Pattern vectors from algebraic graph theory.
Wilson, Richard C; Hancock, Edwin R; Luo, Bin
2005-07-01
Graph structures have proven computationally cumbersome for pattern analysis. The reason for this is that, before graphs can be converted to pattern vectors, correspondences must be established between the nodes of structures which are potentially of different size. To overcome this problem, in this paper, we turn to the spectral decomposition of the Laplacian matrix. We show how the elements of the spectral matrix for the Laplacian can be used to construct symmetric polynomials that are permutation invariants. The coefficients of these polynomials can be used as graph features which can be encoded in a vectorial manner. We extend this representation to graphs in which there are unary attributes on the nodes and binary attributes on the edges by using the spectral decomposition of a Hermitian property matrix that can be viewed as a complex analogue of the Laplacian. To embed the graphs in a pattern space, we explore whether the vectors of invariants can be embedded in a low-dimensional space using a number of alternative strategies, including principal components analysis (PCA), multidimensional scaling (MDS), and locality preserving projection (LPP). Experimentally, we demonstrate that the embeddings result in well-defined graph clusters. Our experiments with the spectral representation involve both synthetic and real-world data. The experiments with synthetic data demonstrate that the distances between spectral feature vectors can be used to discriminate between graphs on the basis of their structure. The real-world experiments show that the method can be used to locate clusters of graphs.
K-theory of locally finite graph C∗-algebras
NASA Astrophysics Data System (ADS)
Iyudu, Natalia
2013-09-01
We calculate the K-theory of the Cuntz-Krieger algebra OE associated with an infinite, locally finite graph, via the Bass-Hashimoto operator. The formulae we get express the Grothendieck group and the Whitehead group in purely graph theoretic terms. We consider the category of finite (black-and-white, bi-directed) subgraphs with certain graph homomorphisms and construct a continuous functor to abelian groups. In this category K0 is an inductive limit of K-groups of finite graphs, which were calculated in Cornelissen et al. (2008) [3]. In the case of an infinite graph with the finite Betti number we obtain the formula for the Grothendieck group K0(OE)=Z, where β(E) is the first Betti number and γ(E) is the valency number of the graph E. We note that in the infinite case the torsion part of K0, which is present in the case of a finite graph, vanishes. The Whitehead group depends only on the first Betti number: K1(OE)=Z. These allow us to provide a counterexample to the fact, which holds for finite graphs, that K1(OE) is the torsion free part of K0(OE).
Chen, J.; Safro, I.
2011-01-01
Measuring the connection strength between a pair of vertices in a graph is one of the most important concerns in many graph applications. Simple measures such as edge weights may not be sufficient for capturing the effects associated with short paths of lengths greater than one. In this paper, we consider an iterative process that smooths an associated value for nearby vertices, and we present a measure of the local connection strength (called the algebraic distance; see [D. Ron, I. Safro, and A. Brandt, Multiscale Model. Simul., 9 (2011), pp. 407-423]) based on this process. The proposed measure is attractive in that the process is simple, linear, and easily parallelized. An analysis of the convergence property of the process reveals that the local neighborhoods play an important role in determining the connectivity between vertices. We demonstrate the practical effectiveness of the proposed measure through several combinatorial optimization problems on graphs and hypergraphs.
Algebraic connectivity and graph robustness.
Feddema, John Todd; Byrne, Raymond Harry; Abdallah, Chaouki T.
2009-07-01
Recent papers have used Fiedler's definition of algebraic connectivity to show that network robustness, as measured by node-connectivity and edge-connectivity, can be increased by increasing the algebraic connectivity of the network. By the definition of algebraic connectivity, the second smallest eigenvalue of the graph Laplacian is a lower bound on the node-connectivity. In this paper we show that for circular random lattice graphs and mesh graphs algebraic connectivity is a conservative lower bound, and that increases in algebraic connectivity actually correspond to a decrease in node-connectivity. This means that the networks are actually less robust with respect to node-connectivity as the algebraic connectivity increases. However, an increase in algebraic connectivity seems to correlate well with a decrease in the characteristic path length of these networks - which would result in quicker communication through the network. Applications of these results are then discussed for perimeter security.
Sanfilippo, Antonio P.
2005-12-27
Graph theory is a branch of discrete combinatorial mathematics that studies the properties of graphs. The theory was pioneered by the Swiss mathematician Leonhard Euler in the 18th century, commenced its formal development during the second half of the 19th century, and has witnessed substantial growth during the last seventy years, with applications in areas as diverse as engineering, computer science, physics, sociology, chemistry and biology. Graph theory has also had a strong impact in computational linguistics by providing the foundations for the theory of features structures that has emerged as one of the most widely used frameworks for the representation of grammar formalisms.
Feynman graph generation and calculations in the Hopf algebra of Feynman graphs
NASA Astrophysics Data System (ADS)
Borinsky, Michael
2014-12-01
Two programs for the computation of perturbative expansions of quantum field theory amplitudes are provided. feyngen can be used to generate Feynman graphs for Yang-Mills, QED and φk theories. Using dedicated graph theoretic tools feyngen can generate graphs of comparatively high loop orders. feyncop implements the Hopf algebra of those Feynman graphs which incorporates the renormalization procedure necessary to calculate finite results in perturbation theory of the underlying quantum field theory. feyngen is validated by comparison to explicit calculations of zero dimensional quantum field theories and feyncop is validated using a combinatorial identity on the Hopf algebra of graphs.
NASA Astrophysics Data System (ADS)
Connes, Alain; Kreimer, Dirk
This paper gives a complete selfcontained proof of our result announced in [6] showing that renormalization in quantum field theory is a special instance of a general mathematical procedure of extraction of finite values based on the Riemann-Hilbert problem. We shall first show that for any quantum field theory, the combinatorics of Feynman graphs gives rise to a Hopf algebra which is commutative as an algebra. It is the dual Hopf algebra of the enveloping algebra of a Lie algebra whose basis is labelled by the one particle irreducible Feynman graphs. The Lie bracket of two such graphs is computed from insertions of one graph in the other and vice versa. The corresponding Lie group G is the group of characters of . We shall then show that, using dimensional regularization, the bare (unrenormalized) theory gives rise to a loop
Computations and generation of elements on the Hopf algebra of Feynman graphs
NASA Astrophysics Data System (ADS)
Borinsky, Michael
2015-05-01
Two programs, feyngen and feyncop, were developed. feyngen is designed to generate high loop order Feynman graphs for Yang-Mills, QED and ϕk theories. feyncop can compute the coproduct of these graphs on the underlying Hopf algebra of Feynman graphs. The programs can be validated by exploiting zero dimensional field theory combinatorics and identities on the Hopf algebra which follow from the renormalizability of the theories. A benchmark for both programs was made.
A graph algebra for scalable visual analytics.
Shaverdian, Anna A; Zhou, Hao; Michailidis, George; Jagadish, Hosagrahar V
2012-01-01
Visual analytics (VA), which combines analytical techniques with advanced visualization features, is fast becoming a standard tool for extracting information from graph data. Researchers have developed many tools for this purpose, suggesting a need for formal methods to guide these tools' creation. Increased data demands on computing requires redesigning VA tools to consider performance and reliability in the context of analysis of exascale datasets. Furthermore, visual analysts need a way to document their analyses for reuse and results justification. A VA graph framework encapsulated in a graph algebra helps address these needs. Its atomic operators include selection and aggregation. The framework employs a visual operator and supports dynamic attributes of data to enable scalable visual exploration of data.
Clifford Algebras, Random Graphs, and Quantum Random Variables
NASA Astrophysics Data System (ADS)
Schott, René; Staples, G. Stacey
2008-08-01
For fixed n > 0, the space of finite graphs on n vertices is canonically associated with an abelian, nilpotent-generated subalgebra of the Clifford algebra {C}l2n,2n which is canonically isomorphic to the 2n-particle fermion algebra. Using the generators of the subalgebra, an algebraic probability space of "Clifford adjacency matrices" associated with finite graphs is defined. Each Clifford adjacency matrix is a quantum random variable whose mth moment corresponds to the number of m-cycles in the graph G. Each matrix admits a canonical "quantum decomposition" into a sum of three algebraic random variables: a = aΔ + aΥ + aΛ, where aΔ is classical while aΥ and aΛ are quantum. Moreover, within the Clifford algebra context the NP problem of cycle enumeration is reduced to matrix multiplication, requiring no more than n4 Clifford (geo-metric) multiplications within the algebra.
ERIC Educational Resources Information Center
Hopkins, Brian
2004-01-01
The interconnected world of actors and movies is a familiar, rich example for graph theory. This paper gives the history of the "Kevin Bacon Game" and makes extensive use of a Web site to analyze the underlying graph. The main content is the classroom development of the weighted average to determine the best choice of "center" for the graph. The…
NASA Technical Reports Server (NTRS)
Iachello, Franco
1995-01-01
An algebraic formulation of quantum mechanics is presented. In this formulation, operators of interest are expanded onto elements of an algebra, G. For bound state problems in nu dimensions the algebra G is taken to be U(nu + 1). Applications to the structure of molecules are presented.
NASA Technical Reports Server (NTRS)
Butler, Ricky W.; Sjogren, Jon A.
1998-01-01
This paper documents the NASA Langley PVS graph theory library. The library provides fundamental definitions for graphs, subgraphs, walks, paths, subgraphs generated by walks, trees, cycles, degree, separating sets, and four notions of connectedness. Theorems provided include Ramsey's and Menger's and the equivalence of all four notions of connectedness.
Activities: Relating to Graphs in Introductory Algebra.
ERIC Educational Resources Information Center
Van Dyke, Frances
1994-01-01
Presents activities designed to help students bridge the gap between the graphical representation of a function and a verbal description. Includes a number of nonstandard graphs and reproducible student worksheets. (MKR)
Symplectic Clifford Algebraic Field Theory.
NASA Astrophysics Data System (ADS)
Dixon, Geoffrey Moore
We develop a mathematical framework on which is built a theory of fermion, scalar, and gauge vector fields. This field theory is shown to be equivalent to the original Weinberg-Salam model of weak and electromagnetic interactions, but since the new framework is more rigid than that on which the original Weinberg-Salam model was built, a concomitant reduction in the number of assumptions lying outside of the framework has resulted. In particular, parity violation is actually hiding within our framework, and with little difficulty we are able to manifest it. The mathematical framework upon which we build our field theory is arrived at along two separate paths. The first is by the marriage of a Clifford algebra and a Lie superalgebra, the result being called a super Clifford algebra. The second is by providing a new characterization for a Clifford algebra employing its generators and a symmetric array of metric coefficients. Subsequently we generalize this characterization to the case of an antisymmetric array of metric coefficients, and we call the algebra which results a symplectic Clifford algebra. It is upon one of these that we build our field theory, and it is shown that this symplectic Clifford algebra is a particular subalgebra of a super Clifford algebra. The final ingredient is the operation of bracketing which involves treating the elements of our algebra as endomorphisms of a particular inner product space, and employing this space and its inner product to provide us with maps from our algebra to the reals. It is this operation which enables us to manifest the parity violation hiding in our algebra.
Algebraic orbifold conformal field theories
Xu, Feng
2000-01-01
The unitary rational orbifold conformal field theories in the algebraic quantum field theory and subfactor theory framework are formulated. Under general conditions, it is shown that the orbifold of a given unitary rational conformal field theory generates a unitary modular category. Many new unitary modular categories are obtained. It is also shown that the irreducible representations of orbifolds of rank one lattice vertex operator algebras give rise to unitary modular categories and determine the corresponding modular matrices, which has been conjectured for some time. PMID:11106383
Graphs and matroids weighted in a bounded incline algebra.
Lu, Ling-Xia; Zhang, Bei
2014-01-01
Firstly, for a graph weighted in a bounded incline algebra (or called a dioid), a longest path problem (LPP, for short) is presented, which can be considered the uniform approach to the famous shortest path problem, the widest path problem, and the most reliable path problem. The solutions for LPP and related algorithms are given. Secondly, for a matroid weighted in a linear matroid, the maximum independent set problem is studied.
Chemical Applications of Graph Theory: Part II. Isomer Enumeration.
ERIC Educational Resources Information Center
Hansen, Peter J.; Jurs, Peter C.
1988-01-01
Discusses the use of graph theory to aid in the depiction of organic molecular structures. Gives a historical perspective of graph theory and explains graph theory terminology with organic examples. Lists applications of graph theory to current research projects. (ML)
Some Applications of Graph Theory to Clustering
ERIC Educational Resources Information Center
Hubert, Lawrence J.
1974-01-01
The connection between graph theory and clustering is reviewed and extended. Major emphasis is on restating, in a graph-theoretic context, selected past work in clustering, and conversely, developing alternative strategies from several standard concepts used in graph theory per se. (Author/RC)
Combinatorial Hopf Algebras in Quantum Field Theory I
NASA Astrophysics Data System (ADS)
Figueroa, Héctor; Gracia-Bondía, José M.
This paper stands at the interface between combinatorial Hopf algebra theory and renormalization theory. Its plan is as follows: Sec. 1.1 is the introduction, and contains an elementary invitation to the subject as well. The rest of Sec. 1 is devoted to the basics of Hopf algebra theory and examples in ascending level of complexity. Section 2 turns around the all-important Faà di Bruno Hopf algebra. Section 2.1 contains a first, direct approach to it. Section 2.2 gives applications of the Faà di Bruno algebra to quantum field theory and Lagrange reversion. Section 2.3 rederives the related Connes-Moscovici algebras. In Sec. 3, we turn to the Connes-Kreimer Hopf algebras of Feynman graphs and, more generally, to incidence bialgebras. In Sec. 3.1, we describe the first. Then in Sec. 3.2, we give a simple derivation of (the properly combinatorial part of) Zimmermann's cancellation-free method, in its original diagrammatic form. In Sec. 3.3, general incidence algebras are introduced, and the Faà di Bruno bialgebras are described as incidence bialgebras. In Sec. 3.4, deeper lore on Rota's incidence algebras allows us to reinterpret Connes-Kreimer algebras in terms of distributive lattices. Next, the general algebraic-combinatorial proof of the cancellation-free formula for antipodes is ascertained. The structure results for commutative Hopf algebras are found in Sec. 4. An outlook section very briefly reviews the coalgebraic aspects of quantization and the Rota-Baxter map in renormalization.
Using Number Theory to Reinforce Elementary Algebra.
ERIC Educational Resources Information Center
Covillion, Jane D.
1995-01-01
Demonstrates that using the elementary number theory in algebra classes helps students to use acquired algebraic skills as well as helping them to more clearly understand concepts that are presented. Discusses factoring, divisibility rules, and number patterns. (AIM)
Flying through Graphs: An Introduction to Graph Theory.
ERIC Educational Resources Information Center
McDuffie, Amy Roth
2001-01-01
Presents an activity incorporating basic terminology, concepts, and solution methods of graph theory in the context of solving problems related to air travel. Discusses prerequisite knowledge and resources and includes a teacher's guide with a student worksheet. (KHR)
From operator algebras to superconformal field theory
Kawahigashi, Yasuyuki
2010-01-15
We survey operator algebraic approach to (super)conformal field theory. We discuss representation theory, classification results, full and boundary conformal field theories, relations to supervertex operator algebras and Moonshine, connections to subfactor theory of Jones, and certain aspects of noncommutative geometry of Connes.
Computer algebra and transport theory.
Warsa, J. S.
2004-01-01
Modern symbolic algebra computer software augments and complements more traditional approaches to transport theory applications in several ways. The first area is in the development and enhancement of numerical solution methods for solving the Boltzmann transport equation. Typically, special purpose computer codes are designed and written to solve specific transport problems in particular ways. Different aspects of the code are often written from scratch and the pitfalls of developing complex computer codes are numerous and well known. Software such as MAPLE and MATLAB can be used to prototype, analyze, verify and determine the suitability of numerical solution methods before a full-scale transport application is written. Once it is written, the relevant pieces of the full-scale code can be verified using the same tools I that were developed for prototyping. Another area is in the analysis of numerical solution methods or the calculation of theoretical results that might otherwise be difficult or intractable. Algebraic manipulations are done easily and without error and the software also provides a framework for any additional numerical calculations that might be needed to complete the analysis. We will discuss several applications in which we have extensively used MAPLE and MATLAB in our work. All of them involve numerical solutions of the S{sub N} transport equation. These applications encompass both of the two main areas in which we have found computer algebra software essential.
On the Primitive Ideal spaces of the C(*) -algebras of graphs
NASA Astrophysics Data System (ADS)
Bates, Teresa
2005-11-01
We characterise the topological spaces which arise as the primitive ideal spaces of the Cuntz-Krieger algebras of graphs satisfying condition (K): directed graphs in which every vertex lying on a loop lies on at least two loops. We deduce that the spaces which arise as Prim;C(*(E)) are precisely the spaces which arise as the primitive ideal spaces of AF-algebras. Finally, we construct a graph wt{E} from E such that C(*(wt{E})) is an AF-algebra and Prim;C(*(E)) and Prim;C(*(wt{E})) are homeomorphic.
Hinz, Andreas M.
2012-01-01
The appropriate mathematical model for the problem space of tower transformation tasks is the state graph representing positions of discs or balls and their moves. Graph theoretical quantities like distance, eccentricities or degrees of vertices and symmetries of graphs support the choice of problems, the selection of tasks and the analysis of performance of subjects whose solution paths can be projected onto the graph. The mathematical model is also at the base of a computerized test tool to administer various types of tower tasks. PMID:22207419
Graph Theory and the High School Student.
ERIC Educational Resources Information Center
Chartrand, Gary; Wall, Curtiss E.
1980-01-01
Graph theory is presented as a tool to instruct high school mathematics students. A variety of real world problems can be modeled which help students recognize the importance and difficulty of applying mathematics. (MP)
ERIC Educational Resources Information Center
Thomas, Ryan Vail
2016-01-01
The goal of this study is to explore and characterize the effects of using a dynamic graphing utility (DGU) on conceptual understanding and attitudes toward mathematics, measured by the responses of college algebra students to an attitude survey and concepts assessment. Two sections of college algebra taught by the primary researcher are included…
Application of Computer Graphics to Graphing in Algebra and Trigonometry. Final Report.
ERIC Educational Resources Information Center
Morris, J. Richard
This project was designed to improve the graphing competency of students in elementary algebra, intermediate algebra, and trigonometry courses at Virginia Commonwealth University. Computer graphics programs were designed using an Apple II Plus computer and implemented using Pascal. The software package is interactive and gives students control…
Quantum graphs and random-matrix theory
NASA Astrophysics Data System (ADS)
Pluhař, Z.; Weidenmüller, H. A.
2015-07-01
For simple connected graphs with incommensurate bond lengths and with unitary symmetry we prove the Bohigas-Giannoni-Schmit (BGS) conjecture in its most general form. Using supersymmetry and taking the limit of infinite graph size, we show that the generating function for every (P,Q) correlation function for both closed and open graphs coincides with the corresponding expression of random-matrix theory. We show that the classical Perron-Frobenius operator is bistochastic and possesses a single eigenvalue +1. In the quantum case that implies the existence of a zero (or massless) mode of the effective action. That mode causes universal fluctuation properties. Avoiding the saddle-point approximation we show that for graphs that are classically mixing (i.e. for which the spectrum of the classical Perron-Frobenius operator possesses a finite gap) and that do not carry a special class of bound states, the zero mode dominates in the limit of infinite graph size.
Many-core graph analytics using accelerated sparse linear algebra routines
NASA Astrophysics Data System (ADS)
Kozacik, Stephen; Paolini, Aaron L.; Fox, Paul; Kelmelis, Eric
2016-05-01
Graph analytics is a key component in identifying emerging trends and threats in many real-world applications. Largescale graph analytics frameworks provide a convenient and highly-scalable platform for developing algorithms to analyze large datasets. Although conceptually scalable, these techniques exhibit poor performance on modern computational hardware. Another model of graph computation has emerged that promises improved performance and scalability by using abstract linear algebra operations as the basis for graph analysis as laid out by the GraphBLAS standard. By using sparse linear algebra as the basis, existing highly efficient algorithms can be adapted to perform computations on the graph. This approach, however, is often less intuitive to graph analytics experts, who are accustomed to vertex-centric APIs such as Giraph, GraphX, and Tinkerpop. We are developing an implementation of the high-level operations supported by these APIs in terms of linear algebra operations. This implementation is be backed by many-core implementations of the fundamental GraphBLAS operations required, and offers the advantages of both the intuitive programming model of a vertex-centric API and the performance of a sparse linear algebra implementation. This technology can reduce the number of nodes required, as well as the run-time for a graph analysis problem, enabling customers to perform more complex analysis with less hardware at lower cost. All of this can be accomplished without the requirement for the customer to make any changes to their analytics code, thanks to the compatibility with existing graph APIs.
The conceptual basis of mathematics in cardiology: (I) algebra, functions and graphs.
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
Algebraic methods in system theory
NASA Technical Reports Server (NTRS)
Brockett, R. W.; Willems, J. C.; Willsky, A. S.
1975-01-01
Investigations on problems of the type which arise in the control of switched electrical networks are reported. The main results concern the algebraic structure and stochastic aspects of these systems. Future reports will contain more detailed applications of these results to engineering studies.
Reducing Abstraction When Learning Graph Theory
ERIC Educational Resources Information Center
Hazzan, Orit; Hadar, Irit
2005-01-01
This article presents research on students' understanding of basic concepts in Graph Theory. Students' understanding is analyzed through the lens of the theoretical framework of reducing abstraction (Hazzan, 1999). As it turns out, in spite of the relative simplicity of the concepts that are introduced in the introductory part of a traditional…
Operator algebra in logarithmic conformal field theory
Nagi, Jasbir
2005-10-15
For some time now, conformal field theories in two dimensions have been studied as integrable systems. Much of the success of these studies is related to the existence of an operator algebra of the theory. In this paper, some of the extensions of this machinery to the logarithmic case are studied and used. More precisely, from Moebius symmetry constraints, the generic three- and four-point functions of logarithmic quasiprimary fields are calculated in closed form for arbitrary Jordan rank. As an example, c=0 disordered systems with nondegenerate vacua are studied. With the aid of two-, three-, and four-point functions, the operator algebra is obtained and associativity of the algebra studied.
Graph Theory-Based Pinning Synchronization of Stochastic Complex Dynamical Networks.
Li, Xiao-Jian; Yang, Guang-Hong
2017-02-01
This paper is concerned with the adaptive pinning synchronization problem of stochastic complex dynamical networks (CDNs). Based on algebraic graph theory and Lyapunov theory, pinning controller design conditions are derived, and the rigorous convergence analysis of synchronization errors in the probability sense is also conducted. Compared with the existing results, the topology structures of stochastic CDN are allowed to be unknown due to the use of graph theory. In particular, it is shown that the selection of nodes for pinning depends on the unknown lower bounds of coupling strengths. Finally, an example on a Chua's circuit network is given to validate the effectiveness of the theoretical results.
The Impact of Handheld Graphing Calculator Use on Student Achievement in Algebra 1
ERIC Educational Resources Information Center
Heller, Joan I.; Curtis, Deborah A.; Jaffe, Rebecca; Verboncoeur, Carol J.
2005-01-01
This study investigated the relationship between instructional use of handheld graphing calculators and student achievement in Algebra 1. Three end-of-course test forms were administered (without calculators) using matrix sampling to 458 high-school students in two suburban school districts in Oregon and Kansas. Test questions on two forms were…
Graph Theory at the Service of Electroencephalograms.
Iakovidou, Nantia D
2017-03-29
The brain is one of the largest and most complex organs in the human body and EEG is a noninvasive electrophysiological monitoring method that is used to record the electrical activity of the brain. Lately, the functional connectivity in human brain has been regarded and studied as a complex network using EEG signals. This means that the brain is studied as a connected system where nodes, or units, represent different specialized brain regions and links, or connections, represent communication pathways between the nodes. Graph theory and theory of complex networks provide a variety of measures, methods, and tools that can be useful to efficiently model, analyze, and study EEG networks. This article is addressed to computer scientists who wish to be acquainted and deal with the study of EEG data and also to neuroscientists who would like to become familiar with graph theoretic approaches and tools to analyze EEG data.
Charge transfer in algebraic quantum field theory
NASA Astrophysics Data System (ADS)
Wright, Jill Dianne
We discuss aspects of the algebraic structure of quantum field theory. We take the view that the superselection structure of a theory should be determinable from the vacuum representation of the observable algebra, and physical properties of the charge. Hence one determines the nature of the charge transfer operations: the automorphisms of the observable algebra corresponding to the movement of charge along space-time paths. New superselection sectors are obtained from the vacuum sector by an automorphism which is a limit of charge transfer operations along paths with an endpoint tending to spacelike infinity. Roberts has shown that for a gauge theory of the first kind, the charge transfer operations for a given charge form a certain kind of 1-cocycle over Minkowski space. The local 1-cohomology group of their equivalence classes corresponds to the superselection structure. The exact definition of the cohomology group depends on the properties of the charge. Using displaced Fock representations of free fields, we develop model field theories which illustrate this structure. The cohomological classification of displaced Fock representations has been elucidated by Araki. For more general representations, explicit determination of the cohomology group is a hard problem. Using our models, we can illustrate ways in which fields with reasonable physical properties depart fromthe abovementioned structure. In 1+1 dimensions, we use the Streater-Wilde model to illustrate explicitly the representation-dependence of the cohomology structure, and the direction-dependence of the limiting charge transfer operation. The cohomology structure may also be representation-dependent in higher-dimensional theories without strict localization of charge, for example the electromagnetic field. The algebraic structure of the electromagnetic field has many other special features, which we discuss in relation to the concept of charge transfer. We also give some indication of the modifications
A Topos for Algebraic Quantum Theory
NASA Astrophysics Data System (ADS)
Heunen, Chris; Landsman, Nicolaas P.; Spitters, Bas
2009-10-01
The aim of this paper is to relate algebraic quantum mechanics to topos theory, so as to construct new foundations for quantum logic and quantum spaces. Motivated by Bohr’s idea that the empirical content of quantum physics is accessible only through classical physics, we show how a noncommutative C*-algebra of observables A induces a topos {mathcal{T}(A)} in which the amalgamation of all of its commutative subalgebras comprises a single commutative C*-algebra {A} . According to the constructive Gelfand duality theorem of Banaschewski and Mulvey, the latter has an internal spectrum {\\underline{Σ}(A)} in {mathcal{T}(A)} , which in our approach plays the role of the quantum phase space of the system. Thus we associate a locale (which is the topos-theoretical notion of a space and which intrinsically carries the intuitionistic logical structure of a Heyting algebra) to a C*-algebra (which is the noncommutative notion of a space). In this setting, states on A become probability measures (more precisely, valuations) on {\\underline{Σ}} , and self-adjoint elements of A define continuous functions (more precisely, locale maps) from {\\underline{Σ}} to Scott’s interval domain. Noting that open subsets of {\\underline{Σ}(A)} correspond to propositions about the system, the pairing map that assigns a (generalized) truth value to a state and a proposition assumes an extremely simple categorical form. Formulated in this way, the quantum theory defined by A is essentially turned into a classical theory, internal to the topos {mathcal{T}(A)}. These results were inspired by the topos-theoretic approach to quantum physics proposed by Butterfield and Isham, as recently generalized by Döring and Isham.
Algebraic Theories and (Infinity,1)-Categories
NASA Astrophysics Data System (ADS)
Cranch, James
2010-11-01
We adapt the classical framework of algebraic theories to work in the setting of (infinity,1)-categories developed by Joyal and Lurie. This gives a suitable approach for describing highly structured objects from homotopy theory. A central example, treated at length, is the theory of E_infinity spaces: this has a tidy combinatorial description in terms of span diagrams of finite sets. We introduce a theory of distributive laws, allowing us to describe objects with two distributing E_infinity stuctures. From this we produce a theory of E_infinity ring spaces. We also study grouplike objects, and produce theories modelling infinite loop spaces (or connective spectra), and infinite loop spaces with coherent multiplicative structure (or connective ring spectra). We use this to construct the units of a grouplike E_infinity ring space in a natural manner. Lastly we provide a speculative pleasant description of the K-theory of monoidal quasicategories and quasicategories with ring-like structures.
Discrete Minimal Surface Algebras
NASA Astrophysics Data System (ADS)
Arnlind, Joakim; Hoppe, Jens
2010-05-01
We consider discrete minimal surface algebras (DMSA) as generalized noncommutative analogues of minimal surfaces in higher dimensional spheres. These algebras appear naturally in membrane theory, where sequences of their representations are used as a regularization. After showing that the defining relations of the algebra are consistent, and that one can compute a basis of the enveloping algebra, we give several explicit examples of DMSAs in terms of subsets of sln (any semi-simple Lie algebra providing a trivial example by itself). A special class of DMSAs are Yang-Mills algebras. The representation graph is introduced to study representations of DMSAs of dimension d ≤ 4, and properties of representations are related to properties of graphs. The representation graph of a tensor product is (generically) the Cartesian product of the corresponding graphs. We provide explicit examples of irreducible representations and, for coinciding eigenvalues, classify all the unitary representations of the corresponding algebras.
Characterising insomnia: A graph spectral theory approach.
Chaparro-Vargas, Ramiro; Ahmed, Beena; Penzel, Thomas; Cvetkovic, Dean
2015-01-01
This paper introduces a computational approach to characterise healthy controls and insomniacs based on graph spectral theory. Based upon expert-generated hypnograms of sleep onset periods, a network of sleep stages transitions is derived to compute four similarity distances amongst subjects' sleeping patterns. A subsequent statistical analysis is performed to differentiate the 16-subject healthy group from a 16-patient disordered cohort. Our findings demonstrated that the similarity distances based on eigenvalues determination, i.e. d1 and d4 were the most reliable and robust measures to characterise insomniacs, discriminating 93% and 87% of the affected population, respectively.
Gnutzmann, Sven; Waltner, Daniel
2016-12-01
We consider exact and asymptotic solutions of the stationary cubic nonlinear Schrödinger equation on metric graphs. We focus on some basic example graphs. The asymptotic solutions are obtained using the canonical perturbation formalism developed in our earlier paper [S. Gnutzmann and D. Waltner, Phys. Rev. E 93, 032204 (2016)2470-004510.1103/PhysRevE.93.032204]. For closed example graphs (interval, ring, star graph, tadpole graph), we calculate spectral curves and show how the description of spectra reduces to known characteristic functions of linear quantum graphs in the low-intensity limit. Analogously for open examples, we show how nonlinear scattering of stationary waves arises and how it reduces to known linear scattering amplitudes at low intensities. In the short-wavelength asymptotics we discuss how genuine nonlinear effects may be described using the leading order of canonical perturbation theory: bifurcation of spectral curves (and the corresponding solutions) in closed graphs and multistability in open graphs.
NASA Astrophysics Data System (ADS)
Gnutzmann, Sven; Waltner, Daniel
2016-12-01
We consider exact and asymptotic solutions of the stationary cubic nonlinear Schrödinger equation on metric graphs. We focus on some basic example graphs. The asymptotic solutions are obtained using the canonical perturbation formalism developed in our earlier paper [S. Gnutzmann and D. Waltner, Phys. Rev. E 93, 032204 (2016), 10.1103/PhysRevE.93.032204]. For closed example graphs (interval, ring, star graph, tadpole graph), we calculate spectral curves and show how the description of spectra reduces to known characteristic functions of linear quantum graphs in the low-intensity limit. Analogously for open examples, we show how nonlinear scattering of stationary waves arises and how it reduces to known linear scattering amplitudes at low intensities. In the short-wavelength asymptotics we discuss how genuine nonlinear effects may be described using the leading order of canonical perturbation theory: bifurcation of spectral curves (and the corresponding solutions) in closed graphs and multistability in open graphs.
Recent Developments in Quantitative Graph Theory: Information Inequalities for Networks
Dehmer, Matthias; Sivakumar, Lavanya
2012-01-01
In this article, we tackle a challenging problem in quantitative graph theory. We establish relations between graph entropy measures representing the structural information content of networks. In particular, we prove formal relations between quantitative network measures based on Shannon's entropy to study the relatedness of those measures. In order to establish such information inequalities for graphs, we focus on graph entropy measures based on information functionals. To prove such relations, we use known graph classes whose instances have been proven useful in various scientific areas. Our results extend the foregoing work on information inequalities for graphs. PMID:22355362
Recent developments in quantitative graph theory: information inequalities for networks.
Dehmer, Matthias; Sivakumar, Lavanya
2012-01-01
In this article, we tackle a challenging problem in quantitative graph theory. We establish relations between graph entropy measures representing the structural information content of networks. In particular, we prove formal relations between quantitative network measures based on Shannon's entropy to study the relatedness of those measures. In order to establish such information inequalities for graphs, we focus on graph entropy measures based on information functionals. To prove such relations, we use known graph classes whose instances have been proven useful in various scientific areas. Our results extend the foregoing work on information inequalities for graphs.
ERIC Educational Resources Information Center
Reznichenko, Nataliya
2012-01-01
Since technology has taken its place in almost all classrooms in schools and colleges across the country, there is a need to know how technology influences the mathematics that is taught and how students learn. In this study, the graphing calculator (GC) (namely the Texas Instruments TI-83) was implemented as a tool to enhance learning of function…
Discovering genetic ancestry using spectral graph theory.
Lee, Ann B; Luca, Diana; Klei, Lambertus; Devlin, Bernie; Roeder, Kathryn
2010-01-01
As one approach to uncovering the genetic underpinnings of complex disease, individuals are measured at a large number of genetic variants (usually SNPs) across the genome and these SNP genotypes are assessed for association with disease status. We propose a new statistical method called Spectral-GEM for the analysis of genome-wide association studies; the goal of Spectral-GEM is to quantify the ancestry of the sample from such genotypic data. Ignoring structure due to differential ancestry can lead to an excess of spurious findings and reduce power. Ancestry is commonly estimated using the eigenvectors derived from principal component analysis (PCA). To develop an alternative to PCA we draw on connections between multidimensional scaling and spectral graph theory. Our approach, based on a spectral embedding derived from the normalized Laplacian of a graph, can produce more meaningful delineation of ancestry than by using PCA. Often the results from Spectral-GEM are straightforward to interpret and therefore useful in association analysis. We illustrate the new algorithm with an analysis of the POPRES data [Nelson et al., 2008].
Graph Theory and Brain Connectivity in Alzheimer's Disease.
delEtoile, Jon; Adeli, Hojjat
2017-04-01
This article presents a review of recent advances in neuroscience research in the specific area of brain connectivity as a potential biomarker of Alzheimer's disease with a focus on the application of graph theory. The review will begin with a brief overview of connectivity and graph theory. Then resent advances in connectivity as a biomarker for Alzheimer's disease will be presented and analyzed.
Decomposition Theory in the Teaching of Elementary Linear Algebra.
ERIC Educational Resources Information Center
London, R. R.; Rogosinski, H. P.
1990-01-01
Described is a decomposition theory from which the Cayley-Hamilton theorem, the diagonalizability of complex square matrices, and functional calculus can be developed. The theory and its applications are based on elementary polynomial algebra. (KR)
Systemic Risk and Complex Systems: A Graph-Theory Analysis
NASA Astrophysics Data System (ADS)
Lautier, Delphine; Raynaud, Franck
This chapter summarizes several empirical studies in finance, undertaken through the prism of the graph theory. In these studies, we built graphs in order to investigate integration and systemic risk in derivative markets. Several classes of underlying assets (i.e. energy products, metals, financial assets, agricultural products) are considered, on a twelve-year period. In such a high dimensional analysis, the graph theory enables us to understand the dynamic behavior of our price system. The dimension of the fully connected graph being high, we rely on a specific type of graphs: Minimum Spanning Trees (MSTs). Such a tree is especially interesting for the study of systemic risk: it can be assimilated into the shortest and most probable path for the propagation of a price shock. We first examine the topology of the MSTs. Then, given the time dependency of our correlation-based graphs, we study their evolution over time and their stability.
Zapata, Francisco; Kreinovich, Vladik; Joslyn, Cliff A.; Hogan, Emilie A.
2013-08-01
To make a decision, we need to compare the values of quantities. In many practical situations, we know the values with interval uncertainty. In such situations, we need to compare intervals. Allen’s algebra describes all possible relations between intervals on the real line, and ordering relations between such intervals are well studied. In this paper, we extend this description to intervals in an arbitrary partially ordered set (poset). In particular, we explicitly describe ordering relations between intervals that generalize relation between points. As auxiliary results, we provide a logical interpretation of the relation between intervals, and extend the results about interval graphs to intervals over posets.
Metric Lie 3-algebras in Bagger-Lambert theory
NASA Astrophysics Data System (ADS)
de Medeiros, Paul; Figueroa-O'Farrill, José; Méndez-Escobar, Elena
2008-08-01
We recast physical properties of the Bagger-Lambert theory, such as shift-symmetry and decoupling of ghosts, the absence of scale and parity invariance, in Lie 3-algebraic terms, thus motivating the study of metric Lie 3-algebras and their Lie algebras of derivations. We prove a structure theorem for metric Lie 3-algebras in arbitrary signature showing that they can be constructed out of the simple and one-dimensional Lie 3-algebras iterating two constructions: orthogonal direct sum and a new construction called a double extension, by analogy with the similar construction for Lie algebras. We classify metric Lie 3-algebras of signature (2, p) and study their Lie algebras of derivations, including those which preserve the conformal class of the inner product. We revisit the 3-algebraic criteria spelt out at the start of the paper and select those algebras with signature (2, p) which satisfy them, as well as indicate the construction of more general metric Lie 3-algebras satisfying the ghost-decoupling criterion.
Graph-based linear scaling electronic structure theory
NASA Astrophysics Data System (ADS)
Niklasson, Anders M. N.; Mniszewski, Susan M.; Negre, Christian F. A.; Cawkwell, Marc J.; Swart, Pieter J.; Mohd-Yusof, Jamal; Germann, Timothy C.; Wall, Michael E.; Bock, Nicolas; Rubensson, Emanuel H.; Djidjev, Hristo
2016-06-01
We show how graph theory can be combined with quantum theory to calculate the electronic structure of large complex systems. The graph formalism is general and applicable to a broad range of electronic structure methods and materials, including challenging systems such as biomolecules. The methodology combines well-controlled accuracy, low computational cost, and natural low-communication parallelism. This combination addresses substantial shortcomings of linear scaling electronic structure theory, in particular with respect to quantum-based molecular dynamics simulations.
Graph-based linear scaling electronic structure theory.
Niklasson, Anders M N; Mniszewski, Susan M; Negre, Christian F A; Cawkwell, Marc J; Swart, Pieter J; Mohd-Yusof, Jamal; Germann, Timothy C; Wall, Michael E; Bock, Nicolas; Rubensson, Emanuel H; Djidjev, Hristo
2016-06-21
We show how graph theory can be combined with quantum theory to calculate the electronic structure of large complex systems. The graph formalism is general and applicable to a broad range of electronic structure methods and materials, including challenging systems such as biomolecules. The methodology combines well-controlled accuracy, low computational cost, and natural low-communication parallelism. This combination addresses substantial shortcomings of linear scaling electronic structure theory, in particular with respect to quantum-based molecular dynamics simulations.
Generative Graph Prototypes from Information Theory.
Han, Lin; Wilson, Richard C; Hancock, Edwin R
2015-10-01
In this paper we present a method for constructing a generative prototype for a set of graphs by adopting a minimum description length approach. The method is posed in terms of learning a generative supergraph model from which the new samples can be obtained by an appropriate sampling mechanism. We commence by constructing a probability distribution for the occurrence of nodes and edges over the supergraph. We encode the complexity of the supergraph using an approximate Von Neumann entropy. A variant of the EM algorithm is developed to minimize the description length criterion in which the structure of the supergraph and the node correspondences between the sample graphs and the supergraph are treated as missing data. To generate new graphs, we assume that the nodes and edges of graphs arise under independent Bernoulli distributions and sample new graphs according to their node and edge occurrence probabilities. Empirical evaluations on real-world databases demonstrate the practical utility of the proposed algorithm and show the effectiveness of the generative model for the tasks of graph classification, graph clustering and generating new sample graphs.
Applications of graph theory to landscape genetics
Garroway, Colin J; Bowman, Jeff; Carr, Denis; Wilson, Paul J
2008-01-01
We investigated the relationships among landscape quality, gene flow, and population genetic structure of fishers (Martes pennanti) in ON, Canada. We used graph theory as an analytical framework considering each landscape as a network node. The 34 nodes were connected by 93 edges. Network structure was characterized by a higher level of clustering than expected by chance, a short mean path length connecting all pairs of nodes, and a resiliency to the loss of highly connected nodes. This suggests that alleles can be efficiently spread through the system and that extirpations and conservative harvest are not likely to affect their spread. Two measures of node centrality were negatively related to both the proportion of immigrants in a node and node snow depth. This suggests that central nodes are producers of emigrants, contain high-quality habitat (i.e., deep snow can make locomotion energetically costly) and that fishers were migrating from high to low quality habitat. A method of community detection on networks delineated five genetic clusters of nodes suggesting cryptic population structure. Our analyses showed that network models can provide system-level insight into the process of gene flow with implications for understanding how landscape alterations might affect population fitness and evolutionary potential. PMID:25567802
Guidelines for the CSMP K-6 Curriculum in Graph Theory.
ERIC Educational Resources Information Center
Deskins, W. E.; And Others
This volume is designed for teachers preparing to teach upper elementary students using the Comprehensive School Mathematics Program (CSMP) curriculum. It begins with a discussion of the importance of graph theory in mathematics and science. A mathematical development of graph-theoretic concepts and theorems is presented, followed by a set of…
Current algebra formulation of M-theory based on E11 Kac-Moody algebra
NASA Astrophysics Data System (ADS)
Sugawara, Hirotaka
2017-02-01
Quantum M-theory is formulated using the current algebra technique. The current algebra is based on a Kac-Moody algebra rather than usual finite dimensional Lie algebra. Specifically, I study the E11 Kac-Moody algebra that was shown recently1‑5 to contain all the ingredients of M-theory. Both the internal symmetry and the external Lorentz symmetry can be realized inside E11, so that, by constructing the current algebra of E11, I obtain both internal gauge theory and gravity theory. The energy-momentum tensor is constructed as the bilinear form of the currents, yielding a system of quantum equations of motion of the currents/fields. Supersymmetry is incorporated in a natural way. The so-called “field-current identity” is built in and, for example, the gravitino field is itself a conserved supercurrent. One unanticipated outcome is that the quantum gravity equation is not identical to the one obtained from the Einstein-Hilbert action.
Symmetric linear systems - An application of algebraic systems theory
NASA Technical Reports Server (NTRS)
Hazewinkel, M.; Martin, C.
1983-01-01
Dynamical systems which contain several identical subsystems occur in a variety of applications ranging from command and control systems and discretization of partial differential equations, to the stability augmentation of pairs of helicopters lifting a large mass. Linear models for such systems display certain obvious symmetries. In this paper, we discuss how these symmetries can be incorporated into a mathematical model that utilizes the modern theory of algebraic systems. Such systems are inherently related to the representation theory of algebras over fields. We will show that any control scheme which respects the dynamical structure either implicitly or explicitly uses the underlying algebra.
ERIC Educational Resources Information Center
Ruthven, Kenneth; Deaney, Rosemary; Hennessy, Sara
2009-01-01
From preliminary analysis of teacher-nominated examples of successful technology-supported practice in secondary-school mathematics, the use of graphing software to teach about algebraic forms was identified as being an important archetype. Employing evidence from lesson observation and teacher interview, such practice was investigated in greater…
Entanglement witnesses for graph states: General theory and examples
Jungnitsch, Bastian; Moroder, Tobias; Guehne, Otfried
2011-09-15
We present a general theory for the construction of witnesses that detect genuine multipartite entanglement in graph states. First, we present explicit witnesses for all graph states of up to six qubits which are better than all criteria so far. Therefore, lower fidelities are required in experiments that aim at the preparation of graph states. Building on these results, we develop analytical methods to construct two different types of entanglement witnesses for general graph states. For many classes of states, these operators exhibit white noise tolerances that converge to 1 when increasing the number of particles. We illustrate our approach for states such as the linear and the 2D cluster state. Finally, we study an entanglement monotone motivated by our approach for graph states.
Applications of graph theory in protein structure identification.
Yan, Yan; Zhang, Shenggui; Wu, Fang-Xiang
2011-10-14
There is a growing interest in the identification of proteins on the proteome wide scale. Among different kinds of protein structure identification methods, graph-theoretic methods are very sharp ones. Due to their lower costs, higher effectiveness and many other advantages, they have drawn more and more researchers' attention nowadays. Specifically, graph-theoretic methods have been widely used in homology identification, side-chain cluster identification, peptide sequencing and so on. This paper reviews several methods in solving protein structure identification problems using graph theory. We mainly introduce classical methods and mathematical models including homology modeling based on clique finding, identification of side-chain clusters in protein structures upon graph spectrum, and de novo peptide sequencing via tandem mass spectrometry using the spectrum graph model. In addition, concluding remarks and future priorities of each method are given.
Three-Dimensional Algebraic Models of the tRNA Code and 12 Graphs for Representing the Amino Acids
José, Marco V.; Morgado, Eberto R.; Guimarães, Romeu Cardoso; Zamudio, Gabriel S.; de Farías, Sávio Torres; Bobadilla, Juan R.; Sosa, Daniela
2014-01-01
Three-dimensional algebraic models, also called Genetic Hotels, are developed to represent the Standard Genetic Code, the Standard tRNA Code (S-tRNA-C), and the Human tRNA code (H-tRNA-C). New algebraic concepts are introduced to be able to describe these models, to wit, the generalization of the 2n-Klein Group and the concept of a subgroup coset with a tail. We found that the H-tRNA-C displayed broken symmetries in regard to the S-tRNA-C, which is highly symmetric. We also show that there are only 12 ways to represent each of the corresponding phenotypic graphs of amino acids. The averages of statistical centrality measures of the 12 graphs for each of the three codes are carried out and they are statistically compared. The phenotypic graphs of the S-tRNA-C display a common triangular prism of amino acids in 10 out of the 12 graphs, whilst the corresponding graphs for the H-tRNA-C display only two triangular prisms. The graphs exhibit disjoint clusters of amino acids when their polar requirement values are used. We contend that the S-tRNA-C is in a frozen-like state, whereas the H-tRNA-C may be in an evolving state. PMID:25370377
Quantifying Riverscape Connectivity with Graph Theory
NASA Astrophysics Data System (ADS)
Carbonneau, P.; Milledge, D.; Sinha, R.; Tandon, S. K.
2013-12-01
Fluvial catchments convey fluxes of water, sediment, nutrients and aquatic biota. At continental scales, crustal topography defines the overall path of channels whilst at local scales depositional and/or erosional features generally determine the exact path of a channel. Furthermore, constructions such as dams, for either water abstraction or hydropower, often have a significant impact on channel networks.The concept of ';connectivity' is commonly invoked when conceptualising the structure of a river network.This concept is easy to grasp but there have been uneven efforts across the environmental sciences to actually quantify connectivity. Currently there have only been a few studies reporting quantitative indices of connectivity in river sciences, notably, in the study of avulsion processes. However, the majority of current work describing some form of environmental connectivity in a quantitative manner is in the field of landscape ecology. Driven by the need to quantify habitat fragmentation, landscape ecologists have returned to graph theory. Within this formal setting, landscape ecologists have successfully developed a range of indices which can model connectivity loss. Such formal connectivity metrics are currently needed for a range of applications in fluvial sciences. One of the most urgent needs relates to dam construction. In the developed world, hydropower development has generally slowed and in many countries, dams are actually being removed. However, this is not the case in the developing world where hydropower is seen as a key element to low-emissions power-security. For example, several dam projects are envisaged in Himalayan catchments in the next 2 decades. This region is already under severe pressure from climate change and urbanisation, and a better understanding of the network fragmentation which can be expected in this system is urgently needed. In this paper, we apply and adapt connectivity metrics from landscape ecology. We then examine the
From string theory to algebraic geometry and back
Brinzanescu, Vasile
2011-02-10
We describe some facts in physics which go up to the modern string theory and the related concepts in algebraic geometry. Then we present some recent results on moduli-spaces of vector bundles on non-Kaehler Calabi-Yau 3-folds and their consequences for heterotic string theory.
A Cohomology Theory of Grading-Restricted Vertex Algebras
NASA Astrophysics Data System (ADS)
Huang, Yi-Zhi
2014-04-01
We introduce a cohomology theory of grading-restricted vertex algebras. To construct the correct cohomologies, we consider linear maps from tensor powers of a grading-restricted vertex algebra to "rational functions valued in the algebraic completion of a module for the algebra," instead of linear maps from tensor powers of the algebra to a module for the algebra. One subtle complication arising from such functions is that we have to carefully address the issue of convergence when we compose these linear maps with vertex operators. In particular, for each , we have an inverse system of nth cohomologies and an additional nth cohomology of a grading-restricted vertex algebra V with coefficients in a V-module W such that is isomorphic to the inverse limit of the inverse system . In the case of n = 2, there is an additional second cohomology denoted by which will be shown in a sequel to the present paper to correspond to what we call square-zero extensions of V and to first order deformations of V when W = V.
On directed information theory and Granger causality graphs.
Amblard, Pierre-Olivier; Michel, Olivier J J
2011-02-01
Directed information theory deals with communication channels with feedback. When applied to networks, a natural extension based on causal conditioning is needed. We show here that measures built from directed information theory in networks can be used to assess Granger causality graphs of stochastic processes. We show that directed information theory includes measures such as the transfer entropy, and that it is the adequate information theoretic framework needed for neuroscience applications, such as connectivity inference problems.
Linear {GLP}-algebras and their elementary theories
NASA Astrophysics Data System (ADS)
Pakhomov, F. N.
2016-12-01
The polymodal provability logic {GLP} was introduced by Japaridze in 1986. It is the provability logic of certain chains of provability predicates of increasing strength. Every polymodal logic corresponds to a variety of polymodal algebras. Beklemishev and Visser asked whether the elementary theory of the free {GLP}-algebra generated by the constants \\mathbf{0}, \\mathbf{1} is decidable [1]. For every positive integer n we solve the corresponding question for the logics {GLP}_n that are the fragments of {GLP} with n modalities. We prove that the elementary theory of the free {GLP}_n-algebra generated by the constants \\mathbf{0}, \\mathbf{1} is decidable for all n. We introduce the notion of a linear {GLP}_n-algebra and prove that all free {GLP}_n-algebras generated by the constants \\mathbf{0}, \\mathbf{1} are linear. We also consider the more general case of the logics {GLP}_α whose modalities are indexed by the elements of a linearly ordered set α: we define the notion of a linear algebra and prove the latter result in this case.
Graph theory findings in the pathophysiology of temporal lobe epilepsy
Chiang, Sharon; Haneef, Zulfi
2014-01-01
Temporal lobe epilepsy (TLE) is the most common form of adult epilepsy. Accumulating evidence has shown that TLE is a disorder of abnormal epileptogenic networks, rather than focal sources. Graph theory allows for a network-based representation of TLE brain networks, and has potential to illuminate characteristics of brain topology conducive to TLE pathophysiology, including seizure initiation and spread. We review basic concepts which we believe will prove helpful in interpreting results rapidly emerging from graph theory research in TLE. In addition, we summarize the current state of graph theory findings in TLE as they pertain its pathophysiology. Several common findings have emerged from the many modalities which have been used to study TLE using graph theory, including structural MRI, diffusion tensor imaging, surface EEG, intracranial EEG, magnetoencephalography, functional MRI, cell cultures, simulated models, and mouse models, involving increased regularity of the interictal network configuration, altered local segregation and global integration of the TLE network, and network reorganization of temporal lobe and limbic structures. As different modalities provide different views of the same phenomenon, future studies integrating data from multiple modalities are needed to clarify findings and contribute to the formation of a coherent theory on the pathophysiology of TLE. PMID:24831083
Graph theory findings in the pathophysiology of temporal lobe epilepsy.
Chiang, Sharon; Haneef, Zulfi
2014-07-01
Temporal lobe epilepsy (TLE) is the most common form of adult epilepsy. Accumulating evidence has shown that TLE is a disorder of abnormal epileptogenic networks, rather than focal sources. Graph theory allows for a network-based representation of TLE brain networks, and has potential to illuminate characteristics of brain topology conducive to TLE pathophysiology, including seizure initiation and spread. We review basic concepts which we believe will prove helpful in interpreting results rapidly emerging from graph theory research in TLE. In addition, we summarize the current state of graph theory findings in TLE as they pertain its pathophysiology. Several common findings have emerged from the many modalities which have been used to study TLE using graph theory, including structural MRI, diffusion tensor imaging, surface EEG, intracranial EEG, magnetoencephalography, functional MRI, cell cultures, simulated models, and mouse models, involving increased regularity of the interictal network configuration, altered local segregation and global integration of the TLE network, and network reorganization of temporal lobe and limbic structures. As different modalities provide different views of the same phenomenon, future studies integrating data from multiple modalities are needed to clarify findings and contribute to the formation of a coherent theory on the pathophysiology of TLE.
Address block localization based on graph theory
NASA Astrophysics Data System (ADS)
Gaceb, Djamel; Eglin, Véronique; Lebourgeois, Frank; Emptoz, Hubert
2008-01-01
An efficient mail sorting system is mainly based on an accurate optical recognition of the addresses on the envelopes. However, the localizing of the address block (ABL) should be done before the OCR recognition process. The location step is very crucial as it has a great impact on the global performance of the system. Currently, a good localizing step leads to a better recognition rate. The limit of current methods is mainly caused by modular linear architectures used for ABL: their performances greatly depend on each independent module performance. We are presenting in this paper a new approach for ABL based on a pyramidal data organization and on a hierarchical graph coloring for classification process. This new approach presents the advantage to guarantee a good coherence between different modules and reduces both the computation time and the rejection rate. The proposed method gives a very satisfying rate of 98% of good locations on a set of 750 envelope images.
Algebraic isomorphism in two-dimensional anomalous gauge theories
Carvalhaes, C.G.; Natividade, C.P.
1997-08-01
The operator solution of the anomalous chiral Schwinger model is discussed on the basis of the general principles of Wightman field theory. Some basic structural properties of the model are analyzed taking a careful control on the Hilbert space associated with the Wightman functions. The isomorphism between gauge noninvariant and gauge invariant descriptions of the anomalous theory is established in terms of the corresponding field algebras. We show that (i) the {Theta}-vacuum representation and (ii) the suggested equivalence of vector Schwinger model and chiral Schwinger model cannot be established in terms of the intrinsic field algebra. {copyright} 1997 Academic Press, Inc.
BootGraph: probabilistic fiber tractography using bootstrap algorithms and graph theory.
Vorburger, Robert S; Reischauer, Carolin; Boesiger, Peter
2013-02-01
Bootstrap methods have recently been introduced to diffusion-weighted magnetic resonance imaging to estimate the measurement uncertainty of ensuing diffusion parameters directly from the acquired data without the necessity to assume a noise model. These methods have been previously combined with deterministic streamline tractography algorithms to allow for the assessment of connection probabilities in the human brain. Thereby, the local noise induced disturbance in the diffusion data is accumulated additively due to the incremental progression of streamline tractography algorithms. Graph based approaches have been proposed to overcome this drawback of streamline techniques. For this reason, the bootstrap method is in the present work incorporated into a graph setup to derive a new probabilistic fiber tractography method, called BootGraph. The acquired data set is thereby converted into a weighted, undirected graph by defining a vertex in each voxel and edges between adjacent vertices. By means of the cone of uncertainty, which is derived using the wild bootstrap, a weight is thereafter assigned to each edge. Two path finding algorithms are subsequently applied to derive connection probabilities. While the first algorithm is based on the shortest path approach, the second algorithm takes all existing paths between two vertices into consideration. Tracking results are compared to an established algorithm based on the bootstrap method in combination with streamline fiber tractography and to another graph based algorithm. The BootGraph shows a very good performance in crossing situations with respect to false negatives and permits incorporating additional constraints, such as a curvature threshold. By inheriting the advantages of the bootstrap method and graph theory, the BootGraph method provides a computationally efficient and flexible probabilistic tractography setup to compute connection probability maps and virtual fiber pathways without the drawbacks of
Applying Directed Graph Theory to Faculty Contact Structure.
ERIC Educational Resources Information Center
House, Ernest R.; Long, John M.
Although recent writings indicate the importance of personal contact structures in diffusing innovations and in determining perceptions, the internal contact structure of the school faculty remains unexamined. This study applies directed graph theory, a new branch of mathematics, to analyzing school contact structure. Sociometric data and…
Surface charge algebra in gauge theories and thermodynamic integrability
NASA Astrophysics Data System (ADS)
Barnich, Glenn; Compère, Geoffrey
2008-04-01
Surface charges and their algebra in interacting Lagrangian gauge field theories are constructed out of the underlying linearized theory using techniques from the variational calculus. In the case of exact solutions and symmetries, the surface charges are interpreted as a Pfaff system. Integrability is governed by Frobenius' theorem and the charges associated with the derived symmetry algebra are shown to vanish. In the asymptotic context, we provide a generalized covariant derivation of the result that the representation of the asymptotic symmetry algebra through charges may be centrally extended. Comparison with Hamiltonian and covariant phase space methods is made. All approaches are shown to agree for exact solutions and symmetries while there are differences in the asymptotic context.
Partial Fractions in Calculus, Number Theory, and Algebra
ERIC Educational Resources Information Center
Yackel, C. A.; Denny, J. K.
2007-01-01
This paper explores the development of the method of partial fraction decomposition from elementary number theory through calculus to its abstraction in modern algebra. This unusual perspective makes the topic accessible and relevant to readers from high school through seasoned calculus instructors.
Algebraic formulation of quantum theory, particle identity and entanglement
NASA Astrophysics Data System (ADS)
Govindarajan, T. R.
2016-08-01
Quantum theory as formulated in conventional framework using statevectors in Hilbert spaces misses the statistical nature of the underlying quantum physics. Formulation using operators 𝒞∗ algebra and density matrices appropriately captures this feature in addition leading to the correct formulation of particle identity. In this framework, Hilbert space is an emergent concept. Problems related to anomalies and quantum epistemology are discussed.
Category of trees in representation theory of quantum algebras
Moskaliuk, N. M.; Moskaliuk, S. S.
2013-10-15
New applications of categorical methods are connected with new additional structures on categories. One of such structures in representation theory of quantum algebras, the category of Kuznetsov-Smorodinsky-Vilenkin-Smirnov (KSVS) trees, is constructed, whose objects are finite rooted KSVS trees and morphisms generated by the transition from a KSVS tree to another one.
Edge covers and independence: Algebraic approach
NASA Astrophysics Data System (ADS)
Kalinina, E. A.; Khitrov, G. M.; Pogozhev, S. V.
2016-06-01
In this paper, linear algebra methods are applied to solve some problems of graph theory. For ordinary connected graphs, edge coverings and independent sets are considered. Some results concerning minimum edge covers and maximum matchings are proved with the help of linear algebraic approach. The problem of finding a maximum matching of a graph is fundamental both practically and theoretically, and has numerous applications, e.g., in computational chemistry and mathematical chemistry.
Universal Algebraic Varieties and Ideals in Physics:. Field Theory on Algebraic Varieties
NASA Astrophysics Data System (ADS)
Iguchi, Kazumoto
A class of universal algebraic varieties in physics is discussed herein using the concepts of determinant ideals in algebraic geometry. It is shown that these algebraic varieties arise with very different physical contexts in many branches of physics and mathematics from high energy physics theory to chaos theory. In these physical systems the models are constructed by using the fields on usual manifolds such as vector fields in a Euclidean space and a Minkowskian space. But there is a universal mathematical aspect of linear algebra for linear vector spaces, where the linear independency and dependency are described using the Gramians of the vectors. These Gramians form a class of hypersurfaces in a higher-dimensional mathematical space: If there exist g vectors vi in an n-dimensional Euclidean space, the Gramian Gg is given as a g × g determinant Gg=Det[xij] with the inner products xij=(vi,vj), and exists in a g(g-1)/2-[g(g+1)/2-] dimensional space if the vectors are (not) normalized, xii=1 (xii ≠ 1). It is also shown that the Gramians are invariant under automorphisms of the vectors. The mathematical structure of the Gramians is revealed to be equivalent to the concepts of determinant ideals Ig(v), each element of which is a g × g determinant constructed from components of an arbitrary N×N matrix with N>n and which have inclusion relation: R=I0(v)⊃ I1(v) ⊃···⊃ Ig(v) ⊃···, and Ig(v)=0 if g>n. In the various physical systems the ideals naturally emerge to give us dynamical flows on the hypersurfaces, and therefore, it is called the field theory on algebraic varieties. This viewpoint provides us a grand viewpoint in physics and mathematics.
Application of chemical graph theory for automated mechanism generation.
Ratkiewicz, Artur; Truong, Thanh N
2003-01-01
We present an application of the chemical graph theory approach for generating elementary reactions of complex systems. Molecular species are naturally represented by graphs, which are identified by their vertices and edges where vertices are atom types and edges are bonds. The mechanism is generated using a set of reaction patterns (sub-graphs). These subgraphs are the internal representations for a given class of reaction thus allowing for the possibility of eliminating unimportant product species a priori. Furthermore, each molecule is canonically represented by a set of topological indices (Connectivity Index, Balaban Index, Schulz TI Index, WID Index, etc.) and thus eliminates the probability for regenerating the same species twice. Theoretical background and test cases on combustion of hydrocarbons are presented.
Dynamic modeling of electrochemical systems using linear graph theory
NASA Astrophysics Data System (ADS)
Dao, Thanh-Son; McPhee, John
An electrochemical cell is a multidisciplinary system which involves complex chemical, electrical, and thermodynamical processes. The primary objective of this paper is to develop a linear graph-theoretical modeling for the dynamic description of electrochemical systems through the representation of the system topologies. After a brief introduction to the topic and a review of linear graphs, an approach to develop linear graphs for electrochemical systems using a circuitry representation is discussed, followed in turn by the use of the branch and chord transformation techniques to generate final dynamic equations governing the system. As an example, the application of linear graph theory to modeling a nickel metal hydride (NiMH) battery will be presented. Results show that not only the number of equations are reduced significantly, but also the linear graph model simulates faster compared to the original lumped parameter model. The approach presented in this paper can be extended to modeling complex systems such as an electric or hybrid electric vehicle where a battery pack is interconnected with other components in many different domains.
Time series characterization via horizontal visibility graph and Information Theory
NASA Astrophysics Data System (ADS)
Gonçalves, Bruna Amin; Carpi, Laura; Rosso, Osvaldo A.; Ravetti, Martín G.
2016-12-01
Complex networks theory have gained wider applicability since methods for transformation of time series to networks were proposed and successfully tested. In the last few years, horizontal visibility graph has become a popular method due to its simplicity and good results when applied to natural and artificially generated data. In this work, we explore different ways of extracting information from the network constructed from the horizontal visibility graph and evaluated by Information Theory quantifiers. Most works use the degree distribution of the network, however, we found alternative probability distributions, more efficient than the degree distribution in characterizing dynamical systems. In particular, we find that, when using distributions based on distances and amplitude values, significant shorter time series are required. We analyze fractional Brownian motion time series, and a paleoclimatic proxy record of ENSO from the Pallcacocha Lake to study dynamical changes during the Holocene.
Discretized Abelian Chern-Simons gauge theory on arbitrary graphs
NASA Astrophysics Data System (ADS)
Sun, Kai; Kumar, Krishna; Fradkin, Eduardo
2015-09-01
In this paper, we show how to discretize the Abelian Chern-Simons gauge theory on generic planar lattices/graphs (with or without translational symmetries) embedded in arbitrary two-dimensional closed orientable manifolds. We find that, as long as a one-to-one correspondence between vertices and faces can be defined on the graph such that each face is paired up with a neighboring vertex (and vice versa), a discretized Abelian Chern-Simons theory can be constructed consistently. We further verify that all the essential properties of the Chern-Simons gauge theory are preserved in the discretized setup. In addition, we find that the existence of such a one-to-one correspondence is not only a sufficient condition for discretizing a Chern-Simons gauge theory but, for the discretized theory to be nonsingular and to preserve some key properties of the topological field theory, this correspondence is also a necessary one. A specific example will then be provided, in which we discretize the Abelian Chern-Simons gauge theory on a tetrahedron.
Operator Algebras and Noncommutative Geometric Aspects in Conformal Field Theory
NASA Astrophysics Data System (ADS)
Longo, Roberto
2010-03-01
The Operator Algebraic approach to Conformal Field Theory has been particularly fruitful in recent years (leading for example to the classification of all local conformal nets on the circle with central charge c < 1, jointly with Y. Kawahigashi). On the other hand the Operator Algebraic viewpoint offers a natural perspective for a Noncommutative Geometric context within Conformal Field Theory. One basic point here is to uncover the relevant structures. In this talk I will explain some of the basic steps in this "Noncommutative Geometrization program" up to the recent construction of a spectral triple associated with certain Ramond representations of the Supersymmetric Virasoro net. So Alain Connes framework enters into play. This is a joint work with S. Carpi, Y. Kawahigashi, and R. Hillier.
Stationary waves on nonlinear quantum graphs: General framework and canonical perturbation theory.
Gnutzmann, Sven; Waltner, Daniel
2016-03-01
In this paper we present a general framework for solving the stationary nonlinear Schrödinger equation (NLSE) on a network of one-dimensional wires modeled by a metric graph with suitable matching conditions at the vertices. A formal solution is given that expresses the wave function and its derivative at one end of an edge (wire) nonlinearly in terms of the values at the other end. For the cubic NLSE this nonlinear transfer operation can be expressed explicitly in terms of Jacobi elliptic functions. Its application reduces the problem of solving the corresponding set of coupled ordinary nonlinear differential equations to a finite set of nonlinear algebraic equations. For sufficiently small amplitudes we use canonical perturbation theory, which makes it possible to extract the leading nonlinear corrections over large distances.
C*-algebraic scattering theory and explicitly solvable quantum field theories
NASA Astrophysics Data System (ADS)
Warchall, Henry A.
1985-06-01
A general theoretical framework is developed for the treatment of a class of quantum field theories that are explicitly exactly solvable, but require the use of C*-algebraic techniques because time-dependent scattering theory cannot be constructed in any one natural representation of the observable algebra. The purpose is to exhibit mechanisms by which inequivalent representations of the observable algebra can arise in quantum field theory, in a setting free of other complications commonly associated with the specification of dynamics. One of two major results is the development of necessary and sufficient conditions for the concurrent unitary implementation of two automorphism groups in a class of quasifree representations of the algebra of the canonical commutation relations (CCR). The automorphism groups considered are induced by one-parameter groups of symplectic transformations on the classical phase space over which the Weyl algebra of the CCR is built; each symplectic group is conjugate by a fixed symplectic transformation to a one-parameter unitary group. The second result, an analog to the Birman-Belopol'skii theorem in two-Hilbert-space scattering theory, gives sufficient conditions for the existence of Mo/ller wave morphisms in theories with time-development automorphism groups of the above type. In a paper which follows, this framework is used to analyze a particular model system for which wave operators fail to exist in any natural representation of the observable algebra, but for which wave morphisms and an associated S matrix are easily constructed.
Fiber tracking of brain white matter based on graph theory.
Lu, Meng
2015-01-01
Brain white matter tractography is reconstructed via diffusion-weighted magnetic resonance images. Due to the complex structure of brain white matter fiber bundles, fiber crossing and fiber branching are abundant in human brain. And regular methods with diffusion tensor imaging (DTI) can't accurately handle this problem. the biggest problems of the brain tractography. Therefore, this paper presented a novel brain white matter tractography method based on graph theory, so the fiber tracking between two voxels is transformed into locating the shortest path in a graph. Besides, the presented method uses Q-ball imaging (QBI) as the source data instead of DTI, because QBI can provide accurate information about multiple fiber crossing and branching in one voxel using orientation distribution function (ODF). Experiments showed that the presented method can accurately handle the problem of brain white matter fiber crossing and branching, and reconstruct brain tractograhpy both in phantom data and real brain data.
Algebraic perturbation theory for dense liquids with discrete potentials.
Adib, Artur B
2007-06-01
A simple theory for the leading-order correction g{1}(r) to the structure of a hard-sphere liquid with discrete (e.g., square-well) potential perturbations is proposed. The theory makes use of a general approximation that effectively eliminates four-particle correlations from g{1}(r) with good accuracy at high densities. For the particular case of discrete perturbations, the remaining three-particle correlations can be modeled with a simple volume-exclusion argument, resulting in an algebraic and surprisingly accurate expression for g{1}(r). The structure of a discrete "core-softened" model for liquids with anomalous thermodynamic properties is reproduced as an application.
Cartan-Weyl 3-algebras and the BLG theory. I: classification of Cartan-Weyl 3-algebras
NASA Astrophysics Data System (ADS)
Chu, Chong-Sun
2010-10-01
As Lie algebras of compact connected Lie groups, semisimple Lie algebras have wide applications in the description of continuous symmetries of physical systems. Mathematically, semisimple Lie algebra admits a Cartan-Weyl basis of generators which consists of a Cartan subalgebra of mutually commuting generators H I and a number of step generators E α that are characterized by a root space of non-degenerate one-forms α. This simple decomposition in terms of the root space allows for a complete classification of semisimple Lie algebras. In this paper, we introduce the analogous concept of a Cartan-Weyl Lie 3-algebra. We analyze their structure and obtain a complete classification of them. Many known examples of metric Lie 3-algebras (e.g. the Lorentzian 3-algebras) are special cases of the Cartan-Weyl 3-algebras. Due to their elegant and simple structure, we speculate that Cartan-Weyl 3-algebras may be useful for describing some kinds of generalized symmetries. As an application, we consider their use in the Bagger-Lambert-Gustavsson (BLG) theory.
Huang, Yu-tin; Johansson, Henrik
2013-04-26
We show that three-dimensional supergravity amplitudes can be obtained as double copies of either three-algebra super-Chern-Simons matter theory or two-algebra super-Yang-Mills theory when either theory is organized to display the color-kinematics duality. We prove that only helicity-conserving four-dimensional gravity amplitudes have nonvanishing descendants when reduced to three dimensions, implying the vanishing of odd-multiplicity S-matrix elements, in agreement with Chern-Simons matter theory. We explicitly verify the double-copy correspondence at four and six points for N = 12,10,8 supergravity theories and discuss its validity for all multiplicity.
Functional connectivity and graph theory in preclinical Alzheimer's disease.
Brier, Matthew R; Thomas, Jewell B; Fagan, Anne M; Hassenstab, Jason; Holtzman, David M; Benzinger, Tammie L; Morris, John C; Ances, Beau M
2014-04-01
Alzheimer's disease (AD) has a long preclinical phase in which amyloid and tau cerebral pathology accumulate without producing cognitive symptoms. Resting state functional connectivity magnetic resonance imaging has demonstrated that brain networks degrade during symptomatic AD. It is unclear to what extent these degradations exist before symptomatic onset. In this study, we investigated graph theory metrics of functional integration (path length), functional segregation (clustering coefficient), and functional distinctness (modularity) as a function of disease severity. Further, we assessed whether these graph metrics were affected in cognitively normal participants with cerebrospinal fluid evidence of preclinical AD. Clustering coefficient and modularity, but not path length, were reduced in AD. Cognitively normal participants who harbored AD biomarker pathology also showed reduced values in these graph measures, demonstrating brain changes similar to, but smaller than, symptomatic AD. Only modularity was significantly affected by age. We also demonstrate that AD has a particular effect on hub-like regions in the brain. We conclude that AD causes large-scale disconnection that is present before onset of symptoms.
1979-09-01
without determinantal divisors, Linear and Multilinear Algebra 7(1979), 107-109. 4. The use of integral operators in number theory (with C. Ryavec and...Gersgorin revisited, to appear in Letters in Linear Algebra. 15. A surprising determinantal inequality for real matrices (with C.R. Johnson), to appear in...Analysis: An Essay Concerning the Limitations of Some Mathematical Methods in the Social , Political and Biological Sciences, David Berlinski, MIT Press
Graph Theory Roots of Spatial Operators for Kinematics and Dynamics
NASA Technical Reports Server (NTRS)
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
Automatic cone photoreceptor segmentation using graph theory and dynamic programming.
Chiu, Stephanie J; Lokhnygina, Yuliya; Dubis, Adam M; Dubra, Alfredo; Carroll, Joseph; Izatt, Joseph A; Farsiu, Sina
2013-06-01
Geometrical analysis of the photoreceptor mosaic can reveal subclinical ocular pathologies. In this paper, we describe a fully automatic algorithm to identify and segment photoreceptors in adaptive optics ophthalmoscope images of the photoreceptor mosaic. This method is an extension of our previously described closed contour segmentation framework based on graph theory and dynamic programming (GTDP). We validated the performance of the proposed algorithm by comparing it to the state-of-the-art technique on a large data set consisting of over 200,000 cones and posted the results online. We found that the GTDP method achieved a higher detection rate, decreasing the cone miss rate by over a factor of five.
Using Combinatorica/Mathematica for Student Projects in Random Graph Theory
ERIC Educational Resources Information Center
Pfaff, Thomas J.; Zaret, Michele
2006-01-01
We give an example of a student project that experimentally explores a topic in random graph theory. We use the "Combinatorica" package in "Mathematica" to estimate the minimum number of edges needed in a random graph to have a 50 percent chance that the graph is connected. We provide the "Mathematica" code and compare it to the known theoretical…
The algebraic theory of latent projectors in lambda matrices
NASA Technical Reports Server (NTRS)
Denman, E. D.; Leyva-Ramos, J.; Jeon, G. J.
1981-01-01
Multivariable systems such as a finite-element model of vibrating structures, control systems, and large-scale systems are often formulated in terms of differential equations which give rise to lambda matrices. The present investigation is concerned with the formulation of the algebraic theory of lambda matrices and the relationship of latent roots, latent vectors, and latent projectors to the eigenvalues, eigenvectors, and eigenprojectors of the companion form. The chain rule for latent projectors and eigenprojectors for the repeated latent root or eigenvalues is given.
AMG (Algebraic Multigrid): Basic Development, Applications and Theory.
1987-01-07
NAME OF RESPONSIBLE INDIVIDUAL 22b. TELEPHONE NUMBER 22c OFFICE SYMBOL I n iude .4 re4 Code Captain Thomas (202) 767-5025 NM DO FORM 1473.83 APR...31 (1977), 333-390, ICASE Report 76-27. (B2) A. Brandt; "Algebraic multigrid: theory", Proc. Int’l M3onf., Copper 1.buntain., C), Aprol, 1983. (B3) A... Copper Mtn., OD, April 1983. (Dl) J.E. Dendy, Jr.; "Black box multigrid," LA-UR-Sl-2337 Los Alamos National Laboratory, Los Alamos, New Mexico, J. Ccn
The elastic theory of shells using geometric algebra
Lasenby, J.; Agarwal, A.
2017-01-01
We present a novel derivation of the elastic theory of shells. We use the language of geometric algebra, which allows us to express the fundamental laws in component-free form, thus aiding physical interpretation. It also provides the tools to express equations in an arbitrary coordinate system, which enhances their usefulness. The role of moments and angular velocity, and the apparent use by previous authors of an unphysical angular velocity, has been clarified through the use of a bivector representation. In the linearized theory, clarification of previous coordinate conventions which have been the cause of confusion is provided, and the introduction of prior strain into the linearized theory of shells is made possible.
n-Nucleotide circular codes in graph theory.
Fimmel, Elena; Michel, Christian J; Strüngmann, Lutz
2016-03-13
The circular code theory proposes that genes are constituted of two trinucleotide codes: the classical genetic code with 61 trinucleotides for coding the 20 amino acids (except the three stop codons {TAA,TAG,TGA}) and a circular code based on 20 trinucleotides for retrieving, maintaining and synchronizing the reading frame. It relies on two main results: the identification of a maximal C(3) self-complementary trinucleotide circular code X in genes of bacteria, eukaryotes, plasmids and viruses (Michel 2015 J. Theor. Biol. 380, 156-177. (doi:10.1016/j.jtbi.2015.04.009); Arquès & Michel 1996 J. Theor. Biol. 182, 45-58. (doi:10.1006/jtbi.1996.0142)) and the finding of X circular code motifs in tRNAs and rRNAs, in particular in the ribosome decoding centre (Michel 2012 Comput. Biol. Chem. 37, 24-37. (doi:10.1016/j.compbiolchem.2011.10.002); El Soufi & Michel 2014 Comput. Biol. Chem. 52, 9-17. (doi:10.1016/j.compbiolchem.2014.08.001)). The univerally conserved nucleotides A1492 and A1493 and the conserved nucleotide G530 are included in X circular code motifs. Recently, dinucleotide circular codes were also investigated (Michel & Pirillo 2013 ISRN Biomath. 2013, 538631. (doi:10.1155/2013/538631); Fimmel et al. 2015 J. Theor. Biol. 386, 159-165. (doi:10.1016/j.jtbi.2015.08.034)). As the genetic motifs of different lengths are ubiquitous in genes and genomes, we introduce a new approach based on graph theory to study in full generality n-nucleotide circular codes X, i.e. of length 2 (dinucleotide), 3 (trinucleotide), 4 (tetranucleotide), etc. Indeed, we prove that an n-nucleotide code X is circular if and only if the corresponding graph [Formula: see text] is acyclic. Moreover, the maximal length of a path in [Formula: see text] corresponds to the window of nucleotides in a sequence for detecting the correct reading frame. Finally, the graph theory of tournaments is applied to the study of dinucleotide circular codes. It has full equivalence between the combinatorics
Use of graph theory measures to identify errors in record linkage.
Randall, Sean M; Boyd, James H; Ferrante, Anna M; Bauer, Jacqueline K; Semmens, James B
2014-07-01
Ensuring high linkage quality is important in many record linkage applications. Current methods for ensuring quality are manual and resource intensive. This paper seeks to determine the effectiveness of graph theory techniques in identifying record linkage errors. A range of graph theory techniques was applied to two linked datasets, with known truth sets. The ability of graph theory techniques to identify groups containing errors was compared to a widely used threshold setting technique. This methodology shows promise; however, further investigations into graph theory techniques are required. The development of more efficient and effective methods of improving linkage quality will result in higher quality datasets that can be delivered to researchers in shorter timeframes.
Hopf algebras of rooted forests, cocyles, and free Rota-Baxter algebras
NASA Astrophysics Data System (ADS)
Zhang, Tianjie; Gao, Xing; Guo, Li
2016-10-01
The Hopf algebra and the Rota-Baxter algebra are the two algebraic structures underlying the algebraic approach of Connes and Kreimer to renormalization of perturbative quantum field theory. In particular, the Hopf algebra of rooted trees serves as the "baby model" of Feynman graphs in their approach and can be characterized by certain universal properties involving a Hochschild 1-cocycle. Decorated rooted trees have also been applied to study Feynman graphs. We will continue the study of universal properties of various spaces of decorated rooted trees with such a 1-cocycle, leading to the concept of a cocycle Hopf algebra. We further apply the universal properties to equip a free Rota-Baxter algebra with the structure of a cocycle Hopf algebra.
Topological insulators and C*-algebras: Theory and numerical practice
Hastings, Matthew B.; Loring, Terry A.
2011-07-15
Research Highlights: > We classify topological insulators using C* algebras. > We present new K-theory invariants. > We develop efficient numerical algorithms based on this technique. > We observe unexpected quantum phase transitions using our algorithm. - Abstract: We apply ideas from C*-algebra to the study of disordered topological insulators. We extract certain almost commuting matrices from the free Fermi Hamiltonian, describing band projected coordinate matrices. By considering topological obstructions to approximating these matrices by exactly commuting matrices, we are able to compute invariants quantifying different topological phases. We generalize previous two dimensional results to higher dimensions; we give a general expression for the topological invariants for arbitrary dimension and several symmetry classes, including chiral symmetry classes, and we present a detailed K-theory treatment of this expression for time reversal invariant three dimensional systems. We can use these results to show non-existence of localized Wannier functions for these systems. We use this approach to calculate the index for time-reversal invariant systems with spin-orbit scattering in three dimensions, on sizes up to 12{sup 3}, averaging over a large number of samples. The results show an interesting separation between the localization transition and the point at which the average index (which can be viewed as an 'order parameter' for the topological insulator) begins to fluctuate from sample to sample, implying the existence of an unsuspected quantum phase transition separating two different delocalized phases in this system. One of the particular advantages of the C*-algebraic technique that we present is that it is significantly faster in practice than other methods of computing the index, allowing the study of larger systems. In this paper, we present a detailed discussion of numerical implementation of our method.
Noncommutative Common Cause Principles in algebraic quantum field theory
Hofer-Szabo, Gabor; Vecsernyes, Peter
2013-04-15
States in algebraic quantum field theory 'typically' establish correlation between spacelike separated events. Reichenbach's Common Cause Principle, generalized to the quantum field theoretical setting, offers an apt tool to causally account for these superluminal correlations. In the paper we motivate first why commutativity between the common cause and the correlating events should be abandoned in the definition of the common cause. Then we show that the Noncommutative Weak Common Cause Principle holds in algebraic quantum field theory with locally finite degrees of freedom. Namely, for any pair of projections A, B supported in spacelike separated regions V{sub A} and V{sub B}, respectively, there is a local projection C not necessarily commuting with A and B such that C is supported within the union of the backward light cones of V{sub A} and V{sub B} and the set {l_brace}C, C{sup Up-Tack }{r_brace} screens off the correlation between A and B.
Optimal multiplexed sensing: bounds, conditions and a graph theory link.
Ratner, Netanel; Schechner, Yoav Y; Goldberg, Felix
2007-12-10
Measuring an array of variables is central to many systems, including imagers (array of pixels), spectrometers (array of spectral bands) and lighting systems. Each of the measurements, however, is prone to noise and potential sensor saturation. It is recognized by a growing number of methods that such problems can be reduced by multiplexing the measured variables. In each measurement, multiple variables (radiation channels) are mixed (multiplexed) by a code. Then, after data acquisition, the variables are decoupled computationally in post processing. Potential benefits of the use of multiplexing include increased signal-to-noise ratio and accommodation of scene dynamic range. However, existing multiplexing schemes, including Hadamard-based codes, are inhibited by fundamental limits set by sensor saturation and Poisson distributed photon noise, which is scene dependent. There is thus a need to find optimal codes that best increase the signal to noise ratio, while accounting for these effects. Hence, this paper deals with the pursuit of such optimal measurements that avoid saturation and account for the signal dependency of noise. The paper derives lower bounds on the mean square error of demultiplexed variables. This is useful for assessing the optimality of numerically-searched multiplexing codes, thus expediting the numerical search. Furthermore, the paper states the necessary conditions for attaining the lower bounds by a general code.We show that graph theory can be harnessed for finding such ideal codes, by the use of strongly regular graphs.
Calculation of exchange energies using algebraic perturbation theory
Burrows, B. L.; Dalgarno, A.; Cohen, M.
2010-04-15
An algebraic perturbation theory is presented for efficient calculations of localized states and hence of exchange energies, which are the differences between low-lying states of the valence electron of a molecule, formed by the collision of an ion Y{sup +} with an atom X. For the case of a homonuclear molecule these are the gerade and ungerade states and the exchange energy is an exponentially decreasing function of the internuclear distance. For such homonuclear systems the theory is used in conjunction with the Herring-Holstein technique to give accurate exchange energies for a range of intermolecular separations R. Since the perturbation parameter is essentially 1/R, this method is suitable for large R. In particular, exchange energies are calculated for X{sub 2}{sup +} systems, where X is H, Li, Na, K, Rb, or Cs.
On a programming language for graph algorithms
NASA Technical Reports Server (NTRS)
Rheinboldt, W. C.; Basili, V. R.; Mesztenyi, C. K.
1971-01-01
An algorithmic language, GRAAL, is presented for describing and implementing graph algorithms of the type primarily arising in applications. The language is based on a set algebraic model of graph theory which defines the graph structure in terms of morphisms between certain set algebraic structures over the node set and arc set. GRAAL is modular in the sense that the user specifies which of these mappings are available with any graph. This allows flexibility in the selection of the storage representation for different graph structures. In line with its set theoretic foundation, the language introduces sets as a basic data type and provides for the efficient execution of all set and graph operators. At present, GRAAL is defined as an extension of ALGOL 60 (revised) and its formal description is given as a supplement to the syntactic and semantic definition of ALGOL. Several typical graph algorithms are written in GRAAL to illustrate various features of the language and to show its applicability.
Quantization of gauge fields, graph polynomials and graph homology
Kreimer, Dirk; Sars, Matthias; Suijlekom, Walter D. van
2013-09-15
We review quantization of gauge fields using algebraic properties of 3-regular graphs. We derive the Feynman integrand at n loops for a non-abelian gauge theory quantized in a covariant gauge from scalar integrands for connected 3-regular graphs, obtained from the two Symanzik polynomials. The transition to the full gauge theory amplitude is obtained by the use of a third, new, graph polynomial, the corolla polynomial. This implies effectively a covariant quantization without ghosts, where all the relevant signs of the ghost sector are incorporated in a double complex furnished by the corolla polynomial–we call it cycle homology–and by graph homology. -- Highlights: •We derive gauge theory Feynman from scalar field theory with 3-valent vertices. •We clarify the role of graph homology and cycle homology. •We use parametric renormalization and the new corolla polynomial.
Alternative Representations for Algebraic Problem Solving: When Are Graphs Better than Equations?
ERIC Educational Resources Information Center
Mielicki, Marta K.; Wiley, Jennifer
2016-01-01
Successful algebraic problem solving entails adaptability of solution methods using different representations. Prior research has suggested that students are more likely to prefer symbolic solution methods (equations) over graphical ones, even when graphical methods should be more efficient. However, this research has not tested how representation…
Graph Theory and Its Application in Educational Research: A Review and Integration.
ERIC Educational Resources Information Center
Tatsuoka, Maurice M.
1986-01-01
A nontechnical exposition of graph theory is presented, followed by survey of the literature on applications of graph theory in research in education and related disciplines. Applications include order-theoretic studies of the dimensionality of data sets, the investigation of hierarchical structures in various domains, and cluster analysis.…
Three-dimensional spin-3 theories based on general kinematical algebras
NASA Astrophysics Data System (ADS)
Bergshoeff, Eric; Grumiller, Daniel; Prohazka, Stefan; Rosseel, Jan
2017-01-01
We initiate the study of non- and ultra-relativistic higher spin theories. For sake of simplicity we focus on the spin-3 case in three dimensions. We classify all kinematical algebras that can be obtained by all possible Inönü-Wigner contraction procedures of the kinematical algebra of spin-3 theory in three dimensional (anti-) de Sitter space-time. We demonstrate how to construct associated actions of Chern-Simons type, directly in the ultra-relativistic case and by suitable algebraic extensions in the non-relativistic case. We show how to give these kinematical algebras an infinite-dimensional lift by imposing suitable boundary conditions in a theory we call "Carroll Gravity", whose asymptotic symmetry algebra turns out to be an infinite-dimensional extension of the Carroll algebra.
The Clifford algebra of physical space and Dirac theory
NASA Astrophysics Data System (ADS)
Vaz, Jayme, Jr.
2016-09-01
The claim found in many textbooks that the Dirac equation cannot be written solely in terms of Pauli matrices is shown to not be completely true. It is only true as long as the term β \\psi in the usual Dirac factorization of the Klein-Gordon equation is assumed to be the product of a square matrix β and a column matrix ψ. In this paper we show that there is another possibility besides this matrix product, in fact a possibility involving a matrix operation, and show that it leads to another possible expression for the Dirac equation. We show that, behind this other possible factorization is the formalism of the Clifford algebra of physical space. We exploit this fact, and discuss several different aspects of Dirac theory using this formalism. In particular, we show that there are four different possible sets of definitions for the parity, time reversal, and charge conjugation operations for the Dirac equation.
Superalgebra realization of the 3-algebras in N=6, 8 Chern-Simons-matter theories
NASA Astrophysics Data System (ADS)
Chen, Fa-Min
2012-01-01
We use superalgebras to realize the 3-algebras used to construct N=6, 8 Chern-Simons-matter (CSM) theories. We demonstrate that the superalgebra realization of the 3-algebras provides a unified framework for classifying the gauge groups of the Nge 5 theories based on 3-algebras. Using this realization, we rederive the ordinary Lie algebra construction of the general N=6 CSM theory from its 3-algebra counterpart and reproduce all known examples as well. In particular, we explicitly construct the Nambu 3-bracket in terms of a double graded commutator of PSU(2|2). The N=8 theory of Bagger, Lambert and Gustavsson (BLG) with SO(4) gauge group is constructed by using several different ways. A quantization scheme for the 3-brackets is proposed by promoting the double graded commutators as quantum mechanical double graded commutators.
Hopf algebras and topological recursion
NASA Astrophysics Data System (ADS)
Esteves, João N.
2015-11-01
We consider a model for topological recursion based on the Hopf algebra of planar binary trees defined by Loday and Ronco (1998 Adv. Math. 139 293-309 We show that extending this Hopf algebra by identifying pairs of nearest neighbor leaves, and thus producing graphs with loops, we obtain the full recursion formula discovered by Eynard and Orantin (2007 Commun. Number Theory Phys. 1 347-452).
Graph theory as a proxy for spatially explicit population models in conservation planning.
Minor, Emily S; Urban, Dean L
2007-09-01
Spatially explicit population models (SEPMs) are often considered the best way to predict and manage species distributions in spatially heterogeneous landscapes. However, they are computationally intensive and require extensive knowledge of species' biology and behavior, limiting their application in many cases. An alternative to SEPMs is graph theory, which has minimal data requirements and efficient algorithms. Although only recently introduced to landscape ecology, graph theory is well suited to ecological applications concerned with connectivity or movement. This paper compares the performance of graph theory to a SEPM in selecting important habitat patches for Wood Thrush (Hylocichla mustelina) conservation. We use both models to identify habitat patches that act as population sources and persistent patches and also use graph theory to identify patches that act as stepping stones for dispersal. Correlations of patch rankings were very high between the two models. In addition, graph theory offers the ability to identify patches that are very important to habitat connectivity and thus long-term population persistence across the landscape. We show that graph theory makes very similar predictions in most cases and in other cases offers insight not available from the SEPM, and we conclude that graph theory is a suitable and possibly preferable alternative to SEPMs for species conservation in heterogeneous landscapes.
Fibonacci Identities, Matrices, and Graphs
ERIC Educational Resources Information Center
Huang, Danrun
2005-01-01
General strategies used to help discover, prove, and generalize identities for Fibonacci numbers are described along with some properties about the determinants of square matrices. A matrix proof for identity (2) that has received immense attention from many branches of mathematics, like linear algebra, dynamical systems, graph theory and others…
Augmenting Conceptual Design Trajectory Tradespace Exploration with Graph Theory
NASA Technical Reports Server (NTRS)
Dees, Patrick D.; Zwack, Mathew R.; Steffens, Michael; Edwards, Stephen
2016-01-01
Within conceptual design changes occur rapidly due to a combination of uncertainty and shifting requirements. To stay relevant in this fluid time, trade studies must also be performed rapidly. In order to drive down analysis time while improving the information gained by these studies, surrogate models can be created to represent the complex output of a tool or tools within a specified tradespace. In order to create this model however, a large amount of data must be collected in a short amount of time. By this method, the historical approach of relying on subject matter experts to generate the data required is schedule infeasible. However, by implementing automation and distributed analysis the required data can be generated in a fraction of the time. Previous work focused on setting up a tool called multiPOST capable of orchestrating many simultaneous runs of an analysis tool assessing these automated analyses utilizing heuristics gleaned from the best practices of current subject matter experts. In this update to the previous work, elements of graph theory are included to further drive down analysis time by leveraging data previously gathered. It is shown to outperform the previous method in both time required, and the quantity and quality of data produced.
Scheduling for indoor visible light communication based on graph theory.
Tao, Yuyang; Liang, Xiao; Wang, Jiaheng; Zhao, Chunming
2015-02-09
Visible light communication (VLC) has drawn much attention in the field of high-rate indoor wireless communication. While most existing works focused on point-to-point VLC technologies, few studies have concerned multiuser VLC, where multiple optical access points (APs) transmit data to multiple user receivers. In such scenarios, inter-user interference constitutes the major factor limiting the system performance. Therefore, a proper scheduling scheme has to be proposed to coordinate the interference and optimize the whole system performance. In this work, we aim to maximize the sum rate of the system while taking into account user fairness by appropriately assigning LED lamps to multiple users. The formulated scheduling problem turns out to be a maximum weighted independent set problem. We then propose a novel and efficient resource allocation method based on graph theory to achieve high sum rates. Moreover, we also introduce proportional fairness into our scheduling scheme to ensure the user fairness. Our proposed scheduling scheme can, with low complexity, achieve more multiplexing gains, higher sum rate, and better fairness than the existing works.
Exploring biomedical ontology mappings with graph theory methods
2017-01-01
Background In the era of semantic web, life science ontologies play an important role in tasks such as annotating biological objects, linking relevant data pieces, and verifying data consistency. Understanding ontology structures and overlapping ontologies is essential for tasks such as ontology reuse and development. We present an exploratory study where we examine structure and look for patterns in BioPortal, a comprehensive publicly available repository of live science ontologies. Methods We report an analysis of biomedical ontology mapping data over time. We apply graph theory methods such as Modularity Analysis and Betweenness Centrality to analyse data gathered at five different time points. We identify communities, i.e., sets of overlapping ontologies, and define similar and closest communities. We demonstrate evolution of identified communities over time and identify core ontologies of the closest communities. We use BioPortal project and category data to measure community coherence. We also validate identified communities with their mutual mentions in scientific literature. Results With comparing mapping data gathered at five different time points, we identified similar and closest communities of overlapping ontologies, and demonstrated evolution of communities over time. Results showed that anatomy and health ontologies tend to form more isolated communities compared to other categories. We also showed that communities contain all or the majority of ontologies being used in narrower projects. In addition, we identified major changes in mapping data after migration to BioPortal Version 4. PMID:28265499
Fragmentation network of doubly charged methionine: Interpretation using graph theory.
Ha, D T; Yamazaki, K; Wang, Y; Alcamí, M; Maeda, S; Kono, H; Martín, F; Kukk, E
2016-09-07
The fragmentation of doubly charged gas-phase methionine (HO2CCH(NH2)CH2CH2SCH3) is systematically studied using the self-consistent charge density functional tight-binding molecular dynamics (MD) simulation method. We applied graph theory to analyze the large number of the calculated MD trajectories, which appears to be a highly effective and convenient means of extracting versatile information from the large data. The present theoretical results strongly concur with the earlier studied experimental ones. Essentially, the dication dissociates into acidic group CO2H and basic group C4NSH10. The former may carry a single or no charge and stays intact in most cases, whereas the latter may hold either a single or a double charge and tends to dissociate into smaller fragments. The decay of the basic group is observed to follow the Arrhenius law. The dissociation pathways to CO2H and C4NSH10 and subsequent fragmentations are also supported by ab initio calculations.
ERIC Educational Resources Information Center
Debnath, Lokenath
2010-01-01
This article is essentially devoted to a brief historical introduction to Euler's formula for polyhedra, topology, theory of graphs and networks with many examples from the real-world. Celebrated Konigsberg seven-bridge problem and some of the basic properties of graphs and networks for some understanding of the macroscopic behaviour of real…
Quantum field theory on toroidal topology: Algebraic structure and applications
NASA Astrophysics Data System (ADS)
Khanna, F. C.; Malbouisson, A. P. C.; Malbouisson, J. M. C.; Santana, A. E.
2014-05-01
The development of quantum theory on a torus has a long history, and can be traced back to the 1920s, with the attempts by Nordström, Kaluza and Klein to define a fourth spatial dimension with a finite size, being curved in the form of a torus, such that Einstein and Maxwell equations would be unified. Many developments were carried out considering cosmological problems in association with particle physics, leading to methods that are useful for areas of physics, in which size effects play an important role. This interest in finite size effect systems has been increasing rapidly over the last decades, due principally to experimental improvements. In this review, the foundations of compactified quantum field theory on a torus are presented in a unified way, in order to consider applications in particle and condensed matter physics. The theory on a torus ΓDd=(S1)d×RD-d is developed from a Lie-group representation and c*c*-algebra formalisms. As a first application, the quantum field theory at finite temperature, in its real- and imaginary-time versions, is addressed by focusing on its topological structure, the torus Γ41. The toroidal quantum-field theory provides the basis for a consistent approach of spontaneous symmetry breaking driven by both temperature and spatial boundaries. Then the superconductivity in films, wires and grains are analyzed, leading to some results that are comparable with experiments. The Casimir effect is studied taking the electromagnetic and Dirac fields on a torus. In this case, the method of analysis is based on a generalized Bogoliubov transformation, that separates the Green function into two parts: one is associated with the empty space-time, while the other describes the impact of compactification. This provides a natural procedure for calculating the renormalized energy-momentum tensor. Self interacting four-fermion systems, described by the Gross-Neveu and Nambu-Jona-Lasinio models, are considered. Then finite size effects on
Chemical Applications of Graph Theory: Part I. Fundamentals and Topological Indices.
ERIC Educational Resources Information Center
Hansen, Peter J.; Jurs, Peter C.
1988-01-01
Explores graph theory and use of topological indices to predict boiling points. Lists three indices: Wiener Number, Randic Branching Index and Molecular Connectivity, and Molecular Identification numbers. Warns of inadequacies with stereochemistry. (ML)
Investigation of Zero Knowledge Proof Approaches Based on Graph Theory
2011-02-01
34 Singapore ICCS/ISITA, 1992: 274-278. LeBodic, P., H. Locteau, S. Adam, P. Heroux, Y. Lecourtier, and A. Knippel. "Symbol Detection Using Region...a particular isomorphism between the two graphs (not just any isomorphism). Outlines the protocol and proves that successful forgery in the...demonstrates a forgery algorithm. GRAPH NON-ISOMORPHISM 21. O. Goldreich. Foundations of Cryptography: Fragments of a Book. Weizmann Institute
Perturbative quantization of Yang-Mills theory with classical double as gauge algebra
NASA Astrophysics Data System (ADS)
Ruiz Ruiz, F.
2016-02-01
Perturbative quantization of Yang-Mills theory with a gauge algebra given by the classical double of a semisimple Lie algebra is considered. The classical double of a real Lie algebra is a nonsemisimple real Lie algebra that admits a nonpositive definite invariant metric, the indefiniteness of the metric suggesting an apparent lack of unitarity. It is shown that the theory is UV divergent at one loop and that there are no radiative corrections at higher loops. One-loop UV divergences are removed through renormalization of the coupling constant, thus introducing a renormalization scale. The terms in the classical action that would spoil unitarity are proved to be cohomologically trivial with respect to the Slavnov-Taylor operator that controls gauge invariance for the quantum theory. Hence they do not contribute gauge invariant radiative corrections to the quantum effective action and the theory is unitary.
SYMBOLIC ALGEBRAIC MANIPULATION BY DIGITAL COMPUTER IN PROBLEMS OF CONTROL THEORY.
shown, using a FORMAC program. The advantages over the conventional root locus method are discussed. Areas of possible future use of FORMAC in algebraic problems of control theory are discussed. (Author)
Spectral Graph Theory Analysis of Software-Defined Networks to Improve Performance and Security
2015-09-01
networks for transmission operations in smart grids,” in the Proc. IEEE PES Innovative Smart Grid Technologies (ISGT), Washington, DC, 2013. [34] D...GRAPH THEORY ANALYSIS OF SOFTWARE-DEFINED NETWORKS TO IMPROVE PERFORMANCE AND SECURITY by Thomas C. Parker September 2015 Dissertation Co...September 2015 3. REPORT TYPE AND DATES COVERED Dissertation 4. TITLE AND SUBTITLE SPECTRAL GRAPH THEORY ANALYSIS OF SOFTWARE-DEFINED NETWORKS
Compactly supported Wannier functions and algebraic K -theory
NASA Astrophysics Data System (ADS)
Read, N.
2017-03-01
In a tight-binding lattice model with n orbitals (single-particle states) per site, Wannier functions are n -component vector functions of position that fall off rapidly away from some location, and such that a set of them in some sense span all states in a given energy band or set of bands; compactly supported Wannier functions are such functions that vanish outside a bounded region. They arise not only in band theory, but also in connection with tensor-network states for noninteracting fermion systems, and for flat-band Hamiltonians with strictly short-range hopping matrix elements. In earlier work, it was proved that for general complex band structures (vector bundles) or general complex Hamiltonians—that is, class A in the tenfold classification of Hamiltonians and band structures—a set of compactly supported Wannier functions can span the vector bundle only if the bundle is topologically trivial, in any dimension d of space, even when use of an overcomplete set of such functions is permitted. This implied that, for a free-fermion tensor network state with a nontrivial bundle in class A, any strictly short-range parent Hamiltonian must be gapless. Here, this result is extended to all ten symmetry classes of band structures without additional crystallographic symmetries, with the result that in general the nontrivial bundles that can arise from compactly supported Wannier-type functions are those that may possess, in each of d directions, the nontrivial winding that can occur in the same symmetry class in one dimension, but nothing else. The results are obtained from a very natural usage of algebraic K -theory, based on a ring of polynomials in e±i kx,e±i ky,..., which occur as entries in the Fourier-transformed Wannier functions.
NASA Astrophysics Data System (ADS)
Debnath, Lokenath
2010-09-01
This article is essentially devoted to a brief historical introduction to Euler's formula for polyhedra, topology, theory of graphs and networks with many examples from the real-world. Celebrated Königsberg seven-bridge problem and some of the basic properties of graphs and networks for some understanding of the macroscopic behaviour of real physical systems are included. We also mention some important and modern applications of graph theory or network problems from transportation to telecommunications. Graphs or networks are effectively used as powerful tools in industrial, electrical and civil engineering, communication networks in the planning of business and industry. Graph theory and combinatorics can be used to understand the changes that occur in many large and complex scientific, technical and medical systems. With the advent of fast large computers and the ubiquitous Internet consisting of a very large network of computers, large-scale complex optimization problems can be modelled in terms of graphs or networks and then solved by algorithms available in graph theory. Many large and more complex combinatorial problems dealing with the possible arrangements of situations of various kinds, and computing the number and properties of such arrangements can be formulated in terms of networks. The Knight's tour problem, Hamilton's tour problem, problem of magic squares, the Euler Graeco-Latin squares problem and their modern developments in the twentieth century are also included.
NASA Technical Reports Server (NTRS)
Byrnes, C. I.
1980-01-01
It is noted that recent work by Kamen (1979) on the stability of half-plane digital filters shows that the problem of the existence of a feedback law also arises for other Banach algebras in applications. This situation calls for a realization theory and stabilizability criteria for systems defined over Banach for Frechet algebra A. Such a theory is developed here, with special emphasis placed on the construction of finitely generated realizations, the existence of coprime factorizations for T(s) defined over A, and the solvability of the quadratic optimal control problem and the associated algebraic Riccati equation over A.
Detecting labor using graph theory on connectivity matrices of uterine EMG.
Al-Omar, S; Diab, A; Nader, N; Khalil, M; Karlsson, B; Marque, C
2015-08-01
Premature labor is one of the most serious health problems in the developed world. One of the main reasons for this is that no good way exists to distinguish true labor from normal pregnancy contractions. The aim of this paper is to investigate if the application of graph theory techniques to multi-electrode uterine EMG signals can improve the discrimination between pregnancy contractions and labor. To test our methods we first applied them to synthetic graphs where we detected some differences in the parameters results and changes in the graph model from pregnancy-like graphs to labor-like graphs. Then, we applied the same methods to real signals. We obtained the best differentiation between pregnancy and labor through the same parameters. Major improvements in differentiating between pregnancy and labor were obtained using a low pass windowing preprocessing step. Results show that real graphs generally became more organized when moving from pregnancy, where the graph showed random characteristics, to labor where the graph became a more small-world like graph.
Description of the human hand grasp using graph theory.
Liu, Xiancan; Zhan, Qiang
2013-07-01
This paper presents a method to describe and analyze the human hand grasp postures so as to indicate which fingers should act during grasping and the required movements of those fingers. The method first describes the human hand with human hand tree graph and incidence matrix, and then the relationship between the human hand and the grasped object is described by grasp contact graph and basic cycle matrix that can be divided into an identity matrix and a Bf12 matrix. The nonzero columns of the Bf12 matrix can be described by a graph called VF-tree, which can indicate which fingers are active while grasping and the required degree of freedom of each finger. The method is validated by describing and analyzing the six basic grasp postures of the human hand.
Time-dependence of graph theory metrics in functional connectivity analysis
Chiang, Sharon; Cassese, Alberto; Guindani, Michele; Vannucci, Marina; Yeh, Hsiang J.; Haneef, Zulfi; Stern, John M.
2016-01-01
Brain graphs provide a useful way to computationally model the network structure of the connectome, and this has led to increasing interest in the use of graph theory to quantitate and investigate the topological characteristics of the healthy brain and brain disorders on the network level. The majority of graph theory investigations of functional connectivity have relied on the assumption of temporal stationarity. However, recent evidence increasingly suggests that functional connectivity fluctuates over the length of the scan. In this study, we investigate the stationarity of brain network topology using a Bayesian hidden Markov model (HMM) approach that estimates the dynamic structure of graph theoretical measures of whole-brain functional connectivity. In addition to extracting the stationary distribution and transition probabilities of commonly employed graph theory measures, we propose two estimators of temporal stationarity: the S-index and N-index. These indexes can be used to quantify different aspects of the temporal stationarity of graph theory measures. We apply the method and proposed estimators to resting-state functional MRI data from healthy controls and patients with temporal lobe epilepsy. Our analysis shows that several graph theory measures, including small-world index, global integration measures, and betweenness centrality, may exhibit greater stationarity over time and therefore be more robust. Additionally, we demonstrate that accounting for subject-level differences in the level of temporal stationarity of network topology may increase discriminatory power in discriminating between disease states. Our results confirm and extend findings from other studies regarding the dynamic nature of functional connectivity, and suggest that using statistical models which explicitly account for the dynamic nature of functional connectivity in graph theory analyses may improve the sensitivity of investigations and consistency across investigations. PMID
Time-dependence of graph theory metrics in functional connectivity analysis.
Chiang, Sharon; Cassese, Alberto; Guindani, Michele; Vannucci, Marina; Yeh, Hsiang J; Haneef, Zulfi; Stern, John M
2016-01-15
Brain graphs provide a useful way to computationally model the network structure of the connectome, and this has led to increasing interest in the use of graph theory to quantitate and investigate the topological characteristics of the healthy brain and brain disorders on the network level. The majority of graph theory investigations of functional connectivity have relied on the assumption of temporal stationarity. However, recent evidence increasingly suggests that functional connectivity fluctuates over the length of the scan. In this study, we investigate the stationarity of brain network topology using a Bayesian hidden Markov model (HMM) approach that estimates the dynamic structure of graph theoretical measures of whole-brain functional connectivity. In addition to extracting the stationary distribution and transition probabilities of commonly employed graph theory measures, we propose two estimators of temporal stationarity: the S-index and N-index. These indexes can be used to quantify different aspects of the temporal stationarity of graph theory measures. We apply the method and proposed estimators to resting-state functional MRI data from healthy controls and patients with temporal lobe epilepsy. Our analysis shows that several graph theory measures, including small-world index, global integration measures, and betweenness centrality, may exhibit greater stationarity over time and therefore be more robust. Additionally, we demonstrate that accounting for subject-level differences in the level of temporal stationarity of network topology may increase discriminatory power in discriminating between disease states. Our results confirm and extend findings from other studies regarding the dynamic nature of functional connectivity, and suggest that using statistical models which explicitly account for the dynamic nature of functional connectivity in graph theory analyses may improve the sensitivity of investigations and consistency across investigations.
Angular momentum algebra for symbolic expansions in atomic structure theory
NASA Astrophysics Data System (ADS)
Matulioniene, Rasa
Computer programs based on multiconfiguration methods have become standard tools in atomic structure theory. Reliable predictions of atomic properties require very large configuration expansions. The computational resources required often exceed the capabilities of conventional computers. There is a need to restructure existing computer programs to take advantage of modern high-performance computational technology. This dissertation deals with one important aspect of the effort to implement two widely used atomic structure packages (MCHF and GRASP92) on distributed memory parallel computers: the method for handling the angular momentum algebra. In the existing algorithms, the angular integrations required for the Hamiltonian matrix elements are computed for each pair of configurations, even though the results may be identical or very similar for all configurations of a given type. This redundancy leads to a significant increase in computer resource requirements, because the angular matrix elements, which are repeatedly reused in the calculation, need to be stored in computer memory or on disk. At present, the size (and, therefore, accuracy) of the calculations is limited by the large amounts of angular data produced. The aim of the research reported in this dissertation is to provide the theoretical basis for a computational method to curtail the growth of stored angular data with the size of the calculation. The multiconfiguration basis is often generated by one- and two-particle replacements from a reference set to correlation orbitals. The redundancy in the stored angular data could be removed by reformulating the algorithm to treat simultaneously all angular matrix elements that differ only in the quantum numbers of the correlation orbitals. To accomplish this, we expand N- electron matrix elements of a general symmetric two-body scalar operator, an example of which is the Hamiltonian, in terms of two-electron matrix elements. Using diagrammatic methods of
A preliminary study on atrial epicardial mapping signals based on Graph Theory.
Sun, Liqian; Yang, Cuiwei; Zhang, Lin; Chen, Ying; Wu, Zhong; Shao, Jun
2014-07-01
In order to get a better understanding of atrial fibrillation, we introduced a method based on Graph Theory to interpret the relations of different parts of the atria. Atrial electrograms under sinus rhythm and atrial fibrillation were collected from eight living mongrel dogs with cholinergic AF model. These epicardial signals were acquired from 95 unipolar electrodes attached to the surface of the atria and four pulmonary veins. Then, we analyzed the electrode correlations using Graph Theory. The topology, the connectivity and the parameters of graphs during different rhythms were studied. Our results showed that the connectivity of graphs varied from sinus rhythm to atrial fibrillation and there were parameter gradients in various parts of the atria. The results provide spatial insight into the interaction between different parts of the atria and the method may have its potential for studying atrial fibrillation.
Linear algebraic theory of partial coherence: discrete fields and measures of partial coherence.
Ozaktas, Haldun M; Yüksel, Serdar; Kutay, M Alper
2002-08-01
A linear algebraic theory of partial coherence is presented that allows precise mathematical definitions of concepts such as coherence and incoherence. This not only provides new perspectives and insights but also allows us to employ the conceptual and algebraic tools of linear algebra in applications. We define several scalar measures of the degree of partial coherence of an optical field that are zero for full incoherence and unity for full coherence. The mathematical definitions are related to our physical understanding of the corresponding concepts by considering them in the context of Young's experiment.
Why plerograms are not used in chemical graph theory? The case of terminal-Wiener index
NASA Astrophysics Data System (ADS)
Gutman, Ivan; Essalih, Mohamed; Marraki, Mohamed El; Furtula, Boris
2013-05-01
The reasons why plerogram-type molecular graphs are almost never used in contemporary chemical graph theory are analyzed. As one more argument against the usage of plerograms, we show that the terminal-Wiener index of the plerogram of an alkane is in an (exact) linear manner related to the ordinary Wiener index, thus having the precisely same structure dependency as the ordinary Wiener index.
Threshold Graph Limits and Random Threshold Graphs
Diaconis, Persi; Holmes, Susan; Janson, Svante
2010-01-01
We study the limit theory of large threshold graphs and apply this to a variety of models for random threshold graphs. The results give a nice set of examples for the emerging theory of graph limits. PMID:20811581
Linear algebraic theory of partial coherence: continuous fields and measures of partial coherence.
Ozaktas, Haldun M; Gulcu, Talha Cihad; Alper Kutay, M
2016-11-01
This work presents a linear algebraic theory of partial coherence for optical fields of continuous variables. This approach facilitates use of linear algebraic techniques and makes it possible to precisely define the concepts of incoherence and coherence in a mathematical way. We have proposed five scalar measures for the degree of partial coherence. These measures are zero for incoherent fields, unity for fully coherent fields, and between zero and one for partially coherent fields.
Algebraic approach to form factors in the complex sinh-Gordon theory
NASA Astrophysics Data System (ADS)
Lashkevich, Michael; Pugai, Yaroslav
2017-01-01
We study form factors of the quantum complex sinh-Gordon theory in the algebraic approach. In the case of exponential fields the form factors can be obtained from the known form factors of the ZN-symmetric Ising model. The algebraic construction also provides an Ansatz for form factors of descendant operators. We obtain generating functions of such form factors and establish their main properties: the cluster factorization and reflection equations.
Poor Textural Image Matching Based on Graph Theory
NASA Astrophysics Data System (ADS)
Chen, Shiyu; Yuan, Xiuxiao; Yuan, Wei; Cai, Yang
2016-06-01
Image matching lies at the heart of photogrammetry and computer vision. For poor textural images, the matching result is affected by low contrast, repetitive patterns, discontinuity or occlusion, few or homogeneous textures. Recently, graph matching became popular for its integration of geometric and radiometric information. Focused on poor textural image matching problem, it is proposed an edge-weight strategy to improve graph matching algorithm. A series of experiments have been conducted including 4 typical landscapes: Forest, desert, farmland, and urban areas. And it is experimentally found that our new algorithm achieves better performance. Compared to SIFT, doubled corresponding points were acquired, and the overall recall rate reached up to 68%, which verifies the feasibility and effectiveness of the algorithm.
Network meta-analysis, electrical networks and graph theory.
Rücker, Gerta
2012-12-01
Network meta-analysis is an active field of research in clinical biostatistics. It aims to combine information from all randomized comparisons among a set of treatments for a given medical condition. We show how graph-theoretical methods can be applied to network meta-analysis. A meta-analytic graph consists of vertices (treatments) and edges (randomized comparisons). We illustrate the correspondence between meta-analytic networks and electrical networks, where variance corresponds to resistance, treatment effects to voltage, and weighted treatment effects to current flows. Based thereon, we then show that graph-theoretical methods that have been routinely applied to electrical networks also work well in network meta-analysis. In more detail, the resulting consistent treatment effects induced in the edges can be estimated via the Moore-Penrose pseudoinverse of the Laplacian matrix. Moreover, the variances of the treatment effects are estimated in analogy to electrical effective resistances. It is shown that this method, being computationally simple, leads to the usual fixed effect model estimate when applied to pairwise meta-analysis and is consistent with published results when applied to network meta-analysis examples from the literature. Moreover, problems of heterogeneity and inconsistency, random effects modeling and including multi-armed trials are addressed. Copyright © 2012 John Wiley & Sons, Ltd.
Higher gauge theories from Lie n-algebras and off-shell covariantization
NASA Astrophysics Data System (ADS)
Carow-Watamura, Ursula; Heller, Marc Andre; Ikeda, Noriaki; Kaneko, Yukio; Watamura, Satoshi
2016-07-01
We analyze higher gauge theories in various dimensions using a supergeometric method based on a differential graded symplectic manifold, called a QP-manifold, which is closely related to the BRST-BV formalism in gauge theories. Extensions of the Lie 2-algebra gauge structure are formulated within the Lie n-algebra induced by the QP-structure. We find that in 5 and 6 dimensions there are special extensions of the gauge algebra. In these cases, a restriction of the gauge symmetry by imposing constraints on the auxiliary gauge fields leads to a covariantized theory. As an example we show that we can obtain an off-shell covariantized higher gauge theory in 5 dimensions, which is similar to the one proposed in [1].
A new application of algebraic geometry to systems theory
NASA Technical Reports Server (NTRS)
Martin, C. F.; Hermann, R.
1976-01-01
Following an introduction to algebraic geometry, the dominant morphism theorem is stated, and the application of this theorem to systems-theoretic problems, such as the feedback problem, is discussed. The Gaussian elimination method used for solving linear equations is shown to be an example of a dominant morphism.
Sacchet, Matthew D.; Prasad, Gautam; Foland-Ross, Lara C.; Thompson, Paul M.; Gotlib, Ian H.
2015-01-01
Recently, there has been considerable interest in understanding brain networks in major depressive disorder (MDD). Neural pathways can be tracked in the living brain using diffusion-weighted imaging (DWI); graph theory can then be used to study properties of the resulting fiber networks. To date, global abnormalities have not been reported in tractography-based graph metrics in MDD, so we used a machine learning approach based on “support vector machines” to differentiate depressed from healthy individuals based on multiple brain network properties. We also assessed how important specific graph metrics were for this differentiation. Finally, we conducted a local graph analysis to identify abnormal connectivity at specific nodes of the network. We were able to classify depression using whole-brain graph metrics. Small-worldness was the most useful graph metric for classification. The right pars orbitalis, right inferior parietal cortex, and left rostral anterior cingulate all showed abnormal network connectivity in MDD. This is the first use of structural global graph metrics to classify depressed individuals. These findings highlight the importance of future research to understand network properties in depression across imaging modalities, improve classification results, and relate network alterations to psychiatric symptoms, medication, and comorbidities. PMID:25762941
Sacchet, Matthew D; Prasad, Gautam; Foland-Ross, Lara C; Thompson, Paul M; Gotlib, Ian H
2015-01-01
Recently, there has been considerable interest in understanding brain networks in major depressive disorder (MDD). Neural pathways can be tracked in the living brain using diffusion-weighted imaging (DWI); graph theory can then be used to study properties of the resulting fiber networks. To date, global abnormalities have not been reported in tractography-based graph metrics in MDD, so we used a machine learning approach based on "support vector machines" to differentiate depressed from healthy individuals based on multiple brain network properties. We also assessed how important specific graph metrics were for this differentiation. Finally, we conducted a local graph analysis to identify abnormal connectivity at specific nodes of the network. We were able to classify depression using whole-brain graph metrics. Small-worldness was the most useful graph metric for classification. The right pars orbitalis, right inferior parietal cortex, and left rostral anterior cingulate all showed abnormal network connectivity in MDD. This is the first use of structural global graph metrics to classify depressed individuals. These findings highlight the importance of future research to understand network properties in depression across imaging modalities, improve classification results, and relate network alterations to psychiatric symptoms, medication, and comorbidities.
Test-retest reliability of graph theory measures of structural brain connectivity.
Dennis, Emily L; Jahanshad, Neda; Toga, Arthur W; McMahon, Katie L; de Zubicaray, Greig I; Martin, Nicholas G; Wright, Margaret J; Thompson, Paul M
2012-01-01
The human connectome has recently become a popular research topic in neuroscience, and many new algorithms have been applied to analyze brain networks. In particular, network topology measures from graph theory have been adapted to analyze network efficiency and 'small-world' properties. While there has been a surge in the number of papers examining connectivity through graph theory, questions remain about its test-retest reliability (TRT). In particular, the reproducibility of structural connectivity measures has not been assessed. We examined the TRT of global connectivity measures generated from graph theory analyses of 17 young adults who underwent two high-angular resolution diffusion (HARDI) scans approximately 3 months apart. Of the measures assessed, modularity had the highest TRT, and it was stable across a range of sparsities (a thresholding parameter used to define which network edges are retained). These reliability measures underline the need to develop network descriptors that are robust to acquisition parameters.
Matched signal detection on graphs: Theory and application to brain imaging data classification.
Hu, Chenhui; Sepulcre, Jorge; Johnson, Keith A; Fakhri, Georges E; Lu, Yue M; Li, Quanzheng
2016-01-15
Motivated by recent progress in signal processing on graphs, we have developed a matched signal detection (MSD) theory for signals with intrinsic structures described by weighted graphs. First, we regard graph Laplacian eigenvalues as frequencies of graph-signals and assume that the signal is in a subspace spanned by the first few graph Laplacian eigenvectors associated with lower eigenvalues. The conventional matched subspace detector can be applied to this case. Furthermore, we study signals that may not merely live in a subspace. Concretely, we consider signals with bounded variation on graphs and more general signals that are randomly drawn from a prior distribution. For bounded variation signals, the test is a weighted energy detector. For the random signals, the test statistic is the difference of signal variations on associated graphs, if a degenerate Gaussian distribution specified by the graph Laplacian is adopted. We evaluate the effectiveness of the MSD on graphs both with simulated and real data sets. Specifically, we apply MSD to the brain imaging data classification problem of Alzheimer's disease (AD) based on two independent data sets: 1) positron emission tomography data with Pittsburgh compound-B tracer of 30 AD and 40 normal control (NC) subjects, and 2) resting-state functional magnetic resonance imaging (R-fMRI) data of 30 early mild cognitive impairment and 20 NC subjects. Our results demonstrate that the MSD approach is able to outperform the traditional methods and help detect AD at an early stage, probably due to the success of exploiting the manifold structure of the data.
Revealing long-range interconnected hubs in human chromatin interaction data using graph theory.
Boulos, R E; Arneodo, A; Jensen, P; Audit, B
2013-09-13
We use graph theory to analyze chromatin interaction (Hi-C) data in the human genome. We show that a key functional feature of the genome--"master" replication origins--corresponds to DNA loci of maximal network centrality. These loci form a set of interconnected hubs both within chromosomes and between different chromosomes. Our results open the way to a fruitful use of graph theory concepts to decipher DNA structural organization in relation to genome functions such as replication and transcription. This quantitative information should prove useful to discriminate between possible polymer models of nuclear organization.
Revealing Long-Range Interconnected Hubs in Human Chromatin Interaction Data Using Graph Theory
NASA Astrophysics Data System (ADS)
Boulos, R. E.; Arneodo, A.; Jensen, P.; Audit, B.
2013-09-01
We use graph theory to analyze chromatin interaction (Hi-C) data in the human genome. We show that a key functional feature of the genome—“master” replication origins—corresponds to DNA loci of maximal network centrality. These loci form a set of interconnected hubs both within chromosomes and between different chromosomes. Our results open the way to a fruitful use of graph theory concepts to decipher DNA structural organization in relation to genome functions such as replication and transcription. This quantitative information should prove useful to discriminate between possible polymer models of nuclear organization.
Insights into the organization of biochemical regulatory networks using graph theory analyses.
Ma'ayan, Avi
2009-02-27
Graph theory has been a valuable mathematical modeling tool to gain insights into the topological organization of biochemical networks. There are two types of insights that may be obtained by graph theory analyses. The first provides an overview of the global organization of biochemical networks; the second uses prior knowledge to place results from multivariate experiments, such as microarray data sets, in the context of known pathways and networks to infer regulation. Using graph analyses, biochemical networks are found to be scale-free and small-world, indicating that these networks contain hubs, which are proteins that interact with many other molecules. These hubs may interact with many different types of proteins at the same time and location or at different times and locations, resulting in diverse biological responses. Groups of components in networks are organized in recurring patterns termed network motifs such as feedback and feed-forward loops. Graph analysis revealed that negative feedback loops are less common and are present mostly in proximity to the membrane, whereas positive feedback loops are highly nested in an architecture that promotes dynamical stability. Cell signaling networks have multiple pathways from some input receptors and few from others. Such topology is reminiscent of a classification system. Signaling networks display a bow-tie structure indicative of funneling information from extracellular signals and then dispatching information from a few specific central intracellular signaling nexuses. These insights show that graph theory is a valuable tool for gaining an understanding of global regulatory features of biochemical networks.
Derive Workshop Matrix Algebra and Linear Algebra.
ERIC Educational Resources Information Center
Townsley Kulich, Lisa; Victor, Barbara
This document presents the course content for a workshop that integrates the use of the computer algebra system Derive with topics in matrix and linear algebra. The first section is a guide to using Derive that provides information on how to write algebraic expressions, make graphs, save files, edit, define functions, differentiate expressions,…
Comparing brain networks of different size and connectivity density using graph theory.
van Wijk, Bernadette C M; Stam, Cornelis J; Daffertshofer, Andreas
2010-10-28
Graph theory is a valuable framework to study the organization of functional and anatomical connections in the brain. Its use for comparing network topologies, however, is not without difficulties. Graph measures may be influenced by the number of nodes (N) and the average degree (k) of the network. The explicit form of that influence depends on the type of network topology, which is usually unknown for experimental data. Direct comparisons of graph measures between empirical networks with different N and/or k can therefore yield spurious results. We list benefits and pitfalls of various approaches that intend to overcome these difficulties. We discuss the initial graph definition of unweighted graphs via fixed thresholds, average degrees or edge densities, and the use of weighted graphs. For instance, choosing a threshold to fix N and k does eliminate size and density effects but may lead to modifications of the network by enforcing (ignoring) non-significant (significant) connections. Opposed to fixing N and k, graph measures are often normalized via random surrogates but, in fact, this may even increase the sensitivity to differences in N and k for the commonly used clustering coefficient and small-world index. To avoid such a bias we tried to estimate the N,k-dependence for empirical networks, which can serve to correct for size effects, if successful. We also add a number of methods used in social sciences that build on statistics of local network structures including exponential random graph models and motif counting. We show that none of the here-investigated methods allows for a reliable and fully unbiased comparison, but some perform better than others.
Topics in Computational Learning Theory and Graph Algorithms.
ERIC Educational Resources Information Center
Board, Raymond Acton
This thesis addresses problems from two areas of theoretical computer science. The first area is that of computational learning theory, which is the study of the phenomenon of concept learning using formal mathematical models. The goal of computational learning theory is to investigate learning in a rigorous manner through the use of techniques…
A Qualitative Analysis Framework Using Natural Language Processing and Graph Theory
ERIC Educational Resources Information Center
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…
ERIC Educational Resources Information Center
Wilks, Clarissa; Meara, Paul
2002-01-01
Examines the implications of the metaphor of the vocabulary network. Takes a formal approach to the exploration of this metaphor by applying the principles of graph theory to word association data to compare the relative densities of the first language and second language lexical networks. (Author/VWL)
What Can Graph Theory Tell Us about Word Learning and Lexical Retrieval?
ERIC Educational Resources Information Center
Vitevitch, Michael S.
2008-01-01
Purpose: Graph theory and the new science of networks provide a mathematically rigorous approach to examine the development and organization of complex systems. These tools were applied to the mental lexicon to examine the organization of words in the lexicon and to explore how that structure might influence the acquisition and retrieval of…
Chen Famin; Wu Yongshi
2010-11-15
We present a superspace formulation of the D=3, N=4, 5 superconformal Chern-Simons Matter theories, with matter supermultiplets valued in a symplectic 3-algebra. We first construct an N=1 superconformal action and then generalize a method used by Gaitto and Witten to enhance the supersymmetry from N=1 to N=5. By decomposing the N=5 supermultiplets and the symplectic 3-algebra properly and proposing a new superpotential term, we construct the N=4 superconformal Chern-Simons matter theories in terms of two sets of generators of a (quaternion) symplectic 3-algebra. The N=4 theories can also be derived by requiring that the supersymmetry transformations are closed on-shell. The relationship between the 3-algebras, Lie superalgebras, Lie algebras, and embedding tensors (proposed in [E. A. Bergshoeff, O. Hohm, D. Roest, H. Samtleben, and E. Sezgin, J. High Energy Phys. 09 (2008) 101.]) is also clarified. The general N=4, 5 superconformal Chern-Simons matter theories in terms of ordinary Lie algebras can be re-derived in our 3-algebra approach. All known N=4, 5 superconformal Chern-Simons matter theories can be recovered in the present superspace formulation for super-Lie algebra realization of symplectic 3-algebras.
Impaired functional integration in multiple sclerosis: a graph theory study.
Rocca, Maria A; Valsasina, Paola; Meani, Alessandro; Falini, Andrea; Comi, Giancarlo; Filippi, Massimo
2016-01-01
Aim of this study was to explore the topological organization of functional brain network connectivity in a large cohort of multiple sclerosis (MS) patients and to assess whether its disruption contributes to disease clinical manifestations. Graph theoretical analysis was applied to resting state fMRI data from 246 MS patients and 55 matched healthy controls (HC). Functional connectivity between 116 cortical and subcortical brain regions was estimated using a bivariate correlation analysis. Global network properties (network degree, global efficiency, hierarchy, path length and assortativity) were abnormal in MS patients vs HC, and contributed to distinguish cognitively impaired MS patients (34%) from HC, but not the main MS clinical phenotypes. Compared to HC, MS patients also showed: (1) a loss of hubs in the superior frontal gyrus, precuneus and anterior cingulum in the left hemisphere; (2) a different lateralization of basal ganglia hubs (mostly located in the left hemisphere in HC, and in the right hemisphere in MS patients); and (3) a formation of hubs, not seen in HC, in the left temporal pole and cerebellum. MS patients also experienced a decreased nodal degree in the bilateral caudate nucleus and right cerebellum. Such a modification of regional network properties contributed to cognitive impairment and phenotypic variability of MS. An impairment of global integration (likely to reflect a reduced competence in information exchange between distant brain areas) occurs in MS and is associated with cognitive deficits. A regional redistribution of network properties contributes to cognitive status and phenotypic variability of these patients.
NASA Astrophysics Data System (ADS)
Schlueter-Kuck, Kristy L.; Dabiri, John O.
2017-01-01
We present a frame-invariant method for detecting coherent structures from Lagrangian flow trajectories that can be sparse in number, as is the case in many fluid mechanics applications of practical interest. The method, based on principles used in graph coloring and spectral graph drawing algorithms, examines a measure of the kinematic dissimilarity of all pairs of fluid trajectories, either measured experimentally, e.g. using particle tracking velocimetry; or numerically, by advecting fluid particles in the Eulerian velocity field. Coherence is assigned to groups of particles whose kinematics remain similar throughout the time interval for which trajectory data is available, regardless of their physical proximity to one another. Through the use of several analytical and experimental validation cases, this algorithm is shown to robustly detect coherent structures using significantly less flow data than is required by existing spectral graph theory methods.
Lamplighter groups, de Brujin graphs, spider-web graphs and their spectra
NASA Astrophysics Data System (ADS)
Grigorchuk, R.; Leemann, P.-H.; Nagnibeda, T.
2016-05-01
We study the infinite family of spider-web graphs \\{{{ S }}k,N,M\\}, k≥slant 2, N≥slant 0 and M≥slant 1, initiated in the 50s in the context of network theory. It was later shown in physical literature that these graphs have remarkable percolation and spectral properties. We provide a mathematical explanation of these properties by putting the spider-web graphs in the context of group theory and algebraic graph theory. Namely, we realize them as tensor products of the well-known de Bruijn graphs \\{{{ B }}k,N\\} with cyclic graphs \\{{C}M\\} and show that these graphs are described by the action of the lamplighter group {{ L }}k={Z}/k{Z}\\wr {Z} on the infinite binary tree. Our main result is the identification of the infinite limit of \\{{{ S }}k,N,M\\}, as N,M\\to ∞ , with the Cayley graph of the lamplighter group {{ L }}k which, in turn, is one of the famous Diestel-Leader graphs {{DL}}k,k. As an application we compute the spectra of all spider-web graphs and show their convergence to the discrete spectral distribution associated with the Laplacian on the lamplighter group.
Khakzad, Nima; Landucci, Gabriele; Reniers, Genserik
2016-11-10
In the present study, we have introduced a methodology based on graph theory and multicriteria decision analysis for cost-effective fire protection of chemical plants subject to fire-induced domino effects. By modeling domino effects in chemical plants as a directed graph, the graph centrality measures such as out-closeness and betweenness scores can be used to identify the installations playing a key role in initiating and propagating potential domino effects. It is demonstrated that active fire protection of installations with the highest out-closeness score and passive fire protection of installations with the highest betweenness score are the most effective strategies for reducing the vulnerability of chemical plants to fire-induced domino effects. We have employed a dynamic graph analysis to investigate the impact of both the availability and the degradation of fire protection measures over time on the vulnerability of chemical plants. The results obtained from the graph analysis can further be prioritized using multicriteria decision analysis techniques such as the method of reference point to find the most cost-effective fire protection strategy.
Xu, Xiayu; Reinhardt, Joseph M; Hu, Qiao; Bakall, Benjamin; Tlucek, Paul S; Bertelsen, Geir; Abràmoff, Michael D
2012-01-01
The retinal vessel width relationship at vessel branch points in fundus images is an important biomarker of retinal and systemic disease. We propose a fully automatic method to measure the vessel widths at branch points in fundus images. The method is a graph-based method, in which a graph construction method based on electric field theory is applied which specifically deals with complex branching patterns. The vessel centerline image is used as the initial segmentation of the graph. Branching points are detected on the vessel centerline image using a set of detection kernels. Crossing points are distinguished from branch points and excluded. The electric field based graph method is applied to construct the graph. This method is inspired by the non-intersecting force lines in an electric field. At last, the method is further improved to give a consistent vessel width measurement for the whole vessel tree. The algorithm was validated on 100 artery branchings and 100 vein branchings selected from 50 fundus images by comparing with vessel width measurements from two human experts.
NASA Astrophysics Data System (ADS)
Orantin, N.
2007-09-01
The 2-matrix model has been introduced to study Ising model on random surfaces. Since then, the link between matrix models and combinatorics of discrete surfaces has strongly tightened. This manuscript aims to investigate these deep links and extend them beyond the matrix models, following my work's evolution. First, I take care to define properly the hermitian 2 matrix model which gives rise to generating functions of discrete surfaces equipped with a spin structure. Then, I show how to compute all the terms in the topological expansion of any observable by using algebraic geometry tools. They are obtained as differential forms on an algebraic curve associated to the model: the spectral curve. In a second part, I show how to define such differentials on any algebraic curve even if it does not come from a matrix model. I then study their numerous symmetry properties under deformations of the algebraic curve. In particular, I show that these objects coincide with the topological expansion of the observable of a matrix model if the algebraic curve is the spectral curve of this model. Finally, I show that fine tuning the parameters ensure that these objects can be promoted to modular invariants and satisfy the holomorphic anomaly equation of the Kodaira-Spencer theory. This gives a new hint that the Dijkgraaf-Vafa conjecture is correct.
Does there exist a sensible quantum theory of an ``algebra-valued'' scalar field\\?
NASA Astrophysics Data System (ADS)
Anco, Stephen C.; Wald, Robert M.
1989-04-01
Consider a scalar field φ in Minkowski spacetime, but let φ be valued in an associative, commutative algebra openA rather than openR. One may view the resulting theory as describing a collection of coupled real scalar fields. At the classical level, theories of this type are completely well behaved and have a global symmetry group which is a nontrivial enlargement of the Poincaré group. (They are analogs of the new class of gauge theories for massless spin-2 fields found recently by one of us, whose gauge group is a nontrivial enlargement of the usual diffeomorphism group.) We investigate the quantization of such scalar field theories here by studying the case of a λφ4 field, with φ valued in the two-dimensional algebra generated by an identity element e and a nilpotent element v satisfying v2=0. The Coleman-Mandula theorem, which states that the symmetry group of a nontrivial quantum field theory cannot be a nontrivial enlargement of the Poincaré group, is evaded here because the finite ``extra'' symmetries of the classical theory fail to be implemented in the quantum theory by unitary operators and the infinitesimal symmetries (which can be represented in the quantum theory by quadratic forms) connect the one-particle Hilbert space to multiparticle states. Nevertheless, we find that the conventional Feynman rules for this theory lead to vacuum decay at the tree level and fail to yield a well-defined S matrix. Some alternative approaches are investigated, but these also appear to fail. Thus, although the classical theory is perfectly well behaved, it seems that there does not exist a sensible quantum theory of an algebra-valued scalar field.
Connections between the Sznajd model with general confidence rules and graph theory.
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).
Connections between the Sznajd model with general confidence rules and graph theory
NASA Astrophysics Data System (ADS)
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).
Algebraic Characterization of the Vacuum in Light-Front Field Theory
NASA Astrophysics Data System (ADS)
Herrmann, Marc; Polyzou, Wayne
2016-03-01
In the light-front formulation of quantum field theory, the vacuum vector of an interacting field theory has a relatively simple relationship to the vacuum of a free field theory. This is a benefit over the usual equal-time formulation where the interacting vacuum vector has infinite norm with respect to the Hilbert space of the free field theory. By describing the vacuum as a positive linear functional on an operator algebra constructed from free fields with two distinct masses, it can be demonstrated that the complications associated with adding dynamics to the vacuum of a free theory are not present in the construction of the light-front vacuum. Instead, the complications are moved into defining a subalgebra of the light-front algebra which corresponds to the physically relevant algebra of local fields. These results can then be applied to interacting fields by first describing them in terms of asymptotic in or out fields. However, in order to treat local operators products, the vacuum functional may need to be modified to include states with zero eigenvalue of the generator of translations in the direction along the light front, x- =1/√(2) >x0-x3. This work supported by DOE contract No. DE-FG02-86ER40286.
Visibility graph analysis on quarterly macroeconomic series of China based on complex network theory
NASA Astrophysics Data System (ADS)
Wang, Na; Li, Dong; Wang, Qiwen
2012-12-01
The visibility graph approach and complex network theory provide a new insight into time series analysis. The inheritance of the visibility graph from the original time series was further explored in the paper. We found that degree distributions of visibility graphs extracted from Pseudo Brownian Motion series obtained by the Frequency Domain algorithm exhibit exponential behaviors, in which the exponential exponent is a binomial function of the Hurst index inherited in the time series. Our simulations presented that the quantitative relations between the Hurst indexes and the exponents of degree distribution function are different for different series and the visibility graph inherits some important features of the original time series. Further, we convert some quarterly macroeconomic series including the growth rates of value-added of three industry series and the growth rates of Gross Domestic Product series of China to graphs by the visibility algorithm and explore the topological properties of graphs associated from the four macroeconomic series, namely, the degree distribution and correlations, the clustering coefficient, the average path length, and community structure. Based on complex network analysis we find degree distributions of associated networks from the growth rates of value-added of three industry series are almost exponential and the degree distributions of associated networks from the growth rates of GDP series are scale free. We also discussed the assortativity and disassortativity of the four associated networks as they are related to the evolutionary process of the original macroeconomic series. All the constructed networks have “small-world” features. The community structures of associated networks suggest dynamic changes of the original macroeconomic series. We also detected the relationship among government policy changes, community structures of associated networks and macroeconomic dynamics. We find great influences of government
Periodic orbit theory and the statistical analysis of scaling quantum graph spectra.
Dabaghian, Yu
2007-05-01
The explicit solution to the spectral problem of quantum graphs found recently by Dabaghian and Blümel [Phys. Rev. E 68, 055201(R) (2003); 70, 046206 (2004); JETP Lett. 77, 530 (2003)] is used to produce an exact periodic orbit theory description for the probability distributions of spectral statistics, including the distribution for the nearest neighbor separations sn = kn - kn-1, and the distribution of the spectral oscillations around the average, deltakn=kn - kn.
Adjusting protein graphs based on graph entropy.
Peng, Sheng-Lung; Tsay, Yu-Wei
2014-01-01
Measuring protein structural similarity attempts to establish a relationship of equivalence between polymer structures based on their conformations. In several recent studies, researchers have explored protein-graph remodeling, instead of looking a minimum superimposition for pairwise proteins. When graphs are used to represent structured objects, the problem of measuring object similarity become one of computing the similarity between graphs. Graph theory provides an alternative perspective as well as efficiency. Once a protein graph has been created, its structural stability must be verified. Therefore, a criterion is needed to determine if a protein graph can be used for structural comparison. In this paper, we propose a measurement for protein graph remodeling based on graph entropy. We extend the concept of graph entropy to determine whether a graph is suitable for representing a protein. The experimental results suggest that when applied, graph entropy helps a conformational on protein graph modeling. Furthermore, it indirectly contributes to protein structural comparison if a protein graph is solid.
Garcia-Ramos, Camille; Lin, Jack J; Kellermann, Tanja S; Bonilha, Leonardo; Prabhakaran, Vivek; Hermann, Bruce P
2016-11-01
The recent revision of the classification of the epilepsies released by the ILAE Commission on Classification and Terminology (2005-2009) has been a major development in the field. Papers in this section of the special issue explore the relevance of other techniques to examine, categorize, and classify cognitive and behavioral comorbidities in epilepsy. In this review, we investigate the applicability of graph theory to understand the impact of epilepsy on cognition compared with controls and, then, the patterns of cognitive development in normally developing children which would set the stage for prospective comparisons of children with epilepsy and controls. The overall goal is to examine the potential utility of this analytic tool and approach to conceptualize the cognitive comorbidities in epilepsy. Given that the major cognitive domains representing cognitive function are interdependent, the associations between neuropsychological abilities underlying these domains can be referred to as a cognitive network. Therefore, the architecture of this cognitive network can be quantified and assessed using graph theory methods, rendering a novel approach to the characterization of cognitive status. We first provide fundamental information about graph theory procedures, followed by application of these techniques to cross-sectional analysis of neuropsychological data in children with epilepsy compared with that of controls, concluding with prospective analysis of neuropsychological development in younger and older healthy controls. This article is part of a Special Issue entitled "The new approach to classification: Rethinking cognition and behavior in epilepsy".
NASA Astrophysics Data System (ADS)
Heckmann, Tobias; Schwanghart, Wolfgang
2013-01-01
Through their relevance for sediment budgets and the sensitivity of geomorphic systems, geomorphic coupling and (sediment) connectivity represent important topics in geomorphology. Since the introduction of the systems perspective to physical geography by Chorley and Kennedy (1971), a catchment has been perceived as consisting of landscape elements (e.g. landforms, subcatchments) that are coupled by geomorphic processes through sediment transport. In this study, we present a novel application of mathematical graph theory to explore the network structure of coarse sediment pathways in a central alpine catchment. Numerical simulation models for rockfall, debris flows, and (hillslope and channel) fluvial processes are used to establish a spatially explicit graph model of sediment sources, pathways and sinks. The raster cells of a digital elevation model form the nodes of this graph, and simulated sediment trajectories represent the corresponding edges. Model results are validated by visual comparison with the field situation and aerial photos. The interaction of sediment pathways, i.e. where the deposits of a geomorphic process form the sources of another process, forms sediment cascades, represented by paths (a succession of edges) in the graph model. We show how this graph can be used to explore upslope (contributing area) and downslope (source to sink) functional connectivity by analysing its nodes, edges and paths. The analysis of the spatial distribution, composition and frequency of sediment cascades yields information on the relative importance of geomorphic processes and their interaction (however regardless of their transport capacity). In the study area, the analysis stresses the importance of mass movements and their interaction, e.g. the linkage of large rockfall source areas to debris flows that potentially enter the channel network. Moreover, it is shown that only a small percentage of the study area is coupled to the channel network which itself is
NASA Astrophysics Data System (ADS)
Heckmann, T.; Schwanghart, W.
2012-04-01
In this study, we demonstrate and discuss the application of graph theory as a methodological framework for the appraisal of sediment connectivity. We understand sediment connectivity as a system property, that is to say the degree to which system components (landforms, river reaches, subcatchments) are coupled, i.e. allow for sediment transfer from one to another. Additionally, connectivity affects system behaviour, e.g. its sensitivity with respect to changes. In this concept, sediment connectivity is governed by (a) the location and topology of the system's components and (b) the activity of geomorphic processes driving sediment transfer. The distinction (and the interaction) of these two modes of connectivity corresponds to the notion of connectivity in other disciplines. In landscape ecology and hydrology, connectivity resulting from the spatial configuration of system components (e.g. habitat patches, landscape units with similar hydrological properties) has been defined as structural connectivity, while a linkage of system components by processes (e.g. migration of individual animals, continuous surface runoff) constitutes functional connectivity. We show how graph theory can be employed to model system components and their linkage for both modes of (sediment) connectivity. With respect to structural connectivity, landscape units, e.g. as delineated on a geomorphological map, are modelled as nodes which are connected by edges if two units share a common boundary and if this boundary can be crossed in the direction of flow (the resulting graph is directed because sediment pathways are governed by gravity and topography). In such a graph, even multiple flow directions could be accounted for. In order to assess functional connectivity, the activity of processes needs to be considered. This can be done by modelling potential sediment pathways (edges) between landscape or even smaller spatial units (forming the nodes); we demonstrate this approach for rockfall
Sparsified-dynamics modeling of discrete point vortices with graph theory
NASA Astrophysics Data System (ADS)
Taira, Kunihiko; Nair, Aditya
2014-11-01
We utilize graph theory to derive a sparsified interaction-based model that captures unsteady point vortex dynamics. The present model builds upon the Biot-Savart law and keeps the number of vortices (graph nodes) intact and reduces the number of inter-vortex interactions (graph edges). We achieve this reduction in vortex interactions by spectral sparsification of graphs. This approach drastically reduces the computational cost to predict the dynamical behavior, sharing characteristics of reduced-order models. Sparse vortex dynamics are illustrated through an example of point vortex clusters interacting amongst themselves. We track the centroids of the individual vortex clusters to evaluate the error in bulk motion of the point vortices in the sparsified setup. To further improve the accuracy in predicting the nonlinear behavior of the vortices, resparsification strategies are employed for the sparsified interaction-based models. The model retains the nonlinearity of the interaction and also conserves the invariants of discrete vortex dynamics; namely the Hamiltonian, linear impulse, and angular impulse as well as circulation. Work supported by US Army Research Office (W911NF-14-1-0386) and US Air Force Office of Scientific Research (YIP: FA9550-13-1-0183).
Energy Minimization of Discrete Protein Titration State Models Using Graph Theory
Purvine, Emilie AH; Monson, Kyle E.; Jurrus, Elizabeth R.; Star, Keith T.; Baker, Nathan A.
2016-09-01
There are several applications in computational biophysics which require the optimization of discrete interacting states; e.g., amino acid titration states, ligand oxidation states, or discrete rotamer angles. Such optimization can be very time-consuming as it scales exponentially in the number of sites to be optimized. In this paper, we describe a new polynomial-time algorithm for optimization of discrete states in macromolecular systems. This algorithm was adapted from image processing and uses techniques from discrete mathematics and graph theory to restate the optimization problem in terms of maximum flow-minimum cut graph analysis. The interaction energy graph, a graph in which vertices (amino acids) and edges (interactions) are weighted with their respective energies, is transformed into a flow network in which the value of the minimum cut in the network equals the minimum free energy of the protein, and the cut itself encodes the state that achieves the minimum free energy. Because of its deterministic nature and polynomial-time performance, this algorithm has the potential to allow for the ionization state of larger proteins to be discovered.
Energy Minimization of Discrete Protein Titration State Models Using Graph Theory.
Purvine, Emilie; Monson, Kyle; Jurrus, Elizabeth; Star, Keith; Baker, Nathan A
2016-08-25
There are several applications in computational biophysics that require the optimization of discrete interacting states, for example, amino acid titration states, ligand oxidation states, or discrete rotamer angles. Such optimization can be very time-consuming as it scales exponentially in the number of sites to be optimized. In this paper, we describe a new polynomial time algorithm for optimization of discrete states in macromolecular systems. This algorithm was adapted from image processing and uses techniques from discrete mathematics and graph theory to restate the optimization problem in terms of "maximum flow-minimum cut" graph analysis. The interaction energy graph, a graph in which vertices (amino acids) and edges (interactions) are weighted with their respective energies, is transformed into a flow network in which the value of the minimum cut in the network equals the minimum free energy of the protein and the cut itself encodes the state that achieves the minimum free energy. Because of its deterministic nature and polynomial time performance, this algorithm has the potential to allow for the ionization state of larger proteins to be discovered.
NASA Astrophysics Data System (ADS)
Zemenkova, M. Yu; Shipovalov, A. N.; Zemenkov, Yu D.
2016-04-01
The main technological equipment of pipeline transport of hydrocarbons are hydraulic machines. During transportation of oil mainly used of centrifugal pumps, designed to work in the “pumping station-pipeline” system. Composition of a standard pumping station consists of several pumps, complex hydraulic piping. The authors have developed a set of models and algorithms for calculating system reliability of pumps. It is based on the theory of reliability. As an example, considered one of the estimation methods with the application of graph theory.
From brain topography to brain topology: relevance of graph theory to functional neuroscience.
Minati, Ludovico; Varotto, Giulia; D'Incerti, Ludovico; Panzica, Ferruccio; Chan, Dennis
2013-07-10
Although several brain regions show significant specialization, higher functions such as cross-modal information integration, abstract reasoning and conscious awareness are viewed as emerging from interactions across distributed functional networks. Analytical approaches capable of capturing the properties of such networks can therefore enhance our ability to make inferences from functional MRI, electroencephalography and magnetoencephalography data. Graph theory is a branch of mathematics that focuses on the formal modelling of networks and offers a wide range of theoretical tools to quantify specific features of network architecture (topology) that can provide information complementing the anatomical localization of areas responding to given stimuli or tasks (topography). Explicit modelling of the architecture of axonal connections and interactions among areas can furthermore reveal peculiar topological properties that are conserved across diverse biological networks, and highly sensitive to disease states. The field is evolving rapidly, partly fuelled by computational developments that enable the study of connectivity at fine anatomical detail and the simultaneous interactions among multiple regions. Recent publications in this area have shown that graph-based modelling can enhance our ability to draw causal inferences from functional MRI experiments, and support the early detection of disconnection and the modelling of pathology spread in neurodegenerative disease, particularly Alzheimer's disease. Furthermore, neurophysiological studies have shown that network topology has a profound link to epileptogenesis and that connectivity indices derived from graph models aid in modelling the onset and spread of seizures. Graph-based analyses may therefore significantly help understand the bases of a range of neurological conditions. This review is designed to provide an overview of graph-based analyses of brain connectivity and their relevance to disease aimed
Sharing Teaching Ideas: Graphing Families of Curves Using Transformations of Reference Graphs
ERIC Educational Resources Information Center
Kukla, David
2007-01-01
This article provides for a fast extremely accurate approach to graphing functions that is based on learning function reference graphs and then applying algebraic transformations to these reference graphs.
Graph theory for feature extraction and classification: a migraine pathology case study.
Jorge-Hernandez, Fernando; Garcia Chimeno, Yolanda; Garcia-Zapirain, Begonya; Cabrera Zubizarreta, Alberto; Gomez Beldarrain, Maria Angeles; Fernandez-Ruanova, Begonya
2014-01-01
Graph theory is also widely used as a representational form and characterization of brain connectivity network, as is machine learning for classifying groups depending on the features extracted from images. Many of these studies use different techniques, such as preprocessing, correlations, features or algorithms. This paper proposes an automatic tool to perform a standard process using images of the Magnetic Resonance Imaging (MRI) machine. The process includes pre-processing, building the graph per subject with different correlations, atlas, relevant feature extraction according to the literature, and finally providing a set of machine learning algorithms which can produce analyzable results for physicians or specialists. In order to verify the process, a set of images from prescription drug abusers and patients with migraine have been used. In this way, the proper functioning of the tool has been proved, providing results of 87% and 92% of success depending on the classifier used.
NASA Astrophysics Data System (ADS)
Grifoni, E.; Legnaioli, S.; Lorenzetti, G.; Pagnotta, S.; Palleschi, V.
2016-04-01
In this paper we present a new approach for unsupervised classification of materials from the spectra obtained using the Laser-Induced Breakdown Spectroscopy technique. The method is based on the calculation of the correlation matrix between the LIBS spectra, which is interpreted as an Adjacency matrix in the framework of Graph theory. A threshold is applied on the edge values, which is determined through maximization of the Modularity of the Graph. The classification of the spectra is done automatically after the calculation of the Modularity parameter. An example of the application of the proposed method is given, based on the study of six bronze standards of known composition. The advantages of the proposed approach with respect to Principal Component Analysis are also discussed.
Does there exist a sensible quantum theory of an ''algebra-valued'' scalar field
Anco, S.C.; Wald, R.M.
1989-04-15
Consider a scalar field phi in Minkowski spacetime, but let phi be valued in an associative, commutative algebra openA rather than openR. One may view the resulting theory as describing a collection of coupled real scalar fields. At the classical level, theories of this type are completely well behaved and have a global symmetry group which is a nontrivial enlargement of the Poincare group. (They are analogs of the new class of gauge theories for massless spin-2 fields found recently by one of us, whose gauge group is a nontrivial enlargement of the usual diffeomorphism group.) We investigate the quantization of such scalar field theories here by studying the case of a lambdaphi/sup 4/ field, with phi valued in the two-dimensional algebra generated by an identity element e and a nilpotent element v satisfying v/sup 2/ = 0. The Coleman-Mandula theorem, which states that the symmetry group of a nontrivial quantum field theory cannot be a nontrivial enlargement of the Poincare group, is evaded here because the finite ''extra'' symmetries of the classical theory fail to be implemented in the quantum theory by unitary operators and the infinitesimal symmetries (which can be represented in the quantum theory by quadratic forms) connect the one-particle Hilbert space to multiparticle states. Nevertheless, we find that the conventional Feynman rules for this theory lead to vacuum decay at the tree level and fail to yield a well-defined S matrix. Some alternative approaches are investigated, but these also appear to fail.
Extending the algebraic formalism for genome rearrangements to include linear chromosomes.
Feijão, Pedro; Meidanis, João
2013-01-01
Algebraic rearrangement theory, as introduced by Meidanis and Dias, focuses on representing the order in which genes appear in chromosomes, and applies to circular chromosomes only. By shifting our attention to genome adjacencies, we introduce the adjacency algebraic theory, extending the original algebraic theory to linear chromosomes in a very natural way, also allowing the original algebraic distance formula to be used to the general multichromosomal case, with both linear and circular chromosomes. The resulting distance, which we call algebraic distance here, is very similar to, but not quite the same as, double-cut-and-join distance. We present linear time algorithms to compute it and to sort genomes. We show how to compute the rearrangement distance from the adjacency graph, for an easier comparison with other rearrangement distances. A thorough discussion on the relationship between the chromosomal and adjacency representation is also given, and we show how all classic rearrangement operations can be modeled using the algebraic theory.
Discovering Authorities and Hubs in Different Topological Web Graph Structures.
ERIC Educational Resources Information Center
Meghabghab, George
2002-01-01
Discussion of citation analysis on the Web considers Web hyperlinks as a source to analyze citations. Topics include basic graph theory applied to Web pages, including matrices, linear algebra, and Web topology; and hubs and authorities, including a search technique called HITS (Hyperlink Induced Topic Search). (Author/LRW)
A novel model for DNA sequence similarity analysis based on graph theory.
Qi, Xingqin; Wu, Qin; Zhang, Yusen; Fuller, Eddie; Zhang, Cun-Quan
2011-01-01
Determination of sequence similarity is one of the major steps in computational phylogenetic studies. As we know, during evolutionary history, not only DNA mutations for individual nucleotide but also subsequent rearrangements occurred. It has been one of major tasks of computational biologists to develop novel mathematical descriptors for similarity analysis such that various mutation phenomena information would be involved simultaneously. In this paper, different from traditional methods (eg, nucleotide frequency, geometric representations) as bases for construction of mathematical descriptors, we construct novel mathematical descriptors based on graph theory. In particular, for each DNA sequence, we will set up a weighted directed graph. The adjacency matrix of the directed graph will be used to induce a representative vector for DNA sequence. This new approach measures similarity based on both ordering and frequency of nucleotides so that much more information is involved. As an application, the method is tested on a set of 0.9-kb mtDNA sequences of twelve different primate species. All output phylogenetic trees with various distance estimations have the same topology, and are generally consistent with the reported results from early studies, which proves the new method's efficiency; we also test the new method on a simulated data set, which shows our new method performs better than traditional global alignment method when subsequent rearrangements happen frequently during evolutionary history.
A graph theory practice on transformed image: a random image steganography.
Thanikaiselvan, V; Arulmozhivarman, P; Subashanthini, S; Amirtharajan, Rengarajan
2013-01-01
Modern day information age is enriched with the advanced network communication expertise but unfortunately at the same time encounters infinite security issues when dealing with secret and/or private information. The storage and transmission of the secret information become highly essential and have led to a deluge of research in this field. In this paper, an optimistic effort has been taken to combine graceful graph along with integer wavelet transform (IWT) to implement random image steganography for secure communication. The implementation part begins with the conversion of cover image into wavelet coefficients through IWT and is followed by embedding secret image in the randomly selected coefficients through graph theory. Finally stegoimage is obtained by applying inverse IWT. This method provides a maximum of 44 dB peak signal to noise ratio (PSNR) for 266646 bits. Thus, the proposed method gives high imperceptibility through high PSNR value and high embedding capacity in the cover image due to adaptive embedding scheme and high robustness against blind attack through graph theoretic random selection of coefficients.
A Graph Theory Practice on Transformed Image: A Random Image Steganography
Thanikaiselvan, V.; Arulmozhivarman, P.; Subashanthini, S.; Amirtharajan, Rengarajan
2013-01-01
Modern day information age is enriched with the advanced network communication expertise but unfortunately at the same time encounters infinite security issues when dealing with secret and/or private information. The storage and transmission of the secret information become highly essential and have led to a deluge of research in this field. In this paper, an optimistic effort has been taken to combine graceful graph along with integer wavelet transform (IWT) to implement random image steganography for secure communication. The implementation part begins with the conversion of cover image into wavelet coefficients through IWT and is followed by embedding secret image in the randomly selected coefficients through graph theory. Finally stegoimage is obtained by applying inverse IWT. This method provides a maximum of 44 dB peak signal to noise ratio (PSNR) for 266646 bits. Thus, the proposed method gives high imperceptibility through high PSNR value and high embedding capacity in the cover image due to adaptive embedding scheme and high robustness against blind attack through graph theoretic random selection of coefficients. PMID:24453857
NASA Astrophysics Data System (ADS)
Yang, Y.; Li, H. T.; Han, Y. S.; Gu, H. Y.
2015-06-01
Image segmentation is the foundation of further object-oriented image analysis, understanding and recognition. It is one of the key technologies in high resolution remote sensing applications. In this paper, a new fast image segmentation algorithm for high resolution remote sensing imagery is proposed, which is based on graph theory and fractal net evolution approach (FNEA). Firstly, an image is modelled as a weighted undirected graph, where nodes correspond to pixels, and edges connect adjacent pixels. An initial object layer can be obtained efficiently from graph-based segmentation, which runs in time nearly linear in the number of image pixels. Then FNEA starts with the initial object layer and a pairwise merge of its neighbour object with the aim to minimize the resulting summed heterogeneity. Furthermore, according to the character of different features in high resolution remote sensing image, three different merging criterions for image objects based on spectral and spatial information are adopted. Finally, compared with the commercial remote sensing software eCognition, the experimental results demonstrate that the efficiency of the algorithm has significantly improved, and the result can maintain good feature boundaries.
NASA Astrophysics Data System (ADS)
Nakata, Yosuke; Urade, Yoshiro; Nakanishi, Toshihiro; Miyamaru, Fumiaki; Takeda, Mitsuo Wada; Kitano, Masao
2016-04-01
We investigate the supersymmetry (SUSY) structures for inductor-capacitor circuit networks on a simple regular graph and its line graph. We show that their eigenspectra must coincide (except, possibly, for the highest eigenfrequency) due to SUSY, which is derived from the topological nature of the circuits. To observe this spectra correspondence in the high-frequency range, we study spoof plasmons on metallic hexagonal and kagomé lattices. The band correspondence between them is predicted by a simulation. Using terahertz time-domain spectroscopy, we demonstrate the band correspondence of fabricated metallic hexagonal and kagomé lattices.
ERIC Educational Resources Information Center
Axtell, M.; Stickles, J.
2010-01-01
The last ten years have seen an explosion of research in the zero-divisor graphs of commutative rings--by professional mathematicians "and" undergraduates. The objective is to find algebraic information within the geometry of these graphs. This topic is approachable by anyone with one or two semesters of abstract algebra. This article gives the…
Dilts, Thomas E.; Weisberg, Peter J.; Leitner, Phillip; Matocq, Marjorie D.; Inman, Richard D.; Nussear, Ken E.; Esque, Todd
2016-01-01
Conservation planning and biodiversity management require information on landscape connectivity across a range of spatial scales from individual home ranges to large regions. Reduction in landscape connectivity due changes in land-use or development is expected to act synergistically with alterations to habitat mosaic configuration arising from climate change. We illustrate a multi-scale connectivity framework to aid habitat conservation prioritization in the context of changing land use and climate. Our approach, which builds upon the strengths of multiple landscape connectivity methods including graph theory, circuit theory and least-cost path analysis, is here applied to the conservation planning requirements of the Mohave ground squirrel. The distribution of this California threatened species, as for numerous other desert species, overlaps with the proposed placement of several utility-scale renewable energy developments in the American Southwest. Our approach uses information derived at three spatial scales to forecast potential changes in habitat connectivity under various scenarios of energy development and climate change. By disentangling the potential effects of habitat loss and fragmentation across multiple scales, we identify priority conservation areas for both core habitat and critical corridor or stepping stone habitats. This approach is a first step toward applying graph theory to analyze habitat connectivity for species with continuously-distributed habitat, and should be applicable across a broad range of taxa.
Dilt, Thomas E; Weisberg, Peter J; Leitner, Philip; Matocq, Marjorie D; Inman, Richard D; Nussear, Kenneth E; Esque, Todd C
2016-06-01
Conservation planning and biodiversity management require information on landscape connectivity across a range of spatial scales from individual home ranges to large regions. Reduction in landscape connectivity due changes in land use or development is expected to act synergistically with alterations to habitat mosaic configuration arising from climate change. We illustrate a multiscale connectivity framework to aid habitat conservation prioritization in the context of changing land use and climate. Our approach, which builds upon the strengths of multiple landscape connectivity methods, including graph theory, circuit theory, and least-cost path analysis, is here applied to the conservation planning requirements of the Mohave ground squirrel. The distribution of this threatened Californian species, as for numerous other desert species, overlaps with the proposed placement of several utility-scale renewable energy developments in the American southwest. Our approach uses information derived at three spatial scales to forecast potential changes in habitat connectivity under various scenarios of energy development and climate change. By disentangling the potential effects of habitat loss and fragmentation across multiple scales, we identify priority conservation areas for both core habitat and critical corridor or stepping stone habitats. This approach is a first step toward applying graph theory to analyze habitat connectivity for species with continuously distributed habitat and should be applicable across a broad range of taxa.
Larocca, Francesco; Chiu, Stephanie J; McNabb, Ryan P; Kuo, Anthony N; Izatt, Joseph A; Farsiu, Sina
2011-06-01
Segmentation of anatomical structures in corneal images is crucial for the diagnosis and study of anterior segment diseases. However, manual segmentation is a time-consuming and subjective process. This paper presents an automatic approach for segmenting corneal layer boundaries in Spectral Domain Optical Coherence Tomography images using graph theory and dynamic programming. Our approach is robust to the low-SNR and different artifact types that can appear in clinical corneal images. We show that our method segments three corneal layer boundaries in normal adult eyes more accurately compared to an expert grader than a second grader-even in the presence of significant imaging outliers.
Li, Qu; Yao, Min; Yang, Jianhua; Xu, Ning
2014-01-01
Online friend recommendation is a fast developing topic in web mining. In this paper, we used SVD matrix factorization to model user and item feature vector and used stochastic gradient descent to amend parameter and improve accuracy. To tackle cold start problem and data sparsity, we used KNN model to influence user feature vector. At the same time, we used graph theory to partition communities with fairly low time and space complexity. What is more, matrix factorization can combine online and offline recommendation. Experiments showed that the hybrid recommendation algorithm is able to recommend online friends with good accuracy.
NASA Astrophysics Data System (ADS)
Vieira, Sidney R.; Vidal Vázquez, Eva; Miranda, José G. V.; Paz Ferreiro, Jorge; Topp, George C.
2010-05-01
Spatial and temporal variability of soil moisture content has been frequently evaluated using statistical and geostatistical methods for several issues. For example, the statistical study of the temporal persistence or temporal stability in spatial patterns of soil moisture content has found interest to improve soil water monitoring strategies and to correct the average soil water content for missing data. Fractal analysis and graph theory are additional tools that can provide information and further insight to assess and to model indirect or hidden interactions in soil moisture content. In fractal analysis the fractal dimension (D) is an indicator of the pattern and extent of spatial and/or temporal variability. Large D values indicate the importance of short-range variation, while small D values reflect the importance of long-range variation when spatial and temporal data sets are analyzed. Moreover, for spatial and temporal variability, D can range from 1 to 2 for a profile and from 2 to 3 for a two dimensional network. Moreover, as the fractal dimension value increases the degree of roughness also increases. Graph theory tools take into account network structure by modelling pair wise relations between objects, which allow considering explicitly spatial-temporal connectivity of a given data set. The objective of this study was to use fractal analysis and graph theory to characterize the pattern of spatial and temporal variability of soil moisture content. The experimental field was located at Ottawa, Canada. Volumetric water content was monitored using Time Domain Reflectometry (TDR) during 34 dates at 164 locations per date. The depth of the TDR probes was 20 cm. The first and last measurements were 21 month apart and no data were taken in winter when the soil was covered by snow. The fractal dimension, D, was estimated from the slope of the regression line of log semivariogram versus distance for each of studied data sets. Using graph theory various
Characterizations of MV-algebras based on the theory of falling shadows.
Yang, Yongwei; Xin, Xiaolong; He, Pengfei
2014-01-01
Based on the falling shadow theory, the concept of falling fuzzy (implicative) ideals as a generalization of that of a T ∧-fuzzy (implicative) ideal is proposed in MV-algebras. The relationships between falling fuzzy (implicative) ideals and T-fuzzy (implicative) ideals are discussed, and conditions for a falling fuzzy (implicative) ideal to be a T ∧-fuzzy (implicative) ideal are provided. Some characterizations of falling fuzzy (implicative) ideals are presented by studying proprieties of them. The product ⊛ and the up product ⊚ operations on falling shadows and the upset of a falling shadow are established, by which T-fuzzy ideals are investigated based on probability spaces.
Li, Zhan-Chao; Lai, Yan-Hua; Chen, Li-Li; Chen, Chao; Xie, Yun; Dai, Zong; Zou, Xiao-Yong
2013-04-05
In the post-genome era, one of the most important and challenging tasks is to identify the subcellular localizations of protein complexes, and further elucidate their functions in human health with applications to understand disease mechanisms, diagnosis and therapy. Although various experimental approaches have been developed and employed to identify the subcellular localizations of protein complexes, the laboratory technologies fall far behind the rapid accumulation of protein complexes. Therefore, it is highly desirable to develop a computational method to rapidly and reliably identify the subcellular localizations of protein complexes. In this study, a novel method is proposed for predicting subcellular localizations of mammalian protein complexes based on graph theory with a random forest algorithm. Protein complexes are modeled as weighted graphs containing nodes and edges, where nodes represent proteins, edges represent protein-protein interactions and weights are descriptors of protein primary structures. Some topological structure features are proposed and adopted to characterize protein complexes based on graph theory. Random forest is employed to construct a model and predict subcellular localizations of protein complexes. Accuracies on a training set by a 10-fold cross-validation test for predicting plasma membrane/membrane attached, cytoplasm and nucleus are 84.78%, 71.30%, and 82.00%, respectively. And accuracies for the independent test set are 81.31%, 69.95% and 81.00%, respectively. These high prediction accuracies exhibit the state-of-the-art performance of the current method. It is anticipated that the proposed method may become a useful high-throughput tool and plays a complementary role to the existing experimental techniques in identifying subcellular localizations of mammalian protein complexes. The source code of Matlab and the dataset can be obtained freely on request from the authors.
Analysis of the enzyme network involved in cattle milk production using graph theory.
Ghorbani, Sholeh; Tahmoorespur, Mojtaba; Masoudi Nejad, Ali; Nasiri, Mohammad; Asgari, Yazdan
2015-06-01
Understanding cattle metabolism and its relationship with milk products is important in bovine breeding. A systemic view could lead to consequences that will result in a better understanding of existing concepts. Topological indices and quantitative characterizations mostly result from the application of graph theory on biological data. In the present work, the enzyme network involved in cattle milk production was reconstructed and analyzed based on available bovine genome information using several public datasets (NCBI, Uniprot, KEGG, and Brenda). The reconstructed network consisted of 3605 reactions named by KEGG compound numbers and 646 enzymes that catalyzed the corresponding reactions. The characteristics of the directed and undirected network were analyzed using Graph Theory. The mean path length was calculated to be4.39 and 5.41 for directed and undirected networks, respectively. The top 11 hub enzymes whose abnormality could harm bovine health and reduce milk production were determined. Therefore, the aim of constructing the enzyme centric network was twofold; first to find out whether such network followed the same properties of other biological networks, and second, to find the key enzymes. The results of the present study can improve our understanding of milk production in cattle. Also, analysis of the enzyme network can help improve the modeling and simulation of biological systems and help design desired phenotypes to increase milk production quality or quantity.
Differences in graph theory functional connectivity in left and right temporal lobe epilepsy
Chiang, Sharon; Stern, John M.; Engel, Jerome; Levin, Harvey S.; Haneef, Zulfi
2016-01-01
Summary Purpose To investigate lateralized differences in limbic system functional connectivity between left and right temporal lobe epilepsy (TLE) using graph theory. Methods Interictal resting state fMRI was performed in 14 left TLE patients, 11 right TLE patients, and 12 controls. Graph theory analysis of 10 bilateral limbic regions of interest was conducted. Changes in edgewise functional connectivity, network topology, and regional topology were quantified, and then left and right TLE were compared. Results Limbic edgewise functional connectivity was predominantly reduced in both left and right TLE. More regional connections were reduced in right TLE, most prominently involving reduced interhemispheric connectivity between the bilateral insula and bilateral hippocampi. A smaller number of limbic connections were increased in TLE, more so in left than in right TLE. Topologically, the most pronounced change was a reduction in average network betweenness centrality and concurrent increase in left hippocampal betweenness centrality in right TLE. In contrast, left TLE exhibited a weak trend toward increased right hippocampal betweenness centrality, with no change in average network betweenness centrality. Conclusion Limbic functional connectivity is predominantly reduced in both left and right TLE, with more pronounced reductions in right TLE. In contrast, left TLE exhibits both edgewise and topological changes that suggest a tendency toward reorganization. Network changes in TLE and lateralized differences thereof may have important diagnostic and prognostic implications. PMID:25445238
Rule-based graph theory to enable exploration of the space system architecture design space
NASA Astrophysics Data System (ADS)
Arney, Dale Curtis
The primary goal of this research is to improve upon system architecture modeling in order to enable the exploration of design space options. A system architecture is the description of the functional and physical allocation of elements and the relationships, interactions, and interfaces between those elements necessary to satisfy a set of constraints and requirements. The functional allocation defines the functions that each system (element) performs, and the physical allocation defines the systems required to meet those functions. Trading the functionality between systems leads to the architecture-level design space that is available to the system architect. The research presents a methodology that enables the modeling of complex space system architectures using a mathematical framework. To accomplish the goal of improved architecture modeling, the framework meets five goals: technical credibility, adaptability, flexibility, intuitiveness, and exhaustiveness. The framework is technically credible, in that it produces an accurate and complete representation of the system architecture under consideration. The framework is adaptable, in that it provides the ability to create user-specified locations, steady states, and functions. The framework is flexible, in that it allows the user to model system architectures to multiple destinations without changing the underlying framework. The framework is intuitive for user input while still creating a comprehensive mathematical representation that maintains the necessary information to completely model complex system architectures. Finally, the framework is exhaustive, in that it provides the ability to explore the entire system architecture design space. After an extensive search of the literature, graph theory presents a valuable mechanism for representing the flow of information or vehicles within a simple mathematical framework. Graph theory has been used in developing mathematical models of many transportation and
ERIC Educational Resources Information Center
Store, Jessie Chitsanzo
2012-01-01
There is ample literature documenting that, for many decades, high school students view algebra as difficult and do not demonstrate understanding of algebraic concepts. Algebraic reasoning in elementary school aims at meaningfully introducing algebra to elementary school students in preparation for higher-level mathematics. While there is research…
PDB2Graph: A toolbox for identifying critical amino acids map in proteins based on graph theory.
Niknam, Niloofar; Khakzad, Hamed; Arab, Seyed Shahriar; Naderi-Manesh, Hossein
2016-05-01
The integrative and cooperative nature of protein structure involves the assessment of topological and global features of constituent parts. Network concept takes complete advantage of both of these properties in the analysis concomitantly. High compatibility to structural concepts or physicochemical properties in addition to exploiting a remarkable simplification in the system has made network an ideal tool to explore biological systems. There are numerous examples in which different protein structural and functional characteristics have been clarified by the network approach. Here, we present an interactive and user-friendly Matlab-based toolbox, PDB2Graph, devoted to protein structure network construction, visualization, and analysis. Moreover, PDB2Graph is an appropriate tool for identifying critical nodes involved in protein structural robustness and function based on centrality indices. It maps critical amino acids in protein networks and can greatly aid structural biologists in selecting proper amino acid candidates for manipulating protein structures in a more reasonable and rational manner. To introduce the capability and efficiency of PDB2Graph in detail, the structural modification of Calmodulin through allosteric binding of Ca(2+) is considered. In addition, a mutational analysis for three well-identified model proteins including Phage T4 lysozyme, Barnase and Ribonuclease HI, was performed to inspect the influence of mutating important central residues on protein activity.
Daianu, Madelaine; Mezher, Adam; Jahanshad, Neda; Hibar, Derrek P; Nir, Talia M; Jack, Clifford R; Weiner, Michael W; Bernstein, Matt A; Thompson, Paul M
2015-04-01
Our understanding of network breakdown in Alzheimer's disease (AD) is likely to be enhanced through advanced mathematical descriptors. Here, we applied spectral graph theory to provide novel metrics of structural connectivity based on 3-Tesla diffusion weighted images in 42 AD patients and 50 healthy controls. We reconstructed connectivity networks using whole-brain tractography and examined, for the first time here, cortical disconnection based on the graph energy and spectrum. We further assessed supporting metrics - link density and nodal strength - to better interpret our results. Metrics were analyzed in relation to the well-known APOE-4 genetic risk factor for late-onset AD. The number of disconnected cortical regions increased with the number of copies of the APOE-4 risk gene in people with AD. Each additional copy of the APOE-4 risk gene may lead to more dysfunctional networks with weakened or abnormal connections, providing evidence for the previously hypothesized "disconnection syndrome".
Stavrakas, Vassilis; Melas, Ioannis N; Sakellaropoulos, Theodore; Alexopoulos, Leonidas G
2015-01-01
Modeling of signal transduction pathways is instrumental for understanding cells' function. People have been tackling modeling of signaling pathways in order to accurately represent the signaling events inside cells' biochemical microenvironment in a way meaningful for scientists in a biological field. In this article, we propose a method to interrogate such pathways in order to produce cell-specific signaling models. We integrate available prior knowledge of protein connectivity, in a form of a Prior Knowledge Network (PKN) with phosphoproteomic data to construct predictive models of the protein connectivity of the interrogated cell type. Several computational methodologies focusing on pathways' logic modeling using optimization formulations or machine learning algorithms have been published on this front over the past few years. Here, we introduce a light and fast approach that uses a breadth-first traversal of the graph to identify the shortest pathways and score proteins in the PKN, fitting the dependencies extracted from the experimental design. The pathways are then combined through a heuristic formulation to produce a final topology handling inconsistencies between the PKN and the experimental scenarios. Our results show that the algorithm we developed is efficient and accurate for the construction of medium and large scale signaling networks. We demonstrate the applicability of the proposed approach by interrogating a manually curated interaction graph model of EGF/TNFA stimulation against made up experimental data. To avoid the possibility of erroneous predictions, we performed a cross-validation analysis. Finally, we validate that the introduced approach generates predictive topologies, comparable to the ILP formulation. Overall, an efficient approach based on graph theory is presented herein to interrogate protein-protein interaction networks and to provide meaningful biological insights.
Hydrogen bond network topology in liquid water and methanol: a graph theory approach.
Bakó, Imre; Bencsura, Akos; Hermannson, Kersti; Bálint, Szabolcs; Grósz, Tamás; Chihaia, Viorel; Oláh, Julianna
2013-09-28
Networks are increasingly recognized as important building blocks of various systems in nature and society. Water is known to possess an extended hydrogen bond network, in which the individual bonds are broken in the sub-picosecond range and still the network structure remains intact. We investigated and compared the topological properties of liquid water and methanol at various temperatures using concepts derived within the framework of graph and network theory (neighbour number and cycle size distribution, the distribution of local cyclic and local bonding coefficients, Laplacian spectra of the network, inverse participation ratio distribution of the eigenvalues and average localization distribution of a node) and compared them to small world and Erdős-Rényi random networks. Various characteristic properties (e.g. the local cyclic and bonding coefficients) of the network in liquid water could be reproduced by small world and/or Erdős-Rényi networks, but the ring size distribution of water is unique and none of the studied graph models could describe it. Using the inverse participation ratio of the Laplacian eigenvectors we characterized the network inhomogeneities found in water and showed that similar phenomena can be observed in Erdős-Rényi and small world graphs. We demonstrated that the topological properties of the hydrogen bond network found in liquid water systematically change with the temperature and that increasing temperature leads to a broader ring size distribution. We applied the studied topological indices to the network of water molecules with four hydrogen bonds, and showed that at low temperature (250 K) these molecules form a percolated or nearly-percolated network, while at ambient or high temperatures only small clusters of four-hydrogen bonded water molecules exist.
NASA Astrophysics Data System (ADS)
Dimitrov, B. G.
2010-02-01
On the base of the distinction between covariant and contravariant metric tensor components, a new (multivariable) cubic algebraic equation for reparametrization invariance of the gravitational Lagrangian has been derived and parametrized with complicated non - elliptic functions, depending on the (elliptic) Weierstrass function and its derivative. This is different from standard algebraic geometry, where only two-dimensional cubic equations are parametrized with elliptic functions and not multivariable ones. Physical applications of the approach have been considered in reference to theories with extra dimensions. The s.c. "length function" l(x) has been introduced and found as a solution of quasilinear differential equations in partial derivatives for two different cases of "compactification + rescaling" and "rescaling + compactification". New physically important relations (inequalities) between the parameters in the action are established, which cannot be derived in the case $l=1$ of the standard gravitational theory, but should be fulfilled also for that case.
A new measure based on degree distribution that links information theory and network graph analysis
2012-01-01
Background Detailed connection maps of human and nonhuman brains are being generated with new technologies, and graph metrics have been instrumental in understanding the general organizational features of these structures. Neural networks appear to have small world properties: they have clustered regions, while maintaining integrative features such as short average pathlengths. Results We captured the structural characteristics of clustered networks with short average pathlengths through our own variable, System Difference (SD), which is computationally simple and calculable for larger graph systems. SD is a Jaccardian measure generated by averaging all of the differences in the connection patterns between any two nodes of a system. We calculated SD over large random samples of matrices and found that high SD matrices have a low average pathlength and a larger number of clustered structures. SD is a measure of degree distribution with high SD matrices maximizing entropic properties. Phi (Φ), an information theory metric that assesses a system’s capacity to integrate information, correlated well with SD - with SD explaining over 90% of the variance in systems above 11 nodes (tested for 4 to 13 nodes). However, newer versions of Φ do not correlate well with the SD metric. Conclusions The new network measure, SD, provides a link between high entropic structures and degree distributions as related to small world properties. PMID:22726594
Exploring the Epileptic Brain Network using Time-Variant Effective Connectivity and Graph Theory.
Storti, Silvia; Boscolo Galazzo, Ilaria; Khan, Sehresh; Manganotti, Paolo; Menegaz, Gloria
2016-09-09
The application of time-varying measures of causality between source time series can be very informative to elucidate the direction of communication among the regions of an epileptic brain. The aim of the study was to identify the dynamic patterns of epileptic networks in focal epilepsy by applying multivariate adaptive directed transfer function (ADTF) analysis and graph-theory to high-density electroencephalographic (hdEEG) recordings. The cortical network was modeled after source reconstruction and topology modulations were detected during interictal spikes. First a distributed linear inverse solution, constrained to the individual grey matter, was applied to the averaged spikes and the mean source activity over 112 regions, as identified by the Harvard-Oxford Atlas, was calculated. Then, ADTF, a dynamic measure of causality, was used to quantify the connectivity strength between pairs of regions acting as nodes in the graph, and the measure of node centrality was derived. The proposed analysis was effective in detecting the focal regions as well as in characterizing the dynamics of the spike propagation, providing evidence of the fact that node centrality is a reliable feature for the identification of the epileptogenic zones. Validation was performed by multi-modal analysis as well as from surgical outcomes. In conclusion, the time-variant connectivity analysis applied to the epileptic patients can distinguish the generator of the abnormal activity from the propagation spread and identify the connectivity pattern over time.
Graph Theory-Based Analysis of the Lymph Node Fibroblastic Reticular Cell Network.
Novkovic, Mario; Onder, Lucas; Bocharov, Gennady; Ludewig, Burkhard
2017-01-01
Secondary lymphoid organs have developed segregated niches that are able to initiate and maintain effective immune responses. Such global organization requires tight control of diverse cellular components, specifically those that regulate lymphocyte trafficking. Fibroblastic reticular cells (FRCs) form a densely interconnected network in lymph nodes and provide key factors necessary for T cell migration and retention, and foster subsequent interactions between T cells and dendritic cells. Development of integrative systems biology approaches has made it possible to elucidate this multilevel complexity of the immune system. Here, we present a graph theory-based analysis of the FRC network in murine lymph nodes, where generation of the network topology is performed using high-resolution confocal microscopy and 3D reconstruction. This approach facilitates the analysis of physical cell-to-cell connectivity, and estimation of topological robustness and global behavior of the network when it is subjected to perturbation in silico.
Chiu, Stephanie J; Toth, Cynthia A; Bowes Rickman, Catherine; Izatt, Joseph A; Farsiu, Sina
2012-05-01
This paper presents a generalized framework for segmenting closed-contour anatomical and pathological features using graph theory and dynamic programming (GTDP). More specifically, the GTDP method previously developed for quantifying retinal and corneal layer thicknesses is extended to segment objects such as cells and cysts. The presented technique relies on a transform that maps closed-contour features in the Cartesian domain into lines in the quasi-polar domain. The features of interest are then segmented as layers via GTDP. Application of this method to segment closed-contour features in several ophthalmic image types is shown. Quantitative validation experiments for retinal pigmented epithelium cell segmentation in confocal fluorescence microscopy images attests to the accuracy of the presented technique.
Hart, Michael G; Ypma, Rolf J F; Romero-Garcia, Rafael; Price, Stephen J; Suckling, John
2016-06-01
Neuroanatomy has entered a new era, culminating in the search for the connectome, otherwise known as the brain's wiring diagram. While this approach has led to landmark discoveries in neuroscience, potential neurosurgical applications and collaborations have been lagging. In this article, the authors describe the ideas and concepts behind the connectome and its analysis with graph theory. Following this they then describe how to form a connectome using resting state functional MRI data as an example. Next they highlight selected insights into healthy brain function that have been derived from connectome analysis and illustrate how studies into normal development, cognitive function, and the effects of synthetic lesioning can be relevant to neurosurgery. Finally, they provide a précis of early applications of the connectome and related techniques to traumatic brain injury, functional neurosurgery, and neurooncology.
Graph theory approach to the eigenvalue problem of large space structures
NASA Technical Reports Server (NTRS)
Reddy, A. S. S. R.; Bainum, P. M.
1981-01-01
Graph theory is used to obtain numerical solutions to eigenvalue problems of large space structures (LSS) characterized by a state vector of large dimensions. The LSS are considered as large, flexible systems requiring both orientation and surface shape control. Graphic interpretation of the determinant of a matrix is employed to reduce a higher dimensional matrix into combinations of smaller dimensional sub-matrices. The reduction is implemented by means of a Boolean equivalent of the original matrices formulated to obtain smaller dimensional equivalents of the original numerical matrix. Computation time becomes less and more accurate solutions are possible. An example is provided in the form of a free-free square plate. Linearized system equations and numerical values of a stiffness matrix are presented, featuring a state vector with 16 components.
The Altered Triple Networks Interaction in Depression under Resting State Based on Graph Theory.
Zheng, Hongna; Xu, Lele; Xie, Fufang; Guo, Xiaojuan; Zhang, Jiacai; Yao, Li; Wu, Xia
2015-01-01
The triple network model (Menon, 2011) has been proposed, which helps with finding a common framework for understanding the dysfunction in core neurocognitive network across multiple disorders. The alteration of the triple networks in the major depression disorder (MDD) is not clear. In our study, the altered interaction of the triple networks, which include default model network (DMN), central executive network (CEN), and salience network (SN), was examined in the MDD by graph theory method. The results showed that the connectivity degree of right anterior insula (rAI) significantly increased in MDD compared with healthy control (HC), and the connectivity degree between DMN and CEN significantly decreased in MDD. These results not only supported the proposal of the triple network model, but also prompted us to understand the dysfunction of neural mechanism in MDD.
A review on molecular topology: applying graph theory to drug discovery and design
NASA Astrophysics Data System (ADS)
Amigó, José María; Gálvez, Jorge; Villar, Vincent M.
2009-07-01
Molecular topology is an application of graph theory and statistics in fields like chemistry, biology, and pharmacology, in which the molecular structure matters. Its scope is the topological characterization of molecules by means of numerical invariants, called topological indices, which are the main ingredients of the molecular topological models. These are statistical models that are instrumental in the discovery of new applications of naturally occurring molecules, as well as in the design of synthetic molecules with specific chemical, biological, or pharmacological properties. In this review, we focus on pharmacology, which is a novel field of application of molecular topology. Besides summarizing some recent developments, we also seek to bring closer this interesting biomedical application of mathematics to an interdisciplinary readership.
Morphological brain network assessed using graph theory and network filtration in deaf adults.
Kim, Eunkyung; Kang, Hyejin; Lee, Hyekyoung; Lee, Hyo-Jeong; Suh, Myung-Whan; Song, Jae-Jin; Oh, Seung-Ha; Lee, Dong Soo
2014-09-01
Prolonged deprivation of auditory input can change brain networks in pre- and postlingual deaf adults by brain-wide reorganization. To investigate morphological changes in these brains voxel-based morphometry, voxel-wise correlation with the primary auditory cortex, and whole brain network analyses using morphological covariance were performed in eight prelingual deaf, eleven postlingual deaf, and eleven hearing adults. Network characteristics based on graph theory and network filtration based on persistent homology were examined. Gray matter density in the primary auditor cortex was preserved in prelingual deafness, while it tended to decrease in postlingual deafness. Unlike postlingual, prelingual deafness showed increased bilateral temporal connectivity of the primary auditory cortex compared to the hearing adults. Of the graph theory-based characteristics, clustering coefficient, betweenness centrality, and nodal efficiency all increased in prelingual deafness, while all the parameters of postlingual deafness were similar to the hearing adults. Patterns of connected components changing during network filtration were different between prelingual deafness and hearing adults according to the barcode, dendrogram, and single linkage matrix representations, while these were the same in postlingual deafness. Nodes in fronto-limbic and left temporal components were closely coupled, and nodes in the temporo-parietal component were loosely coupled, in prelingual deafness. Patterns of connected components changing in postlingual deafness were the same as hearing adults. We propose that the preserved density of auditory cortex associated with increased connectivity in prelingual deafness, and closer coupling between certain brain areas, represent distinctive reorganization of auditory and related cortices compared with hearing or postlingual deaf adults. The differential network reorganization in the prelingual deaf adults could be related to the absence of auditory speech
What graph theory actually tells us about resting state interictal MEG epileptic activity.
Niso, Guiomar; Carrasco, Sira; Gudín, María; Maestú, Fernando; Del-Pozo, Francisco; Pereda, Ernesto
2015-01-01
Graph theory provides a useful framework to study functional brain networks from neuroimaging data. In epilepsy research, recent findings suggest that it offers unique insight into the fingerprints of this pathology on brain dynamics. Most studies hitherto have focused on seizure activity during focal epilepsy, but less is known about functional epileptic brain networks during interictal activity in frontal focal and generalized epilepsy. Besides, it is not clear yet which measures are most suitable to characterize these networks. To address these issues, we recorded magnetoencephalographic (MEG) data using two orthogonal planar gradiometers from 45 subjects from three groups (15 healthy controls (7 males, 24 ± 6 years), 15 frontal focal (8 male, 32 ± 16 years) and 15 generalized epileptic (6 male, 27 ± 7 years) patients) during interictal resting state with closed eyes. Then, we estimated the total and relative spectral power of the largest principal component of the gradiometers, and the degree of phase synchronization between each sensor site in the frequency range [0.5-40 Hz]. We further calculated a comprehensive battery of 15 graph-theoretic measures and used the affinity propagation clustering algorithm to elucidate the minimum set of them that fully describe these functional brain networks. The results show that differences in spectral power between the control and the other two groups have a distinctive pattern: generalized epilepsy presents higher total power for all frequencies except the alpha band over a widespread set of sensors; frontal focal epilepsy shows higher relative power in the beta band bilaterally in the fronto-central sensors. Moreover, all network indices can be clustered into three groups, whose exemplars are the global network efficiency, the eccentricity and the synchronizability. Again, the patterns of differences were clear: the brain network of the generalized epilepsy patients presented greater efficiency and lower
NASA Astrophysics Data System (ADS)
Adhikari, C. M.; Debierre, V.; Jentschura, U. D.
2017-01-01
We analyze, in general terms, the evolution of energy levels in quantum mechanics, as a function of a coupling parameter, and demonstrate the possibility of level crossings in systems described by irreducible matrices. In long-range interactions, the coupling parameter is the interatomic distance. We demonstrate the utility of adjacency matrices and adjacency graphs in the analysis of "hidden" symmetries of a problem; these allow us to break reducible matrices into irreducible subcomponents. A possible breakdown of the no-crossing theorem for higher-dimensional irreducible matrices is indicated, and an application to the 2 S-2 S interaction in hydrogen is briefly described. The analysis of interatomic interactions in this system is important for further progress on optical measurements of the 2 S hyperfine splitting.
7D bosonic higher spin gauge theory: symmetry algebra and linearized constraints
NASA Astrophysics Data System (ADS)
Sezgin, E.; Sundell, P.
2002-07-01
We construct the minimal bosonic higher spin extension of the 7D AdS algebra SO(6,2), which we call hs(8 ∗) . The generators, which have spin s=1,3,5,… , are realized as monomials in Grassmann even spinor oscillators. Irreducibility, in the form of tracelessness, is achieved by modding out an infinite-dimensional ideal containing the traces. In this a key role is played by the tree bilinear traces which form an SU(2) K algebra. We show that gauging of hs(8 ∗) yields a spectrum of physical fields with spin s=0,2,4,… which make up a UIR of hs(8 ∗) isomorphic to the symmetric tensor product of two 6D scalar doubletons. The scalar doubleton is the unique SU(2) K invariant 6D doubleton. The spin s⩾2 sector comes from an hs(8 ∗) -valued one-form which also contains the auxiliary gauge fields required for writing the curvature constraints in covariant form. The physical spin s=0 field arises in a separate zero-form in a 'quasi-adjoint' representation of hs(8 ∗) . This zero-form also contains the spin s⩾2 Weyl tensors, i.e., the curvatures which are non-vanishing on-shell. We suggest that the hs(8 ∗) gauge theory describes the minimal bosonic, massless truncation of M-theory on AdS7× S4 in an unbroken phase where the holographic dual is given by N free (2,0) tensor multiplets for large N.
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
Hul, Oleh; Seba, Petr; Sirko, Leszek
2009-06-01
Parameter-dependent statistical properties of spectra of totally connected irregular quantum graphs with Neumann boundary conditions are studied. The autocorrelation functions of level velocities c(x) and c[over ](omega,x) as well as the distributions of level curvatures and avoided crossing gaps are calculated. The numerical results are compared with the predictions of random matrix theory for Gaussian orthogonal ensemble (GOE) and for coupled GOE matrices. The application of coupled GOE matrices was justified by studying localization phenomena in graphs' wave functions Psi(x) using the inverse participation ratio and the amplitude distribution P(Psi(x)) .
NASA Astrophysics Data System (ADS)
Robertsson, L.
2016-12-01
A simple method is proposed for the evaluation of comparative measurements when closed measurement loops are present in the data. Using a graph theory representation of the data a basis for the residual subspace is readily identified that together with loop closure values provide the input to a matrix equation of the same rank as the number of independent loops in the graph. As an example, ultra-high-precision frequency ratio measurements including optical standards are considered and small corrections to the experimental data are derived, yielding a list of consistent frequency ratios for work on precision frequency standards and discussions relating to a possible future redefinition of the SI second.
Zhang, Yang; Máté, Gabriell; Müller, Patrick; Hillebrandt, Sabina; Krufczik, Matthias; Bach, Margund; Kaufmann, Rainer; Hausmann, Michael; Heermann, Dieter W
2015-01-01
It has been well established that the architecture of chromatin in cell nuclei is not random but functionally correlated. Chromatin damage caused by ionizing radiation raises complex repair machineries. This is accompanied by local chromatin rearrangements and structural changes which may for instance improve the accessibility of damaged sites for repair protein complexes. Using stably transfected HeLa cells expressing either green fluorescent protein (GFP) labelled histone H2B or yellow fluorescent protein (YFP) labelled histone H2A, we investigated the positioning of individual histone proteins in cell nuclei by means of high resolution localization microscopy (Spectral Position Determination Microscopy = SPDM). The cells were exposed to ionizing radiation of different doses and aliquots were fixed after different repair times for SPDM imaging. In addition to the repair dependent histone protein pattern, the positioning of antibodies specific for heterochromatin and euchromatin was separately recorded by SPDM. The present paper aims to provide a quantitative description of structural changes of chromatin after irradiation and during repair. It introduces a novel approach to analyse SPDM images by means of statistical physics and graph theory. The method is based on the calculation of the radial distribution functions as well as edge length distributions for graphs defined by a triangulation of the marker positions. The obtained results show that through the cell nucleus the different chromatin re-arrangements as detected by the fluorescent nucleosomal pattern average themselves. In contrast heterochromatic regions alone indicate a relaxation after radiation exposure and re-condensation during repair whereas euchromatin seemed to be unaffected or behave contrarily. SPDM in combination with the analysis techniques applied allows the systematic elucidation of chromatin re-arrangements after irradiation and during repair, if selected sub-regions of nuclei are
Cavanaugh, Kyle C; Siegel, David A; Raimondi, Peter T; Alberto, Filipe
2014-02-01
The manner in which patches are delineated in spatially realistic metapopulation models will influence the size, connectivity, and extinction and recolonization dynamics of those patches. Most commonly used patch-definition methods focus on identifying discrete, contiguous patches of habitat from a single temporal observation of species occurrence or from a model of habitat suitability. However, these approaches are not suitable for many metapopulation systems where entire patches may not be fully colonized at a given time. For these metapopulation systems, a single large patch of habitat may actually support multiple, interacting subpopulations. The interactions among these subpopulations will be ignored if the patch is treated as a single unit, a situation we term the "mega-patch problem." Mega-patches are characterized by variable intra-patch synchrony, artificially low inter-patch connectivity, and low extinction rates. One way to detect this problem is by using time series data to calculate demographic synchrony within mega-patches. We present a framework for identifying subpopulations in mega-patches using a combination of spatial autocorrelation and graph theory analyses. We apply our approach to southern California giant kelp (Macrocystis pyrifera) forests using a new, long-term (27 years), satellite-based data set of giant kelp canopy biomass. We define metapopulation patches using our method as well as several other commonly used patch delineation methodologies and examine the colonization and extinction dynamics of the metapopulation under each approach. We find that the relationships between patch characteristics such as area and connectivity and the demographic processes of colonizations and extinctions vary among the different patch-definition methods. Our spatial-analysis/graph-theoretic framework produces results that match theoretical expectations better than the other methods. This approach can be used to identify subpopulations in metapopulations
Understanding cancer complexome using networks, spectral graph theory and multilayer framework
Rai, Aparna; Pradhan, Priodyuti; Nagraj, Jyothi; Lohitesh, K.; Chowdhury, Rajdeep; Jalan, Sarika
2017-01-01
Cancer complexome comprises a heterogeneous and multifactorial milieu that varies in cytology, physiology, signaling mechanisms and response to therapy. The combined framework of network theory and spectral graph theory along with the multilayer analysis provides a comprehensive approach to analyze the proteomic data of seven different cancers, namely, breast, oral, ovarian, cervical, lung, colon and prostate. Our analysis demonstrates that the protein-protein interaction networks of the normal and the cancerous tissues associated with the seven cancers have overall similar structural and spectral properties. However, few of these properties implicate unsystematic changes from the normal to the disease networks depicting difference in the interactions and highlighting changes in the complexity of different cancers. Importantly, analysis of common proteins of all the cancer networks reveals few proteins namely the sensors, which not only occupy significant position in all the layers but also have direct involvement in causing cancer. The prediction and analysis of miRNAs targeting these sensor proteins hint towards the possible role of these proteins in tumorigenesis. This novel approach helps in understanding cancer at the fundamental level and provides a clue to develop promising and nascent concept of single drug therapy for multiple diseases as well as personalized medicine. PMID:28155908
Understanding cancer complexome using networks, spectral graph theory and multilayer framework
NASA Astrophysics Data System (ADS)
Rai, Aparna; Pradhan, Priodyuti; Nagraj, Jyothi; Lohitesh, K.; Chowdhury, Rajdeep; Jalan, Sarika
2017-02-01
Cancer complexome comprises a heterogeneous and multifactorial milieu that varies in cytology, physiology, signaling mechanisms and response to therapy. The combined framework of network theory and spectral graph theory along with the multilayer analysis provides a comprehensive approach to analyze the proteomic data of seven different cancers, namely, breast, oral, ovarian, cervical, lung, colon and prostate. Our analysis demonstrates that the protein-protein interaction networks of the normal and the cancerous tissues associated with the seven cancers have overall similar structural and spectral properties. However, few of these properties implicate unsystematic changes from the normal to the disease networks depicting difference in the interactions and highlighting changes in the complexity of different cancers. Importantly, analysis of common proteins of all the cancer networks reveals few proteins namely the sensors, which not only occupy significant position in all the layers but also have direct involvement in causing cancer. The prediction and analysis of miRNAs targeting these sensor proteins hint towards the possible role of these proteins in tumorigenesis. This novel approach helps in understanding cancer at the fundamental level and provides a clue to develop promising and nascent concept of single drug therapy for multiple diseases as well as personalized medicine.
Understanding cancer complexome using networks, spectral graph theory and multilayer framework.
Rai, Aparna; Pradhan, Priodyuti; Nagraj, Jyothi; Lohitesh, K; Chowdhury, Rajdeep; Jalan, Sarika
2017-02-03
Cancer complexome comprises a heterogeneous and multifactorial milieu that varies in cytology, physiology, signaling mechanisms and response to therapy. The combined framework of network theory and spectral graph theory along with the multilayer analysis provides a comprehensive approach to analyze the proteomic data of seven different cancers, namely, breast, oral, ovarian, cervical, lung, colon and prostate. Our analysis demonstrates that the protein-protein interaction networks of the normal and the cancerous tissues associated with the seven cancers have overall similar structural and spectral properties. However, few of these properties implicate unsystematic changes from the normal to the disease networks depicting difference in the interactions and highlighting changes in the complexity of different cancers. Importantly, analysis of common proteins of all the cancer networks reveals few proteins namely the sensors, which not only occupy significant position in all the layers but also have direct involvement in causing cancer. The prediction and analysis of miRNAs targeting these sensor proteins hint towards the possible role of these proteins in tumorigenesis. This novel approach helps in understanding cancer at the fundamental level and provides a clue to develop promising and nascent concept of single drug therapy for multiple diseases as well as personalized medicine.
1991-01-01
Editors, p-Adic methods in number theory and algebraic geometry , 1992 132 Mark Gotay, Jerrold Marsden, and Vincent Moncrief, Editors, Mathematical...Lange and Shengwang Wang, New approaches in spectral decomposition, 1992 127 Vladimir Oliker and Andrejs Treibergs, Editors, Geometry and nonlinear...algebraic geometry , 1992 125 F. Thomas Bruss, Thomas S. Ferguson, and Stephen M. Samuels, Editors, Strategies for sequential search and selection in real
Anticipation-related brain connectivity in bipolar and unipolar depression: a graph theory approach.
Manelis, Anna; Almeida, Jorge R C; Stiffler, Richelle; Lockovich, Jeanette C; Aslam, Haris A; Phillips, Mary L
2016-09-01
Bipolar disorder is often misdiagnosed as major depressive disorder, which leads to inadequate treatment. Depressed individuals versus healthy control subjects, show increased expectation of negative outcomes. Due to increased impulsivity and risk for mania, however, depressed individuals with bipolar disorder may differ from those with major depressive disorder in neural mechanisms underlying anticipation processes. Graph theory methods for neuroimaging data analysis allow the identification of connectivity between multiple brain regions without prior model specification, and may help to identify neurobiological markers differentiating these disorders, thereby facilitating development of better therapeutic interventions. This study aimed to compare brain connectivity among regions involved in win/loss anticipation in depressed individuals with bipolar disorder (BDD) versus depressed individuals with major depressive disorder (MDD) versus healthy control subjects using graph theory methods. The study was conducted at the University of Pittsburgh Medical Center and included 31 BDD, 39 MDD, and 36 healthy control subjects. Participants were scanned while performing a number guessing reward task that included the periods of win and loss anticipation. We first identified the anticipatory network across all 106 participants by contrasting brain activation during all anticipation periods (win anticipation + loss anticipation) versus baseline, and win anticipation versus loss anticipation. Brain connectivity within the identified network was determined using the Independent Multiple sample Greedy Equivalence Search (IMaGES) and Linear non-Gaussian Orientation, Fixed Structure (LOFS) algorithms. Density of connections (the number of connections in the network), path length, and the global connectivity direction ('top-down' versus 'bottom-up') were compared across groups (BDD/MDD/healthy control subjects) and conditions (win/loss anticipation). These analyses showed that
ERIC Educational Resources Information Center
Senarat, Somprasong; Tayraukham, Sombat; Piyapimonsit, Chatsiri; Tongkhambanjong, Sakesan
2013-01-01
The purpose of this research is to develop a multidimensional computerized adaptive test for diagnosing the cognitive process of grade 7 students in learning algebra by applying multidimensional item response theory. The research is divided into 4 steps: 1) the development of item bank of algebra, 2) the development of the multidimensional…
NASA Astrophysics Data System (ADS)
Setare, M. R.; Adami, H.
2017-01-01
In this paper we study the near horizon symmetry algebra of the non-extremal black hole solutions of the Chern-Simons-like theories of gravity, which are stationary but are not necessarily spherically symmetric. We define the extended off-shell ADT current which is an extension of the generalized ADT current. We use the extended off-shell ADT current to define quasi-local conserved charges such that they are conserved for Killing vectors and asymptotically Killing vectors which depend on dynamical fields of the considered theory. We apply this formalism to the Generalized Minimal Massive Gravity (GMMG) and obtain conserved charges of a spacetime which describes near horizon geometry of non-extremal black holes. Eventually, we find the algebra of conserved charges in Fourier modes. It is interesting that, similar to the Einstein gravity in the presence of negative cosmological constant, for the GMMG model also we obtain the Heisenberg algebra as the near horizon symmetry algebra of the black flower solutions. Also the vacuum state and all descendants of the vacuum have the same energy. Thus these zero energy excitations on the horizon appear as soft hairs on the black hole.
Morphology and Performance of Polymer Solar Cell Characterized by DPD Simulation and Graph Theory.
Du, Chunmiao; Ji, Yujin; Xue, Junwei; Hou, Tingjun; Tang, Jianxin; Lee, Shuit-Tong; Li, Youyong
2015-11-19
The morphology of active layers in the bulk heterojunction (BHJ) solar cells is critical to the performance of organic photovoltaics (OPV). Currently, there is limited information for the morphology from transmission electron microscopy (TEM) techniques. Meanwhile, there are limited approaches to predict the morphology /efficiency of OPV. Here we use Dissipative Particle Dynamics (DPD) to determine 3D morphology of BHJ solar cells and show DPD to be an efficient approach to predict the 3D morphology. Based on the 3D morphology, we estimate the performance indicator of BHJ solar cells by using graph theory. Specifically, we study poly (3-hexylthiophene)/[6, 6]-phenyl-C61butyric acid methyl ester (P3HT/PCBM) BHJ solar cells. We find that, when the volume fraction of PCBM is in the region 0.4 ∼ 0.5, P3HT/PCBM will show bi-continuous morphology and optimum performance, consistent with experimental results. Further, the optimum temperature (413 K) for the morphology and performance of P3HT/PCBM is in accord with annealing results. We find that solvent additive plays a critical role in the desolvation process of P3HT/PCBM BHJ solar cell. Our approach provides a direct method to predict dynamic 3D morphology and performance indicator for BHJ solar cells.
Morphology and Performance of Polymer Solar Cell Characterized by DPD Simulation and Graph Theory
NASA Astrophysics Data System (ADS)
Du, Chunmiao; Ji, Yujin; Xue, Junwei; Hou, Tingjun; Tang, Jianxin; Lee, Shuit-Tong; Li, Youyong
2015-11-01
The morphology of active layers in the bulk heterojunction (BHJ) solar cells is critical to the performance of organic photovoltaics (OPV). Currently, there is limited information for the morphology from transmission electron microscopy (TEM) techniques. Meanwhile, there are limited approaches to predict the morphology /efficiency of OPV. Here we use Dissipative Particle Dynamics (DPD) to determine 3D morphology of BHJ solar cells and show DPD to be an efficient approach to predict the 3D morphology. Based on the 3D morphology, we estimate the performance indicator of BHJ solar cells by using graph theory. Specifically, we study poly (3-hexylthiophene)/[6, 6]-phenyl-C61butyric acid methyl ester (P3HT/PCBM) BHJ solar cells. We find that, when the volume fraction of PCBM is in the region 0.4 ∼ 0.5, P3HT/PCBM will show bi-continuous morphology and optimum performance, consistent with experimental results. Further, the optimum temperature (413 K) for the morphology and performance of P3HT/PCBM is in accord with annealing results. We find that solvent additive plays a critical role in the desolvation process of P3HT/PCBM BHJ solar cell. Our approach provides a direct method to predict dynamic 3D morphology and performance indicator for BHJ solar cells.
Morphology and Performance of Polymer Solar Cell Characterized by DPD Simulation and Graph Theory
Du, Chunmiao; Ji, Yujin; Xue, Junwei; Hou, Tingjun; Tang, Jianxin; Lee, Shuit-Tong; Li, Youyong
2015-01-01
The morphology of active layers in the bulk heterojunction (BHJ) solar cells is critical to the performance of organic photovoltaics (OPV). Currently, there is limited information for the morphology from transmission electron microscopy (TEM) techniques. Meanwhile, there are limited approaches to predict the morphology /efficiency of OPV. Here we use Dissipative Particle Dynamics (DPD) to determine 3D morphology of BHJ solar cells and show DPD to be an efficient approach to predict the 3D morphology. Based on the 3D morphology, we estimate the performance indicator of BHJ solar cells by using graph theory. Specifically, we study poly (3-hexylthiophene)/[6, 6]-phenyl-C61butyric acid methyl ester (P3HT/PCBM) BHJ solar cells. We find that, when the volume fraction of PCBM is in the region 0.4 ∼ 0.5, P3HT/PCBM will show bi-continuous morphology and optimum performance, consistent with experimental results. Further, the optimum temperature (413 K) for the morphology and performance of P3HT/PCBM is in accord with annealing results. We find that solvent additive plays a critical role in the desolvation process of P3HT/PCBM BHJ solar cell. Our approach provides a direct method to predict dynamic 3D morphology and performance indicator for BHJ solar cells. PMID:26581407
Roos, Annerine; Fouche, Jean-Paul; Stein, Dan J
2016-10-18
Evidence suggests that women who suffer from intimate partner violence (IPV) and posttraumatic stress disorder (PTSD) have structural and functional alterations in specific brain regions. Yet, little is known about how brain connectivity may be altered in individuals with IPV, but without PTSD. Women exposed to IPV (n = 18) and healthy controls (n = 18) underwent structural brain imaging using a Siemens 3T MRI. Global and regional brain network connectivity measures were determined, using graph theory analyses. Structural covariance networks were created using volumetric and cortical thickness data after controlling for intracranial volume, age and alcohol use. Nonparametric permutation tests were used to investigate group differences. Findings revealed altered connectivity on a global and regional level in the IPV group of regions involved in cognitive-emotional control, with principal involvement of the caudal anterior cingulate, the middle temporal gyrus, left amygdala and ventral diencephalon that includes the thalamus. To our knowledge, this is the first evidence showing different brain network connectivity in global and regional networks in women exposed to IPV, and without PTSD. Altered cognitive-emotional control in IPV may underlie adaptive neural mechanisms in environments characterized by potentially dangerous cues.
Increased segregation of brain networks in focal epilepsy: An fMRI graph theory finding.
Pedersen, Mangor; Omidvarnia, Amir H; Walz, Jennifer M; Jackson, Graeme D
2015-01-01
Focal epilepsy is conceived of as activating local areas of the brain as well as engaging regional brain networks. Graph theory represents a powerful quantitative framework for investigation of brain networks. Here we investigate whether functional network changes are present in extratemporal focal epilepsy. Task-free functional magnetic resonance imaging data from 15 subjects with extratemporal epilepsy and 26 age and gender matched healthy controls were used for analysis. Local network properties were calculated using local efficiency, clustering coefficient and modularity metrics. Global network properties were assessed with global efficiency and betweenness centrality metrics. Cost-efficiency of the networks at both local and global levels was evaluated by estimating the physical distance between functionally connected nodes, in addition to the overall numbers of connections in the network. Clustering coefficient, local efficiency and modularity were significantly higher in individuals with focal epilepsy than healthy control subjects, while global efficiency and betweenness centrality were not significantly different between the two groups. Local network properties were also highly efficient, at low cost, in focal epilepsy subjects compared to healthy controls. Our results show that functional networks in focal epilepsy are altered in a way that the nodes of the network are more isolated. We postulate that network regularity, or segregation of the nodes of the networks, may be an adaptation that inhibits the conversion of the interictal state to seizures. It remains possible that this may be part of the epileptogenic process or an effect of medications.
Functional Brain Dysfunction in Patients with Benign Childhood Epilepsy as Revealed by Graph Theory.
Adebimpe, Azeez; Aarabi, Ardalan; Bourel-Ponchel, Emilie; Mahmoudzadeh, Mahdi; Wallois, Fabrice
2015-01-01
There is growing evidence that brain networks are altered in epileptic subjects. In this study, we investigated the functional connectivity and brain network properties of benign childhood epilepsy with centrotemporal spikes using graph theory. Benign childhood epilepsy with centrotemporal spikes is the most common form of idiopathic epilepsy in young children under the age of 16 years. High-density EEG data were recorded from patients and controls in resting state with eyes closed. Data were preprocessed and spike and spike-free segments were selected for analysis. Phase locking value was calculated for all paired combinations of channels and for five frequency bands (δ, θ, α, β1 and β2). We computed the degree and small-world parameters--clustering coefficient (C) and path length (L)--and compared the two patient conditions to controls. A higher degree at epileptic zones during interictal epileptic spikes (IES) was observed in all frequency bands. Both patient conditions reduced connection at the occipital and right frontal regions close to the epileptic zone in the α band. The "small-world" features (high C and short L) were deviated in patients compared to controls. A changed from an ordered network in the δ band to a more randomly organized network in the α band was observed in patients compared to healthy controls. These findings show that the benign epileptic brain network is disrupted not only at the epileptic zone, but also in other brain regions especially frontal regions.
Zhou, Yongxia; Yu, Fang; Duong, Timothy
2014-01-01
This study employed graph theory and machine learning analysis of multiparametric MRI data to improve characterization and prediction in autism spectrum disorders (ASD). Data from 127 children with ASD (13.5±6.0 years) and 153 age- and gender-matched typically developing children (14.5±5.7 years) were selected from the multi-center Functional Connectome Project. Regional gray matter volume and cortical thickness increased, whereas white matter volume decreased in ASD compared to controls. Small-world network analysis of quantitative MRI data demonstrated decreased global efficiency based on gray matter cortical thickness but not with functional connectivity MRI (fcMRI) or volumetry. An integrative model of 22 quantitative imaging features was used for classification and prediction of phenotypic features that included the autism diagnostic observation schedule, the revised autism diagnostic interview, and intelligence quotient scores. Among the 22 imaging features, four (caudate volume, caudate-cortical functional connectivity and inferior frontal gyrus functional connectivity) were found to be highly informative, markedly improving classification and prediction accuracy when compared with the single imaging features. This approach could potentially serve as a biomarker in prognosis, diagnosis, and monitoring disease progression. PMID:24922325
Applications of automata and graphs: Labeling operators in Hilbert space. II
Cho, Ilwoo; Jorgensen, Palle E. T.
2009-06-15
We introduced a family of infinite graphs directly associated with a class of von Neumann automaton model A{sub G}. These are finite state models used in symbolic dynamics: stimuli models and in control theory. In the context of groupoid von Neumann algebras, and an associated fractal group, we prove a classification theorem for representations of automata.
Graph and circuit theory connectivity models of conservation biological control agents.
Koh, Insu; Rowe, Helen I; Holland, Jeffrey D
2013-10-01
The control of agricultural pests is an important ecosystem service provided by predacious insects. In Midwestern USA, areas of remnant tallgrass prairie and prairie restorations may serve as relatively undisturbed sources of natural predators, and smaller areas of non-crop habitats such as seminatural areas and conservation plantings (CP) may serve as stepping stones across landscapes dominated by intensive agriculture. However, little is known about the flow of beneficial insects across large habitat networks. We measured abundance of soybean aphids and predators in 15 CP and adjacent soybean fields. We tested two hypotheses: (1) landscape connectivity enhances the flow of beneficial insects; and (2) prairies act as a source of sustaining populations of beneficial insects in well-connected habitats, by using adaptations of graph and circuit theory, respectively. For graph connectivity, incoming fluxes to the 15 CP from connected habitats were measured using an area- and distance-weighted flux metric with a range of negative exponential dispersal kernels. Distance was weighted by the percentage of seminatural area within ellipse-shaped landscapes, the shape of which was determined with correlated random walks. For circuit connectivity, effective conductance from the prairie to the individual 15 CP was measured by regarding the flux as conductance in a circuit. We used these two connectivity measures to predict the abundance of natural enemies in the selected sites. The most abundant predators were Anthocoridae, followed by exotic Coccinellidae, and native Coccinellidae. Predator abundances were explained well by aphid abundance. However, only native Coccinellidae were influenced by the flux and conductance. Interestingly, exotic Coccinellidae were negatively related to the flux, and native Coccinellidae were highly influenced by the interaction between exotic Coccinellidae and aphids. Our area- and distance-weighted flux and the conductance variables showed better
Synchronization Patterns and Related Problems in Combinatorial Analysis and Graph Theory
1981-06-01
APPENDIX 86 REFERENCES 90 Li ~1 ivF LIST OF FIGURES Figure Page 1-1 The Leech Tree 6 2-1 The Graph L and the Graph R 9 2-2 The Dodecahedron and Two...forest F with 4 nodes, but in this case Ithe direct count to find the number takes more work. D Figure 2-2. The Dodecahedron and Two Copies of the...with n - 2m nodes, girth > 4, and d - 3, then G m - 2 Om-4) T -(Zm 3 24m + 100m 147). The dodecahedron , D, and one graph, P, consisting of two copies
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.
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. PMID:24416311
New morphometric properties for channel network classification using the graph theory and DEM
NASA Astrophysics Data System (ADS)
Moussa, R.; Colin, F.; Rabotin, M.; A; Crabit
2012-04-01
The channel network controls the spatial pattern of hydrological processes within a catchment. Hence the identification of key hydrological features characterising the channel network can contribute to a rational classification of catchments. This presentation aims to investigate morphometric properties of the channel network derived from DEM using the graph theory, and estimate whether these properties can be used as similarity indices for the classification of channel networks. The graph theory was used in order to represent the contributing drainage area which has properties of a scale free network, and was subsequently characterised by highly connected nodes called hubs. The method involves ranking the hubs of a channel network according to the contributing drainage area and the distance to the outlet. The hubs' characteristics can be considered as morphometric descriptors of the channel network and are used to compare and classify channel network. Applications were conducted on 788 French catchments with the same area (between 100 and 105 km2) and on 18 catchments having an area between 43 and 116450 km2. First, we present some newly found invariance properties of headwater subcatchments and show that some invariant morphometric properties characterize only natural channel networks verifying Optimal Channel Networks (OCN) properties, but are not verified for non-OCN (Moussa et al., 2011, Water Resources Research, 47, W08518). A new empirical model based on self-affine properties was developed in order to calculate the number N and the total headwater area H as a function of the cutoff area S used to extract the channel network from DEM. Results show that H(S) / S0 (S0 being the catchment area) is independent from S and seems constant (0.29 +/- 0.03) for various shapes and sizes of channel networks, and consequently can be considered as invariant general descriptor of natural channel networks. On the contrary, this is not the case when the approach is applied
Agreement Problems in Networks with Directed Graphs and Switching Topology
2003-02-15
C–3613. References [1] N. Biggs. Algebraic Graph Theory. Cambridge Tracks in Mathematics. Cambridge University Press , 1974. 15 [2] J. P. Desai, J. P...Springer, 2001. [9] R. A. Horn and C. R. Johnson. Matrix Analysis. Cambridge University Press , 1987. [10] A. Jadbabaie, J. Lin, and S. A. Morse...Networks of Dynamic Agents. Proc. of the American Control Conference, June 2003. [15] F. Paganini , J. Doyle, and S Low. Scalable laws for stable
Connectomics and graph theory analyses: Novel insights into network abnormalities in epilepsy.
Gleichgerrcht, Ezequiel; Kocher, Madison; Bonilha, Leonardo
2015-11-01
The assessment of neural networks in epilepsy has become increasingly relevant in the context of translational research, given that localized forms of epilepsy are more likely to be related to abnormal function within specific brain networks, as opposed to isolated focal brain pathology. It is notable that variability in clinical outcomes from epilepsy treatment may be a reflection of individual patterns of network abnormalities. As such, network endophenotypes may be important biomarkers for the diagnosis and treatment of epilepsy. Despite its exceptional potential, measuring abnormal networks in translational research has been thus far constrained by methodologic limitations. Fortunately, recent advancements in neuroscience, particularly in the field of connectomics, permit a detailed assessment of network organization, dynamics, and function at an individual level. Data from the personal connectome can be assessed using principled forms of network analyses based on graph theory, which may disclose patterns of organization that are prone to abnormal dynamics and epileptogenesis. Although the field of connectomics is relatively new, there is already a rapidly growing body of evidence to suggest that it can elucidate several important and fundamental aspects of abnormal networks to epilepsy. In this article, we provide a review of the emerging evidence from connectomics research regarding neural network architecture, dynamics, and function related to epilepsy. We discuss how connectomics may bring together pathophysiologic hypotheses from conceptual and basic models of epilepsy and in vivo biomarkers for clinical translational research. By providing neural network information unique to each individual, the field of connectomics may help to elucidate variability in clinical outcomes and open opportunities for personalized medicine approaches to epilepsy. Connectomics involves complex and rich data from each subject, thus collaborative efforts to enable the
Enhancing multiple-point geostatistical modeling: 1. Graph theory and pattern adjustment
NASA Astrophysics Data System (ADS)
Tahmasebi, Pejman; Sahimi, Muhammad
2016-03-01
In recent years, higher-order geostatistical methods have been used for modeling of a wide variety of large-scale porous media, such as groundwater aquifers and oil reservoirs. Their popularity stems from their ability to account for qualitative data and the great flexibility that they offer for conditioning the models to hard (quantitative) data, which endow them with the capability for generating realistic realizations of porous formations with very complex channels, as well as features that are mainly a barrier to fluid flow. One group of such models consists of pattern-based methods that use a set of data points for generating stochastic realizations by which the large-scale structure and highly-connected features are reproduced accurately. The cross correlation-based simulation (CCSIM) algorithm, proposed previously by the authors, is a member of this group that has been shown to be capable of simulating multimillion cell models in a matter of a few CPU seconds. The method is, however, sensitive to pattern's specifications, such as boundaries and the number of replicates. In this paper the original CCSIM algorithm is reconsidered and two significant improvements are proposed for accurately reproducing large-scale patterns of heterogeneities in porous media. First, an effective boundary-correction method based on the graph theory is presented by which one identifies the optimal cutting path/surface for removing the patchiness and discontinuities in the realization of a porous medium. Next, a new pattern adjustment method is proposed that automatically transfers the features in a pattern to one that seamlessly matches the surrounding patterns. The original CCSIM algorithm is then combined with the two methods and is tested using various complex two- and three-dimensional examples. It should, however, be emphasized that the methods that we propose in this paper are applicable to other pattern-based geostatistical simulation methods.
Disrupted Brain Functional Organization in Epilepsy Revealed by Graph Theory Analysis.
Song, Jie; Nair, Veena A; Gaggl, Wolfgang; Prabhakaran, Vivek
2015-06-01
The human brain is a complex and dynamic system that can be modeled as a large-scale brain network to better understand the reorganizational changes secondary to epilepsy. In this study, we developed a brain functional network model using graph theory methods applied to resting-state fMRI data acquired from a group of epilepsy patients and age- and gender-matched healthy controls. A brain functional network model was constructed based on resting-state functional connectivity. A minimum spanning tree combined with proportional thresholding approach was used to obtain sparse connectivity matrices for each subject, which formed the basis of brain networks. We examined the brain reorganizational changes in epilepsy thoroughly at the level of the whole brain, the functional network, and individual brain regions. At the whole-brain level, local efficiency was significantly decreased in epilepsy patients compared with the healthy controls. However, global efficiency was significantly increased in epilepsy due to increased number of functional connections between networks (although weakly connected). At the functional network level, there were significant proportions of newly formed connections between the default mode network and other networks and between the subcortical network and other networks. There was a significant proportion of decreasing connections between the cingulo-opercular task control network and other networks. Individual brain regions from different functional networks, however, showed a distinct pattern of reorganizational changes in epilepsy. These findings suggest that epilepsy alters brain efficiency in a consistent pattern at the whole-brain level, yet alters brain functional networks and individual brain regions differently.
Panzica, Ferruccio; Varotto, Giulia; Rotondi, Fabio; Spreafico, Roberto; Franceschetti, Silvana
2013-11-06
In the context of focal drug-resistant epilepsies, the surgical resection of the epileptogenic zone (EZ), the cortical region responsible for the onset, early seizures organization, and propagation, may be the only therapeutic option for reducing or suppressing seizures. The rather high rate of failure in epilepsy surgery of extra-temporal epilepsies highlights that the precise identification of the EZ, mandatory objective to achieve seizure freedom, is still an unsolved problem that requires more sophisticated methods of investigation. Despite the wide range of non-invasive investigations, intracranial stereo-EEG (SEEG) recordings still represent, in many patients, the gold standard for the EZ identification. In this contest, the EZ localization is still based on visual analysis of SEEG, inevitably affected by the drawback of subjectivity and strongly time-consuming. Over the last years, considerable efforts have been made to develop advanced signal analysis techniques able to improve the identification of the EZ. Particular attention has been paid to those methods aimed at quantifying and characterizing the interactions and causal relationships between neuronal populations, since is nowadays well assumed that epileptic phenomena are associated with abnormal changes in brain synchronization mechanisms, and initial evidence has shown the suitability of this approach for the EZ localization. The aim of this review is to provide an overview of the different EEG signal processing methods applied to study connectivity between distinct brain cortical regions, namely in focal epilepsies. In addition, with the aim of localizing the EZ, the approach based on graph theory will be described, since the study of the topological properties of the networks has strongly improved the study of brain connectivity mechanisms.
ERIC Educational Resources Information Center
Sokol, William
In this autoinstructional packet, the student is given an experimental situation which introduces him to the process of graphing. The lesson is presented for secondary school students in chemistry. Algebra I and a Del Mod System program (indicated as SE 018 020) are suggested prerequisites for the use of this program. Behavioral objectives are…
ERIC Educational Resources Information Center
Buerman, Margaret
2007-01-01
Finding real-world examples for middle school algebra classes can be difficult but not impossible. As we strive to accomplish teaching our students how to solve and graph equations, we neglect to teach the big ideas of algebra. One of those big ideas is functions. This article gives three examples of functions that are found in Arches National…
Quantum cluster algebras and quantum nilpotent algebras
Goodearl, Kenneth R.; Yakimov, Milen T.
2014-01-01
A major direction in the theory of cluster algebras is to construct (quantum) cluster algebra structures on the (quantized) coordinate rings of various families of varieties arising in Lie theory. We prove that all algebras in a very large axiomatically defined class of noncommutative algebras possess canonical quantum cluster algebra structures. Furthermore, they coincide with the corresponding upper quantum cluster algebras. We also establish analogs of these results for a large class of Poisson nilpotent algebras. Many important families of coordinate rings are subsumed in the class we are covering, which leads to a broad range of applications of the general results to the above-mentioned types of problems. As a consequence, we prove the Berenstein–Zelevinsky conjecture [Berenstein A, Zelevinsky A (2005) Adv Math 195:405–455] for the quantized coordinate rings of double Bruhat cells and construct quantum cluster algebra structures on all quantum unipotent groups, extending the theorem of Geiß et al. [Geiß C, et al. (2013) Selecta Math 19:337–397] for the case of symmetric Kac–Moody groups. Moreover, we prove that the upper cluster algebras of Berenstein et al. [Berenstein A, et al. (2005) Duke Math J 126:1–52] associated with double Bruhat cells coincide with the corresponding cluster algebras. PMID:24982197
Integration of graph theory and quantum chemistry for structure-activity relationships.
Balasubramanian, K
1994-01-01
The objective of this article is to outline both graph-theoretically based and quantum chemically based structural indices of potential use in quantitative structure activity correlations. We consider graph-theoretical indices such as the connectivity index, topological index, Wiener index and molecular ID indices. Several structural and geometry-dependent indices can be derived from semiempirical and ab initio quantum calculations based on the charge densities, overlap matrices, frontier orbitals, molecular hardness, free valence, density matrices, quantum spectral difference indices, quantum spectral indices and bond matrices. Finally, the use of electrostatic potentials and charge densities for the prediction of reactive sites will be discussed.
Permutation centralizer algebras and multimatrix invariants
NASA Astrophysics Data System (ADS)
Mattioli, Paolo; Ramgoolam, Sanjaye
2016-03-01
We introduce a class of permutation centralizer algebras which underly the combinatorics of multimatrix gauge-invariant observables. One family of such noncommutative algebras is parametrized by two integers. Its Wedderburn-Artin decomposition explains the counting of restricted Schur operators, which were introduced in the physics literature to describe open strings attached to giant gravitons and were subsequently used to diagonalize the Gaussian inner product for gauge invariants of two-matrix models. The structure of the algebra, notably its dimension, its center and its maximally commuting subalgebra, is related to Littlewood-Richardson numbers for composing Young diagrams. It gives a precise characterization of the minimal set of charges needed to distinguish arbitrary matrix gauge invariants, which are related to enhanced symmetries in gauge theory. The algebra also gives a star product for matrix invariants. The center of the algebra allows efficient computation of a sector of multimatrix correlators. These generate the counting of a certain class of bicoloured ribbon graphs with arbitrary genus.
An Application of Cartesian Graphing to Seismic Exploration.
ERIC Educational Resources Information Center
Robertson, Douglas Frederick
1992-01-01
Describes how college students enrolled in a course in elementary algebra apply graphing and algebra to data collected from a seismic profile to uncover the structure of a subterranean rock formation. Includes steps guiding the activity. (MDH)
dos Santos Siqueira, Anderson; Biazoli Junior, Claudinei Eduardo; Comfort, William Edgar; Rohde, Luis Augusto; Sato, João Ricardo
2014-01-01
The framework of graph theory provides useful tools for investigating the neural substrates of neuropsychiatric disorders. Graph description measures may be useful as predictor variables in classification procedures. Here, we consider several centrality measures as predictor features in a classification algorithm to identify nodes of resting-state networks containing predictive information that can discriminate between typical developing children and patients with attention-deficit/hyperactivity disorder (ADHD). The prediction was based on a support vector machines classifier. The analyses were performed in a multisite and publicly available resting-state fMRI dataset of healthy children and ADHD patients: the ADHD-200 database. Network centrality measures contained little predictive information for the discrimination between ADHD patients and healthy subjects. However, the classification between inattentive and combined ADHD subtypes was more promising, achieving accuracies higher than 65% (balance between sensitivity and specificity) in some sites. Finally, brain regions were ranked according to the amount of discriminant information and the most relevant were mapped. As hypothesized, we found that brain regions in motor, frontoparietal, and default mode networks contained the most predictive information. We concluded that the functional connectivity estimations are strongly dependent on the sample characteristics. Thus different acquisition protocols and clinical heterogeneity decrease the predictive values of the graph descriptors.
dos Santos Siqueira, Anderson; Biazoli Junior, Claudinei Eduardo; Comfort, William Edgar; Rohde, Luis Augusto; Sato, João Ricardo
2014-01-01
The framework of graph theory provides useful tools for investigating the neural substrates of neuropsychiatric disorders. Graph description measures may be useful as predictor variables in classification procedures. Here, we consider several centrality measures as predictor features in a classification algorithm to identify nodes of resting-state networks containing predictive information that can discriminate between typical developing children and patients with attention-deficit/hyperactivity disorder (ADHD). The prediction was based on a support vector machines classifier. The analyses were performed in a multisite and publicly available resting-state fMRI dataset of healthy children and ADHD patients: the ADHD-200 database. Network centrality measures contained little predictive information for the discrimination between ADHD patients and healthy subjects. However, the classification between inattentive and combined ADHD subtypes was more promising, achieving accuracies higher than 65% (balance between sensitivity and specificity) in some sites. Finally, brain regions were ranked according to the amount of discriminant information and the most relevant were mapped. As hypothesized, we found that brain regions in motor, frontoparietal, and default mode networks contained the most predictive information. We concluded that the functional connectivity estimations are strongly dependent on the sample characteristics. Thus different acquisition protocols and clinical heterogeneity decrease the predictive values of the graph descriptors. PMID:25309910
Sound Off! A Dialogue between Calculator and Algebra
ERIC Educational Resources Information Center
Wade, William R.
2006-01-01
This article illustrates the fact that unless tempered by algebraic reasoning, a graphing calculator can lead one to erroneous conclusions. It also demonstrates that some problems can be solved by combining technology with algebra.
Cluster Consensus of Nonlinearly Coupled Multi-Agent Systems in Directed Graphs
NASA Astrophysics Data System (ADS)
Lu, Xiao-Qing; Francis, Austin; Chen, Shi-Hua
2010-05-01
We investigate the cluster consensus problem in directed networks of nonlinearly coupled multi-agent systems by using pinning control. Depending on the community structure generated by the group partition of the underlying digraph, various clusters can be made coherently independent by applying feedback injections to a fraction of the agents. Sufficient conditions for cluster consensus are obtained using algebraic graph theory and matrix theory and some simulations results are included to illustrate the method.
ERIC Educational Resources Information Center
Bair, Sherry L.; Rich, Beverly S.
2011-01-01
This article characterizes the development of a deep and connected body of mathematical knowledge categorized by Ball and Bass' (2003b) model of Mathematical Knowledge for Teaching (MKT), as Specialized Content Knowledge for Teaching (SCK) in algebraic reasoning and number sense. The research employed multiple cases across three years from two…
Student Reactions to Learning Theory Based Curriculum Materials in Linear Algebra--A Survey Analysis
ERIC Educational Resources Information Center
Cooley, Laurel; Vidakovic, Draga; Martin, William O.; Dexter, Scott; Suzuki, Jeff
2016-01-01
In this report we examine students' perceptions of the implementation of carefully designed curriculum materials (called modules) in linear algebra courses at three different universities. The curricular materials were produced collaboratively by STEM and mathematics education faculty as members of a professional learning community (PLC) over…
NASA Astrophysics Data System (ADS)
Pilipchuk, L. A.; Pilipchuk, A. S.
2016-12-01
For constructing of the solutions of the sparse linear systems we propose effective methods, technologies and their implementation in Wolfram Mathematica. Sparse systems of these types appear in generalized network flow programming problems in the form of restrictions and can be characterized as systems with a large sparse sub-matrix representing the embedded network structure. In addition, such systems arise in estimating traffic in the generalized graph or multigraph on its unobservable part. For computing of each vector of the basis solution space with linear estimate in the worst case we propose effective algorithms and data structures in the case when a support of the multigraph or graph for the sparse systems contains a cycles.
Kocevar, Gabriel; Stamile, Claudio; Hannoun, Salem; Cotton, François; Vukusic, Sandra; Durand-Dubief, Françoise; Sappey-Marinier, Dominique
2016-01-01
Purpose: In this work, we introduce a method to classify Multiple Sclerosis (MS) patients into four clinical profiles using structural connectivity information. For the first time, we try to solve this question in a fully automated way using a computer-based method. The main goal is to show how the combination of graph-derived metrics with machine learning techniques constitutes a powerful tool for a better characterization and classification of MS clinical profiles. Materials and Methods: Sixty-four MS patients [12 Clinical Isolated Syndrome (CIS), 24 Relapsing Remitting (RR), 24 Secondary Progressive (SP), and 17 Primary Progressive (PP)] along with 26 healthy controls (HC) underwent MR examination. T1 and diffusion tensor imaging (DTI) were used to obtain structural connectivity matrices for each subject. Global graph metrics, such as density and modularity, were estimated and compared between subjects' groups. These metrics were further used to classify patients using tuned Support Vector Machine (SVM) combined with Radial Basic Function (RBF) kernel. Results: When comparing MS patients to HC subjects, a greater assortativity, transitivity, and characteristic path length as well as a lower global efficiency were found. Using all graph metrics, the best F-Measures (91.8, 91.8, 75.6, and 70.6%) were obtained for binary (HC-CIS, CIS-RR, RR-PP) and multi-class (CIS-RR-SP) classification tasks, respectively. When using only one graph metric, the best F-Measures (83.6, 88.9, and 70.7%) were achieved for modularity with previous binary classification tasks. Conclusion: Based on a simple DTI acquisition associated with structural brain connectivity analysis, this automatic method allowed an accurate classification of different MS patients' clinical profiles. PMID:27826224
Kocevar, Gabriel; Stamile, Claudio; Hannoun, Salem; Cotton, François; Vukusic, Sandra; Durand-Dubief, Françoise; Sappey-Marinier, Dominique
2016-01-01
Purpose: In this work, we introduce a method to classify Multiple Sclerosis (MS) patients into four clinical profiles using structural connectivity information. For the first time, we try to solve this question in a fully automated way using a computer-based method. The main goal is to show how the combination of graph-derived metrics with machine learning techniques constitutes a powerful tool for a better characterization and classification of MS clinical profiles. Materials and Methods: Sixty-four MS patients [12 Clinical Isolated Syndrome (CIS), 24 Relapsing Remitting (RR), 24 Secondary Progressive (SP), and 17 Primary Progressive (PP)] along with 26 healthy controls (HC) underwent MR examination. T1 and diffusion tensor imaging (DTI) were used to obtain structural connectivity matrices for each subject. Global graph metrics, such as density and modularity, were estimated and compared between subjects' groups. These metrics were further used to classify patients using tuned Support Vector Machine (SVM) combined with Radial Basic Function (RBF) kernel. Results: When comparing MS patients to HC subjects, a greater assortativity, transitivity, and characteristic path length as well as a lower global efficiency were found. Using all graph metrics, the best F-Measures (91.8, 91.8, 75.6, and 70.6%) were obtained for binary (HC-CIS, CIS-RR, RR-PP) and multi-class (CIS-RR-SP) classification tasks, respectively. When using only one graph metric, the best F-Measures (83.6, 88.9, and 70.7%) were achieved for modularity with previous binary classification tasks. Conclusion: Based on a simple DTI acquisition associated with structural brain connectivity analysis, this automatic method allowed an accurate classification of different MS patients' clinical profiles.
Classification of pregnancy and labor contractions using a graph theory based analysis.
Nader, N; Hassan, M; Falou, W; Diab, A; Al-Omar, S; Khalil, M; Marque, C
2015-08-01
In this paper, we propose a new framework to characterize the electrohysterographic (EHG) signals recorded during pregnancy and labor. The approach is based on the analysis of the propagation of the uterine electrical activity. The processing pipeline includes i) the estimation of the statistical dependencies between the different recorded EHG signals, ii) the characterization of the obtained connectivity matrices using network measures and iii) the use of these measures in clinical application: the classification between pregnancy and labor. Due to its robustness to volume conductor, we used the imaginary part of coherence in order to produce the connectivity matrix which is then transformed into a graph. We evaluate the performance of several graph measures. We also compare the results with the parameter mostly used in the literature: the peak frequency combined with the propagation velocity (PV +PF). Our results show that the use of the network measures is a promising tool to classify labor and pregnancy contractions with a small superiority of the graph strength over PV+PF.
Toppi, J; Ciaramidaro, A; Vogel, P; Mattia, D; Babiloni, F; Siniatchkin, M; Astolfi, L
2015-08-01
Hyperscanning consists in the simultaneous recording of hemodynamic or neuroelectrical signals from two or more subjects acting in a social context. Well-established methodologies for connectivity estimation have already been adapted to hyperscanning purposes. The extension of graph theory approach to multi-subjects case is still a challenging issue. In the present work we aim to test the ability of the currently used graph theory global indices in describing the properties of a network given by two interacting subjects. The testing was conducted first on surrogate brain-to-brain networks reproducing typical social scenarios and then on real EEG hyperscanning data recorded during a Joint Action task. The results of the simulation study highlighted the ability of all the investigated indexes in modulating their values according to the level of interaction between subjects. However, only global efficiency and path length indexes demonstrated to be sensitive to an asymmetry in the communication between the two subjects. Such results were, then, confirmed by the application on real EEG data. Global efficiency modulated, in fact, their values according to the inter-brain density, assuming higher values in the social condition with respect to the non-social condition.
ERIC Educational Resources Information Center
Ozgun-Koca, S. Ash
2010-01-01
Although growing numbers of secondary school mathematics teachers and students use calculators to study graphs, they mainly rely on paper-and-pencil when manipulating algebraic symbols. However, the Computer Algebra Systems (CAS) on computers or handheld calculators create new possibilities for teaching and learning algebraic manipulation. This…
Zhang, Chunmei; Li, Wenxue; Wang, Ke
2015-08-01
In this paper, a novel class of stochastic coupled systems with Lévy noise on networks (SCSLNNs) is presented. Both white noise and Lévy noise are considered in the networks. By exploiting graph theory and Lyapunov stability theory, criteria ensuring p th moment exponential stability and stability in probability of these SCSLNNs are established, respectively. These principles are closely related to the topology of the network and the perturbation intensity of white noise and Lévy noise. Moreover, to verify the theoretical results, stochastic coupled oscillators with Lévy noise on a network and stochastic Volterra predator-prey system with Lévy noise are performed. Finally, a numerical example about oscillators' network is provided to illustrate the feasibility of our analytical results.
Teaching Algebra without Algebra
ERIC Educational Resources Information Center
Kalman, Richard S.
2008-01-01
Algebra is, among other things, a shorthand way to express quantitative reasoning. This article illustrates ways for the classroom teacher to convert algebraic solutions to verbal problems into conversational solutions that can be understood by students in the lower grades. Three reasonably typical verbal problems that either appeared as or…
Noncommutative Riemannian geometry on graphs
NASA Astrophysics Data System (ADS)
Majid, Shahn
2013-07-01
We show that arising out of noncommutative geometry is a natural family of edge Laplacians on the edges of a graph. The family includes a canonical edge Laplacian associated to the graph, extending the usual graph Laplacian on vertices, and we find its spectrum. We show that for a connected graph its eigenvalues are strictly positive aside from one mandatory zero mode, and include all the vertex degrees. Our edge Laplacian is not the graph Laplacian on the line graph but rather it arises as the noncommutative Laplace-Beltrami operator on differential 1-forms, where we use the language of differential algebras to functorially interpret a graph as providing a 'finite manifold structure' on the set of vertices. We equip any graph with a canonical 'Euclidean metric' and a canonical bimodule connection, and in the case of a Cayley graph we construct a metric compatible connection for the Euclidean metric. We make use of results on bimodule connections on inner calculi on algebras, which we prove, including a general relation between zero curvature and the braid relations.
Mean Curvature, Threshold Dynamics, and Phase Field Theory on Finite Graphs
2013-06-28
to the uniform distribution [SBXD04]. Due to this property, λ−12 is sometimes referred to as the mixing- time for a graph. The second eigenvalue λ2 is...discrete sequence of sets Sn = S(nðt), n ∈ N, that correspond to the state of the system at discrete time steps. Our construction follows the well-known...stationary under an (MBOτ ) iteration. To prove (31), we write the solution to the heat equation at time τ , u(τ) = e−τ∆χS = χS − ∫ τ 0 ∆u(t)dt. This
Saad, Tony; Sutherland, James C.
2016-05-04
To address the coding and software challenges of modern hybrid architectures, we propose an approach to multiphysics code development for high-performance computing. This approach is based on using a Domain Specific Language (DSL) in tandem with a directed acyclic graph (DAG) representation of the problem to be solved that allows runtime algorithm generation. When coupled with a large-scale parallel framework, the result is a portable development framework capable of executing on hybrid platforms and handling the challenges of multiphysics applications. In addition, we share our experience developing a code in such an environment – an effort that spans an interdisciplinary team of engineers and computer scientists.
Jordan Algebraic Quantum Categories
NASA Astrophysics Data System (ADS)
Graydon, Matthew; Barnum, Howard; Ududec, Cozmin; Wilce, Alexander
2015-03-01
State cones in orthodox quantum theory over finite dimensional complex Hilbert spaces enjoy two particularly essential features: homogeneity and self-duality. Orthodox quantum theory is not, however, unique in that regard. Indeed, all finite dimensional formally real Jordan algebras -- arenas for generalized quantum theories with close algebraic kinship to the orthodox theory -- admit homogeneous self-dual positive cones. We construct categories wherein these theories are unified. The structure of composite systems is cast from universal tensor products of the universal C*-algebras enveloping ambient spaces for the constituent state cones. We develop, in particular, a notion of composition that preserves the local distinction of constituent systems in quaternionic quantum theory. More generally, we explicitly derive the structure of hybrid quantum composites with subsystems of arbitrary Jordan algebraic type.
Network representation of protein interactions: Theory of graph description and analysis.
Kurzbach, Dennis
2016-09-01
A methodological framework is presented for the graph theoretical interpretation of NMR data of protein interactions. The proposed analysis generalizes the idea of network representations of protein structures by expanding it to protein interactions. This approach is based on regularization of residue-resolved NMR relaxation times and chemical shift data and subsequent construction of an adjacency matrix that represents the underlying protein interaction as a graph or network. The network nodes represent protein residues. Two nodes are connected if two residues are functionally correlated during the protein interaction event. The analysis of the resulting network enables the quantification of the importance of each amino acid of a protein for its interactions. Furthermore, the determination of the pattern of correlations between residues yields insights into the functional architecture of an interaction. This is of special interest for intrinsically disordered proteins, since the structural (three-dimensional) architecture of these proteins and their complexes is difficult to determine. The power of the proposed methodology is demonstrated at the example of the interaction between the intrinsically disordered protein osteopontin and its natural ligand heparin.
A study of chemical systems using signal flow graph theory: application to Neptune
NASA Astrophysics Data System (ADS)
Dobrijevic, M.; Parisot, J. P.; Dutour, I.
1995-02-01
Photochemistry of giant planets and their satellites is characterized by numerous reactions involving many chemical species. In the present paper, chemical systems are modeled by signal flow graphs. Such a technique evaluates the transmission of any input into the system (solar flux, electrons…) and gives access to the identification of the most important mechanisms in the chemical system. For a given chemical system, we first evaluate rate coefficients. Then, in order to obtain concentrations of each compound, we integrate the set of continuity equations by Gear's method. Gear's method is chosen rather than another classical method because it is recommended for a system of stiff equations due to the existence of greatly differing time constants. Finally, the technique of signal flow graphs is used. This method is applied to the production of hydrocarbons in the atmospheres of giant planets. In particular, the production of C 2H 6 in the atmosphere of Neptune from the photodissociation of CH 4 is investigated. Different paths of dissociation of CH 4 are possible from L α radiations. A chemical system containing 14 species and 30 reactions including these different paths of dissociation is integrated. The main mechanism of production of C 2H 6 is identified and evaluated for each model of dissociation. The importance of various reaction paths as a function of time is discussed.
ERIC Educational Resources Information Center
Jackson, David F.; And Others
Recent research has demonstrated the promise of graphing software as an aid to teaching graphs in two content areas: line graphs of aspects of motion and graphs of algebraic functions. This study attempted to generalize the idea of computer-assisted graphing to include the use of several kinds of graphs to solve a wider range of problems. A unit…
Network analysis for a network disorder: The emerging role of graph theory in the study of epilepsy.
Bernhardt, Boris C; Bonilha, Leonardo; Gross, Donald W
2015-09-01
Recent years have witnessed a paradigm shift in the study and conceptualization of epilepsy, which is increasingly understood as a network-level disorder. An emblematic case is temporal lobe epilepsy (TLE), the most common drug-resistant epilepsy that is electroclinically defined as a focal epilepsy and pathologically associated with hippocampal sclerosis. In this review, we will summarize histopathological, electrophysiological, and neuroimaging evidence supporting the concept that the substrate of TLE is not limited to the hippocampus alone, but rather is broadly distributed across multiple brain regions and interconnecting white matter pathways. We will introduce basic concepts of graph theory, a formalism to quantify topological properties of complex systems that has recently been widely applied to study networks derived from brain imaging and electrophysiology. We will discuss converging graph theoretical evidence indicating that networks in TLE show marked shifts in their overall topology, providing insight into the neurobiology of TLE as a network-level disorder. Our review will conclude by discussing methodological challenges and future clinical applications of this powerful analytical approach.
Saad, Tony; Sutherland, James C.
2016-05-04
To address the coding and software challenges of modern hybrid architectures, we propose an approach to multiphysics code development for high-performance computing. This approach is based on using a Domain Specific Language (DSL) in tandem with a directed acyclic graph (DAG) representation of the problem to be solved that allows runtime algorithm generation. When coupled with a large-scale parallel framework, the result is a portable development framework capable of executing on hybrid platforms and handling the challenges of multiphysics applications. In addition, we share our experience developing a code in such an environment – an effort that spans an interdisciplinarymore » team of engineers and computer scientists.« less
Poza, Jesus; Garcia, Maria; Gomez, Carlos; Bachiller, Alejandro; Carreres, Alicia; Hornero, Roberto
2013-01-01
The aim of this research was to study the changes that Alzheimer's disease (AD) elicits in the organization of brain networks. For this task, the electroencephalographic (EEG) activity from 32 AD patients and 25 healthy controls was analyzed. In a first step, a disequilibrium measure, the Euclidean distance (ED), was used to estimate the similarity between the spectral content of each pair of electrodes. In a second step, the similarity matrices were used to generate the corresponding graphs, from which two parameters were computed to characterize the network structure: the mean clustering coefficient and the mean path length. Results revealed significant changes (p<0.05) in ED values, as well as in the mean clustering coefficient and the mean path length, though they depend on the specific frequency band. Our findings suggest that AD is accompanied by a significant frequency-dependent alteration of brain network organization.
Application of the algebraic RNG model for transition simulation. [renormalization group theory
NASA Technical Reports Server (NTRS)
Lund, Thomas S.
1990-01-01
The algebraic form of the RNG model of Yakhot and Orszag (1986) is investigated as a transition model for the Reynolds averaged boundary layer equations. It is found that the cubic equation for the eddy viscosity contains both a jump discontinuity and one spurious root. A yet unpublished transformation to a quartic equation is shown to remove the numerical difficulties associated with the discontinuity, but only at the expense of merging both the physical and spurious root of the cubic. Jumps between the branches of the resulting multiple-valued solution are found to lead to oscillations in flat plate transition calculations. Aside from the oscillations, the transition behavior is qualitatively correct.
Dynamical basis sets for algebraic variational calculations in quantum-mechanical scattering theory
NASA Technical Reports Server (NTRS)
Sun, Yan; Kouri, Donald J.; Truhlar, Donald G.; Schwenke, David W.
1990-01-01
New basis sets are proposed for linear algebraic variational calculations of transition amplitudes in quantum-mechanical scattering problems. These basis sets are hybrids of those that yield the Kohn variational principle (KVP) and those that yield the generalized Newton variational principle (GNVP) when substituted in Schlessinger's stationary expression for the T operator. Trial calculations show that efficiencies almost as great as that of the GNVP and much greater than the KVP can be obtained, even for basis sets with the majority of the members independent of energy.
Shojaedini, Seyed Vahab; Heydari, Masoud
2014-10-01
Shape and movement features of sperms are important parameters for infertility study and treatment. In this article, a new method is introduced for characterization sperms in microscopic videos. In this method, first a hypothesis framework is defined to distinguish sperms from other particles in captured video. Then decision about each hypothesis is done in following steps: Selecting some primary regions as candidates for sperms by watershed-based segmentation, pruning of some false candidates during successive frames using graph theory concept and finally confirming correct sperms by using their movement trajectories. Performance of the proposed method is evaluated on real captured images belongs to semen with high density of sperms. The obtained results show the proposed method may detect 97% of sperms in presence of 5% false detections and track 91% of moving sperms. Furthermore, it can be shown that better characterization of sperms in proposed algorithm doesn't lead to extracting more false sperms compared to some present approaches.
Li, Zhigang; Shi, Zhongping; Li, Xin
2014-05-01
Several fermentations with consecutively feeding of acetate/butyrate were conducted in a 7 L fermentor and the results indicated that exogenous acetate/butyrate enhanced solvents productivities by 47.1% and 39.2% respectively, and changed butyrate/acetate ratios greatly. Then extracellular butyrate/acetate ratios were utilized for calculation of acids rates and the results revealed that acetate and butyrate formation pathways were almost blocked by corresponding acids feeding. In addition, models for acetate/butyrate feeding fermentations were constructed by graph theory based on calculation results and relevant reports. Solvents concentrations and butanol/acetone ratios of these fermentations were also calculated and the results of models calculation matched fermentation data accurately which demonstrated that models were constructed in a reasonable way.
Fatakia, Sarosh N.; Costanzi, Stefano; Chow, Carson C.
2009-01-01
G protein-coupled receptors (GPCRs) are a superfamily of seven transmembrane-spanning proteins involved in a wide array of physiological functions and are the most common targets of pharmaceuticals. This study aims to identify a cohort or clique of positions that share high mutual information. Using a multiple sequence alignment of the transmembrane (TM) domains, we calculated the mutual information between all inter-TM pairs of aligned positions and ranked the pairs by mutual information. A mutual information graph was constructed with vertices that corresponded to TM positions and edges between vertices were drawn if the mutual information exceeded a threshold of statistical significance. Positions with high degree (i.e. had significant mutual information with a large number of other positions) were found to line a well defined inter-TM ligand binding cavity for class A as well as class C GPCRs. Although the natural ligands of class C receptors bind to their extracellular N-terminal domains, the possibility of modulating their activity through ligands that bind to their helical bundle has been reported. Such positions were not found for class B GPCRs, in agreement with the observation that there are not known ligands that bind within their TM helical bundle. All identified key positions formed a clique within the MI graph of interest. For a subset of class A receptors we also considered the alignment of a portion of the second extracellular loop, and found that the two positions adjacent to the conserved Cys that bridges the loop with the TM3 qualified as key positions. Our algorithm may be useful for localizing topologically conserved regions in other protein families. PMID:19262747
Rees algebras, Monomial Subrings and Linear Optimization Problems
NASA Astrophysics Data System (ADS)
Dupont, Luis A.
2010-06-01
In this thesis we are interested in studying algebraic properties of monomial algebras, that can be linked to combinatorial structures, such as graphs and clutters, and to optimization problems. A goal here is to establish bridges between commutative algebra, combinatorics and optimization. We study the normality and the Gorenstein property-as well as the canonical module and the a-invariant-of Rees algebras and subrings arising from linear optimization problems. In particular, we study algebraic properties of edge ideals and algebras associated to uniform clutters with the max-flow min-cut property or the packing property. We also study algebraic properties of symbolic Rees algebras of edge ideals of graphs, edge ideals of clique clutters of comparability graphs, and Stanley-Reisner rings.
ERIC Educational Resources Information Center
Skurnick, Ronald; Davi, Charles; Skurnick, Mia
2005-01-01
Since 1952, several well-known graph theorists have proven numerous results regarding Hamiltonian graphs. In fact, many elementary graph theory textbooks contain the theorems of Ore, Bondy and Chvatal, Chvatal and Erdos, Posa, and Dirac, to name a few. In this note, the authors state and prove some propositions of their own concerning Hamiltonian…
Schulz, Martin; Arnold, Dorian
2007-06-12
GraphLib is a support library used by other tools to create, manipulate, store, and export graphs. It provides a simple interface to specifS arbitrary directed and undirected graphs by adding nodes and edges. Each node and edge can be associated with a set of attributes describing size, color, and shape. Once created, graphs can be manipulated using a set of graph analysis algorithms, including merge, prune, and path coloring operations. GraphLib also has the ability to export graphs into various open formats such as DOT and GML.
Highlighting the structure-function relationship of the brain with the Ising model and graph theory.
Das, T K; Abeyasinghe, P M; Crone, J S; Sosnowski, A; Laureys, S; Owen, A M; Soddu, A
2014-01-01
With the advent of neuroimaging techniques, it becomes feasible to explore the structure-function relationships in the brain. When the brain is not involved in any cognitive task or stimulated by any external output, it preserves important activities which follow well-defined spatial distribution patterns. Understanding the self-organization of the brain from its anatomical structure, it has been recently suggested to model the observed functional pattern from the structure of white matter fiber bundles. Different models which study synchronization (e.g., the Kuramoto model) or global dynamics (e.g., the Ising model) have shown success in capturing fundamental properties of the brain. In particular, these models can explain the competition between modularity and specialization and the need for integration in the brain. Graphing the functional and structural brain organization supports the model and can also highlight the strategy used to process and organize large amount of information traveling between the different modules. How the flow of information can be prevented or partially destroyed in pathological states, like in severe brain injured patients with disorders of consciousness or by pharmacological induction like in anaesthesia, will also help us to better understand how global or integrated behavior can emerge from local and modular interactions.
Abnormal brain white matter network in young smokers: a graph theory analysis study.
Zhang, Yajuan; Li, Min; Wang, Ruonan; Bi, Yanzhi; Li, Yangding; Yi, Zhang; Liu, Jixin; Yu, Dahua; Yuan, Kai
2017-03-13
Previous diffusion tensor imaging (DTI) studies had investigated the white matter (WM) integrity abnormalities in some specific fiber bundles in smokers. However, little is known about the changes in topological organization of WM structural network in young smokers. In current study, we acquired DTI datasets from 58 male young smokers and 51 matched nonsmokers and constructed the WM networks by the deterministic fiber tracking approach. Graph theoretical analysis was used to compare the topological parameters of WM network (global and nodal) and the inter-regional fractional anisotropy (FA) weighted WM connections between groups. The results demonstrated that both young smokers and nonsmokers had small-world topology in WM network. Further analysis revealed that the young smokers exhibited the abnormal topological organization, i.e., increased network strength, global efficiency, and decreased shortest path length. In addition, the increased nodal efficiency predominately was located in frontal cortex, striatum and anterior cingulate gyrus (ACG) in smokers. Moreover, based on network-based statistic (NBS) approach, the significant increased FA-weighted WM connections were mainly found in the PFC, ACG and supplementary motor area (SMA) regions. Meanwhile, the network parameters were correlated with the nicotine dependence severity (FTND) scores, and the nodal efficiency of orbitofrontal cortex was positive correlation with the cigarette per day (CPD) in young smokers. We revealed the abnormal topological organization of WM network in young smokers, which may improve our understanding of the neural mechanism of young smokers form WM topological organization level.
Vecchio, F; Miraglia, F; Quaranta, D; Granata, G; Romanello, R; Marra, C; Bramanti, P; Rossini, P M
2016-03-01
Functional brain abnormalities including memory loss are found to be associated with pathological changes in connectivity and network neural structures. Alzheimer's disease (AD) interferes with memory formation from the molecular level, to synaptic functions and neural networks organization. Here, we determined whether brain connectivity of resting-state networks correlate with memory in patients affected by AD and in subjects with mild cognitive impairment (MCI). One hundred and forty-four subjects were recruited: 70 AD (MMSE Mini Mental State Evaluation 21.4), 50 MCI (MMSE 25.2) and 24 healthy subjects (MMSE 29.8). Undirected and weighted cortical brain network was built to evaluate graph core measures to obtain Small World parameters. eLORETA lagged linear connectivity as extracted by electroencephalogram (EEG) signals was used to weight the network. A high statistical correlation between Small World and memory performance was found. Namely, higher Small World characteristic in EEG gamma frequency band during the resting state, better performance in short-term memory as evaluated by the digit span tests. Such Small World pattern might represent a biomarker of working memory impairment in older people both in physiological and pathological conditions.
A Systematic Composite Service Design Modeling Method Using Graph-Based Theory
Elhag, Arafat Abdulgader Mohammed; Mohamad, Radziah; Aziz, Muhammad Waqar; Zeshan, Furkh
2015-01-01
The composite service design modeling is an essential process of the service-oriented software development life cycle, where the candidate services, composite services, operations and their dependencies are required to be identified and specified before their design. However, a systematic service-oriented design modeling method for composite services is still in its infancy as most of the existing approaches provide the modeling of atomic services only. For these reasons, a new method (ComSDM) is proposed in this work for modeling the concept of service-oriented design to increase the reusability and decrease the complexity of system while keeping the service composition considerations in mind. Furthermore, the ComSDM method provides the mathematical representation of the components of service-oriented design using the graph-based theoryto facilitate the design quality measurement. To demonstrate that the ComSDM method is also suitable for composite service design modeling of distributed embedded real-time systems along with enterprise software development, it is implemented in the case study of a smart home. The results of the case study not only check the applicability of ComSDM, but can also be used to validate the complexity and reusability of ComSDM. This also guides the future research towards the design quality measurement such as using the ComSDM method to measure the quality of composite service design in service-oriented software system. PMID:25928358
A systematic composite service design modeling method using graph-based theory.
Elhag, Arafat Abdulgader Mohammed; Mohamad, Radziah; Aziz, Muhammad Waqar; Zeshan, Furkh
2015-01-01
The composite service design modeling is an essential process of the service-oriented software development life cycle, where the candidate services, composite services, operations and their dependencies are required to be identified and specified before their design. However, a systematic service-oriented design modeling method for composite services is still in its infancy as most of the existing approaches provide the modeling of atomic services only. For these reasons, a new method (ComSDM) is proposed in this work for modeling the concept of service-oriented design to increase the reusability and decrease the complexity of system while keeping the service composition considerations in mind. Furthermore, the ComSDM method provides the mathematical representation of the components of service-oriented design using the graph-based theoryto facilitate the design quality measurement. To demonstrate that the ComSDM method is also suitable for composite service design modeling of distributed embedded real-time systems along with enterprise software development, it is implemented in the case study of a smart home. The results of the case study not only check the applicability of ComSDM, but can also be used to validate the complexity and reusability of ComSDM. This also guides the future research towards the design quality measurement such as using the ComSDM method to measure the quality of composite service design in service-oriented software system.
A Study towards Building An Optimal Graph Theory Based Model For The Design of Tourism Website
NASA Astrophysics Data System (ADS)
Panigrahi, Goutam; Das, Anirban; Basu, Kajla
2010-10-01
Effective tourism website is a key to attract tourists from different parts of the world. Here we identify the factors of improving the effectiveness of website by considering it as a graph, where web pages including homepage are the nodes and hyperlinks are the edges between the nodes. In this model, the design constraints for building a tourism website are taken into consideration. Our objectives are to build a framework of an effective tourism website providing adequate level of information, service and also to enable the users to reach to the desired page by spending minimal loading time. In this paper an information hierarchy specifying the upper limit of outgoing link of a page has also been proposed. Following the hierarchy, the web developer can prepare an effective tourism website. Here loading time depends on page size and network traffic. We have assumed network traffic as uniform and the loading time is directly proportional with page size. This approach is done by quantifying the link structure of a tourism website. In this approach we also propose a page size distribution pattern of a tourism website.
NASA Astrophysics Data System (ADS)
Schlueter, Kristy; Dabiri, John
2016-11-01
Coherent structure identification is important in many fluid dynamics applications, including transport phenomena in ocean flows and mixing and diffusion in turbulence. However, many of the techniques currently available for measuring such flows, including ocean drifter datasets and particle tracking velocimetry, only result in sparse velocity data. This is often insufficient for the use of current coherent structure detection algorithms based on analysis of the deformation gradient. Here, we present a frame-invariant method for detecting coherent structures from Lagrangian flow trajectories that can be sparse in number. The method, based on principles used in graph coloring algorithms, examines a measure of the kinematic dissimilarity of all pairs of flow trajectories, either measured experimentally, e.g. using particle tracking velocimetry; or numerically, by advecting fluid particles in the Eulerian velocity field. Coherence is assigned to groups of particles whose kinematics remain similar throughout the time interval for which trajectory data is available, regardless of their physical proximity to one another. Through the use of several analytical and experimental validation cases, this algorithm is shown to robustly detect coherent structures using significantly less flow data than is required by existing methods. This research was supported by the Department of Defense (DoD) through the National Defense Science & Engineering Graduate Fellowship (NDSEG) Program.
Higher-order graph wavelets and sparsity on circulant graphs
NASA Astrophysics Data System (ADS)
Kotzagiannidis, Madeleine S.; Dragotti, Pier Luigi
2015-08-01
The notion of a graph wavelet gives rise to more advanced processing of data on graphs due to its ability to operate in a localized manner, across newly arising data-dependency structures, with respect to the graph signal and underlying graph structure, thereby taking into consideration the inherent geometry of the data. In this work, we tackle the problem of creating graph wavelet filterbanks on circulant graphs for a sparse representation of certain classes of graph signals. The underlying graph can hereby be data-driven as well as fixed, for applications including image processing and social network theory, whereby clusters can be modelled as circulant graphs, respectively. We present a set of novel graph wavelet filter-bank constructions, which annihilate higher-order polynomial graph signals (up to a border effect) defined on the vertices of undirected, circulant graphs, and are localised in the vertex domain. We give preliminary results on their performance for non-linear graph signal approximation and denoising. Furthermore, we provide extensions to our previously developed segmentation-inspired graph wavelet framework for non-linear image approximation, by incorporating notions of smoothness and vanishing moments, which further improve performance compared to traditional methods.
Graphing as a Problem-Solving Strategy.
ERIC Educational Resources Information Center
Cohen, Donald
1984-01-01
The focus is on how line graphs can be used to approximate solutions to rate problems and to suggest equations that offer exact algebraic solutions to the problem. Four problems requiring progressively greater graphing sophistication are presented plus four exercises. (MNS)
Zhou, Chaoyang; Hu, Xiaofei; Hu, Jun; Liang, Minglong; Yin, Xuntao; Chen, Lin; Zhang, Jiuquan; Wang, Jian
2016-01-01
Amyotrophic lateral sclerosis (ALS) is a rare degenerative disorder characterized by loss of upper and lower motor neurons. Neuroimaging has provided noticeable evidence that ALS is a complex disease, and shown that anatomical and functional lesions extend beyond precentral cortices and corticospinal tracts, to include the corpus callosum; frontal, sensory, and premotor cortices; thalamus; and midbrain. The aim of this study is to investigate graph theory-based functional network abnormalities at voxel-wise level in ALS patients on a whole brain scale. Forty-three ALS patients and 44 age- and sex-matched healthy volunteers were enrolled. The voxel-wise network degree centrality (DC), a commonly employed graph-based measure of network organization, was used to characterize the alteration of whole brain functional network. Compared with the controls, the ALS patients showed significant increase of DC in the left cerebellum posterior lobes, bilateral cerebellum crus, bilateral occipital poles, right orbital frontal lobe, and bilateral prefrontal lobes; significant decrease of DC in the bilateral primary motor cortex, bilateral sensory motor region, right prefrontal lobe, left bilateral precuneus, bilateral lateral temporal lobes, left cingulate cortex, and bilateral visual processing cortex. The DC's z-scores of right inferior occipital gyrus were significant negative correlated with the ALSFRS-r scores. Our findings confirm that the regions with abnormal network DC in ALS patients were located in multiple brain regions including primary motor, somatosensory and extra-motor areas, supporting the concept that ALS is a multisystem disorder. Specifically, our study found that DC in the visual areas was altered and ALS patients with higher DC in right inferior occipital gyrus have more severity of disease. The result demonstrated that the altered DC value in this region can probably be used to assess severity of ALS.
Raberto, Marco; Rapallo, Fabio; Scalas, Enrico
2011-01-01
In this paper, we outline a model of graph (or network) dynamics based on two ingredients. The first ingredient is a Markov chain on the space of possible graphs. The second ingredient is a semi-Markov counting process of renewal type. The model consists in subordinating the Markov chain to the semi-Markov counting process. In simple words, this means that the chain transitions occur at random time instants called epochs. The model is quite rich and its possible connections with algebraic geometry are briefly discussed. Moreover, for the sake of simplicity, we focus on the space of undirected graphs with a fixed number of nodes. However, in an example, we present an interbank market model where it is meaningful to use directed graphs or even weighted graphs. PMID:21887245
Lattices of processes in graphs with inputs
Shakhbazyan, K.V.
1995-09-01
This article is a continuation of others work, presenting a detailed analysis of finite lattices of processes in graphs with input nodes. Lattices of processes in such graphs are studied by representing the lattices in the form of an algebra of pairs. We define the algebra of pairs somewhat generalizing the definition. Let K and D be bounded distributive lattices. A sublattice {delta} {contained_in} K x D is called an algebra of pairs if for all K {element_of} K we have (K, 1{sub D}) {element_of} {delta} and for all d {element_of} D we have (O{sub K}).
Completeness and regularity of generalized fuzzy graphs.
Samanta, Sovan; Sarkar, Biswajit; Shin, Dongmin; Pal, Madhumangal
2016-01-01
Fuzzy graphs are the backbone of many real systems like networks, image, scheduling, etc. But, due to some restriction on edges, fuzzy graphs are limited to represent for some systems. Generalized fuzzy graphs are appropriate to avoid such restrictions. In this study generalized fuzzy graphs are introduced. In this study, matrix representation of generalized fuzzy graphs is described. Completeness and regularity are two important parameters of graph theory. Here, regular and complete generalized fuzzy graphs are introduced. Some properties of them are discussed. After that, effective regular graphs are exemplified.
I Teach Economics, Not Algebra and Calculus
ERIC Educational Resources Information Center
Hey, John D.
2005-01-01
Most people learn to drive without knowing how the engine works. In a similar vein, the author believes that students can learn economics without knowing the algebra and calculus underlying the results. If instructors follow the philosophy of other economics courses in using graphs to illustrate the results, and draw the graphs accurately, then…
Evolutionary stability on graphs
Ohtsuki, Hisashi; Nowak, Martin A.
2008-01-01
Evolutionary stability is a fundamental concept in evolutionary game theory. A strategy is called an evolutionarily stable strategy (ESS), if its monomorphic population rejects the invasion of any other mutant strategy. Recent studies have revealed that population structure can considerably affect evolutionary dynamics. Here we derive the conditions of evolutionary stability for games on graphs. We obtain analytical conditions for regular graphs of degree k > 2. Those theoretical predictions are compared with computer simulations for random regular graphs and for lattices. We study three different update rules: birth-death (BD), death-birth (DB), and imitation (IM) updating. Evolutionary stability on sparse graphs does not imply evolutionary stability in a well-mixed population, nor vice versa. We provide a geometrical interpretation of the ESS condition on graphs. PMID:18295801
Introduction to the Graph Theory and Structural Balance Approaches to International Relations
1971-11-01
suakmrazinSe and ILHtrary, Robert Z Norman, and Dorin Cartwright , Structurc-2 Models: ;n Introduction to the Theory of Directed Gr~h.NwYork: John Wiley...The not-ion was re-worked to apply strictly to inter-actor relations bj Dorwin Cart- uright and Frank Harary in "St ructural Balance: A...Science, 4, 4 (Octo- ber 1959), 316-323; Harary, Norman, and Cartwright , StructuralModels, Chapter 13, pp. 339-362; and Paul 11. Holland Saridel Lein
On computational complexity of Clifford algebra
NASA Astrophysics Data System (ADS)
Budinich, Marco
2009-05-01
After a brief discussion of the computational complexity of Clifford algebras, we present a new basis for even Clifford algebra Cl(2m) that simplifies greatly the actual calculations and, without resorting to the conventional matrix isomorphism formulation, obtains the same complexity. In the last part we apply these results to the Clifford algebra formulation of the NP-complete problem of the maximum clique of a graph introduced by Budinich and Budinich ["A spinorial formulation of the maximum clique problem of a graph," J. Math. Phys. 47, 043502 (2006)].
NASA Astrophysics Data System (ADS)
Foufoula-Georgiou, Efi; Schwenk, Jon; Tejedor, Alejandro
2015-04-01
Are the dynamics of meandering rivers non-linear? What information does the shape of an oxbow lake carry about its forming process? How to characterize self-dissimilar landscapes carrying the signature of larger-scale geologic or tectonic controls? Do we have proper frameworks for quantifying the topology and dynamics of deltaic systems? What can the structural complexity of river networks (erosional and depositional) reveal about their vulnerability and response to change? Can the structure and dynamics of river networks reveal potential hotspots of geomorphic change? All of the above problems are at the heart of understanding landscape evolution, relating process to structure and form, and developing methodologies for inferring how a system might respond to future changes. We argue that a new surge of rigorous methodologies is needed to address these problems. The innovations introduced herein are: (1) gradual wavelet reconstruction for depicting threshold nonlinearity (due to cutoffs) versus inherent nonlinearity (due to underlying dynamics) in river meandering, (2) graph theory for studying the topology and dynamics of deltaic river networks and their response to change, and (3) Lagrangian approaches combined with topology and non-linear dynamics for inferring sediment-driven hotspots of geomorphic change.
Algebraic Activities Aid Discovery Lessons
ERIC Educational Resources Information Center
Wallace-Gomez, Patricia
2013-01-01
After a unit on the rules for positive and negative numbers and the order of operations for evaluating algebraic expressions, many students believe that they understand these principles well enough, but they really do not. They clearly need more practice, but not more of the same kind of drill. Wallace-Gomez provides three graphing activities that…
Spectral fluctuations of quantum graphs
Pluhař, Z.; Weidenmüller, H. A.
2014-10-15
We prove the Bohigas-Giannoni-Schmit conjecture in its most general form for completely connected simple graphs with incommensurate bond lengths. We show that for graphs that are classically mixing (i.e., graphs for which the spectrum of the classical Perron-Frobenius operator possesses a finite gap), the generating functions for all (P,Q) correlation functions for both closed and open graphs coincide (in the limit of infinite graph size) with the corresponding expressions of random-matrix theory, both for orthogonal and for unitary symmetry.
Central extensions of Lax operator algebras
NASA Astrophysics Data System (ADS)
Schlichenmaier, M.; Sheinman, O. K.
2008-08-01
Lax operator algebras were introduced by Krichever and Sheinman as a further development of Krichever's theory of Lax operators on algebraic curves. These are almost-graded Lie algebras of current type. In this paper local cocycles and associated almost-graded central extensions of Lax operator algebras are classified. It is shown that in the case when the corresponding finite-dimensional Lie algebra is simple the two-cohomology space is one-dimensional. An important role is played by the action of the Lie algebra of meromorphic vector fields on the Lax operator algebra via suitable covariant derivatives.
The algebra of physical observables in non-linearly realized gauge theories
NASA Astrophysics Data System (ADS)
Quadri, Andrea
2010-11-01
We classify the physical observables in spontaneously broken non-linearly realized gauge theories in the recently proposed loopwise expansion governed by the Weak Power-Counting (WPC) and the Local Functional Equation. The latter controls the non-trivial quantum deformation of the classical non-linearly realized gauge symmetry, to all orders in the loop expansion. The Batalin-Vilkovisky (BV) formalism is used. We show that the dependence of the vertex functional on the Goldstone fields is obtained via a canonical transformation w.r.t. the BV bracket associated with the BRST symmetry of the model. We also compare the WPC with strict power-counting renormalizability in linearly realized gauge theories. In the case of the electroweak group we find that the tree-level Weinberg relation still holds if power-counting renormalizability is weakened to the WPC condition.
Example of a quantum field theory based on a nonlinear Lie algebra
Schoutens, K. . Inst. for Theoretical Physics); Sevrin, A. ); van Nieuwenhuizen, P. . Theory Div.)
1991-11-01
In this contribution to Tini Veltman's Festschrift we shall give a paedagogical account of our work on a new class of gauge theories called W gravities. They contain higher spin gauge fields, but the usual no-go theorems for interacting field theories with spins exceeding two do not apply since these theories are in two dimensions. It is, of course, well known that ghost-free interacting massless spin 2 fields ( the metric') are gauge fields, and correspond to the geometrical notion of general coordinate transformations in general relativity, but it is yet unknown what extension of these ideas is introduced by the presence of massless higher spin gauge fields. A parallel with supergravity may be drawn: there the presence of massless spin 3/2 fields (gravitinos) corresponds to local fermi-bose symmetries of which these gravitinos are the gauge fields. Their geometrical meaning becomes only clear if one introduces superspace (with bosonic and fermionic coordinates): they correspond to local transformations of the fermionic coordinates. For W gravity one might speculate on a kind of W-superspace with extra bosonic coordinates.
Example of a quantum field theory based on a nonlinear Lie algebra
Schoutens, K.; Sevrin, A.; van Nieuwenhuizen, P.
1991-11-01
In this contribution to Tini Veltman`s Festschrift we shall give a paedagogical account of our work on a new class of gauge theories called W gravities. They contain higher spin gauge fields, but the usual no-go theorems for interacting field theories with spins exceeding two do not apply since these theories are in two dimensions. It is, of course, well known that ghost-free interacting massless spin 2 fields (`the metric`) are gauge fields, and correspond to the geometrical notion of general coordinate transformations in general relativity, but it is yet unknown what extension of these ideas is introduced by the presence of massless higher spin gauge fields. A parallel with supergravity may be drawn: there the presence of massless spin 3/2 fields (gravitinos) corresponds to local fermi-bose symmetries of which these gravitinos are the gauge fields. Their geometrical meaning becomes only clear if one introduces superspace (with bosonic and fermionic coordinates): they correspond to local transformations of the fermionic coordinates. For W gravity one might speculate on a kind of W-superspace with extra bosonic coordinates.
Mears, David; Pollard, Harvey B
2016-06-01
Over the past 15 years, the emerging field of network science has revealed the key features of brain networks, which include small-world topology, the presence of highly connected hubs, and hierarchical modularity. The value of network studies of the brain is underscored by the range of network alterations that have been identified in neurological and psychiatric disorders, including epilepsy, depression, Alzheimer's disease, schizophrenia, and many others. Here we briefly summarize the concepts of graph theory that are used to quantify network properties and describe common experimental approaches for analysis of brain networks of structural and functional connectivity. These range from tract tracing to functional magnetic resonance imaging, diffusion tensor imaging, electroencephalography, and magnetoencephalography. We then summarize the major findings from the application of graph theory to nervous systems ranging from Caenorhabditis elegans to more complex primate brains, including man. Focusing, then, on studies involving the amygdala, a brain region that has attracted intense interest as a center for emotional processing, fear, and motivation, we discuss the features of the amygdala in brain networks for fear conditioning and emotional perception. Finally, to highlight the utility of graph theory for studying dysfunction of the amygdala in mental illness, we review data with regard to changes in the hub properties of the amygdala in brain networks of patients with depression. We suggest that network studies of the human brain may serve to focus attention on regions and connections that act as principal drivers and controllers of brain function in health and disease.
García-Moyano, Antonio; Aguirre, Jacobo; Cruz-Gil, Patricia; Palacín, Arantxa; van Heerden, Esta; Parro, Víctor
2014-01-01
In the South African deep mines, a variety of biofilms growing in mine corridor walls as water seeps from intersections or from fractures represents excellent proxies for deep-subsurface environments. However, they may be greatly affected by the oxygen inputs through the galleries of mining activities. As a consequence, the interaction between the anaerobic water coming out from the walls with the oxygen inputs creates new conditions that support rich microbial communities. The inherent difficulties for sampling these delicate habitats, together with transport and storage conditions may alter the community features and composition. Therefore, the development of in situ monitoring methods would be desirable for quick evaluation of the microbial community. In this work, we report the usefulness of an antibody-microarray (EMChip66) immunoassay for a quick check of the microbial diversity of biofilms located at 1.3 km below surface within the Beatrix deep gold mine (South Africa). In addition, a deconvolution method, previously described and used for environmental monitoring, based on graph theory and applied on antibody cross-reactivity was used to interpret the immunoassay results. The results were corroborated and further expanded by 16S rRNA gene sequencing analysis. Both culture-independent techniques coincided in detecting features related to aerobic sulfur-oxidizers, aerobic chemoorganotrophic Alphaproteobacteria and metanotrophic Gammaproteobacteria. 16S rRNA gene sequencing detected phylotypes related to nitrate-reducers and anaerobic sulfur-oxidizers, whereas the EMChip66 detected immunological features from methanogens and sulfate-reducers. The results reveal a diverse microbial community with syntrophic metabolisms both anaerobic (fermentation, methanogenesis, sulphate and nitrate reduction) and aerobic (methanotrophy, sulphur oxidation). The presence of oxygen-scavenging microbes might indicate that the system is modified by the artificial oxygen inputs
Butler, William E; Atai, Nadia; Carter, Bob; Hochberg, Fred
2014-01-01
The Richard Floor Biorepository supports collaborative studies of extracellular vesicles (EVs) found in human fluids and tissue specimens. The current emphasis is on biomarkers for central nervous system neoplasms but its structure may serve as a template for collaborative EV translational studies in other fields. The informatic system provides specimen inventory tracking with bar codes assigned to specimens and containers and projects, is hosted on globalized cloud computing resources, and embeds a suite of shared documents, calendars, and video-conferencing features. Clinical data are recorded in relation to molecular EV attributes and may be tagged with terms drawn from a network of externally maintained ontologies thus offering expansion of the system as the field matures. We fashioned the graphical user interface (GUI) around a web-based data visualization package. This system is now in an early stage of deployment, mainly focused on specimen tracking and clinical, laboratory, and imaging data capture in support of studies to optimize detection and analysis of brain tumour-specific mutations. It currently includes 4,392 specimens drawn from 611 subjects, the majority with brain tumours. As EV science evolves, we plan biorepository changes which may reflect multi-institutional collaborations, proteomic interfaces, additional biofluids, changes in operating procedures and kits for specimen handling, novel procedures for detection of tumour-specific EVs, and for RNA extraction and changes in the taxonomy of EVs. We have used an ontology-driven data model and web-based architecture with a graph theory-driven GUI to accommodate and stimulate the semantic web of EV science.
Twining characters and orbit Lie algebras
Fuchs, Jurgen; Ray, Urmie; Schellekens, Bert; Schweigert, Christoph
1996-12-05
We associate to outer automorphisms of generalized Kac-Moody algebras generalized character-valued indices, the twining characters. A character formula for twining characters is derived which shows that they coincide with the ordinary characters of some other generalized Kac-Moody algebra, the so-called orbit Lie algebra. Some applications to problems in conformal field theory, algebraic geometry and the theory of sporadic simple groups are sketched.
NASA Astrophysics Data System (ADS)
Blanchard, Philippe; Hellmich, Mario; Ługiewicz, Piotr; Olkiewicz, Robert
Quantum mechanics is the greatest revision of our conception of the character of the physical world since Newton. Consequently, David Hilbert was very interested in quantum mechanics. He and John von Neumann discussed it frequently during von Neumann's residence in Göttingen. He published in 1932 his book Mathematical Foundations of Quantum Mechanics. In Hilbert's opinion it was the first exposition of quantum mechanics in a mathematically rigorous way. The pioneers of quantum mechanics, Heisenberg and Dirac, neither had use for rigorous mathematics nor much interest in it. Conceptually, quantum theory as developed by Bohr and Heisenberg is based on the positivism of Mach as it describes only observable quantities. It first emerged as a result of experimental data in the form of statistical observations of quantum noise, the basic concept of quantum probability.
Guturu, Parthasarathy; Dantu, Ram
2008-06-01
Many graph- and set-theoretic problems, because of their tremendous application potential and theoretical appeal, have been well investigated by the researchers in complexity theory and were found to be NP-hard. Since the combinatorial complexity of these problems does not permit exhaustive searches for optimal solutions, only near-optimal solutions can be explored using either various problem-specific heuristic strategies or metaheuristic global-optimization methods, such as simulated annealing, genetic algorithms, etc. In this paper, we propose a unified evolutionary algorithm (EA) to the problems of maximum clique finding, maximum independent set, minimum vertex cover, subgraph and double subgraph isomorphism, set packing, set partitioning, and set cover. In the proposed approach, we first map these problems onto the maximum clique-finding problem (MCP), which is later solved using an evolutionary strategy. The proposed impatient EA with probabilistic tabu search (IEA-PTS) for the MCP integrates the best features of earlier successful approaches with a number of new heuristics that we developed to yield a performance that advances the state of the art in EAs for the exploration of the maximum cliques in a graph. Results of experimentation with the 37 DIMACS benchmark graphs and comparative analyses with six state-of-the-art algorithms, including two from the smaller EA community and four from the larger metaheuristics community, indicate that the IEA-PTS outperforms the EAs with respect to a Pareto-lexicographic ranking criterion and offers competitive performance on some graph instances when individually compared to the other heuristic algorithms. It has also successfully set a new benchmark on one graph instance. On another benchmark suite called Benchmarks with Hidden Optimal Solutions, IEA-PTS ranks second, after a very recent algorithm called COVER, among its peers that have experimented with this suite.
Hincapié, Doracelly; Ospina, Juan
2011-01-01
Algebraic analysis of social networks exhibited by SARS-Beijing-2003 and AH1N1 flu-México-2009 was realized. The main tools were the Tutte polynomials and Maple package Graph-Theory. The topological structures like graphs and networks were represented by invariant polynomials. The evolution of a given social network was represented like an evolution of the algebraic complexity of the corresponding Tutte polynomial. The reduction of a given social network was described like an involution of the algebraic complexity of the associated Tutte polynomial. The outbreaks of SARS and AH1N1 Flu were considered like represented by a reduction of previously existing contact networks via the control measures executed by health authorities. From Tutte polynomials were derived numerical indicators about efficiency of control measures.
D p-branes, NS5-branes and U-duality from nonabelian (2,0) theory with Lie 3-algebra
NASA Astrophysics Data System (ADS)
Honma, Yoshinori; Ogawa, Morirou; Shiba, Shotaro
2011-04-01
We derive the super Yang-Mills action of D p-branes on a torus T p-4 from the nonabelian (2, 0) theory with Lie 3-algebra [1]. Our realization is based on Lie 3-algebra with pairs of Lorentzian metric generators. The resultant theory then has negative norm modes, but it results in a unitary theory by setting VEV's of these modes. This procedure corresponds to the torus compactification, therefore by taking a transformation which is equivalent to T-duality, the D p-brane action is obtained. We also study type IIA/IIB NS5brane and Kaluza-Klein monopole systems by taking other VEV assignments. Such various compactifications can be realized in the nonabelian (2, 0) theory, since both longitudinal and transverse directions can be compactified, which is different from the BLG theory. We finally discuss U-duality among these branes, and show that most of the moduli parameters in U-duality group are recovered. Especially in D5-brane case, the whole U-duality relation is properly reproduced.
NASA Astrophysics Data System (ADS)
Setare, M. R.; Adami, H.
2016-08-01
The Chern-Simons-like theories of gravity (CSLTG) are formulated at first order formalism. In this formalism, the derivation of the entropy of a black hole on bifurcation surface, as a quasi-local conserved charge is problematic. In this paper we overcome these problems by considering the concept of total variation and the Lorentz-Lie derivative. We firstly find an expression for the ADT conserved current in the context of the CSLTG which is based on the concept of the Killing vector fields. Then, we generalize it to be conserved for all diffeomorphism generators. Thus, we can extract an off-shell conserved charge for any vector field which generates a diffeomorphism. The formalism presented here is based on the concept of quasi-local conserved charges which are off-shell. The charges can be calculated on any codimension two space-like surface surrounding a black hole and the results are independent of the chosen surface. By using the off-shell quasi-local conserved charge, we investigate the Virasoro algebra and find a formula to calculate the central extension term. We apply the formalism to the BTZ black hole solution in the context of the Einstein gravity and the Generalized massive gravity, then we find the eigenvalues of their Virasoro generators as well as the corresponding central charges. Eventually, we calculate the entropy of the BTZ black hole by the Cardy formula and we show that the result exactly matches the one obtained by the concept of the off-shell conserved charges.
Array algebra estimation in signal processing
NASA Astrophysics Data System (ADS)
Rauhala, U. A.
A general theory of linear estimators called array algebra estimation is interpreted in some terms of multidimensional digital signal processing, mathematical statistics, and numerical analysis. The theory has emerged during the past decade from the new field of a unified vector, matrix and tensor algebra called array algebra. The broad concepts of array algebra and its estimation theory cover several modern computerized sciences and technologies converting their established notations and terminology into one common language. Some concepts of digital signal processing are adopted into this language after a review of the principles of array algebra estimation and its predecessors in mathematical surveying sciences.
ERIC Educational Resources Information Center
Connery, Keely Flynn
2007-01-01
Graphing predictions is especially important in classes where relationships between variables need to be explored and derived. In this article, the author describes how his students sketch the graphs of their predictions before they begin their investigations on two laboratory activities: Distance Versus Time Cart Race Lab and Resistance; and…
Clifford Algebras and Their Decomposition into Conjugate Fermionic Heisenberg Algebras
NASA Astrophysics Data System (ADS)
Catto, Sultan; Gürcan, Yasemin; Khalfan, Amish; Kurt, Levent; Kato La, V.
2016-10-01
We discuss a construction scheme for Clifford numbers of arbitrary dimension. The scheme is based upon performing direct products of the Pauli spin and identity matrices. Conjugate fermionic algebras can then be formed by considering linear combinations of the Clifford numbers and the Hermitian conjugates of such combinations. Fermionic algebras are important in investigating systems that follow Fermi-Dirac statistics. We will further comment on the applications of Clifford algebras to Fueter analyticity, twistors, color algebras, M-theory and Leech lattice as well as unification of ancient and modern geometries through them.
Algebraic approach to small-world network models
NASA Astrophysics Data System (ADS)
Rudolph-Lilith, Michelle; Muller, Lyle E.
2014-01-01
We introduce an analytic model for directed Watts-Strogatz small-world graphs and deduce an algebraic expression of its defining adjacency matrix. The latter is then used to calculate the small-world digraph's asymmetry index and clustering coefficient in an analytically exact fashion, valid nonasymptotically for all graph sizes. The proposed approach is general and can be applied to all algebraically well-defined graph-theoretical measures, thus allowing for an analytical investigation of finite-size small-world graphs.
Relativity on Rotated Graph Paper
NASA Astrophysics Data System (ADS)
Salgado, Roberto
2011-11-01
We present visual calculations in special relativity using spacetime diagrams drawn on graph paper that has been rotated by 45 degrees. The rotated lines represent lightlike directions in Minkowski spacetime, and the boxes in the grid (called light-clock diamonds) represent ticks of an inertial observer's lightclock. We show that many quantitative results can be read off a spacetime diagram by counting boxes, using a minimal amount of algebra.
Scenario Graphs and Attack Graphs
2004-04-14
46 6.1 Vulnerability Analysis of a Network . . . . . . . . . . . . . . . . . . . . . . . . . 53 6.2 Sandia Red Team Attack Graph...asymptotic bound. The test machine was a 1Ghz Pentium III with 1GB of RAM, running Red Hat Linux 7.3. Figure 4.1(a) plots running time of the implemen...host scanning tools network information vulnerability Attack Graph network Red
Arnaud-Haond, Sophie; Moalic, Yann; Barnabé, Christian; Ayala, Francisco José; Tibayrenc, Michel
2014-01-01
Micropathogens (viruses, bacteria, fungi, parasitic protozoa) share a common trait, which is partial clonality, with wide variance in the respective influence of clonality and sexual recombination on the dynamics and evolution of taxa. The discrimination of distinct lineages and the reconstruction of their phylogenetic history are key information to infer their biomedical properties. However, the phylogenetic picture is often clouded by occasional events of recombination across divergent lineages, limiting the relevance of classical phylogenetic analysis and dichotomic trees. We have applied a network analysis based on graph theory to illustrate the relationships among genotypes of Trypanosoma cruzi, the parasitic protozoan responsible for Chagas disease, to identify major lineages and to unravel their past history of divergence and possible recombination events. At the scale of T. cruzi subspecific diversity, graph theory-based networks applied to 22 isoenzyme loci (262 distinct Multi-Locus-Enzyme-Electrophoresis -MLEE) and 19 microsatellite loci (66 Multi-Locus-Genotypes -MLG) fully confirms the high clustering of genotypes into major lineages or "near-clades". The release of the dichotomic constraint associated with phylogenetic reconstruction usually applied to Multilocus data allows identifying putative hybrids and their parental lineages. Reticulate topology suggests a slightly different history for some of the main "near-clades", and a possibly more complex origin for the putative hybrids than hitherto proposed. Finally the sub-network of the near-clade T. cruzi I (28 MLG) shows a clustering subdivision into three differentiated lesser near-clades ("Russian doll pattern"), which confirms the hypothesis recently proposed by other investigators. The present study broadens and clarifies the hypotheses previously obtained from classical markers on the same sets of data, which demonstrates the added value of this approach. This underlines the potential of graph
Cyclic homology for Hom-associative algebras
NASA Astrophysics Data System (ADS)
Hassanzadeh, Mohammad; Shapiro, Ilya; Sütlü, Serkan
2015-12-01
In the present paper we investigate the noncommutative geometry of a class of algebras, called the Hom-associative algebras, whose associativity is twisted by a homomorphism. We define the Hochschild, cyclic, and periodic cyclic homology and cohomology for this class of algebras generalizing these theories from the associative to the Hom-associative setting.
Li, Jing; Hong, Wenxue
2014-12-01
The feature extraction and feature selection are the important issues in pattern recognition. Based on the geometric algebra representation of vector, a new feature extraction method using blade coefficient of geometric algebra was proposed in this study. At the same time, an improved differential evolution (DE) feature selection method was proposed to solve the elevated high dimension issue. The simple linear discriminant analysis was used as the classifier. The result of the 10-fold cross-validation (10 CV) classification of public breast cancer biomedical dataset was more than 96% and proved superior to that of the original features and traditional feature extraction method.
NASA Astrophysics Data System (ADS)
Reinisch, Elena C.; Cardiff, Michael; Feigl, Kurt L.
2017-01-01
Graph theory is useful for analyzing time-dependent model parameters estimated from interferometric synthetic aperture radar (InSAR) data in the temporal domain. Plotting acquisition dates (epochs) as vertices and pair-wise interferometric combinations as edges defines an incidence graph. The edge-vertex incidence matrix and the normalized edge Laplacian matrix are factors in the covariance matrix for the pair-wise data. Using empirical measures of residual scatter in the pair-wise observations, we estimate the relative variance at each epoch by inverting the covariance of the pair-wise data. We evaluate the rank deficiency of the corresponding least-squares problem via the edge-vertex incidence matrix. We implement our method in a MATLAB software package called GraphTreeTA available on GitHub (https://github.com/feigl/gipht). We apply temporal adjustment to the data set described in Lu et al. (Geophys Res Solid Earth 110, 2005) at Okmok volcano, Alaska, which erupted most recently in 1997 and 2008. The data set contains 44 differential volumetric changes and uncertainties estimated from interferograms between 1997 and 2004. Estimates show that approximately half of the magma volume lost during the 1997 eruption was recovered by the summer of 2003. Between June 2002 and September 2003, the estimated rate of volumetric increase is (6.2 ± 0.6) × 10^6 m^3/year . Our preferred model provides a reasonable fit that is compatible with viscoelastic relaxation in the five years following the 1997 eruption. Although we demonstrate the approach using volumetric rates of change, our formulation in terms of incidence graphs applies to any quantity derived from pair-wise differences, such as range change, range gradient, or atmospheric delay.
ERIC Educational Resources Information Center
Cavanagh, Sean
2009-01-01
As educators and policymakers search for ways to prepare students for the rigors of algebra, teachers in the Helena, Montana, school system are starting early by attempting to nurture students' algebraic-reasoning ability, as well as their basic number skills, in early elementary school, rather than waiting until middle or early high school.…
Algebraic mesh quality metrics
KNUPP,PATRICK
2000-04-24
Quality metrics for structured and unstructured mesh generation are placed within an algebraic framework to form a mathematical theory of mesh quality metrics. The theory, based on the Jacobian and related matrices, provides a means of constructing, classifying, and evaluating mesh quality metrics. The Jacobian matrix is factored into geometrically meaningful parts. A nodally-invariant Jacobian matrix can be defined for simplicial elements using a weight matrix derived from the Jacobian matrix of an ideal reference element. Scale and orientation-invariant algebraic mesh quality metrics are defined. the singular value decomposition is used to study relationships between metrics. Equivalence of the element condition number and mean ratio metrics is proved. Condition number is shown to measure the distance of an element to the set of degenerate elements. Algebraic measures for skew, length ratio, shape, volume, and orientation are defined abstractly, with specific examples given. Combined metrics for shape and volume, shape-volume-orientation are algebraically defined and examples of such metrics are given. Algebraic mesh quality metrics are extended to non-simplical elements. A series of numerical tests verify the theoretical properties of the metrics defined.
ERIC Educational Resources Information Center
Hagerty, Gary; Smith, Stanley; Goodwin, Danielle
2010-01-01
In 2001, Black Hills State University (BHSU) redesigned college algebra to use the computer-based mastery learning program, Assessment and Learning in Knowledge Spaces [1], historical development of concepts modules, whole class discussions, cooperative activities, relevant applications problems, and many fewer lectures. This resulted in a 21%…
Glassman, Robert B
2003-04-15
Cognitive experimentation suggests that at any single instant only three or four items ("chunks") are simultaneously prominent as a working memory (WM) trace, if we disregard the rehearsal component of WM. The reason for small WM capacity may concern combinatorial manageability. How might the neural representations of these few coactive chunks occupy a spatially distributed set of areas of the sheet-like cortex, while providing both order and flexibility to associate items in WM? Each attribute of each simultaneously active WM item must have broad access to the representational facilities of the cortical sheet, comprising tens of thousands of modular "cortical columns." The two hypothesized neural levels of WM during any moment of cognition comprise (a) "binding" together of many distributed attribute representations within each respective WM chunk, and (b) combinatorial play among three or four WM chunk-representations. Anatomical and functional evidence of cortical unity through its depth suggests that cortex may be viewed as essentially planar in its distribution of activations. Thus, a moment's WM is hypothesized here to reside in myriad activated cortical planar "patches," each subdivided into up to four amoeboid "subpatches." Two different lines of topological reasoning suggest orderly associations of such representations. (1) The four-color principle of map topology, and the related K(4) is planar theorem of graph theory, imply that if a small cortical area is dynamically subdivided into no more than four, discretely bounded planar subareas, then each such segment has ample free access to each of the others. (2) A hypothetical alternative to such associative adjacency of simultaneously active cortical representations of chunk-attributes is associative overlap, whereby, in dense cortical neuropil, activated subpatches behave like Venn diagrams of intersecting sets. As the number of Venn-like coactive subpatches within a patch increases, maintaining ad hoc
Quraan, Maher A; McCormick, Cornelia; Cohn, Melanie; Valiante, Taufik A; McAndrews, Mary Pat
2013-01-01
Despite a wealth of EEG epilepsy data that accumulated for over half a century, our ability to understand brain dynamics associated with epilepsy remains limited. Using EEG data from 15 controls and 9 left temporal lobe epilepsy (LTLE) patients, in this study we characterize how the dynamics of the healthy brain differ from the "dynamically balanced" state of the brain of epilepsy patients treated with anti-epileptic drugs in the context of resting state. We show that such differences can be observed in band power, synchronization and network measures, as well as deviations from the small world network (SWN) architecture of the healthy brain. The θ (4-7 Hz) and high α (10-13 Hz) bands showed the biggest deviations from healthy controls across various measures. In particular, patients demonstrated significantly higher power and synchronization than controls in the θ band, but lower synchronization and power in the high α band. Furthermore, differences between controls and patients in graph theory metrics revealed deviations from a SWN architecture. In the θ band epilepsy patients showed deviations toward an orderly network, while in the high α band they deviated toward a random network. These findings show that, despite the focal nature of LTLE, the epileptic brain differs in its global network characteristics from the healthy brain. To our knowledge, this is the only study to encompass power, connectivity and graph theory metrics to investigate the reorganization of resting state functional networks in LTLE patients.
Quraan, Maher A.; McCormick, Cornelia; Cohn, Melanie; Valiante, Taufik A.; McAndrews, Mary Pat
2013-01-01
Despite a wealth of EEG epilepsy data that accumulated for over half a century, our ability to understand brain dynamics associated with epilepsy remains limited. Using EEG data from 15 controls and 9 left temporal lobe epilepsy (LTLE) patients, in this study we characterize how the dynamics of the healthy brain differ from the “dynamically balanced” state of the brain of epilepsy patients treated with anti-epileptic drugs in the context of resting state. We show that such differences can be observed in band power, synchronization and network measures, as well as deviations from the small world network (SWN) architecture of the healthy brain. The θ (4–7 Hz) and high α (10–13 Hz) bands showed the biggest deviations from healthy controls across various measures. In particular, patients demonstrated significantly higher power and synchronization than controls in the θ band, but lower synchronization and power in the high α band. Furthermore, differences between controls and patients in graph theory metrics revealed deviations from a SWN architecture. In the θ band epilepsy patients showed deviations toward an orderly network, while in the high α band they deviated toward a random network. These findings show that, despite the focal nature of LTLE, the epileptic brain differs in its global network characteristics from the healthy brain. To our knowledge, this is the only study to encompass power, connectivity and graph theory metrics to investigate the reorganization of resting state functional networks in LTLE patients. PMID:23922658
Jucys-Murphy elements for Birman-Murakami-Wenzl algebras
NASA Astrophysics Data System (ADS)
Isaev, A. P.; Ogievetsky, O. V.
2011-05-01
The Burman-Wenzl-Murakami algebra, considered as the quotient of the braid group algebra, possesses the commutative set of Jucys-Murphy elements. We show that the set of Jucys-Murphy elements is maximal commutative for the generic Birman-Wenzl-Murakami algebra and reconstruct the representation theory of the tower of Birman-Wenzl-Murakami algebras.
Constructing Graphs over with Small Prescribed Mean-Curvature
NASA Astrophysics Data System (ADS)
Carley, Holly; Kiessling, Michael K.-H.
2015-12-01
In this paper nonlinear Hodge theory and Banach algebra estimates are employed to construct a convergent series expansion which solves the prescribed mean curvature equation for n-dimensional hypersurfaces in (+ sign) and (- sign) which are graphs of a smooth function , and whose mean curvature function H is α-Hölder continuous and integrable, with small norm. The radius of convergence is estimated explicitly from below. Our approach is inspired by, and applied to, the Maxwell-Born-Infeld theory of electromagnetism in , for which our method yields the first systematic way of explicitly computing the electrostatic potential for regular charge densities and small Born parameter, with explicit error estimates at any order of truncation of the series. In particular, our results level the ground for a controlled computation of Born-Infeld effects on the Hydrogen spectrum.
A Graph Syntax for Processes and Services
NASA Astrophysics Data System (ADS)
Bruni, Roberto; Gadducci, Fabio; Lafuente, Alberto Lluch
We propose a class of hierarchical graphs equipped with a simple algebraic syntax as a convenient way to describe configurations in languages with inherently hierarchical features such as sessions, fault- handling scopes or transactions. The graph syntax can be seen as an intermediate representation language, that facilitates the encoding of structured specifications and, in particular, of process calculi, since it provides primitives for nesting, name restriction and parallel composition. The syntax is based on an algebraic presentation that faithfully characterises families of hierarchical graphs, meaning that each term of the language uniquely identifies an equivalence class of graphs (modulo graph isomorphism). Proving soundness and completeness of an encoding (i.e. proving that structurally equivalent processes are mapped to isomorphic graphs) is then facilitated and can be done by structural induction. Summing up, the graph syntax facilitates the definition of faithful encodings, yet allowing a precise visual representation. We illustrate our work with an application to a workflow language and a service-oriented calculus.
Graph Mining Meets the Semantic Web
Lee, Sangkeun; Sukumar, Sreenivas R; Lim, Seung-Hwan
2015-01-01
The Resource Description Framework (RDF) and SPARQL Protocol and RDF Query Language (SPARQL) were introduced about a decade ago to enable flexible schema-free data interchange on the Semantic Web. Today, data scientists use the framework as a scalable graph representation for integrating, querying, exploring and analyzing data sets hosted at different sources. With increasing adoption, the need for graph mining capabilities for the Semantic Web has emerged. We address that need through implementation of three popular iterative Graph Mining algorithms (Triangle count, Connected component analysis, and PageRank). We implement these algorithms as SPARQL queries, wrapped within Python scripts. We evaluate the performance of our implementation on 6 real world data sets and show graph mining algorithms (that have a linear-algebra formulation) can indeed be unleashed on data represented as RDF graphs using the SPARQL query interface.
NASA Astrophysics Data System (ADS)
Durka, R.
2017-04-01
The S-expansion framework is analyzed in the context of a freedom in closing the multiplication tables for the abelian semigroups. Including the possibility of the zero element in the resonant decomposition, and associating the Lorentz generator with the semigroup identity element, leads to a wide class of the expanded Lie algebras introducing interesting modifications to the gauge gravity theories. Among the results, we find all the Maxwell algebras of type {{B}m} , {{C}m} , and the recently introduced {{D}m} . The additional new examples complete the resulting generalization of the bosonic enlargements for an arbitrary number of the Lorentz-like and translational-like generators. Some further prospects concerning enlarging the algebras are discussed, along with providing all the necessary constituents for constructing the gravity actions based on the obtained results.
Linear game non-contextuality and Bell inequalities—a graph-theoretic approach
NASA Astrophysics Data System (ADS)
Rosicka, M.; Ramanathan, R.; Gnaciński, P.; Horodecki, K.; Horodecki, M.; Horodecki, P.; Severini, S.
2016-04-01
We study the classical and quantum values of a class of one- and two-party unique games, that generalizes the well-known XOR games to the case of non-binary outcomes. In the bipartite case the generalized XOR (XOR-d) games we study are a subclass of the well-known linear games. We introduce a ‘constraint graph’ associated to such a game, with the constraints defining the game represented by an edge-coloring of the graph. We use the graph-theoretic characterization to relate the task of finding equivalent games to the notion of signed graphs and switching equivalence from graph theory. We relate the problem of computing the classical value of single-party anti-correlation XOR games to finding the edge bipartization number of a graph, which is known to be MaxSNP hard, and connect the computation of the classical value of XOR-d games to the identification of specific cycles in the graph. We construct an orthogonality graph of the game from the constraint graph and study its Lovász theta number as a general upper bound on the quantum value even in the case of single-party contextual XOR-d games. XOR-d games possess appealing properties for use in device-independent applications such as randomness of the local correlated outcomes in the optimal quantum strategy. We study the possibility of obtaining quantum algebraic violation of these games, and show that no finite XOR-d game possesses the property of pseudo-telepathy leaving the frequently used chained Bell inequalities as the natural candidates for such applications. We also show this lack of pseudo-telepathy for multi-party XOR-type inequalities involving two-body correlation functions.
Test of Graphing and Graph Interpretation Skills.
ERIC Educational Resources Information Center
Hermann, J.
This monograph is a test of graphing and graph interpretation skills which assesses performance on all the skills of graphing which are contained in the AAAS program, Science - A Process Approach. The testing includes construction of bar graphs, interpreting information on graphs, the use of the Cartesian coordinate system, making predictions from…
NASA Astrophysics Data System (ADS)
Prager, Stefan; Zech, Alexander; Aquilante, Francesco; Dreuw, Andreas; Wesolowski, Tomasz A.
2016-05-01
The combination of Frozen Density Embedding Theory (FDET) and the Algebraic Diagrammatic Construction (ADC) scheme for the polarization propagator for describing environmental effects on electronically excited states is presented. Two different ways of interfacing and expressing the so-called embedding operator are introduced. The resulting excited states are compared with supermolecular calculations of the total system at the ADC(2) level of theory. Molecular test systems were chosen to investigate molecule-environment interactions of varying strength from dispersion interaction up to multiple hydrogen bonds. The overall difference between the supermolecular and the FDE-ADC calculations in excitation energies is lower than 0.09 eV (max) and 0.032 eV in average, which is well below the intrinsic error of the ADC(2) method itself.
Maheswaran, Ravi; Craigs, Cheryl; Read, Simon; Bath, Peter A; Willett, Peter
2009-01-01
Background Graph theoretical methods are extensively used in the field of computational chemistry to search datasets of compounds to see if they contain particular molecular sub-structures or patterns. We describe a preliminary application of a graph theoretical method, developed in computational chemistry, to geographical epidemiology in relation to testing a prior hypothesis. We tested the methodology on the hypothesis that if a socioeconomically deprived neighbourhood is situated in a wider deprived area, then that neighbourhood would experience greater adverse effects on mortality compared with a similarly deprived neighbourhood which is situated in a wider area with generally less deprivation. Methods We used the Trent Region Health Authority area for this study, which contained 10,665 census enumeration districts (CED). Graphs are mathematical representations of objects and their relationships and within the context of this study, nodes represented CEDs and edges were determined by whether or not CEDs were neighbours (shared a common boundary). The overall area in this study was represented by one large graph comprising all CEDs in the region, along with their adjacency information. We used mortality data from 1988–1998, CED level population estimates and the Townsend Material Deprivation Index as an indicator of neighbourhood level deprivation. We defined deprived CEDs as those in the top 20% most deprived in the Region. We then set out to classify these deprived CEDs into seven groups defined by increasing deprivation levels in the neighbouring CEDs. 506 (24.2%) of the deprived CEDs had five adjacent CEDs and we limited pattern development and searching to these CEDs. We developed seven query patterns and used the RASCAL (Rapid Similarity Calculator) program to carry out the search for each of the query patterns. This program used a maximum common subgraph isomorphism method which was modified to handle geographical data. Results Of the 506 deprived CEDs
Bertti, Poliana; Tejada, Julian; Martins, Ana Paula Pinheiro; Dal-Cól, Maria Luiza Cleto; Terra, Vera Cristina; de Oliveira, José Antônio Cortes; Velasco, Tonicarlo Rodrigues; Sakamoto, Américo Ceiki; Garcia-Cairasco, Norberto
2014-09-01
Epileptic syndromes and seizures are the expression of complex brain systems. Because no analysis of complexity has been applied to epileptic seizure semiology, our goal was to apply neuroethology and graph analysis to the study of the complexity of behavioral manifestations of epileptic seizures in human frontal lobe epilepsy (FLE) and temporal lobe epilepsy (TLE). We analyzed the video recordings of 120 seizures of 18 patients with FLE and 28 seizures of 28 patients with TLE. All patients were seizure-free >1 year after surgery (Engel Class I). All patients' behavioral sequences were analyzed by means of a glossary containing all behaviors and analyzed for neuroethology (Ethomatic software). The same series were used for graph analysis (CYTOSCAPE). Behaviors, displayed as nodes, were connected by edges to other nodes according to their temporal sequence of appearance. Using neuroethology analysis, we confirmed data in the literature such as in FLE: brief/frequent seizures, complex motor behaviors, head and eye version, unilateral/bilateral tonic posturing, speech arrest, vocalization, and rapid postictal recovery and in the case of TLE: presence of epigastric aura, lateralized dystonias, impairment of consciousness/speech during ictal and postictal periods, and development of secondary generalization. Using graph analysis metrics of FLE and TLE confirmed data from flowcharts. However, because of the algorithms we used, they highlighted more powerfully the connectivity and complex associations among behaviors in a quite selective manner, depending on the origin of the seizures. The algorithms we used are commonly employed to track brain connectivity from EEG and MRI sources, which makes our study very promising for future studies of complexity in this field.
NASA Astrophysics Data System (ADS)
Beeken, Paul
2014-11-01
Graphing is an essential skill that forms the foundation of any physical science.1 Understanding the relationships between measurements ultimately determines which modeling equations are successful in predicting observations.2 Over the years, science and math teachers have approached teaching this skill with a variety of techniques. For secondary school instruction, the job of graphing skills falls heavily on physics teachers. By virtue of the nature of the topics we cover, it is our mission to develop this skill to the fine art that it is.
Phillips, David J; McGlaughlin, Alec; Ruth, David; Jager, Leah R; Soldan, Anja
2015-01-01
Graph theory is increasingly being used to study brain connectivity across the spectrum of Alzheimer's disease (AD), but prior findings have been inconsistent, likely reflecting methodological differences. We systematically investigated how methods of graph creation (i.e., type of correlation matrix and edge weighting) affect structural network properties and group differences. We estimated the structural connectivity of brain networks based on correlation maps of cortical thickness obtained from MRI. Four groups were compared: 126 cognitively normal older adults, 103 individuals with Mild Cognitive Impairment (MCI) who retained MCI status for at least 3 years (stable MCI), 108 individuals with MCI who progressed to AD-dementia within 3 years (progressive MCI), and 105 individuals with AD-dementia. Small-world measures of connectivity (characteristic path length and clustering coefficient) differed across groups, consistent with prior studies. Groups were best discriminated by the Randić index, which measures the degree to which highly connected nodes connect to other highly connected nodes. The Randić index differentiated the stable and progressive MCI groups, suggesting that it might be useful for tracking and predicting the progression of AD. Notably, however, the magnitude and direction of group differences in all three measures were dependent on the method of graph creation, indicating that it is crucial to take into account how graphs are constructed when interpreting differences across diagnostic groups and studies. The algebraic connectivity measures showed few group differences, independent of the method of graph construction, suggesting that global connectivity as it relates to node degree is not altered in early AD.
Gusfield, Dan
2010-03-01
The Multi-State Perfect Phylogeny Problem is an extension of the Binary Perfect Phylogeny Problem, allowing characters to take on more than two states. In this article, we consider three problems that extend the utility of the multi-state perfect phylogeny model: (1) the Missing Data (MD) Problem, where some entries in the input are missing and the question is whether (bounded) values for the missing data can be imputed so that the resulting data has a multi-state perfect phylogeny; (2) the Character-Removal (CR) Problem, where we want to minimize the number of characters to remove from the data so that the resulting data has a multi-state perfect phylogeny; and (3) the Missing-Data Character-Removal (MDCR) Problem, where the input has missing data and we want to impute values for the missing data to minimize the solution to the resulting Character-Removal Problem. We discuss Integer Linear Programming (ILP) solutions to these problems for the special case of three, four, and five permitted states per character, and we report on extensive empirical testing of these solutions. Then we develop a general theory to solve the MD problem for an arbitrary number of permitted states, using chordal graph theory and results on minimal triangulation of non-chordal graphs. This establishes new necessary and sufficient conditions for the existence of a perfect phylogeny with (or without) missing data. We implement the general theory using integer linear programming, although other optimization methods are possible. We extensively explore the empirical behavior of the general solution, showing that the methods are very practical for data of size and complexity that is characteristic of many current applications in phylogenetics. Some of the empirical results for the MD problem with an arbitrary number of permitted states are very surprising, suggesting the existence of additional combinatorial structure in multi-state perfect phylogenies. Finally, we note some relationships
Vanicek, Thomas; Hahn, Andreas; Traub-Weidinger, Tatjana; Hilger, Eva; Spies, Marie; Wadsak, Wolfgang; Lanzenberger, Rupert; Pataraia, Ekaterina; Asenbaum-Nan, Susanne
2016-01-01
The human brain exhibits marked hemispheric differences, though it is not fully understood to what extent lateralization of the epileptic focus is relevant. Preoperative [18F]FDG-PET depicts lateralization of seizure focus in patients with temporal lobe epilepsy and reveals dysfunctional metabolic brain connectivity. The aim of the present study was to compare metabolic connectivity, inferred from inter-regional [18F]FDG PET uptake correlations, in right-sided (RTLE; n = 30) and left-sided TLE (LTLE; n = 32) with healthy controls (HC; n = 31) using graph theory based network analysis. Comparing LTLE and RTLE and patient groups separately to HC, we observed higher lobar connectivity weights in RTLE compared to LTLE for connections of the temporal and the parietal lobe of the contralateral hemisphere (CH). Moreover, especially in RTLE compared to LTLE higher local efficiency were found in the temporal cortices and other brain regions of the CH. The results of this investigation implicate altered metabolic networks in patients with TLE specific to the lateralization of seizure focus, and describe compensatory mechanisms especially in the CH of patients with RTLE. We propose that graph theoretical analysis of metabolic connectivity using [18F]FDG-PET offers an important additional modality to explore brain networks. PMID:27349503
Vanicek, Thomas; Hahn, Andreas; Traub-Weidinger, Tatjana; Hilger, Eva; Spies, Marie; Wadsak, Wolfgang; Lanzenberger, Rupert; Pataraia, Ekaterina; Asenbaum-Nan, Susanne
2016-06-28
The human brain exhibits marked hemispheric differences, though it is not fully understood to what extent lateralization of the epileptic focus is relevant. Preoperative [(18)F]FDG-PET depicts lateralization of seizure focus in patients with temporal lobe epilepsy and reveals dysfunctional metabolic brain connectivity. The aim of the present study was to compare metabolic connectivity, inferred from inter-regional [(18)F]FDG PET uptake correlations, in right-sided (RTLE; n = 30) and left-sided TLE (LTLE; n = 32) with healthy controls (HC; n = 31) using graph theory based network analysis. Comparing LTLE and RTLE and patient groups separately to HC, we observed higher lobar connectivity weights in RTLE compared to LTLE for connections of the temporal and the parietal lobe of the contralateral hemisphere (CH). Moreover, especially in RTLE compared to LTLE higher local efficiency were found in the temporal cortices and other brain regions of the CH. The results of this investigation implicate altered metabolic networks in patients with TLE specific to the lateralization of seizure focus, and describe compensatory mechanisms especially in the CH of patients with RTLE. We propose that graph theoretical analysis of metabolic connectivity using [(18)F]FDG-PET offers an important additional modality to explore brain networks.
Zhang, Feng; Liao, Xiangke; Peng, Shaoliang; Cui, Yingbo; Wang, Bingqiang; Zhu, Xiaoqian; Liu, Jie
2016-06-01
' The de novo assembly of DNA sequences is increasingly important for biological researches in the genomic era. After more than one decade since the Human Genome Project, some challenges still exist and new solutions are being explored to improve de novo assembly of genomes. String graph assembler (SGA), based on the string graph theory, is a new method/tool developed to address the challenges. In this paper, based on an in-depth analysis of SGA we prove that the SGA-based sequence de novo assembly is an NP-complete problem. According to our analysis, SGA outperforms other similar methods/tools in memory consumption, but costs much more time, of which 60-70 % is spent on the index construction. Upon this analysis, we introduce a hybrid parallel optimization algorithm and implement this algorithm in the TianHe-2's parallel framework. Simulations are performed with different datasets. For data of small size the optimized solution is 3.06 times faster than before, and for data of middle size it's 1.60 times. The results demonstrate an evident performance improvement, with the linear scalability for parallel FM-index construction. This results thus contribute significantly to improving the efficiency of de novo assembly of DNA sequences.
ERIC Educational Resources Information Center
Beeken, Paul
2014-01-01
Graphing is an essential skill that forms the foundation of any physical science. Understanding the relationships between measurements ultimately determines which modeling equations are successful in predicting observations. Over the years, science and math teachers have approached teaching this skill with a variety of techniques. For secondary…
Sukumar, Sreenivas R.; Hong, Seokyong; Lee, Sangkeun; Lim, Seung-Hwan
2016-06-01
GraphBench is a benchmark suite for graph pattern mining and graph analysis systems. The benchmark suite is a significant addition to conducting apples-apples comparison of graph analysis software (databases, in-memory tools, triple stores, etc.)
ERIC Educational Resources Information Center
Xi, Xiaoming
2010-01-01
Motivated by cognitive theories of graph comprehension, this study systematically manipulated characteristics of a line graph description task in a speaking test in ways to mitigate the influence of graph familiarity, a potential source of construct-irrelevant variance. It extends Xi (2005), which found that the differences in holistic scores on…
Learning Activity Package, Algebra-Trigonometry.
ERIC Educational Resources Information Center
Holland, Bill
A series of ten teacher-prepared Learning Activity Packages (LAPs) in advanced algebra and trigonometry, the units cover logic; absolute value, inequalities, exponents, and complex numbers; functions; higher degree equations and the derivative; the trigonometric function; graphs and applications of the trigonometric functions; sequences and…
NASA Technical Reports Server (NTRS)
Mulligan, Jeffrey B.
2017-01-01
A color algebra refers to a system for computing sums and products of colors, analogous to additive and subtractive color mixtures. We would like it to match the well-defined algebra of spectral functions describing lights and surface reflectances, but an exact correspondence is impossible after the spectra have been projected to a three-dimensional color space, because of metamerism physically different spectra can produce the same color sensation. Metameric spectra are interchangeable for the purposes of addition, but not multiplication, so any color algebra is necessarily an approximation to physical reality. Nevertheless, because the majority of naturally-occurring spectra are well-behaved (e.g., continuous and slowly-varying), color algebras can be formulated that are largely accurate and agree well with human intuition. Here we explore the family of algebras that result from associating each color with a member of a three-dimensional manifold of spectra. This association can be used to construct a color product, defined as the color of the spectrum of the wavelength-wise product of the spectra associated with the two input colors. The choice of the spectral manifold determines the behavior of the resulting system, and certain special subspaces allow computational efficiencies. The resulting systems can be used to improve computer graphic rendering techniques, and to model various perceptual phenomena such as color constancy.
On N = 2 compactifications of M-theory to AdS{sub 3} using geometric algebra techniques
Babalic, E. M.; Coman, I. A.; Condeescu, C.; Micu, A.; Lazaroiu, C. I.
2013-11-13
We investigate the most general warped compactification of eleven-dimensional supergravity on eight-dimensional manifolds to AdS{sub 3} spaces (in the presence of non-vanishing four-form flux) which preserves N = 2 supersymmetry in three dimensions. Without imposing any restrictions on the chirality of the internal part of the supersymmetry generators, we use geometric algebra techniques to study some implications of the supersymmetry constraints. In particular, we discuss the Lie bracket of certain vector fields constructed as pinor bilinears on the compactification manifold.
NASA Astrophysics Data System (ADS)
Liu, Cheng-Wei; Polkovnikov, Anatoli; Sandvik, Anders W.
2015-04-01
We discuss an Ising spin glass where each S =1 /2 spin is coupled antiferromagnetically to three other spins (3-regular graphs). Inducing quantum fluctuations by a time-dependent transverse field, we use out-of-equilibrium quantum Monte Carlo simulations to study dynamic scaling at the quantum glass transition. Comparing the dynamic exponent and other critical exponents with those of the classical (temperature-driven) transition, we conclude that quantum annealing is less efficient than classical simulated annealing in bringing the system into the glass phase. Quantum computing based on the quantum annealing paradigm is therefore inferior to classical simulated annealing for this class of problems. We also comment on previous simulations where a parameter is changed with the simulation time, which is very different from the true Hamiltonian dynamics simulated here.
Liu, Cheng-Wei; Polkovnikov, Anatoli; Sandvik, Anders W
2015-04-10
We discuss an Ising spin glass where each S=1/2 spin is coupled antiferromagnetically to three other spins (3-regular graphs). Inducing quantum fluctuations by a time-dependent transverse field, we use out-of-equilibrium quantum Monte Carlo simulations to study dynamic scaling at the quantum glass transition. Comparing the dynamic exponent and other critical exponents with those of the classical (temperature-driven) transition, we conclude that quantum annealing is less efficient than classical simulated annealing in bringing the system into the glass phase. Quantum computing based on the quantum annealing paradigm is therefore inferior to classical simulated annealing for this class of problems. We also comment on previous simulations where a parameter is changed with the simulation time, which is very different from the true Hamiltonian dynamics simulated here.
NASA Astrophysics Data System (ADS)
Warchalowski, Wiktor; Krawczyk, Malgorzata J.
2017-03-01
We found the Lindenmayer systems for line graphs built on selected fractals. We show that the fractal dimension of such obtained graphs in all analysed cases is the same as for their original graphs. Both for the original graphs and for their line graphs we identified classes of nodes which reflect symmetry of the graph.
Proving relations between modular graph functions
NASA Astrophysics Data System (ADS)
Basu, Anirban
2016-12-01
We consider modular graph functions that arise in the low energy expansion of the four graviton amplitude in type II string theory. The vertices of these graphs are the positions of insertions of vertex operators on the toroidal worldsheet, while the links are the scalar Green functions connecting the vertices. Graphs with four and five links satisfy several non-trivial relations, which have been proved recently. We prove these relations by using elementary properties of Green functions and the details of the graphs. We also prove a relation between modular graph functions with six links.
A Ring Construction Using Finite Directed Graphs
ERIC Educational Resources Information Center
Bardzell, Michael
2012-01-01
In this paper we discuss an interesting class of noncommutative rings which can be constructed using finite directed graphs. This construction also creates a vector space. These structures provide undergraduate students connections between ring theory and graph theory and, among other things, allow them to see a ring unity element that looks quite…
NASA Astrophysics Data System (ADS)
Mikhalev, A. V.; Pinchuk, I. A.
2005-06-01
The structure of Steinberg conformal algebras is studied; these are analogues of Steinberg groups (algebras, superalgebras).A Steinberg conformal algebra is defined as an abstract algebra by a system of generators and relations between the generators. It is proved that a Steinberg conformal algebra is the universal central extension of the corresponding conformal Lie algebra; the kernel of this extension is calculated.
Generalized graph states based on Hadamard matrices
Cui, Shawn X.; Yu, Nengkun; Zeng, Bei
2015-07-15
Graph states are widely used in quantum information theory, including entanglement theory, quantum error correction, and one-way quantum computing. Graph states have a nice structure related to a certain graph, which is given by either a stabilizer group or an encoding circuit, both can be directly given by the graph. To generalize graph states, whose stabilizer groups are abelian subgroups of the Pauli group, one approach taken is to study non-abelian stabilizers. In this work, we propose to generalize graph states based on the encoding circuit, which is completely determined by the graph and a Hadamard matrix. We study the entanglement structures of these generalized graph states and show that they are all maximally mixed locally. We also explore the relationship between the equivalence of Hadamard matrices and local equivalence of the corresponding generalized graph states. This leads to a natural generalization of the Pauli (X, Z) pairs, which characterizes the local symmetries of these generalized graph states. Our approach is also naturally generalized to construct graph quantum codes which are beyond stabilizer codes.
ERIC Educational Resources Information Center
Capani, Antonio; De Dominicis, Gabriel
This paper proposes a model for a general interface between people and Computer Algebra Systems (CAS). The main features in the CAS interface are data navigation and the possibility of accessing powerful remote machines. This model is based on the idea of session management, in which the main engine of the tool enables interactions with the…
Vecchio, Fabrizio; Miraglia, Francesca; Curcio, Giuseppe; Altavilla, Riccardo; Scrascia, Federica; Giambattistelli, Federica; Quattrocchi, Carlo Cosimo; Bramanti, Placido; Vernieri, Fabrizio; Rossini, Paolo Maria
2015-01-01
A relatively new approach to brain function in neuroscience is the "functional connectivity", namely the synchrony in time of activity in anatomically-distinct but functionally-collaborating brain regions. On the other hand, diffusion tensor imaging (DTI) is a recently developed magnetic resonance imaging (MRI)-based technique with the capability to detect brain structural connection with fractional anisotropy (FA) identification. FA decrease has been observed in the corpus callosum of subjects with Alzheimer's disease (AD) and mild cognitive impairment (MCI, an AD prodromal stage). Corpus callosum splenium DTI abnormalities are thought to be associated with functional disconnections among cortical areas. This study aimed to investigate possible correlations between structural damage, measured by MRI-DTI, and functional abnormalities of brain integration, measured by characteristic path length detected in resting state EEG source activity (40 participants: 9 healthy controls, 10 MCI, 10 mild AD, 11 moderate AD). For each subject, undirected and weighted brain network was built to evaluate graph core measures. eLORETA lagged linear connectivity values were used as weight of the edges of the network. Results showed that callosal FA reduction is associated to a loss of brain interhemispheric functional connectivity characterized by increased delta and decreased alpha path length. These findings suggest that "global" (average network shortest path length representing an index of how efficient is the information transfer between two parts of the network) functional measure can reflect the reduction of fiber connecting the two hemispheres as revealed by DTI analysis and also anticipate in time this structural loss.
A note on derivations of Murray–von Neumann algebras
Kadison, Richard V.; Liu, Zhe
2014-01-01
A Murray–von Neumann algebra is the algebra of operators affiliated with a finite von Neumann algebra. In this article, we first present a brief introduction to the theory of derivations of operator algebras from both the physical and mathematical points of view. We then describe our recent work on derivations of Murray–von Neumann algebras. We show that the “extended derivations” of a Murray–von Neumann algebra, those that map the associated finite von Neumann algebra into itself, are inner. In particular, we prove that the only derivation that maps a Murray–von Neumann algebra associated with a factor of type II1 into that factor is 0. Those results are extensions of Singer’s seminal result answering a question of Kaplansky, as applied to von Neumann algebras: The algebra may be noncommutative and may even contain unbounded elements. PMID:24469831
ERIC Educational Resources Information Center
Barnes, Julie; Jaqua, Kathy
2011-01-01
A kinesthetic approach to developing ideas of function transformations can get students physically and intellectually involved. This article presents low- or no-cost activities which use kinesthetics to support high school students' mathematical understanding of transformations of function graphs. The important point of these activities is to help…
Fast Approximate Quadratic Programming for Graph Matching
Vogelstein, Joshua T.; Conroy, John M.; Lyzinski, Vince; Podrazik, Louis J.; Kratzer, Steven G.; Harley, Eric T.; Fishkind, Donniell E.; Vogelstein, R. Jacob; Priebe, Carey E.
2015-01-01
Quadratic assignment problems arise in a wide variety of domains, spanning operations research, graph theory, computer vision, and neuroscience, to name a few. The graph matching problem is a special case of the quadratic assignment problem, and graph matching is increasingly important as graph-valued data is becoming more prominent. With the aim of efficiently and accurately matching the large graphs common in big data, we present our graph matching algorithm, the Fast Approximate Quadratic assignment algorithm. We empirically demonstrate that our algorithm is faster and achieves a lower objective value on over 80% of the QAPLIB benchmark library, compared with the previous state-of-the-art. Applying our algorithm to our motivating example, matching C. elegans connectomes (brain-graphs), we find that it efficiently achieves performance. PMID:25886624
Fast approximate quadratic programming for graph matching.
Vogelstein, Joshua T; Conroy, John M; Lyzinski, Vince; Podrazik, Louis J; Kratzer, Steven G; Harley, Eric T; Fishkind, Donniell E; Vogelstein, R Jacob; Priebe, Carey E
2015-01-01
Quadratic assignment problems arise in a wide variety of domains, spanning operations research, graph theory, computer vision, and neuroscience, to name a few. The graph matching problem is a special case of the quadratic assignment problem, and graph matching is increasingly important as graph-valued data is becoming more prominent. With the aim of efficiently and accurately matching the large graphs common in big data, we present our graph matching algorithm, the Fast Approximate Quadratic assignment algorithm. We empirically demonstrate that our algorithm is faster and achieves a lower objective value on over 80% of the QAPLIB benchmark library, compared with the previous state-of-the-art. Applying our algorithm to our motivating example, matching C. elegans connectomes (brain-graphs), we find that it efficiently achieves performance.
Representations and module-extensions of 3-hom-Lie algebras
NASA Astrophysics Data System (ADS)
Liu, Yan; Chen, Liangyun; Ma, Yao
2015-12-01
In this paper, we study the representations and module-extensions of 3-hom-Lie algebras. We show that a linear map between 3-hom-Lie algebras is a morphism if and only if its graph is a hom subalgebra and show that the set of derivations of a 3-hom-Lie algebra is a Lie algebra. Moreover, we introduce the definition of Tθ-extensions and Tθ∗ -extensions of 3-hom-Lie algebras in terms of modules, providing the necessary and sufficient conditions for a 2 k-dimensional metric 3-hom-Lie algebra to be isometric to a Tθ∗ -extension.
Replica methods for loopy sparse random graphs
NASA Astrophysics Data System (ADS)
Coolen, ACC
2016-03-01
I report on the development of a novel statistical mechanical formalism for the analysis of random graphs with many short loops, and processes on such graphs. The graphs are defined via maximum entropy ensembles, in which both the degrees (via hard constraints) and the adjacency matrix spectrum (via a soft constraint) are prescribed. The sum over graphs can be done analytically, using a replica formalism with complex replica dimensions. All known results for tree-like graphs are recovered in a suitable limit. For loopy graphs, the emerging theory has an appealing and intuitive structure, suggests how message passing algorithms should be adapted, and what is the structure of theories describing spin systems on loopy architectures. However, the formalism is still largely untested, and may require further adjustment and refinement. This paper is dedicated to the memory of our colleague and friend Jun-Ichi Inoue, with whom the author has had the great pleasure and privilege of collaborating.
Multibody graph transformations and analysis
2013-01-01
This two-part paper uses graph transformation methods to develop methods for partitioning, aggregating, and constraint embedding for multibody systems. This first part focuses on tree-topology systems and reviews the key notion of spatial kernel operator (SKO) models for such systems. It develops systematic and rigorous techniques for partitioning SKO models in terms of the SKO models of the component subsystems based on the path-induced property of the component subgraphs. It shows that the sparsity structure of key matrix operators and the mass matrix for the multibody system can be described using partitioning transformations. Subsequently, the notions of node contractions and subgraph aggregation and their role in coarsening graphs are discussed. It is shown that the tree property of a graph is preserved after subgraph aggregation if and only if the subgraph satisfies an aggregation condition. These graph theory ideas are used to develop SKO models for the aggregated tree multibody systems. PMID:24288438
Spectral correlations of individual quantum graphs
Gnutzmann, Sven; Altland, Alexander
2005-11-01
We investigate the spectral properties of chaotic quantum graphs. We demonstrate that the energy-average over the spectrum of individual graphs can be traded for the functional average over a supersymmetric nonlinear {sigma}-model action. This proves that spectral correlations of individual quantum graphs behave according to the predictions of Wigner-Dyson random matrix theory. We explore the stability of the universal random matrix behavior with regard to perturbations, and discuss the crossover between different types of symmetries.
Partitioning sparse matrices with eigenvectors of graphs
NASA Technical Reports Server (NTRS)
Pothen, Alex; Simon, Horst D.; Liou, Kang-Pu
1990-01-01
The problem of computing a small vertex separator in a graph arises in the context of computing a good ordering for the parallel factorization of sparse, symmetric matrices. An algebraic approach for computing vertex separators is considered in this paper. It is shown that lower bounds on separator sizes can be obtained in terms of the eigenvalues of the Laplacian matrix associated with a graph. The Laplacian eigenvectors of grid graphs can be computed from Kronecker products involving the eigenvectors of path graphs, and these eigenvectors can be used to compute good separators in grid graphs. A heuristic algorithm is designed to compute a vertex separator in a general graph by first computing an edge separator in the graph from an eigenvector of the Laplacian matrix, and then using a maximum matching in a subgraph to compute the vertex separator. Results on the quality of the separators computed by the spectral algorithm are presented, and these are compared with separators obtained from other algorithms for computing separators. Finally, the time required to compute the Laplacian eigenvector is reported, and the accuracy with which the eigenvector must be computed to obtain good separators is considered. The spectral algorithm has the advantage that it can be implemented on a medium-size multiprocessor in a straightforward manner.
Computational Complexity Reductions Using Clifford Algebras
NASA Astrophysics Data System (ADS)
Schott, René; Staples, G. Stacey
Given a computing architecture based on Clifford algebras, a natural context for determining an algorithm's time complexity is in terms of the number of geometric (Clifford) operations required. In this paper the existence of such a processor is assumed, and a number of graph-theoretical problems are considered. This paper is an extension of previous work, in which the authors defined the "nilpotent adjacency matrix" associated with a finite graph and showed that a number of graph problems of complexity class NP are polynomial in the number of Clifford operations required. Previous results are recalled and illustrated with Mathematica examples. New results are obtained, and old results are improved by the development of new techniques. In particular, a matrix-free approach is developed to count matchings, compute girth, and enumerate proper cycle covers of finite graphs. These new results and techniques are also illustrated with Mathematica examples.
A note on probabilistic models over strings: the linear algebra approach.
Bouchard-Côté, Alexandre
2013-12-01
Probabilistic models over strings have played a key role in developing methods that take into consideration indels as phylogenetically informative events. There is an extensive literature on using automata and transducers on phylogenies to do inference on these probabilistic models, in which an important theoretical question is the complexity of computing the normalization of a class of string-valued graphical models. This question has been investigated using tools from combinatorics, dynamic programming, and graph theory, and has practical applications in Bayesian phylogenetics. In this work, we revisit this theoretical question from a different point of view, based on linear algebra. The main contribution is a set of results based on this linear algebra view that facilitate the analysis and design of inference algorithms on string-valued graphical models. As an illustration, we use this method to give a new elementary proof of a known result on the complexity of inference on the "TKF91" model, a well-known probabilistic model over strings. Compared to previous work, our proving method is easier to extend to other models, since it relies on a novel weak condition, triangular transducers, which is easy to establish in practice. The linear algebra view provides a concise way of describing transducer algorithms and their compositions, opens the possibility of transferring fast linear algebra libraries (for example, based on GPUs), as well as low rank matrix approximation methods, to string-valued inference problems.
The operator algebra approach to quantum groups
Kustermans, Johan; Vaes, Stefaan
2000-01-01
A relatively simple definition of a locally compact quantum group in the C*-algebra setting will be explained as it was recently obtained by the authors. At the same time, we put this definition in the historical and mathematical context of locally compact groups, compact quantum groups, Kac algebras, multiplicative unitaries, and duality theory. PMID:10639116
Some Applications of Algebraic System Solving
ERIC Educational Resources Information Center
Roanes-Lozano, Eugenio
2011-01-01
Technology and, in particular, computer algebra systems, allows us to change both the way we teach mathematics and the mathematical curriculum. Curiously enough, unlike what happens with linear system solving, algebraic system solving is not widely known. The aim of this paper is to show that, although the theory lying behind the "exact…
ON CLUSTERING TECHNIQUES OF CITATION GRAPHS.
ERIC Educational Resources Information Center
CHIEN, R.T.; PREPARATA, F.P.
ONE OF THE PROBLEMS ENCOUNTERED IN CLUSTERING TECHNIQUES AS APPLIED TO DOCUMENT RETRIEVAL SYSTEMS USING BIBLIOGRAPHIC COUPLING DEVICES IS THAT THE COMPUTATIONAL EFFORT REQUIRED GROWS ROUGHLY AS THE SQUARE OF THE COLLECTION SIZE. IN THIS STUDY GRAPH THEORY IS APPLIED TO THIS PROBLEM BY FIRST MAPPING THE CITATION GRAPH OF THE DOCUMENT COLLECTION…
Universal spectral statistics in quantum graphs.
Gnutzmann, Sven; Altland, Alexander
2004-11-05
We prove that the spectrum of an individual chaotic quantum graph shows universal spectral correlations, as predicted by random-matrix theory. The stability of these correlations with regard to nonuniversal corrections is analyzed in terms of the linear operator governing the classical dynamics on the graph.
Sequential visibility-graph motifs
NASA Astrophysics Data System (ADS)
Iacovacci, Jacopo; Lacasa, Lucas
2016-04-01
Visibility algorithms transform time series into graphs and encode dynamical information in their topology, paving the way for graph-theoretical time series analysis as well as building a bridge between nonlinear dynamics and network science. In this work we introduce and study the concept of sequential visibility-graph motifs, smaller substructures of n consecutive nodes that appear with characteristic frequencies. We develop a theory to compute in an exact way the motif profiles associated with general classes of deterministic and stochastic dynamics. We find that this simple property is indeed a highly informative and computationally efficient feature capable of distinguishing among different dynamics and robust against noise contamination. We finally confirm that it can be used in practice to perform unsupervised learning, by extracting motif profiles from experimental heart-rate series and being able, accordingly, to disentangle meditative from other relaxation states. Applications of this general theory include the automatic classification and description of physical, biological, and financial time series.
Classification of central extensions of Lax operator algebras
Schlichenmaier, Martin
2008-11-18
Lax operator algebras were introduced by Krichever and Sheinman as further developments of Krichever's theory of Lax operators on algebraic curves. They are infinite dimensional Lie algebras of current type with meromorphic objects on compact Riemann surfaces (resp. algebraic curves) as elements. Here we report on joint work with Oleg Sheinman on the classification of their almost-graded central extensions. It turns out that in case that the finite-dimensional Lie algebra on which the Lax operator algebra is based on is simple there is a unique almost-graded central extension up to equivalence and rescaling of the central element.
Linearizing W2,4 and WB2 algebras
NASA Astrophysics Data System (ADS)
Bellucci, S.; Krivonos, S.; Sorin, A.
1995-02-01
It has recently been shown that the W3 and W3(2) algebras can be considered as subalgebras in some linear conformal algebras. In this paper we show that the nonlinear algebras W2,4 and WB2 as well as Zamolodchikov's spin {5}/{2} superalgebra also can be embedded as subalgebras into some linear conformal algebras with a finite set of currents. These linear algebras give rise to new realizations of the nonlinear algebras which could be suitable in the construction of W-string theories.
Classification of central extensions of Lax operator algebras
NASA Astrophysics Data System (ADS)
Schlichenmaier, Martin
2008-11-01
Lax operator algebras were introduced by Krichever and Sheinman as further developments of Krichever's theory of Lax operators on algebraic curves. They are infinite dimensional Lie algebras of current type with meromorphic objects on compact Riemann surfaces (resp. algebraic curves) as elements. Here we report on joint work with Oleg Sheinman on the classification of their almost-graded central extensions. It turns out that in case that the finite-dimensional Lie algebra on which the Lax operator algebra is based on is simple there is a unique almost-graded central extension up to equivalence and rescaling of the central element.
Ito, Kengo; Tsutsumi, Yu; Date, Yasuhiro; Kikuchi, Jun
2016-04-15
The abundant observation of chemical fragment information for molecular complexities is a major advantage of biological NMR analysis. Thus, the development of a novel technique for NMR signal assignment and metabolite identification may offer new possibilities for exploring molecular complexities. We propose a new signal assignment approach for metabolite mixtures by assembling H-H, H-C, C-C, and Q-C fragmental information obtained by multidimensional NMR, followed by the application of graph and network theory. High-speed experiments and complete automatic signal assignments were achieved for 12 combined mixtures of (13)C-labeled standards. Application to a (13)C-labeled seaweed extract showed 66 H-C, 60 H-H, 326 C-C, and 28 Q-C correlations, which were successfully assembled to 18 metabolites by the automatic assignment. The validity of automatic assignment was supported by quantum chemical calculations. This new approach can predict entire metabolite structures from peak networks of biological extracts.
Banach Algebras Associated to Lax Pairs
NASA Astrophysics Data System (ADS)
Glazebrook, James F.
2015-04-01
Lax pairs featuring in the theory of integrable systems are known to be constructed from a commutative algebra of formal pseudodifferential operators known as the Burchnall- Chaundy algebra. Such pairs induce the well known KP flows on a restricted infinite-dimensional Grassmannian. The latter can be exhibited as a Banach homogeneous space constructed from a Banach *-algebra. It is shown that this commutative algebra of operators generating Lax pairs can be associated with a commutative C*-subalgebra in the C*-norm completion of the *-algebra. In relationship to the Bose-Fermi correspondence and the theory of vertex operators, this C*-algebra has an association with the CAR algebra of operators as represented on Fermionic Fock space by the Gelfand-Naimark-Segal construction. Instrumental is the Plücker embedding of the restricted Grassmannian into the projective space of the associated Hilbert space. The related Baker and tau-functions provide a connection between these two C*-algebras, following which their respective state spaces and Jordan-Lie-Banach algebras structures can be compared.
Chromatic polynomials of random graphs
NASA Astrophysics Data System (ADS)
Van Bussel, Frank; Ehrlich, Christoph; Fliegner, Denny; Stolzenberg, Sebastian; Timme, Marc
2010-04-01
Chromatic polynomials and related graph invariants are central objects in both graph theory and statistical physics. Computational difficulties, however, have so far restricted studies of such polynomials to graphs that were either very small, very sparse or highly structured. Recent algorithmic advances (Timme et al 2009 New J. Phys. 11 023001) now make it possible to compute chromatic polynomials for moderately sized graphs of arbitrary structure and number of edges. Here we present chromatic polynomials of ensembles of random graphs with up to 30 vertices, over the entire range of edge density. We specifically focus on the locations of the zeros of the polynomial in the complex plane. The results indicate that the chromatic zeros of random graphs have a very consistent layout. In particular, the crossing point, the point at which the chromatic zeros with non-zero imaginary part approach the real axis, scales linearly with the average degree over most of the density range. While the scaling laws obtained are purely empirical, if they continue to hold in general there are significant implications: the crossing points of chromatic zeros in the thermodynamic limit separate systems with zero ground state entropy from systems with positive ground state entropy, the latter an exception to the third law of thermodynamics.
1973-10-01
The theory of strongly regular graphs was introduced by Bose r7 1 in 1963, in connection with partial geometries and 2 class association schemes. One...non adjacent vertices is constant and equal to ~. We shall denote by ~(p) (reap.r(p)) the set of vertices adjacent (resp.non adjacent) to a vertex p...is the complement of .2’ if the set of vertices of ~ is the set of vertices of .2’ and if two vertices in .2’ are adjacent if and only if they were
NASA Astrophysics Data System (ADS)
Vaninsky, Alexander
2011-04-01
This article introduces a trigonometric field (TF) that extends the field of real numbers by adding two new elements: sin and cos - satisfying an axiom sin2 + cos2 = 1. It is shown that by assigning meaningful names to particular elements of the field, all known trigonometric identities may be introduced and proved. Two different interpretations of the TF are discussed with many others potentially possible. The main objective of this article is to introduce a broader view of trigonometry that can serve as motivation for mathematics students and teachers to study and teach abstract algebraic structures.
Some Recent Results on Graph Matching,
1987-06-01
extendable graphs In Section 2 of this paper we saw how the brick decomposition procedure can be carried out on an arbitrary 1-extendable graph and that...in fact, the procedure is "canonical" in the sense that the final list of bricks so obtained is an invariant of the graph. Furthermore, we saw how...JACKSON, P. KATERINIS and A. SAITO, Toughness and the existence of k-factors, J. Graph Theory 9, 1985, 87-95. [G1] T. GALLAI, Kritische Graphen II
Algebraic Lattices in QFT Renormalization
NASA Astrophysics Data System (ADS)
Borinsky, Michael
2016-07-01
The structure of overlapping subdivergences, which appear in the perturbative expansions of quantum field theory, is analyzed using algebraic lattice theory. It is shown that for specific QFTs the sets of subdivergences of Feynman diagrams form algebraic lattices. This class of QFTs includes the standard model. In kinematic renormalization schemes, in which tadpole diagrams vanish, these lattices are semimodular. This implies that the Hopf algebra of Feynman diagrams is graded by the coradical degree or equivalently that every maximal forest has the same length in the scope of BPHZ renormalization. As an application of this framework, a formula for the counter terms in zero-dimensional QFT is given together with some examples of the enumeration of primitive or skeleton diagrams.
NASA Astrophysics Data System (ADS)
Toner, John; Tu, Yu-Hai
2002-05-01
We have developed a new continuum dynamical model for the collective motion of large "flocks" of biological organisms (e.g., flocks of birds, schools of fish, herds of wildebeest, hordes of bacteria, slime molds, etc.) . This model does for flocks what the Navier-Stokes equation does for fluids. The model predicts that, unlike simple fluids, flocks show huge fluctuation effects in spatial dimensions d < 4 that radically change their behavior. In d=2, it is only these effects that make it possible for the flock to move coherently at all. This explains why a million wildebeest can march together across the Serengeti plain, despite the fact that a million physicists gathered on the same plane could NOT all POINT in the same direction. Detailed quantitative predictions of this theory agree beautifully with computer simulations of flock motion.
NASA Technical Reports Server (NTRS)
Cleaveland, Rance; Luettgen, Gerald; Natarajan, V.
1999-01-01
This paper surveys the semantic ramifications of extending traditional process algebras with notions of priority that allow for some transitions to be given precedence over others. These enriched formalisms allow one to model system features such as interrupts, prioritized choice, or real-time behavior. Approaches to priority in process algebras can be classified according to whether the induced notion of preemption on transitions is global or local and whether priorities are static or dynamic. Early work in the area concentrated on global pre-emption and static priorities and led to formalisms for modeling interrupts and aspects of real-time, such as maximal progress, in centralized computing environments. More recent research has investigated localized notions of pre-emption in which the distribution of systems is taken into account, as well as dynamic priority approaches, i.e., those where priority values may change as systems evolve. The latter allows one to model behavioral phenomena such as scheduling algorithms and also enables the efficient encoding of real-time semantics. Technically, this paper studies the different models of priorities by presenting extensions of Milner's Calculus of Communicating Systems (CCS) with static and dynamic priority as well as with notions of global and local pre- emption. In each case the operational semantics of CCS is modified appropriately, behavioral theories based on strong and weak bisimulation are given, and related approaches for different process-algebraic settings are discussed.
Structural features of algebraic quantum notations
NASA Astrophysics Data System (ADS)
Gire, Elizabeth; Price, Edward
2015-12-01
[This paper is part of the Focused Collection on Upper Division Physics Courses.] The formalism of quantum mechanics includes a rich collection of representations for describing quantum systems, including functions, graphs, matrices, histograms of probabilities, and Dirac notation. The varied features of these representations affect how computations are performed. For example, identifying probabilities of measurement outcomes for a state described in Dirac notation may involve identifying expansion coefficients by inspection, but if the state is described as a function, identifying those expansion coefficients often involves performing integrals. In this study, we focus on three notational systems: Dirac notation, algebraic wave-function notation, and matrix notation. These quantum notations must include information about basis states and their associated complex probability amplitudes. In this theory paper, we identify four structural features of quantum notations, which we term individuation, degree of externalization, compactness, and symbolic support for computational rules. We illustrate how student reasoning interacts with these structural features with episodes from interviews with advanced undergraduate physics majors reasoning about a superposition state of an infinite square well system. We find evidence of the students coordinating different notations through the use of Dirac notation, using an expression in Dirac notation to guide their work in another notation. These uses are supported by the high degree of individuation, compactness, and symbolic support for computation and the moderate degree of externalization provided by Dirac notation.
Algebras with convergent star products and their representations in Hilbert spaces
NASA Astrophysics Data System (ADS)
Soloviev, M. A.
2013-07-01
We study star product algebras of analytic functions for which the power series defining the products converge absolutely. Such algebras arise naturally in deformation quantization theory and in noncommutative quantum field theory. We consider different star products in a unifying way and present results on the structure and basic properties of these algebras, which are useful for applications. Special attention is given to the Hilbert space representation of the algebras and to the exact description of their corresponding operator algebras.
Unsupervised spectral mesh segmentation driven by heterogeneous graphs.
Theologou, Panagiotis; Pratikakis, Ioannis; Theoharis, Theoharis
2016-03-21
A fully automatic mesh segmentation scheme using heterogeneous graphs is presented. We introduce a spectral framework where local geometry affinities are coupled with surface patch affinities. A heterogeneous graph is constructed combining two distinct graphs: a weighted graph based on adjacency of patches of an initial over-segmentation, and the weighted dual mesh graph. The partitioning relies on processing each eigenvector of the heterogeneous graph Laplacian individually, taking into account the nodal set and nodal domain theory. Experiments on standard datasets show that the proposed unsupervised approach outperforms the state-of-the-art unsupervised methodologies and is comparable to the best supervised approaches.
Unsupervised Spectral Mesh Segmentation Driven by Heterogeneous Graphs.
Theologou, Panagiotis; Pratikakis, Ioannis; Theoharis, Theoharis
2017-02-01
A fully automatic mesh segmentation scheme using heterogeneous graphs is presented. We introduce a spectral framework where local geometry affinities are coupled with surface patch affinities. A heterogeneous graph is constructed combining two distinct graphs: a weighted graph based on adjacency of patches of an initial over-segmentation, and the weighted dual mesh graph. The partitioning relies on processing each eigenvector of the heterogeneous graph Laplacian individually, taking into account the nodal set and nodal domain theory. Experiments on standard datasets show that the proposed unsupervised approach outperforms the state-of-the-art unsupervised methodologies and is comparable to the best supervised approaches.
ERIC Educational Resources Information Center
Lawes, Jonathan F.
2013-01-01
Graphing polar curves typically involves a combination of three traditional techniques, all of which can be time-consuming and tedious. However, an alternative method--graphing the polar function on a rectangular plane--simplifies graphing, increases student understanding of the polar coordinate system, and reinforces graphing techniques learned…
Celestial mechanics with geometric algebra
NASA Technical Reports Server (NTRS)
Hestenes, D.
1983-01-01
Geometric algebra is introduced as a general tool for Celestial Mechanics. A general method for handling finite rotations and rotational kinematics is presented. The constants of Kepler motion are derived and manipulated in a new way. A new spinor formulation of perturbation theory is developed.
Sequential motif profile of natural visibility graphs
NASA Astrophysics Data System (ADS)
Iacovacci, Jacopo; Lacasa, Lucas
2016-11-01
The concept of sequential visibility graph motifs—subgraphs appearing with characteristic frequencies in the visibility graphs associated to time series—has been advanced recently along with a theoretical framework to compute analytically the motif profiles associated to horizontal visibility graphs (HVGs). Here we develop a theory to compute the profile of sequential visibility graph motifs in the context of natural visibility graphs (VGs). This theory gives exact results for deterministic aperiodic processes with a smooth invariant density or stochastic processes that fulfill the Markov property and have a continuous marginal distribution. The framework also allows for a linear time numerical estimation in the case of empirical time series. A comparison between the HVG and the VG case (including evaluation of their robustness for short series polluted with measurement noise) is also presented.
Vortices and superfields on a graph
NASA Astrophysics Data System (ADS)
Kan, Nahomi; Kobayashi, Koichiro; Shiraishi, Kiyoshi
2009-08-01
We extend the dimensional deconstruction by utilizing the knowledge of graph theory. In the dimensional deconstruction, one uses the moose diagram to exhibit the structure of the “theory space.” We generalize the moose diagram to a general graph with oriented edges. In the present paper, we consider only the U(1) gauge symmetry. We also introduce supersymmetry into our model by use of superfields. We suppose that vector superfields reside at the vertices and chiral superfields at the edges of a given graph. Then we can consider multivector, multi-Higgs models. In our model, [U(1)]p (where p is the number of vertices) is broken to a single U(1). Therefore, for specific graphs, we get vortexlike classical solutions in our model. We show some examples of the graphs admitting the vortex solutions of simple structure as the Bogomolnyi solution.
Vortices and superfields on a graph
Kan, Nahomi; Kobayashi, Koichiro; Shiraishi, Kiyoshi
2009-08-15
We extend the dimensional deconstruction by utilizing the knowledge of graph theory. In the dimensional deconstruction, one uses the moose diagram to exhibit the structure of the 'theory space'. We generalize the moose diagram to a general graph with oriented edges. In the present paper, we consider only the U(1) gauge symmetry. We also introduce supersymmetry into our model by use of superfields. We suppose that vector superfields reside at the vertices and chiral superfields at the edges of a given graph. Then we can consider multivector, multi-Higgs models. In our model, [U(1)]{sup p} (where p is the number of vertices) is broken to a single U(1). Therefore, for specific graphs, we get vortexlike classical solutions in our model. We show some examples of the graphs admitting the vortex solutions of simple structure as the Bogomolnyi solution.
Graph mining: procedure, application to drug discovery and recent advances.
Takigawa, Ichigaku; Mamitsuka, Hiroshi
2013-01-01
Combinatorial chemistry has generated chemical libraries and databases with a huge number of chemical compounds, which include prospective drugs. Chemical structures of compounds can be molecular graphs, to which a variety of graph-based techniques in computer science, specifically graph mining, can be applied. The most basic way for analyzing molecular graphs is using structural fragments, so-called subgraphs in graph theory. The mainstream technique in graph mining is frequent subgraph mining, by which we can retrieve essential subgraphs in given molecular graphs. In this article we explain the idea and procedure of mining frequent subgraphs from given molecular graphs, raising some real applications, and we describe the recent advances of graph mining.
On Ramsey (3K2, K3) - minimal graphs
NASA Astrophysics Data System (ADS)
Wijaya, Kristiana; Baskoro, Edy Tri; Assiyatun, Hilda; Suprijanto, Djoko
2016-02-01
The Ramsey graph theory has many interesting applications, such as in the fields of communications, information retrieval, and decision making. One of growing topics in Ramsey theory is Ramsey minimal graph. For any given graphs G and H, find graphs F such that any red-blue coloring of all edges of F contains either a red copy of G or a blue copy of H. If this condition is not satisfied by the graph F - e, then we call the graph F as a Ramsey (G, H) - minimal. In this paper, we derive the properties of (3K2, K3) - minimal graphs. We, then, characterize all Ramsey (3K2, K3) - minimal graphs.
Algebraic complexities and algebraic curves over finite fields
Chudnovsky, D. V.; Chudnovsky, G. V.
1987-01-01
We consider the problem of minimal (multiplicative) complexity of polynomial multiplication and multiplication in finite extensions of fields. For infinite fields minimal complexities are known [Winograd, S. (1977) Math. Syst. Theory 10, 169-180]. We prove lower and upper bounds on minimal complexities over finite fields, both linear in the number of inputs, using the relationship with linear coding theory and algebraic curves over finite fields. PMID:16593816
Gauged Ads-Maxwell Algebra and Gravity
NASA Astrophysics Data System (ADS)
Durka, R.; Kowalski-Glikman, J.; Szczachor, M.
We deform the anti-de Sitter algebra by adding additional generators {Z}ab, forming in this way the negative cosmological constant counterpart of the Maxwell algebra. We gauge this algebra and construct a dynamical model with the help of a constrained BF theory. It turns out that the resulting theory is described by the Einstein-Cartan action with Holst term, and the gauge fields associated with the Maxwell generators {Z}ab appear only in topological terms that do not influence dynamical field equations. We briefly comment on the extension of this construction, which would lead to a nontrivial Maxwell fields dynamics.
Students' Interpretation of a Function Associated with a Real-Life Problem from Its Graph
ERIC Educational Resources Information Center
Mahir, Nevin
2010-01-01
The properties of a function such as limit, continuity, derivative, growth, or concavity can be determined more easily from its graph than by doing any algebraic operation. For this reason, it is important for students of mathematics to interpret some of the properties of a function from its graph. In this study, we investigated the competence of…
Some results on the spectra of strongly regular graphs
NASA Astrophysics Data System (ADS)
Vieira, Luís António de Almeida; Mano, Vasco Moço
2016-06-01
Let G be a strongly regular graph whose adjacency matrix is A. We associate a real finite dimensional Euclidean Jordan algebra 𝒱, of rank three to the strongly regular graph G, spanned by I and the natural powers of A, endowed with the Jordan product of matrices and with the inner product as being the usual trace of matrices. Finally, by the analysis of the binomial Hadamard series of an element of 𝒱, we establish some inequalities on the parameters and on the spectrum of a strongly regular graph like those established in theorems 3 and 4.
Crist, Michele R.; Knick, Steven T.; Hanser, Steven E.
2017-01-01
The delineation of priority areas in western North America for managing Greater Sage-Grouse (Centrocercus urophasianus) represents a broad-scale experiment in conservation biology. The strategy of limiting spatial disturbance and focusing conservation actions within delineated areas may benefit the greatest proportion of Greater Sage-Grouse. However, land use under normal restrictions outside priority areas potentially limits dispersal and gene flow, which can isolate priority areas and lead to spatially disjunct populations. We used graph theory, representing priority areas as spatially distributed nodes interconnected by movement corridors, to understand the capacity of priority areas to function as connected networks in the Bi-State, Central, and Washington regions of the Greater Sage-Grouse range. The Bi-State and Central networks were highly centralized; the dominant pathways and shortest linkages primarily connected a small number of large and centrally located priority areas. These priority areas are likely strongholds for Greater Sage-Grouse populations and might also function as refugia and sources. Priority areas in the Central network were more connected than those in the Bi-State and Washington networks. Almost 90% of the priority areas in the Central network had ≥2 pathways to other priority areas when movement through the landscape was set at an upper threshold (effective resistance, ER12). At a lower threshold (ER4), 83 of 123 priority areas in the Central network were clustered in 9 interconnected subgroups. The current conservation strategy has risks; 45 of 61 priority areas in the Bi-State network, 68 of 123 in the Central network, and all 4 priority areas in the Washington network had ≤1 connection to another priority area at the lower ER4threshold. Priority areas with few linkages also averaged greater environmental resistance to movement along connecting pathways. Without maintaining corridors to larger priority areas or a clustered group
Algebraic Methods to Design Signals
2015-08-27
group theory are employed to investigate the theory of their construction methods leading to new families of these arrays and some generalizations...sequences and arrays with desirable correlation properties. The methods used are very algebraic and number theoretic. Many new families of sequences...context of optical quantum computing, we prove that infinite families of anticirculant block weighing matrices can be obtained from generic weighing
ERIC Educational Resources Information Center
Yildiz Ulus, Aysegul
2013-01-01
This paper examines experimental and algorithmic contributions of advanced calculators (graphing and computer algebra system, CAS) in teaching the concept of "diagonalization," one of the key topics in Linear Algebra courses taught at the undergraduate level. Specifically, the proposed hypothesis of this study is to assess the effective…
Introducing Algebra through the Graphical Representation of Functions: A Study among LD Students
ERIC Educational Resources Information Center
Sauriol, Jennifer
2013-01-01
This longitudinal study evaluates the impact of a new Algebra 1 course at a High School for language-based learning-disabled (LD) students. The new course prioritized the teaching of relationship graphs and functions as an introduction to algebra. Across three studies, the dissertation documents and evaluates the progress made by LD high school…
Bandlimited graph signal reconstruction by diffusion operator
NASA Astrophysics Data System (ADS)
Yang, Lishan; You, Kangyong; Guo, Wenbin
2016-12-01
Signal processing on graphs extends signal processing concepts and methodologies from the classical signal processing theory to data indexed by general graphs. For a bandlimited graph signal, the unknown data associated with unsampled vertices can be reconstructed from the sampled data by exploiting the spatial relationship of graph signal. In this paper, we propose a generalized analytical framework of unsampled graph signal and introduce a concept of diffusion operator which consists of local-mean and global-bias diffusion operator. Then, a diffusion operator-based iterative algorithm is proposed to reconstruct bandlimited graph signal from sampled data. In each iteration, the reconstructed residuals associated with the sampled vertices are diffused to all the unsampled vertices for accelerating the convergence. We then prove that the proposed reconstruction strategy converges to the original graph signal. The simulation results demonstrate the effectiveness of the proposed reconstruction strategy with various downsampling patterns, fluctuation of graph cut-off frequency, robustness on the classic graph structures, and noisy scenarios.
Intrinsic graph structure estimation using graph Laplacian.
Noda, Atsushi; Hino, Hideitsu; Tatsuno, Masami; Akaho, Shotaro; Murata, Noboru
2014-07-01
A graph is a mathematical representation of a set of variables where some pairs of the variables are connected by edges. Common examples of graphs are railroads, the Internet, and neural networks. It is both theoretically and practically important to estimate the intensity of direct connections between variables. In this study, a problem of estimating the intrinsic graph structure from observed data is considered. The observed data in this study are a matrix with elements representing dependency between nodes in the graph. The dependency represents more than direct connections because it includes influences of various paths. For example, each element of the observed matrix represents a co-occurrence of events at two nodes or a correlation of variables corresponding to two nodes. In this setting, spurious correlations make the estimation of direct connection difficult. To alleviate this difficulty, a digraph Laplacian is used for characterizing a graph. A generative model of this observed matrix is proposed, and a parameter estimation algorithm for the model is also introduced. The notable advantage of the proposed method is its ability to deal with directed graphs, while conventional graph structure estimation methods such as covariance selections are applicable only to undirected graphs. The algorithm is experimentally shown to be able to identify the intrinsic graph structure.
The Algebra of Lexical Semantics
NASA Astrophysics Data System (ADS)
Kornai, András
The current generative theory of the lexicon relies primarily on tools from formal language theory and mathematical logic. Here we describe how a different formal apparatus, taken from algebra and automata theory, resolves many of the known problems with the generative lexicon. We develop a finite state theory of word meaning based on machines in the sense of Eilenberg [11], a formalism capable of describing discrepancies between syntactic type (lexical category) and semantic type (number of arguments). This mechanism is compared both to the standard linguistic approaches and to the formalisms developed in AI/KR.
Generalizing a Categorization of Students' Interpretations of Linear Kinematics Graphs
ERIC Educational Resources Information Center
Bollen, Laurens; De Cock, Mieke; Zuza, Kristina; Guisasola, Jenaro; van Kampen, Paul
2016-01-01
We have investigated whether and how a categorization of responses to questions on linear distance-time graphs, based on a study of Irish students enrolled in an algebra-based course, could be adopted and adapted to responses from students enrolled in calculus-based physics courses at universities in Flanders, Belgium (KU Leuven) and the Basque…
Ecosystem Simulations and Chaos on the Graphing Calculator
ERIC Educational Resources Information Center
Sinn, Robb
2007-01-01
An eighth grade algebra class used graphing calculators to simulate ecosystems. One simulation introduced mathematical chaos. The activities exposed the students to nonlinear patterns and modeling. The rate-of-change investigations related the ideas of intercept and slope to the changing equilibrium. The chaotic model intrigued them and was useful…
Improving Student Knowledge of the Graphing Calculator's Capabilities.
ERIC Educational Resources Information Center
Hubbard, Donna
This paper describes an intervention in two Algebra II classes in which the graphing calculator was incorporated into the curriculum as often as possible. The targeted population consisted of high school students in a growing middle to upper class community located in a suburb of a large city. The problem of a lack of understanding of the…
NASA Astrophysics Data System (ADS)
Paech, Martin; Apel, Walter; Kalinowski, Eva; Jeckelmann, Eric
2014-12-01
We present a large-scale combinatorial-diagrammatic computation of high-order contributions to the strong-coupling Kato-Takahashi perturbation series for the Hubbard model in high dimensions. The ground-state energy of the Mott-insulating phase is determined exactly up to the 15th order in 1 /U . The perturbation expansion is extrapolated to infinite order and the critical behavior is determined using the Domb-Sykes method. We compare the perturbative results with two dynamical mean-field theory (DMFT) calculations using a quantum Monte Carlo method and a density-matrix renormalization group method as impurity solvers. The comparison demonstrates the excellent agreement and accuracy of both extrapolated strong-coupling perturbation theory and quantum Monte Carlo based DMFT, even close to the critical coupling where the Mott insulator becomes unstable.
Private quantum subsystems and quasiorthogonal operator algebras
NASA Astrophysics Data System (ADS)
Levick, Jeremy; Jochym-O'Connor, Tomas; Kribs, David W.; Laflamme, Raymond; Pereira, Rajesh
2016-03-01
We generalize a recently discovered example of a private quantum subsystem to find private subsystems for Abelian subgroups of the n-qubit Pauli group, which exist in the absence of private subspaces. In doing so, we also connect these quantum privacy investigations with the theory of quasiorthogonal operator algebras through the use of tools from group theory and operator theory.
Relativity on rotated graph paper
NASA Astrophysics Data System (ADS)
Salgado, Roberto B.
2016-05-01
We demonstrate a method for constructing spacetime diagrams for special relativity on graph paper that has been rotated by 45°. The diagonal grid lines represent light-flash worldlines in Minkowski spacetime, and the boxes in the grid (called "clock diamonds") represent units of measurement corresponding to the ticks of an inertial observer's light clock. We show that many quantitative results can be read off a spacetime diagram simply by counting boxes, with very little algebra. In particular, we show that the squared interval between two events is equal to the signed area of the parallelogram on the grid (called the "causal diamond") with opposite vertices corresponding to those events. We use the Doppler effect—without explicit use of the Doppler formula—to motivate the method.
On molecular graph comparison.
Melo, Jenny A; Daza, Edgar
2011-06-01
Since the last half of the nineteenth century, molecular graphs have been present in several branches of chemistry. When used for molecular structure representation, they have been compared after mapping the corresponding graphs into mathematical objects. However, direct molecular comparison of molecular graphs is a research field less explored. The goal of this mini-review is to show some distance and similarity coefficients which were proposed to directly compare molecular graphs or which could be useful to do so.
Writing to Promote and Assess Conceptual Understanding in College Algebra
ERIC Educational Resources Information Center
Gay, A. Susan; Peterson, Ingrid
2014-01-01
Concept-focused quiz questions required College Algebra students to write about their understanding. The questions can be viewed in three broad categories: a focus on sense-making, a focus on describing a mathematical object such as a graph or an equation, and a focus on understanding vocabulary. Student responses from 10 classes were analyzed.…
Learning Activity Package, Algebra 124, LAPs 46-55.
ERIC Educational Resources Information Center
Holland, Bill
A series of 10 teacher-prepared Learning Activity Packages (LAPs) in advanced algebra and trigonometry, these units cover absolute value, inequalities, exponents, radicals, and complex numbers; functions; higher degree equations and the derivative; the trigonometric functions; graphs and applications of the trigonometric functions; sequences and…
SAYD Modules over Lie-Hopf Algebras
NASA Astrophysics Data System (ADS)
Rangipour, Bahram; Sütlü, Serkan
2012-11-01
In this paper a general van Est type isomorphism is proved. The isomorphism is between the Lie algebra cohomology of a bicrossed sum Lie algebra and the Hopf cyclic cohomology of its Hopf algebra. We first prove a one to one correspondence between stable-anti-Yetter-Drinfeld (SAYD) modules over the total Lie algebra and those modules over the associated Hopf algebra. In contrast to the non-general case done in our previous work, here the van Est isomorphism is proved at the first level of a natural spectral sequence, rather than at the level of complexes. It is proved that the Connes-Moscovici Hopf algebras do not admit any finite dimensional SAYD modules except the unique one-dimensional one found by Connes-Moscovici in 1998. This is done by extending our techniques to work with the infinite dimensional Lie algebra of formal vector fields. At the end, the one to one correspondence is applied to construct a highly nontrivial four dimensional SAYD module over the Schwarzian Hopf algebra. We then illustrate the whole theory on this example. Finally explicit representative cocycles of the cohomology classes for this example are calculated.
NASA Astrophysics Data System (ADS)
Sati, Hisham; Schreiber, Urs
2017-03-01
We uncover higher algebraic structures on Noether currents and BPS charges. It is known that equivalence classes of conserved currents form a Lie algebra. We show that at least for target space symmetries of higher parameterized WZW-type sigma-models this naturally lifts to a Lie ( p + 1)-algebra structure on the Noether currents themselves. Applied to the Green-Schwarz-type action functionals for super p-brane sigma-models this yields super Lie ( p+1)-algebra refinements of the traditional BPS brane charge extensions of supersymmetry algebras. We discuss this in the generality of higher differential geometry, where it applies also to branes with (higher) gauge fields on their worldvolume. Applied to the M5-brane sigma-model we recover and properly globalize the M-theory super Lie algebra extension of 11-dimensional superisometries by 2-brane and 5-brane charges. Passing beyond the infinitesimal Lie theory we find cohomological corrections to these charges in higher analogy to the familiar corrections for D-brane charges as they are lifted from ordinary cohomology to twisted K-theory. This supports the proposal that M-brane charges live in a twisted cohomology theory.
Graphing Inequalities, Connecting Meaning
ERIC Educational Resources Information Center
Switzer, J. Matt
2014-01-01
Students often have difficulty with graphing inequalities (see Filloy, Rojano, and Rubio 2002; Drijvers 2002), and J. Matt Switzer's students were no exception. Although students can produce graphs for simple inequalities, they often struggle when the format of the inequality is unfamiliar. Even when producing a correct graph of an…
ERIC Educational Resources Information Center
Reading Teacher, 2012
2012-01-01
The "Toolbox" column features content adapted from ReadWriteThink.org lesson plans and provides practical tools for classroom teachers. This issue's column features a lesson plan adapted from "Graphing Plot and Character in a Novel" by Lisa Storm Fink and "Bio-graph: Graphing Life Events" by Susan Spangler. Students retell biographic events…
Twisted vertex algebras, bicharacter construction and boson-fermion correspondences
NASA Astrophysics Data System (ADS)
Anguelova, Iana I.
2013-12-01
The boson-fermion correspondences are an important phenomena on the intersection of several areas in mathematical physics: representation theory, vertex algebras and conformal field theory, integrable systems, number theory, cohomology. Two such correspondences are well known: the types A and B (and their super extensions). As a main result of this paper we present a new boson-fermion correspondence of type D-A. Further, we define a new concept of twisted vertex algebra of order N, which generalizes super vertex algebra. We develop the bicharacter construction which we use for constructing classes of examples of twisted vertex algebras, as well as for deriving formulas for the operator product expansions, analytic continuations, and normal ordered products. By using the underlying Hopf algebra structure we prove general bicharacter formulas for the vacuum expectation values for two important groups of examples. We show that the correspondences of types B, C, and D-A are isomorphisms of twisted vertex algebras.
Conformal current algebra in two dimensions
NASA Astrophysics Data System (ADS)
Ashok, Sujay K.; Benichou, Raphael; Troost, Jan
2009-06-01
We construct a non-chiral current algebra in two dimensions consistent with conformal invariance. We show that the conformal current algebra is realized in non-linear sigma-models on supergroup manifolds with vanishing Killing form, with or without a Wess-Zumino term. The current algebra is computed using two distinct methods. First we exploit special algebraic properties of supergroups to compute the exact two- and three-point functions of the currents and from them we infer the current algebra. The algebra is also calculated by using conformal perturbation theory about the Wess-Zumino-Witten point and resumming the perturbation series. We also prove that these models realize a non-chiral Kac-Moody algebra and construct an infinite set of commuting operators that is closed under the action of the Kac-Moody generators. The supergroup models that we consider include models with applications to statistical mechanics, condensed matter and string theory. In particular, our results may help to systematically solve and clarify the quantum integrability of PSU(n|n) models and their cosets, which appear prominently in string worldsheet models on anti-deSitter spaces.
Special issue on cluster algebras in mathematical physics
NASA Astrophysics Data System (ADS)
Di Francesco, Philippe; Gekhtman, Michael; Kuniba, Atsuo; Yamazaki, Masahito
2013-10-01
This is a call for contributions to a special issue of Journal of Physics A: Mathematical and Theoretical dedicated to cluster algebras in mathematical physics. Over the ten years since their introduction by Fomin and Zelevinsky, the theory of cluster algebras has witnessed a spectacular growth, first and foremost due to the many links that have been discovered with a wide range of subjects in mathematics and, increasingly, theoretical and mathematical physics. The main motivation of this special issue is to gather together reviews, recent developments and open problems, mainly from a mathematical physics viewpoint, into a single comprehensive issue. We expect that such a special issue will become a valuable reference for the broad scientific community working in mathematical and theoretical physics. The issue will consist of invited review articles and contributed papers containing new results on the interplays of cluster algebras with mathematical physics. Editorial policy The Guest Editors for this issue are Philippe Di Francesco, Michael Gekhtman, Atsuo Kuniba and Masahito Yamazaki. The areas and topics for this issue include, but are not limited to: discrete integrable systems arising from cluster mutations cluster structure on Poisson varieties cluster algebras and soliton interactions cluster positivity conjecture Y-systems in the thermodynamic Bethe ansatz and Zamolodchikov's periodicity conjecture T-system of transfer matrices of integrable lattice models dilogarithm identities in conformal field theory wall crossing in 4d N = 2 supersymmetric gauge theories 4d N = 1 quiver gauge theories described by networks scattering amplitudes of 4d N = 4 theories 3d N = 2 gauge theories described by flat connections on 3-manifolds integrability of dimer/Ising models on graphs. All contributions will be refereed and processed according to the usual procedure of the journal. Guidelines for preparation of contributions The deadline for contributed papers is 31 March
Special issue on cluster algebras in mathematical physics
NASA Astrophysics Data System (ADS)
Di Francesco, Philippe; Gekhtman, Michael; Kuniba, Atsuo; Yamazaki, Masahito
2013-11-01
This is a call for contributions to a special issue of Journal of Physics A: Mathematical and Theoretical dedicated to cluster algebras in mathematical physics. Over the ten years since their introduction by Fomin and Zelevinsky, the theory of cluster algebras has witnessed a spectacular growth, first and foremost due to the many links that have been discovered with a wide range of subjects in mathematics and, increasingly, theoretical and mathematical physics. The main motivation of this special issue is to gather together reviews, recent developments and open problems, mainly from a mathematical physics viewpoint, into a single comprehensive issue. We expect that such a special issue will become a valuable reference for the broad scientific community working in mathematical and theoretical physics. The issue will consist of invited review articles and contributed papers containing new results on the interplays of cluster algebras with mathematical physics. Editorial policy The Guest Editors for this issue are Philippe Di Francesco, Michael Gekhtman, Atsuo Kuniba and Masahito Yamazaki. The areas and topics for this issue include, but are not limited to: discrete integrable systems arising from cluster mutations cluster structure on Poisson varieties cluster algebras and soliton interactions cluster positivity conjecture Y-systems in the thermodynamic Bethe ansatz and Zamolodchikov's periodicity conjecture T-system of transfer matrices of integrable lattice models dilogarithm identities in conformal field theory wall crossing in 4d N = 2 supersymmetric gauge theories 4d N = 1 quiver gauge theories described by networks scattering amplitudes of 4d N = 4 theories 3d N = 2 gauge theories described by flat connections on 3-manifolds integrability of dimer/Ising models on graphs. All contributions will be refereed and processed according to the usual procedure of the journal. Guidelines for preparation of contributions The deadline for contributed papers is 31 March
Special issue on cluster algebras in mathematical physics
NASA Astrophysics Data System (ADS)
Di Francesco, Philippe; Gekhtman, Michael; Kuniba, Atsuo; Yamazaki, Masahito
2014-02-01
This is a call for contributions to a special issue of Journal of Physics A: Mathematical and Theoretical dedicated to cluster algebras in mathematical physics. Over the ten years since their introduction by Fomin and Zelevinsky, the theory of cluster algebras has witnessed a spectacular growth, first and foremost due to the many links that have been discovered with a wide range of subjects in mathematics and, increasingly, theoretical and mathematical physics. The main motivation of this special issue is to gather together reviews, recent developments and open problems, mainly from a mathematical physics viewpoint, into a single comprehensive issue. We expect that such a special issue will become a valuable reference for the broad scientific community working in mathematical and theoretical physics. The issue will consist of invited review articles and contributed papers containing new results on the interplays of cluster algebras with mathematical physics. Editorial policy The Guest Editors for this issue are Philippe Di Francesco, Michael Gekhtman, Atsuo Kuniba and Masahito Yamazaki. The areas and topics for this issue include, but are not limited to: discrete integrable systems arising from cluster mutations cluster structure on Poisson varieties cluster algebras and soliton interactions cluster positivity conjecture Y-systems in the thermodynamic Bethe ansatz and Zamolodchikov's periodicity conjecture T-system of transfer matrices of integrable lattice models dilogarithm identities in conformal field theory wall crossing in 4d N = 2 supersymmetric gauge theories 4d N = 1 quiver gauge theories described by networks scattering amplitudes of 4d N = 4 theories 3d N = 2 gauge theories described by flat connections on 3-manifolds integrability of dimer/Ising models on graphs. All contributions will be refereed and processed according to the usual procedure of the journal. Guidelines for preparation of contributions The deadline for contributed papers is 31 March
Special issue on cluster algebras in mathematical physics
NASA Astrophysics Data System (ADS)
Di Francesco, Philippe; Gekhtman, Michael; Kuniba, Atsuo; Yamazaki, Masahito
2013-12-01
This is a call for contributions to a special issue of Journal of Physics A: Mathematical and Theoretical dedicated to cluster algebras in mathematical physics. Over the ten years since their introduction by Fomin and Zelevinsky, the theory of cluster algebras has witnessed a spectacular growth, first and foremost due to the many links that have been discovered with a wide range of subjects in mathematics and, increasingly, theoretical and mathematical physics. The main motivation of this special issue is to gather together reviews, recent developments and open problems, mainly from a mathematical physics viewpoint, into a single comprehensive issue. We expect that such a special issue will become a valuable reference for the broad scientific community working in mathematical and theoretical physics. The issue will consist of invited review articles and contributed papers containing new results on the interplays of cluster algebras with mathematical physics. Editorial policy The Guest Editors for this issue are Philippe Di Francesco, Michael Gekhtman, Atsuo Kuniba and Masahito Yamazaki. The areas and topics for this issue include, but are not limited to: discrete integrable systems arising from cluster mutations cluster structure on Poisson varieties cluster algebras and soliton interactions cluster positivity conjecture Y-systems in the thermodynamic Bethe ansatz and Zamolodchikov's periodicity conjecture T-system of transfer matrices of integrable lattice models dilogarithm identities in conformal field theory wall crossing in 4d N = 2 supersymmetric gauge theories 4d N = 1 quiver gauge theories described by networks scattering amplitudes of 4d N = 4 theories 3d N = 2 gauge theories described by flat connections on 3-manifolds integrability of dimer/Ising models on graphs. All contributions will be refereed and processed according to the usual procedure of the journal. Guidelines for preparation of contributions The deadline for contributed papers is 31 March
Algebraic complementarity in quantum theory
Petz, Denes
2010-01-15
This paper is an overview of the concept of complementarity, the relation to state estimation, to Connes-Stoermer conditional (or relative) entropy, and to uncertainty relation. Complementary Abelian and noncommutative subalgebras are analyzed. All the known results about complementary decompositions are described and several open questions are included. The paper contains only few proofs, typically references are given.
Wong, Pak C.; Mackey, Patrick S.; Perrine, Kenneth A.; Foote, Harlan P.; Thomas, James J.
2008-12-23
Methods for visualizing a graph by automatically drawing elements of the graph as labels are disclosed. In one embodiment, the method comprises receiving node information and edge information from an input device and/or communication interface, constructing a graph layout based at least in part on that information, wherein the edges are automatically drawn as labels, and displaying the graph on a display device according to the graph layout. In some embodiments, the nodes are automatically drawn as labels instead of, or in addition to, the label-edges.
NASA Astrophysics Data System (ADS)
Miao, Qian; Hu, Xiao-Rui; Chen, Yong
2014-02-01
We present a Maple computer algebra package, ONEOptimal, which can calculate one-dimensional optimal system of finite dimensional Lie algebra for nonlinear equations automatically based on Olver's theory. The core of this theory is viewing the Killing form of the Lie algebra as an invariant for the adjoint representation. Some examples are given to demonstrate the validity and efficiency of the program.
Vague Congruences and Quotient Lattice Implication Algebras
Qin, Xiaoyan; Xu, Yang
2014-01-01
The aim of this paper is to further develop the congruence theory on lattice implication algebras. Firstly, we introduce the notions of vague similarity relations based on vague relations and vague congruence relations. Secondly, the equivalent characterizations of vague congruence relations are investigated. Thirdly, the relation between the set of vague filters and the set of vague congruences is studied. Finally, we construct a new lattice implication algebra induced by a vague congruence, and the homomorphism theorem is given. PMID:25133207
Interacting particle systems on graphs
NASA Astrophysics Data System (ADS)
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
Profiles of Algebraic Competence
ERIC Educational Resources Information Center
Humberstone, J.; Reeve, R.A.
2008-01-01
The algebraic competence of 72 12-year-old female students was examined to identify profiles of understanding reflecting different algebraic knowledge states. Beginning algebraic competence (mapping abilities: word-to-symbol and vice versa, classifying, and solving equations) was assessed. One week later, the nature of assistance required to map…
ERIC Educational Resources Information Center
Miller, L. Diane; England, David A.
1989-01-01
Describes a study in a large metropolitan high school to ascertain what influence the use of regular writing in algebra classes would have on students' attitudes towards algebra and their skills in algebra. Reports the simpler and more direct the writing topics the better. (MVL)
Applied Algebra Curriculum Modules.
ERIC Educational Resources Information Center
Texas State Technical Coll., Marshall.
This collection of 11 applied algebra curriculum modules can be used independently as supplemental modules for an existing algebra curriculum. They represent diverse curriculum styles that should stimulate the teacher's creativity to adapt them to other algebra concepts. The selected topics have been determined to be those most needed by students…
Connecting Arithmetic to Algebra
ERIC Educational Resources Information Center
Darley, Joy W.; Leapard, Barbara B.
2010-01-01
Algebraic thinking is a top priority in mathematics classrooms today. Because elementary school teachers lay the groundwork to develop students' capacity to think algebraically, it is crucial for teachers to have a conceptual understanding of the connections between arithmetic and algebra and be confident in communicating these connections. Many…
Ternary Virasoro - Witt algebra.
Zachos, C.; Curtright, T.; Fairlie, D.; High Energy Physics; Univ. of Miami; Univ. of Durham
2008-01-01
A 3-bracket variant of the Virasoro-Witt algebra is constructed through the use of su(1,1) enveloping algebra techniques. The Leibniz rules for 3-brackets acting on other 3-brackets in the algebra are discussed and verified in various situations.
Product of bipolar fuzzy graphs and their degree
NASA Astrophysics Data System (ADS)
Rashmanlou, Hossein; Samanta, Sovan; Pal, Madhumangal; Borzooei, Rajab Ali
2016-01-01
The concepts of graph theory are applied in many areas of computer science including data mining, image segmentation, clustering, image capturing and networking. It is also known that lots of uncertainties occur in these areas. To handle the uncertainty that occurs in graph theory, fuzzy graph theory is successfully used in many problems. A bipolar fuzzy set is a generalization of the fuzzy set. In this paper, two new operations on bipolar fuzzy graphs, viz. normal product and tensor product, are defined. Also, the degrees of the vertices of the resultant graphs which are obtained from two given bipolar fuzzy graphs ? and ? using the operations Cartesian product, composition, tensor and normal product are determined.
Riemannian manifolds as Lie-Rinehart algebras
NASA Astrophysics Data System (ADS)
Pessers, Victor; van der Veken, Joeri
2016-07-01
In this paper, we show how Lie-Rinehart algebras can be applied to unify and generalize the elementary theory of Riemannian geometry. We will first review some necessary theory on a.o. modules, bilinear forms and derivations. We will then translate some classical theory on Riemannian geometry to the setting of Rinehart spaces, a special kind of Lie-Rinehart algebras. Some generalized versions of classical results will be obtained, such as the existence of a unique Levi-Civita connection, inducing a Levi-Civita connection on a submanifold, and the construction of spaces with constant sectional curvature.
Computer algebra and operators
NASA Technical Reports Server (NTRS)
Fateman, Richard; Grossman, Robert
1989-01-01
The symbolic computation of operator expansions is discussed. Some of the capabilities that prove useful when performing computer algebra computations involving operators are considered. These capabilities may be broadly divided into three areas: the algebraic manipulation of expressions from the algebra generated by operators; the algebraic manipulation of the actions of the operators upon other mathematical objects; and the development of appropriate normal forms and simplification algorithms for operators and their actions. Brief descriptions are given of the computer algebra computations that arise when working with various operators and their actions.
Giusti, Chad; Ghrist, Robert; Bassett, Danielle S
2016-08-01
The language of graph theory, or network science, has proven to be an exceptional tool for addressing myriad problems in neuroscience. Yet, the use of networks is predicated on a critical simplifying assumption: that the quintessential unit of interest in a brain is a dyad - two nodes (neurons or brain regions) connected by an edge. While rarely mentioned, this fundamental assumption inherently limits the types of neural structure and function that graphs can be used to model. Here, we describe a generalization of graphs that overcomes these limitations, thereby offering a broad range of new possibilities in terms of modeling and measuring neural phenomena. Specifically, we explore the use of simplicial complexes: a structure developed in the field of mathematics known as algebraic topology, of increasing applicability to real data due to a rapidly growing computational toolset. We review the underlying mathematical formalism as well as the budding literature applying simplicial complexes to neural data, from electrophysiological recordings in animal models to hemodynamic fluctuations in humans. Based on the exceptional flexibility of the tools and recent ground-breaking insights into neural function, we posit that this framework has the potential to eclipse graph theory in unraveling the fundamental mysteries of cognition.
Lie algebras and linear differential equations.
NASA Technical Reports Server (NTRS)
Brockett, R. W.; Rahimi, A.
1972-01-01
Certain symmetry properties possessed by the solutions of linear differential equations are examined. For this purpose, some basic ideas from the theory of finite dimensional linear systems are used together with the work of Wei and Norman on the use of Lie algebraic methods in differential equation theory.
Modules as Learning Tools in Linear Algebra
ERIC Educational Resources Information Center
Cooley, Laurel; Vidakovic, Draga; Martin, William O.; Dexter, Scott; Suzuki, Jeff; Loch, Sergio
2014-01-01
This paper reports on the experience of STEM and mathematics faculty at four different institutions working collaboratively to integrate learning theory with curriculum development in a core undergraduate linear algebra context. The faculty formed a Professional Learning Community (PLC) with a focus on learning theories in mathematics and…
Conformal manifolds in four dimensions and chiral algebras
NASA Astrophysics Data System (ADS)
Buican, Matthew; Nishinaka, Takahiro
2016-11-01
Any { N }=2 superconformal field theory (SCFT) in four dimensions has a sector of operators related to a two-dimensional chiral algebra containing a Virasoro sub-algebra. Moreover, there are well-known examples of isolated SCFTs whose chiral algebra is a Virasoro algebra. In this note, we consider the chiral algebras associated with interacting { N }=2 SCFTs possessing an exactly marginal deformation that can be interpreted as a gauge coupling (i.e., at special points on the resulting conformal manifolds, free gauge fields appear that decouple from isolated SCFT building blocks). At any point on these conformal manifolds, we argue that the associated chiral algebras possess at least three generators. In addition, we show that there are examples of SCFTs realizing such a minimal chiral algebra: they are certain points on the conformal manifold obtained by considering the low-energy limit of type IIB string theory on the three complex-dimensional hypersurface singularity {x}13+{x}23+{x}33+α {x}1{x}2{x}3+{w}2=0. The associated chiral algebra is the { A }(6) theory of Feigin, Feigin, and Tipunin. As byproducts of our work, we argue that (i) a collection of isolated theories can be conformally gauged only if there is a SUSY moduli space associated with the corresponding symmetry current moment maps in each sector, and (ii) { N }=2 SCFTs with a≥slant c have hidden fermionic symmetries (in the sense of fermionic chiral algebra generators).
Supersymmetry in physics: an algebraic overview
Ramond, P.
1983-01-01
In 1970, while attempting to generalize the Veneziano model (string model) to include fermions, I introduced a new algebraic structure which turned out to be a graded Lie algebra; it was used as a spectrum-generating algebra. This approach was soon after generalized to include interactions, yielding a complete model of fermions and boson (RNS model). In an unrelated work in the Soviet Union, it was shown how to generalize the Poincare group to include fermionic charges. However it was not until 1974 that an interacting field theory invariant under the Graded Poincare group in 3 + 1 dimensions was built (WZ model). Supersymmetric field theories turned out to have less divergent ultraviolet behavior than non-supersymmetric field theories. Gravity was generalized to include supersymmetry, to a theory called supergravity. By now many interacting local field theories exhibiting supersymmetry have been built and studied from 1 + 1 to 10 + 1 dimensions. Supersymmetric local field theories in less than 9 + 1 dimensions, can be understood as limits of multilocal (string) supersymmetric theories, in 9 + 1 dimensions. On the other hand, graded Lie algebras have been used in non-relativistic physics as approximate symmetries of Hamiltonians. The most striking such use so far helps comparing even and odd nuclei energy levels. It is believed that graded Lie algebras can be used whenever paired and unpaired fermions excitations can coexist. In this overview of a tremendously large field, I will only survey finite graded Lie algebras and their representations. For non-relativistic applications, all of GLA are potentially useful, while for relativistic applications, only these which include the Poincare group are to be considered.
Object Discovery: Soft Attributed Graph Mining.
Zhang, Quanshi; Song, Xuan; Shao, Xiaowei; Zhao, Huijing; Shibasaki, Ryosuke
2016-03-01
We categorize this research in terms of its contribution to both graph theory and computer vision. From the theoretical perspective, this study can be considered as the first attempt to formulate the idea of mining maximal frequent subgraphs in the challenging domain of messy visual data, and as a conceptual extension to the unsupervised learning of graph matching. We define a soft attributed pattern (SAP) to represent the common subgraph pattern among a set of attributed relational graphs (ARGs), considering both their structure and attributes. Regarding the differences between ARGs with fuzzy attributes and conventional labeled graphs, we propose a new mining strategy that directly extracts the SAP with the maximal graph size without applying node enumeration. Given an initial graph template and a number of ARGs, we develop an unsupervised method to modify the graph template into the maximal-size SAP. From a practical perspective, this research develops a general platform for learning the category model (i.e., the SAP) from cluttered visual data (i.e., the ARGs) without labeling "what is where," thereby opening the possibility for a series of applications in the era of big visual data. Experiments demonstrate the superior performance of the proposed method on RGB/RGB-D images and videos.
Approximate von Neumann entropy for directed graphs.
Ye, Cheng; Wilson, Richard C; Comin, César H; Costa, Luciano da F; Hancock, Edwin R
2014-05-01
In this paper, we develop an entropy measure for assessing the structural complexity of directed graphs. Although there are many existing alternative measures for quantifying the structural properties of undirected graphs, there are relatively few corresponding measures for directed graphs. To fill this gap in the literature, we explore an alternative technique that is applicable to directed graphs. We commence by using Chung's generalization of the Laplacian of a directed graph to extend the computation of von Neumann entropy from undirected to directed graphs. We provide a simplified form of the entropy which can be expressed in terms of simple node in-degree and out-degree statistics. Moreover, we find approximate forms of the von Neumann entropy that apply to both weakly and strongly directed graphs, and that can be used to characterize network structure. We illustrate the usefulness of these simplified entropy forms defined in this paper on both artificial and real-world data sets, including structures from protein databases and high energy physics theory citation networks.
The automorphisms of Novikov algebras in low dimensions
NASA Astrophysics Data System (ADS)
Bai, Chengming; Meng, Daoji
2003-07-01
Novikov algebras were introduced in connection with Poisson brackets of hydrodynamic type and Hamiltonian operators in the formal variational calculus. They also correspond to a class of vertex algebras. An automorphism of a Novikov algebra is a linear isomorphism varphi satisfying varphi(xy) = varphi(x)varphi(y) which keeps the algebraic structure. The set of automorphisms of a Novikov algebra is a Lie group whose Lie algebra is just the Novikov algebra's derivation algebra. The theory of automorphisms plays an important role in the study of Novikov algebras. In this paper, we study the automorphisms of Novikov algebras. We get some results on their properties and classification in low dimensions. These results are fundamental in a certain sense, and they will serve as a guide for further development. Moreover, we apply these results to classify Gel'fand-Dorfman bialgebras and Novikov-Poisson algebras. These results also can be used to study certain phase spaces and geometric classical r-matrices.
NASA Technical Reports Server (NTRS)
Lieberman, R. N.
1972-01-01
Given a directed graph, a natural topology is defined and relationships between standard topological properties and graph theoretical concepts are studied. In particular, the properties of connectivity and separatedness are investigated. A metric is introduced which is shown to be related to separatedness. The topological notions of continuity and homeomorphism. A class of maps is studied which preserve both graph and topological properties. Applications involving strong maps and contractions are also presented.
Graph distance for complex networks
NASA Astrophysics Data System (ADS)
Shimada, Yutaka; Hirata, Yoshito; Ikeguchi, Tohru; Aihara, Kazuyuki
2016-10-01
Networks are widely used as a tool for describing diverse real complex systems and have been successfully applied to many fields. The distance between networks is one of the most fundamental concepts for properly classifying real networks, detecting temporal changes in network structures, and effectively predicting their temporal evolution. However, this distance has rarely been discussed in the theory of complex networks. Here, we propose a graph distance between networks based on a Laplacian matrix that reflects the structural and dynamical properties of networked dynamical systems. Our results indicate that the Laplacian-based graph distance effectively quantifies the structural difference between complex networks. We further show that our approach successfully elucidates the temporal properties underlying temporal networks observed in the context of face-to-face human interactions.
Graph distance for complex networks
Shimada, Yutaka; Hirata, Yoshito; Ikeguchi, Tohru; Aihara, Kazuyuki
2016-01-01
Networks are widely used as a tool for describing diverse real complex systems and have been successfully applied to many fields. The distance between networks is one of the most fundamental concepts for properly classifying real networks, detecting temporal changes in network structures, and effectively predicting their temporal evolution. However, this distance has rarely been discussed in the theory of complex networks. Here, we propose a graph distance between networks based on a Laplacian matrix that reflects the structural and dynamical properties of networked dynamical systems. Our results indicate that the Laplacian-based graph distance effectively quantifies the structural difference between complex networks. We further show that our approach successfully elucidates the temporal properties underlying temporal networks observed in the context of face-to-face human interactions. PMID:27725690
Recognition of Probe Ptolemaic Graphs
NASA Astrophysics Data System (ADS)
Chang, Maw-Shang; Hung, Ling-Ju
Let G denote a graph class. An undirected graph G is called a probe G graph if one can make G a graph in G by adding edges between vertices in some independent set of G. By definition graph class G is a subclass of probe G graphs. Ptolemaic graphs are chordal and induced gem free. They form a subclass of both chordal graphs and distance-hereditary graphs. Many problems NP-hard on chordal graphs can be solved in polynomial time on ptolemaic graphs. We proposed an O(nm)-time algorithm to recognize probe ptolemaic graphs where n and m are the numbers of vertices and edges of the input graph respectively.
Lothian, Josh; Powers, Sarah S; Sullivan, Blair D; Baker, Matthew B; Schrock, Jonathan; Poole, Stephen W
2013-12-01
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 dierent 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.
Frishman, Yaniv; Tal, Ayellet
2008-01-01
This paper presents an algorithm for drawing a sequence of graphs online. The algorithm strives to maintain the global structure of the graph and thus the user's mental map, while allowing arbitrary modifications between consecutive layouts. The algorithm works online and uses various execution culling methods in order to reduce the layout time and handle large dynamic graphs. Techniques for representing graphs on the GPU allow a speedup by a factor of up to 17 compared to the CPU implementation. The scalability of the algorithm across GPU generations is demonstrated. Applications of the algorithm to the visualization of discussion threads in Internet sites and to the visualization of social networks are provided.
Constrained Markovian Dynamics of Random Graphs
NASA Astrophysics Data System (ADS)
Coolen, A. C. C.; de Martino, A.; Annibale, A.
2009-09-01
We introduce a statistical mechanics formalism for the study of constrained graph evolution as a Markovian stochastic process, in analogy with that available for spin systems, deriving its basic properties and highlighting the role of the `mobility' (the number of allowed moves for any given graph). As an application of the general theory we analyze the properties of degree-preserving Markov chains based on elementary edge switchings. We give an exact yet simple formula for the mobility in terms of the graph's adjacency matrix and its spectrum. This formula allows us to define acceptance probabilities for edge switchings, such that the Markov chains become controlled Glauber-type detailed balance processes, designed to evolve to any required invariant measure (representing the asymptotic frequencies with which the allowed graphs are visited during the process). As a corollary we also derive a condition in terms of simple degree statistics, sufficient to guarantee that, in the limit where the number of nodes diverges, even for state-independent acceptance probabilities of proposed moves the invariant measure of the process will be uniform. We test our theory on synthetic graphs and on realistic larger graphs as studied in cellular biology, showing explicitly that, for instances where the simple edge swap dynamics fails to converge to the uniform measure, a suitably modified Markov chain instead generates the correct phase space sampling.
Dynamic molecular graphs: "hopping" structures.
Cortés-Guzmán, Fernando; Rocha-Rinza, Tomas; Guevara-Vela, José Manuel; Cuevas, Gabriel; Gómez, Rosa María
2014-05-05
This work aims to contribute to the discussion about the suitability of bond paths and bond-critical points as indicators of chemical bonding defined within the theoretical framework of the quantum theory of atoms in molecules. For this purpose, we consider the temporal evolution of the molecular structure of [Fe{C(CH2 )3 }(CO)3 ] throughout Born-Oppenheimer molecular dynamics (BOMD), which illustrates the changing behaviour of the molecular graph (MG) of an electronic system. Several MGs with significant lifespans are observed across the BOMD simulations. The bond paths between the trimethylenemethane and the metallic core are uninterruptedly formed and broken. This situation is reminiscent of a "hopping" ligand over the iron atom. The molecular graph wherein the bonding between trimethylenemethane and the iron atom takes place only by means of the tertiary carbon atom has the longest lifespan of all the considered structures, which is consistent with the MG found by X-ray diffraction experiments and quantum chemical calculations. In contrast, the η(4) complex predicted by molecular-orbital theory has an extremely brief lifetime. The lifespan of different molecular structures is related to bond descriptors on the basis of the topology of the electron density such as the ellipticities at the FeCH2 bond-critical points and electron delocalisation indices. This work also proposes the concept of a dynamic molecular graph composed of the different structures found throughout the BOMD trajectories in analogy to a resonance hybrid of Lewis structures. It is our hope that the notion of dynamic molecular graphs will prove useful in the discussion of electronic systems, in particular for those in which analysis on the basis of static structures leads to controversial conclusions.
Graphs, matrices, and the GraphBLAS: Seven good reasons
Kepner, Jeremy; Bader, David; Buluç, Aydın; ...
2015-01-01
The analysis of graphs has become increasingly important to a wide range of applications. Graph analysis presents a number of unique challenges in the areas of (1) software complexity, (2) data complexity, (3) security, (4) mathematical complexity, (5) theoretical analysis, (6) serial performance, and (7) parallel performance. Implementing graph algorithms using matrix-based approaches provides a number of promising solutions to these challenges. The GraphBLAS standard (istcbigdata.org/GraphBlas) is being developed to bring the potential of matrix based graph algorithms to the broadest possible audience. The GraphBLAS mathematically defines a core set of matrix-based graph operations that can be used to implementmore » a wide class of graph algorithms in a wide range of programming environments. This paper provides an introduction to the GraphBLAS and describes how the GraphBLAS can be used to address many of the challenges associated with analysis of graphs.« less
Graphs, matrices, and the GraphBLAS: Seven good reasons
Kepner, Jeremy; Bader, David; Buluç, Aydın; Gilbert, John; Mattson, Timothy; Meyerhenke, Henning
2015-01-01
The analysis of graphs has become increasingly important to a wide range of applications. Graph analysis presents a number of unique challenges in the areas of (1) software complexity, (2) data complexity, (3) security, (4) mathematical complexity, (5) theoretical analysis, (6) serial performance, and (7) parallel performance. Implementing graph algorithms using matrix-based approaches provides a number of promising solutions to these challenges. The GraphBLAS standard (istcbigdata.org/GraphBlas) is being developed to bring the potential of matrix based graph algorithms to the broadest possible audience. The GraphBLAS mathematically defines a core set of matrix-based graph operations that can be used to implement a wide class of graph algorithms in a wide range of programming environments. This paper provides an introduction to the GraphBLAS and describes how the GraphBLAS can be used to address many of the challenges associated with analysis of graphs.
Prediction of Algebraic Instabilities
NASA Astrophysics Data System (ADS)
Zaretzky, Paula; King, Kristina; Hill, Nicole; Keithley, Kimberlee; Barlow, Nathaniel; Weinstein, Steven; Cromer, Michael
2016-11-01
A widely unexplored type of hydrodynamic instability is examined - large-time algebraic growth. Such growth occurs on the threshold of (exponentially) neutral stability. A new methodology is provided for predicting the algebraic growth rate of an initial disturbance, when applied to the governing differential equation (or dispersion relation) describing wave propagation in dispersive media. Several types of algebraic instabilities are explored in the context of both linear and nonlinear waves.
Tadić, Bosiljka; Andjelković, Miroslav; Boshkoska, Biljana Mileva; Levnajić, Zoran
2016-01-01
Human behaviour in various circumstances mirrors the corresponding brain connectivity patterns, which are suitably represented by functional brain networks. While the objective analysis of these networks by graph theory tools deepened our understanding of brain functions, the multi-brain structures and connections underlying human social behaviour remain largely unexplored. In this study, we analyse the aggregate graph that maps coordination of EEG signals previously recorded during spoken communications in two groups of six listeners and two speakers. Applying an innovative approach based on the algebraic topology of graphs, we analyse higher-order topological complexes consisting of mutually interwoven cliques of a high order to which the identified functional connections organise. Our results reveal that the topological quantifiers provide new suitable measures for differences in the brain activity patterns and inter-brain synchronisation between speakers and listeners. Moreover, the higher topological complexity correlates with the listener’s concentration to the story, confirmed by self-rating, and closeness to the speaker’s brain activity pattern, which is measured by network-to-network distance. The connectivity structures of the frontal and parietal lobe consistently constitute distinct clusters, which extend across the listener’s group. Formally, the topology quantifiers of the multi-brain communities exceed the sum of those of the participating individuals and also reflect the listener’s rated attributes of the speaker and the narrated subject. In the broader context, the presented study exposes the relevance of higher topological structures (besides standard graph measures) for characterising functional brain networks under different stimuli. PMID:27880802
Random graphs with arbitrary degree distributions and their applications
NASA Astrophysics Data System (ADS)
Newman, M. E. J.; Strogatz, S. H.; Watts, D. J.
2001-08-01
Recent work on the structure of social networks and the internet has focused attention on graphs with distributions of vertex degree that are significantly different from the Poisson degree distributions that have been widely studied in the past. In this paper we develop in detail the theory of random graphs with arbitrary degree distributions. In addition to simple undirected, unipartite graphs, we examine the properties of directed and bipartite graphs. Among other results, we derive exact expressions for the position of the phase transition at which a giant component first forms, the mean component size, the size of the giant component if there is one, the mean number of vertices a certain distance away from a randomly chosen vertex, and the average vertex-vertex distance within a graph. We apply our theory to some real-world graphs, including the world-wide web and collaboration graphs of scientists and Fortune 1000 company directors. We demonstrate that in some cases random graphs with appropriate distributions of vertex degree predict with surprising accuracy the behavior of the real world, while in others there is a measurable discrepancy between theory and reality, perhaps indicating the presence of additional social structure in the network that is not captured by the random graph.
Connecting Algebra and Chemistry.
ERIC Educational Resources Information Center
O'Connor, Sean
2003-01-01
Correlates high school chemistry curriculum with high school algebra curriculum and makes the case for an integrated approach to mathematics and science instruction. Focuses on process integration. (DDR)
ERIC Educational Resources Information Center
Hirsch, Christian R.
1975-01-01
Using a set of worksheets, students will discover and apply Euler's formula regarding connected planar graphs and play and analyze the game of Sprouts. One sheet leads to the discovery of Euler's formula; another concerns traversability of a graph; another gives an example and a game involving these ideas. (Author/KM)
ERIC Educational Resources Information Center
Lind, Joy; Narayan, Darren
2009-01-01
We present the topic of graph connectivity along with a famous theorem of Menger in the real-world setting of the national computer network infrastructure of "National LambdaRail". We include a set of exercises where students reinforce their understanding of graph connectivity by analysing the "National LambdaRail" network. Finally, we give…
Caetano, Tibério S; McAuley, Julian J; Cheng, Li; Le, Quoc V; Smola, Alex J
2009-06-01
As a fundamental problem in pattern recognition, graph matching has applications in a variety of fields, from computer vision to computational biology. In graph matching, patterns are modeled as graphs and pattern recognition amounts to finding a correspondence between the nodes of different graphs. Many formulations of this problem can be cast in general as a quadratic assignment problem, where a linear term in the objective function encodes node compatibility and a quadratic term encodes edge compatibility. The main research focus in this theme is about designing efficient algorithms for approximately solving the quadratic assignment problem, since it is NP-hard. In this paper we turn our attention to a different question: how to estimate compatibility functions such that the solution of the resulting graph matching problem best matches the expected solution that a human would manually provide. We present a method for learning graph matching: the training examples are pairs of graphs and the 'labels' are matches between them. Our experimental results reveal that learning can substantially improve the performance of standard graph matching algorithms. In particular, we find that simple linear assignment with such a learning scheme outperforms Graduated Assignment with bistochastic normalisation, a state-of-the-art quadratic assignment relaxation algorithm.
ERIC Educational Resources Information Center
Petrosino, Anthony
2012-01-01
This article responds to arguments by Skidmore and Thompson (this issue of "Educational Researcher") that a graph published more than 10 years ago was erroneously reproduced and "gratuitously damaged" perceptions of the quality of education research. After describing the purpose of the original graph, the author counters assertions that the graph…
ERIC Educational Resources Information Center
Shen, Ji
2009-01-01
In the Walking Out Graphs Lesson described here, students experience several types of representations used to describe motion, including words, sentences, equations, graphs, data tables, and actions. The most important theme of this lesson is that students have to understand the consistency among these representations and form the habit of…
Graphing from Everyday Experience.
ERIC Educational Resources Information Center
Carraher, David; Schliemann, Analucia; Nemirousky, Ricardo
1995-01-01
Discusses the importance of teaching grounded in the everyday experiences and concerns of the learners. Studies how people with limited school experience can understand graphs and concludes that individuals with limited academic education can clarify the role of everyday experiences in learning about graphs. (ASK)
ERIC Educational Resources Information Center
Johnson, Millie
1997-01-01
Graphs from media sources and questions developed from them can be used in the middle school mathematics classroom. Graphs depict storage temperature on a milk carton; air pressure measurements on a package of shock absorbers; sleep-wake patterns of an infant; a dog's breathing patterns; and the angle, velocity, and radius of a leaning bicyclist…
ERIC Educational Resources Information Center
Doto, Julianne; Golbeck, Susan
2007-01-01
Collecting data and analyzing the results of experiments is difficult for children. The authors found a surprising way to help their third graders make graphs and draw conclusions from their data: digital photographs. The pictures bridged the gap between an abstract graph and the plants it represented. With the support of the photos, students…
Directed Abelian algebras and their application to stochastic models
NASA Astrophysics Data System (ADS)
Alcaraz, F. C.; Rittenberg, V.
2008-10-01
With each directed acyclic graph (this includes some D -dimensional lattices) one can associate some Abelian algebras that we call directed Abelian algebras (DAAs). On each site of the graph one attaches a generator of the algebra. These algebras depend on several parameters and are semisimple. Using any DAA, one can define a family of Hamiltonians which give the continuous time evolution of a stochastic process. The calculation of the spectra and ground-state wave functions (stationary state probability distributions) is an easy algebraic exercise. If one considers D -dimensional lattices and chooses Hamiltonians linear in the generators, in finite-size scaling the Hamiltonian spectrum is gapless with a critical dynamic exponent z=D . One possible application of the DAA is to sandpile models. In the paper we present this application, considering one- and two-dimensional lattices. In the one-dimensional case, when the DAA conserves the number of particles, the avalanches belong to the random walker universality class (critical exponent στ=3/2 ). We study the local density of particles inside large avalanches, showing a depletion of particles at the source of the avalanche and an enrichment at its end. In two dimensions we did extensive Monte-Carlo simulations and found στ=1.780±0.005 .
Linear Algebra Revisited: An Attempt to Understand Students' Conceptual Difficulties
ERIC Educational Resources Information Center
Britton, Sandra; Henderson, Jenny
2009-01-01
This article looks at some of the conceptual difficulties that students have in a linear algebra course. An overview of previous research in this area is given, and the various theories that have been espoused regarding the reasons that students find linear algebra so difficult are discussed. Student responses to two questions testing the ability…
Neural Population Dynamics Modeled by Mean-Field Graphs
NASA Astrophysics Data System (ADS)
Kozma, Robert; Puljic, Marko
2011-09-01
In this work we apply random graph theory approach to describe neural population dynamics. There are important advantages of using random graph theory approach in addition to ordinary and partial differential equations. The mathematical theory of large-scale random graphs provides an efficient tool to describe transitions between high- and low-dimensional spaces. Recent advances in studying neural correlates of higher cognition indicate the significance of sudden changes in space-time neurodynamics, which can be efficiently described as phase transitions in the neuropil medium. Phase transitions are rigorously defined mathematically on random graph sequences and they can be naturally generalized to a class of percolation processes called neuropercolation. In this work we employ mean-field graphs with given vertex degree distribution and edge strength distribution. We demonstrate the emergence of collective oscillations in the style of brains.
Thinking Graphically: Connecting Vision and Cognition during Graph Comprehension
ERIC Educational Resources Information Center
Ratwani, Raj M.; Trafton, J. Gregory; Boehm-Davis, Deborah A.
2008-01-01
Task analytic theories of graph comprehension account for the perceptual and conceptual processes required to extract specific information from graphs. Comparatively, the processes underlying information integration have received less attention. We propose a new framework for information integration that highlights visual integration and cognitive…
On Graph Isomorphism and the PageRank Algorithm
2008-09-01
International Conference on Robotics and Automation, IEEE, 2004. [Chu94] F. Chung . Spectral Graph Theory, Regional Conference Series in Mathematics, vol. 92...Enumeration of cospectral graphs”, European Journal of Combinatorics, 25:199–211, 2004. [HeL93] B. Hendrickson and R. Leland . “An improved spectral graph
An algebra of discrete event processes
NASA Technical Reports Server (NTRS)
Heymann, Michael; Meyer, George
1991-01-01
This report deals with an algebraic framework for modeling and control of discrete event processes. The report consists of two parts. The first part is introductory, and consists of a tutorial survey of the theory of concurrency in the spirit of Hoare's CSP, and an examination of the suitability of such an algebraic framework for dealing with various aspects of discrete event control. To this end a new concurrency operator is introduced and it is shown how the resulting framework can be applied. It is further shown that a suitable theory that deals with the new concurrency operator must be developed. In the second part of the report the formal algebra of discrete event control is developed. At the present time the second part of the report is still an incomplete and occasionally tentative working paper.
Algebraic quantum gravity (AQG): II. Semiclassical analysis
NASA Astrophysics Data System (ADS)
Giesel, K.; Thiemann, T.
2007-05-01
In the previous paper (Giesel and Thiemann 2006 Conceptual setup Preprint gr-qc/0607099) a new combinatorial and thus purely algebraical approach to quantum gravity, called algebraic quantum gravity (AQG), was introduced. In the framework of AQG, existing semiclassical tools can be applied to operators that encode the dynamics of AQG such as the master constraint operator. In this paper, we will analyse the semiclassical limit of the (extended) algebraic master constraint operator and show that it reproduces the correct infinitesimal generators of general relativity. Therefore, the question of whether general relativity is included in the semiclassical sector of the theory, which is still an open problem in LQG, can be significantly improved in the framework of AQG. For the calculations, we will substitute SU(2) with U(1)3. That this substitution is justified will be demonstrated in the third paper (Giesel and Thiemann 2006 Semiclassical perturbation theory Preprint gr-qc/0607101) of this series.
Graph representation of protein free energy landscape
Li, Minghai; Duan, Mojie; Fan, Jue; Huo, Shuanghong; Han, Li
2013-11-14
The thermodynamics and kinetics of protein folding and protein conformational changes are governed by the underlying free energy landscape. However, the multidimensional nature of the free energy landscape makes it difficult to describe. We propose to use a weighted-graph approach to depict the free energy landscape with the nodes on the graph representing the conformational states and the edge weights reflecting the free energy barriers between the states. Our graph is constructed from a molecular dynamics trajectory and does not involve projecting the multi-dimensional free energy landscape onto a low-dimensional space defined by a few order parameters. The calculation of free energy barriers was based on transition-path theory using the MSMBuilder2 package. We compare our graph with the widely used transition disconnectivity graph (TRDG) which is constructed from the same trajectory and show that our approach gives more accurate description of the free energy landscape than the TRDG approach even though the latter can be organized into a simple tree representation. The weighted-graph is a general approach and can be used on any complex system.
Graph representation of protein free energy landscape.
Li, Minghai; Duan, Mojie; Fan, Jue; Han, Li; Huo, Shuanghong
2013-11-14
The thermodynamics and kinetics of protein folding and protein conformational changes are governed by the underlying free energy landscape. However, the multidimensional nature of the free energy landscape makes it difficult to describe. We propose to use a weighted-graph approach to depict the free energy landscape with the nodes on the graph representing the conformational states and the edge weights reflecting the free energy barriers between the states. Our graph is constructed from a molecular dynamics trajectory and does not involve projecting the multi-dimensional free energy landscape onto a low-dimensional space defined by a few order parameters. The calculation of free energy barriers was based on transition-path theory using the MSMBuilder2 package. We compare our graph with the widely used transition disconnectivity graph (TRDG) which is constructed from the same trajectory and show that our approach gives more accurate description of the free energy landscape than the TRDG approach even though the latter can be organized into a simple tree representation. The weighted-graph is a general approach and can be used on any complex system.
From time series to complex networks: the visibility graph.
Lacasa, Lucas; Luque, Bartolo; Ballesteros, Fernando; Luque, Jordi; Nuño, Juan Carlos
2008-04-01
In this work we present a simple and fast computational method, the visibility algorithm, that converts a time series into a graph. The constructed graph inherits several properties of the series in its structure. Thereby, periodic series convert into regular graphs, and random series do so into random graphs. Moreover, fractal series convert into scale-free networks, enhancing the fact that power law degree distributions are related to fractality, something highly discussed recently. Some remarkable examples and analytical tools are outlined to test the method's reliability. Many different measures, recently developed in the complex network theory, could by means of this new approach characterize time series from a new point of view.
Bicovariant quantum algebras and quantum Lie algebras
NASA Astrophysics Data System (ADS)
Schupp, Peter; Watts, Paul; Zumino, Bruno
1993-10-01
A bicovariant calculus of differential operators on a quantum group is constructed in a natural way, using invariant maps from Fun(mathfrak{G}_q ) to U q g, given by elements of the pure braid group. These operators—the “reflection matrix” Y≡L + SL - being a special case—generate algebras that linearly close under adjoint actions, i.e. they form generalized Lie algebras. We establish the connection between the Hopf algebra formulation of the calculus and a formulation in compact matrix form which is quite powerful for actual computations and as applications we find the quantum determinant and an orthogonality relation for Y in SO q (N).
ERIC Educational Resources Information Center
Cavanagh, Sean
2008-01-01
A popular humorist and avowed mathphobe once declared that in real life, there's no such thing as algebra. Kathie Wilson knows better. Most of the students in her 8th grade class will be thrust into algebra, the definitive course that heralds the beginning of high school mathematics, next school year. The problem: Many of them are about three…
Parastatistics Algebras and Combinatorics
NASA Astrophysics Data System (ADS)
Popov, T.
2005-03-01
We consider the algebras spanned by the creation parafermionic and parabosonic operators which give rise to generalized parastatistics Fock spaces. The basis of such a generalized Fock space can be labelled by Young tableaux which are combinatorial objects. By means of quantum deformations a nice combinatorial structure of the algebra of the plactic monoid that lies behind the parastatistics is revealed.
Algebraic Reasoning through Patterns
ERIC Educational Resources Information Center
Rivera, F. D.; Becker, Joanne Rossi
2009-01-01
This article presents the results of a three-year study that explores students' performance on patterning tasks involving prealgebra and algebra. The findings, insights, and issues drawn from the study are intended to help teach prealgebra and algebra. In the remainder of the article, the authors take a more global view of the three-year study on…
Learning Activity Package, Algebra.
ERIC Educational Resources Information Center
Evans, Diane
A set of ten teacher-prepared Learning Activity Packages (LAPs) in beginning algebra and nine in intermediate algebra, these units cover sets, properties of operations, number systems, open expressions, solution sets of equations and inequalities in one and two variables, exponents, factoring and polynomials, relations and functions, radicals,…
NASA Technical Reports Server (NTRS)
Lawson, C. L.; Krogh, F. T.; Gold, S. S.; Kincaid, D. R.; Sullivan, J.; Williams, E.; Hanson, R. J.; Haskell, K.; Dongarra, J.; Moler, C. B.
1982-01-01
The Basic Linear Algebra Subprograms (BLAS) library is a collection of 38 FORTRAN-callable routines for performing basic operations of numerical linear algebra. BLAS library is portable and efficient source of basic operations for designers of programs involving linear algebriac computations. BLAS library is supplied in portable FORTRAN and Assembler code versions for IBM 370, UNIVAC 1100 and CDC 6000 series computers.
ERIC Educational Resources Information Center
Levy, Alissa Beth
2012-01-01
The California Department of Education (CDE) has long asserted that success Algebra I by Grade 8 is the goal for all California public school students. In fact, the state's accountability system penalizes schools that do not require all of their students to take the Algebra I end-of-course examination by Grade 8 (CDE, 2009). In this dissertation,…
ERIC Educational Resources Information Center
Merlin, Ethan M.
2013-01-01
This article describes how the author has developed tasks for students that address the missed "essence of the matter" of algebraic transformations. Specifically, he has found that having students practice "perceiving" algebraic structure--by naming the "glue" in the expressions, drawing expressions using…