing power as its discrete-time counterpart, in particular it cannot distinguish strongly regular graphs with the same family parameters with the same efficiency. (paper)

Interaction graphs

DEFF Research Database (Denmark)

Seiller, Thomas

2016-01-01

Interaction graphs were introduced as a general, uniform, construction of dynamic models of linear logic, encompassing all Geometry of Interaction (GoI) constructions introduced so far. This series of work was inspired from Girard's hyperfinite GoI, and develops a quantitative approach that should...... be understood as a dynamic version of weighted relational models. Until now, the interaction graphs framework has been shown to deal with exponentials for the constrained system ELL (Elementary Linear Logic) while keeping its quantitative aspect. Adapting older constructions by Girard, one can clearly define...... "full" exponentials, but at the cost of these quantitative features. We show here that allowing interpretations of proofs to use continuous (yet finite in a measure-theoretic sense) sets of states, as opposed to earlier Interaction Graphs constructions were these sets of states were discrete (and finite...

Edge Cover Domination in Mangoldt Graph

African Journals Online (AJOL)

Bheema

Department of Applied Mathematics, Y.V. University, Kadapa, Andhra Pradesh, India. 2. Department of Mathematics, Sri Padmavati Mahila University, Tirupati, ...... arithmetic graphs, Ph.D Thesis, Sri Venkateswara University, Tirupati, India.

Geometry, structure and randomness in combinatorics

CERN Document Server

Nešetřil, Jaroslav; Pellegrini, Marco

2014-01-01

This book collects some surveys on current trends in discrete mathematics and discrete geometry. The areas covered include:  graph representations, structural graphs theory, extremal graph theory, Ramsey theory and constrained satisfaction problems.

Test bank for precalculus functions & graphs

CERN Document Server

Kolman, Bernard; Levitan, Michael L

1984-01-01

Test Bank for Precalculus: Functions & Graphs is a supplementary material for the text, Precalculus: Functions & Graphs. The book is intended for use by mathematics teachers.The book contains standard tests for each chapter in the textbook. Each set of test focuses on gauging the level of knowledge the student has achieved during the course. The answers for each chapter test and the final exam are found at the end of the book.Mathematics teachers teaching calculus will find the book extremely useful.

Quantum information processing with graph states

International Nuclear Information System (INIS)

Schlingemann, Dirk-Michael

2005-04-01

Graph states are multiparticle states which are associated with graphs. Each vertex of the graph corresponds to a single system or particle. The links describe quantum correlations (entanglement) between pairs of connected particles. Graph states were initiated independently by two research groups: On the one hand, graph states were introduced by Briegel and Raussendorf as a resource for a new model of one-way quantum computing, where algorithms are implemented by a sequence of measurements at single particles. On the other hand, graph states were developed by the author of this thesis and ReinhardWerner in Braunschweig, as a tool to build quantum error correcting codes, called graph codes. The connection between the two approaches was fully realized in close cooperation of both research groups. This habilitation thesis provides a survey of the theory of graph codes, focussing mainly, but not exclusively on the author's own research work. We present the theoretical and mathematical background for the analysis of graph codes. The concept of one-way quantum computing for general graph states is discussed. We explicitly show how to realize the encoding and decoding device of a graph code on a one-way quantum computer. This kind of implementation is to be seen as a mathematical description of a quantum memory device. In addition to that, we investigate interaction processes, which enable the creation of graph states on very large systems. Particular graph states can be created, for instance, by an Ising type interaction between next neighbor particles which sits at the points of an infinitely extended cubic lattice. Based on the theory of quantum cellular automata, we give a constructive characterization of general interactions which create a translationally invariant graph state. (orig.)

The conceptual basis of mathematics in cardiology: (I) algebra, functions and graphs.

Science.gov (United States)

Bates, Jason H T; Sobel, Burton E

2003-02-01

This is the first in a series of four articles developed for the readers of. Without language ideas cannot be articulated. What may not be so immediately obvious is that they cannot be formulated either. One of the essential languages of cardiology is mathematics. Unfortunately, medical education does not emphasize, and in fact, often neglects empowering physicians to think mathematically. Reference to statistics, conditional probability, multicompartmental modeling, algebra, calculus and transforms is common but often without provision of genuine conceptual understanding. At the University of Vermont College of Medicine, Professor Bates developed a course designed to address these deficiencies. The course covered mathematical principles pertinent to clinical cardiovascular and pulmonary medicine and research. It focused on fundamental concepts to facilitate formulation and grasp of ideas. This series of four articles was developed to make the material available for a wider audience. The articles will be published sequentially in Coronary Artery Disease. Beginning with fundamental axioms and basic algebraic manipulations they address algebra, function and graph theory, real and complex numbers, calculus and differential equations, mathematical modeling, linear system theory and integral transforms and statistical theory. The principles and concepts they address provide the foundation needed for in-depth study of any of these topics. Perhaps of even more importance, they should empower cardiologists and cardiovascular researchers to utilize the language of mathematics in assessing the phenomena of immediate pertinence to diagnosis, pathophysiology and therapeutics. The presentations are interposed with queries (by Coronary Artery Disease, abbreviated as CAD) simulating the nature of interactions that occurred during the course itself. Each article concludes with one or more examples illustrating application of the concepts covered to cardiovascular medicine and

A first course in graph theory and combinatorics

CERN Document Server

Cioabă, Sebastian M

2009-01-01

The concept of a graph is fundamental in mathematics since it conveniently encodes diverse relations and facilitates combinatorial analysis of many complicated counting problems. In this book, the authors have traced the origins of graph theory from its humble beginnings of recreational mathematics to its modern setting for modeling communication networks as is evidenced by the World Wide Web graph used by many Internet search engines. This book is an introduction to graph theory and combinatorial analysis. It is based on courses given by the second author at Queen's University at Kingston, Ontario, Canada between 2002 and 2008. The courses were aimed at students in their final year of their undergraduate program.

A Characterization of 2-Tree Probe Interval Graphs

Directory of Open Access Journals (Sweden)

Brown David E.

2014-08-01

Full Text Available A graph is a probe interval graph if its vertices correspond to some set of intervals of the real line and can be partitioned into sets P and N so that vertices are adjacent if and only if their corresponding intervals intersect and at least one belongs to P. We characterize the 2-trees which are probe interval graphs and extend a list of forbidden induced subgraphs for such graphs created by Pržulj and Corneil in [2-tree probe interval graphs have a large obstruction set, Discrete Appl. Math. 150 (2005 216-231

On the diameter of dot-critical graphs

Directory of Open Access Journals (Sweden)

Doost Ali Mojdeh

2009-01-01

Full Text Available A graph G is \\(k\\-dot-critical (totaly \\(k\\-dot-critical if \\(G\\ is dot-critical (totaly dot-critical and the domination number is \\(k\\. In the paper [T. Burtona, D. P. Sumner, Domination dot-critical graphs, Discrete Math, 306 (2006, 11-18] the following question is posed: What are the best bounds for the diameter of a \\(k\\-dot-critical graph and a totally \\(k\\-dot-critical graph \\(G\\ with no critical vertices for \\(k \\geq 4\\? We find the best bound for the diameter of a \\(k\\-dot-critical graph, where \\(k \\in\\{4,5,6\\}\\ and we give a family of \\(k\\-dot-critical graphs (with no critical vertices with sharp diameter \\(2k-3\\ for even \\(k \\geq 4\\.

Discrete mathematics, formal methods, the Z schema and the software life cycle

Science.gov (United States)

Bown, Rodney L.

1991-01-01

The proper role and scope for the use of discrete mathematics and formal methods in support of engineering the security and integrity of components within deployed computer systems are discussed. It is proposed that the Z schema can be used as the specification language to capture the precise definition of system and component interfaces. This can be accomplished with an object oriented development paradigm.

The Use of Graphing Technology to Promote Transfer of Learning: the Interpretation of Graphs in Physics.

Science.gov (United States)

Nichols, Jeri Ann

This study examined the relationship between mathematics background and performance on graph-related problems in physics before and after instruction on the graphical analysis of motion and several microcomputer-based laboratory experiences. Students identified as either having or not having a graphing technology enhanced precalculus mathematics background were further categorized into one of four groups according to mathematics placement at the university. The performances of these groups were compared to identity differences. Pre- and Post-test data were collected from 589 students and 312 students during Autumn Quarter 1990 and Winter Quarter 1991 respectively. Background information was collected from each student. Significant differences were found between students with the technology enhanced mathematics background and those without when considering the entire populations both quarters. The students with the technology background were favored Autumn quarter and students without the technology background were favored Winter quarter. However, the entire population included an underrepresentation of students at the highest and lowest placements; hence, these were eliminated from the analyses. No significant differences were found between the technology/no technology groups after the elimination of the underrepresented groups. All categories of students increased their mean scores from pretest to post-test; the average increase was 8.23 points Autumn Quarter and 11.41 points Winter Quarter. Males consistently outperformed females on both the pretest and the post-test Autumn 1990. All students found questions involving the concept of acceleration more difficult than questions involving velocity or distance. Questions requiring students to create graphs were more difficult than questions requiring students to interpret graphs. Further research involving a qualitative component is recommended to identify the specific skills students use when solving graph

Surviving Rates of Graphs with Bounded Treewidth for the Firefighter Problem

DEFF Research Database (Denmark)

Cai, Leizhen; Cheng, Yongxi; Verbin, Elad

2010-01-01

The firefighter problem is the following discrete-time game on a graph. Initially, a fire starts at a vertex of the graph. In each round, a firefighter protects one vertex not yet on fire, and then the fire spreads to all unprotected neighbors of the vertices on fire. The objective of the firefig...... of Cai and Wang [SIAM J. Discrete Math., 23 (2009), pp. 1814-1826] in affirmative....

A mathematical approach for evaluating Markov models in continuous time without discrete-event simulation.

Science.gov (United States)

van Rosmalen, Joost; Toy, Mehlika; O'Mahony, James F

2013-08-01

Markov models are a simple and powerful tool for analyzing the health and economic effects of health care interventions. These models are usually evaluated in discrete time using cohort analysis. The use of discrete time assumes that changes in health states occur only at the end of a cycle period. Discrete-time Markov models only approximate the process of disease progression, as clinical events typically occur in continuous time. The approximation can yield biased cost-effectiveness estimates for Markov models with long cycle periods and if no half-cycle correction is made. The purpose of this article is to present an overview of methods for evaluating Markov models in continuous time. These methods use mathematical results from stochastic process theory and control theory. The methods are illustrated using an applied example on the cost-effectiveness of antiviral therapy for chronic hepatitis B. The main result is a mathematical solution for the expected time spent in each state in a continuous-time Markov model. It is shown how this solution can account for age-dependent transition rates and discounting of costs and health effects, and how the concept of tunnel states can be used to account for transition rates that depend on the time spent in a state. The applied example shows that the continuous-time model yields more accurate results than the discrete-time model but does not require much computation time and is easily implemented. In conclusion, continuous-time Markov models are a feasible alternative to cohort analysis and can offer several theoretical and practical advantages.

On the partition dimension of two-component graphs

Indian Academy of Sciences (India)

D O Haryeni

2017-11-17

Nov 17, 2017 ... Partition dimension; disconnected graph; component. 2010 Mathematics Subject Classification. 05C12, 05C15. 1. Introduction. The study of the partition dimension for graphs was initiated by Chartrand et al. [2] aimed at finding a new way to solve the problem in metric dimensions of graphs. Many results.

The toughness of split graphs

NARCIS (Netherlands)

Woeginger, G.J.

1998-01-01

In this short note we argue that the toughness of split graphs can be computed in polynomial time. This solves an open problem from a recent paper by Kratsch et al. (Discrete Math. 150 (1996) 231–245).

Central limit theorems for large graphs: Method of quantum decomposition

International Nuclear Information System (INIS)

Hashimoto, Yukihiro; Hora, Akihito; Obata, Nobuaki

2003-01-01

A new method is proposed for investigating spectral distribution of the combinatorial Laplacian (adjacency matrix) of a large regular graph on the basis of quantum decomposition and quantum central limit theorem. General results are proved for Cayley graphs of discrete groups and for distance-regular graphs. The Coxeter groups and the Johnson graphs are discussed in detail by way of illustration. In particular, the limit distributions obtained from the Johnson graphs are characterized by the Meixner polynomials which form a one-parameter deformation of the Laguerre polynomials

Lectures on financial mathematics discrete asset pricing

CERN Document Server

Anderson, Greg

2010-01-01

This is a short book on the fundamental concepts of the no-arbitrage theory of pricing financial derivatives. Its scope is limited to the general discrete setting of models for which the set of possible states is finite and so is the set of possible trading times--this includes the popular binomial tree model. This setting has the advantage of being fairly general while not requiring a sophisticated understanding of analysis at the graduate level. Topics include understanding the several variants of "arbitrage", the fundamental theorems of asset pricing in terms of martingale measures, and applications to forwards and futures. The authors' motivation is to present the material in a way that clarifies as much as possible why the often confusing basic facts are true. Therefore the ideas are organized from a mathematical point of view with the emphasis on understanding exactly what is under the hood and how it works. Every effort is made to include complete explanations and proofs, and the reader is encouraged t...

Graph theory

CERN Document Server

Diestel, Reinhard

2017-01-01

This standard textbook of modern graph theory, now in its fifth edition, combines the authority of a classic with the engaging freshness of style that is the hallmark of active mathematics. It covers the core material of the subject with concise yet reliably complete proofs, while offering glimpses of more advanced methods in each field by one or two deeper results, again with proofs given in full detail. The book can be used as a reliable text for an introductory course, as a graduate text, and for self-study. From the reviews: “This outstanding book cannot be substituted with any other book on the present textbook market. It has every chance of becoming the standard textbook for graph theory.”Acta Scientiarum Mathematiciarum “Deep, clear, wonderful. This is a serious book about the heart of graph theory. It has depth and integrity. ”Persi Diaconis & Ron Graham, SIAM Review “The book has received a very enthusiastic reception, which it amply deserves. A masterly elucidation of modern graph theo...

Essential spectra of difference operators on Zn-periodic graphs

International Nuclear Information System (INIS)

Rabinovich, Vladimir S; Roch, Steffen

2007-01-01

Let (X, ρ) be a discrete metric space. We suppose that the group Z n acts freely on X and that the number of orbits of X with respect to this action is finite. Then we call X a Z n -periodic discrete metric space. We examine the Fredholm property and essential spectra of band-dominated operators on l p (X) when 1 n and their limit operators. In the case where X is the set of vertices of a combinatorial graph, the graph structure defines a Schroedinger operator on l p (X) in a natural way. We illustrate our approach by determining the essential spectra of Schroedinger operators with slowly oscillating potential both on zig-zag and on hexagonal graphs, the latter being related to nano-structures

Discrete algorithmic mathematics

CERN Document Server

Maurer, Stephen B

2005-01-01

The exposition is self-contained, complemented by diverse exercises and also accompanied by an introduction to mathematical reasoning … this book is an excellent textbook for a one-semester undergraduate course and it includes a lot of additional material to choose from.-EMS, March 2006In a textbook, it is necessary to select carefully the statements and difficulty of the problems … in this textbook, this is fully achieved … This review considers this book an excellent one.-The Mathematical Gazette, March 2006

On the number of subgraphs of the Barabasi-Albert random graph

Energy Technology Data Exchange (ETDEWEB)

Ryabchenko, Aleksandr A; Samosvat, Egor A [Moscow Institute of Physics and Technology (State University), Dolgoprudnyi, Moscow Region, Russian Frderation (Russian Federation)

2012-06-30

We study a model of a random graph of the type of the Barabasi-Albert preferential attachment model. We develop a technique that makes it possible to estimate the mathematical expectation for a fairly wide class of random variables in the model under consideration. We use this technique to prove a theorem on the asymptotics of the mathematical expectation of the number of subgraphs isomorphic to a certain fixed graph in the random graphs of this model.

Two-colorable graph states with maximal Schmidt measure

International Nuclear Information System (INIS)

Severini, Simone

2006-01-01

The Schmidt measure was introduced by Eisert and Briegel for quantifying the degree of entanglement of multipartite quantum systems [J. Eisert, H.-J. Briegel, Phys. Rev. A 64 (2001) 22306]. For two-colorable graph states, the Schmidt measure is related to the spectrum of the associated graph. We observe that almost all two-colorable graph states have maximal Schmidt measure and we construct specific examples. By making appeal to a result of Ehrenfeucht et al. [A. Ehrenfeucht, T. Harju, G. Rozenberg, Discrete Math. 278 (2004) 45], we point out that the graph operations called local complementation and switching form a transitive group acting on the set of all graph states of a given dimension

Groups, graphs and random walks

CERN Document Server

Salvatori, Maura; Sava-Huss, Ecaterina

2017-01-01

An accessible and panoramic account of the theory of random walks on groups and graphs, stressing the strong connections of the theory with other branches of mathematics, including geometric and combinatorial group theory, potential analysis, and theoretical computer science. This volume brings together original surveys and research-expository papers from renowned and leading experts, many of whom spoke at the workshop 'Groups, Graphs and Random Walks' celebrating the sixtieth birthday of Wolfgang Woess in Cortona, Italy. Topics include: growth and amenability of groups; Schrödinger operators and symbolic dynamics; ergodic theorems; Thompson's group F; Poisson boundaries; probability theory on buildings and groups of Lie type; structure trees for edge cuts in networks; and mathematical crystallography. In what is currently a fast-growing area of mathematics, this book provides an up-to-date and valuable reference for both researchers and graduate students, from which future research activities will undoubted...

Cognitive and attitudinal predictors related to graphing achievement among pre-service elementary teachers

Science.gov (United States)

Szyjka, Sebastian P.

The purpose of this study was to determine the extent to which six cognitive and attitudinal variables predicted pre-service elementary teachers' performance on line graphing. Predictors included Illinois teacher education basic skills sub-component scores in reading comprehension and mathematics, logical thinking performance scores, as well as measures of attitudes toward science, mathematics and graphing. This study also determined the strength of the relationship between each prospective predictor variable and the line graphing performance variable, as well as the extent to which measures of attitude towards science, mathematics and graphing mediated relationships between scores on mathematics, reading, logical thinking and line graphing. Ninety-four pre-service elementary education teachers enrolled in two different elementary science methods courses during the spring 2009 semester at Southern Illinois University Carbondale participated in this study. Each subject completed five different instruments designed to assess science, mathematics and graphing attitudes as well as logical thinking and graphing ability. Sixty subjects provided copies of primary basic skills score reports that listed subset scores for both reading comprehension and mathematics. The remaining scores were supplied by a faculty member who had access to a database from which the scores were drawn. Seven subjects, whose scores could not be found, were eliminated from final data analysis. Confirmatory factor analysis (CFA) was conducted in order to establish validity and reliability of the Questionnaire of Attitude Toward Line Graphs in Science (QALGS) instrument. CFA tested the statistical hypothesis that the five main factor structures within the Questionnaire of Attitude Toward Statistical Graphs (QASG) would be maintained in the revised QALGS. Stepwise Regression Analysis with backward elimination was conducted in order to generate a parsimonious and precise predictive model. This

Graphs of groups on surfaces interactions and models

CERN Document Server

White, AT

2001-01-01

The book, suitable as both an introductory reference and as a text book in the rapidly growing field of topological graph theory, models both maps (as in map-coloring problems) and groups by means of graph imbeddings on sufaces. Automorphism groups of both graphs and maps are studied. In addition connections are made to other areas of mathematics, such as hypergraphs, block designs, finite geometries, and finite fields. There are chapters on the emerging subfields of enumerative topological graph theory and random topological graph theory, as well as a chapter on the composition of English

A generalization of zero divisor graphs associated to commutative ...

Indian Academy of Sciences (India)

M. Afkhami

2018-03-19

Mar 19, 2018 ... R . We also determine all isomorphic classes of finite commutative rings whose generalized zero divisor graphs have genus at most three. Keywords. Zero divisor graph; lower triangular matrix; genus; complete graph. 2010 Mathematics Subject Classification. 15B33, 05C10, 05C25, 05C45. 1. Introduction.

Chromatic Roots and Limits of Dense Graphs

Czech Academy of Sciences Publication Activity Database

Csikvári, P.; Frenkel, E.; Hladký, Jan; Hubai, T.

2017-01-01

Roč. 340, č. 5 (2017), s. 1129-1135 ISSN 0012-365X Institutional support: RVO:67985807 Keywords : chromatic root * graph limit * holomorphic moment Subject RIV: BA - General Mathematics OBOR OECD: Pure mathematics Impact factor: 0.639, year: 2016

Graphs, Ideal Flow, and the Transportation Network

OpenAIRE

Teknomo, Kardi

2016-01-01

This lecture discusses the mathematical relationship between network structure and network utilization of transportation network. Network structure means the graph itself. Network utilization represent the aggregation of trajectories of agents in using the network graph. I show the similarity and relationship between the structural pattern of the network and network utilization.

Mechatronic modeling and simulation using bond graphs

CERN Document Server

Das, Shuvra

2009-01-01

Introduction to Mechatronics and System ModelingWhat Is Mechatronics?What Is a System and Why Model Systems?Mathematical Modeling Techniques Used in PracticeSoftwareBond Graphs: What Are They?Engineering SystemsPortsGeneralized VariablesBond GraphsBasic Components in SystemsA Brief Note about Bond Graph Power DirectionsSummary of Bond Direction RulesDrawing Bond Graphs for Simple Systems: Electrical and MechanicalSimplification Rules for Junction StructureDrawing Bond Graphs for Electrical SystemsDrawing Bond Graphs for Mechanical SystemsCausalityDrawing Bond Graphs for Hydraulic and Electronic Components and SystemsSome Basic Properties and Concepts for FluidsBond Graph Model of Hydraulic SystemsElectronic SystemsDeriving System Equations from Bond GraphsSystem VariablesDeriving System EquationsTackling Differential CausalityAlgebraic LoopsSolution of Model Equations and Their InterpretationZeroth Order SystemsFirst Order SystemsSecond Order SystemTransfer Functions and Frequency ResponsesNumerical Solution ...

No hexavalent half-arc-transitive graphs of order twice a prime ...

Indian Academy of Sciences (India)

Mi-Mi Zhang

2018-03-19

Mar 19, 2018 ... Wang and Feng (Discrete. Math. 310 (2010) 1721–1724) proved that there exists no tetravalent half-arc-transitive graphs of order 2p2. In this paper, we extend this result to prove that no hexavalent half-arc-transitive graphs of order 2p2 exist. Keywords. Half-arc-transitive; bi-Cayley graph; vertex transitive; ...

Connections between the Sznajd model with general confidence rules and graph theory

Science.gov (United States)

Timpanaro, André M.; Prado, Carmen P. C.

2012-10-01

The Sznajd model is a sociophysics model that is used to model opinion propagation and consensus formation in societies. Its main feature is that its rules favor bigger groups of agreeing people. In a previous work, we generalized the bounded confidence rule in order to model biases and prejudices in discrete opinion models. In that work, we applied this modification to the Sznajd model and presented some preliminary results. The present work extends what we did in that paper. We present results linking many of the properties of the mean-field fixed points, with only a few qualitative aspects of the confidence rule (the biases and prejudices modeled), finding an interesting connection with graph theory problems. More precisely, we link the existence of fixed points with the notion of strongly connected graphs and the stability of fixed points with the problem of finding the maximal independent sets of a graph. We state these results and present comparisons between the mean field and simulations in Barabási-Albert networks, followed by the main mathematical ideas and appendices with the rigorous proofs of our claims and some graph theory concepts, together with examples. We also show that there is no qualitative difference in the mean-field results if we require that a group of size q>2, instead of a pair, of agreeing agents be formed before they attempt to convince other sites (for the mean field, this would coincide with the q-voter model).

The Harary index of a graph

CERN Document Server

Xu, Kexiang; Trinajstić, Nenad

2015-01-01

This is the first book to focus on the topological index, the Harary index, of a graph, including its mathematical properties, chemical applications and some related and attractive open problems. This book is dedicated to Professor Frank Harary (1921—2005), the grandmaster of graph theory and its applications. It has be written by experts in the field of graph theory and its applications. For a connected graph G, as an important distance-based topological index, the Harary index H(G) is defined as the sum of the reciprocals of the distance between any two unordered vertices of the graph G. In this book, the authors report on the newest results on the Harary index of a graph. These results mainly concern external graphs with respect to the Harary index; the relations to other topological indices; its properties and applications to pure graph theory and chemical graph theory; and two significant variants, i.e., additively and multiplicatively weighted Harary indices. In the last chapter, we present a number o...

Colour Mathematics: With Graphs and Numbers

Science.gov (United States)

LoPresto, Michael C.

2009-01-01

The different combinations involved in additive and subtractive colour mixing can often be difficult for students to remember. Using transmission graphs for filters of the primary colours and a numerical scheme to write out the relationships are good exercises in analytical thinking that can help students recall the combinations rather than just…

Generating Realistic Labelled, Weighted Random Graphs

Directory of Open Access Journals (Sweden)

Michael Charles Davis

2015-12-01

Full Text Available Generative algorithms for random graphs have yielded insights into the structure and evolution of real-world networks. Most networks exhibit a well-known set of properties, such as heavy-tailed degree distributions, clustering and community formation. Usually, random graph models consider only structural information, but many real-world networks also have labelled vertices and weighted edges. In this paper, we present a generative model for random graphs with discrete vertex labels and numeric edge weights. The weights are represented as a set of Beta Mixture Models (BMMs with an arbitrary number of mixtures, which are learned from real-world networks. We propose a Bayesian Variational Inference (VI approach, which yields an accurate estimation while keeping computation times tractable. We compare our approach to state-of-the-art random labelled graph generators and an earlier approach based on Gaussian Mixture Models (GMMs. Our results allow us to draw conclusions about the contribution of vertex labels and edge weights to graph structure.

On The Roman Domination Stable Graphs

Directory of Open Access Journals (Sweden)

Hajian Majid

2017-11-01

Full Text Available A Roman dominating function (or just RDF on a graph G = (V,E is a function f : V → {0, 1, 2} satisfying the condition that every vertex u for which f(u = 0 is adjacent to at least one vertex v for which f(v = 2. The weight of an RDF f is the value f(V (G = Pu2V (G f(u. The Roman domination number of a graph G, denoted by R(G, is the minimum weight of a Roman dominating function on G. A graph G is Roman domination stable if the Roman domination number of G remains unchanged under removal of any vertex. In this paper we present upper bounds for the Roman domination number in the class of Roman domination stable graphs, improving bounds posed in [V. Samodivkin, Roman domination in graphs: the class RUV R, Discrete Math. Algorithms Appl. 8 (2016 1650049].

Disease management research using event graphs.

Science.gov (United States)

Allore, H G; Schruben, L W

2000-08-01

Event Graphs, conditional representations of stochastic relationships between discrete events, simulate disease dynamics. In this paper, we demonstrate how Event Graphs, at an appropriate abstraction level, also extend and organize scientific knowledge about diseases. They can identify promising treatment strategies and directions for further research and provide enough detail for testing combinations of new medicines and interventions. Event Graphs can be enriched to incorporate and validate data and test new theories to reflect an expanding dynamic scientific knowledge base and establish performance criteria for the economic viability of new treatments. To illustrate, an Event Graph is developed for mastitis, a costly dairy cattle disease, for which extensive scientific literature exists. With only a modest amount of imagination, the methodology presented here can be seen to apply modeling to any disease, human, plant, or animal. The Event Graph simulation presented here is currently being used in research and in a new veterinary epidemiology course. Copyright 2000 Academic Press.

Precalculus Teachers' Perspectives on Using Graphing Calculators: An Example from One Curriculum

Science.gov (United States)

Karadeniz, Ilyas; Thompson, Denisse R.

2018-01-01

Graphing calculators are hand-held technological tools currently used in mathematics classrooms. Teachers' perspectives on using graphing calculators are important in terms of exploring what teachers think about using such technology in advanced mathematics courses, particularly precalculus courses. A descriptive intrinsic case study was conducted…

Introduction to graph theory

CERN Document Server

Wilson, Robin J

1985-01-01

Graph Theory has recently emerged as a subject in its own right, as well as being an important mathematical tool in such diverse subjects as operational research, chemistry, sociology and genetics. This book provides a comprehensive introduction to the subject.

Algorithmic mathematics

CERN Document Server

Hougardy, Stefan

2016-01-01

Algorithms play an increasingly important role in nearly all fields of mathematics. This book allows readers to develop basic mathematical abilities, in particular those concerning the design and analysis of algorithms as well as their implementation. It presents not only fundamental algorithms like the sieve of Eratosthenes, the Euclidean algorithm, sorting algorithms, algorithms on graphs, and Gaussian elimination, but also discusses elementary data structures, basic graph theory, and numerical questions. In addition, it provides an introduction to programming and demonstrates in detail how to implement algorithms in C++. This textbook is suitable for students who are new to the subject and covers a basic mathematical lecture course, complementing traditional courses on analysis and linear algebra. Both authors have given this "Algorithmic Mathematics" course at the University of Bonn several times in recent years.

Theoretical issues in quantum computing: Graph isomorphism, PageRank, and Hamiltonian determination

Science.gov (United States)

Rudinger, Kenneth Michael

This thesis explores several theoretical questions pertaining to quantum computing. First we examine several questions regarding multi-particle quantum random walk-based algorithms for the graph isomorphism problem. We find that there exists a non-trivial difference between continuous-time walks of one and two non-interacting particles as compared to non-interacting walks of three or more particles, in that the latter are able to distinguish many strongly regular graphs (SRGs), a class of graphs with many graph pairs that are difficult to distinguish. We demonstrate analytically where this distinguishing power comes from, and we show numerically that three-particle and four-particle non-interacting continuous-time walks can distinguish many pairs of strongly regular graphs. We additionally show that this distinguishing power, while it grows with particle number, is bounded, so that no continuous-time non-interacting walk of fixed particle number can distinguish all strongly regular graphs. We then investigate the relationship between continuous-time and discrete-time walks, in the context of the graph isomorphism problem. While it has been previously demonstrated numerically that discrete-time walks of non-interacting particles can distinguish some SRGs, we demonstrate where this distinguishing power comes from. We also show that while no continuous-time non-interacting walk of fixed particle number can distinguish SRGs, it remains a possibility that such a discrete-time walk could, leaving open the possibility of a non-trivial difference between discrete-time and continuous-time walks. The last piece of our work on graph isomorphism examines limitations on certain kinds of continuous-time walk-based algorithms for distinguishing graphs. We show that a very general class of continuous-time walk algorithms, with a broad class of allowable interactions, cannot distinguish all graphs. We next consider a previously-proposed quantum adiabatic algorithm for computing the

Discrete Calculus by Analogy

CERN Document Server

Izadi, F A; Bagirov, G

2009-01-01

With its origins stretching back several centuries, discrete calculus is now an increasingly central methodology for many problems related to discrete systems and algorithms. The topics covered here usually arise in many branches of science and technology, especially in discrete mathematics, numerical analysis, statistics and probability theory as well as in electrical engineering, but our viewpoint here is that these topics belong to a much more general realm of mathematics; namely calculus and differential equations because of the remarkable analogy of the subject to this branch of mathemati

The boundary value problem for discrete analytic functions

KAUST Repository

Skopenkov, Mikhail

2013-01-01

This paper is on further development of discrete complex analysis introduced by R.Isaacs, J.Ferrand, R.Duffin, and C.Mercat. We consider a graph lying in the complex plane and having quadrilateral faces. A function on the vertices is called discrete

Contracts for Cross-organizational Workflows as Timed Dynamic Condition Response Graphs

DEFF Research Database (Denmark)

Hildebrandt, Thomas; Mukkamala, Raghava Rao; Slaats, Tijs

2013-01-01

We conservatively extend the declarative Dynamic Condition Response (DCR) Graph process model, introduced in the PhD thesis of the second author, to allow for discrete time deadlines. We prove that safety and liveness properties can be verified by mapping finite timed DCR Graphs to finite state...

Discrete optimization in architecture extremely modular systems

CERN Document Server

Zawidzki, Machi

2017-01-01

This book is comprised of two parts, both of which explore modular systems: Pipe-Z (PZ) and Truss-Z (TZ), respectively. It presents several methods of creating PZ and TZ structures subjected to discrete optimization. The algorithms presented employ graph-theoretic and heuristic methods. The underlying idea of both systems is to create free-form structures using the minimal number of types of modular elements. PZ is more conceptual, as it forms single-branch mathematical knots with a single type of module. Conversely, TZ is a skeletal system for creating free-form pedestrian ramps and ramp networks among any number of terminals in space. In physical space, TZ uses two types of modules that are mirror reflections of each other. The optimization criteria discussed include: the minimal number of units, maximal adherence to the given guide paths, etc.

Pristine transfinite graphs and permissive electrical networks

CERN Document Server

Zemanian, Armen H

2001-01-01

A transfinite graph or electrical network of the first rank is obtained conceptually by connecting conventionally infinite graphs and networks together at their infinite extremities. This process can be repeated to obtain a hierarchy of transfiniteness whose ranks increase through the countable ordinals. This idea, which is of recent origin, has enriched the theories of graphs and networks with radically new constructs and research problems. The book provides a more accessible introduction to the subject that, though sacrificing some generality, captures the essential ideas of transfiniteness for graphs and networks. Thus, for example, some results concerning discrete potentials and random walks on transfinite networks can now be presented more concisely. Conversely, the simplifications enable the development of many new results that were previously unavailable. Topics and features: *A simplified exposition provides an introduction to transfiniteness for graphs and networks.*Various results for conventional g...

Mathematical biodescriptors of proteomics maps: background and applications.

Science.gov (United States)

Basak, Subhash C; Gute, Brian D

2008-05-01

This article reviews recent developments in the formulation and application of biodescriptors to characterize proteomics maps. Such biodescriptors can be derived by applying techniques from discrete mathematics (graph theory, linear algebra and information theory). This review focuses on the development of biodescriptors for proteomics maps derived from 2D gel electrophoresis. Preliminary results demonstrated that such descriptors have a reasonable ability to differentiate between proteomics patterns that result from exposure to closely related individual chemicals and complex mixtures, such as the jet fuel JP-8. Further research is required to evaluate the utility of these proteomics-based biodescriptors for drug discovery and predictive toxicology.

Quantum walks on quotient graphs

International Nuclear Information System (INIS)

Krovi, Hari; Brun, Todd A.

2007-01-01

A discrete-time quantum walk on a graph Γ is the repeated application of a unitary evolution operator to a Hilbert space corresponding to the graph. If this unitary evolution operator has an associated group of symmetries, then for certain initial states the walk will be confined to a subspace of the original Hilbert space. Symmetries of the original graph, given by its automorphism group, can be inherited by the evolution operator. We show that a quantum walk confined to the subspace corresponding to this symmetry group can be seen as a different quantum walk on a smaller quotient graph. We give an explicit construction of the quotient graph for any subgroup H of the automorphism group and illustrate it with examples. The automorphisms of the quotient graph which are inherited from the original graph are the original automorphism group modulo the subgroup H used to construct it. The quotient graph is constructed by removing the symmetries of the subgroup H from the original graph. We then analyze the behavior of hitting times on quotient graphs. Hitting time is the average time it takes a walk to reach a given final vertex from a given initial vertex. It has been shown in earlier work [Phys. Rev. A 74, 042334 (2006)] that the hitting time for certain initial states of a quantum walks can be infinite, in contrast to classical random walks. We give a condition which determines whether the quotient graph has infinite hitting times given that they exist in the original graph. We apply this condition for the examples discussed and determine which quotient graphs have infinite hitting times. All known examples of quantum walks with hitting times which are short compared to classical random walks correspond to systems with quotient graphs much smaller than the original graph; we conjecture that the existence of a small quotient graph with finite hitting times is necessary for a walk to exhibit a quantum speedup

On the number of subgraphs of the Barabási-Albert random graph

International Nuclear Information System (INIS)

Ryabchenko, Aleksandr A; Samosvat, Egor A

2012-01-01

We study a model of a random graph of the type of the Barabási-Albert preferential attachment model. We develop a technique that makes it possible to estimate the mathematical expectation for a fairly wide class of random variables in the model under consideration. We use this technique to prove a theorem on the asymptotics of the mathematical expectation of the number of subgraphs isomorphic to a certain fixed graph in the random graphs of this model.

Entanglement in coined quantum walks on regular graphs

International Nuclear Information System (INIS)

Carneiro, Ivens; Loo, Meng; Xu, Xibai; Girerd, Mathieu; Kendon, Viv; Knight, Peter L

2005-01-01

Quantum walks, both discrete (coined) and continuous time, form the basis of several recent quantum algorithms. Here we use numerical simulations to study the properties of discrete, coined quantum walks. We investigate the variation in the entanglement between the coin and the position of the particle by calculating the entropy of the reduced density matrix of the coin. We consider both dynamical evolution and asymptotic limits for coins of dimensions from two to eight on regular graphs. For low coin dimensions, quantum walks which spread faster (as measured by the mean square deviation of their distribution from uniform) also exhibit faster convergence towards the asymptotic value of the entanglement between the coin and particle's position. For high-dimensional coins, the DFT coin operator is more efficient at spreading than the Grover coin. We study the entanglement of the coin on regular finite graphs such as cycles, and also show that on complete bipartite graphs, a quantum walk with a Grover coin is always periodic with period four. We generalize the 'glued trees' graph used by Childs et al (2003 Proc. STOC, pp 59-68) to higher branching rate (fan out) and verify that the scaling with branching rate and with tree depth is polynomial

On characterizing terrain visibility graphs

Directory of Open Access Journals (Sweden)

William Evans

2015-06-01

Full Text Available A terrain is an $x$-monotone polygonal line in the $xy$-plane. Two vertices of a terrain are mutually visible if and only if there is no terrain vertex on or above the open line segment connecting them. A graph whose vertices represent terrain vertices and whose edges represent mutually visible pairs of terrain vertices is called a terrain visibility graph. We would like to find properties that are both necessary and sufficient for a graph to be a terrain visibility graph; that is, we would like to characterize terrain visibility graphs.Abello et al. [Discrete and Computational Geometry, 14(3:331--358, 1995] showed that all terrain visibility graphs are “persistent”. They showed that the visibility information of a terrain point set implies some ordering requirements on the slopes of the lines connecting pairs of points in any realization, and as a step towards showing sufficiency, they proved that for any persistent graph $M$ there is a total order on the slopes of the (pseudo lines in a generalized configuration of points whose visibility graph is $M$.We give a much simpler proof of this result by establishing an orientation to every triple of vertices, reflecting some slope ordering requirements that are consistent with $M$ being the visibility graph, and prove that these requirements form a partial order. We give a faster algorithm to construct a total order on the slopes. Our approach attempts to clarify the implications of the graph theoretic properties on the ordering of the slopes, and may be interpreted as defining properties on an underlying oriented matroid that we show is a restricted type of $3$-signotope.

Multiple graph regularized nonnegative matrix factorization

KAUST Repository

Wang, Jim Jing-Yan

2013-10-01

Non-negative matrix factorization (NMF) has been widely used as a data representation method based on components. To overcome the disadvantage of NMF in failing to consider the manifold structure of a data set, graph regularized NMF (GrNMF) has been proposed by Cai et al. by constructing an affinity graph and searching for a matrix factorization that respects graph structure. Selecting a graph model and its corresponding parameters is critical for this strategy. This process is usually carried out by cross-validation or discrete grid search, which are time consuming and prone to overfitting. In this paper, we propose a GrNMF, called MultiGrNMF, in which the intrinsic manifold is approximated by a linear combination of several graphs with different models and parameters inspired by ensemble manifold regularization. Factorization metrics and linear combination coefficients of graphs are determined simultaneously within a unified object function. They are alternately optimized in an iterative algorithm, thus resulting in a novel data representation algorithm. Extensive experiments on a protein subcellular localization task and an Alzheimer\\'s disease diagnosis task demonstrate the effectiveness of the proposed algorithm. © 2013 Elsevier Ltd. All rights reserved.

Discrete mathematics with applications

CERN Document Server

Koshy, Thomas

2003-01-01

This approachable text studies discrete objects and the relationsips that bind them. It helps students understand and apply the power of discrete math to digital computer systems and other modern applications. It provides excellent preparation for courses in linear algebra, number theory, and modern/abstract algebra and for computer science courses in data structures, algorithms, programming languages, compilers, databases, and computation.* Covers all recommended topics in a self-contained, comprehensive, and understandable format for students and new professionals * Emphasizes problem-solving techniques, pattern recognition, conjecturing, induction, applications of varying nature, proof techniques, algorithm development and correctness, and numeric computations* Weaves numerous applications into the text* Helps students learn by doing with a wealth of examples and exercises: - 560 examples worked out in detail - More than 3,700 exercises - More than 150 computer assignments - More than 600 writing projects*...

An original approach to the mathematical concept of graph from braid crafts

Directory of Open Access Journals (Sweden)

Albanese Veronica

2016-01-01

Full Text Available In previous researches we found that a community of Argentinean artisans models its own practices of braiding using graphs. Inspired by these findings, we designed an educational activity to introduce the concept of graphs. The study of graphs helps students to develop combinatorial and systematic thinking as well as skills to model reality and abstract and generalize patterns from particular situations. The tasks proposed aim to construct the concept of graphs, then identify characteristics that allow some graphs to be models of braids and finally use them to invent more graphs for new braids. The activity performed in a secondary school teachers’ educational course, had quite satisfactory results due to the number of braids invented and the small amount of mistakes made by the participants.

Process Modeling for Energy Usage in “Smart House” System with a Help of Markov Discrete Chain

Directory of Open Access Journals (Sweden)

Victor Kravets

2016-05-01

Full Text Available Method for evaluating economic efficiency of technical systems using discrete Markov chains modelling illustrated by the system of “Smart house”, consisting, for example, of the three independently functioning elements. Dynamic model of a random power consumption process in the form of a symmetrical state graph of heterogeneous discrete Markov chain is built. The corresponding mathematical model of a random Markov process of power consumption in the “smart house” system in recurrent matrix form is being developed. Technique of statistical determination of probability of random transition elements of the system and the corresponding to the transition probability matrix of the discrete inhomogeneous Markov chain are developed. Statistically determined random transitions of system elements power consumption and the corresponding distribution laws are introduced. The matrix of transition prices, expectations for the possible states of a system price transition and, eventually, the cost of Markov process of power consumption throughout the day.

Handbook on modelling for discrete optimization

CERN Document Server

Pitsoulis, Leonidas; Williams, H

2006-01-01

The primary objective underlying the Handbook on Modelling for Discrete Optimization is to demonstrate and detail the pervasive nature of Discrete Optimization. While its applications cut across an incredibly wide range of activities, many of the applications are only known to specialists. It is the aim of this handbook to correct this. It has long been recognized that "modelling" is a critically important mathematical activity in designing algorithms for solving these discrete optimization problems. Nevertheless solving the resultant models is also often far from straightforward. In recent years it has become possible to solve many large-scale discrete optimization problems. However, some problems remain a challenge, even though advances in mathematical methods, hardware, and software technology have pushed the frontiers forward. This handbook couples the difficult, critical-thinking aspects of mathematical modeling with the hot area of discrete optimization. It will be done in an academic handbook treatment...

An application of discrete mathematics in the design of an open pit mine

Energy Technology Data Exchange (ETDEWEB)

Caccetta, L.; Giannini, L.M.

1988-09-01

The determination of the 'optimum pit limit' of a mine is considered to be a fundamental problem in mine planning as it provides information which is essential in the evaluation of the economic potential of a mineral deposit, and in the formulation of long-, intermediate-, and short-range mine plans. A number of mathematical techniques have been proposed to solve this problem, some of the more elaborate ones posing considerable computational problems. In this paper we discuss the development and implementation of a graph-theoretic technique originally proposed by Lerchs and Grossman. Our implementation strategy involves the use of a dynamic programming technique to 'bound' the optimum. 19 refs., 4 figs.

Using graph theory for automated electric circuit solving

International Nuclear Information System (INIS)

Toscano, L; Stella, S; Milotti, E

2015-01-01

Graph theory plays many important roles in modern physics and in many different contexts, spanning diverse topics such as the description of scale-free networks and the structure of the universe as a complex directed graph in causal set theory. Graph theory is also ideally suited to describe many concepts in computer science. Therefore it is increasingly important for physics students to master the basic concepts of graph theory. Here we describe a student project where we develop a computational approach to electric circuit solving which is based on graph theoretic concepts. This highly multidisciplinary approach combines abstract mathematics, linear algebra, the physics of circuits, and computer programming to reach the ambitious goal of implementing automated circuit solving. (paper)

Generating random networks and graphs

CERN Document Server

Coolen, Ton; Roberts, Ekaterina

2017-01-01

This book supports researchers who need to generate random networks, or who are interested in the theoretical study of random graphs. The coverage includes exponential random graphs (where the targeted probability of each network appearing in the ensemble is specified), growth algorithms (i.e. preferential attachment and the stub-joining configuration model), special constructions (e.g. geometric graphs and Watts Strogatz models) and graphs on structured spaces (e.g. multiplex networks). The presentation aims to be a complete starting point, including details of both theory and implementation, as well as discussions of the main strengths and weaknesses of each approach. It includes extensive references for readers wishing to go further. The material is carefully structured to be accessible to researchers from all disciplines while also containing rigorous mathematical analysis (largely based on the techniques of statistical mechanics) to support those wishing to further develop or implement the theory of rand...

Equitable Colorings Of Corona Multiproducts Of Graphs

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Furmánczyk Hanna

2017-11-01

Full Text Available A graph is equitably k-colorable if its vertices can be partitioned into k independent sets in such a way that the numbers of vertices in any two sets differ by at most one. The smallest k for which such a coloring exists is known as the equitable chromatic number of G and denoted by =(G. It is known that the problem of computation of =(G is NP-hard in general and remains so for corona graphs. In this paper we consider the same model of coloring in the case of corona multiproducts of graphs. In particular, we obtain some results regarding the equitable chromatic number for the l-corona product G ◦l H, where G is an equitably 3- or 4-colorable graph and H is an r-partite graph, a cycle or a complete graph. Our proofs are mostly constructive in that they lead to polynomial algorithms for equitable coloring of such graph products provided that there is given an equitable coloring of G. Moreover, we confirm the Equitable Coloring Conjecture for corona products of such graphs. This paper extends the results from [H. Furmánczyk, K. Kaliraj, M. Kubale and V.J. Vivin, Equitable coloring of corona products of graphs, Adv. Appl. Discrete Math. 11 (2013 103–120].

Cliques in dense inhomogenous random graphs

Czech Academy of Sciences Publication Activity Database

Doležal, Martin; Hladký, Jan; Máthé, A.

2017-01-01

Roč. 51, č. 2 (2017), s. 275-314 ISSN 1042-9832 R&D Projects: GA ČR GA16-07378S EU Projects: European Commission(XE) 628974 - PAECIDM Institutional support: RVO:67985840 Keywords : inhomogeneous random graphs * clique number Subject RIV: BA - General Mathematics OBOR OECD: Pure mathematics Impact factor: 1.243, year: 2016 http://onlinelibrary.wiley.com/doi/10.1002/ rsa .20715/abstract

Cliques in dense inhomogenous random graphs

Czech Academy of Sciences Publication Activity Database

Doležal, Martin; Hladký, Jan; Máthé, A.

2017-01-01

Roč. 51, č. 2 (2017), s. 275-314 ISSN 1042-9832 R&D Projects: GA ČR GA16-07378S EU Projects: European Commission(XE) 628974 - PAECIDM Institutional support: RVO:67985840 Keywords : inhomogeneous random graphs * clique number Subject RIV: BA - General Mathematics OBOR OECD: Pure mathematics Impact factor: 1.243, year: 2016 http://onlinelibrary.wiley.com/doi/10.1002/rsa.20715/abstract

Chromatic roots and limits of dense graphs

Czech Academy of Sciences Publication Activity Database

Csikvári, P.; Frenkel, P. E.; Hladký, Jan; Hubai, T.

2017-01-01

Roč. 340, č. 5 (2017), s. 1129-1135 ISSN 0012-365X EU Projects: European Commission(XE) 628974 - PAECIDM Institutional support: RVO:67985840 Keywords : chromatic root * graph limit * holomorphic moment Subject RIV: BA - General Mathematics OBOR OECD: Pure mathematics Impact factor: 0.639, year: 2016 http://www.sciencedirect.com/science/article/pii/S0012365X16303661

The Use of Spatial Cognition in Graph Interpretation

Science.gov (United States)

2007-08-01

Mathematics has emphasized the importance of proactively teaching students of all ages to interpret graphs and use them to make inferences ( NCTM ... Mathematics . Reston, VA: National Council of Teachers of Mathematics . Oh, S., & Kim, M. (2004). The role of spatial working memory in visual...in learning science (Schunn et al, in press). Not coincidentally, in developing its recent national standards, the National Council of Teachers of

System dynamics and control with bond graph modeling

CERN Document Server

Kypuros, Javier

2013-01-01

Part I Dynamic System ModelingIntroduction to System DynamicsIntroductionSystem Decomposition and Model ComplexityMathematical Modeling of Dynamic SystemsAnalysis and Design of Dynamic SystemsControl of Dynamic SystemsDiagrams of Dynamic SystemsA Graph-Centered Approach to ModelingSummaryPracticeExercisesBasic Bond Graph ElementsIntroductionPower and Energy VariablesBasic 1-Port ElementsBasic 2-Ports ElementsJunction ElementsSimple Bond Graph ExamplesSummaryPracticeExercisesBond Graph Synthesis and Equation DerivationIntroductionGeneral GuidelinesMechanical TranslationMechanical RotationElectrical CircuitsHydraulic CircuitsMixed SystemsState Equation DerivationState-Space RepresentationsAlgebraic Loops and Derivative CausalitySummaryPracticeExercisesImpedance Bond GraphsIntroductionLaplace Transform of the State-Space EquationBasic 1-Port ImpedancesImpedance Bond Graph SynthesisJunctions, Transformers, and GyratorsEffort and Flow DividersSign ChangesTransfer Function DerivationAlternative Derivation of Transf...

The investigation of social networks based on multi-component random graphs

Science.gov (United States)

Zadorozhnyi, V. N.; Yudin, E. B.

2018-01-01

The methods of non-homogeneous random graphs calibration are developed for social networks simulation. The graphs are calibrated by the degree distributions of the vertices and the edges. The mathematical foundation of the methods is formed by the theory of random graphs with the nonlinear preferential attachment rule and the theory of Erdôs-Rényi random graphs. In fact, well-calibrated network graph models and computer experiments with these models would help developers (owners) of the networks to predict their development correctly and to choose effective strategies for controlling network projects.

Trace formulae and spectral statistics for discrete Laplacians on regular graphs (I)

Energy Technology Data Exchange (ETDEWEB)

Oren, Idan; Godel, Amit; Smilansky, Uzy [Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 76100 (Israel)], E-mail: idan.oren@weizmann.ac.il, E-mail: amit.godel@weizmann.ac.il, E-mail: uzy.smilansky@weizmann.ac.il

2009-10-16

Trace formulae for d-regular graphs are derived and used to express the spectral density in terms of the periodic walks on the graphs under consideration. The trace formulae depend on a parameter w which can be tuned continuously to assign different weights to different periodic orbit contributions. At the special value w = 1, the only periodic orbits which contribute are the non-back-scattering orbits, and the smooth part in the trace formula coincides with the Kesten-McKay expression. As w deviates from unity, non-vanishing weights are assigned to the periodic walks with backscatter, and the smooth part is modified in a consistent way. The trace formulae presented here are the tools to be used in the second paper in this sequence, for showing the connection between the spectral properties of d-regular graphs and the theory of random matrices.

Ordering non-bipartite unicyclic graphs with pendant vertices by the least Q-eigenvalue

Directory of Open Access Journals (Sweden)

Shu-Guang Guo

2016-05-01

Full Text Available Abstract A unicyclic graph is a connected graph whose number of edges is equal to the number of vertices. Fan et al. (Discrete Math. 313:903-909, 2013 and Liu et al. (Electron. J. Linear Algebra 26:333-344, 2013 determined, independently, the unique unicyclic graph whose least Q-eigenvalue attains the minimum among all non-bipartite unicyclic graphs of order n with k pendant vertices. In this paper, we extend their results and determine the first three non-bipartite unicyclic graphs of order n with k pendant vertices ordering by least Q-eigenvalue.

Infinite Random Graphs as Statistical Mechanical Models

DEFF Research Database (Denmark)

Durhuus, Bergfinnur Jøgvan; Napolitano, George Maria

2011-01-01

We discuss two examples of infinite random graphs obtained as limits of finite statistical mechanical systems: a model of two-dimensional dis-cretized quantum gravity defined in terms of causal triangulated surfaces, and the Ising model on generic random trees. For the former model we describe a ...

Adaptive Discrete Hypergraph Matching.

Science.gov (United States)

Yan, Junchi; Li, Changsheng; Li, Yin; Cao, Guitao

2018-02-01

This paper addresses the problem of hypergraph matching using higher-order affinity information. We propose a solver that iteratively updates the solution in the discrete domain by linear assignment approximation. The proposed method is guaranteed to converge to a stationary discrete solution and avoids the annealing procedure and ad-hoc post binarization step that are required in several previous methods. Specifically, we start with a simple iterative discrete gradient assignment solver. This solver can be trapped in an -circle sequence under moderate conditions, where is the order of the graph matching problem. We then devise an adaptive relaxation mechanism to jump out this degenerating case and show that the resulting new path will converge to a fixed solution in the discrete domain. The proposed method is tested on both synthetic and real-world benchmarks. The experimental results corroborate the efficacy of our method.

Secondary School Teachers' Conceptions and Their Teaching Practices Using Graphing Calculators

Science.gov (United States)

Lee, Jane A.; McDougall, Douglas E.

2010-01-01

This article investigates secondary school teachers' conceptions of mathematics and their teaching practices in the use of graphing calculators in their mathematics classrooms. Case studies on three teacher participants were developed using quantitative and qualitative data that consisted of self-assessments on beliefs in mathematics,…

Outer-2-independent domination in graphs

Indian Academy of Sciences (India)

Outer-2-independent domination in graphs. MARCIN KRZYWKOWSKI1,2,∗, DOOST ALI MOJDEH3 and MARYEM RAOOFI4. 1Department of Pure and Applied Mathematics, University of Johannesburg,. Johannesburg, South Africa. 2Faculty of Electronics, Telecommunications and Informatics, Gdansk University.

Using graph approach for managing connectivity in integrative landscape modelling

Science.gov (United States)

Rabotin, Michael; Fabre, Jean-Christophe; Libres, Aline; Lagacherie, Philippe; Crevoisier, David; Moussa, Roger

2013-04-01

FLUID-landr library has been developed in order i) to be used with no GIS expert skills needed (common gis formats can be read and simplified spatial management is provided), ii) to easily develop adapted rules of landscape discretization and graph creation to follow spatialized model requirements and iii) to allow model developers to manage dynamic and complex spatial topology. Graph management in OpenFLUID are shown with i) examples of hydrological modelizations on complex farmed landscapes and ii) the new implementation of Geo-MHYDAS tool based on the OpenFLUID-landr library, which allows to discretize a landscape and create graph structure for the MHYDAS model requirements.

Distributed graph coloring fundamentals and recent developments

CERN Document Server

Barenboim, Leonid

2013-01-01

The focus of this monograph is on symmetry breaking problems in the message-passing model of distributed computing. In this model a communication network is represented by a n-vertex graph G = (V,E), whose vertices host autonomous processors. The processors communicate over the edges of G in discrete rounds. The goal is to devise algorithms that use as few rounds as possible.A typical symmetry-breaking problem is the problem of graph coloring. Denote by ? the maximum degree of G. While coloring G with ? + 1 colors is trivial in the centralized setting, the problem becomes much more challenging

Discrete frequency identification using the HP 5451B Fourier analyser

International Nuclear Information System (INIS)

Holland, L.; Barry, P.

1977-01-01

The frequency analysis by the HP5451B discrete frequency Fourier analyser is studied. The advantages of cross correlation analysis to identify discrete frequencies in a background noise are discussed in conjuction with the elimination of aliasing and wraparound error. Discrete frequency identification is illustrated by a series of graphs giving the results of analysing 'electrical' and 'acoustical' white noise and sinusoidal signals [pt

Derivation and computation of discrete-delay and continuous-delay SDEs in mathematical biology.

Science.gov (United States)

Allen, Edward J

2014-06-01

Stochastic versions of several discrete-delay and continuous-delay differential equations, useful in mathematical biology, are derived from basic principles carefully taking into account the demographic, environmental, or physiological randomness in the dynamic processes. In particular, stochastic delay differential equation (SDDE) models are derived and studied for Nicholson's blowflies equation, Hutchinson's equation, an SIS epidemic model with delay, bacteria/phage dynamics, and glucose/insulin levels. Computational methods for approximating the SDDE models are described. Comparisons between computational solutions of the SDDEs and independently formulated Monte Carlo calculations support the accuracy of the derivations and of the computational methods.

C7-Decompositions of the Tensor Product of Complete Graphs

Directory of Open Access Journals (Sweden)

Manikandan R.S.

2017-08-01

Full Text Available In this paper we consider a decomposition of Km × Kn, where × denotes the tensor product of graphs, into cycles of length seven. We prove that for m, n ≥ 3, cycles of length seven decompose the graph Km × Kn if and only if (1 either m or n is odd and (2 14 | m(m − 1n(n − 1. The results of this paper together with the results of [Cp-Decompositions of some regular graphs, Discrete Math. 306 (2006 429–451] and [C5-Decompositions of the tensor product of complete graphs, Australasian J. Combinatorics 37 (2007 285–293], give necessary and sufficient conditions for the existence of a p-cycle decomposition, where p ≥ 5 is a prime number, of the graph Km × Kn.

Generating hierarchial scale-free graphs from fractals

Energy Technology Data Exchange (ETDEWEB)

Komjathy, Julia, E-mail: komyju@math.bme.hu [Department of Stochastics, Institute of Mathematics, Technical University of Budapest, H-1529 P.O. Box 91 (Hungary); Simon, Karoly, E-mail: simonk@math.bme.hu [Department of Stochastics, Institute of Mathematics, Technical University of Budapest, H-1529 P.O. Box 91 (Hungary)

2011-08-15

Highlights: > We generate deterministic scale-free networks using graph-directed self similar IFS. > Our model exhibits similar clustering, power law decay properties to real networks. > The average length of shortest path and the diameter of the graph are determined. > Using this model, we generate random graphs with prescribed power law exponent. - Abstract: Motivated by the hierarchial network model of E. Ravasz, A.-L. Barabasi, and T. Vicsek, we introduce deterministic scale-free networks derived from a graph directed self-similar fractal {Lambda}. With rigorous mathematical results we verify that our model captures some of the most important features of many real networks: the scale-free and the high clustering properties. We also prove that the diameter is the logarithm of the size of the system. We point out a connection between the power law exponent of the degree distribution and some intrinsic geometric measure theoretical properties of the underlying fractal. Using our (deterministic) fractal {Lambda} we generate random graph sequence sharing similar properties.

Resolvent expansion for the Schrödinger operator on a graph with infinite rays

DEFF Research Database (Denmark)

Ito, Kenichi; Jensen, Arne

2018-01-01

We consider the Schrödinger operator on a combinatorial graph consisting of a finite graph and a finite number of discrete half-lines, all jointed together, and compute an asymptotic expansion of its resolvent around the threshold 0. Precise expressions are obtained for the first few coefficients...

Precalculus teachers' perspectives on using graphing calculators: an example from one curriculum

Science.gov (United States)

Karadeniz, Ilyas; Thompson, Denisse R.

2018-01-01

Graphing calculators are hand-held technological tools currently used in mathematics classrooms. Teachers' perspectives on using graphing calculators are important in terms of exploring what teachers think about using such technology in advanced mathematics courses, particularly precalculus courses. A descriptive intrinsic case study was conducted to analyse the perspectives of 11 teachers using graphing calculators with potential Computer Algebra System (CAS) capability while teaching Functions, Statistics, and Trigonometry, a precalculus course for 11th-grade students developed by the University of Chicago School Mathematics Project. Data were collected from multiple sources as part of a curriculum evaluation study conducted during the 2007-2008 school year. Although all teachers were using the same curriculum that integrated CAS into the instructional materials, teachers had mixed views about the technology. Graphing calculator features were used much more than CAS features, with many teachers concerned about the use of CAS because of pressures from external assessments. In addition, several teachers found it overwhelming to learn a new technology at the same time they were learning a new curriculum. The results have implications for curriculum developers and others working with teachers to update curriculum and the use of advanced technologies simultaneously.

Laplacian eigenvectors of graphs Perron-Frobenius and Faber-Krahn type theorems

CERN Document Server

Biyikoğu, Türker; Stadler, Peter F

2007-01-01

Eigenvectors of graph Laplacians have not, to date, been the subject of expository articles and thus they may seem a surprising topic for a book. The authors propose two motivations for this new LNM volume: (1) There are fascinating subtle differences between the properties of solutions of Schrödinger equations on manifolds on the one hand, and their discrete analogs on graphs. (2) "Geometric" properties of (cost) functions defined on the vertex sets of graphs are of practical interest for heuristic optimization algorithms. The observation that the cost functions of quite a few of the well-studied combinatorial optimization problems are eigenvectors of associated graph Laplacians has prompted the investigation of such eigenvectors. The volume investigates the structure of eigenvectors and looks at the number of their sign graphs ("nodal domains"), Perron components, graphs with extremal properties with respect to eigenvectors. The Rayleigh quotient and rearrangement of graphs form the main methodology.

Application of Bond Graph Modeling for Photovoltaic Module Simulation

Directory of Open Access Journals (Sweden)

Madi S.

2016-01-01

Full Text Available In this paper, photovoltaic generator is represented using the bond-graph methodology. Starting from the equivalent circuit the bond graph and the block diagram of the photovoltaic generator have been derived. Upon applying bond graph elements and rules a mathematical model of the photovoltaic generator is obtained. Simulation results of this obtained model using real recorded data (irradiation and temperature at the Renewable Energies Development Centre in Bouzaréah – Algeria are obtained using MATLAB/SMULINK software. The results have compared with datasheet of the photovoltaic generator for validation purposes.

Graphs in kinematics—a need for adherence to principles of algebraic functions

Science.gov (United States)

Sokolowski, Andrzej

2017-11-01

Graphs in physics are central to the analysis of phenomena and to learning about a system’s behavior. The ways students handle graphs are frequently researched. Students’ misconceptions are highlighted, and methods of improvement suggested. While kinematics graphs are to represent a real motion, they are also algebraic entities that must satisfy conditions for being algebraic functions. To be algebraic functions, they must pass certain tests before they can be used to infer more about motion. A preliminary survey of some physics resources has revealed that little attention is paid to verifying if the position, velocity and acceleration versus time graphs, that are to depict real motion, satisfy the most critical condition for being an algebraic function; the vertical line test. The lack of attention to this adherence shows as vertical segments in piecewise graphs. Such graphs generate unrealistic interpretations and may confuse students. A group of 25 college physics students was provided with such a graph and asked to analyse its adherence to reality. The majority of the students (N  =  16, 64%) questioned the graph’s validity. It is inferred that such graphs might not only jeopardize the function principles studied in mathematics but also undermine the purpose of studying these principles. The aim of this study was to bring this idea forth and suggest a better alignment of physics and mathematics methods.

Local adjacency metric dimension of sun graph and stacked book graph

Science.gov (United States)

Yulisda Badri, Alifiah; Darmaji

2018-03-01

A graph is a mathematical system consisting of a non-empty set of nodes and a set of empty sides. One of the topics to be studied in graph theory is the metric dimension. Application in the metric dimension is the navigation robot system on a path. Robot moves from one vertex to another vertex in the field by minimizing the errors that occur in translating the instructions (code) obtained from the vertices of that location. To move the robot must give different instructions (code). In order for the robot to move efficiently, the robot must be fast to translate the code of the nodes of the location it passes. so that the location vertex has a minimum distance. However, if the robot must move with the vertex location on a very large field, so the robot can not detect because the distance is too far.[6] In this case, the robot can determine its position by utilizing location vertices based on adjacency. The problem is to find the minimum cardinality of the required location vertex, and where to put, so that the robot can determine its location. The solution to this problem is the dimension of adjacency metric and adjacency metric bases. Rodrguez-Velzquez and Fernau combine the adjacency metric dimensions with local metric dimensions, thus becoming the local adjacency metric dimension. In the local adjacency metric dimension each vertex in the graph may have the same adjacency representation as the terms of the vertices. To obtain the local metric dimension of values in the graph of the Sun and the stacked book graph is used the construction method by considering the representation of each adjacent vertex of the graph.

On the Additively Weighted Harary Index of Some Composite Graphs

Directory of Open Access Journals (Sweden)

Behrooz Khosravi

2017-03-01

Full Text Available The Harary index is defined as the sum of reciprocals of distances between all pairs of vertices of a connected graph. The additively weighted Harary index H A ( G is a modification of the Harary index in which the contributions of vertex pairs are weighted by the sum of their degrees. This new invariant was introduced in (Alizadeh, Iranmanesh and Došlić. Additively weighted Harary index of some composite graphs, Discrete Math, 2013 and they posed the following question: What is the behavior of H A ( G when G is a composite graph resulting for example by: splice, link, corona and rooted product? We investigate the additively weighted Harary index for these standard graph products. Then we obtain lower and upper bounds for some of them.

Graph Structure in Three National Academic Webs: Power Laws with Anomalies.

Science.gov (United States)

Thelwall, Mike; Wilkinson, David

2003-01-01

Explains how the Web can be modeled as a mathematical graph and analyzes the graph structures of three national university publicly indexable Web sites from Australia, New Zealand, and the United Kingdom. Topics include commercial search engines and academic Web link research; method-analysis environment and data sets; and power laws. (LRW)

Laplacian Estrada and normalized Laplacian Estrada indices of evolving graphs.

Science.gov (United States)

Shang, Yilun

2015-01-01

Large-scale time-evolving networks have been generated by many natural and technological applications, posing challenges for computation and modeling. Thus, it is of theoretical and practical significance to probe mathematical tools tailored for evolving networks. In this paper, on top of the dynamic Estrada index, we study the dynamic Laplacian Estrada index and the dynamic normalized Laplacian Estrada index of evolving graphs. Using linear algebra techniques, we established general upper and lower bounds for these graph-spectrum-based invariants through a couple of intuitive graph-theoretic measures, including the number of vertices or edges. Synthetic random evolving small-world networks are employed to show the relevance of the proposed dynamic Estrada indices. It is found that neither the static snapshot graphs nor the aggregated graph can approximate the evolving graph itself, indicating the fundamental difference between the static and dynamic Estrada indices.

An introduction to grids, graphs, and networks

CERN Document Server

Pozrikidis, C

2014-01-01

An Introduction to Grids, Graphs, and Networks aims to provide a concise introduction to graphs and networks at a level that is accessible to scientists, engineers, and students. In a practical approach, the book presents only the necessary theoretical concepts from mathematics and considers a variety of physical and conceptual configurations as prototypes or examples. The subject is timely, as the performance of networks is recognized as an important topic in the study of complex systems with applications in energy, material, and information grid transport (epitomized by the internet). The bo

Inferring gene networks from discrete expression data

KAUST Repository

Zhang, L.

2013-07-18

The modeling of gene networks from transcriptional expression data is an important tool in biomedical research to reveal signaling pathways and to identify treatment targets. Current gene network modeling is primarily based on the use of Gaussian graphical models applied to continuous data, which give a closedformmarginal likelihood. In this paper,we extend network modeling to discrete data, specifically data from serial analysis of gene expression, and RNA-sequencing experiments, both of which generate counts of mRNAtranscripts in cell samples.We propose a generalized linear model to fit the discrete gene expression data and assume that the log ratios of the mean expression levels follow a Gaussian distribution.We restrict the gene network structures to decomposable graphs and derive the graphs by selecting the covariance matrix of the Gaussian distribution with the hyper-inverse Wishart priors. Furthermore, we incorporate prior network models based on gene ontology information, which avails existing biological information on the genes of interest. We conduct simulation studies to examine the performance of our discrete graphical model and apply the method to two real datasets for gene network inference. © The Author 2013. Published by Oxford University Press. All rights reserved.

A mathematical model for generating bipartite graphs and its application to protein networks

International Nuclear Information System (INIS)

Nacher, J C; Ochiai, T; Hayashida, M; Akutsu, T

2009-01-01

Complex systems arise in many different contexts from large communication systems and transportation infrastructures to molecular biology. Most of these systems can be organized into networks composed of nodes and interacting edges. Here, we present a theoretical model that constructs bipartite networks with the particular feature that the degree distribution can be tuned depending on the probability rate of fundamental processes. We then use this model to investigate protein-domain networks. A protein can be composed of up to hundreds of domains. Each domain represents a conserved sequence segment with specific functional tasks. We analyze the distribution of domains in Homo sapiens and Arabidopsis thaliana organisms and the statistical analysis shows that while (a) the number of domain types shared by k proteins exhibits a power-law distribution, (b) the number of proteins composed of k types of domains decays as an exponential distribution. The proposed mathematical model generates bipartite graphs and predicts the emergence of this mixing of (a) power-law and (b) exponential distributions. Our theoretical and computational results show that this model requires (1) growth process and (2) copy mechanism.

A mathematical model for generating bipartite graphs and its application to protein networks

Science.gov (United States)

Nacher, J. C.; Ochiai, T.; Hayashida, M.; Akutsu, T.

2009-12-01

Complex systems arise in many different contexts from large communication systems and transportation infrastructures to molecular biology. Most of these systems can be organized into networks composed of nodes and interacting edges. Here, we present a theoretical model that constructs bipartite networks with the particular feature that the degree distribution can be tuned depending on the probability rate of fundamental processes. We then use this model to investigate protein-domain networks. A protein can be composed of up to hundreds of domains. Each domain represents a conserved sequence segment with specific functional tasks. We analyze the distribution of domains in Homo sapiens and Arabidopsis thaliana organisms and the statistical analysis shows that while (a) the number of domain types shared by k proteins exhibits a power-law distribution, (b) the number of proteins composed of k types of domains decays as an exponential distribution. The proposed mathematical model generates bipartite graphs and predicts the emergence of this mixing of (a) power-law and (b) exponential distributions. Our theoretical and computational results show that this model requires (1) growth process and (2) copy mechanism.

A mathematical model for generating bipartite graphs and its application to protein networks

Energy Technology Data Exchange (ETDEWEB)

Nacher, J C [Department of Complex Systems, Future University-Hakodate (Japan); Ochiai, T [Faculty of Engineering, Toyama Prefectural University (Japan); Hayashida, M; Akutsu, T [Bioinformatics Center, Institute for Chemical Research, Kyoto University (Japan)

2009-12-04

Complex systems arise in many different contexts from large communication systems and transportation infrastructures to molecular biology. Most of these systems can be organized into networks composed of nodes and interacting edges. Here, we present a theoretical model that constructs bipartite networks with the particular feature that the degree distribution can be tuned depending on the probability rate of fundamental processes. We then use this model to investigate protein-domain networks. A protein can be composed of up to hundreds of domains. Each domain represents a conserved sequence segment with specific functional tasks. We analyze the distribution of domains in Homo sapiens and Arabidopsis thaliana organisms and the statistical analysis shows that while (a) the number of domain types shared by k proteins exhibits a power-law distribution, (b) the number of proteins composed of k types of domains decays as an exponential distribution. The proposed mathematical model generates bipartite graphs and predicts the emergence of this mixing of (a) power-law and (b) exponential distributions. Our theoretical and computational results show that this model requires (1) growth process and (2) copy mechanism.

Discrete calculus applied analysis on graphs for computational science

CERN Document Server

Grady, Leo J

2010-01-01

This unique text brings together into a single framework current research in the three areas of discrete calculus, complex networks, and algorithmic content extraction. Many example applications from several fields of computational science are provided.

Clique Relaxations in Biological and Social Network Analysis Foundations and Algorithms

Science.gov (United States)

2015-10-26

NUMBER 9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) Optimization and Discrete Mathematics Mathematics , Information and Life Sciences Air...Business Media, 2013, pp.149-174. J. Pattillo, Y. Wang, and S. Butenko. Approximating 2-cliques in unit disk graphs. Discrete Applied Mathematics 166 (2014...for developing effective combinatorial branch-and-bound strategies for detection of maximum clique relaxation structures. The concepts of weak heredity

Partition function expansion on region graphs and message-passing equations

International Nuclear Information System (INIS)

Zhou, Haijun; Wang, Chuang; Xiao, Jing-Qing; Bi, Zedong

2011-01-01

Disordered and frustrated graphical systems are ubiquitous in physics, biology, and information science. For models on complete graphs or random graphs, deep understanding has been achieved through the mean-field replica and cavity methods. But finite-dimensional 'real' systems remain very challenging because of the abundance of short loops and strong local correlations. A statistical mechanics theory is constructed in this paper for finite-dimensional models based on the mathematical framework of the partition function expansion and the concept of region graphs. Rigorous expressions for the free energy and grand free energy are derived. Message-passing equations on the region graph, such as belief propagation and survey propagation, are also derived rigorously. (letter)

The Transition to Formal Thinking in Mathematics

Science.gov (United States)

Tall, David

2008-01-01

This paper focuses on the changes in thinking involved in the transition from school mathematics to formal proof in pure mathematics at university. School mathematics is seen as a combination of visual representations, including geometry and graphs, together with symbolic calculations and manipulations. Pure mathematics in university shifts…

RNA graph partitioning for the discovery of RNA modularity: a novel application of graph partition algorithm to biology.

Directory of Open Access Journals (Sweden)

Namhee Kim

Full Text Available Graph representations have been widely used to analyze and design various economic, social, military, political, and biological networks. In systems biology, networks of cells and organs are useful for understanding disease and medical treatments and, in structural biology, structures of molecules can be described, including RNA structures. In our RNA-As-Graphs (RAG framework, we represent RNA structures as tree graphs by translating unpaired regions into vertices and helices into edges. Here we explore the modularity of RNA structures by applying graph partitioning known in graph theory to divide an RNA graph into subgraphs. To our knowledge, this is the first application of graph partitioning to biology, and the results suggest a systematic approach for modular design in general. The graph partitioning algorithms utilize mathematical properties of the Laplacian eigenvector (µ2 corresponding to the second eigenvalues (λ2 associated with the topology matrix defining the graph: λ2 describes the overall topology, and the sum of µ2's components is zero. The three types of algorithms, termed median, sign, and gap cuts, divide a graph by determining nodes of cut by median, zero, and largest gap of µ2's components, respectively. We apply these algorithms to 45 graphs corresponding to all solved RNA structures up through 11 vertices (∼ 220 nucleotides. While we observe that the median cut divides a graph into two similar-sized subgraphs, the sign and gap cuts partition a graph into two topologically-distinct subgraphs. We find that the gap cut produces the best biologically-relevant partitioning for RNA because it divides RNAs at less stable connections while maintaining junctions intact. The iterative gap cuts suggest basic modules and assembly protocols to design large RNA structures. Our graph substructuring thus suggests a systematic approach to explore the modularity of biological networks. In our applications to RNA structures, subgraphs

Laplacian Estrada and normalized Laplacian Estrada indices of evolving graphs.

Directory of Open Access Journals (Sweden)

Yilun Shang

Full Text Available Large-scale time-evolving networks have been generated by many natural and technological applications, posing challenges for computation and modeling. Thus, it is of theoretical and practical significance to probe mathematical tools tailored for evolving networks. In this paper, on top of the dynamic Estrada index, we study the dynamic Laplacian Estrada index and the dynamic normalized Laplacian Estrada index of evolving graphs. Using linear algebra techniques, we established general upper and lower bounds for these graph-spectrum-based invariants through a couple of intuitive graph-theoretic measures, including the number of vertices or edges. Synthetic random evolving small-world networks are employed to show the relevance of the proposed dynamic Estrada indices. It is found that neither the static snapshot graphs nor the aggregated graph can approximate the evolving graph itself, indicating the fundamental difference between the static and dynamic Estrada indices.

A Universal Concept for Robust Solving of Shortest Path Problems in Dynamically Reconfigurable Graphs

Directory of Open Access Journals (Sweden)

Jean Chamberlain Chedjou

2015-01-01

Full Text Available This paper develops a flexible analytical concept for robust shortest path detection in dynamically reconfigurable graphs. The concept is expressed by a mathematical model representing the shortest path problem solver. The proposed mathematical model is characterized by three fundamental parameters expressing (a the graph topology (through the “incidence matrix”, (b the edge weights (with dynamic external weights’ setting capability, and (c the dynamic reconfigurability through external input(s of the source-destination nodes pair. In order to demonstrate the universality of the developed concept, a general algorithm is proposed to determine the three fundamental parameters (of the mathematical model developed for all types of graphs regardless of their topology, magnitude, and size. It is demonstrated that the main advantage of the developed concept is that arc costs, the origin-destination pair setting, and the graph topology are dynamically provided by external commands, which are inputs of the shortest path solver model. This enables high flexibility and full reconfigurability of the developed concept, without any retraining need. To validate the concept developed, benchmarking is performed leading to a comparison of its performance with the performances of two well-known concepts based on neural networks.

Graph theory and binary alloys passivated by nickel

International Nuclear Information System (INIS)

McCafferty, E.

2005-01-01

The passivity of a nickel binary alloy is considered in terms of a network of -Ni-O-Ni- bridges in the oxide film, where Ni is the component of the binary alloy which produces passivity. The structure of the oxide is represented by a mathematical graph, and graph theory is used to calculate the connectivity of the oxide, given by the product of the number of edges in the graph and the Randic index. A stochastic calculation is employed to insert ions of the second metal into the oxide film so as to disrupt the connectivity of the -Ni-O-Ni- network. This disruption occurs at a critical ionic concentration of the oxide film. Mathematical relationships are developed for the introduction of a general ion B +n into the oxide film, and critical ionic compositions are calculated for oxide films on the nickel binary alloys. The notation B refers to any metal B which produces B +n ions in the oxide film, where +n is the oxidation number of the ion. The results of this analysis for Fe-Ni and Cu-Ni binary alloys are in good agreement with experimental results

Non-Lipschitz Dynamics Approach to Discrete Event Systems

Science.gov (United States)

Zak, M.; Meyers, R.

1995-01-01

This paper presents and discusses a mathematical formalism for simulation of discrete event dynamics (DED) - a special type of 'man- made' system designed to aid specific areas of information processing. A main objective is to demonstrate that the mathematical formalism for DED can be based upon the terminal model of Newtonian dynamics which allows one to relax Lipschitz conditions at some discrete points.

Random graph states, maximal flow and Fuss-Catalan distributions

International Nuclear Information System (INIS)

Collins, BenoIt; Nechita, Ion; Zyczkowski, Karol

2010-01-01

For any graph consisting of k vertices and m edges we construct an ensemble of random pure quantum states which describe a system composed of 2m subsystems. Each edge of the graph represents a bipartite, maximally entangled state. Each vertex represents a random unitary matrix generated according to the Haar measure, which describes the coupling between subsystems. Dividing all subsystems into two parts, one may study entanglement with respect to this partition. A general technique to derive an expression for the average entanglement entropy of random pure states associated with a given graph is presented. Our technique relies on Weingarten calculus and flow problems. We analyze the statistical properties of spectra of such random density matrices and show for which cases they are described by the free Poissonian (Marchenko-Pastur) distribution. We derive a discrete family of generalized, Fuss-Catalan distributions and explicitly construct graphs which lead to ensembles of random states characterized by these novel distributions of eigenvalues.

Discrimination Power of Polynomial-Based Descriptors for Graphs by Using Functional Matrices.

Science.gov (United States)

Dehmer, Matthias; Emmert-Streib, Frank; Shi, Yongtang; Stefu, Monica; Tripathi, Shailesh

2015-01-01

In this paper, we study the discrimination power of graph measures that are based on graph-theoretical matrices. The paper generalizes the work of [M. Dehmer, M. Moosbrugger. Y. Shi, Encoding structural information uniquely with polynomial-based descriptors by employing the Randić matrix, Applied Mathematics and Computation, 268(2015), 164-168]. We demonstrate that by using the new functional matrix approach, exhaustively generated graphs can be discriminated more uniquely than shown in the mentioned previous work.

Quantum superposition of the state discrete spectrum of mathematical correlation molecule for small samples of biometric data

Directory of Open Access Journals (Sweden)

Vladimir I. Volchikhin

2017-06-01

Full Text Available Introduction: The study promotes to decrease a number of errors of calculating the correlation coefficient in small test samples. Materials and Methods: We used simulation tool for the distribution functions of the density values of the correlation coefficient in small samples. A method for quantization of the data, allows obtaining a discrete spectrum states of one of the varieties of correlation functional. This allows us to consider the proposed structure as a mathematical correlation molecule, described by some analogue continuous-quantum Schrödinger equation. Results: The chi-squared Pearson’s molecule on small samples allows enhancing power of classical chi-squared test to 20 times. A mathematical correlation molecule described in the article has similar properties. It allows in the future reducing calculation errors of the classical correlation coefficients in small samples. Discussion and Conclusions: The authors suggest that there are infinitely many mathematical molecules are similar in their properties to the actual physical molecules. Schrödinger equations are not unique, their analogues can be constructed for each mathematical molecule. You can expect a mathematical synthesis of molecules for a large number of known statistical tests and statistical moments. All this should make it possible to reduce calculation errors due to quantum effects that occur in small test samples.

Mathematical Model Taking into Account Nonlocal Effects of Plasmonic Structures on the Basis of the Discrete Source Method

Science.gov (United States)

Eremin, Yu. A.; Sveshnikov, A. G.

2018-04-01

The discrete source method is used to develop and implement a mathematical model for solving the problem of scattering electromagnetic waves by a three-dimensional plasmonic scatterer with nonlocal effects taken into account. Numerical results are presented whereby the features of the scattering properties of plasmonic particles with allowance for nonlocal effects are demonstrated depending on the direction and polarization of the incident wave.

Lower bounds on the independence number of certain graphs of odd girth at least seven

DEFF Research Database (Denmark)

Pedersen, A. S.; Rautenbach, D.; Regen, F.

2011-01-01

Heckman and Thomas [C.C. Heckman, R. Thomas, A new proof of the independence ratio of triangle-free cubic graphs, Discrete Math. 233 (2001) 233-237] proved that every connected subcubic triangle-free graph G has an independent set of order at least (4n(G) - m(G) - 1)/7 where n(G) and m(G) denote...

Discrete control systems

CERN Document Server

Okuyama, Yoshifumi

2014-01-01

Discrete Control Systems establishes a basis for the analysis and design of discretized/quantized control systemsfor continuous physical systems. Beginning with the necessary mathematical foundations and system-model descriptions, the text moves on to derive a robust stability condition. To keep a practical perspective on the uncertain physical systems considered, most of the methods treated are carried out in the frequency domain. As part of the design procedure, modified Nyquist–Hall and Nichols diagrams are presented and discretized proportional–integral–derivative control schemes are reconsidered. Schemes for model-reference feedback and discrete-type observers are proposed. Although single-loop feedback systems form the core of the text, some consideration is given to multiple loops and nonlinearities. The robust control performance and stability of interval systems (with multiple uncertainties) are outlined. Finally, the monograph describes the relationship between feedback-control and discrete ev...

Simulating continuous-time Hamiltonian dynamics by way of a discrete-time quantum walk

International Nuclear Information System (INIS)

Schmitz, A.T.; Schwalm, W.A.

2016-01-01

Much effort has been made to connect the continuous-time and discrete-time quantum walks. We present a method for making that connection for a general graph Hamiltonian on a bigraph. Furthermore, such a scheme may be adapted for simulating discretized quantum models on a quantum computer. A coin operator is found for the discrete-time quantum walk which exhibits the same dynamics as the continuous-time evolution. Given the spectral decomposition of the graph Hamiltonian and certain restrictions, the discrete-time evolution is solved for explicitly and understood at or near important values of the parameters. Finally, this scheme is connected to past results for the 1D chain. - Highlights: • A discrete-time quantum walk is purposed which approximates a continuous-time quantum walk. • The purposed quantum walk could be used to simulate Hamiltonian dynamics on a quantum computer. • Given the spectra decomposition of the Hamiltonian, the quantum walk is solved explicitly. • The method is demonstrated and connected to previous work done on the 1D chain.

The complexity of proving that a graph is Ramsey

Czech Academy of Sciences Publication Activity Database

Lauria, M.; Pudlák, Pavel; Rödl, V.; Thapen, Neil

2017-01-01

Roč. 37, č. 2 (2017), s. 253-268 ISSN 0209-9683 R&D Projects: GA AV ČR IAA100190902; GA ČR GBP202/12/G061 Institutional support: RVO:67985840 Keywords : complexity * c-Ramsey graphs Subject RIV: BA - General Mathematics OBOR OECD: Pure mathematics Impact factor: 1.048, year: 2016 http://link.springer.com/article/10.1007%2Fs00493-015-3193-9

Discrete Tomography and Imaging of Polycrystalline Structures

DEFF Research Database (Denmark)

Alpers, Andreas

High resolution transmission electron microscopy is commonly considered as the standard application for discrete tomography. While this has yet to be technically realized, new applications with a similar flavor have emerged in materials science. In our group at Ris� DTU (Denmark's National...... Laboratory for Sustainable Energy), for instance, we study polycrystalline materials via synchrotron X-ray diffraction. Several reconstruction problems arise, most of them exhibit inherently discrete aspects. In this talk I want to give a concise mathematical introduction to some of these reconstruction...... problems. Special focus is on their relationship to classical discrete tomography. Several open mathematical questions will be mentioned along the way....

Pathfinding in graph-theoretic sabotage models. I. Simultaneous attack by several teams

International Nuclear Information System (INIS)

Hulme, B.L.

1976-07-01

Graph models are developed for fixed-site safeguards systems. The problem of finding optimal routes for several sabotage teams is cast as a problem of finding shortest paths in a graph. The motivation, rationale, and interpretation of the mathematical models are discussed in detail, and an algorithm for efficiently solving the associated path problem is described

Generating hierarchical scale free-graphs from fractals

NARCIS (Netherlands)

Komjáthy, J.; Simon, K.

2011-01-01

Motivated by the hierarchial network model of E. Ravasz, A.-L. Barabási, and T. Vicsek, we introduce deterministic scale-free networks derived from a graph directed self-similar fractal ¿. With rigorous mathematical results we verify that our model captures some of the most important features of

Exclusivity structures and graph representatives of local complementation orbits

Science.gov (United States)

Cabello, Adán; Parker, Matthew G.; Scarpa, Giannicola; Severini, Simone

2013-07-01

We describe a construction that maps any connected graph G on three or more vertices into a larger graph, H(G), whose independence number is strictly smaller than its Lovász number which is equal to its fractional packing number. The vertices of H(G) represent all possible events consistent with the stabilizer group of the graph state associated with G, and exclusive events are adjacent. Mathematically, the graph H(G) corresponds to the orbit of G under local complementation. Physically, the construction translates into graph-theoretic terms the connection between a graph state and a Bell inequality maximally violated by quantum mechanics. In the context of zero-error information theory, the construction suggests a protocol achieving the maximum rate of entanglement-assisted capacity, a quantum mechanical analogue of the Shannon capacity, for each H(G). The violation of the Bell inequality is expressed by the one-shot version of this capacity being strictly larger than the independence number. Finally, given the correspondence between graphs and exclusivity structures, we are able to compute the independence number for certain infinite families of graphs with the use of quantum non-locality, therefore highlighting an application of quantum theory in the proof of a purely combinatorial statement.

Navigability of Random Geometric Graphs in the Universe and Other Spacetimes.

Science.gov (United States)

Cunningham, William; Zuev, Konstantin; Krioukov, Dmitri

2017-08-18

Random geometric graphs in hyperbolic spaces explain many common structural and dynamical properties of real networks, yet they fail to predict the correct values of the exponents of power-law degree distributions observed in real networks. In that respect, random geometric graphs in asymptotically de Sitter spacetimes, such as the Lorentzian spacetime of our accelerating universe, are more attractive as their predictions are more consistent with observations in real networks. Yet another important property of hyperbolic graphs is their navigability, and it remains unclear if de Sitter graphs are as navigable as hyperbolic ones. Here we study the navigability of random geometric graphs in three Lorentzian manifolds corresponding to universes filled only with dark energy (de Sitter spacetime), only with matter, and with a mixture of dark energy and matter. We find these graphs are navigable only in the manifolds with dark energy. This result implies that, in terms of navigability, random geometric graphs in asymptotically de Sitter spacetimes are as good as random hyperbolic graphs. It also establishes a connection between the presence of dark energy and navigability of the discretized causal structure of spacetime, which provides a basis for a different approach to the dark energy problem in cosmology.

Introduction of hypermatrix and operator notation into a discrete mathematics simulation model of malignant tumour response to therapeutic schemes in vivo. Some operator properties.

Science.gov (United States)

Stamatakos, Georgios S; Dionysiou, Dimitra D

2009-10-21

The tremendous rate of accumulation of experimental and clinical knowledge pertaining to cancer dictates the development of a theoretical framework for the meaningful integration of such knowledge at all levels of biocomplexity. In this context our research group has developed and partly validated a number of spatiotemporal simulation models of in vivo tumour growth and in particular tumour response to several therapeutic schemes. Most of the modeling modules have been based on discrete mathematics and therefore have been formulated in terms of rather complex algorithms (e.g. in pseudocode and actual computer code). However, such lengthy algorithmic descriptions, although sufficient from the mathematical point of view, may render it difficult for an interested reader to readily identify the sequence of the very basic simulation operations that lie at the heart of the entire model. In order to both alleviate this problem and at the same time provide a bridge to symbolic mathematics, we propose the introduction of the notion of hypermatrix in conjunction with that of a discrete operator into the already developed models. Using a radiotherapy response simulation example we demonstrate how the entire model can be considered as the sequential application of a number of discrete operators to a hypermatrix corresponding to the dynamics of the anatomic area of interest. Subsequently, we investigate the operators' commutativity and outline the "summarize and jump" strategy aiming at efficiently and realistically address multilevel biological problems such as cancer. In order to clarify the actual effect of the composite discrete operator we present further simulation results which are in agreement with the outcome of the clinical study RTOG 83-02, thus strengthening the reliability of the model developed.

Discrete-Time Nonlinear Control of VSC-HVDC System

Directory of Open Access Journals (Sweden)

TianTian Qian

2015-01-01

Full Text Available Because VSC-HVDC is a kind of strong nonlinear, coupling, and multi-input multioutput (MIMO system, its control problem is always attracting much attention from scholars. And a lot of papers have done research on its control strategy in the continuous-time domain. But the control system is implemented through the computer discrete sampling in practical engineering. It is necessary to study the mathematical model and control algorithm in the discrete-time domain. The discrete mathematical model based on output feedback linearization and discrete sliding mode control algorithm is proposed in this paper. And to ensure the effectiveness of the control system in the quasi sliding mode state, the fast output sampling method is used in the output feedback. The results from simulation experiment in MATLAB/SIMULINK prove that the proposed discrete control algorithm can make the VSC-HVDC system have good static, dynamic, and robust characteristics in discrete-time domain.

From Specific Information Extraction to Inferences: A Hierarchical Framework of Graph Comprehension

Science.gov (United States)

2004-09-01

The skill to interpret the information displayed in graphs is so important to have, the National Council of Teachers of Mathematics has created...guidelines to ensure that students learn these skills ( NCTM : Standards for Mathematics , 2003). These guidelines are based primarily on the extraction of...graphical perception. Human Computer Interaction, 8, 353-388. NCTM : Standards for Mathematics . (2003, 2003). Peebles, D., & Cheng, P. C.-H. (2002

A discrete control model of PLANT

Science.gov (United States)

Mitchell, C. M.

1985-01-01

A model of the PLANT system using the discrete control modeling techniques developed by Miller is described. Discrete control models attempt to represent in a mathematical form how a human operator might decompose a complex system into simpler parts and how the control actions and system configuration are coordinated so that acceptable overall system performance is achieved. Basic questions include knowledge representation, information flow, and decision making in complex systems. The structure of the model is a general hierarchical/heterarchical scheme which structurally accounts for coordination and dynamic focus of attention. Mathematically, the discrete control model is defined in terms of a network of finite state systems. Specifically, the discrete control model accounts for how specific control actions are selected from information about the controlled system, the environment, and the context of the situation. The objective is to provide a plausible and empirically testable accounting and, if possible, explanation of control behavior.

A characterization of horizontal visibility graphs and combinatorics on words

Science.gov (United States)

Gutin, Gregory; Mansour, Toufik; Severini, Simone

2011-06-01

A Horizontal Visibility Graph (HVG) is defined in association with an ordered set of non-negative reals. HVGs realize a methodology in the analysis of time series, their degree distribution being a good discriminator between randomness and chaos Luque et al. [B. Luque, L. Lacasa, F. Ballesteros, J. Luque, Horizontal visibility graphs: exact results for random time series, Phys. Rev. E 80 (2009), 046103]. We prove that a graph is an HVG if and only if it is outerplanar and has a Hamilton path. Therefore, an HVG is a noncrossing graph, as defined in algebraic combinatorics Flajolet and Noy [P. Flajolet, M. Noy, Analytic combinatorics of noncrossing configurations, Discrete Math., 204 (1999) 203-229]. Our characterization of HVGs implies a linear time recognition algorithm. Treating ordered sets as words, we characterize subfamilies of HVGs highlighting various connections with combinatorial statistics and introducing the notion of a visible pair. With this technique, we determine asymptotically the average number of edges of HVGs.

Principles of discrete time mechanics

CERN Document Server

Jaroszkiewicz, George

2014-01-01

Could time be discrete on some unimaginably small scale? Exploring the idea in depth, this unique introduction to discrete time mechanics systematically builds the theory up from scratch, beginning with the historical, physical and mathematical background to the chronon hypothesis. Covering classical and quantum discrete time mechanics, this book presents all the tools needed to formulate and develop applications of discrete time mechanics in a number of areas, including spreadsheet mechanics, classical and quantum register mechanics, and classical and quantum mechanics and field theories. A consistent emphasis on contextuality and the observer-system relationship is maintained throughout.

Dynamic Programming and Graph Algorithms in Computer Vision*

Science.gov (United States)

Felzenszwalb, Pedro F.; Zabih, Ramin

2013-01-01

Optimization is a powerful paradigm for expressing and solving problems in a wide range of areas, and has been successfully applied to many vision problems. Discrete optimization techniques are especially interesting, since by carefully exploiting problem structure they often provide non-trivial guarantees concerning solution quality. In this paper we briefly review dynamic programming and graph algorithms, and discuss representative examples of how these discrete optimization techniques have been applied to some classical vision problems. We focus on the low-level vision problem of stereo; the mid-level problem of interactive object segmentation; and the high-level problem of model-based recognition. PMID:20660950

How Symmetric Are Real-World Graphs? A Large-Scale Study

Directory of Open Access Journals (Sweden)

Fabian Ball

2018-01-01

Full Text Available The analysis of symmetry is a main principle in natural sciences, especially physics. For network sciences, for example, in social sciences, computer science and data science, only a few small-scale studies of the symmetry of complex real-world graphs exist. Graph symmetry is a topic rooted in mathematics and is not yet well-received and applied in practice. This article underlines the importance of analyzing symmetry by showing the existence of symmetry in real-world graphs. An analysis of over 1500 graph datasets from the meta-repository networkrepository.com is carried out and a normalized version of the “network redundancy” measure is presented. It quantifies graph symmetry in terms of the number of orbits of the symmetry group from zero (no symmetries to one (completely symmetric, and improves the recognition of asymmetric graphs. Over 70% of the analyzed graphs contain symmetries (i.e., graph automorphisms, independent of size and modularity. Therefore, we conclude that real-world graphs are likely to contain symmetries. This contribution is the first larger-scale study of symmetry in graphs and it shows the necessity of handling symmetry in data analysis: The existence of symmetries in graphs is the cause of two problems in graph clustering we are aware of, namely, the existence of multiple equivalent solutions with the same value of the clustering criterion and, secondly, the inability of all standard partition-comparison measures of cluster analysis to identify automorphic partitions as equivalent.

Constitutive hybrid processes: A process-algebraic semantics for hybrid bond graphs

NARCIS (Netherlands)

Cuijpers, P.J.L.; Broenink, J.F.; Mosterman, P.J.

2008-01-01

Models of physical systems have to be based on physical principles such as conservation of energy and continuity of power. These principles are inherently enforced by the bond graph modeling formalism. Often, however, physical components may be best modeled as piecewise continuous with discrete mode

Constitutive Hybrid Processes: a Process-Algebraic Semantics for Hybrid Bond Graphs

NARCIS (Netherlands)

Cuijpers, P.J.L.; Broenink, Johannes F.; Mosterman, P.J.

Models of physical systems have to be based on physical principles such as conservation of energy and continuity of power. These principles are inherently enforced by the bond graph modeling formalism. Often, however, physical components may be best modeled as piecewise continuous with discrete mode

Constitutive Hybrid Processes: a Process-Algebraic Semantics for Hybrid Bond Graphs

NARCIS (Netherlands)

Cuijpers, Pieter J.L.; Broenink, Johannes F.; Mosterman, Pieter J.

2008-01-01

Models of physical systems have to be based on physical principles such as conservation of energy and continuity of power. These principles are inherently enforced by the bond graph modeling formalism. Often, however, physical components may be best modeled as piecewise continuous with discrete mode

A Mathematics Software Database Update.

Science.gov (United States)

Cunningham, R. S.; Smith, David A.

1987-01-01

Contains an update of an earlier listing of software for mathematics instruction at the college level. Topics are: advanced mathematics, algebra, calculus, differential equations, discrete mathematics, equation solving, general mathematics, geometry, linear and matrix algebra, logic, statistics and probability, and trigonometry. (PK)

A simple method for finding the scattering coefficients of quantum graphs

International Nuclear Information System (INIS)

Cottrell, Seth S.

2015-01-01

Quantum walks are roughly analogous to classical random walks, and similar to classical walks they have been used to find new (quantum) algorithms. When studying the behavior of large graphs or combinations of graphs, it is useful to find the response of a subgraph to signals of different frequencies. In doing so, we can replace an entire subgraph with a single vertex with variable scattering coefficients. In this paper, a simple technique for quickly finding the scattering coefficients of any discrete-time quantum graph will be presented. These scattering coefficients can be expressed entirely in terms of the characteristic polynomial of the graph’s time step operator. This is a marked improvement over previous techniques which have traditionally required finding eigenstates for a given eigenvalue, which is far more computationally costly. With the scattering coefficients we can easily derive the “impulse response” which is the key to predicting the response of a graph to any signal. This gives us a powerful set of tools for rapidly understanding the behavior of graphs or for reducing a large graph into its constituent subgraphs regardless of how they are connected

A librarian's guide to graphs, data and the semantic web

CERN Document Server

Powell, James

2015-01-01

Graphs are about connections, and are an important part of our connected and data-driven world. A Librarian's Guide to Graphs, Data and the Semantic Web is geared toward library and information science professionals, including librarians, software developers and information systems architects who want to understand the fundamentals of graph theory, how it is used to represent and explore data, and how it relates to the semantic web. This title provides a firm grounding in the field at a level suitable for a broad audience, with an emphasis on open source solutions and what problems these tools solve at a conceptual level, with minimal emphasis on algorithms or mathematics. The text will also be of special interest to data science librarians and data professionals, since it introduces many graph theory concepts by exploring data-driven networks from various scientific disciplines. The first two chapters consider graphs in theory and the science of networks, before the following chapters cover networks in vario...

Mathematical games, abstract games

CERN Document Server

Neto, Joao Pedro

2013-01-01

User-friendly, visually appealing collection offers both new and classic strategic board games. Includes abstract games for two and three players and mathematical games such as Nim and games on graphs.

Introduction of Hypermatrix and Operator Notation into a Discrete Mathematics Simulation Model of Malignant Tumour Response to Therapeutic Schemes In Vivo. Some Operator Properties

Directory of Open Access Journals (Sweden)

Georgios S. Stamatakos

2009-10-01

Full Text Available The tremendous rate of accumulation of experimental and clinical knowledge pertaining to cancer dictates the development of a theoretical framework for the meaningful integration of such knowledge at all levels of biocomplexity. In this context our research group has developed and partly validated a number of spatiotemporal simulation models of in vivo tumour growth and in particular tumour response to several therapeutic schemes. Most of the modeling modules have been based on discrete mathematics and therefore have been formulated in terms of rather complex algorithms (e.g. in pseudocode and actual computer code. However, such lengthy algorithmic descriptions, although sufficient from the mathematical point of view, may render it difficult for an interested reader to readily identify the sequence of the very basic simulation operations that lie at the heart of the entire model. In order to both alleviate this problem and at the same time provide a bridge to symbolic mathematics, we propose the introduction of the notion of hypermatrix in conjunction with that of a discrete operator into the already developed models. Using a radiotherapy response simulation example we demonstrate how the entire model can be considered as the sequential application of a number of discrete operators to a hypermatrix corresponding to the dynamics of the anatomic area of interest. Subsequently, we investigate the operators’ commutativity and outline the “summarize and jump” strategy aiming at efficiently and realistically address multilevel biological problems such as cancer. In order to clarify the actual effect of the composite discrete operator we present further simulation results which are in agreement with the outcome of the clinical study RTOG 83–02, thus strengthening the reliability of the model developed.

Discrete thoughts essays on mathematics, science, and philosophy

CERN Document Server

Kac, Mark; Schwartz, Jacob T

1992-01-01

This is a volume of essays and reviews that delightfully explore mathematics in all its moods — from the light and the witty, and humorous to serious, rational, and cerebral. Topics include: logic, combinatorics, statistics, economics, artificial intelligence, computer science, and applications of mathematics broadly. You will also find history and philosophy covered, including discussion of the work of Ulam, Kant, Heidegger among others. "...these papers reflect on mathematics and its influence on human society. They can help the specialist to notice what is going on around him, and they may lead educated people from other domains to a better understanding of mathematics. Many of these papers can advise educators how to form a modern mathematics education, which develops approved ideas and institutions...I admire the stimulating perspectives of the authors."---American Mathematical Society "‘Mathematicians, like Proust and everyone else, are at their best when writing about their first love’ … They a...

The Effects of Constructivist Learning Environment on Prospective Mathematics Teachers' Opinions

Science.gov (United States)

Narli, Serkan; Baser, Nes'e

2010-01-01

To explore the effects of constructivist learning environment on prospective teachers' opinions about "mathematics, department of mathematics, discrete mathematics, countable and uncountable infinity" taught under the subject of Cantorian Set Theory in discrete mathematics class, 60 first-year students in the Division of Mathematics…

Data assimilation a mathematical introduction

CERN Document Server

Law, Kody; Zygalakis, Konstantinos

2015-01-01

This book provides a systematic treatment of the mathematical underpinnings of work in data assimilation, covering both theoretical and computational approaches. Specifically the authors develop a unified mathematical framework in which a Bayesian formulation of the problem provides the bedrock for the derivation, development and analysis of algorithms; the many examples used in the text, together with the algorithms which are introduced and discussed, are all illustrated by the MATLAB software detailed in the book and made freely available online. The book is organized into nine chapters: the first contains a brief introduction to the mathematical tools around which the material is organized; the next four are concerned with discrete time dynamical systems and discrete time data; the last four are concerned with continuous time dynamical systems and continuous time data and are organized analogously to the corresponding discrete time chapters. This book is aimed at mathematical researchers interested in a sy...

Variational and PDE-Based Methods for Big Data Analysis, Classification and Image Processing Using Graphs

Science.gov (United States)

2015-01-01

Assistant for Calculus (winter 2011) xii CHAPTER 1 Introduction We present several methods, outlined in Chapters 3-5, for image processing and data...local calculus formulation [103] to generalize the continuous formulation to a (non-local) discrete setting, while other non-local versions for...graph-based model based on the Ginzburg-Landau functional in their work [9]. To define the functional on a graph, the spatial gradient is replaced by a

Computing the Gromov hyperbolicity constant of a discrete metric space

KAUST Repository

Ismail, Anas

2012-01-01

, and many other areas of research. The Gromov hyperbolicity constant of several families of graphs and geometric spaces has been determined. However, so far, the only known algorithm for calculating the Gromov hyperbolicity constant δ of a discrete metric

Mathematics without boundaries surveys in pure mathematics

CERN Document Server

Pardalos, Panos

2014-01-01

The contributions in this volume have been written by eminent scientists from the international mathematical community and present significant advances in several theories, methods and problems of Mathematical Analysis, Discrete Mathematics, Geometry and their Applications. The chapters focus on both old and recent developments in Functional Analysis, Harmonic Analysis, Complex Analysis, Operator Theory, Combinatorics, Functional Equations, Differential Equations as well as a variety of Applications. The book also contains some review works, which could prove particularly useful for a broader audience of readers in Mathematical Sciences, and especially to graduate students looking for the  latest information.

A Note on Longest Paths in Circular Arc Graphs

Directory of Open Access Journals (Sweden)

Joos Felix

2015-08-01

Full Text Available As observed by Rautenbach and Sereni [SIAM J. Discrete Math. 28 (2014 335-341] there is a gap in the proof of the theorem of Balister et al. [Combin. Probab. Comput. 13 (2004 311-317], which states that the intersection of all longest paths in a connected circular arc graph is nonempty. In this paper we close this gap.

Interacting particle systems on graphs

Science.gov (United States)

Sood, Vishal

In this dissertation, the dynamics of socially or biologically interacting populations are investigated. The individual members of the population are treated as particles that interact via links on a social or biological network represented as a graph. The effect of the structure of the graph on the properties of the interacting particle system is studied using statistical physics techniques. In the first chapter, the central concepts of graph theory and social and biological networks are presented. Next, interacting particle systems that are drawn from physics, mathematics and biology are discussed in the second chapter. In the third chapter, the random walk on a graph is studied. The mean time for a random walk to traverse between two arbitrary sites of a random graph is evaluated. Using an effective medium approximation it is found that the mean first-passage time between pairs of sites, as well as all moments of this first-passage time, are insensitive to the density of links in the graph. The inverse of the mean-first passage time varies non-monotonically with the density of links near the percolation transition of the random graph. Much of the behavior can be understood by simple heuristic arguments. Evolutionary dynamics, by which mutants overspread an otherwise uniform population on heterogeneous graphs, are studied in the fourth chapter. Such a process underlies' epidemic propagation, emergence of fads, social cooperation or invasion of an ecological niche by a new species. The first part of this chapter is devoted to neutral dynamics, in which the mutant genotype does not have a selective advantage over the resident genotype. The time to extinction of one of the two genotypes is derived. In the second part of this chapter, selective advantage or fitness is introduced such that the mutant genotype has a higher birth rate or a lower death rate. This selective advantage leads to a dynamical competition in which selection dominates for large populations

Using Graph and Vertex Entropy to Compare Empirical Graphs with Theoretical Graph Models

Directory of Open Access Journals (Sweden)

Tomasz Kajdanowicz

2016-09-01

Full Text Available Over the years, several theoretical graph generation models have been proposed. Among the most prominent are: the Erdős–Renyi random graph model, Watts–Strogatz small world model, Albert–Barabási preferential attachment model, Price citation model, and many more. Often, researchers working with real-world data are interested in understanding the generative phenomena underlying their empirical graphs. They want to know which of the theoretical graph generation models would most probably generate a particular empirical graph. In other words, they expect some similarity assessment between the empirical graph and graphs artificially created from theoretical graph generation models. Usually, in order to assess the similarity of two graphs, centrality measure distributions are compared. For a theoretical graph model this means comparing the empirical graph to a single realization of a theoretical graph model, where the realization is generated from the given model using an arbitrary set of parameters. The similarity between centrality measure distributions can be measured using standard statistical tests, e.g., the Kolmogorov–Smirnov test of distances between cumulative distributions. However, this approach is both error-prone and leads to incorrect conclusions, as we show in our experiments. Therefore, we propose a new method for graph comparison and type classification by comparing the entropies of centrality measure distributions (degree centrality, betweenness centrality, closeness centrality. We demonstrate that our approach can help assign the empirical graph to the most similar theoretical model using a simple unsupervised learning method.

Using Sorting Networks for Skill Building and Reasoning

Science.gov (United States)

Andre, Robert; Wiest, Lynda R.

2007-01-01

Sorting networks, used in graph theory, have instructional value as a skill- building tool as well as an interesting exploration in discrete mathematics. Students can practice mathematics facts and develop reasoning and logic skills with this topic. (Contains 4 figures.)

A mathematical analysis of the effects of Hebbian learning rules on the dynamics and structure of discrete-time random recurrent neural networks.

Science.gov (United States)

Siri, Benoît; Berry, Hugues; Cessac, Bruno; Delord, Bruno; Quoy, Mathias

2008-12-01

We present a mathematical analysis of the effects of Hebbian learning in random recurrent neural networks, with a generic Hebbian learning rule, including passive forgetting and different timescales, for neuronal activity and learning dynamics. Previous numerical work has reported that Hebbian learning drives the system from chaos to a steady state through a sequence of bifurcations. Here, we interpret these results mathematically and show that these effects, involving a complex coupling between neuronal dynamics and synaptic graph structure, can be analyzed using Jacobian matrices, which introduce both a structural and a dynamical point of view on neural network evolution. Furthermore, we show that sensitivity to a learned pattern is maximal when the largest Lyapunov exponent is close to 0. We discuss how neural networks may take advantage of this regime of high functional interest.

Magical mathematics the mathematical ideas that animate great magic tricks

CERN Document Server

Diaconis, Persi

2012-01-01

Magical Mathematics reveals the secrets of amazing, fun-to-perform card tricks--and the profound mathematical ideas behind them--that will astound even the most accomplished magician. Persi Diaconis and Ron Graham provide easy, step-by-step instructions for each trick, explaining how to set up the effect and offering tips on what to say and do while performing it. Each card trick introduces a new mathematical idea, and varying the tricks in turn takes readers to the very threshold of today's mathematical knowledge. For example, the Gilbreath Principle--a fantastic effect where the cards remain in control despite being shuffled--is found to share an intimate connection with the Mandelbrot set. Other card tricks link to the mathematical secrets of combinatorics, graph theory, number theory, topology, the Riemann hypothesis, and even Fermat's last theorem.

The Spectrum of Mathematical Models.

Science.gov (United States)

Karplus, Walter J.

1983-01-01

Mathematical modeling problems encountered in many disciplines are discussed in terms of the modeling process and applications of models. The models are classified according to three types of abstraction: continuous-space-continuous-time, discrete-space-continuous-time, and discrete-space-discrete-time. Limitations in different kinds of modeling…

Discrete and computational geometry

CERN Document Server

Devadoss, Satyan L

2011-01-01

Discrete geometry is a relatively new development in pure mathematics, while computational geometry is an emerging area in applications-driven computer science. Their intermingling has yielded exciting advances in recent years, yet what has been lacking until now is an undergraduate textbook that bridges the gap between the two. Discrete and Computational Geometry offers a comprehensive yet accessible introduction to this cutting-edge frontier of mathematics and computer science. This book covers traditional topics such as convex hulls, triangulations, and Voronoi diagrams, as well as more recent subjects like pseudotriangulations, curve reconstruction, and locked chains. It also touches on more advanced material, including Dehn invariants, associahedra, quasigeodesics, Morse theory, and the recent resolution of the Poincaré conjecture. Connections to real-world applications are made throughout, and algorithms are presented independently of any programming language. This richly illustrated textbook also fe...

Advances in discrete differential geometry

CERN Document Server

2016-01-01

This is one of the first books on a newly emerging field of discrete differential geometry and an excellent way to access this exciting area. It surveys the fascinating connections between discrete models in differential geometry and complex analysis, integrable systems and applications in computer graphics. The authors take a closer look at discrete models in differential geometry and dynamical systems. Their curves are polygonal, surfaces are made from triangles and quadrilaterals, and time is discrete. Nevertheless, the difference between the corresponding smooth curves, surfaces and classical dynamical systems with continuous time can hardly be seen. This is the paradigm of structure-preserving discretizations. Current advances in this field are stimulated to a large extent by its relevance for computer graphics and mathematical physics. This book is written by specialists working together on a common research project. It is about differential geometry and dynamical systems, smooth and discrete theories, ...

Mathematics and Computer Science: Exploring a Symbiotic Relationship

Science.gov (United States)

Bravaco, Ralph; Simonson, Shai

2004-01-01

This paper describes a "learning community" designed for sophomore computer science majors who are simultaneously studying discrete mathematics. The learning community consists of three courses: Discrete Mathematics, Data Structures and an Integrative Seminar/Lab. The seminar functions as a link that integrates the two disciplines. Participation…

Identification of parameters of discrete-continuous models

International Nuclear Information System (INIS)

Cekus, Dawid; Warys, Pawel

2015-01-01

In the paper, the parameters of a discrete-continuous model have been identified on the basis of experimental investigations and formulation of optimization problem. The discrete-continuous model represents a cantilever stepped Timoshenko beam. The mathematical model has been formulated and solved according to the Lagrange multiplier formalism. Optimization has been based on the genetic algorithm. The presented proceeding’s stages make the identification of any parameters of discrete-continuous systems possible

Identification of parameters of discrete-continuous models

Energy Technology Data Exchange (ETDEWEB)

Cekus, Dawid, E-mail: cekus@imipkm.pcz.pl; Warys, Pawel, E-mail: warys@imipkm.pcz.pl [Institute of Mechanics and Machine Design Foundations, Czestochowa University of Technology, Dabrowskiego 73, 42-201 Czestochowa (Poland)

2015-03-10

In the paper, the parameters of a discrete-continuous model have been identified on the basis of experimental investigations and formulation of optimization problem. The discrete-continuous model represents a cantilever stepped Timoshenko beam. The mathematical model has been formulated and solved according to the Lagrange multiplier formalism. Optimization has been based on the genetic algorithm. The presented proceeding’s stages make the identification of any parameters of discrete-continuous systems possible.

The origin of discrete particles

CERN Document Server

Bastin, T

2009-01-01

This book is a unique summary of the results of a long research project undertaken by the authors on discreteness in modern physics. In contrast with the usual expectation that discreteness is the result of mathematical tools for insertion into a continuous theory, this more basic treatment builds up the world from the discrimination of discrete entities. This gives an algebraic structure in which certain fixed numbers arise. As such, one agrees with the measured value of the fine-structure constant to one part in 10,000,000 (10 7 ). Sample Chapter(s). Foreword (56 KB). Chapter 1: Introduction

A scheme for designing extreme multistable discrete dynamical ...

Indian Academy of Sciences (India)

A scheme for designing extreme multistable discrete dynamical systems ... Abstract. In this paper, we propose a scheme for designing discrete extreme multistable systems coupling two identical dynamical systems. Existence ... Department of Applied Mathematics, University of Calcutta, 92 APC Road, Kolkata 700 009, India ...

On domination multisubdivision number of unicyclic graphs

Directory of Open Access Journals (Sweden)

Joanna Raczek

2018-01-01

Full Text Available The paper continues the interesting study of the domination subdivision number and the domination multisubdivision number. On the basis of the constructive characterization of the trees with the domination subdivision number equal to 3 given in [H. Aram, S.M. Sheikholeslami, O. Favaron, Domination subdivision number of trees, Discrete Math. 309 (2009, 622-628], we constructively characterize all connected unicyclic graphs with the domination multisubdivision number equal to 3. We end with further questions and open problems.

Approximation properties of fine hyperbolic graphs

Indian Academy of Sciences (India)

2016-08-26

Aug 26, 2016 ... In this paper, we propose a definition of approximation property which is called the metric invariant translation approximation property for a countable discrete metric space. Moreover, we use ... Department of Applied Mathematics, Shanghai Finance University, Shanghai 201209, People's Republic of China ...

Go Figure: Calculus Students' Use of Figures and Graphs in Technical Report Writing

Directory of Open Access Journals (Sweden)

Thomas J. Pfaff

2011-01-01

Full Text Available Understanding how to read and use graphs to communicate scientific and mathematical information is critical for STEM majors, as well as an important part of quantitative literacy. Our study suggests that first-semester calculus students do not know how to use graphs in a technical report without explicit instruction. Although not a surprising result, it leaves us wondering about when such skills are developed, and if calculus I is a place to start. Our work is now exploring the potential benefit on students' use of graphs by having them formally evaluate other students' reports.

建构主义教学法在离散数学教学中的应用初探%Application of Constructivism Teaching Method in Discrete Ma-thematics

Institute of Scientific and Technical Information of China (English)

刘峰; 唐存琛

2016-01-01

以建构主义教学理论为依据，采用抛锚方式应用于高校离散数学教学实践。通过对三个不同年级的跟踪实验，总结出建构主义教学法在离散数学教学中的优势。%The constructivism is introduced into Discrete Mathema-tics in this paper, which use the anchor teaching method to present the teaching contents. It is concluded that the constructivism theory can perform well in Discrete Mathematics by three experiments on diff erent grades.

Graph Theory Roots of Spatial Operators for Kinematics and Dynamics

Science.gov (United States)

Jain, Abhinandan

2011-01-01

Spatial operators have been used to analyze the dynamics of robotic multibody systems and to develop novel computational dynamics algorithms. Mass matrix factorization, inversion, diagonalization, and linearization are among several new insights obtained using such operators. While initially developed for serial rigid body manipulators, the spatial operators and the related mathematical analysis have been shown to extend very broadly including to tree and closed topology systems, to systems with flexible joints, links, etc. This work uses concepts from graph theory to explore the mathematical foundations of spatial operators. The goal is to study and characterize the properties of the spatial operators at an abstract level so that they can be applied to a broader range of dynamics problems. The rich mathematical properties of the kinematics and dynamics of robotic multibody systems has been an area of strong research interest for several decades. These properties are important to understand the inherent physical behavior of systems, for stability and control analysis, for the development of computational algorithms, and for model development of faithful models. Recurring patterns in spatial operators leads one to ask the more abstract question about the properties and characteristics of spatial operators that make them so broadly applicable. The idea is to step back from the specific application systems, and understand more deeply the generic requirements and properties of spatial operators, so that the insights and techniques are readily available across different kinematics and dynamics problems. In this work, techniques from graph theory were used to explore the abstract basis for the spatial operators. The close relationship between the mathematical properties of adjacency matrices for graphs and those of spatial operators and their kernels were established. The connections hold across very basic requirements on the system topology, the nature of the component

A Numerical Approach to Long Cycles in Graphs and Digraphs

Czech Academy of Sciences Publication Activity Database

Fiedler, Miroslav

2001-01-01

Roč. 235, - (2001), s. 233-236 ISSN 0012-365X R&D Projects: GA ČR GA201/98/0222 Institutional research plan: AV0Z1030915 Keywords : graph * diagraph * cycle * Hamilton cycle Subject RIV: BA - General Mathematics Impact factor: 0.301, year: 2001

Application of direct discrete method (DDM) to multigroup neutron transport problems

International Nuclear Information System (INIS)

Vosoughi, Naser; Salehi, Ali Akbar; Shahriari, Majid

2003-01-01

The Direct Discrete Method (DDM), which produced excellent results for one-group neutron transport problems, has been developed for multigroup energy. A multigroup neutron transport discrete equation has been produced for a cylindrical shape fuel element with and without associated coolant regions with two boundary conditions. The calculations are illustrated for two-group energy by graphs showing the fast and thermal fluxes. The validity of the results are tested against the results obtained by the ANISN code. (author)

A graph rewriting programming language for graph drawing

OpenAIRE

Rodgers, Peter

1998-01-01

This paper describes Grrr, a prototype visual graph drawing tool. Previously there were no visual languages for programming graph drawing algorithms despite the inherently visual nature of the process. The languages which gave a diagrammatic view of graphs were not computationally complete and so could not be used to implement complex graph drawing algorithms. Hence current graph drawing tools are all text based. Recent developments in graph rewriting systems have produced computationally com...

Evolutionary dynamics on graphs: Efficient method for weak selection

Science.gov (United States)

Fu, Feng; Wang, Long; Nowak, Martin A.; Hauert, Christoph

2009-04-01

Investigating the evolutionary dynamics of game theoretical interactions in populations where individuals are arranged on a graph can be challenging in terms of computation time. Here, we propose an efficient method to study any type of game on arbitrary graph structures for weak selection. In this limit, evolutionary game dynamics represents a first-order correction to neutral evolution. Spatial correlations can be empirically determined under neutral evolution and provide the basis for formulating the game dynamics as a discrete Markov process by incorporating a detailed description of the microscopic dynamics based on the neutral correlations. This framework is then applied to one of the most intriguing questions in evolutionary biology: the evolution of cooperation. We demonstrate that the degree heterogeneity of a graph impedes cooperation and that the success of tit for tat depends not only on the number of rounds but also on the degree of the graph. Moreover, considering the mutation-selection equilibrium shows that the symmetry of the stationary distribution of states under weak selection is skewed in favor of defectors for larger selection strengths. In particular, degree heterogeneity—a prominent feature of scale-free networks—generally results in a more pronounced increase in the critical benefit-to-cost ratio required for evolution to favor cooperation as compared to regular graphs. This conclusion is corroborated by an analysis of the effects of population structures on the fixation probabilities of strategies in general 2×2 games for different types of graphs. Computer simulations confirm the predictive power of our method and illustrate the improved accuracy as compared to previous studies.

Symmetric discrete coherent states for n-qubits

International Nuclear Information System (INIS)

Muñoz, C; Klimov, A B; Sánchez-Soto, L L

2012-01-01

We put forward a method of constructing discrete coherent states for n qubits. After establishing appropriate displacement operators, the coherent states appear as displaced versions of a fiducial vector that is fixed by imposing a number of natural symmetry requirements on its Q-function. Using these coherent states, we establish a partial order in the discrete phase space, which allows us to picture some n-qubit states as apparent distributions. We also analyze correlations in terms of sums of squared Q-functions. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Coherent states: mathematical and physical aspects’. (paper)

The local limit of the uniform spanning tree on dense graphs

Czech Academy of Sciences Publication Activity Database

Hladký, Jan; Nachmias, A.; Tran, Tuan

First Online: 10 January (2018) ISSN 0022-4715 R&D Projects: GA ČR GJ16-07822Y Keywords : uniform spanning tree * graph limits * Benjamini-Schramm convergence * graphon * branching process Subject RIV: BA - General Mathematics Impact factor: 1.349, year: 2016

Modeling flow and transport in fracture networks using graphs

Science.gov (United States)

Karra, S.; O'Malley, D.; Hyman, J. D.; Viswanathan, H. S.; Srinivasan, G.

2018-03-01

Fractures form the main pathways for flow in the subsurface within low-permeability rock. For this reason, accurately predicting flow and transport in fractured systems is vital for improving the performance of subsurface applications. Fracture sizes in these systems can range from millimeters to kilometers. Although modeling flow and transport using the discrete fracture network (DFN) approach is known to be more accurate due to incorporation of the detailed fracture network structure over continuum-based methods, capturing the flow and transport in such a wide range of scales is still computationally intractable. Furthermore, if one has to quantify uncertainty, hundreds of realizations of these DFN models have to be run. To reduce the computational burden, we solve flow and transport on a graph representation of a DFN. We study the accuracy of the graph approach by comparing breakthrough times and tracer particle statistical data between the graph-based and the high-fidelity DFN approaches, for fracture networks with varying number of fractures and degree of heterogeneity. Due to our recent developments in capabilities to perform DFN high-fidelity simulations on fracture networks with large number of fractures, we are in a unique position to perform such a comparison. We show that the graph approach shows a consistent bias with up to an order of magnitude slower breakthrough when compared to the DFN approach. We show that this is due to graph algorithm's underprediction of the pressure gradients across intersections on a given fracture, leading to slower tracer particle speeds between intersections and longer travel times. We present a bias correction methodology to the graph algorithm that reduces the discrepancy between the DFN and graph predictions. We show that with this bias correction, the graph algorithm predictions significantly improve and the results are very accurate. The good accuracy and the low computational cost, with O (104) times lower times than

Vision in elementary mathematics

CERN Document Server

Sawyer, W W

2003-01-01

Sure-fire techniques of visualizing, dramatizing, and analyzing numbers promise to attract and retain students' attention and understanding. Topics include basic multiplication and division, algebra, word problems, graphs, negative numbers, fractions, many other practical applications of elementary mathematics. 1964 ed. Answers to Problems.

The heat kernel as the pagerank of a graph

Science.gov (United States)

Chung, Fan

2007-01-01

The concept of pagerank was first started as a way for determining the ranking of Web pages by Web search engines. Based on relations in interconnected networks, pagerank has become a major tool for addressing fundamental problems arising in general graphs, especially for large information networks with hundreds of thousands of nodes. A notable notion of pagerank, introduced by Brin and Page and denoted by PageRank, is based on random walks as a geometric sum. In this paper, we consider a notion of pagerank that is based on the (discrete) heat kernel and can be expressed as an exponential sum of random walks. The heat kernel satisfies the heat equation and can be used to analyze many useful properties of random walks in a graph. A local Cheeger inequality is established, which implies that, by focusing on cuts determined by linear orderings of vertices using the heat kernel pageranks, the resulting partition is within a quadratic factor of the optimum. This is true, even if we restrict the volume of the small part separated by the cut to be close to some specified target value. This leads to a graph partitioning algorithm for which the running time is proportional to the size of the targeted volume (instead of the size of the whole graph).

Algorithmic Principles of Mathematical Programming

NARCIS (Netherlands)

Faigle, Ulrich; Kern, Walter; Still, Georg

2002-01-01

Algorithmic Principles of Mathematical Programming investigates the mathematical structures and principles underlying the design of efficient algorithms for optimization problems. Recent advances in algorithmic theory have shown that the traditionally separate areas of discrete optimization, linear

Mathematical Background of Public Key Cryptography

DEFF Research Database (Denmark)

Frey, Gerhard; Lange, Tanja

2005-01-01

The two main systems used for public key cryptography are RSA and protocols based on the discrete logarithm problem in some cyclic group. We focus on the latter problem and state cryptographic protocols and mathematical background material.......The two main systems used for public key cryptography are RSA and protocols based on the discrete logarithm problem in some cyclic group. We focus on the latter problem and state cryptographic protocols and mathematical background material....

A Novel Discrete Differential Evolution Algorithm for the Vehicle Routing Problem in B2C E-Commerce

Science.gov (United States)

Xia, Chao; Sheng, Ying; Jiang, Zhong-Zhong; Tan, Chunqiao; Huang, Min; He, Yuanjian

2015-12-01

In this paper, a novel discrete differential evolution (DDE) algorithm is proposed to solve the vehicle routing problems (VRP) in B2C e-commerce, in which VRP is modeled by the incomplete graph based on the actual urban road system. First, a variant of classical VRP is described and a mathematical programming model for the variant is given. Second, the DDE is presented, where individuals are represented as the sequential encoding scheme, and a novel reparation operator is employed to repair the infeasible solutions. Furthermore, a FLOYD operator for dealing with the shortest route is embedded in the proposed DDE. Finally, an extensive computational study is carried out in comparison with the predatory search algorithm and genetic algorithm, and the results show that the proposed DDE is an effective algorithm for VRP in B2C e-commerce.

Discrete pseudo-integrals

Czech Academy of Sciences Publication Activity Database

Mesiar, Radko; Li, J.; Pap, E.

2013-01-01

Roč. 54, č. 3 (2013), s. 357-364 ISSN 0888-613X R&D Projects: GA ČR GAP402/11/0378 Institutional support: RVO:67985556 Keywords : concave integral * pseudo-addition * pseudo-multiplication Subject RIV: BA - General Mathematics Impact factor: 1.977, year: 2013 http://library.utia.cas.cz/separaty/2013/E/mesiar-discrete pseudo-integrals.pdf

Bond graph modeling and LQG/LTR controller design of magnetically levitation systems

International Nuclear Information System (INIS)

Kim, Jong Shik; Park, Jeon Soo

1991-01-01

A logical and systematic procedure to derive a mathematical model for magnetically levitation (MAGLEV) systems with a combined lift and guidance is developed by using bond graph modeling techniques. First, bond graph is contructed for the 1 st -dimensional MAGLEV system in which three subsystems (energy feeding, track and vehicle) are considered. And, the 2 nd -dimensional MAGLEV system in which lift and guidance dynamics are coupled is modeled by using the concept of multi-port field in bond graph languages. Finally, the LQG/LTR control system is designed for a multivariable MAGLEV system with stagger configuration type. In this paper, it has been shown that the bond graph is an excellent effective method for modeling multi-energy domain systems such as MAGLEV systems with uncertainties such as mass variations, track irregularities and wind gusts. (Author)

Bond graph modeling and LQG/LTR controller design of magnetically levitation systems

Energy Technology Data Exchange (ETDEWEB)

Kim, Jong Shik; Park, Jeon Soo [Busan National Univ. (Korea, Republic of)

1991-09-01

A logical and systematic procedure to derive a mathematical model for magnetically levitation (MAGLEV) systems with a combined lift and guidance is developed by using bond graph modeling techniques. First, bond graph is contructed for the 1{sup st}-dimensional MAGLEV system in which three subsystems (energy feeding, track and vehicle) are considered. And, the 2{sup nd}-dimensional MAGLEV system in which lift and guidance dynamics are coupled is modeled by using the concept of multi-port field in bond graph languages. Finally, the LQG/LTR control system is designed for a multivariable MAGLEV system with stagger configuration type. In this paper, it has been shown that the bond graph is an excellent effective method for modeling multi-energy domain systems such as MAGLEV systems with uncertainties such as mass variations, track irregularities and wind gusts. (Author).

Examining Graphing Calculator Affordances in Learning Pre-Calculus among Undergraduate Students

Science.gov (United States)

Nzuki, Francis

2016-01-01

This study examines graphing calculator affordances in learning mathematics among college precalculus students. The study draws from the Cognitive Load Theory (CLT) and the "Intelligent Technology" theoretical framework proposed by Salomon, Perkins, and Globerson (1991). From these perspectives the effects "with" the graphing…

Fostering Positive Attitude in Probability Learning Using Graphing Calculator

Science.gov (United States)

Tan, Choo-Kim; Harji, Madhubala Bava; Lau, Siong-Hoe

2011-01-01

Although a plethora of research evidence highlights positive and significant outcomes of the incorporation of the Graphing Calculator (GC) in mathematics education, its use in the teaching and learning process appears to be limited. The obvious need to revisit the teaching and learning of Probability has resulted in this study, i.e. to incorporate…

Graph Aggregation

NARCIS (Netherlands)

Endriss, U.; Grandi, U.

Graph aggregation is the process of computing a single output graph that constitutes a good compromise between several input graphs, each provided by a different source. One needs to perform graph aggregation in a wide variety of situations, e.g., when applying a voting rule (graphs as preference

Problem solving through recreational mathematics

CERN Document Server

Averbach, Bonnie

1999-01-01

Historically, many of the most important mathematical concepts arose from problems that were recreational in origin. This book takes advantage of that fact, using recreational mathematics - problems, puzzles and games - to teach students how to think critically. Encouraging active participation rather than just observation, the book focuses less on mathematical results than on how these results can be applied to thinking about problems and solving them. Each chapter contains a diverse array of problems in such areas as logic, number and graph theory, two-player games of strategy, solitaire ga

Evolutionary stability of mixed strategies on graphs

International Nuclear Information System (INIS)

Li, Yan; Liu, Xinsheng; Claussen, Jens Christian

2016-01-01

Up to the present time, the study of evolutionary dynamics mostly focused on pure strategy games in finite discrete strategy space, either in well-mixed or structured populations. In this paper, we study mixed strategy games in continuous strategy space on graphs of degree k . Each player is arranged on a vertex of the graph. The edges denote the interaction between two individuals. In the limit of weak selection, we first derive the payoff functions of two mixed strategies under three different updating rules, named birth–death, death–birth and imitation. Then we obtain the conditions for a strategy being a continuously stable strategy (CSS), and we also confirm that the equilibrium distribution corresponding to the CSS is neighborhood attracting and strongly uninvadable. Finally, we apply our theory to the prisoner’s dilemma and the snowdrift game to obtain possible CSS. Simulations are performed for the two special games and the results are well consistent with the conclusions we made. (paper)

Graph-based geometric-iconic guide-wire tracking.

Science.gov (United States)

Honnorat, Nicolas; Vaillant, Régis; Paragios, Nikos

2011-01-01

In this paper we introduce a novel hybrid graph-based approach for Guide-wire tracking. The image support is captured by steerable filters and improved through tensor voting. Then, a graphical model is considered that represents guide-wire extraction/tracking through a B-spline control-point model. Points with strong geometric interest (landmarks) are automatically determined and anchored to such a representation. Tracking is then performed through discrete MRFs that optimize the spatio-temporal positions of the control points while establishing landmark temporal correspondences. Promising results demonstrate the potentials of our method.

Modelling and nonlinear shock waves for binary gas mixtures by the discrete Boltzmann equation with multiple collisions

International Nuclear Information System (INIS)

Bianchi, M.P.

1991-01-01

The discrete Boltzmann equation is a mathematical model in the kinetic theory of gases which defines the time and space evolution of a system of gas particles with a finite number of selected velocities. Discrete kinetic theory is an interesting field of research in mathematical physics and applied mathematics for several reasons. One of the relevant fields of application of the discrete Boltzmann equation is the analysis of nonlinear shock wave phenomena. Here, a new multiple collision regular plane model for binary gas mixtures is proposed within the discrete theory of gases and applied to the analysis of the classical problems of shock wave propagation

Integrating concepts and skills: Slope and kinematics graphs

Science.gov (United States)

Tonelli, Edward P., Jr.

The concept of force is a foundational idea in physics. To predict the results of applying forces to objects, a student must be able to interpret data representing changes in distance, time, speed, and acceleration. Comprehension of kinematics concepts requires students to interpret motion graphs, where rates of change are represented as slopes of line segments. Studies have shown that majorities of students who show proficiency with mathematical concepts fail accurately to interpret motion graphs. The primary aim of this study was to examine how students apply their knowledge of slope when interpreting kinematics graphs. To answer the research questions a mixed methods research design, which included a survey and interviews, was adopted. Ninety eight (N=98) high school students completed surveys which were quantitatively analyzed along with qualitative information collected from interviews of students (N=15) and teachers ( N=2). The study showed that students who recalled methods for calculating slopes and speeds calculated slopes accurately, but calculated speeds inaccurately. When comparing the slopes and speeds, most students resorted to calculating instead of visual inspection. Most students recalled and applied memorized rules. Students who calculated slopes and speeds inaccurately failed to recall methods of calculating slopes and speeds, but when comparing speeds, these students connected the concepts of distance and time to the line segments and the rates of change they represented. This study's findings will likely help mathematics and science educators to better assist their students to apply their knowledge of the definition of slope and skills in kinematics concepts.

Tailored graph ensembles as proxies or null models for real networks II: results on directed graphs

International Nuclear Information System (INIS)

Roberts, E S; Coolen, A C C; Schlitt, T

2011-01-01

We generate new mathematical tools with which to quantify the macroscopic topological structure of large directed networks. This is achieved via a statistical mechanical analysis of constrained maximum entropy ensembles of directed random graphs with prescribed joint distributions for in- and out-degrees and prescribed degree-degree correlation functions. We calculate exact and explicit formulae for the leading orders in the system size of the Shannon entropies and complexities of these ensembles, and for information-theoretic distances. The results are applied to data on gene regulation networks.

Mathematics for electronic technology

CERN Document Server

Howson, D P

1975-01-01

Mathematics for Electronic Technology is a nine-chapter book that begins with the elucidation of the introductory concepts related to use of mathematics in electronic engineering, including differentiation, integration, partial differentiation, infinite series, vectors, vector algebra, and surface, volume and line integrals. Subsequent chapters explore the determinants, differential equations, matrix analysis, complex variable, topography, graph theory, and numerical analysis used in this field. The use of Fourier method for harmonic analysis and the Laplace transform is also described. The ma

Proxy Graph: Visual Quality Metrics of Big Graph Sampling.

Science.gov (United States)

Nguyen, Quan Hoang; Hong, Seok-Hee; Eades, Peter; Meidiana, Amyra

2017-06-01

Data sampling has been extensively studied for large scale graph mining. Many analyses and tasks become more efficient when performed on graph samples of much smaller size. The use of proxy objects is common in software engineering for analysis and interaction with heavy objects or systems. In this paper, we coin the term 'proxy graph' and empirically investigate how well a proxy graph visualization can represent a big graph. Our investigation focuses on proxy graphs obtained by sampling; this is one of the most common proxy approaches. Despite the plethora of data sampling studies, this is the first evaluation of sampling in the context of graph visualization. For an objective evaluation, we propose a new family of quality metrics for visual quality of proxy graphs. Our experiments cover popular sampling techniques. Our experimental results lead to guidelines for using sampling-based proxy graphs in visualization.

Global spectral graph wavelet signature for surface analysis of carpal bones

Science.gov (United States)

Masoumi, Majid; Rezaei, Mahsa; Ben Hamza, A.

2018-02-01

Quantitative shape comparison is a fundamental problem in computer vision, geometry processing and medical imaging. In this paper, we present a spectral graph wavelet approach for shape analysis of carpal bones of the human wrist. We employ spectral graph wavelets to represent the cortical surface of a carpal bone via the spectral geometric analysis of the Laplace-Beltrami operator in the discrete domain. We propose global spectral graph wavelet (GSGW) descriptor that is isometric invariant, efficient to compute, and combines the advantages of both low-pass and band-pass filters. We perform experiments on shapes of the carpal bones of ten women and ten men from a publicly-available database of wrist bones. Using one-way multivariate analysis of variance (MANOVA) and permutation testing, we show through extensive experiments that the proposed GSGW framework gives a much better performance compared to the global point signature embedding approach for comparing shapes of the carpal bones across populations.

An Out-of-Math Experience: Einstein, Relativity, and the Developmental Mathematics Student.

Science.gov (United States)

Fiore, Greg

2000-01-01

Discusses Einstein's special relativity theory and some of the developmental mathematics involved. Presents motivational classroom materials used in discussing relative-motion problems, evaluating a radical expression, graphing with asymptotes, interpreting a graph, studying variation, and solving literal and radical equations. (KHR)

An analysis of the Simpson Discrete Hartley transform | Ramsunder ...

African Journals Online (AJOL)

The relatively new Simpson Discrete Hartley Transform (SDHT) has interesting mathematical properties, which are crucial for applications. These are developed and proved in this paper. This analysis gives one a comprehensive understanding of the transform. Mathematics Subject Classication (2010): 43A32. Key words: ...

A Parallel Framework with Block Matrices of a Discrete Fourier Transform for Vector-Valued Discrete-Time Signals

Directory of Open Access Journals (Sweden)

Pablo Soto-Quiros

2015-01-01

Full Text Available This paper presents a parallel implementation of a kind of discrete Fourier transform (DFT: the vector-valued DFT. The vector-valued DFT is a novel tool to analyze the spectra of vector-valued discrete-time signals. This parallel implementation is developed in terms of a mathematical framework with a set of block matrix operations. These block matrix operations contribute to analysis, design, and implementation of parallel algorithms in multicore processors. In this work, an implementation and experimental investigation of the mathematical framework are performed using MATLAB with the Parallel Computing Toolbox. We found that there is advantage to use multicore processors and a parallel computing environment to minimize the high execution time. Additionally, speedup increases when the number of logical processors and length of the signal increase.

Benchmarking Measures of Network Controllability on Canonical Graph Models

Science.gov (United States)

Wu-Yan, Elena; Betzel, Richard F.; Tang, Evelyn; Gu, Shi; Pasqualetti, Fabio; Bassett, Danielle S.

2018-03-01

The control of networked dynamical systems opens the possibility for new discoveries and therapies in systems biology and neuroscience. Recent theoretical advances provide candidate mechanisms by which a system can be driven from one pre-specified state to another, and computational approaches provide tools to test those mechanisms in real-world systems. Despite already having been applied to study network systems in biology and neuroscience, the practical performance of these tools and associated measures on simple networks with pre-specified structure has yet to be assessed. Here, we study the behavior of four control metrics (global, average, modal, and boundary controllability) on eight canonical graphs (including Erdős-Rényi, regular, small-world, random geometric, Barábasi-Albert preferential attachment, and several modular networks) with different edge weighting schemes (Gaussian, power-law, and two nonparametric distributions from brain networks, as examples of real-world systems). We observe that differences in global controllability across graph models are more salient when edge weight distributions are heavy-tailed as opposed to normal. In contrast, differences in average, modal, and boundary controllability across graph models (as well as across nodes in the graph) are more salient when edge weight distributions are less heavy-tailed. Across graph models and edge weighting schemes, average and modal controllability are negatively correlated with one another across nodes; yet, across graph instances, the relation between average and modal controllability can be positive, negative, or nonsignificant. Collectively, these findings demonstrate that controllability statistics (and their relations) differ across graphs with different topologies and that these differences can be muted or accentuated by differences in the edge weight distributions. More generally, our numerical studies motivate future analytical efforts to better understand the mathematical

Approximation properties of fine hyperbolic graphs

Indian Academy of Sciences (India)

2010 Mathematics Subject Classification. 46L07. 1. Introduction. Given a countable discrete group G, some nice approximation properties for the reduced. C∗-algebras C∗ r (G) can give us the approximation properties of G. For example, Lance. [7] proved that the nuclearity of C∗ r (G) is equivalent to the amenability of G; ...

New Challenges in the Teaching of Mathematics.

Science.gov (United States)

Bourguignon, Jean Pierre

The manifold but discrete presence of mathematics in many objects or services imposes new constraints to the teaching of mathematics. If citizens need to be comfortable in various situations with a variety of mathematical tools, the learning of mathematics requires that one starts with simple concepts. This paper proposes some solutions to solve…

The approximate Loebl-Komlós-Sós Conjecture II: The rough structure of LKS graphs

Czech Academy of Sciences Publication Activity Database

Hladký, Jan; Komlós, J.; Piguet, Diana; Simonovits, M.; Stein, M.; Szemerédi, E.

2017-01-01

Roč. 31, č. 2 (2017), s. 983-1016 ISSN 0895-4801 R&D Projects: GA MŠk(CZ) 1M0545 EU Projects: European Commission(XE) 628974 - PAECIDM Institutional support: RVO:67985840 ; RVO:67985807 Keywords : extremal graph theory * Loebl–Komlós–Sós conjecture * regularity lemma Subject RIV: BA - General Mathematics; BA - General Mathematics (UIVT-O) OBOR OECD: Pure mathematics; Pure mathematics (UIVT-O) Impact factor: 0.755, year: 2016 http://epubs.siam.org/doi/10.1137/140982854

The approximate Loebl-Komlós-Sós Conjecture II: The rough structure of LKS graphs

Czech Academy of Sciences Publication Activity Database

Hladký, Jan; Komlós, J.; Piguet, Diana; Simonovits, M.; Stein, M.; Szemerédi, E.

2017-01-01

Roč. 31, č. 2 (2017), s. 983-1016 ISSN 0895-4801 R&D Projects: GA MŠk(CZ) 1M0545 EU Projects: European Commission(XE) 628974 - PAECIDM Institutional support: RVO:67985840 ; RVO:67985807 Keywords : extremal graph theory * Loebl–Komlós–Sós conjecture * regularity lemma Subject RIV: BA - General Mathematics ; BA - General Mathematics (UIVT-O) OBOR OECD: Pure mathematics ; Pure mathematics (UIVT-O) Impact factor: 0.755, year: 2016 http://epubs.siam.org/doi/10.1137/140982854

Resistance and relatedness on an evolutionary graph

Science.gov (United States)

Maciejewski, Wes

2012-01-01

When investigating evolution in structured populations, it is often convenient to consider the population as an evolutionary graph—individuals as nodes, and whom they may act with as edges. There has, in recent years, been a surge of interest in evolutionary graphs, especially in the study of the evolution of social behaviours. An inclusive fitness framework is best suited for this type of study. A central requirement for an inclusive fitness analysis is an expression for the genetic similarity between individuals residing on the graph. This has been a major hindrance for work in this area as highly technical mathematics are often required. Here, I derive a result that links genetic relatedness between haploid individuals on an evolutionary graph to the resistance between vertices on a corresponding electrical network. An example that demonstrates the potential computational advantage of this result over contemporary approaches is provided. This result offers more, however, to the study of population genetics than strictly computationally efficient methods. By establishing a link between gene transfer and electric circuit theory, conceptualizations of the latter can enhance understanding of the former. PMID:21849384

Chromatic graph theory

CERN Document Server

Chartrand, Gary; Rosen, Kenneth H

2008-01-01

Beginning with the origin of the four color problem in 1852, the field of graph colorings has developed into one of the most popular areas of graph theory. Introducing graph theory with a coloring theme, Chromatic Graph Theory explores connections between major topics in graph theory and graph colorings as well as emerging topics. This self-contained book first presents various fundamentals of graph theory that lie outside of graph colorings, including basic terminology and results, trees and connectivity, Eulerian and Hamiltonian graphs, matchings and factorizations, and graph embeddings. The remainder of the text deals exclusively with graph colorings. It covers vertex colorings and bounds for the chromatic number, vertex colorings of graphs embedded on surfaces, and a variety of restricted vertex colorings. The authors also describe edge colorings, monochromatic and rainbow edge colorings, complete vertex colorings, several distinguishing vertex and edge colorings, and many distance-related vertex coloring...

Better bounds for incremental frequency allocation in bipartite graphs

Czech Academy of Sciences Publication Activity Database

Chrobak, M.; Jeż, Łukasz; Sgall, J.

2013-01-01

Roč. 514, 25 November (2013), s. 75-83 ISSN 0304-3975 R&D Projects: GA AV ČR IAA100190902; GA ČR GBP202/12/G061 Institutional support: RVO:67985840 Keywords : online algorithms * frequency allocation * graph algorithms Subject RIV: BA - General Mathematics Impact factor: 0.516, year: 2013 http://www.sciencedirect.com/science/article/pii/S0304397512004781

Mathematics for the liberal arts

CERN Document Server

Brown, Jason I

2014-01-01

The Math in Your Life Health, Safety, and Mathematics Found in Translation The Essentials of Conversion Making Sense of Your World with Statistics Summarizing Data with a Few Good Numbers Estimating Unknowns Leading You Down the Garden Path with Statistics Visualizing with Mathematics Seeing Data A Graph Is Worth a Thousand Words Money and Risk Money - Now or Later Risk Taking and Probability The Life in Your Math! Deciding to Make the Best Decisions Making the Right Choices for You Game Theory - Coming Out on Top Making Joint Decisions Art Imitating Math The Math that Makes the Art Believing What You See (or Not) The Mathematics of Sound (and the Sound of Mathematics) The Mathematics of Listening The Mathematics of Composing Solving Musical Mysteries with MSI (Math Scene Investigations) Late Night Mathematics - Humor and Philosophy Laughing with Mathematics The Limits of Mathematics Bibliography Index Review questions appear at the end of each chapter.

Coloring graphs from lists with bounded size of their union

Czech Academy of Sciences Publication Activity Database

Král´, D.; Sgall, Jiří

2005-01-01

Roč. 49, č. 3 (2005), s. 177-186 ISSN 0364-9024 R&D Projects: GA ČR(CZ) GA201/01/1195; GA MŠk(CZ) LN00A056 Institutional research plan: CEZ:AV0Z10190503 Keywords : graph coloring * list coloring Subject RIV: BA - General Mathematics Impact factor: 0.319, year: 2005

Proceedings – Mathematical Sciences | Indian Academy of Sciences

Indian Academy of Sciences (India)

Home; Journals; Proceedings – Mathematical Sciences. José M Sigarreta. Articles written in Proceedings – Mathematical Sciences. Volume 120 Issue 5 November 2010 pp 593-609. Gromov Hyperbolicity in Cartesian Product Graphs · Junior Michel José M Rodríguez José M Sigarreta María Villeta · More Details Abstract ...

Quantum dynamics via Planck-scale-stepped action-carrying 'Graph Paths'

CERN Document Server

Chew, Geoffrey Foucar

2003-01-01

A divergence-free, parameter-free, path-based discrete-time quantum dynamics is designed to not only enlarge the achievements of general relativity and the standard particle model, by approximations at spacetime scales far above Planck scale while far below Hubble scale, but to allow tackling of hitherto inaccessible questions. ''Path space'' is larger than and precursor to Hilbert-space basis. The wave-function-propagating paths are action-carrying structured graphs-cubic and quartic structured vertices connected by structured ''fermionic'' or ''bosonic'' ''particle'' and ''nonparticle'' arcs. A Planck-scale path step determines the gravitational constant while controlling all graph structure. The basis of the theory's (zero-rest-mass) elementary-particle Hilbert space (which includes neither gravitons nor scalar bosons) resides in particle arcs. Nonparticle arcs within a path are responsible for energy and rest mass.

Radiative transfer on discrete spaces

CERN Document Server

Preisendorfer, Rudolph W; Stark, M; Ulam, S

1965-01-01

Pure and Applied Mathematics, Volume 74: Radiative Transfer on Discrete Spaces presents the geometrical structure of natural light fields. This book describes in detail with mathematical precision the radiometric interactions of light-scattering media in terms of a few well established principles.Organized into four parts encompassing 15 chapters, this volume begins with an overview of the derivations of the practical formulas and the arrangement of formulas leading to numerical solution procedures of radiative transfer problems in plane-parallel media. This text then constructs radiative tran

Current problems in applied mathematics and mathematical physics

Science.gov (United States)

Samarskii, A. A.

Papers are presented on such topics as mathematical models in immunology, mathematical problems of medical computer tomography, classical orthogonal polynomials depending on a discrete variable, and boundary layer methods for singular perturbation problems in partial derivatives. Consideration is also given to the computer simulation of supernova explosion, nonstationary internal waves in a stratified fluid, the description of turbulent flows by unsteady solutions of the Navier-Stokes equations, and the reduced Galerkin method for external diffraction problems using the spline approximation of fields.

Mathematics make microbes beautiful, beneficial, and bountiful.

Science.gov (United States)

Jungck, John R

2012-01-01

Microbiology is a rich area for visualizing the importance of mathematics in terms of designing experiments, data mining, testing hypotheses, and visualizing relationships. Historically, Nobel Prizes have acknowledged the close interplay between mathematics and microbiology in such examples as the fluctuation test and mutation rates using Poisson statistics by Luria and Delbrück and the use of graph theory of polyhedra by Caspar and Klug. More and more contemporary microbiology journals feature mathematical models, computational algorithms and heuristics, and multidimensional visualizations. While revolutions in research have driven these initiatives, a commensurate effort needs to be made to incorporate much more mathematics into the professional preparation of microbiologists. In order not to be daunting to many educators, a Bloom-like "Taxonomy of Quantitative Reasoning" is shared with explicit examples of microbiological activities for engaging students in (a) counting, measuring, calculating using image analysis of bacterial colonies and viral infections on variegated leaves, measurement of fractal dimensions of beautiful colony morphologies, and counting vertices, edges, and faces on viral capsids and using graph theory to understand self assembly; (b) graphing, mapping, ordering by applying linear, exponential, and logistic growth models of public health and sanitation problems, revisiting Snow's epidemiological map of cholera with computational geometry, and using interval graphs to do complementation mapping, deletion mapping, food webs, and microarray heatmaps; (c) problem solving by doing gene mapping and experimental design, and applying Boolean algebra to gene regulation of operons; (d) analysis of the "Bacterial Bonanza" of microbial sequence and genomic data using bioinformatics and phylogenetics; (e) hypothesis testing-again with phylogenetic trees and use of Poisson statistics and the Luria-Delbrück fluctuation test; and (f) modeling of

A discretized algorithm for the solution of a constrained, continuous ...

African Journals Online (AJOL)

A discretized algorithm for the solution of a constrained, continuous quadratic control problem. ... The results obtained show that the Discretized constrained algorithm (DCA) is much more accurate and more efficient than some of these techniques, particularly the FSA. Journal of the Nigerian Association of Mathematical ...

PUZZLES – A CREATIVE WAY OF DEVELOPMENT OF LOGICAL THINKING

Directory of Open Access Journals (Sweden)

Milková, Eva

2011-12-01

Full Text Available Logical thinking of students should be enhanced at all levels of their studies. There are many possibilities how to achieve it. In the paper one possible way within the subjects “Discrete Mathematics” and “Discrete Methods and Optimization” dealing with graph theory and combinatorial optimization will be presented. These mathematical disciplines are powerful tools for teachers allowing them to develop logical thinking of students, increase their imagination and make them familiar with solutions to various problems. Thanks the knowledge gained within the subjects students should be able to describe various practical situations with the aid of graphs, solve the given problem expressed by the graph, and translate the solution back into the initial situation. Student engagement is crucial for successful education. Practical tasks and puzzles attract students to know more about the explained subject matter and to apply gained knowledge. There are an endless number of enjoyable tasks, puzzles and logic problems in books like “Mathematics is Fun”, in riddles magazines and on the Internet. In the paper, as an inspiration, four puzzles developing logical thinking appropriate to be solved using graph theory and combinatorial optimization will be introduced. On these puzzles of different level of difficulty the students’ ability to find out the appropriate graph-representation of the given task and solve it will be discussed as well. The author of the paper has been prepared with her students various multimedia applications dealing with objects appropriate to subject matter for more than 15 years. In the paper we also discuss a benefit of multimedia applications used as a support of subjects “Discrete Mathematics” and “Discrete Methods and Optimization”.

Proceedings – Mathematical Sciences | Indian Academy of Sciences

Indian Academy of Sciences (India)

Home; Journals; Proceedings – Mathematical Sciences; Volume 123; Issue 2. Notes on Discrete Subgroups of Möbius Transformations ... Department of Applied Mathematics, Hunan University, Changsha 410082, People's Republic of China; School of Mathematics and Computational Science, Wuyi University, Jiangmen, ...

Mathematical modelling

CERN Document Server

2016-01-01

This book provides a thorough introduction to the challenge of applying mathematics in real-world scenarios. Modelling tasks rarely involve well-defined categories, and they often require multidisciplinary input from mathematics, physics, computer sciences, or engineering. In keeping with this spirit of modelling, the book includes a wealth of cross-references between the chapters and frequently points to the real-world context. The book combines classical approaches to modelling with novel areas such as soft computing methods, inverse problems, and model uncertainty. Attention is also paid to the interaction between models, data and the use of mathematical software. The reader will find a broad selection of theoretical tools for practicing industrial mathematics, including the analysis of continuum models, probabilistic and discrete phenomena, and asymptotic and sensitivity analysis.

Graph sampling

OpenAIRE

Zhang, L.-C.; Patone, M.

2017-01-01

We synthesise the existing theory of graph sampling. We propose a formal definition of sampling in finite graphs, and provide a classification of potential graph parameters. We develop a general approach of Horvitz–Thompson estimation to T-stage snowball sampling, and present various reformulations of some common network sampling methods in the literature in terms of the outlined graph sampling theory.

A HYBRID ALGORITHM FOR THE ROBUST GRAPH COLORING PROBLEM

Directory of Open Access Journals (Sweden)

Román Anselmo Mora Gutiérrez

2016-08-01

Full Text Available A hybridalgorithm which combines mathematical programming techniques (Kruskal’s algorithm and the strategy of maintaining arc consistency to solve constraint satisfaction problem “CSP” and heuristic methods (musical composition method and DSATUR to resolve the robust graph coloring problem (RGCP is proposed in this paper. Experimental result shows that this algorithm is better than the other algorithms presented on the literature.

Fast Discrete Fourier Transform Computations Using the Reduced Adder Graph Technique

Directory of Open Access Journals (Sweden)

Andrew G. Dempster

2007-01-01

Full Text Available It has recently been shown that the n-dimensional reduced adder graph (RAG-n technique is beneficial for many DSP applications such as for FIR and IIR filters, where multipliers can be grouped in multiplier blocks. This paper highlights the importance of DFT and FFT as DSP objects and also explores how the RAG-n technique can be applied to these algorithms. This RAG-n DFT will be shown to be of low complexity and possess an attractively regular VLSI data flow when implemented with the Rader DFT algorithm or the Bluestein chirp-z algorithm. ASIC synthesis data are provided and demonstrate the low complexity and high speed of the design when compared to other alternatives.

Fast Discrete Fourier Transform Computations Using the Reduced Adder Graph Technique

Directory of Open Access Journals (Sweden)

Dempster Andrew G

2007-01-01

Full Text Available It has recently been shown that the -dimensional reduced adder graph (RAG- technique is beneficial for many DSP applications such as for FIR and IIR filters, where multipliers can be grouped in multiplier blocks. This paper highlights the importance of DFT and FFT as DSP objects and also explores how the RAG- technique can be applied to these algorithms. This RAG- DFT will be shown to be of low complexity and possess an attractively regular VLSI data flow when implemented with the Rader DFT algorithm or the Bluestein chirp- algorithm. ASIC synthesis data are provided and demonstrate the low complexity and high speed of the design when compared to other alternatives.

On the discrete reconciliation of relativity and quantum mechanics

International Nuclear Information System (INIS)

Noyes, H.P.

1987-03-01

A way is sketched to replace physics based on arbitrary units of mass, length, and time by counting in terms of these quantized values and to replace continuum mathematical physics by computer science. The consequences of such a discrete physics are summarized. These are obtained by postulating finiteness, discreteness, finite computability, absolute non-uniqueness, and additivity

Algorithms for Planar Graphs and Graphs in Metric Spaces

DEFF Research Database (Denmark)

Wulff-Nilsen, Christian

structural properties that can be exploited. For instance, a road network or a wire layout on a microchip is typically (near-)planar and distances in the network are often defined w.r.t. the Euclidean or the rectilinear metric. Specialized algorithms that take advantage of such properties are often orders...... of magnitude faster than the corresponding algorithms for general graphs. The first and main part of this thesis focuses on the development of efficient planar graph algorithms. The most important contributions include a faster single-source shortest path algorithm, a distance oracle with subquadratic...... for geometric graphs and graphs embedded in metric spaces. Roughly speaking, the stretch factor is a real value expressing how well a (geo-)metric graph approximates the underlying complete graph w.r.t. distances. We give improved algorithms for computing the stretch factor of a given graph and for augmenting...

The sharp bounds on general sum-connectivity index of four operations on graphs

Directory of Open Access Journals (Sweden)

Shehnaz Akhter

2016-09-01

Full Text Available Abstract The general sum-connectivity index χ α ( G $\\chi_{\\alpha}(G$ , for a (molecular graph G, is defined as the sum of the weights ( d G ( a 1 + d G ( a 2 α $(d_{G}(a_{1}+d_{G}(a_{2}^{\\alpha}$ of all a 1 a 2 ∈ E ( G $a_{1}a_{2}\\in E(G$ , where d G ( a 1 $d_{G}(a_{1}$ (or d G ( a 2 $d_{G}(a_{2}$ denotes the degree of a vertex a 1 $a_{1}$ (or a 2 $a_{2}$ in the graph G; E ( G $E(G$ denotes the set of edges of G, and α is an arbitrary real number. Eliasi and Taeri (Discrete Appl. Math. 157:794-803, 2009 introduced four new operations based on the graphs S ( G $S(G$ , R ( G $R(G$ , Q ( G $Q(G$ , and T ( G $T(G$ , and they also computed the Wiener index of these graph operations in terms of W ( F ( G $W(F(G$ and W ( H $W(H$ , where F is one of the symbols S, R, Q, T. The aim of this paper is to obtain sharp bounds on the general sum-connectivity index of the four operations on graphs.

Proceedings – Mathematical Sciences | Indian Academy of Sciences

Indian Academy of Sciences (India)

In this paper, we propose a definition of approximation property which is called the metric invariant translation approximation property for a countable discrete metric space. Moreover, we use the techniques of Ozawa's to prove that a fine hyperbolic graph has the metric invariant translation approximation property.

Modeling discrete and continuous entities with fractions and decimals.

Science.gov (United States)

Rapp, Monica; Bassok, Miriam; DeWolf, Melissa; Holyoak, Keith J

2015-03-01

When people use mathematics to model real-life situations, their use of mathematical expressions is often mediated by semantic alignment (Bassok, Chase, & Martin, 1998): The entities in a problem situation evoke semantic relations (e.g., tulips and vases evoke the functionally asymmetric "contain" relation), which people align with analogous mathematical relations (e.g., the noncommutative division operation, tulips/vases). Here we investigate the possibility that semantic alignment is also involved in the comprehension and use of rational numbers (fractions and decimals). A textbook analysis and results from two experiments revealed that both mathematic educators and college students tend to align the discreteness versus continuity of the entities in word problems (e.g., marbles vs. distance) with distinct symbolic representations of rational numbers--fractions versus decimals, respectively. In addition, fractions and decimals tend to be used with nonmetric units and metric units, respectively. We discuss the importance of the ontological distinction between continuous and discrete entities to mathematical cognition, the role of symbolic notations, and possible implications of our findings for the teaching of rational numbers. PsycINFO Database Record (c) 2015 APA, all rights reserved.

On the application of Discrete Time Optimal Control Concepts to ...

African Journals Online (AJOL)

On the application of Discrete Time Optimal Control Concepts to Economic Problems. ... Journal of the Nigerian Association of Mathematical Physics ... Abstract. An extension of the use of the maximum principle to solve Discrete-time Optimal Control Problems (DTOCP), in which the state equations are in the form of general ...

Exact discretization of Schrödinger equation

Energy Technology Data Exchange (ETDEWEB)

Tarasov, Vasily E., E-mail: tarasov@theory.sinp.msu.ru

2016-01-08

There are different approaches to discretization of the Schrödinger equation with some approximations. In this paper we derive a discrete equation that can be considered as exact discretization of the continuous Schrödinger equation. The proposed discrete equation is an equation with difference of integer order that is represented by infinite series. We suggest differences, which are characterized by power-law Fourier transforms. These differences can be considered as exact discrete analogs of derivatives of integer orders. Physically the suggested discrete equation describes a chain (or lattice) model with long-range interaction of power-law form. Mathematically it is a uniquely highlighted difference equation that exactly corresponds to the continuous Schrödinger equation. Using the Young's inequality for convolution, we prove that suggested differences are operators on the Hilbert space of square-summable sequences. We prove that the wave functions, which are exact discrete analogs of the free particle and harmonic oscillator solutions of the continuous Schrödinger equations, are solutions of the suggested discrete Schrödinger equations. - Highlights: • Exact discretization of the continuous Schrödinger equation is suggested. • New long-range interactions of power-law form are suggested. • Solutions of discrete Schrödinger equation are exact discrete analogs of continuous solutions.

Exact discretization of Schrödinger equation

International Nuclear Information System (INIS)

Tarasov, Vasily E.

2016-01-01

There are different approaches to discretization of the Schrödinger equation with some approximations. In this paper we derive a discrete equation that can be considered as exact discretization of the continuous Schrödinger equation. The proposed discrete equation is an equation with difference of integer order that is represented by infinite series. We suggest differences, which are characterized by power-law Fourier transforms. These differences can be considered as exact discrete analogs of derivatives of integer orders. Physically the suggested discrete equation describes a chain (or lattice) model with long-range interaction of power-law form. Mathematically it is a uniquely highlighted difference equation that exactly corresponds to the continuous Schrödinger equation. Using the Young's inequality for convolution, we prove that suggested differences are operators on the Hilbert space of square-summable sequences. We prove that the wave functions, which are exact discrete analogs of the free particle and harmonic oscillator solutions of the continuous Schrödinger equations, are solutions of the suggested discrete Schrödinger equations. - Highlights: • Exact discretization of the continuous Schrödinger equation is suggested. • New long-range interactions of power-law form are suggested. • Solutions of discrete Schrödinger equation are exact discrete analogs of continuous solutions.

Mathematics Textbooks and Their Potential Role in Supporting Misconceptions

Science.gov (United States)

Kajander, Ann; Lovric, Miroslav

2009-01-01

As a fundamental resource, textbooks shape the way we teach and learn mathematics. Based on examination of secondary school and university textbooks, we describe to what extent, and how, the presentation of mathematics material--in our case study, the concept of the line tangent to the graph of a function--could contribute to creation and…

Control of discrete-event systems with modular or distributed structure

Czech Academy of Sciences Publication Activity Database

Komenda, Jan; van Schuppen, J. H.

2007-01-01

Roč. 388, č. 3 (2007), s. 199-226 ISSN 0304-3975 R&D Projects: GA AV ČR(CZ) KJB100190609 Institutional research plan: CEZ:AV0Z10190503 Keywords : supervisory control * modular discrete-event system * distributed discrete-event system Subject RIV: BA - General Mathematics Impact factor: 0.735, year: 2007

Degree Associated Edge Reconstruction Number of Graphs with Regular Pruned Graph

Directory of Open Access Journals (Sweden)

P. Anusha Devi

2015-10-01

Full Text Available An ecard of a graph $G$ is a subgraph formed by deleting an edge. A da-ecard specifies the degree of the deleted edge along with the ecard. The degree associated edge reconstruction number of a graph $G,~dern(G,$ is the minimum number of da-ecards that uniquely determines $G.$  The adversary degree associated edge reconstruction number of a graph $G, adern(G,$ is the minimum number $k$ such that every collection of $k$ da-ecards of $G$ uniquely determines $G.$ The maximal subgraph without end vertices of a graph $G$ which is not a tree is the pruned graph of $G.$ It is shown that $dern$ of complete multipartite graphs and some connected graphs with regular pruned graph is $1$ or $2.$ We also determine $dern$ and $adern$ of corona product of standard graphs.

Transversals in non-discrete groups

Indian Academy of Sciences (India)

Transversals in non-discrete groups. RAMJI LAL and R P SHUKLA. Department of Mathematics, University of Allahabad, Allahabad 211 002, India. E-mail: ramjilal@mri.ernet.in; rps@mri.ernet.in. MS received 2 August 2004; revised 4 August 2005. Abstract. The concept of 'topological right transversal' is introduced to study ...

A Comparison of Approaches for Solving Hard Graph-Theoretic Problems

Science.gov (United States)

2015-04-29

and Search”, in Discrete Mathematics and Its Applications, Book 7, CRC Press (1998): Boca Raton. [6] A. Lucas, “Ising Formulations of Many NP Problems...owner. 14. ABSTRACT In order to formulate mathematical conjectures likely to be true, a number of base cases must be determined. However, many... combinatorial problems are NP-hard and the computational complexity makes this research approach difficult using a standard brute force approach on a

Improving Control Efficiency of Dynamic Street Lighting by Utilizing the Dual Graph Grammar Concept

Directory of Open Access Journals (Sweden)

Igor Wojnicki

2018-02-01

Full Text Available The paper introduces a definition of dual graph grammar. It enables two graphs to share information in a synchronized way. A smart city example application, which is an outdoor lighting control system utilizing the dual graph grammar, is also demonstrated. The system controls dimming of street lights which is based on traffic intensity. Each luminaire’s light level is adjusted individually to comply with the lighting norms to ensure safety. Benefits of applying the dual graph grammar are twofold. First, it increases expressive power of the mathematical model that the system uses. It becomes possible to take into account complex geographical distribution of sensors and logical dependencies among them. Second, it increases the system’s efficiency by reducing the problem size during run-time. Experimental results show a reduction of the computation time by a factor of 2.8. The approach has been verified in practice.

Graph Theory. 2. Vertex Descriptors and Graph Coloring

Directory of Open Access Journals (Sweden)

Lorentz JÄNTSCHI

2002-12-01

Full Text Available This original work presents the construction of a set of ten sequence matrices and their applications for ordering vertices in graphs. For every sequence matrix three ordering criteria are applied: lexicographic ordering, based on strings of numbers, corresponding to every vertex, extracted as rows from sequence matrices; ordering by the sum of path lengths from a given vertex; and ordering by the sum of paths, starting from a given vertex. We also examine a graph that has different orderings for the above criteria. We then proceed to demonstrate that every criterion induced its own partition of graph vertex. We propose the following theoretical result: both LAVS and LVDS criteria generate identical partitioning of vertices in any graph. Finally, a coloring of graph vertices according to introduced ordering criteria was proposed.

On an edge partition and root graphs of some classes of line graphs

Directory of Open Access Journals (Sweden)

K Pravas

2017-04-01

Full Text Available The Gallai and the anti-Gallai graphs of a graph $G$ are complementary pairs of spanning subgraphs of the line graph of $G$. In this paper we find some structural relations between these graph classes by finding a partition of the edge set of the line graph of a graph $G$ into the edge sets of the Gallai and anti-Gallai graphs of $G$. Based on this, an optimal algorithm to find the root graph of a line graph is obtained. Moreover, root graphs of diameter-maximal, distance-hereditary, Ptolemaic and chordal graphs are also discussed.

Graphs and matrices

CERN Document Server

Bapat, Ravindra B

2014-01-01

This new edition illustrates the power of linear algebra in the study of graphs. The emphasis on matrix techniques is greater than in other texts on algebraic graph theory. Important matrices associated with graphs (for example, incidence, adjacency and Laplacian matrices) are treated in detail. Presenting a useful overview of selected topics in algebraic graph theory, early chapters of the text focus on regular graphs, algebraic connectivity, the distance matrix of a tree, and its generalized version for arbitrary graphs, known as the resistance matrix. Coverage of later topics include Laplacian eigenvalues of threshold graphs, the positive definite completion problem and matrix games based on a graph. Such an extensive coverage of the subject area provides a welcome prompt for further exploration. The inclusion of exercises enables practical learning throughout the book. In the new edition, a new chapter is added on the line graph of a tree, while some results in Chapter 6 on Perron-Frobenius theory are reo...

Graphs cospectral with a friendship graph or its complement

Directory of Open Access Journals (Sweden)

Alireza Abdollahi

2013-12-01

Full Text Available Let $n$ be any positive integer and let $F_n$ be the friendship (or Dutch windmill graph with $2n+1$ vertices and $3n$ edges. Here we study graphs with the same adjacency spectrum as the $F_n$. Two graphs are called cospectral if the eigenvalues multiset of their adjacency matrices are the same. Let $G$ be a graph cospectral with $F_n$. Here we prove that if $G$ has no cycle of length $4$ or $5$, then $Gcong F_n$. Moreover if $G$ is connected and planar then $Gcong F_n$.All but one of connected components of $G$ are isomorphic to $K_2$.The complement $overline{F_n}$ of the friendship graph is determined by its adjacency eigenvalues, that is, if $overline{F_n}$ is cospectral with a graph $H$, then $Hcong overline{F_n}$.

Representations of classical groups on the lattice and its application to the field theory on discrete space-time

OpenAIRE

Lorente, M.

2003-01-01

We explore the mathematical consequences of the assumption of a discrete space-time. The fundamental laws of physics have to be translated into the language of discrete mathematics. We find integral transformations that leave the lattice of any dimension invariant and apply these transformations to field equations.

A Qualitative Analysis Framework Using Natural Language Processing and Graph Theory

Science.gov (United States)

Tierney, Patrick J.

2012-01-01

This paper introduces a method of extending natural language-based processing of qualitative data analysis with the use of a very quantitative tool--graph theory. It is not an attempt to convert qualitative research to a positivist approach with a mathematical black box, nor is it a "graphical solution". Rather, it is a method to help qualitative…

DISCRETE FIXED POINT THEOREMS AND THEIR APPLICATION TO NASH EQUILIBRIUM

OpenAIRE

Sato, Junichi; Kawasaki, Hidefumi

2007-01-01

Fixed point theorems are powerful tools in not only mathematics but also economic. In some economic problems, we need not real-valued but integer-valued equilibriums. However, classical fixed point theorems guarantee only real-valued equilibria. So we need discrete fixed point theorems in order to get discrete equilibria. In this paper, we first provide discrete fixed point theorems, next apply them to a non-cooperative game and prove the existence of a Nash equilibrium of pure strategies.

Core-Plus Mathematics. What Works Clearinghouse Intervention Report

Science.gov (United States)

What Works Clearinghouse, 2010

2010-01-01

"Core-Plus Mathematics" is a four-year curriculum that replaces the traditional sequence with courses that each feature interwoven strands of algebra and functions, statistics and probability, geometry and trigonometry, and discrete mathematics. The first three courses in the series provide a common core of broadly useful mathematics,…

Discrete expansions of continuum wave functions

International Nuclear Information System (INIS)

Bang, J.; Ershov, S.N.; Gareev, F.A.; Kazacha, G.S.

1980-01-01

Different methods of expanding continuum wave functions in terms of discrete basis sets are discussed. The convergence properties of these expansions are investigated, both from a mathematical and a numerical point of view, for the case of potentials of Woods-Saxon and square well type. (orig.)

Young Children Use Graphs to Build Mathematical Reasoning

Science.gov (United States)

Larson, Mark J.; Whitin, David J.

2010-01-01

Mathematical, scientific, and technological knowledge is critical for people in a 21st Century world that is dependent upon a global interconnectedness and a knowledge-based economy. This is the kind of knowledge that will power innovations and drive decision making in the years ahead. Schools are therefore being called upon to devise a…

Modelo de dinámica lateral de vehículo mediante bond graph

Directory of Open Access Journals (Sweden)

Juan Carlos Parra Márquez

2008-07-01

Full Text Available Este trabajo presenta los resultados de la investigación, cuyo objetivo es obtener un modelo matemático que permita determinar la dinámica lateral de un vehículo mediante el uso de Bond Graph. Este modelo es válido para robótica móvil. Los análisis de comportamiento del modelo han sido probados con simulaciones típicas del movimiento lateral de un vehículo. Finalmente, este modelo ha sido obtenido e implementado mediante el software 20-Sim. This paper presents the results of a research whose objective was to find a mathematical model in order to determine the lateral dynamic of Vehicle by means of the use of Bond Graph. This model is valid also for mobile robotics. The analyses of behavior of the model were realized across typical simulations of a vehicle in lateral movement. Finally, this mathematical model was obtained and implemented across the software 20-Sim.

Symmetries In Graphs, Maps, And Polytopes Workshop 2014

CERN Document Server

Jajcay, Robert

2016-01-01

This volume contains seventeen of the best papers delivered at the SIGMAP Workshop 2014, representing the most recent advances in the field of symmetries of discrete objects and structures, with a particular emphasis on connections between maps, Riemann surfaces and dessins d’enfant. Providing the global community of researchers in the field with the opportunity to gather, converse and present their newest findings and advances, the Symmetries In Graphs, Maps, and Polytopes Workshop 2014 was the fifth in a series of workshops. The initial workshop, organized by Steve Wilson in Flagstaff, Arizona, in 1998, was followed in 2002 and 2006 by two meetings held in Aveiro, Portugal, organized by Antonio Breda d’Azevedo, and a fourth workshop held in Oaxaca, Mexico, organized by Isabel Hubard in 2010. This book should appeal to both specialists and those seeking a broad overview of what is happening in the area of symmetries of discrete objects and structures.

Graph embedding with rich information through heterogeneous graph

KAUST Repository

Sun, Guolei

2017-11-12

Graph embedding, aiming to learn low-dimensional representations for nodes in graphs, has attracted increasing attention due to its critical application including node classification, link prediction and clustering in social network analysis. Most existing algorithms for graph embedding only rely on the topology information and fail to use the copious information in nodes as well as edges. As a result, their performance for many tasks may not be satisfactory. In this thesis, we proposed a novel and general framework for graph embedding with rich text information (GERI) through constructing a heterogeneous network, in which we integrate node and edge content information with graph topology. Specially, we designed a novel biased random walk to explore the constructed heterogeneous network with the notion of flexible neighborhood. Our sampling strategy can compromise between BFS and DFS local search on heterogeneous graph. To further improve our algorithm, we proposed semi-supervised GERI (SGERI), which learns graph embedding in an discriminative manner through heterogeneous network with label information. The efficacy of our method is demonstrated by extensive comparison experiments with 9 baselines over multi-label and multi-class classification on various datasets including Citeseer, Cora, DBLP and Wiki. It shows that GERI improves the Micro-F1 and Macro-F1 of node classification up to 10%, and SGERI improves GERI by 5% in Wiki.

Topics in graph theory graphs and their Cartesian product

CERN Document Server

Imrich, Wilfried; Rall, Douglas F

2008-01-01

From specialists in the field, you will learn about interesting connections and recent developments in the field of graph theory by looking in particular at Cartesian products-arguably the most important of the four standard graph products. Many new results in this area appear for the first time in print in this book. Written in an accessible way, this book can be used for personal study in advanced applications of graph theory or for an advanced graph theory course.

Study of Chromatic parameters of Line, Total, Middle graphs and Graph operators of Bipartite graph

Science.gov (United States)

Nagarathinam, R.; Parvathi, N.

2018-04-01

Chromatic parameters have been explored on the basis of graph coloring process in which a couple of adjacent nodes receives different colors. But the Grundy and b-coloring executes maximum colors under certain restrictions. In this paper, Chromatic, b-chromatic and Grundy number of some graph operators of bipartite graph has been investigat

The 6th International Conference on Computer Science and Computational Mathematics (ICCSCM 2017)

Science.gov (United States)

2017-09-01

The ICCSCM 2017 (The 6th International Conference on Computer Science and Computational Mathematics) has aimed to provide a platform to discuss computer science and mathematics related issues including Algebraic Geometry, Algebraic Topology, Approximation Theory, Calculus of Variations, Category Theory; Homological Algebra, Coding Theory, Combinatorics, Control Theory, Cryptology, Geometry, Difference and Functional Equations, Discrete Mathematics, Dynamical Systems and Ergodic Theory, Field Theory and Polynomials, Fluid Mechanics and Solid Mechanics, Fourier Analysis, Functional Analysis, Functions of a Complex Variable, Fuzzy Mathematics, Game Theory, General Algebraic Systems, Graph Theory, Group Theory and Generalizations, Image Processing, Signal Processing and Tomography, Information Fusion, Integral Equations, Lattices, Algebraic Structures, Linear and Multilinear Algebra; Matrix Theory, Mathematical Biology and Other Natural Sciences, Mathematical Economics and Financial Mathematics, Mathematical Physics, Measure Theory and Integration, Neutrosophic Mathematics, Number Theory, Numerical Analysis, Operations Research, Optimization, Operator Theory, Ordinary and Partial Differential Equations, Potential Theory, Real Functions, Rings and Algebras, Statistical Mechanics, Structure Of Matter, Topological Groups, Wavelets and Wavelet Transforms, 3G/4G Network Evolutions, Ad-Hoc, Mobile, Wireless Networks and Mobile Computing, Agent Computing & Multi-Agents Systems, All topics related Image/Signal Processing, Any topics related Computer Networks, Any topics related ISO SC-27 and SC- 17 standards, Any topics related PKI(Public Key Intrastructures), Artifial Intelligences(A.I.) & Pattern/Image Recognitions, Authentication/Authorization Issues, Biometric authentication and algorithms, CDMA/GSM Communication Protocols, Combinatorics, Graph Theory, and Analysis of Algorithms, Cryptography and Foundation of Computer Security, Data Base(D.B.) Management & Information

A refresher course in mathematics

CERN Document Server

Camm, F J

2003-01-01

Readers wishing to renew and extend their acquaintance with a variety of branches of mathematics will find this volume a practical companion. Geared toward those who already possess some familiarity with its subjects, the easy-to-follow explanations and straightforward tone make this book highly accessible. The contents are arranged logically and in order of difficulty: fractions, decimals, square and cube root, the metric system, algebra, quadratic and cubic equations, graphs, and the calculus are among the topics. Explanations of mathematical principles are followed by worked examples, and t

Characterization of inclusion neighbourhood in terms of the essential graph

Czech Academy of Sciences Publication Activity Database

Studený, Milan

2005-01-01

Roč. 38, č. 3 (2005), s. 283-309 ISSN 0888-613X R&D Projects: GA ČR GA201/04/0393; GA AV ČR IAA1075104 Institutional research plan: CEZ:AV0Z10750506 Keywords : learning Bayesian networks * inclusion neighbourhood * essential graph Subject RIV: BA - General Mathematics Impact factor: 0.959, year: 2005 http://library.utia.cas.cz/separaty/historie/studeny-0411275.pdf

The Geometry Conference

CERN Document Server

Bárány, Imre; Vilcu, Costin

2016-01-01

This volume presents easy-to-understand yet surprising properties obtained using topological, geometric and graph theoretic tools in the areas covered by the Geometry Conference that took place in Mulhouse, France from September 7–11, 2014 in honour of Tudor Zamfirescu on the occasion of his 70th anniversary. The contributions address subjects in convexity and discrete geometry, in distance geometry or with geometrical flavor in combinatorics, graph theory or non-linear analysis. Written by top experts, these papers highlight the close connections between these fields, as well as ties to other domains of geometry and their reciprocal influence. They offer an overview on recent developments in geometry and its border with discrete mathematics, and provide answers to several open questions. The volume addresses a large audience in mathematics, including researchers and graduate students interested in geometry and geometrical problems.

Two-dimensional quantum gravity - a laboratory for fluctuating graphs and quenched connectivity disorder

Directory of Open Access Journals (Sweden)

W.Janke

2006-01-01

Full Text Available This paper gives a brief introduction to using two-dimensional discrete and Euclidean quantum gravity approaches as a laboratory for studying the properties of fluctuating and frozen random graphs in interaction with "matter fields" represented by simple spin or vertex models. Due to the existence of numerous exact analytical results and predictions for comparison with simulational work, this is an interesting and useful enterprise.

Handbook of graph grammars and computing by graph transformation

CERN Document Server

Engels, G; Kreowski, H J; Rozenberg, G

1999-01-01

Graph grammars originated in the late 60s, motivated by considerations about pattern recognition and compiler construction. Since then, the list of areas which have interacted with the development of graph grammars has grown quite impressively. Besides the aforementioned areas, it includes software specification and development, VLSI layout schemes, database design, modeling of concurrent systems, massively parallel computer architectures, logic programming, computer animation, developmental biology, music composition, visual languages, and many others.The area of graph grammars and graph tran

Engineering system dynamics a unified graph-centered approach

CERN Document Server

Brown, Forbes T

2006-01-01

For today's students, learning to model the dynamics of complex systems is increasingly important across nearly all engineering disciplines. First published in 2001, Forbes T. Brown's Engineering System Dynamics: A Unified Graph-Centered Approach introduced students to a unique and highly successful approach to modeling system dynamics using bond graphs. Updated with nearly one-third new material, this second edition expands this approach to an even broader range of topics. What's New in the Second Edition? In addition to new material, this edition was restructured to build students' competence in traditional linear mathematical methods before they have gone too far into the modeling that still plays a pivotal role. New topics include magnetic circuits and motors including simulation with magnetic hysteresis; extensive new material on the modeling, analysis, and simulation of distributed-parameter systems; kinetic energy in thermodynamic systems; and Lagrangian and Hamiltonian methods. MATLAB(R) figures promi...

Mathematical Concepts and Proofs from Nicole Oresme: Using the History of Calculus to Teach Mathematics

Science.gov (United States)

Babb, Jeff

2005-01-01

This paper examines the mathematical work of the French bishop, Nicole Oresme (c. 1323-1382), and his contributions towards the development of the concept of graphing functions and approaches to investigating infinite series. The historical importance and pedagogical value of his work will be considered in the context of an undergraduate course on…

On the absence of absolutely continuous spectra for Schrodinger operators on radial tree graphs

Czech Academy of Sciences Publication Activity Database

Exner, Pavel; Lipovský, Jiří

2010-01-01

Roč. 51, č. 12 (2010), 122107/1-122107/19 ISSN 0022-2488 R&D Projects: GA MŠk LC06002 Institutional research plan: CEZ:AV0Z10480505 Keywords : QUANTUM GRAPHS * METRIC TREES Subject RIV: BA - General Mathematics Impact factor: 1.291, year: 2010

An Application of Graph Theory in Markov Chains Reliability Analysis

Directory of Open Access Journals (Sweden)

Pavel Skalny

2014-01-01

Full Text Available The paper presents reliability analysis which was realized for an industrial company. The aim of the paper is to present the usage of discrete time Markov chains and the flow in network approach. Discrete Markov chains a well-known method of stochastic modelling describes the issue. The method is suitable for many systems occurring in practice where we can easily distinguish various amount of states. Markov chains are used to describe transitions between the states of the process. The industrial process is described as a graph network. The maximal flow in the network corresponds to the production. The Ford-Fulkerson algorithm is used to quantify the production for each state. The combination of both methods are utilized to quantify the expected value of the amount of manufactured products for the given time period.

Synchronization Of Parallel Discrete Event Simulations

Science.gov (United States)

Steinman, Jeffrey S.

1992-01-01

Adaptive, parallel, discrete-event-simulation-synchronization algorithm, Breathing Time Buckets, developed in Synchronous Parallel Environment for Emulation and Discrete Event Simulation (SPEEDES) operating system. Algorithm allows parallel simulations to process events optimistically in fluctuating time cycles that naturally adapt while simulation in progress. Combines best of optimistic and conservative synchronization strategies while avoiding major disadvantages. Algorithm processes events optimistically in time cycles adapting while simulation in progress. Well suited for modeling communication networks, for large-scale war games, for simulated flights of aircraft, for simulations of computer equipment, for mathematical modeling, for interactive engineering simulations, and for depictions of flows of information.

Compartmentalization analysis using discrete fracture network models

Energy Technology Data Exchange (ETDEWEB)

La Pointe, P.R.; Eiben, T.; Dershowitz, W. [Golder Associates, Redmond, VA (United States); Wadleigh, E. [Marathon Oil Co., Midland, TX (United States)

1997-08-01

This paper illustrates how Discrete Fracture Network (DFN) technology can serve as a basis for the calculation of reservoir engineering parameters for the development of fractured reservoirs. It describes the development of quantitative techniques for defining the geometry and volume of structurally controlled compartments. These techniques are based on a combination of stochastic geometry, computational geometry, and graph the theory. The parameters addressed are compartment size, matrix block size and tributary drainage volume. The concept of DFN models is explained and methodologies to compute these parameters are demonstrated.

Multiplicative Attribute Graph Model of Real-World Networks

Energy Technology Data Exchange (ETDEWEB)

Kim, Myunghwan [Stanford Univ., CA (United States); Leskovec, Jure [Stanford Univ., CA (United States)

2010-10-20

Large scale real-world network data, such as social networks, Internet andWeb graphs, is ubiquitous in a variety of scientific domains. The study of such social and information networks commonly finds patterns and explain their emergence through tractable models. In most networks, especially in social networks, nodes also have a rich set of attributes (e.g., age, gender) associatedwith them. However, most of the existing network models focus only on modeling the network structure while ignoring the features of nodes in the network. Here we present a class of network models that we refer to as the Multiplicative Attribute Graphs (MAG), which naturally captures the interactions between the network structure and node attributes. We consider a model where each node has a vector of categorical features associated with it. The probability of an edge between a pair of nodes then depends on the product of individual attributeattribute similarities. The model yields itself to mathematical analysis as well as fit to real data. We derive thresholds for the connectivity, the emergence of the giant connected component, and show that the model gives rise to graphs with a constant diameter. Moreover, we analyze the degree distribution to show that the model can produce networks with either lognormal or power-law degree distribution depending on certain conditions.

Centrosymmetric Graphs And A Lower Bound For Graph Energy Of Fullerenes

Directory of Open Access Journals (Sweden)

Katona Gyula Y.

2014-11-01

Full Text Available The energy of a molecular graph G is defined as the summation of the absolute values of the eigenvalues of adjacency matrix of a graph G. In this paper, an infinite class of fullerene graphs with 10n vertices, n ≥ 2, is considered. By proving centrosymmetricity of the adjacency matrix of these fullerene graphs, a lower bound for its energy is given. Our method is general and can be extended to other class of fullerene graphs.

Graphs and Homomorphisms

CERN Document Server

Hell, Pavol

2004-01-01

This is a book about graph homomorphisms. Graph theory is now an established discipline but the study of graph homomorphisms has only recently begun to gain wide acceptance and interest. The subject gives a useful perspective in areas such as graph reconstruction, products, fractional and circular colourings, and has applications in complexity theory, artificial intelligence, telecommunication, and, most recently, statistical physics.Based on the authors' lecture notes for graduate courses, this book can be used as a textbook for a second course in graph theory at 4th year or master's level an

Summer School Mathematical Foundations of Complex Networked Information Systems

CERN Document Server

Fosson, Sophie; Ravazzi, Chiara

2015-01-01

Introducing the reader to the mathematics beyond complex networked systems, these lecture notes investigate graph theory, graphical models, and methods from statistical physics. Complex networked systems play a fundamental role in our society, both in everyday life and in scientific research, with applications ranging from physics and biology to economics and finance. The book is self-contained, and requires only an undergraduate mathematical background.

The Approximate Loebl-Komlos-Sos Conjecture III: The Finer Structure of LKS Graphs

Czech Academy of Sciences Publication Activity Database

Hladký, J.; Komlós, J.; Piguet, Diana; Simonovits, M.; Stein, M.; Szemerédi, E.

2017-01-01

Roč. 31, č. 2 (2017), s. 1017-1071 ISSN 0895-4801 R&D Projects: GA MŠk(CZ) 1M0545; GA ČR GJ16-07822Y Grant - others:EPRSC(GB) EP/D063191/1; EPRSC(GB) EP/J501414/1; FP7(XE) PIEF-GA-2009-253925; GA MŠK(CZ) CZ.1.05/1.1.00/02.0090 Institutional support: RVO:67985807 Keywords : extremal graph theory * Loebl–Komlós–Sós conjecture * regularity lemma Subject RIV: BA - General Mathematics OBOR OECD: Pure mathematics Impact factor: 0.755, year: 2016

Discrete Maximum Principle for Higher-Order Finite Elements in 1D

Czech Academy of Sciences Publication Activity Database

Vejchodský, Tomáš; Šolín, Pavel

2007-01-01

Roč. 76, č. 260 (2007), s. 1833-1846 ISSN 0025-5718 R&D Projects: GA ČR GP201/04/P021 Institutional research plan: CEZ:AV0Z10190503; CEZ:AV0Z20760514 Keywords : discrete maximum principle * discrete Grren´s function * higher-order elements Subject RIV: BA - General Mathematics Impact factor: 1.230, year: 2007

Decomposing Oriented Graphs into Six Locally Irregular Oriented Graphs

DEFF Research Database (Denmark)

Bensmail, Julien; Renault, Gabriel

2016-01-01

An undirected graph G is locally irregular if every two of its adjacent vertices have distinct degrees. We say that G is decomposable into k locally irregular graphs if there exists a partition E1∪E2∪⋯∪Ek of the edge set E(G) such that each Ei induces a locally irregular graph. It was recently co...

Spreading speed and travelling waves for a spatially discrete SIS epidemic model

International Nuclear Information System (INIS)

Zhang, Kate Fang; Zhao Xiaoqiang

2008-01-01

This paper is devoted to the study of the asymptotic speed of spread and travelling waves for a spatially discrete SIS epidemic model. By appealing to the theory of spreading speeds and travelling waves for monotonic semiflows, we establish the existence of asymptotic speed of spread and show that it coincides with the minimal wave speed for monotonic travelling waves. This also gives an affirmative answer to an open problem presented by Rass and Radcliffe (2003 Spatial Deterministic Epidemics (Mathematical Surveys and Monographs vol 102) (Providence, RI: American Mathematical Society)) in the case of discrete spatial habitat

A New Type of Graphical Passwords Based on Odd-Elegant Labelled Graphs

Directory of Open Access Journals (Sweden)

Hongyu Wang

2018-01-01

Full Text Available Graphical password (GPW is one of various passwords used in information communication. The QR code, which is widely used in the current world, is one of GPWs. Topsnut-GPWs are new-type GPWs made by topological structures (also, called graphs and number theory, but the existing GPWs use pictures/images almost. We design new Topsnut-GPWs by means of a graph labelling, called odd-elegant labelling. The new Topsnut-GPWs will be constructed by Topsnut-GPWs having smaller vertex numbers; in other words, they are compound Topsnut-GPWs such that they are more robust to deciphering attacks. Furthermore, the new Topsnut-GPWs can induce some mathematical problems and conjectures.

Supporting Generative Thinking about Number Lines, the Cartesian Plane, and Graphs of Linear Functions

Science.gov (United States)

Earnest, Darrell Steven

2012-01-01

This dissertation explores fifth and eighth grade students' interpretations of three kinds of mathematical representations: number lines, the Cartesian plane, and graphs of linear functions. Two studies were conducted. In Study 1, I administered the paper-and-pencil Linear Representations Assessment (LRA) to examine students'…

Classroom Proven Motivational Mathematics Games, Monograph No. 1.

Science.gov (United States)

Michigan Council of Teachers of Mathematics.

This collection includes 50 mathematical games and puzzles for classroom use at all grade levels. Also included is a wide variety of activities with cubes, flash cards, graphs, dots, number patterns, geometric shapes, cross-number puzzles, and magic squares. (MM)

Non-heuristic reduction of the graph in graph-cut optimization

International Nuclear Information System (INIS)

Malgouyres, François; Lermé, Nicolas

2012-01-01

During the last ten years, graph cuts had a growing impact in shape optimization. In particular, they are commonly used in applications of shape optimization such as image processing, computer vision and computer graphics. Their success is due to their ability to efficiently solve (apparently) difficult shape optimization problems which typically involve the perimeter of the shape. Nevertheless, solving problems with a large number of variables remains computationally expensive and requires a high memory usage since underlying graphs sometimes involve billion of nodes and even more edges. Several strategies have been proposed in the literature to improve graph-cuts in this regards. In this paper, we give a formal statement which expresses that a simple and local test performed on every node before its construction permits to avoid the construction of useless nodes for the graphs typically encountered in image processing and vision. A useless node is such that the value of the maximum flow in the graph does not change when removing the node from the graph. Such a test therefore permits to limit the construction of the graph to a band of useful nodes surrounding the final cut.

Chemical graph-theoretic cluster expansions

International Nuclear Information System (INIS)

Klein, D.J.

1986-01-01

A general computationally amenable chemico-graph-theoretic cluster expansion method is suggested as a paradigm for incorporation of chemical structure concepts in a systematic manner. The cluster expansion approach is presented in a formalism general enough to cover a variety of empirical, semiempirical, and even ab initio applications. Formally such approaches for the utilization of chemical structure-related concepts may be viewed as discrete analogues of Taylor series expansions. The efficacy of the chemical structure concepts then is simply bound up in the rate of convergence of the cluster expansions. In many empirical applications, e.g., boiling points, chromatographic separation coefficients, and biological activities, this rate of convergence has been observed to be quite rapid. More note will be made here of quantum chemical applications. Relations to questions concerning size extensivity of energies and size consistency of wave functions are addressed

Graph embedding with rich information through heterogeneous graph

KAUST Repository

Sun, Guolei

2017-01-01

Graph embedding, aiming to learn low-dimensional representations for nodes in graphs, has attracted increasing attention due to its critical application including node classification, link prediction and clustering in social network analysis. Most

Spaces of fractional quotients, discrete operators, and their applications. II

International Nuclear Information System (INIS)

Lifanov, I K; Poltavskii, L N

1999-01-01

The theory of discrete operators in spaces of fractional quotients is developed. A theorem on the stability of discrete operators under smooth perturbations is proved. On this basis, using special quadrature formulae of rectangular kind, the convergence of approximate solutions of hypersingular integral equations to their exact solutions is demonstrated and a mathematical substantiation of the method of closed discrete vortex frameworks is obtained. The same line of argument is also applied to difference equations arising in the solution of the homogeneous Dirichlet problem for a general second-order elliptic equation with variable coefficients

Exploring mathematics problem-solving and proof

CERN Document Server

Grieser, Daniel

2018-01-01

Have you ever faced a mathematical problem and had no idea how to approach it? Or perhaps you had an idea but got stuck halfway through? This book guides you in developing your creativity, as it takes you on a voyage of discovery into mathematics. Readers will not only learn strategies for solving problems and logical reasoning, but they will also learn about the importance of proofs and various proof techniques. Other topics covered include recursion, mathematical induction, graphs, counting, elementary number theory, and the pigeonhole, extremal and invariance principles. Designed to help students make the transition from secondary school to university level, this book provides readers with a refreshing look at mathematics and deep insights into universal principles that are valuable far beyond the scope of this book. Aimed especially at undergraduate and secondary school students as well as teachers, this book will appeal to anyone interested in mathematics. Only basic secondary school mathematics is requi...

Graphing trillions of triangles.

Science.gov (United States)

Burkhardt, Paul

2017-07-01

The increasing size of Big Data is often heralded but how data are transformed and represented is also profoundly important to knowledge discovery, and this is exemplified in Big Graph analytics. Much attention has been placed on the scale of the input graph but the product of a graph algorithm can be many times larger than the input. This is true for many graph problems, such as listing all triangles in a graph. Enabling scalable graph exploration for Big Graphs requires new approaches to algorithms, architectures, and visual analytics. A brief tutorial is given to aid the argument for thoughtful representation of data in the context of graph analysis. Then a new algebraic method to reduce the arithmetic operations in counting and listing triangles in graphs is introduced. Additionally, a scalable triangle listing algorithm in the MapReduce model will be presented followed by a description of the experiments with that algorithm that led to the current largest and fastest triangle listing benchmarks to date. Finally, a method for identifying triangles in new visual graph exploration technologies is proposed.

Attractors for discrete periodic dynamical systems

Science.gov (United States)

John E. Franke; James F. Selgrade

2003-01-01

A mathematical framework is introduced to study attractors of discrete, nonautonomous dynamical systems which depend periodically on time. A structure theorem for such attractors is established which says that the attractor of a time-periodic dynamical system is the unin of attractors of appropriate autonomous maps. If the nonautonomous system is a perturbation of an...

JSXGraph--Dynamic Mathematics with JavaScript

Science.gov (United States)

Gerhauser, Michael; Valentin, Bianca; Wassermann, Alfred

2010-01-01

Since Java applets seem to be on the retreat in web application, other approaches for displaying interactive mathematics in the web browser are needed. One such alternative could be our open-source project JSXGraph. It is a cross-browser library for displaying interactive geometry, function plotting, graphs, and data visualization in a web…

Coloring mixed hypergraphs

CERN Document Server

Voloshin, Vitaly I

2002-01-01

The theory of graph coloring has existed for more than 150 years. Historically, graph coloring involved finding the minimum number of colors to be assigned to the vertices so that adjacent vertices would have different colors. From this modest beginning, the theory has become central in discrete mathematics with many contemporary generalizations and applications. Generalization of graph coloring-type problems to mixed hypergraphs brings many new dimensions to the theory of colorings. A main feature of this book is that in the case of hypergraphs, there exist problems on both the minimum and th

Using genre pedagogy to promote student proficiency in the language required for interpreting line graphs

Science.gov (United States)

Smit, Jantien; Bakker, Arthur; van Eerde, Dolly; Kuijpers, Maggie

2016-09-01

The importance of language in mathematics learning has been widely acknowledged. However, little is known about how to make this insight productive in the design and enactment of language-oriented mathematics education. In a design-based research project, we explored how language-oriented mathematics education can be designed and enacted. We drew on genre pedagogy to promote student proficiency in the language required for interpreting line graphs. In the intervention, the teacher used scaffolding strategies to focus students' attention on the structure and linguistic features of the language involved in this particular domain. The research question addressed in this paper is how student proficiency in this language may be promoted. The study comprised nine lessons involving 22 students in grades 5 and 6 (aged 10-12); of these students, 19 had a migrant background. In light of the research aim, we first describe the rationale behind our design. Next, we illustrate how the design was enacted by means of a case study focusing on one student in the classroom practice of developing proficiency in the language required for interpreting line graphs. On the basis of pre- and posttest scores, we conclude that overall their proficiency has increased. Together, the results indicate that and how genre pedagogy may be used to help students become more proficient in the language required in a mathematical domain.

Adaptive Graph Convolutional Neural Networks

OpenAIRE

Li, Ruoyu; Wang, Sheng; Zhu, Feiyun; Huang, Junzhou

2018-01-01

Graph Convolutional Neural Networks (Graph CNNs) are generalizations of classical CNNs to handle graph data such as molecular data, point could and social networks. Current filters in graph CNNs are built for fixed and shared graph structure. However, for most real data, the graph structures varies in both size and connectivity. The paper proposes a generalized and flexible graph CNN taking data of arbitrary graph structure as input. In that way a task-driven adaptive graph is learned for eac...

Decentralized Observer with a Consensus Filter for Distributed Discrete-Time Linear Systems

Science.gov (United States)

Acikmese, Behcet; Mandic, Milan

2011-01-01

This paper presents a decentralized observer with a consensus filter for the state observation of a discrete-time linear distributed systems. In this setup, each agent in the distributed system has an observer with a model of the plant that utilizes the set of locally available measurements, which may not make the full plant state detectable. This lack of detectability is overcome by utilizing a consensus filter that blends the state estimate of each agent with its neighbors' estimates. We assume that the communication graph is connected for all times as well as the sensing graph. It is proven that the state estimates of the proposed observer asymptotically converge to the actual plant states under arbitrarily changing, but connected, communication and sensing topologies. As a byproduct of this research, we also obtained a result on the location of eigenvalues, the spectrum, of the Laplacian for a family of graphs with self-loops.

High Dimensional Spectral Graph Theory and Non-backtracking Random Walks on Graphs

Science.gov (United States)

Kempton, Mark

This thesis has two primary areas of focus. First we study connection graphs, which are weighted graphs in which each edge is associated with a d-dimensional rotation matrix for some fixed dimension d, in addition to a scalar weight. Second, we study non-backtracking random walks on graphs, which are random walks with the additional constraint that they cannot return to the immediately previous state at any given step. Our work in connection graphs is centered on the notion of consistency, that is, the product of rotations moving from one vertex to another is independent of the path taken, and a generalization called epsilon-consistency. We present higher dimensional versions of the combinatorial Laplacian matrix and normalized Laplacian matrix from spectral graph theory, and give results characterizing the consistency of a connection graph in terms of the spectra of these matrices. We generalize several tools from classical spectral graph theory, such as PageRank and effective resistance, to apply to connection graphs. We use these tools to give algorithms for sparsification, clustering, and noise reduction on connection graphs. In non-backtracking random walks, we address the question raised by Alon et. al. concerning how the mixing rate of a non-backtracking random walk to its stationary distribution compares to the mixing rate for an ordinary random walk. Alon et. al. address this question for regular graphs. We take a different approach, and use a generalization of Ihara's Theorem to give a new proof of Alon's result for regular graphs, and to extend the result to biregular graphs. Finally, we give a non-backtracking version of Polya's Random Walk Theorem for 2-dimensional grids.

On the physical relevance of the discrete Fourier transform

CSIR Research Space (South Africa)

Greben, JM

1991-11-01

Full Text Available This paper originated from the author's dissatisfaction with the way the discrete Fourier transform is usually presented in the literature. Although mathematically correct, the physical meaning of the common representation is unsatisfactory...

X-Graphs: Language and Algorithms for Heterogeneous Graph Streams

Science.gov (United States)

2017-09-01

are widely used by academia and industry. 15. SUBJECT TERMS Data Analytics, Graph Analytics, High-Performance Computing 16. SECURITY CLASSIFICATION...form the core of the DeepDive Knowledge Construction System. 2 INTRODUCTION The goal of the X-Graphs project was to develop computational techniques...memory multicore machine. Ringo is based on Snap.py and SNAP, and uses Python . Ringo now allows the integration of Delite DSL Framework Graph

On the sizes of expander graphs and minimum distances of graph codes

DEFF Research Database (Denmark)

Høholdt, Tom; Justesen, Jørn

2014-01-01

We give lower bounds for the minimum distances of graph codes based on expander graphs. The bounds depend only on the second eigenvalue of the graph and the parameters of the component codes. We also give an upper bound on the size of a degree regular graph with given second eigenvalue....

Essentials engineering mathematics

CERN Document Server

Jeffrey, Alan

2004-01-01

Real numbers, inequalities and intervalsFunction, domain and rangeBasic coordinate geometryPolar coordinatesMathematical inductionBinomial theoremCombination of functionsSymmetry in functions and graphsInverse functionsComplex numbers; real and imaginary formsGeometry of complex analysisModulus-argument form of a complex numberRoots of complex numbersLimitsOne-sided limitsDerivativesLeibniz's formulaDifferentialsDifferentiation of inverse trigonometric functionsImplicit differentiationParametrically defined curves and parametric differentiationThe exponential functionThe logarithmic functionHy

Tailored graph ensembles as proxies or null models for real networks I: tools for quantifying structure

International Nuclear Information System (INIS)

Annibale, A; Coolen, A C C; Fernandes, L P; Fraternali, F; Kleinjung, J

2009-01-01

We study the tailoring of structured random graph ensembles to real networks, with the objective of generating precise and practical mathematical tools for quantifying and comparing network topologies macroscopically, beyond the level of degree statistics. Our family of ensembles can produce graphs with any prescribed degree distribution and any degree-degree correlation function; its control parameters can be calculated fully analytically, and as a result we can calculate (asymptotically) formulae for entropies and complexities and for information-theoretic distances between networks, expressed directly and explicitly in terms of their measured degree distribution and degree correlations.

Proceedings – Mathematical Sciences | Indian Academy of Sciences

Indian Academy of Sciences (India)

Home; Journals; Proceedings – Mathematical Sciences; Volume 118; Issue 2 ... Quantum random walk; quantum Lévy process; discrete approximation. ... route de Gray, 25 030 Besançon cedex, France; Graduate School of Information Sciences, Tohoku University, Sendai 980-8579, Japan; Department of Mathematics, ...

Similarity Measure of Graphs

Directory of Open Access Journals (Sweden)

Amine Labriji

2017-07-01

Full Text Available The topic of identifying the similarity of graphs was considered as highly recommended research field in the Web semantic, artificial intelligence, the shape recognition and information research. One of the fundamental problems of graph databases is finding similar graphs to a graph query. Existing approaches dealing with this problem are usually based on the nodes and arcs of the two graphs, regardless of parental semantic links. For instance, a common connection is not identified as being part of the similarity of two graphs in cases like two graphs without common concepts, the measure of similarity based on the union of two graphs, or the one based on the notion of maximum common sub-graph (SCM, or the distance of edition of graphs. This leads to an inadequate situation in the context of information research. To overcome this problem, we suggest a new measure of similarity between graphs, based on the similarity measure of Wu and Palmer. We have shown that this new measure satisfies the properties of a measure of similarities and we applied this new measure on examples. The results show that our measure provides a run time with a gain of time compared to existing approaches. In addition, we compared the relevance of the similarity values obtained, it appears that this new graphs measure is advantageous and  offers a contribution to solving the problem mentioned above.

Simple and Intuitive Mathematics for Learning Elementary Physics

Science.gov (United States)

Kobayashi, Yukio

Mathematics is the language of physics and simple and intuitive mathematics is effective for imaging physical pictures of phenomena. This is important because geometrical viewpoints inspire ideas in physics. For example, some problems on the motion of a particle in a uniform gravitational field can be well illustrated by simple diagrams. Calculus is not only a way of calculating but is also closely related to the law of inertia through slope on a position-time graph. As such, cross-curricular study between mathematics and physics is effective for broadly developing thinking power at the high school and college levels.

Mathematical Biology Modules Based on Modern Molecular Biology and Modern Discrete Mathematics

Science.gov (United States)

Robeva, Raina; Davies, Robin; Hodge, Terrell; Enyedi, Alexander

2010-01-01

We describe an ongoing collaborative curriculum materials development project between Sweet Briar College and Western Michigan University, with support from the National Science Foundation. We present a collection of modules under development that can be used in existing mathematics and biology courses, and we address a critical national need to…

Spectra of Graphs

NARCIS (Netherlands)

Brouwer, A.E.; Haemers, W.H.

2012-01-01

This book gives an elementary treatment of the basic material about graph spectra, both for ordinary, and Laplace and Seidel spectra. The text progresses systematically, by covering standard topics before presenting some new material on trees, strongly regular graphs, two-graphs, association

A necessary condition for dispersal driven growth of populations with discrete patch dynamics.

Science.gov (United States)

Guiver, Chris; Packman, David; Townley, Stuart

2017-07-07

We revisit the question of when can dispersal-induced coupling between discrete sink populations cause overall population growth? Such a phenomenon is called dispersal driven growth and provides a simple explanation of how dispersal can allow populations to persist across discrete, spatially heterogeneous, environments even when individual patches are adverse or unfavourable. For two classes of mathematical models, one linear and one non-linear, we provide necessary conditions for dispersal driven growth in terms of the non-existence of a common linear Lyapunov function, which we describe. Our approach draws heavily upon the underlying positive dynamical systems structure. Our results apply to both discrete- and continuous-time models. The theory is illustrated with examples and both biological and mathematical conclusions are drawn. Copyright © 2017 The Authors. Published by Elsevier Ltd.. All rights reserved.

On approximation of Lie groups by discrete subgroups

Indian Academy of Sciences (India)

1Department of Mathematics, Faculty of Sciences at Sfax, University of Sfax,. Route Soukra ... Let S (G) denote the space of discrete co-compact subgroup of a Lie group G. We ..... For example, it suffices to apply the following fact: The mapping.

Graph theory enables drug repurposing--how a mathematical model can drive the discovery of hidden mechanisms of action.

Science.gov (United States)

Gramatica, Ruggero; Di Matteo, T; Giorgetti, Stefano; Barbiani, Massimo; Bevec, Dorian; Aste, Tomaso

2014-01-01

We introduce a methodology to efficiently exploit natural-language expressed biomedical knowledge for repurposing existing drugs towards diseases for which they were not initially intended. Leveraging on developments in Computational Linguistics and Graph Theory, a methodology is defined to build a graph representation of knowledge, which is automatically analysed to discover hidden relations between any drug and any disease: these relations are specific paths among the biomedical entities of the graph, representing possible Modes of Action for any given pharmacological compound. We propose a measure for the likeliness of these paths based on a stochastic process on the graph. This measure depends on the abundance of indirect paths between a peptide and a disease, rather than solely on the strength of the shortest path connecting them. We provide real-world examples, showing how the method successfully retrieves known pathophysiological Mode of Action and finds new ones by meaningfully selecting and aggregating contributions from known bio-molecular interactions. Applications of this methodology are presented, and prove the efficacy of the method for selecting drugs as treatment options for rare diseases.

Graph theory enables drug repurposing--how a mathematical model can drive the discovery of hidden mechanisms of action.

Directory of Open Access Journals (Sweden)

Ruggero Gramatica

Full Text Available We introduce a methodology to efficiently exploit natural-language expressed biomedical knowledge for repurposing existing drugs towards diseases for which they were not initially intended. Leveraging on developments in Computational Linguistics and Graph Theory, a methodology is defined to build a graph representation of knowledge, which is automatically analysed to discover hidden relations between any drug and any disease: these relations are specific paths among the biomedical entities of the graph, representing possible Modes of Action for any given pharmacological compound. We propose a measure for the likeliness of these paths based on a stochastic process on the graph. This measure depends on the abundance of indirect paths between a peptide and a disease, rather than solely on the strength of the shortest path connecting them. We provide real-world examples, showing how the method successfully retrieves known pathophysiological Mode of Action and finds new ones by meaningfully selecting and aggregating contributions from known bio-molecular interactions. Applications of this methodology are presented, and prove the efficacy of the method for selecting drugs as treatment options for rare diseases.

ASPECTS OF MATHEMATICAL MODELING AND INTERPRETATION OF A MANUFACTURING SYSTEM

Directory of Open Access Journals (Sweden)

Mihaela ALDEA

2013-05-01

Full Text Available In the paper developing we started from a model that allows a detailed decoding of causalrelationships and getting the laws that determine the evolution of the phenomenon.The model chosen for the study is a discrete event system applicable to optimize the transport systemused in pottery. In order to simulate the manufacturing process we chose Matlab package that contains pntoollibrary, by which can be realized modeling of analyzed graphs. Since the timings of manufacture are very highand the process simulation is conducted with difficulty, we divided the graph according to the transport system.

Open problems in mathematics

CERN Document Server

Nash, Jr, John Forbes

2016-01-01

The goal in putting together this unique compilation was to present the current status of the solutions to some of the most essential open problems in pure and applied mathematics. Emphasis is also given to problems in interdisciplinary research for which mathematics plays a key role. This volume comprises highly selected contributions by some of the most eminent mathematicians in the international mathematical community on longstanding problems in very active domains of mathematical research. A joint preface by the two volume editors is followed by a personal farewell to John F. Nash, Jr. written by Michael Th. Rassias. An introduction by Mikhail Gromov highlights some of Nash’s legendary mathematical achievements. The treatment in this book includes open problems in the following fields: algebraic geometry, number theory, analysis, discrete mathematics, PDEs, differential geometry, topology, K-theory, game theory, fluid mechanics, dynamical systems and ergodic theory, cryptography, theoretical computer sc...

Pattern graph rewrite systems

Directory of Open Access Journals (Sweden)

Aleks Kissinger

2014-03-01

Full Text Available String diagrams are a powerful tool for reasoning about physical processes, logic circuits, tensor networks, and many other compositional structures. Dixon, Duncan and Kissinger introduced string graphs, which are a combinatoric representations of string diagrams, amenable to automated reasoning about diagrammatic theories via graph rewrite systems. In this extended abstract, we show how the power of such rewrite systems can be greatly extended by introducing pattern graphs, which provide a means of expressing infinite families of rewrite rules where certain marked subgraphs, called !-boxes ("bang boxes", on both sides of a rule can be copied any number of times or removed. After reviewing the string graph formalism, we show how string graphs can be extended to pattern graphs and how pattern graphs and pattern rewrite rules can be instantiated to concrete string graphs and rewrite rules. We then provide examples demonstrating the expressive power of pattern graphs and how they can be applied to study interacting algebraic structures that are central to categorical quantum mechanics.

Graph Theoretical Analysis Reveals: Women's Brains Are Better Connected than Men's.

Directory of Open Access Journals (Sweden)

Balázs Szalkai

Full Text Available Deep graph-theoretic ideas in the context with the graph of the World Wide Web led to the definition of Google's PageRank and the subsequent rise of the most popular search engine to date. Brain graphs, or connectomes, are being widely explored today. We believe that non-trivial graph theoretic concepts, similarly as it happened in the case of the World Wide Web, will lead to discoveries enlightening the structural and also the functional details of the animal and human brains. When scientists examine large networks of tens or hundreds of millions of vertices, only fast algorithms can be applied because of the size constraints. In the case of diffusion MRI-based structural human brain imaging, the effective vertex number of the connectomes, or brain graphs derived from the data is on the scale of several hundred today. That size facilitates applying strict mathematical graph algorithms even for some hard-to-compute (or NP-hard quantities like vertex cover or balanced minimum cut. In the present work we have examined brain graphs, computed from the data of the Human Connectome Project, recorded from male and female subjects between ages 22 and 35. Significant differences were found between the male and female structural brain graphs: we show that the average female connectome has more edges, is a better expander graph, has larger minimal bisection width, and has more spanning trees than the average male connectome. Since the average female brain weighs less than the brain of males, these properties show that the female brain has better graph theoretical properties, in a sense, than the brain of males. It is known that the female brain has a smaller gray matter/white matter ratio than males, that is, a larger white matter/gray matter ratio than the brain of males; this observation is in line with our findings concerning the number of edges, since the white matter consists of myelinated axons, which, in turn, roughly correspond to the connections

Computer-Based Mathematics Instructions for Engineering Students

Science.gov (United States)

Khan, Mustaq A.; Wall, Curtiss E.

1996-01-01

Almost every engineering course involves mathematics in one form or another. The analytical process of developing mathematical models is very important for engineering students. However, the computational process involved in the solution of some mathematical problems may be very tedious and time consuming. There is a significant amount of mathematical software such as Mathematica, Mathcad, and Maple designed to aid in the solution of these instructional problems. The use of these packages in classroom teaching can greatly enhance understanding, and save time. Integration of computer technology in mathematics classes, without de-emphasizing the traditional analytical aspects of teaching, has proven very successful and is becoming almost essential. Sample computer laboratory modules are developed for presentation in the classroom setting. This is accomplished through the use of overhead projectors linked to graphing calculators and computers. Model problems are carefully selected from different areas.

The ultimatum game: Discrete vs. continuous offers

Science.gov (United States)

Dishon-Berkovits, Miriam; Berkovits, Richard

2014-09-01

In many experimental setups in social-sciences, psychology and economy the subjects are requested to accept or dispense monetary compensation which is usually given in discrete units. Using computer and mathematical modeling we show that in the framework of studying the dynamics of acceptance of proposals in the ultimatum game, the long time dynamics of acceptance of offers in the game are completely different for discrete vs. continuous offers. For discrete values the dynamics follow an exponential behavior. However, for continuous offers the dynamics are described by a power-law. This is shown using an agent based computer simulation as well as by utilizing an analytical solution of a mean-field equation describing the model. These findings have implications to the design and interpretation of socio-economical experiments beyond the ultimatum game.

What Mathematical Competencies Are Needed for Success in College.

Science.gov (United States)

Garofalo, Joe

1990-01-01

Identifies requisite math skills for a microeconomics course, offering samples of supply curves, demand curves, equilibrium prices, elasticity, and complex graph problems. Recommends developmental mathematics competencies, including problem solving, reasoning, connections, communication, number and operation sense, algebra, relationships,…

Trends in contemporary mathematics

CERN Document Server

Strickland, Elisabetta

2014-01-01

This book covers a wide spectrum of hot topics and current trends in mathematics, including noncommutative algebra via deformation theory,  optimal transportation, nonlinear potential theory, kinetic theory and gas dynamics, geometric numerical integration, finite simple groups of small essential dimension, optimal control problems, extended Dynkin diagrams, spin glasses, aspherical closed manifolds, Boltzmann systems, birational geometry of projective varieties and directed graphs, nonlinear diffusion, geometric constructions of extremal metrics on complex manifolds, and Pell’s equation in polynomials. The book comprises a selection of contributions by leading international mathematicians who were speakers at the "INdAM Day", an initiative dating back to 2004 at which the most recent developments in contemporary mathematics are presented.

On middle cube graphs

Directory of Open Access Journals (Sweden)

C. Dalfo

2015-10-01

Full Text Available We study a family of graphs related to the $n$-cube. The middle cube graph of parameter k is the subgraph of $Q_{2k-1}$ induced by the set of vertices whose binary representation has either $k-1$ or $k$ number of ones. The middle cube graphs can be obtained from the well-known odd graphs by doubling their vertex set. Here we study some of the properties of the middle cube graphs in the light of the theory of distance-regular graphs. In particular, we completely determine their spectra (eigenvalues and their multiplicities, and associated eigenvectors.

Foundations of a discrete physics

International Nuclear Information System (INIS)

McGoveran, D.; Noyes, P.

1988-01-01

Starting from the principles of finiteness, discreteness, finite computability and absolute nonuniqueness, we develop the ordering operator calculus, a strictly constructive mathematical system having the empirical properties required by quantum mechanical and special relativistic phenomena. We show how to construct discrete distance functions, and both rectangular and spherical coordinate systems(with a discrete version of ''π''). The richest discrete space constructible without a preferred axis and preserving translational and rotational invariance is shown to be a discrete 3-space with the usual symmetries. We introduce a local ordering parameter with local (proper) time-like properties and universal ordering parameters with global (cosmological) time-like properties. Constructed ''attribute velocities'' connect ensembles with attributes that are invariant as the appropriate time-like parameter increases. For each such attribute, we show how to construct attribute velocities which must satisfy the '' relativistic Doppler shift'' and the ''relativistic velocity composition law,'' as well as the Lorentz transformations. By construction, these velocities have finite maximum and minimum values. In the space of all attributes, the minimum of these maximum velocities will predominate in all multiple attribute computations, and hence can be identified as a fundamental limiting velocity, General commutation relations are constructed which under the physical interpretation are shown to reduce to the usual quantum mechanical commutation relations. 50 refs., 18 figs

Distributed-observer-based cooperative control for synchronization of linear discrete-time multi-agent systems.

Science.gov (United States)

Liang, Hongjing; Zhang, Huaguang; Wang, Zhanshan

2015-11-01

This paper considers output synchronization of discrete-time multi-agent systems with directed communication topologies. The directed communication graph contains a spanning tree and the exosystem as its root. Distributed observer-based consensus protocols are proposed, based on the relative outputs of neighboring agents. A multi-step algorithm is presented to construct the observer-based protocols. In light of the discrete-time algebraic Riccati equation and internal model principle, synchronization problem is completed. At last, numerical simulation is provided to verify the effectiveness of the theoretical results. Copyright © 2015 ISA. Published by Elsevier Ltd. All rights reserved.

A hybrid discrete-continuum mathematical model of pattern prediction in the developing retinal vasculature.

Science.gov (United States)

McDougall, S R; Watson, M G; Devlin, A H; Mitchell, C A; Chaplain, M A J

2012-10-01

Pathological angiogenesis has been extensively explored by the mathematical modelling community over the past few decades, specifically in the contexts of tumour-induced vascularisation and wound healing. However, there have been relatively few attempts to model angiogenesis associated with normal development, despite the availability of animal models with experimentally accessible and highly ordered vascular topologies: for example, growth and development of the vascular plexus layers in the murine retina. The current study aims to address this issue through the development of a hybrid discrete-continuum mathematical model of the developing retinal vasculature in neonatal mice that is closely coupled with an ongoing experimental programme. The model of the functional vasculature is informed by a range of morphological and molecular data obtained over a period of several days, from 6 days prior to birth to approximately 8 days after birth. The spatio-temporal formation of the superficial retinal vascular plexus (RVP) in wild-type mice occurs in a well-defined sequence. Prior to birth, astrocytes migrate from the optic nerve over the surface of the inner retina in response to a chemotactic gradient of PDGF-A, formed at an earlier stage by migrating retinal ganglion cells (RGCs). Astrocytes express a variety of chemotactic and haptotactic proteins, including VEGF and fibronectin (respectively), which subsequently induce endothelial cell sprouting and modulate growth of the RVP. The developing RVP is not an inert structure; however, the vascular bed adapts and remodels in response to a wide variety of metabolic and biomolecular stimuli. The main focus of this investigation is to understand how these interacting cellular, molecular, and metabolic cues regulate RVP growth and formation. In an earlier one-dimensional continuum model of astrocyte and endothelial migration, we showed that the measured frontal velocities of the two cell types could be accurately reproduced

Graph visualization (Invited talk)

NARCIS (Netherlands)

Wijk, van J.J.; Kreveld, van M.J.; Speckmann, B.

2012-01-01

Black and white node link diagrams are the classic method to depict graphs, but these often fall short to give insight in large graphs or when attributes of nodes and edges play an important role. Graph visualization aims obtaining insight in such graphs using interactive graphical representations.

Bond Graph Model of Cerebral Circulation: Toward Clinically Feasible Systemic Blood Flow Simulations

Science.gov (United States)

Safaei, Soroush; Blanco, Pablo J.; Müller, Lucas O.; Hellevik, Leif R.; Hunter, Peter J.

2018-01-01

We propose a detailed CellML model of the human cerebral circulation that runs faster than real time on a desktop computer and is designed for use in clinical settings when the speed of response is important. A lumped parameter mathematical model, which is based on a one-dimensional formulation of the flow of an incompressible fluid in distensible vessels, is constructed using a bond graph formulation to ensure mass conservation and energy conservation. The model includes arterial vessels with geometric and anatomical data based on the ADAN circulation model. The peripheral beds are represented by lumped parameter compartments. We compare the hemodynamics predicted by the bond graph formulation of the cerebral circulation with that given by a classical one-dimensional Navier-Stokes model working on top of the whole-body ADAN model. Outputs from the bond graph model, including the pressure and flow signatures and blood volumes, are compared with physiological data. PMID:29551979

The discretized Schroedinger equation and simple models for semiconductor quantum wells

International Nuclear Information System (INIS)

Boykin, Timothy B; Klimeck, Gerhard

2004-01-01

The discretized Schroedinger equation is one of the most commonly employed methods for solving one-dimensional quantum mechanics problems on the computer, yet many of its characteristics remain poorly understood. The differences with the continuous Schroedinger equation are generally viewed as shortcomings of the discrete model and are typically described in purely mathematical terms. This is unfortunate since the discretized equation is more productively viewed from the perspective of solid-state physics, which naturally links the discrete model to realistic semiconductor quantum wells and nanoelectronic devices. While the relationship between the discrete model and a one-dimensional tight-binding model has been known for some time, the fact that the discrete Schroedinger equation admits analytic solutions for quantum wells has gone unnoted. Here we present a solution to this new analytically solvable problem. We show that the differences between the discrete and continuous models are due to their fundamentally different bandstructures, and present evidence for our belief that the discrete model is the more physically reasonable one

Distance-regular graphs

NARCIS (Netherlands)

van Dam, Edwin R.; Koolen, Jack H.; Tanaka, Hajime

2016-01-01

This is a survey of distance-regular graphs. We present an introduction to distance-regular graphs for the reader who is unfamiliar with the subject, and then give an overview of some developments in the area of distance-regular graphs since the monograph 'BCN'[Brouwer, A.E., Cohen, A.M., Neumaier,

Optimal perturbations for nonlinear systems using graph-based optimal transport

Science.gov (United States)

Grover, Piyush; Elamvazhuthi, Karthik

2018-06-01

We formulate and solve a class of finite-time transport and mixing problems in the set-oriented framework. The aim is to obtain optimal discrete-time perturbations in nonlinear dynamical systems to transport a specified initial measure on the phase space to a final measure in finite time. The measure is propagated under system dynamics in between the perturbations via the associated transfer operator. Each perturbation is described by a deterministic map in the measure space that implements a version of Monge-Kantorovich optimal transport with quadratic cost. Hence, the optimal solution minimizes a sum of quadratic costs on phase space transport due to the perturbations applied at specified times. The action of the transport map is approximated by a continuous pseudo-time flow on a graph, resulting in a tractable convex optimization problem. This problem is solved via state-of-the-art solvers to global optimality. We apply this algorithm to a problem of transport between measures supported on two disjoint almost-invariant sets in a chaotic fluid system, and to a finite-time optimal mixing problem by choosing the final measure to be uniform. In both cases, the optimal perturbations are found to exploit the phase space structures, such as lobe dynamics, leading to efficient global transport. As the time-horizon of the problem is increased, the optimal perturbations become increasingly localized. Hence, by combining the transfer operator approach with ideas from the theory of optimal mass transportation, we obtain a discrete-time graph-based algorithm for optimal transport and mixing in nonlinear systems.

The mathematics of the compact disc (Chapter 2)

NARCIS (Netherlands)

van Lint, J.H.; Aigner, M.; Behrends, E.

2010-01-01

Everyone uses compact discs nowadays. But why is the musical transfer to a CD purer than that of the traditional vinyl disc? The answer, to adapt a popular slogan, is: There is mathematics inside! More precisely, a branch of discrete mathematics, namely the theory of error correcting codes. This

On the strong metric dimension of generalized butterfly graph, starbarbell graph, and {C}_{m}\\odot {P}_{n} graph

Science.gov (United States)

Yunia Mayasari, Ratih; Atmojo Kusmayadi, Tri

2018-04-01

Let G be a connected graph with vertex set V(G) and edge set E(G). For every pair of vertices u,v\\in V(G), the interval I[u, v] between u and v to be the collection of all vertices that belong to some shortest u ‑ v path. A vertex s\\in V(G) strongly resolves two vertices u and v if u belongs to a shortest v ‑ s path or v belongs to a shortest u ‑ s path. A vertex set S of G is a strong resolving set of G if every two distinct vertices of G are strongly resolved by some vertex of S. The strong metric basis of G is a strong resolving set with minimal cardinality. The strong metric dimension sdim(G) of a graph G is defined as the cardinality of strong metric basis. In this paper we determine the strong metric dimension of a generalized butterfly graph, starbarbell graph, and {C}mȯ {P}n graph. We obtain the strong metric dimension of generalized butterfly graph is sdim(BFn ) = 2n ‑ 2. The strong metric dimension of starbarbell graph is sdim(S{B}{m1,{m}2,\\ldots,{m}n})={\\sum }i=1n({m}i-1)-1. The strong metric dimension of {C}mȯ {P}n graph are sdim({C}mȯ {P}n)=2m-1 for m > 3 and n = 2, and sdim({C}mȯ {P}n)=2m-2 for m > 3 and n > 2.

Subgraph detection using graph signals

KAUST Repository

Chepuri, Sundeep Prabhakar

2017-03-06

In this paper we develop statistical detection theory for graph signals. In particular, given two graphs, namely, a background graph that represents an usual activity and an alternative graph that represents some unusual activity, we are interested in answering the following question: To which of the two graphs does the observed graph signal fit the best? To begin with, we assume both the graphs are known, and derive an optimal Neyman-Pearson detector. Next, we derive a suboptimal detector for the case when the alternative graph is not known. The developed theory is illustrated with numerical experiments.

Subgraph detection using graph signals

KAUST Repository

Chepuri, Sundeep Prabhakar; Leus, Geert

2017-01-01

In this paper we develop statistical detection theory for graph signals. In particular, given two graphs, namely, a background graph that represents an usual activity and an alternative graph that represents some unusual activity, we are interested in answering the following question: To which of the two graphs does the observed graph signal fit the best? To begin with, we assume both the graphs are known, and derive an optimal Neyman-Pearson detector. Next, we derive a suboptimal detector for the case when the alternative graph is not known. The developed theory is illustrated with numerical experiments.

A non-linear discrete transform for pattern recognition of discrete chaotic systems

International Nuclear Information System (INIS)

Karanikas, C.; Proios, G.

2003-01-01

It is shown, by an invertible non-linear discrete transform that any finite sequence or any collection of strings of any length can be presented as a random walk on trees. These transforms create the mathematical background for coding any information, for exploring its local variability and diversity. With the underlying computational algorithms, with several examples and applications we propose that these transforms can be used for pattern recognition of immune type. In other words we propose a mathematical platform for detecting self and non-self strings of any alphabet, based on a negative selection algorithms, for scouting data's periodicity and self-similarity and for measuring the diversity of chaotic strings with fractal dimension methods. In particular we estimate successfully the entropy and the ratio of chaotic data with self similarity. Moreover we give some applications of a non-linear denoising filter

A non-linear discrete transform for pattern recognition of discrete chaotic systems

CERN Document Server

Karanikas, C

2003-01-01

It is shown, by an invertible non-linear discrete transform that any finite sequence or any collection of strings of any length can be presented as a random walk on trees. These transforms create the mathematical background for coding any information, for exploring its local variability and diversity. With the underlying computational algorithms, with several examples and applications we propose that these transforms can be used for pattern recognition of immune type. In other words we propose a mathematical platform for detecting self and non-self strings of any alphabet, based on a negative selection algorithms, for scouting data's periodicity and self-similarity and for measuring the diversity of chaotic strings with fractal dimension methods. In particular we estimate successfully the entropy and the ratio of chaotic data with self similarity. Moreover we give some applications of a non-linear denoising filter.

Using Self-Monitoring of Performance with Self-Graphing to Increase Academic Productivity in Math

Science.gov (United States)

Wells, Jenny C.; Sheehey, Patricia H.; Sheehey, Michael

2017-01-01

Self-regulation skills have been found to be an important predictor of achievement in mathematics. Teaching a student to regulate his or her behavior during independent math work sessions using self-monitoring of performance with self-graphing focuses him or her on academic performance and results in increases in productivity and math proficiency.…

Graph spectrum

NARCIS (Netherlands)

Brouwer, A.E.; Haemers, W.H.; Brouwer, A.E.; Haemers, W.H.

2012-01-01

This chapter presents some simple results on graph spectra.We assume the reader is familiar with elementary linear algebra and graph theory. Throughout, J will denote the all-1 matrix, and 1 is the all-1 vector.

Pragmatic Graph Rewriting Modifications

OpenAIRE

Rodgers, Peter; Vidal, Natalia

1999-01-01

We present new pragmatic constructs for easing programming in visual graph rewriting programming languages. The first is a modification to the rewriting process for nodes the host graph, where nodes specified as 'Once Only' in the LHS of a rewrite match at most once with a corresponding node in the host graph. This reduces the previously common use of tags to indicate the progress of matching in the graph. The second modification controls the application of LHS graphs, where those specified a...

The boundary value problem for discrete analytic functions

KAUST Repository

Skopenkov, Mikhail

2013-06-01

This paper is on further development of discrete complex analysis introduced by R.Isaacs, J.Ferrand, R.Duffin, and C.Mercat. We consider a graph lying in the complex plane and having quadrilateral faces. A function on the vertices is called discrete analytic, if for each face the difference quotients along the two diagonals are equal.We prove that the Dirichlet boundary value problem for the real part of a discrete analytic function has a unique solution. In the case when each face has orthogonal diagonals we prove that this solution uniformly converges to a harmonic function in the scaling limit. This solves a problem of S.Smirnov from 2010. This was proved earlier by R.Courant-K.Friedrichs-H.Lewy and L.Lusternik for square lattices, by D.Chelkak-S.Smirnov and implicitly by P.G.Ciarlet-P.-A.Raviart for rhombic lattices.In particular, our result implies uniform convergence of the finite element method on Delaunay triangulations. This solves a problem of A.Bobenko from 2011. The methodology is based on energy estimates inspired by alternating-current network theory. © 2013 Elsevier Ltd.

Complex networks and SOA: Mathematical modelling of granularity ...

Indian Academy of Sciences (India)

We employ statistical and graph-theoretic meth- ... While the interdisciplinary characteristics of network science continues to .... mathematical language and clustering it, even with known methods, can be a daunting task here, let alone ...... Greenacre M J and Balasius J 1994 Correspondence analysis in the social sciences.

The STAPL Parallel Graph Library

KAUST Repository

Harshvardhan,

2013-01-01

This paper describes the stapl Parallel Graph Library, a high-level framework that abstracts the user from data-distribution and parallelism details and allows them to concentrate on parallel graph algorithm development. It includes a customizable distributed graph container and a collection of commonly used parallel graph algorithms. The library introduces pGraph pViews that separate algorithm design from the container implementation. It supports three graph processing algorithmic paradigms, level-synchronous, asynchronous and coarse-grained, and provides common graph algorithms based on them. Experimental results demonstrate improved scalability in performance and data size over existing graph libraries on more than 16,000 cores and on internet-scale graphs containing over 16 billion vertices and 250 billion edges. © Springer-Verlag Berlin Heidelberg 2013.

Mathematical and Computational Aspects Related to Soil Modeling and Simulation

Science.gov (United States)

2017-09-26

and simulation challenges at the interface of applied math (homogenization, handling of discontinuous behavior, discrete vs. continuum representations...topics: a) Visco-elasto-plastic continuum models of geo-surface materials b) Discrete models of geo-surface materials (rocks/gravel/sand) c) Mixed...continuum- discrete representations. Coarse-graining and fine-graining mathematical formulations d) Multi-physics aspects related to the modeling of

Signed star (k,k-domatic number of a graph

Directory of Open Access Journals (Sweden)

S. M. Sheikholeslami

2014-01-01

Full Text Available Let \\(G\\ be a simple graph without isolated vertices with vertex set \\(V(G\\ and edge set \\(E(G\\ and let \\(k\\ be a positive integer. A function \\(f:E(G\\longrightarrow \\{-1, 1\\}\\ is said to be a signed star \\(k\\-dominating function on \\(G\\ if \\(\\sum_{e\\in E(v}f(e\\ge k\\ for every vertex \\(v\\ of \\(G\\, where \\(E(v=\\{uv\\in E(G\\mid u\\in N(v\\}\\. A set \\(\\{f_1,f_2,\\ldots,f_d\\}\\ of signed star \\(k\\-dominating functions on \\(G\\ with the property that \\(\\sum_{i=1}^df_i(e\\le k\\ for each \\(e\\in E(G\\, is called a signed star \\((k,k\\-dominating family (of functions on \\(G\\. The maximum number of functions in a signed star \\((k,k\\-dominating family on \\(G\\ is the signed star \\((k,k\\-domatic number of \\(G\\, denoted by \\(d^{(k,k}_{SS}(G\\. In this paper we study properties of the signed star \\((k,k\\-domatic number \\(d_{SS}^{(k,k}(G\\. In particular, we present bounds on \\(d_{SS}^{(k,k}(G\\, and we determine the signed \\((k,k\\-domatic number of some regular graphs. Some of our results extend these given by Atapour, Sheikholeslami, Ghameslou and Volkmann [Signed star domatic number of a graph, Discrete Appl. Math. 158 (2010, 213-218] for the signed star domatic number.

An introduction to mathematical cryptography

CERN Document Server

Hoffstein, Jeffrey; Silverman, Joseph H

2014-01-01

This self-contained introduction to modern cryptography emphasizes the mathematics behind the theory of public key cryptosystems and digital signature schemes. The book focuses on these key topics while developing the mathematical tools needed for the construction and security analysis of diverse cryptosystems. Only basic linear algebra is required of the reader; techniques from algebra, number theory, and probability are introduced and developed as required. This text provides an ideal introduction for mathematics and computer science students to the mathematical foundations of modern cryptography. The book includes an extensive bibliography and index; supplementary materials are available online. The book covers a variety of topics that are considered central to mathematical cryptography. Key topics include: classical cryptographic constructions, such as Diffie–Hellmann key exchange, discrete logarithm-based cryptosystems, the RSA cryptosystem, and digital signatures; fundamental mathematical tools for cr...

Topic Model for Graph Mining.

Science.gov (United States)

Xuan, Junyu; Lu, Jie; Zhang, Guangquan; Luo, Xiangfeng

2015-12-01

Graph mining has been a popular research area because of its numerous application scenarios. Many unstructured and structured data can be represented as graphs, such as, documents, chemical molecular structures, and images. However, an issue in relation to current research on graphs is that they cannot adequately discover the topics hidden in graph-structured data which can be beneficial for both the unsupervised learning and supervised learning of the graphs. Although topic models have proved to be very successful in discovering latent topics, the standard topic models cannot be directly applied to graph-structured data due to the "bag-of-word" assumption. In this paper, an innovative graph topic model (GTM) is proposed to address this issue, which uses Bernoulli distributions to model the edges between nodes in a graph. It can, therefore, make the edges in a graph contribute to latent topic discovery and further improve the accuracy of the supervised and unsupervised learning of graphs. The experimental results on two different types of graph datasets show that the proposed GTM outperforms the latent Dirichlet allocation on classification by using the unveiled topics of these two models to represent graphs.

Proceedings – Mathematical Sciences | Indian Academy of Sciences

Indian Academy of Sciences (India)

The discreteness of isometry groups in complex hyperbolic space is a fundamental problem. In this paper, the discreteness criteria of a -dimensional subgroup of S U ( n , 1 ) are investigated by using a test map which may not be in . Author Affiliations. Huani Qin1 2 Yueping Jiang1. College of Applied Mathematics, ...

A graph edit dictionary for correcting errors in roof topology graphs reconstructed from point clouds

Science.gov (United States)

Xiong, B.; Oude Elberink, S.; Vosselman, G.

2014-07-01

In the task of 3D building model reconstruction from point clouds we face the problem of recovering a roof topology graph in the presence of noise, small roof faces and low point densities. Errors in roof topology graphs will seriously affect the final modelling results. The aim of this research is to automatically correct these errors. We define the graph correction as a graph-to-graph problem, similar to the spelling correction problem (also called the string-to-string problem). The graph correction is more complex than string correction, as the graphs are 2D while strings are only 1D. We design a strategy based on a dictionary of graph edit operations to automatically identify and correct the errors in the input graph. For each type of error the graph edit dictionary stores a representative erroneous subgraph as well as the corrected version. As an erroneous roof topology graph may contain several errors, a heuristic search is applied to find the optimum sequence of graph edits to correct the errors one by one. The graph edit dictionary can be expanded to include entries needed to cope with errors that were previously not encountered. Experiments show that the dictionary with only fifteen entries already properly corrects one quarter of erroneous graphs in about 4500 buildings, and even half of the erroneous graphs in one test area, achieving as high as a 95% acceptance rate of the reconstructed models.

Learning Bayesian network structure: towards the essential graph by integer linear programming tools

Czech Academy of Sciences Publication Activity Database

Studený, Milan; Haws, D.

2014-01-01

Roč. 55, č. 4 (2014), s. 1043-1071 ISSN 0888-613X R&D Projects: GA ČR GA13-20012S Institutional support: RVO:67985556 Keywords : learning Bayesian network structure * integer linear programming * characteristic imset * essential graph Subject RIV: BA - General Mathematics Impact factor: 2.451, year: 2014 http://library.utia.cas.cz/separaty/2014/MTR/studeny-0427002.pdf

Discrete conservation of nonnegativity for elliptic problems solved by the hp-FEM

Czech Academy of Sciences Publication Activity Database

Šolín, P.; Vejchodský, Tomáš; Araiza, R.

2007-01-01

Roč. 76, 1-3 (2007), s. 205-210 ISSN 0378-4754 R&D Projects: GA ČR GP201/04/P021 Institutional research plan: CEZ:AV0Z10190503 Keywords : discrete nonnegativity conservation * discrete Green's function * elliptic problems * hp-FEM * higher-order finite element methods * Poisson equation * numerical experimetns Subject RIV: BA - General Mathematics Impact factor: 0.738, year: 2007

Graph Generator Survey

Energy Technology Data Exchange (ETDEWEB)

Lothian, Joshua [Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States); Powers, Sarah S. [Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States); Sullivan, Blair D. [Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States); Baker, Matthew B. [Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States); Schrock, Jonathan [Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States); Poole, Stephen W. [Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)

2013-10-01

The benchmarking effort within the Extreme Scale Systems Center at Oak Ridge National Laboratory seeks to provide High Performance Computing benchmarks and test suites of interest to the DoD sponsor. The work described in this report is a part of the effort focusing on graph generation. A previously developed benchmark, SystemBurn, allowed the emulation of different application behavior profiles within a single framework. To complement this effort, similar capabilities are desired for graph-centric problems. This report examines existing synthetic graph generator implementations in preparation for further study on the properties of their generated synthetic graphs.

Functions and graphs

CERN Document Server

Gelfand, I M; Shnol, E E

1969-01-01

The second in a series of systematic studies by a celebrated mathematician I. M. Gelfand and colleagues, this volume presents students with a well-illustrated sequence of problems and exercises designed to illuminate the properties of functions and graphs. Since readers do not have the benefit of a blackboard on which a teacher constructs a graph, the authors abandoned the customary use of diagrams in which only the final form of the graph appears; instead, the book's margins feature step-by-step diagrams for the complete construction of each graph. The first part of the book employs simple fu

Loose Graph Simulations