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Sample records for algebraic number theory

  1. 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.

  2. Quantum Algorithms for Problems in Number Theory, Algebraic Geometry, and Group Theory

    NASA Astrophysics Data System (ADS)

    van Dam, Wim; Sasaki, Yoshitaka

    2013-09-01

    Quantum computers can execute algorithms that sometimes dramatically outperform classical computation. Undoubtedly the best-known example of this is Shor's discovery of an efficient quantum algorithm for factoring integers, whereas the same problem appears to be intractable on classical computers. Understanding what other computational problems can be solved significantly faster using quantum algorithms is one of the major challenges in the theory of quantum computation, and such algorithms motivate the formidable task of building a large-scale quantum computer. This article will review the current state of quantum algorithms, focusing on algorithms for problems with an algebraic flavor that achieve an apparent superpolynomial speedup over classical computation.

  3. Characterizing the Development of Specialized Mathematical Content Knowledge for Teaching in Algebraic Reasoning and Number Theory

    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…

  4. Algebraic Mean Field Theory

    NASA Astrophysics Data System (ADS)

    Dankova, T. S.; Rosensteel, G.

    1998-10-01

    Mean field theory has an unexpected group theoretic mathematical foundation. Instead of representation theory which applies to most group theoretic quantum models, Hartree-Fock and Hartree-Fock-Bogoliubov have been formulated in terms of coadjoint orbits for the groups U(n) and O(2n). The general theory of mean fields is formulated for an arbitrary Lie algebra L of fermion operators. The moment map provides the correspondence between the Hilbert space of microscopic wave functions and the dual space L^* of densities. The coadjoint orbits of the group in the dual space are phase spaces on which time-dependent mean field theory is equivalent to a classical Hamiltonian dynamical system. Indeed it forms a finite-dimensional Lax system. The mean field theories for the Elliott SU(3) and symplectic Sp(3,R) algebras are constructed explicitly in the coadjoint orbit framework.

  5. Expansion of real numbers by algebraic numbers

    NASA Astrophysics Data System (ADS)

    Hajime, Kaneko

    2008-01-01

    In this paper we represent the fractional part of ξαn, where ξ is a nonzero real number and α is an algebraic number. By using this representation, we give new lower bounds for the distance from ξαn to the nearest integer.

  6. Second-Order Algebraic Theories

    NASA Astrophysics Data System (ADS)

    Fiore, Marcelo; Mahmoud, Ola

    Fiore and Hur [10] recently introduced a conservative extension of universal algebra and equational logic from first to second order. Second-order universal algebra and second-order equational logic respectively provide a model theory and a formal deductive system for languages with variable binding and parameterised metavariables. This work completes the foundations of the subject from the viewpoint of categorical algebra. Specifically, the paper introduces the notion of second-order algebraic theory and develops its basic theory. Two categorical equivalences are established: at the syntactic level, that of second-order equational presentations and second-order algebraic theories; at the semantic level, that of second-order algebras and second-order functorial models. Our development includes a mathematical definition of syntactic translation between second-order equational presentations. This gives the first formalisation of notions such as encodings and transforms in the context of languages with variable binding.

  7. Algebraic theory of molecules

    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.

  8. Characteristic Numbers of Matrix Lie Algebras

    NASA Astrophysics Data System (ADS)

    Zhang, Yu-Feng; Fan, En-Gui

    2008-04-01

    A notion of characteristic number of matrix Lie algebras is defined, which is devoted to distinguishing various Lie algebras that are used to generate integrable couplings of soliton equations. That is, the exact classification of the matrix Lie algebras by using computational formulas is given. Here the characteristic numbers also describe the relations between soliton solutions of the stationary zero curvature equations expressed by various Lie algebras.

  9. On the binary expansions of algebraic numbers

    SciTech Connect

    Bailey, David H.; Borwein, Jonathan M.; Crandall, Richard E.; Pomerance, Carl

    2003-07-01

    Employing concepts from additive number theory, together with results on binary evaluations and partial series, we establish bounds on the density of 1's in the binary expansions of real algebraic numbers. A central result is that if a real y has algebraic degree D > 1, then the number {number_sign}(|y|, N) of 1-bits in the expansion of |y| through bit position N satisfies {number_sign}(|y|, N) > CN{sup 1/D} for a positive number C (depending on y) and sufficiently large N. This in itself establishes the transcendency of a class of reals {summation}{sub n{ge}0} 1/2{sup f(n)} where the integer-valued function f grows sufficiently fast; say, faster than any fixed power of n. By these methods we re-establish the transcendency of the Kempner--Mahler number {summation}{sub n{ge}0}1/2{sup 2{sup n}}, yet we can also handle numbers with a substantially denser occurrence of 1's. Though the number z = {summation}{sub n{ge}0}1/2{sup n{sup 2}} has too high a 1's density for application of our central result, we are able to invoke some rather intricate number-theoretical analysis and extended computations to reveal aspects of the binary structure of z{sup 2}.

  10. Algebraic structures of sequences of numbers

    NASA Astrophysics Data System (ADS)

    Huang, I.-Chiau

    2012-09-01

    For certain sequences of numbers, commutative rings with a module structure over a non-commutative ring are constructed. Identities of these numbers are considered as realizations of algebraic relations.

  11. Computer algebra and transport theory.

    SciTech Connect

    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.

  12. L∞-algebra models and higher Chern-Simons theories

    NASA Astrophysics Data System (ADS)

    Ritter, Patricia; Sämann, Christian

    2016-10-01

    We continue our study of zero-dimensional field theories in which the fields take values in a strong homotopy Lie algebra. In the first part, we review in detail how higher Chern-Simons theories arise in the AKSZ-formalism. These theories form a universal starting point for the construction of L∞-algebra models. We then show how to describe superconformal field theories and how to perform dimensional reductions in this context. In the second part, we demonstrate that Nambu-Poisson and multisymplectic manifolds are closely related via their Heisenberg algebras. As a byproduct of our discussion, we find central Lie p-algebra extensions of 𝔰𝔬(p + 2). Finally, we study a number of L∞-algebra models which are physically interesting and which exhibit quantized multisymplectic manifolds as vacuum solutions.

  13. Algebraic independence of p-adic numbers

    NASA Astrophysics Data System (ADS)

    Nesterenko, Yu V.

    2008-06-01

    We prove lower bounds for the transcendence degree of fields generated by values of the p-adic exponential function. In particular, we estimate the transcendence degree of the field \\mathbb Q(e^{\\alpha_1},\\dots,e^{\\alpha_d}), where \\alpha_1,\\dots,\\alpha_d are algebraic (over the field of rational numbers) p-adic numbers that form a basis of a finite extension of \\mathbb Q.

  14. Formal scattering theory by an algebraic approach

    NASA Astrophysics Data System (ADS)

    Alhassid, Y.; Levine, R. D.

    1985-02-01

    Formal scattering theory is recast in a Lie-algebraic form. The central result is an algebraic Lippmann-Schwinger equation for the wave operator from which an algebraic form of the Born series (containing only linked terms) is obtained. When a finite Lie algebra is sufficient, The Mo/ller wave operator, on the energy shell, can be solved for explicitly as an element of the corresponding group. The method is illustrated for the separable potential whose relevant algebra is found to be U(1,1).

  15. Geometric Algebra Software for Teaching Complex Numbers, Vectors and Spinors.

    ERIC Educational Resources Information Center

    Lounesto, Pertti; And Others

    1990-01-01

    Presents a calculator-type computer program, CLICAL, in conjunction with complex number, vector, and other geometric algebra computations. Compares the CLICAL with other symbolic programs for algebra. (Author/YP)

  16. Fourier theory and C∗-algebras

    NASA Astrophysics Data System (ADS)

    Bédos, Erik; Conti, Roberto

    2016-07-01

    We discuss a number of results concerning the Fourier series of elements in reduced twisted group C∗-algebras of discrete groups, and, more generally, in reduced crossed products associated to twisted actions of discrete groups on unital C∗-algebras. A major part of the article gives a review of our previous work on this topic, but some new results are also included.

  17. Imperfect Cloning Operations in Algebraic Quantum Theory

    NASA Astrophysics Data System (ADS)

    Kitajima, Yuichiro

    2015-01-01

    No-cloning theorem says that there is no unitary operation that makes perfect clones of non-orthogonal quantum states. The objective of the present paper is to examine whether an imperfect cloning operation exists or not in a C*-algebraic framework. We define a universal -imperfect cloning operation which tolerates a finite loss of fidelity in the cloned state, and show that an individual system's algebra of observables is abelian if and only if there is a universal -imperfect cloning operation in the case where the loss of fidelity is less than . Therefore in this case no universal -imperfect cloning operation is possible in algebraic quantum theory.

  18. 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.

  19. On two notions of complexity of algebraic numbers

    NASA Astrophysics Data System (ADS)

    Bugeaud, Yann; Evertse, Jan-Hendrik

    we derive new, improved lower bounds for the block complexity of an irrational algebraic number and for the number of digit changes in the b-ary expansion of an irrational algebraic number. To this end, we apply a quantitative version of the Subspace Theorem due to Evertse and Schlickewei (2002).

  20. Shifted genus expanded W ∞ algebra and shifted Hurwitz numbers

    NASA Astrophysics Data System (ADS)

    Zheng, Quan

    2016-05-01

    We construct the shifted genus expanded W ∞ algebra, which is isomorphic to the central subalgebra A ∞ of infinite symmetric group algebra and to the shifted Schur symmetrical function algebra Λ* defined by Okounkov and Olshanskii. As an application, we get some differential equations for the generating functions of the shifted Hurwitz numbers; thus, we can express the generating functions in terms of the shifted genus expanded cut-and-join operators.

  1. Integrable maps from Galois differential algebras, Borel transforms and number sequences

    NASA Astrophysics Data System (ADS)

    Tempesta, Piergiulio

    A new class of integrable maps, obtained as lattice versions of polynomial dynamical systems is introduced. These systems are obtained by means of a discretization procedure that preserves several analytic and algebraic properties of a given differential equation, in particular symmetries and integrability (see Tempesta, 2010 [40]). Our approach is based on the properties of a suitable Galois differential algebra, that we shall call a Rota algebra. A formulation of the procedure in terms of category theory is proposed. In order to render the lattice dynamics confined, a Borel regularization is also adopted. As a byproduct of the theory, a connection between number sequences and integrability is discussed.

  2. Dual number coefficient octonion algebra, field equations and conservation laws

    NASA Astrophysics Data System (ADS)

    Chanyal, B. C.; Chanyal, S. K.

    2016-08-01

    Starting with octonion algebra, we develop the dual number coefficient octonion (DNCO) algebra having sixteen components. DNCO forms of generalized potential, field and current equations are discussed in consistent manner. We have made an attempt to write the DNCO form of generalized Dirac-Maxwell's equations in presence of electric and magnetic charges (dyons). Accordingly, we demonstrate the work-energy theorem of classical mechanics reproducing the continuity equation for dyons in terms of DNCO algebra. Further, we discuss the DNCO form of linear momentum conservation law for dyons.

  3. Geometric and Algebraic Approaches in the Concept of Complex Numbers

    ERIC Educational Resources Information Center

    Panaoura, A.; Elia, I.; Gagatsis, A.; Giatilis, G.-P.

    2006-01-01

    This study explores pupils' performance and processes in tasks involving equations and inequalities of complex numbers requiring conversions from a geometric representation to an algebraic representation and conversions in the reverse direction, and also in complex numbers problem solving. Data were collected from 95 pupils of the final grade from…

  4. On the number of connected components of random algebraic hypersurfaces

    NASA Astrophysics Data System (ADS)

    Fyodorov, Yan V.; Lerario, Antonio; Lundberg, Erik

    2015-09-01

    We study the expectation of the number of components b0(X) of a random algebraic hypersurface X defined by the zero set in projective space RPn of a random homogeneous polynomial f of degree d. Specifically, we consider invariant ensembles, that is Gaussian ensembles of polynomials that are invariant under an orthogonal change of variables. Fixing n, under some rescaling assumptions on the family of ensembles (as d → ∞), we prove that Eb0(X) has the same order of growth as [ Eb0(X ∩ RP1) ] n. This relates the average number of components of X to the classical problem of M. Kac (1943) on the number of zeros of the random univariate polynomial f|RP1. The proof requires an upper bound for Eb0(X), which we obtain by counting extrema using Random Matrix Theory methods from Fyodorov (2013), and it also requires a lower bound, which we obtain by a modification of the barrier method from Lerario and Lundberg (2015) and Nazarov and Sodin (2009). We also provide quantitative upper bounds on implied constants; for the real Fubini-Study model these estimates provide super-exponential decay (as n → ∞) of the leading coefficient (in d) of Eb0(X) .

  5. Misconceptions in Rational Numbers, Probability, Algebra, and Geometry

    ERIC Educational Resources Information Center

    Rakes, Christopher R.

    2010-01-01

    In this study, the author examined the relationship of probability misconceptions to algebra, geometry, and rational number misconceptions and investigated the potential of probability instruction as an intervention to address misconceptions in all 4 content areas. Through a review of literature, 5 fundamental concepts were identified that, if…

  6. Australian Curriculum Linked Lessons: Reasoning in Number and Algebra

    ERIC Educational Resources Information Center

    Day, Lorraine

    2014-01-01

    The Reasoning Proficiency in number and algebra is about children making sense of the mathematics by explaining their thinking, giving reasons for their decisions and describing mathematical situations and concepts. Lorraine Day notes, children need to be able to speak, read and write the language of mathematics while investigating pattern and…

  7. 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)

  8. Do malaria parasites follow the algebra of sex ratio theory?

    PubMed

    Schall, Jos J

    2009-03-01

    The ratio of male to female gametocytes seen in infections of Plasmodium and related haemosporidian parasites varies substantially, both within and among parasite species. Sex ratio theory, a mainstay of evolutionary biology, accounts for this variation. The theory provides an algebraic solution for the optimal sex ratio that will maximize parasite fitness. A crucial term in this solution is the probability of selfing by clone-mates within the vector (based on the clone number and their relative abundance). Definitive tests of the theory have proven elusive because of technical challenges in measuring clonal diversity within infections. Newly developed molecular methods now provide opportunities to test the theory with an exquisite precision. PMID:19201653

  9. 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.

  10. Exceptional algebraic relations for reciprocal sums of Fibonacci and Lucas numbers

    NASA Astrophysics Data System (ADS)

    Elsner, Carsten; Shimomura, Shun; Shiokawa, Iekata

    2011-09-01

    We discuss algebraic relations for reciprocal sums of Fibonacci and Lucas numbers. For a certain set of 12 such sums, we show that any two numbers are algebraically independent, and that any three are algebraically independent except for those in 22 exceptional triplets. We explicitly present algebraic relations for some of these exceptional cases.

  11. The Q-deformed Oscillator Algebra with an Integer Number Eigenvalue and a Half Odd Integer Number Eigenvalue

    NASA Astrophysics Data System (ADS)

    Chung, Won Sang

    2014-07-01

    In this paper a new q-deformed oscillator algebra with an integer number eigenvalue and a half odd integer number eigenvalue is proposed. For this algebra, the associated energy spectrum and thermodynamic behavior is discussed.

  12. The algebra of bipartite graphs and Hurwitz numbers of seamed surfaces

    NASA Astrophysics Data System (ADS)

    Alekseevskii, A. V.; Natanzon, S. M.

    2008-08-01

    We extend the definition of Hurwitz numbers to the case of seamed surfaces, which arise in new models of mathematical physics, and prove that they form a system of correlators for a Klein topological field theory in the sense defined in [1]. We find the corresponding Cardy-Frobenius algebras, which yield a method for calculating the Hurwitz numbers. As a by-product, we prove that the vector space generated by the bipartite graphs with n edges possesses a natural binary operation that makes this space into a non-commutative Frobenius algebra isomorphic to the algebra of intertwining operators for a representation of the symmetric group S_n on the space generated by the set of all partitions of a set of n elements.

  13. 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.

  14. Geometric representation of interval exchange maps over algebraic number fields

    NASA Astrophysics Data System (ADS)

    Poggiaspalla, G.; Lowenstein, J. H.; Vivaldi, F.

    2008-01-01

    This paper is concerned with the restriction of interval exchange transformations (IETs) to algebraic number fields, which leads to maps on lattices. We characterize renormalizability arithmetically and study its relationships with a geometrical quantity that we call the drift vector. We exhibit some examples of renormalizable IETs with zero and non-zero drift vectors and carry out some investigations of their properties. In particular we look for evidence of the finite decomposition property on a family of IETs extending the example studied in Lowenstein et al (2007 Dyn. Syst. 22 73-106).

  15. Aspects of Coding. Applications of Linear Algebra to Communication and Information Science. [and] A Double-Error Correcting Code. Applications of Algebra to Information Theory. Modules and Monographs in Undergraduate Mathematics and Its Applications Project. UMAP Units 336 and 337.

    ERIC Educational Resources Information Center

    Cohen, Simon; Sherman, Gary J.

    These two modules cover aspects of the coding process and algebraic coding theory. The first unit defines coding as a branch of information and communication science, which draws extensively upon many diverse mathematical fields, primarily abstract and linear algebra, number theory, probability and statistics, and combinatorial theory. Aspects of…

  16. Topological insulators and C∗-algebras: Theory and numerical practice

    NASA Astrophysics Data System (ADS)

    Hastings, Matthew B.; Loring, Terry A.

    2011-07-01

    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 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.

  17. From string theory to algebraic geometry and back

    SciTech Connect

    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.

  18. An Arithmetic-Algebraic Work Space for the Promotion of Arithmetic and Algebraic Thinking: Triangular Numbers

    ERIC Educational Resources Information Center

    Hitt, Fernando; Saboya, Mireille; Cortés Zavala, Carlos

    2016-01-01

    This paper presents an experiment that attempts to mobilise an arithmetic-algebraic way of thinking in order to articulate between arithmetic thinking and the early algebraic thinking, which is considered a prelude to algebraic thinking. In the process of building this latter way of thinking, researchers analysed pupils' spontaneous production…

  19. Classification of hypergeometric identities for pi and other logarithms of algebraic numbers.

    PubMed

    Chudnovsky, D V; Chudnovsky, G V

    1998-03-17

    This paper provides transcendental and algebraic framework for the classification of identities expressing pi and other logarithms of algebraic numbers as rapidly convergent generalized hypergeometric series in rational parameters. Algebraic and arithmetic relations between values of p+1Fp hypergeometric functions and their values are analyzed. The existing identities are explained, and new exhaustive classes of new ones are presented.

  20. Algebraic isomorphism in two-dimensional anomalous gauge theories

    SciTech Connect

    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.

  1. 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).

  2. 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.

  3. Category of trees in representation theory of quantum algebras

    SciTech Connect

    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.

  4. Polytopes, Fibonacci numbers, Hopf algebras, and quasi-symmetric functions

    NASA Astrophysics Data System (ADS)

    Buchstaber, Viktor M.; Erokhovets, Nikolai Yu

    2011-04-01

    This survey is devoted to the classical problem of flag numbers of convex polytopes, and contains an exposition of results obtained on the basis of connections between the theory of convex polytopes and a number of modern directions of research. Bibliography: 62 titles.

  5. On the b-ary expansions of algebraic irrational numbers (survey)

    NASA Astrophysics Data System (ADS)

    Kaneko, Hajime

    2011-09-01

    In this paper we consider the digits of real numbers in integral base. In particular, we introduce lower bounds of the number of nonzero digits of algebraic irrational numbers. As applications, we give criteria for algebraic independence of special values of power series.

  6. Vortex lattice theory: A linear algebra approach

    NASA Astrophysics Data System (ADS)

    Chamoun, George C.

    Vortex lattices are prevalent in a large class of physical settings that are characterized by different mathematical models. We present a coherent and generalized Hamiltonian fluid mechanics-based formulation that reduces all vortex lattices into a classic problem in linear algebra for a non-normal matrix A. Via Singular Value Decomposition (SVD), the solution lies in the null space of the matrix (i.e., we require nullity( A) > 0) as well as the distribution of its singular values. We demonstrate that this approach provides a good model for various types of vortex lattices, and makes it possible to extract a rich amount of information on them. The contributions of this thesis can be classified into four main points. The first is asymmetric equilibria. A 'Brownian ratchet' construct was used which converged to asymmetric equilibria via a random walk scheme that utilized the smallest singular value of A. Distances between configurations and equilibria were measured using the Frobenius norm ||·||F and 2-norm ||·||2, and conclusions were made on the density of equilibria within the general configuration space. The second contribution used Shannon Entropy, which we interpret as a scalar measure of the robustness, or likelihood of lattices to occur in a physical setting. Third, an analytic model was produced for vortex street patterns on the sphere by using SVD in conjunction with expressions for the center of vorticity vector and angular velocity. Equilibrium curves within the configuration space were presented as a function of the geometry, and pole vortices were shown to have a critical role in the formation and destruction of vortex streets. The fourth contribution entailed a more complete perspective of the streamline topology of vortex streets, linking the bifurcations to critical points on the equilibrium curves.

  7. On the complexity of the binary expansions of algebraic irrational numbers (survey)

    NASA Astrophysics Data System (ADS)

    Kaneko, Hajime

    2010-07-01

    Borel conjectured that all irrational numbers are normal in any integral base α. For each positive number ξ and integer α greater than 1, ξ is normal in base α if and only if the sequence ξαn (n = 0,1,…) is uniformly distributed modulo 1. In this paper we survey not only the digit of algebraic irrational numbers in integral base but also the fractional parts of geometric progressions whose common ratios are algebraic numbers greater than 1. In our main results, we give new lower bounds for the number of digit changes in the binary expansions of algebraic irrational numbers.

  8. From rational numbers to algebra: separable contributions of decimal magnitude and relational understanding of fractions.

    PubMed

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

    2015-05-01

    To understand the development of mathematical cognition and to improve instructional practices, it is critical to identify early predictors of difficulty in learning complex mathematical topics such as algebra. Recent work has shown that performance with fractions on a number line estimation task predicts algebra performance, whereas performance with whole numbers on similar estimation tasks does not. We sought to distinguish more specific precursors to algebra by measuring multiple aspects of knowledge about rational numbers. Because fractions are the first numbers that are relational expressions to which students are exposed, we investigated how understanding the relational bipartite format (a/b) of fractions might connect to later algebra performance. We presented middle school students with a battery of tests designed to measure relational understanding of fractions, procedural knowledge of fractions, and placement of fractions, decimals, and whole numbers onto number lines as well as algebra performance. Multiple regression analyses revealed that the best predictors of algebra performance were measures of relational fraction knowledge and ability to place decimals (not fractions or whole numbers) onto number lines. These findings suggest that at least two specific components of knowledge about rational numbers--relational understanding (best captured by fractions) and grasp of unidimensional magnitude (best captured by decimals)--can be linked to early success with algebraic expressions. PMID:25744594

  9. From rational numbers to algebra: separable contributions of decimal magnitude and relational understanding of fractions.

    PubMed

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

    2015-05-01

    To understand the development of mathematical cognition and to improve instructional practices, it is critical to identify early predictors of difficulty in learning complex mathematical topics such as algebra. Recent work has shown that performance with fractions on a number line estimation task predicts algebra performance, whereas performance with whole numbers on similar estimation tasks does not. We sought to distinguish more specific precursors to algebra by measuring multiple aspects of knowledge about rational numbers. Because fractions are the first numbers that are relational expressions to which students are exposed, we investigated how understanding the relational bipartite format (a/b) of fractions might connect to later algebra performance. We presented middle school students with a battery of tests designed to measure relational understanding of fractions, procedural knowledge of fractions, and placement of fractions, decimals, and whole numbers onto number lines as well as algebra performance. Multiple regression analyses revealed that the best predictors of algebra performance were measures of relational fraction knowledge and ability to place decimals (not fractions or whole numbers) onto number lines. These findings suggest that at least two specific components of knowledge about rational numbers--relational understanding (best captured by fractions) and grasp of unidimensional magnitude (best captured by decimals)--can be linked to early success with algebraic expressions.

  10. 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.

  11. Equivalent D = 3 supergravity amplitudes from double copies of three-algebra and two-algebra gauge theories.

    PubMed

    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.

  12. Distribution of real algebraic numbers of arbitrary degree in short intervals

    NASA Astrophysics Data System (ADS)

    Bernik, V. I.; Götze, F.

    2015-02-01

    We consider real algebraic numbers α of degree \\operatorname{deg}α=n and height H=H(α). There are intervals I\\subset{R} of length \\vert I\\vert whose interiors contain no real algebraic numbers α of any degree with H(α)\\lt\\frac12\\vert I\\vert-1. We prove that one can always find a constant c_1=c_1(n) such that if Q is a positive integer and Q \\gt c_1\\vert I\\vert-1, then the interior of I contains at least c_2(n)Qn+1\\vert I\\vert real algebraic numbers α with \\operatorname{deg}α=n and H(α)≤ Q. We use this result to solve a problem of Bugeaud on the regularity of the set of real algebraic numbers in short intervals.

  13. The Number Crunch game: a simple vehicle for building algebraic reasoning skills

    NASA Astrophysics Data System (ADS)

    Sugden, Steve

    2012-03-01

    A newspaper numbers game based on simple arithmetic relationships is discussed. Its potential to give students of elementary algebra practice in semi-ad hoc reasoning and to build general arithmetic reasoning skills is explored.

  14. Bagger-Lambert theory for general Lie algebras

    NASA Astrophysics Data System (ADS)

    Gomis, Jaume; Milanesi, Giuseppe; Russo, Jorge G.

    2008-06-01

    We construct the totally antisymmetric structure constants fABCD of a 3-algebra with a Lorentzian bi-invariant metric starting from an arbitrary semi-simple Lie algebra. The structure constants fABCD can be used to write down a maximally superconformal 3d theory that incorporates the expected degrees of freedom of multiple M2 branes, including the ``center-of-mass" mode described by free scalar and fermion fields. The gauge field sector reduces to a three dimensional BF term, which underlies the gauge symmetry of the theory. We comment on the issue of unitarity of the quantum theory, which is problematic, despite the fact that the specific form of the interactions prevent the ghost fields from running in the internal lines of any Feynman diagram. Giving an expectation value to one of the scalar fields leads to the maximally supersymmetric 3d Yang-Mills Lagrangian with the addition of two U(1) multiplets, one of them ghost-like, which is decoupled at large gYM.

  15. Number-theory dark matter

    NASA Astrophysics Data System (ADS)

    Nakayama, Kazunori; Takahashi, Fuminobu; Yanagida, Tsutomu T.

    2011-05-01

    We propose that the stability of dark matter is ensured by a discrete subgroup of the U(1)B-L gauge symmetry, Z(B-L). We introduce a set of chiral fermions charged under the U(1)B-L in addition to the right-handed neutrinos, and require the anomaly-cancellation conditions associated with the U(1)B-L gauge symmetry. We find that the possible number of fermions and their charges are tightly constrained, and that non-trivial solutions appear when at least five additional chiral fermions are introduced. The Fermat theorem in the number theory plays an important role in this argument. Focusing on one of the solutions, we show that there is indeed a good candidate for dark matter, whose stability is guaranteed by Z(B-L).

  16. The role of difficulty and gender in numbers, algebra, geometry and mathematics achievement

    NASA Astrophysics Data System (ADS)

    Rabab'h, Belal Sadiq Hamed; Veloo, Arsaythamby; Perumal, Selvan

    2015-05-01

    This study aims to identify the role of difficulty and gender in numbers, algebra, geometry and mathematics achievement among secondary schools students in Jordan. The respondent of the study were 337 students from eight public secondary school in Alkoura district by using stratified random sampling. The study comprised of 179 (53%) males and 158 (47%) females students. The mathematics test comprises of 30 items which has eight items for numbers, 14 items for algebra and eight items for geometry. Based on difficulties among male and female students, the findings showed that item 4 (fractions - 0.34) was most difficult for male students and item 6 (square roots - 0.39) for females in numbers. For the algebra, item 11 (inequality - 0.23) was most difficult for male students and item 6 (algebraic expressions - 0.35) for female students. In geometry, item 3 (reflection - 0.34) was most difficult for male students and item 8 (volume - 0.33) for female students. Based on gender differences, female students showed higher achievement in numbers and algebra compare to male students. On the other hand, there was no differences between male and female students achievement in geometry test. This study suggest that teachers need to give more attention on numbers and algebra when teaching mathematics.

  17. Noncommutative Common Cause Principles in algebraic quantum field theory

    SciTech Connect

    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.

  18. Calculation of exchange energies using algebraic perturbation theory

    SciTech Connect

    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.

  19. Algebraic expression of the IBM3 hamiltonian in terms of various quantum numbers

    NASA Astrophysics Data System (ADS)

    Hasegawa, M.

    1991-04-01

    The properties of the IBM3 hamiltonian are algebraically studied. The IBM3 hamiltonian determined microscopically has as characteristic that the isospin T, rmrather than the spin J, is essential to classifying the energy spectra. The T-dependence of the two-body boson interactions is expressed in terms of the Casimir operators or quantum numbers of various groups. This algebraic approach makes preparations for phenomenological understanding of light nuclei with definite isospin.

  20. Counter Conjectures: Using Manipulatives to Scaffold the Development of Number Sense and Algebra

    ERIC Educational Resources Information Center

    West, John

    2016-01-01

    This article takes the position that teachers can use simple manipulative materials to model relatively complex situations and in doing so scaffold the development of students' number sense and early algebra skills. While students' early experiences are usually dominated by the cardinal aspect of number (i.e., counting the number of items in a…

  1. An Algebraic Construction of Boundary Quantum Field Theory

    NASA Astrophysics Data System (ADS)

    Longo, Roberto; Witten, Edward

    2011-04-01

    We build up local, time translation covariant Boundary Quantum Field Theory nets of von Neumann algebras {mathcal A_V} on the Minkowski half-plane M + starting with a local conformal net {mathcal A} of von Neumann algebras on {mathbb R} and an element V of a unitary semigroup {mathcal E(mathcal A)} associated with {mathcal A}. The case V = 1 reduces to the net {mathcal A_+} considered by Rehren and one of the authors; if the vacuum character of {mathcal A} is summable, {mathcal A_V} is locally isomorphic to {mathcal A_+}. We discuss the structure of the semigroup {mathcal E(mathcal A)}. By using a one-particle version of Borchers theorem and standard subspace analysis, we provide an abstract analog of the Beurling-Lax theorem that allows us to describe, in particular, all unitaries on the one-particle Hilbert space whose second quantization promotion belongs to {mathcal E(mathcal A^{(0)})} with {mathcal A^{(0)}} the U(1)-current net. Each such unitary is attached to a scattering function or, more generally, to a symmetric inner function. We then obtain families of models via any Buchholz-Mack-Todorov extension of {mathcal A^{(0)}}. A further family of models comes from the Ising model.

  2. K-theory of the chair tiling via AF-algebras

    NASA Astrophysics Data System (ADS)

    Julien, Antoine; Savinien, Jean

    2016-08-01

    We compute the K-theory groups of the groupoid C∗-algebra of the chair tiling, using a new method. We use exact sequences of Putnam to compute these groups from the K-theory groups of the AF-algebras of the substitution and the induced lower dimensional substitutions on edges and vertices.

  3. Constructing Rotation Symmetric Boolean Functions with Maximum Algebraic Immunity on an Odd Number of Variables

    NASA Astrophysics Data System (ADS)

    Peng, Jie; Kan, Haibin

    It is well known that Boolean functions used in stream and block ciphers should have high algebraic immunity to resist algebraic attacks. Up to now, there have been many constructions of Boolean functions achieving the maximum algebraic immunity. In this paper, we present several constructions of rotation symmetric Boolean functions with maximum algebraic immunity on an odd number of variables which are not symmetric, via a study of invertible cyclic matrices over the binary field. In particular, we generalize the existing results and introduce a new method to construct all the rotation symmetric Boolean functions that differ from the majority function on two orbits. Moreover, we prove that their nonlinearities are upper bounded by 2^{n-1}-\\binom{n-1}{\\lfloor\\frac{n}{2}\\rfloor}+2(n-6).

  4. Algebraic independence of real numbers with low density of nonzero digits (survey)

    NASA Astrophysics Data System (ADS)

    Kaneko, Hajime

    2010-07-01

    We study transcendence and algebraic independence of the value f(z;w(n)) = ∑n = 1∞zw(n), where z is an algebraic number with 0<|z|<1 and w(n) (n = 1,2,…) is a strictly increasing sequence of nonnegative integers. First we survey the case where w(n) is lacunary. In our main results, we give criteria for algebraic independence of the values f(z;w(n)) in the case where z = 1/α for some integer α greater than 1 and w(n) is not lacunary. In particular, using our criteria, we deduce that the uncountable set {ηl = ∑n = 1∞α-[fl(n)]|l≥1} is algebraically independent, where fl(n) = exp((log n)1+l).

  5. Random partitions and asymptotic theory of symmetric groups, Hecke algebras and finite Chevalley groups

    NASA Astrophysics Data System (ADS)

    Méliot, Pierre-Loïc

    2010-12-01

    In this thesis, we investigate the asymptotics of random partitions chosen according to probability measures coming from the representation theory of the symmetric groups S_n and of the finite Chevalley groups GL(n,F_q) and Sp(2n,F_q). More precisely, we prove laws of large numbers and central limit theorems for the q-Plancherel measures of type A and B, the Schur-Weyl measures and the Gelfand measures. Using the RSK algorithm, it also gives results on longest increasing subsequences in random words. We develop a technique of moments (and cumulants) for random partitions, thereby using the polynomial functions on Young diagrams in the sense of Kerov and Olshanski. The algebra of polynomial functions, or observables of Young diagrams is isomorphic to the algebra of partial permutations; in the last part of the thesis, we try to generalize this beautiful construction.

  6. 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.

  7. 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.

  8. Developing Meaning for Algebraic Procedures: An Exploration of the Connections Undergraduate Students Make between Algebraic Rational Expressions and Basic Number Properties

    ERIC Educational Resources Information Center

    Yantz, Jennifer

    2013-01-01

    The attainment and retention of later algebra skills in high school has been identified as a factor significantly impacting the postsecondary success of students majoring in STEM fields. Researchers maintain that learners develop meaning for algebraic procedures by forming connections to the basic number system properties. The present study…

  9. The Hilbert polynomial and linear forms in the logarithms of algebraic numbers

    NASA Astrophysics Data System (ADS)

    Aleksentsev, Yu M.

    2008-12-01

    We prove a new estimate for homogeneous linear forms with integer coefficients in the logarithms of algebraic numbers. We obtain a qualitative improvement of the estimate depending on the coefficients of the linear form and the best value of the constant in the estimate in the case when the number of logarithms is not too large.

  10. Introduction to Number Systems, Boolean Algebra, Logic Circuits. Navy Electricity and Electronics Training Series. Module 13.

    ERIC Educational Resources Information Center

    Naval Education and Training Program Development Center, Pensacola, FL.

    This textbook is one of a series of publications designed to provide information needed by Navy personnel whose duties require an elementary and general knowledge of the fundamental concepts of number systems, logic circuits, and Boolean algebra. Topic 1, Number Systems, describes the radix; the positional notation; the decimal, binary, octal, and…

  11. Algebraic Jacobi-Perron algorithm for biquadratic numbers

    NASA Astrophysics Data System (ADS)

    Tamura, Jun-ichi; Yasutomi, Shin-ichi

    2010-07-01

    We introduced a new algorithm [6] which is something like the modified Jacobi-Perron algorithm and we conjectured that the expansion obtained by our algorithm for a¯ = (α1,…,as)∈Ks (with some natural conditions on a¯) becomes periodic for any real number field K as far as s+1 = degQ(K)≤4. We announce Theorem and some computer experiments for certain biquadratic real number fields K = Q(√m ,√n ) which support our conjecture.

  12. Generalized Heisenberg algebras and k-generalized Fibonacci numbers

    NASA Astrophysics Data System (ADS)

    Schork, Matthias

    2007-04-01

    It is shown how some of the recent results of de Souza et al (2006 J. Phys. A: Math. Gen. 39 10415) can be generalized to describe Hamiltonians whose eigenvalues are given as k-generalized Fibonacci numbers. Here k is an arbitrary integer and the cases considered by de Souza et al correspond to k = 2.

  13. The "Number Crunch" Game: A Simple Vehicle for Building Algebraic Reasoning Skills

    ERIC Educational Resources Information Center

    Sugden, Steve

    2012-01-01

    A newspaper numbers game based on simple arithmetic relationships is discussed. Its potential to give students of elementary algebra practice in semi-"ad hoc" reasoning and to build general arithmetic reasoning skills is explored. (Contains 3 figures, 7 tables and 3 notes.)

  14. Teaching of Real Numbers by Using the Archimedes-Cantor Approach and Computer Algebra Systems

    ERIC Educational Resources Information Center

    Vorob'ev, Evgenii M.

    2015-01-01

    Computer technologies and especially computer algebra systems (CAS) allow students to overcome some of the difficulties they encounter in the study of real numbers. The teaching of calculus can be considerably more effective with the use of CAS provided the didactics of the discipline makes it possible to reveal the full computational potential of…

  15. Algebra for Babies: Exploring Natural Numbers in Simple Arrays. Occasional Paper Five

    ERIC Educational Resources Information Center

    Fluellen, Jerry E., Jr.

    2008-01-01

    In 12 audio taped sessions, three kindergarten children engaged algebra in a teaching for understanding, thematic project. Toni, Asa, and Cornel had one-on-one lessons dealing with simple natural numbers, patterns, and relationships. Along the way, each child studied one of Toni Morrison's Who's got game books to explore repetition patterns in…

  16. Realization theory and quadratic optimal controllers for systems defined over Banach and Frechet algebras

    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.

  17. Number Theory in the High School Classroom.

    ERIC Educational Resources Information Center

    Dence, Thomas

    1999-01-01

    Demonstrates some of the usefulness of number theory to students on the high school setting in four areas: Fibonacci numbers, Diophantine equations, continued fractions, and algorithms for computing pi. (ASK)

  18. Relativistic theory of tidal Love numbers

    SciTech Connect

    Binnington, Taylor; Poisson, Eric

    2009-10-15

    In Newtonian gravitational theory, a tidal Love number relates the mass multipole moment created by tidal forces on a spherical body to the applied tidal field. The Love number is dimensionless, and it encodes information about the body's internal structure. We present a relativistic theory of Love numbers, which applies to compact bodies with strong internal gravities; the theory extends and completes a recent work by Flanagan and Hinderer, which revealed that the tidal Love number of a neutron star can be measured by Earth-based gravitational-wave detectors. We consider a spherical body deformed by an external tidal field, and provide precise and meaningful definitions for electric-type and magnetic-type Love numbers; and these are computed for polytropic equations of state. The theory applies to black holes as well, and we find that the relativistic Love numbers of a nonrotating black hole are all zero.

  19. Teaching of real numbers by using the Archimedes-Cantor approach and computer algebra systems

    NASA Astrophysics Data System (ADS)

    Vorob'ev, Evgenii M.

    2015-11-01

    Computer technologies and especially computer algebra systems (CAS) allow students to overcome some of the difficulties they encounter in the study of real numbers. The teaching of calculus can be considerably more effective with the use of CAS provided the didactics of the discipline makes it possible to reveal the full computational potential of CAS. In the case of real numbers, the Archimedes-Cantor approach satisfies this requirement. The name of Archimedes brings back the exhaustion method. Cantor's name reminds us of the use of Cauchy rational sequences to represent real numbers. The usage of CAS with the Archimedes-Cantor approach enables the discussion of various representations of real numbers such as graphical, decimal, approximate decimal with precision estimates, and representation as points on a straight line. Exercises with numbers such as e, π, the golden ratio ϕ, and algebraic irrational numbers can help students better understand the real numbers. The Archimedes-Cantor approach also reveals a deep and close relationship between real numbers and continuity, in particular the continuity of functions.

  20. Numerical algebraic geometry: a new perspective on gauge and string theories

    NASA Astrophysics Data System (ADS)

    Mehta, Dhagash; He, Yang-Hui; Hauensteine, Jonathan D.

    2012-07-01

    There is a rich interplay between algebraic geometry and string and gauge theories which has been recently aided immensely by advances in computational algebra. However, symbolic (Gröbner) methods are severely limited by algorithmic issues such as exponential space complexity and being highly sequential. In this paper, we introduce a novel paradigm of numerical algebraic geometry which in a plethora of situations overcomes these shortcomings. The so-called `embarrassing parallelizability' allows us to solve many problems and extract physical information which elude symbolic methods. We describe the method and then use it to solve various problems arising from physics which could not be otherwise solved.

  1. Towards a classification of modular invariant partition functions for theories based on N=4 superconformal algebras

    NASA Astrophysics Data System (ADS)

    Taormina, Anne

    1993-05-01

    The representation theory of the doubly extended N=4 superconformal algebra is reviewed. The modular properties of the corresponding characters can be derived, using characters sumrules for coset realizations of these N=4 algebras. Some particular combinations of massless characters are shown to transform as affine SU(2) characters under S and T, a fact used to completely classify the massless sector of the partition function.

  2. Analysis of Computer Algebra System Tutorials Using Cognitive Load Theory

    ERIC Educational Resources Information Center

    May, Patricia

    2004-01-01

    Most research in the area of Computer Algebra Systems (CAS) has been designed to compare the effectiveness of instructional technology to traditional lecture-based formats. While results are promising, research also indicates evidence of the steep learning curve imposed by the technology. Yet no studies have been conducted to investigate this…

  3. 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].

  4. Evaluation of Tachiya's algebraic infinite products involving Fibonacci and Lucas numbers

    NASA Astrophysics Data System (ADS)

    Sondow, Jonathan

    2011-09-01

    In 2007, Tachiya gave necessary and sufficient conditions for the transcendence of certain infinite products involving Fibonacci numbers Fk and Lucas numbers Lk. In the present note, we explicitly evaluate two classes of his algebraic examples. Special cases are ∏ n = 1∞(1+1F2n+1) = 3/φ, ∏ n = 1/∞(1+1L2n+1) = 3-φ, where φ = (1+√5 )/2 is the golden ratio.

  5. Reconstruction of the number and positions of dipoles and quadrupoles using an algebraic method

    NASA Astrophysics Data System (ADS)

    Nara, Takaaki

    2008-11-01

    Localization of dipoles and quadrupoles is important in inverse potential analysis, since they can effectively express spatially extended sources with a small number of parmeters. This paper proposes an algebraic method for reconstruction of pole positions as well as the number of dipole-quadrupoles without providing an initial parameter guess or iterative computing forward solutions. It is also shown that a magnetoencephalography inverse problem with a source model of dipole-quadrupoles in 3D space is reduced into the same problem as in 2D space.

  6. An Application of Number Theory to Cryptology.

    ERIC Educational Resources Information Center

    Snow, Joanne R.

    1989-01-01

    Discussed is an application of number theory to cryptology that can be used with secondary school students. Background on the topics is given first, followed by an explanation for use of the topic. (MNS)

  7. The Extension of the Natural-Number Domain to the Integers in the Transition from Arithmetic to Algebra.

    ERIC Educational Resources Information Center

    Gallardo, Aurora

    2002-01-01

    Analyzes from an historical perspective the extension of the natural-number domain to integers in students' transition from arithmetic to algebra in the context of word problems. Extracts four levels of acceptance of these numbers--subtrahend, relative number, isolated number and formal negative number--from historical texts. The first three…

  8. Real forms of very extended Kac-Moody algebras and theories with eight supersymmetries

    NASA Astrophysics Data System (ADS)

    Riccioni, Fabio; West, Peter; Van Proeyen, Antoine

    2008-05-01

    We consider all theories with eight supersymmetries whose reduction to three dimensions gives rise to scalars that parametrise symmetric manifolds. We conjecture that these theories are non-linear realisations of very-extended Kac-Moody algebras for suitable choices of real forms. We show for the most interesting cases that the bosonic sector of the supersymmetric theory is precisely reproduced by the corresponding non-linear realisation.

  9. Algebraic and group treatments to nonlinear displaced number states and their nonclassicality features: A new approach

    NASA Astrophysics Data System (ADS)

    N Asili, Firouzabadi; M, K. Tavassoly; M, J. Faghihi

    2015-06-01

    Recently, nonlinear displaced number states (NDNSs) have been manually introduced, in which the deformation function f(n) has been artificially added to the previously well-known displaced number states (DNSs). Indeed, just a simple comparison has been performed between the standard coherent state and nonlinear coherent state for the formation of NDNSs. In the present paper, after expressing enough physical motivation of our procedure, four distinct classes of NDNSs are presented by applying algebraic and group treatments. To achieve this purpose, by considering the DNSs and recalling the nonlinear coherent states formalism, the NDNSs are logically defined through an algebraic consideration. In addition, by using a particular class of Gilmore-Perelomov-type of SU(1, 1) and a class of SU(2) coherent states, the NDNSs are introduced via group-theoretical approach. Then, in order to examine the nonclassical behavior of these states, sub-Poissonian statistics by evaluating Mandel parameter and Wigner quasi-probability distribution function associated with the obtained NDNSs are discussed, in detail.

  10. Superspace formulation in a three-algebra approach to D=3, N=4, 5 superconformal Chern-Simons matter theories

    SciTech Connect

    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.

  11. Superspace formulation in a three-algebra approach to D=3, N=4, 5 superconformal Chern-Simons matter theories

    NASA Astrophysics Data System (ADS)

    Chen, Fa-Min; Wu, Yong-Shi

    2010-11-01

    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.JHEPFG1029-8479 09 (2008) 101.10.1088/1126-6708/2008/09/101]) 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.

  12. Vantage Theory, VT2, and Number.

    ERIC Educational Resources Information Center

    Allan, Keith

    2002-01-01

    Reviews vantage theory and makes a claim that it does not replace, but coexists with a semantics for color terms. Identifies basic facts about countability in English, and presents further evidence of the fact that the grammar of number and quantification in English is exploited to reveal different conceptualizations of what is spoken of. Claims…

  13. Proofs in Number Theory: History and Heresy.

    ERIC Educational Resources Information Center

    Rowland, Tim

    The domain of number theory lends itself particularly well to generic argument, presented with the intention of conveying the force and structure of a conventional generalized argument through the medium of a particular case. The potential of generic examples as a didactic tool is virtually unrecognized. Although the use of such examples has good…

  14. Seniority Number in Valence Bond Theory.

    PubMed

    Chen, Zhenhua; Zhou, Chen; Wu, Wei

    2015-09-01

    In this work, a hierarchy of valence bond (VB) methods based on the concept of seniority number, defined as the number of singly occupied orbitals in a determinant or an orbital configuration, is proposed and applied to the studies of the potential energy curves (PECs) of H8, N2, and C2 molecules. It is found that the seniority-based VB expansion converges more rapidly toward the full configuration interaction (FCI) or complete active space self-consistent field (CASSCF) limit and produces more accurate PECs with smaller nonparallelity errors than its molecular orbital (MO) theory-based analogue. Test results reveal that the nonorthogonal orbital-based VB theory provides a reverse but more efficient way to truncate the complete active Hilbert space by seniority numbers.

  15. Algebraic independence results for reciprocal sums of Fibonacci and Lucas numbers

    NASA Astrophysics Data System (ADS)

    Stein, Martin

    2011-09-01

    Let Fn and Ln denote the Fibonacci and Lucas numbers, respectively. D. Duverney, Ke. Nishioka, Ku. Nishioka and I. Shiokawa proved that the values of the Fibonacci zeta function ζF(2s) = Σn = 1∞Fn-2s are transcendental for any s∈N using Nesterenko's theorem on Ramanujan functions P(q), Q(q), and R(q). They obtained similar results for the Lucas zeta function ζL(2s) = Σn = 1∞Ln-2s and some related series. Later, C. Elsner, S. Shimomura and I. Shiokawa found conditions for the algebraic independence of these series. In my PhD thesis I generalized their approach and treated the following problem: We investigate all subsets of { ∑ n = 1∞1/Fn2s1, ∑ n = 1∞(-1)n+1/Fn2s2, ∑ n = 1∞1/Ln2s3, ∑ n = 1∞(-1)n+1/Ln2s4:s1,s2,s3,s4∈N} and decide on their algebraic independence over Q. Actually this is a special case of a more general theorem for reciprocal sums of binary recurrent sequences.

  16. From matrix models' topological expansion to topological string theories: counting surfaces with algebraic geometry

    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.

  17. A new approach on electromagnetism with dual number coefficient octonion algebra

    NASA Astrophysics Data System (ADS)

    Chanyal, Bhupesh Chandra; Chanyal, Sunil Kumar; Bektaş, Özcan; Yüce, Salim

    2016-08-01

    Dual number coefficient octonion (DNCO) is one of the kind of octonion, it has 16 components with an additional dual unit ɛ. Starting with DNCO algebra, we develop the generalized electromagnetic field equations of dyons regarding the DNCOS spaces, which has two octonionic space-times namely the octonionic internal space-time and the octonionic external space-time. Besides, the generalized four-potential components of dyons have been expressed with respect to the dual octonion form. Furthermore, we obtain the symmetrical form of Dirac-Maxwell equations, and the generalized potential wave equations for dyons in terms of the dual octonion. Finally, we conclude that dual octonion formulation is compact and simpler like octonion formulation.

  18. Supersymmetry and the discrete light-cone quantization limit of the Lie 3-algebra model of M theory

    NASA Astrophysics Data System (ADS)

    Sato, Matsuo

    2012-02-01

    In M. Sato, J. High Energy Phys.JHEPFG1029-8479 07 (2010) 02610.1007/JHEP07(2010)026, we proposed two models of M theory, the Hermitian 3-algebra model and Lie 3-algebra model. In this paper, we study the Lie 3-algebra model with a Lorentzian Lie 3-algebra. This model is ghost-free despite the Lorentzian 3-algebra. We show that our model satisfies two criteria as a model of M theory. First, we show that the model possesses N=1 supersymmetry in 11 dimensions. Second, we show the model reduces to Banks-Fischler-Shenker-Susskind matrix theory with finite size matrices in a discrete light-cone quantization limit.

  19. Small numbers in supersymmetric theories of nature

    SciTech Connect

    Graesser, Michael L.

    1999-05-01

    The Standard Model of particle interactions is a successful theory for describing the interactions of quarks, leptons and gauge bosons at microscopic distance scales. Despite these successes, the theory contains many unsatisfactory features. The origin of particle masses is a central mystery that has eluded experimental elucidation. In the Standard Model the known particles obtain their mass from the condensate of the so-called Higgs particle. Quantum corrections to the Higgs mass require an unnatural fine tuning in the Higgs mass of one part in 10{sup {minus}32} to obtain the correct mass scale of electroweak physics. In addition, the origin of the vast hierarchy between the mass scales of the electroweak and quantum gravity physics is not explained in the current theory. Supersymmetric extensions to the Standard Model are not plagued by this fine tuning issue and may therefore be relevant in Nature. In the minimal supersymmetric Standard Model there is also a natural explanation for electroweak symmetry breaking. Supersymmetric Grand Unified Theories also correctly predict a parameter of the Standard Model. This provides non-trivial indirect evidence for these theories. The most general supersymmetric extension to the Standard Model however, is excluded by many physical processes, such as rare flavor changing processes, and the non-observation of the instability of the proton. These processes provide important information about the possible structure such a theory. In particular, certain parameters in this theory must be rather small. A physics explanation for why this is the case would be desirable. It is striking that the gauge couplings of the Standard Model unify if there is supersymmetry close to the weak scale. This suggests that at high energies Nature is described by a supersymmetric Grand Unified Theory. But the mass scale of unification must be introduced into the theory since it does not coincide with the probable mass scale of strong quantum gravity

  20. The Casimir Effect from the Point of View of Algebraic Quantum Field Theory

    NASA Astrophysics Data System (ADS)

    Dappiaggi, Claudio; Nosari, Gabriele; Pinamonti, Nicola

    2016-06-01

    We consider a region of Minkowski spacetime bounded either by one or by two parallel, infinitely extended plates orthogonal to a spatial direction and a real Klein-Gordon field satisfying Dirichlet boundary conditions. We quantize these two systems within the algebraic approach to quantum field theory using the so-called functional formalism. As a first step we construct a suitable unital ∗-algebra of observables whose generating functionals are characterized by a labelling space which is at the same time optimal and separating and fulfils the F-locality property. Subsequently we give a definition for these systems of Hadamard states and we investigate explicit examples. In the case of a single plate, it turns out that one can build algebraic states via a pull-back of those on the whole Minkowski spacetime, moreover inheriting from them the Hadamard property. When we consider instead two plates, algebraic states can be put in correspondence with those on flat spacetime via the so-called method of images, which we translate to the algebraic setting. For a massless scalar field we show that this procedure works perfectly for a large class of quasi-free states including the Poincaré vacuum and KMS states. Eventually Wick polynomials are introduced. Contrary to the Minkowski case, the extended algebras, built in globally hyperbolic subregions can be collected in a global counterpart only after a suitable deformation which is expressed locally in terms of a *-isomorphism. As a last step, we construct explicitly the two-point function and the regularized energy density, showing, moreover, that the outcome is consistent with the standard results of the Casimir effect.

  1. Scale-adaptive tensor algebra for local many-body methods of electronic structure theory

    SciTech Connect

    Liakh, Dmitry I

    2014-01-01

    While the formalism of multiresolution analysis (MRA), based on wavelets and adaptive integral representations of operators, is actively progressing in electronic structure theory (mostly on the independent-particle level and, recently, second-order perturbation theory), the concepts of multiresolution and adaptivity can also be utilized within the traditional formulation of correlated (many-particle) theory which is based on second quantization and the corresponding (generally nonorthogonal) tensor algebra. In this paper, we present a formalism called scale-adaptive tensor algebra (SATA) which exploits an adaptive representation of tensors of many-body operators via the local adjustment of the basis set quality. Given a series of locally supported fragment bases of a progressively lower quality, we formulate the explicit rules for tensor algebra operations dealing with adaptively resolved tensor operands. The formalism suggested is expected to enhance the applicability and reliability of local correlated many-body methods of electronic structure theory, especially those directly based on atomic orbitals (or any other localized basis functions).

  2. The theory of Enceladus and Dione: An application of computerized algebra in dynamical astronomy

    NASA Technical Reports Server (NTRS)

    Jefferys, W. H.; Ries, L. M.

    1974-01-01

    A theory of Saturn's satellites Enceladus and Dione is discussed which is literal (all constants of integration appear explicitly), canonically invariant (the Hori-Lie method is used), and which correctly handles the eccentricity-type resonance between the two satellites. Algebraic manipulations are designed to be performed using the TRIGMAN formula manipulation language, and computer programs were developed so that, with minor modifications, they can be used on the Mimas-Tethys and Titan-Hyperion systems.

  3. What Kinds of Numbers Do Students Assign to Literal Symbols? Aspects of the Transition from Arithmetic to Algebra

    ERIC Educational Resources Information Center

    Christou, Konstantinos P.; Vosniadou, Stella

    2012-01-01

    Three experiments used multiple methods--open-ended assessments, multiple-choice questionnaires, and interviews--to investigate the hypothesis that the development of students' understanding of the concept of real variable in algebra may be influenced in fundamental ways by their initial concept of number, which seems to be organized around the…

  4. Algebraic solutions for UB F(5 ) -OB F(6 ) quantum phase transition in odd-mass-number nuclei

    NASA Astrophysics Data System (ADS)

    Jafarizadeh, M. A.; Ghapanvari, M.; Fouladi, N.

    2015-11-01

    The spherical to γ -unstable nuclei shape-phase transition in odd-A nuclei is investigated by using the dual algebraic structures and the affine SU (1 ,1 ) ̂ Lie algebra within the framework of the interacting boson-fermion model. The new algebraic solution for odd-A nuclei is introduced. In this model, single j =1 /2 and 3/2 fermions are coupled with an even-even boson core. Energy spectra, quadrupole electromagnetic transitions, and an expectation value of the d -boson number operator are presented. Experimental evidence for the UB F(5 ) -OB F(6 ) transition in odd-A Ba and Rh isotopes is presented. The low-states energy spectra and B (E 2 ) values for these nuclei are also calculated and compared with the experimental data.

  5. True orbit simulation of piecewise linear and linear fractional maps of arbitrary dimension using algebraic numbers

    NASA Astrophysics Data System (ADS)

    Saito, Asaki; Yasutomi, Shin-ichi; Tamura, Jun-ichi; Ito, Shunji

    2015-06-01

    We introduce a true orbit generation method enabling exact simulations of dynamical systems defined by arbitrary-dimensional piecewise linear fractional maps, including piecewise linear maps, with rational coefficients. This method can generate sufficiently long true orbits which reproduce typical behaviors (inherent behaviors) of these systems, by properly selecting algebraic numbers in accordance with the dimension of the target system, and involving only integer arithmetic. By applying our method to three dynamical systems—that is, the baker's transformation, the map associated with a modified Jacobi-Perron algorithm, and an open flow system—we demonstrate that it can reproduce their typical behaviors that have been very difficult to reproduce with conventional simulation methods. In particular, for the first two maps, we show that we can generate true orbits displaying the same statistical properties as typical orbits, by estimating the marginal densities of their invariant measures. For the open flow system, we show that an obtained true orbit correctly converges to the stable period-1 orbit, which is inherently possessed by the system.

  6. True orbit simulation of piecewise linear and linear fractional maps of arbitrary dimension using algebraic numbers.

    PubMed

    Saito, Asaki; Yasutomi, Shin-ichi; Tamura, Jun-ichi; Ito, Shunji

    2015-06-01

    We introduce a true orbit generation method enabling exact simulations of dynamical systems defined by arbitrary-dimensional piecewise linear fractional maps, including piecewise linear maps, with rational coefficients. This method can generate sufficiently long true orbits which reproduce typical behaviors (inherent behaviors) of these systems, by properly selecting algebraic numbers in accordance with the dimension of the target system, and involving only integer arithmetic. By applying our method to three dynamical systems-that is, the baker's transformation, the map associated with a modified Jacobi-Perron algorithm, and an open flow system-we demonstrate that it can reproduce their typical behaviors that have been very difficult to reproduce with conventional simulation methods. In particular, for the first two maps, we show that we can generate true orbits displaying the same statistical properties as typical orbits, by estimating the marginal densities of their invariant measures. For the open flow system, we show that an obtained true orbit correctly converges to the stable period-1 orbit, which is inherently possessed by the system.

  7. Computing with impure numbers - Automatic consistency checking and units conversion using computer algebra

    NASA Technical Reports Server (NTRS)

    Stoutemyer, D. R.

    1977-01-01

    The computer algebra language MACSYMA enables the programmer to include symbolic physical units in computer calculations, and features automatic detection of dimensionally-inhomogeneous formulas and conversion of inconsistent units in a dimensionally homogeneous formula. Some examples illustrate these features.

  8. Representations of Conformal Nets, Universal C*-Algebras and K-Theory

    NASA Astrophysics Data System (ADS)

    Carpi, Sebastiano; Conti, Roberto; Hillier, Robin; Weiner, Mihály

    2013-05-01

    We study the representation theory of a conformal net {{A}} on S 1 from a K-theoretical point of view using its universal C*-algebra {C^*({A})}. We prove that if {{A}} satisfies the split property then, for every representation π of {{A}} with finite statistical dimension, {π(C^*({A}))} is weakly closed and hence a finite direct sum of type I∞ factors. We define the more manageable locally normal universal C*-algebra {C_ln^*({A})} as the quotient of {C^*({A})} by its largest ideal vanishing in all locally normal representations and we investigate its structure. In particular, if {{A}} is completely rational with n sectors, then {C_ln^*({A})} is a direct sum of n type I∞ factors. Its ideal {{K}_{A}} of compact operators has nontrivial K-theory, and we prove that the DHR endomorphisms of {C^*({A})} with finite statistical dimension act on {{K}_{A}}, giving rise to an action of the fusion semiring of DHR sectors on {K_0({K}_{A})}. Moreover, we show that this action corresponds to the regular representation of the associated fusion algebra.

  9. Promoting Number Theory in High Schools or Birthday Problem and Number Theory

    ERIC Educational Resources Information Center

    Srinivasan, V. K.

    2010-01-01

    The author introduces the birthday problem in this article. This can amuse willing members of any birthday party. This problem can also be used as the motivational first day lecture in number theory for the gifted students in high schools or in community colleges or in undergraduate classes in colleges.

  10. S-duality and the prepotential in N={2}^{star } theories (I): the ADE algebras

    NASA Astrophysics Data System (ADS)

    Billó, M.; Frau, M.; Fucito, F.; Lerda, A.; Morales, J. F.

    2015-11-01

    The prepotential of N={2}^{star } supersymmetric theories with unitary gauge groups in an Ω background satisfies a modular anomaly equation that can be recursively solved order by order in an expansion for small mass. By requiring that S-duality acts on the prepotential as a Fourier transform we generalise this result to N={2}^{star } theories with gauge algebras of the D and E type and show that their prepotentials can be written in terms of quasi-modular forms of SL(2, {Z}) . The results are checked against microscopic multi-instanton calculus based on localization for the A and D series and reproduce the known 1-instanton prepotential of the pure N=2 theories for any gauge group of ADE type. Our results can also be used to obtain the multi-instanton terms in the exceptional theories for which the microscopic instanton calculus and the ADHM construction are not available.

  11. A novel realization of the Virasoro algebra in number state space

    NASA Astrophysics Data System (ADS)

    Rasetti, M.; Marletto, C.

    2009-08-01

    The group of diffeomorphisms is crucial in quantum computing. Representing it by vector fields over a d-manifold, d⩾2, accounting for both projective action and conformal symmetry at the quantum mechanical level, requires the direct-sum decomposition of tensor product for non-compact algebras, viable only for su(1,1). As a step towards the solution, a realization of the ( d=1) Virasoro algebra Vir∼Diff(S) in the universal envelope of su(1,1) (and h(1)) is presented, which is simple in the discrete positive series irreducible unitary representation Dκ(+) of su(1,1).

  12. On Algebraic Singularities, Finite Graphs and D-Brane Gauge Theories: A String Theoretic Perspective

    NASA Astrophysics Data System (ADS)

    He, Yang-Hui

    2002-09-01

    In this writing we shall address certain beautiful inter-relations between the construction of 4-dimensional supersymmetric gauge theories and resolution of algebraic singularities, from the perspective of String Theory. We review in some detail the requisite background in both the mathematics, such as orbifolds, symplectic quotients and quiver representations, as well as the physics, such as gauged linear sigma models, geometrical engineering, Hanany-Witten setups and D-brane probes. We investigate aspects of world-volume gauge dynamics using D-brane resolutions of various Calabi-Yau singularities, notably Gorenstein quotients and toric singularities. Attention will be paid to the general methodology of constructing gauge theories for these singular backgrounds, with and without the presence of the NS-NS B-field, as well as the T-duals to brane setups and branes wrapping cycles in the mirror geometry. Applications of such diverse and elegant mathematics as crepant resolution of algebraic singularities, representation of finite groups and finite graphs, modular invariants of affine Lie algebras, etc. will naturally arise. Various viewpoints and generalisations of McKay's Correspondence will also be considered. The present work is a transcription of excerpts from the first three volumes of the author's PhD thesis which was written under the direction of Prof. A. Hanany - to whom he is much indebted - at the Centre for Theoretical Physics of MIT, and which, at the suggestion of friends, he posts to the ArXiv pro hac vice; it is his sincerest wish that the ensuing pages might be of some small use to the beginning student.

  13. On a new type of \\ell-adic regulator for algebraic number fields (the \\ell-adic regulator without logarithms)

    NASA Astrophysics Data System (ADS)

    Kuz'min, L. V.

    2015-02-01

    For an algebraic number field K such that a prime \\ell splits completely in K, we define a regulator R_\\ell(K)\\in Z_\\ell that characterizes the subgroup of universal norms from the cyclotomic Z_\\ell-extension of K in the completed group of S-units of K, where S consists of all prime divisors of \\ell. We prove that the inequality R_\\ell(K)\

  14. The theory of Enceladus and Dione - An application of computerized algebra in dynamical astronomy

    NASA Technical Reports Server (NTRS)

    Jefferys, W. H.; Ries, L. M.

    1975-01-01

    The orbits of the satellites of the outer planets are poorly known, due to lack of attention over the past half century. We have been developing a new theory of Saturn's satellites Enceladus and Dione which is literal (all constants of integration appear explicitly), canonically invariant (the Hori-Lie method is used), and which correctly handles the eccentricity-type resonance between the two satellites. The algebraic manipulations are being performed using the TRIGMAN formula manipulation language, and the programs have been developed so that with minor modifications they can be used on the Mimas-Tethys and Titan-Hyperion systems.

  15. Longer Bars for Bigger Numbers? Children's Usage and Understanding of Graphical Representations of Algebraic Problems

    ERIC Educational Resources Information Center

    Lee, Kerry; Khng, Kiat Hui; Ng, Swee Fong; Ng Lan Kong, Jeremy

    2013-01-01

    In Singapore, primary school students are taught to use bar diagrams to represent known and unknown values in algebraic word problems. However, little is known about students' understanding of these graphical representations. We investigated whether students use and think of the bar diagrams in a concrete or a more abstract fashion. We also…

  16. The Minimum Number of Inputs Required for the Controllability of Linear Differential Algebraic Equations

    NASA Astrophysics Data System (ADS)

    Shcheglova, A. A.

    2009-09-01

    Linear control differential algebraic equations are considered. The issue of minimum dimension of the control vector necessitated for complete controllability of the system on any closed interval from the domain of definition is investigated. The problem is analyzed in connection with the time invariant systems having regular matrix pencils and also systems with real-analytic or smooth coefficients, which possess some structural forms.

  17. Conceptualizing Routines of Practice That Support Algebraic Reasoning in Elementary Schools: A Constructivist Grounded Theory

    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…

  18. Kiddie Algebra

    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.…

  19. Classical and quantum Kummer shape algebras

    NASA Astrophysics Data System (ADS)

    Odzijewicz, A.; Wawreniuk, E.

    2016-07-01

    We study a family of integrable systems of nonlinearly coupled harmonic oscillators on the classical and quantum levels. We show that the integrability of these systems follows from their symmetry characterized by algebras, here called Kummer shape algebras. The resolution of identity for a wide class of reproducing kernels is found. A number of examples, illustrating this theory, are also presented.

  20. Affine.m—Mathematica package for computations in representation theory of finite-dimensional and affine Lie algebras

    NASA Astrophysics Data System (ADS)

    Nazarov, Anton

    2012-11-01

    In this paper we present Affine.m-a program for computations in representation theory of finite-dimensional and affine Lie algebras and describe implemented algorithms. The algorithms are based on the properties of weights and Weyl symmetry. Computation of weight multiplicities in irreducible and Verma modules, branching of representations and tensor product decomposition are the most important problems for us. These problems have numerous applications in physics and we provide some examples of these applications. The program is implemented in the popular computer algebra system Mathematica and works with finite-dimensional and affine Lie algebras. Catalogue identifier: AENA_v1_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AENB_v1_0.html Program obtainable from: CPC Program Library, Queen’s University, Belfast, UK Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 24 844 No. of bytes in distributed program, including test data, etc.: 1 045 908 Distribution format: tar.gz Programming language: Mathematica. Computer: i386-i686, x86_64. Operating system: Linux, Windows, Mac OS, Solaris. RAM: 5-500 Mb Classification: 4.2, 5. Nature of problem: Representation theory of finite-dimensional Lie algebras has many applications in different branches of physics, including elementary particle physics, molecular physics, nuclear physics. Representations of affine Lie algebras appear in string theories and two-dimensional conformal field theory used for the description of critical phenomena in two-dimensional systems. Also Lie symmetries play a major role in a study of quantum integrable systems. Solution method: We work with weights and roots of finite-dimensional and affine Lie algebras and use Weyl symmetry extensively. Central problems which are the computations of weight multiplicities, branching and fusion coefficients are solved using one general recurrent

  1. The algebras of large N matrix mechanics

    SciTech Connect

    Halpern, M.B.; Schwartz, C.

    1999-09-16

    Extending early work, we formulate the large N matrix mechanics of general bosonic, fermionic and supersymmetric matrix models, including Matrix theory: The Hamiltonian framework of large N matrix mechanics provides a natural setting in which to study the algebras of the large N limit, including (reduced) Lie algebras, (reduced) supersymmetry algebras and free algebras. We find in particular a broad array of new free algebras which we call symmetric Cuntz algebras, interacting symmetric Cuntz algebras, symmetric Bose/Fermi/Cuntz algebras and symmetric Cuntz superalgebras, and we discuss the role of these algebras in solving the large N theory. Most important, the interacting Cuntz algebras are associated to a set of new (hidden!) local quantities which are generically conserved only at large N. A number of other new large N phenomena are also observed, including the intrinsic nonlocality of the (reduced) trace class operators of the theory and a closely related large N field identification phenomenon which is associated to another set (this time nonlocal) of new conserved quantities at large N.

  2. 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).

  3. Classification of two-dimensional conformal supergravity theories with finite-dimensional algebras

    SciTech Connect

    McCabe, J.; Velikson, B.

    1989-07-15

    We present a list of all the finite supersymmetric extensions of thetwo-dimensional (2D) conformal algebra SO(2,2), which could lead to 2D superconformal gravity theories. The on-shell matter multiplets of the algebrasallowing canonical spin-0 and -1/2 matter are constructed. Usingthese multiplets the Weyl and Yang-Mills anomalies are calculated. There aremany new models. One new model is free of both Yang-Mills and Weyl anomalies,SU(1,1)/times/SU(2/vert bar/1,1), but only the algebrasSU(1,1)/sup 2/, OSP(1/vert bar/2)/sup 2/,OSP(2/vert bar/2)/sup 2/, SU(1,1)/times/OSP(1/vert bar/2),OSP(1/vert bar/2)/times/OSP(2/vert bar/2), andSU(1,1)/times/OSP(2/vert bar/2) lead to models free of all anomalies. Thesemodels correspond to the known string models.

  4. A possible framework of the Lipkin model obeying the SU(n) algebra in arbitrary fermion number. I: The SU(2) algebras extended from the conventional fermion pair and determination of the minimum weight states

    NASA Astrophysics Data System (ADS)

    Tsue, Yasuhiko; Providência, Constança; Providência, João da; Yamamura, Masatoshi

    2016-08-01

    The minimum weight states of the Lipkin model consisting of n single-particle levels and obeying the SU(n) algebra are investigated systematically. The basic idea is to use the SU(2) algebra, which is independent of the SU(n) algebra. This idea has already been presented by the present authors in the case of the conventional Lipkin model consisting of two single-particle levels and obeying the SU(2) algebra. If this idea is followed, the minimum weight states are determined for any fermion number appropriately occupying n single-particle levels. Naturally, the conventional minimum weight state is included: all fermions occupy energetically the lowest single-particle level in the absence of interaction. The cases n=2, 3, 4, and 5 are discussed in some detail.

  5. A Geometrical Application of Number Theory

    ERIC Educational Resources Information Center

    Srinivasan, V. K.

    2013-01-01

    Any quadruple of natural numbers {a, b, c, d} is called a "Pythagorean quadruple" if it satisfies the relationship "a[superscript 2] + b[superscript 2] + c[superscript 2]". This "Pythagorean quadruple" can always be identified with a rectangular box of dimensions "a greater than 0," "b greater than…

  6. Reichenbach's Common Cause Principle in Algebraic Quantum Field Theory with Locally Finite Degrees of Freedom

    NASA Astrophysics Data System (ADS)

    Hofer-Szabó, Gábor; Vecsernyés, Péter

    2012-02-01

    In the paper it will be shown that Reichenbach's Weak Common Cause Principle is not valid in algebraic quantum field theory with locally finite degrees of freedom in general. Namely, for any pair of projections A, B supported in spacelike separated double cones {mathcal{O}}a and {mathcal{O}}b, respectively, a correlating state can be given for which there is no nontrivial common cause (system) located in the union of the backward light cones of {mathcal{O}}a and {mathcal{O}}b and commuting with the both A and B. Since noncommuting common cause solutions are presented in these states the abandonment of commutativity can modulate this result: noncommutative Common Cause Principles might survive in these models.

  7. Algebraic connectivity analysis in molecular electronic structure theory II: total exponential formulation of second-quantised correlated methods

    NASA Astrophysics Data System (ADS)

    Lyakh, Dmitry I.; Bartlett, Rodney J.

    2014-01-01

    The fundamentality of the exponential representation of a second-quantised correlated wave function is emphasised with an accent on the physical sense of cluster amplitudes as cumulants of the correlated ansatz. Three main wave function formalisms, namely, the configuration-interaction theory, the coupled-cluster approach, and the many-body perturbation theory (as well as their extensions, e.g. the equation-of-motion coupled-cluster method, multireference schemes, etc.), are represented in an exponential form, leading to a formulation of the working equations in terms of cluster amplitudes. By expressing the corresponding many-body tensor equations in terms of cluster amplitudes, we could unambiguously check connectivity types and the asymptotic behaviour of all tensors/scalars involved (in the formal limit of an infinite number of correlated particles). In particular, the appearance of disconnected cluster amplitudes corresponds to unphysical correlations. Besides, we demonstrate that the equation-of-motion coupled-cluster approach, as well as certain excited-state configuration-interaction methods, can be recast in a fully connected (exponential) form, thus breaking the common belief that all truncated configuration-interaction methods violate connectivity. Our work is based on the recently developed algebraic framework which can be viewed as a complement to the classical diagrammatic analysis.

  8. Number Worlds: Visual and Experimental Access to Elementary Number Theory Concepts

    ERIC Educational Resources Information Center

    Sinclair, Nathalie; Zazkis, Rina; Liljedahl, Peter

    2004-01-01

    Recent research demonstrates that many issues related to the structure of natural numbers and the relationship among numbers are not well grasped by students. In this article, we describe a computer-based learning environment called "Number Worlds" that was designed to support the exploration of elementary number theory concepts by making the…

  9. A Richer Understanding of Algebra

    ERIC Educational Resources Information Center

    Foy, Michelle

    2008-01-01

    Algebra is one of those hard-to-teach topics where pupils seem to struggle to see it as more than a set of rules to learn, but this author recently used the software "Grid Algebra" from ATM, which engaged her Year 7 pupils in exploring algebraic concepts for themselves. "Grid Algebra" allows pupils to experience number, pre-algebra, and algebra…

  10. Development of a Computerized Adaptive Testing for Diagnosing the Cognitive Process of Grade 7 Students in Learning Algebra, Using Multidimensional Item Response Theory

    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…

  11. Orientation in operator algebras

    PubMed Central

    Alfsen, Erik M.; Shultz, Frederic W.

    1998-01-01

    A concept of orientation is relevant for the passage from Jordan structure to associative structure in operator algebras. The research reported in this paper bridges the approach of Connes for von Neumann algebras and ourselves for C*-algebras in a general theory of orientation that is of geometric nature and is related to dynamics. PMID:9618457

  12. Lie algebra extensions of current algebras on S3

    NASA Astrophysics Data System (ADS)

    Kori, Tosiaki; Imai, Yuto

    2015-06-01

    An affine Kac-Moody algebra is a central extension of the Lie algebra of smooth mappings from S1 to the complexification of a Lie algebra. In this paper, we shall introduce a central extension of the Lie algebra of smooth mappings from S3 to the quaternization of a Lie algebra and investigate its root space decomposition. We think this extension of current algebra might give a mathematical tool for four-dimensional conformal field theory as Kac-Moody algebras give it for two-dimensional conformal field theory.

  13. An Integrated Theory of Whole Number and Fractions Development

    ERIC Educational Resources Information Center

    Siegler, Robert S.; Thompson, Clarissa A.; Schneider, Michael

    2011-01-01

    This article proposes an integrated theory of acquisition of knowledge about whole numbers and fractions. Although whole numbers and fractions differ in many ways that influence their development, an important commonality is the centrality of knowledge of numerical magnitudes in overall understanding. The present findings with 11- and 13-year-olds…

  14. A Coursewriter II Function (FCALC) For the Manipulation of Numerical and Algebraic Expressions. Systems Memo Number One.

    ERIC Educational Resources Information Center

    Smith, Authella; And Others

    Documentation of the Coursewriter II Function FCALC is provided. The function is designed for use on the IBM 1500 instructional system and has three major applications: 1) comparison of a numeric expression in buffer 5 with a numeric expression in buffer 0; 2) comparison of an algebraic expression in buffer 5 with an algebraic expression in buffer…

  15. Open-Closed Homotopy Algebras and Strong Homotopy Leibniz Pairs Through Koszul Operad Theory

    NASA Astrophysics Data System (ADS)

    Hoefel, Eduardo; Livernet, Muriel

    2012-08-01

    Open-closed homotopy algebras (OCHA) and strong homotopy Leibniz pairs (SHLP) were introduced by Kajiura and Stasheff in 2004. In an appendix to their paper, Markl observed that an SHLP is equivalent to an algebra over the minimal model of a certain operad, without showing that the operad is Koszul. In the present paper, we show that both OCHA and SHLP are algebras over the minimal model of the zeroth homology of two versions of the Swiss-cheese operad and prove that these two operads are Koszul. As an application, we show that the OCHA operad is non-formal as a 2-colored operad but is formal as an algebra in the category of 2-collections.

  16. Theory of equivalence systems for describing algebraic closures of a generalized estimation model. II

    NASA Astrophysics Data System (ADS)

    D'Yakonov, A. G.

    2011-03-01

    Characteristic matrices and metrics of equivalence systems are studied that help give an efficient description of conjunctions of equivalence systems. Using these results, families of correct polynomials in the algebraic approach to classification are described.

  17. N = 2 Maxwell-Einstein Supergravity theories: their compact and non-compact gaugings and Jordan algebras

    SciTech Connect

    Guenaydin, M.; Sierra, G.; Townsend, P.K.

    1985-01-01

    In this talk we give a review of our work on the construction and classification of N = 2 Maxwell-Einstein Supergravity theories (MESGT), study of the underlying algebraical and geometrical structure of these theories, and their compact and non-compact gaugings. We begin by summarizing our construction of the N = 2 MESGT's in five dimensions and give a geometrical interpretation to various scalar dependent quantities in the Lagrangian, based on the constraiants implied by supersymmetry. This is followed by a complete classification of the N = 2 MESGT's whose target manifolds parametrized by the scalar fields are symmetric spaces. 39 refs.

  18. Developing Thinking in Algebra

    ERIC Educational Resources Information Center

    Mason, John; Graham, Alan; Johnson-Wilder, Sue

    2005-01-01

    This book is for people with an interest in algebra whether as a learner, or as a teacher, or perhaps as both. It is concerned with the "big ideas" of algebra and what it is to understand the process of thinking algebraically. The book has been structured according to a number of pedagogic principles that are exposed and discussed along the way,…

  19. A Non-Econometric Analysis with Algebraic Models to Forecast the Numbers of Newly Hired and Retirement of Public Primary School Teachers in Taiwan

    ERIC Educational Resources Information Center

    Lung-Hsing, Kuo; Hung-Jen, Yang; Ying-Wen, Lin; Shang-Ming, Su

    2011-01-01

    In recent years, the "street teachers" issue has caused social concern in Taiwan. This study estimates the retirement of and needs for newly hired and public primary school teachers in 2010 using an algebraic model from the paper by Husssar (1999). This recursive methodology predicts the number of newly hired public primary school teachers due to…

  20. An algebraic variational multiscale-multigrid method for large-eddy simulation of turbulent variable-density flow at low Mach number

    NASA Astrophysics Data System (ADS)

    Gravemeier, Volker; Wall, Wolfgang A.

    2010-08-01

    An algebraic variational multiscale-multigrid method is proposed for large-eddy simulation of turbulent variable-density flow at low Mach number. Scale-separating operators generated by level-transfer operators from plain aggregation algebraic multigrid methods enable the application of modeling terms to selected scale groups (here, the smaller of the resolved scales) in a purely algebraic way. Thus, for scale separation, no additional discretization besides the basic one is required, in contrast to earlier approaches based on geometric multigrid methods. The proposed method is thoroughly validated via three numerical test cases of increasing complexity: a Rayleigh-Taylor instability, turbulent channel flow with a heated and a cooled wall, and turbulent flow past a backward-facing step with heating. Results obtained with the algebraic variational multiscale-multigrid method are compared to results obtained with residual-based variational multiscale methods as well as reference results from direct numerical simulation, experiments and LES published elsewhere. Particularly, mean and various second-order velocity and temperature results obtained for turbulent channel flow with a heated and a cooled wall indicate the higher prediction quality achievable when adding a small-scale subgrid-viscosity term within the algebraic multigrid framework instead of residual-based terms accounting for the subgrid-scale part of the non-linear convective term.

  1. Avogadro's number and the kinetic theory of gases

    NASA Astrophysics Data System (ADS)

    Bryan, Ronald

    2000-02-01

    Since the rms speed vrms of gas molecules in a container does not depend on the number of molecules but only on the pressure, the volume, and the total mass of the gas, Bernoulli probably knew that his kinetic theory required an atmospheric vrms is approximately equal to 1100 mi/hr at STP. Concerns over such a high speed were no doubt lessened with measurement of Avogardo's number some 170 years later.

  2. A Successful Senior Seminar: Unsolved Problems in Number Theory

    ERIC Educational Resources Information Center

    Styer, Robert

    2014-01-01

    The "Unsolved Problems in Number Theory" book by Richard Guy provides nice problems suitable for a typical math major. We give examples of problems that have worked well in our senior seminar course and some nice results that senior math majors can obtain.

  3. Negative energy, debts, and disinformation from the viewpoint of analytic number theory

    NASA Astrophysics Data System (ADS)

    Maslov, V. P.

    2016-07-01

    The number zero and negative numbers are added to analytical number theory which includes transcendents. New solutions of Diophantine equations are applied to thermodynamics, information theory and biology.

  4. Plethystic algebras and vector symmetric functions.

    PubMed Central

    Rota, G C; Stein, J A

    1994-01-01

    An isomorphism is established between the plethystic Hopf algebra Pleth(Super[L]) and the algebra of vector symmetric functions. The Hall inner product of symmetric function theory is extended to the Hopf algebra Pleth(Super[L]). PMID:11607504

  5. The BCS-Bogoliubov and the su(2)-Algebraic Approach to the Pairing Model in Many-Fermion System --- The Quasiparticle in the Conservation of the Fermion Number ---

    NASA Astrophysics Data System (ADS)

    Tsue, Y.; Providência, C.; Providência, J. d.; Yamamura, M.

    2012-10-01

    The relation between two approaches to the su(2)-algebraic many-fermion model is discussed: (1) the BCS-Bogoliubov approach in terms of the use of the quasiparticles representing all the degrees of freedom except those related to the Cooper-pairs and (2) the conventional algebraic approach in terms of the use of the minimum weight states, from which the Cooper-pairs are excluded. In order to arrive at the goal, the idea of the quasiparticles is brought up in the conservation of the fermion number. Under the c-number replacement for the three su(2)-generators, the quasiparticles suggested in this paper are reduced to those in the BCS-Bogoliubov approach. It is also shown that the two approaches are equivalent through the c-number replacement. Further, a certain modification of the BCS-Bogoliubov approach is discussed.

  6. Automorphisms of Order Structures of Abelian Parts of Operator Algebras and Their Role in Quantum Theory

    NASA Astrophysics Data System (ADS)

    Hamhalter, Jan; Turilova, Ekaterina

    2014-10-01

    It is shown that any order isomorphism between the structures of unital associative JB subalgebras of JB algebras is given naturally by a partially linear Jordan isomorphism. The same holds for nonunital subalgebras and order isomorphisms preserving the unital subalgebra. Finally, we recover usual action of time evolution group on a von Neumann factor from group of automorphisms of the structure of Abelian subalgebras.

  7. Combinatorics of n-point functions via Hopf algebra in quantum field theory

    SciTech Connect

    Mestre, Angela; Oeckl, Robert

    2006-05-15

    We use a coproduct on the time-ordered algebra of field operators to derive simple relations between complete, connected and 1-particle irreducible n-point functions. Compared to traditional functional methods our approach is much more intrinsic and leads to efficient algorithms suitable for concrete computations. It may also be used to efficiently perform tree level computations.

  8. 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…

  9. 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.

  10. Lepton number violation in theories with a large number of standard model copies

    SciTech Connect

    Kovalenko, Sergey; Schmidt, Ivan; Paes, Heinrich

    2011-03-01

    We examine lepton number violation (LNV) in theories with a saturated black hole bound on a large number of species. Such theories have been advocated recently as a possible solution to the hierarchy problem and an explanation of the smallness of neutrino masses. On the other hand, the violation of the lepton number can be a potential phenomenological problem of this N-copy extension of the standard model as due to the low quantum gravity scale black holes may induce TeV scale LNV operators generating unacceptably large rates of LNV processes. We show, however, that this issue can be avoided by introducing a spontaneously broken U{sub 1(B-L)}. Then, due to the existence of a specific compensation mechanism between contributions of different Majorana neutrino states, LNV processes in the standard model copy become extremely suppressed with rates far beyond experimental reach.

  11. 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…

  12. Very high Mach number shocks - Theory. [in space plasmas

    NASA Technical Reports Server (NTRS)

    Quest, Kevin B.

    1986-01-01

    The theory and simulation of collisionless perpendicular supercritical shock structure is reviewed, with major emphasis on recent research results. The primary tool of investigation is the hybrid simulation method, in which the Newtonian orbits of a large number of ion macroparticles are followed numerically, and in which the electrons are treated as a charge neutralizing fluid. The principal results include the following: (1) electron resistivity is not required to explain the observed quasi-stationarity of the earth's bow shock, (2) the structure of the perpendicular shock at very high Mach numbers depends sensitively on the upstream value of beta (the ratio of the thermal to magnetic pressure) and electron resistivity, (3) two-dimensional turbulence will become increasingly important as the Mach number is increased, and (4) nonadiabatic bulk electron heating will result when a thermal electron cannot complete a gyrorbit while transiting the shock.

  13. Algebra Readiness for Students with Learning Difficulties in Grades 4-8: Support through the Study of Number

    ERIC Educational Resources Information Center

    Ketterlin-Geller, Leanne R.; Chard, David J.

    2011-01-01

    Developing proficiency in algebra is the focus of instruction in high school mathematics courses and is a minimum expectation for high school completion for all students including those with learning difficulties. However, the foundation for success is laid in grades 4-8 (aged 9-14). In this paper, we assert that students' development of algebraic…

  14. Algebraic multigrid

    NASA Technical Reports Server (NTRS)

    Ruge, J. W.; Stueben, K.

    1987-01-01

    The state of the art in algebraic multgrid (AMG) methods is discussed. The interaction between the relaxation process and the coarse grid correction necessary for proper behavior of the solution probes is discussed in detail. Sufficient conditions on relaxation and interpolation for the convergence of the V-cycle are given. The relaxation used in AMG, what smoothing means in an algebraic setting, and how it relates to the existing theory are considered. Some properties of the coarse grid operator are discussed, and results on the convergence of two-level and multilevel convergence are given. Details of an algorithm particularly studied for problems obtained by discretizing a single elliptic, second order partial differential equation are given. Results of experiments with such problems using both finite difference and finite element discretizations are presented.

  15. Geometric Algebra for Physicists

    NASA Astrophysics Data System (ADS)

    Doran, Chris; Lasenby, Anthony

    2007-11-01

    Preface; Notation; 1. Introduction; 2. Geometric algebra in two and three dimensions; 3. Classical mechanics; 4. Foundations of geometric algebra; 5. Relativity and spacetime; 6. Geometric calculus; 7. Classical electrodynamics; 8. Quantum theory and spinors; 9. Multiparticle states and quantum entanglement; 10. Geometry; 11. Further topics in calculus and group theory; 12. Lagrangian and Hamiltonian techniques; 13. Symmetry and gauge theory; 14. Gravitation; Bibliography; Index.

  16. Transportation optimization with fuzzy trapezoidal numbers based on possibility theory.

    PubMed

    He, Dayi; Li, Ran; Huang, Qi; Lei, Ping

    2014-01-01

    In this paper, a parametric method is introduced to solve fuzzy transportation problem. Considering that parameters of transportation problem have uncertainties, this paper develops a generalized fuzzy transportation problem with fuzzy supply, demand and cost. For simplicity, these parameters are assumed to be fuzzy trapezoidal numbers. Based on possibility theory and consistent with decision-makers' subjectiveness and practical requirements, the fuzzy transportation problem is transformed to a crisp linear transportation problem by defuzzifying fuzzy constraints and objectives with application of fractile and modality approach. Finally, a numerical example is provided to exemplify the application of fuzzy transportation programming and to verify the validity of the proposed methods.

  17. 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.

  18. A Linear Algebra Measure of Cluster Quality.

    ERIC Educational Resources Information Center

    Mather, Laura A.

    2000-01-01

    Discussion of models for information retrieval focuses on an application of linear algebra to text clustering, namely, a metric for measuring cluster quality based on the theory that cluster quality is proportional to the number of terms that are disjoint across the clusters. Explains term-document matrices and clustering algorithms. (Author/LRW)

  19. An extension of the classical theory of algebraic invariants to pseudo-Riemannian geometry and Hamiltonian mechanics

    NASA Astrophysics Data System (ADS)

    McLenaghan, Raymond G.; Smirnov, Roman G.; The, Dennis

    2004-03-01

    We develop a new approach to the study of Killing tensors defined in pseudo-Riemannian spaces of constant curvature that is ideologically close to the classical theory of invariants. The main idea, which provides the foundation of the new approach, is to treat a Killing tensor as an algebraic object determined by a set of parameters of the corresponding vector space of Killing tensors under the action of the isometry group. The spaces of group invariants and conformal group invariants of valence two Killing tensors defined in the Minkowski plane are described. The group invariants, which are the generators of the space of invariants, are applied to the problem of classification of orthogonally separable Hamiltonian systems defined in the Minkowski plane. Transformation formulas to separable coordinates expressed in terms of the parameters of the corresponding space of Killing tensors are presented. The results are applied to the problem of orthogonal separability of the Drach superintegrable potentials.

  20. Algebraic Semantics for Narrative

    ERIC Educational Resources Information Center

    Kahn, E.

    1974-01-01

    This paper uses discussion of Edmund Spenser's "The Faerie Queene" to present a theoretical framework for explaining the semantics of narrative discourse. The algebraic theory of finite automata is used. (CK)

  1. Linear algebraic calculation of the Green's function for large-scale electronic structure theory

    NASA Astrophysics Data System (ADS)

    Takayama, R.; Hoshi, T.; Sogabe, T.; Zhang, S.-L.; Fujiwara, T.

    2006-04-01

    A linear algebraic method named the shifted conjugate-orthogonal conjugate-gradient method is introduced for large-scale electronic structure calculation. The method gives an iterative solver algorithm of the Green’s function and the density matrix without calculating eigenstates. The problem is reduced to independent linear equations at many energy points and the calculation is actually carried out only for a single energy point. The method is robust against the round-off error and the calculation can reach the machine accuracy. With the observation of residual vectors, the accuracy can be controlled, microscopically, independently for each element of the Green’s function, and dynamically, at each step in dynamical simulations. The method is applied to both a semiconductor and a metal.

  2. Constraint algebra in bigravity

    SciTech Connect

    Soloviev, V. O.

    2015-07-15

    The number of degrees of freedom in bigravity theory is found for a potential of general form and also for the potential proposed by de Rham, Gabadadze, and Tolley (dRGT). This aim is pursued via constructing a Hamiltonian formalismand studying the Poisson algebra of constraints. A general potential leads to a theory featuring four first-class constraints generated by general covariance. The vanishing of the respective Hessian is a crucial property of the dRGT potential, and this leads to the appearance of two additional second-class constraints and, hence, to the exclusion of a superfluous degree of freedom—that is, the Boulware—Deser ghost. The use of a method that permits avoiding an explicit expression for the dRGT potential is a distinctive feature of the present study.

  3. Constraint algebra in bigravity

    NASA Astrophysics Data System (ADS)

    Soloviev, V. O.

    2015-07-01

    The number of degrees of freedom in bigravity theory is found for a potential of general form and also for the potential proposed by de Rham, Gabadadze, and Tolley (dRGT). This aim is pursued via constructing a Hamiltonian formalismand studying the Poisson algebra of constraints. A general potential leads to a theory featuring four first-class constraints generated by general covariance. The vanishing of the respective Hessian is a crucial property of the dRGT potential, and this leads to the appearance of two additional second-class constraints and, hence, to the exclusion of a superfluous degree of freedom—that is, the Boulware—Deser ghost. The use of a method that permits avoiding an explicit expression for the dRGT potential is a distinctive feature of the present study.

  4. Renormalization in Quantum Field Theory and the Riemann-Hilbert Problem I: The Hopf Algebra Structure of Graphs and the Main Theorem

    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 where C is a small circle of complex dimensions around the integer dimension D of space-time. Our main result is that the renormalized theory is just the evaluation at z=D of the holomorphic part γ+ of the Birkhoff decomposition of γ. We begin to analyse the group G and show that it is a semi-direct product of an easily understood abelian group by a highly non-trivial group closely tied up with groups of diffeomorphisms. The analysis of this latter group as well as the interpretation of the renormalization group and of anomalous dimensions are the content of our second paper with the same overall title.

  5. Algebraic trigonometry

    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.

  6. A modified large number theory with constant G

    NASA Astrophysics Data System (ADS)

    Recami, Erasmo

    1983-03-01

    The inspiring “numerology” uncovered by Dirac, Eddington, Weyl, et al. can be explained and derived when it is slightly modified so to connect the “gravitational world” (cosmos) with the “strong world” (hadron), rather than with the electromagnetic one. The aim of this note is to show the following. In the present approach to the “Large Number Theory,” cosmos and hadrons are considered to be (finite) similar systems, so that the ratio{{bar R} / {{bar R} {bar r}} of the cosmos typical lengthbar R to the hadron typical lengthbar r is constant in time (for instance, if both cosmos and hadrons undergo an expansion/contraction cycle—according to the “cyclical bigbang” hypothesis—thenbar R andbar r can be chosen to be the maximum radii, or the average radii). As a consequence, then gravitational constant G results to be independent of time. The present note is based on work done in collaboration with P. Caldirola, G. D. Maccarrone, and M. Pavšič.

  7. On Fusion Algebras and Modular Matrices

    NASA Astrophysics Data System (ADS)

    Gannon, T.; Walton, M. A.

    We consider the fusion algebras arising in e.g. Wess-Zumino-Witten conformal field theories, affine Kac-Moody algebras at positive integer level, and quantum groups at roots of unity. Using properties of the modular matrix S, we find small sets of primary fields (equivalently, sets of highest weights) which can be identified with the variables of a polynomial realization of the Ar fusion algebra at level k. We prove that for many choices of rank r and level k, the number of these variables is the minimum possible, and we conjecture that it is in fact minimal for most r and k. We also find new, systematic sources of zeros in the modular matrix S. In addition, we obtain a formula relating the entries of S at fixed points, to entries of S at smaller ranks and levels. Finally, we identify the number fields generated over the rationals by the entries of S, and by the fusion (Verlinde) eigenvalues.

  8. Computer Algebra.

    ERIC Educational Resources Information Center

    Pavelle, Richard; And Others

    1981-01-01

    Describes the nature and use of computer algebra and its applications to various physical sciences. Includes diagrams illustrating, among others, a computer algebra system and flow chart of operation of the Euclidean algorithm. (SK)

  9. Some remarks on the \\ell-adic regulator. V. Growth of the \\ell-adic regulator in the cyclotomic Z_\\ell-extension of an algebraic number field

    NASA Astrophysics Data System (ADS)

    Kuz'min, Leonid V.

    2009-10-01

    For an algebraic number field k that is either a field of CM-type (real or imaginary) or a field having Abelian completions at all places over \\ell and satisfying the feeble conjecture on the \\ell-adic regulator [1] and its cyclotomic \\mathbb{Z}_\\ell-extension k_\\infty, we obtain formulae that represent for all sufficiently large n the \\ell-adic exponent of the number R_\\ell(k_{n+1})/R_\\ell(k_n), where R_\\ell(k_n) is the \\ell-adic regulator in the sense of [1]. We discuss the meaning of the Iwasawa invariants occurring in these formulae and the resemblance between our results and the Brauer-Siegel theorem.

  10. Bethe Ansatz and the Spectral Theory of Affine Lie algebra-Valued Connections II: The Non Simply-Laced Case

    NASA Astrophysics Data System (ADS)

    Masoero, Davide; Raimondo, Andrea; Valeri, Daniele

    2016-09-01

    We assess the ODE/IM correspondence for the quantum g -KdV model, for a non-simply laced Lie algebra g. This is done by studying a meromorphic connection with values in the Langlands dual algebra of the affine Lie algebra g^{(1)} , and constructing the relevant {Ψ} -system among subdominant solutions. We then use the {Ψ} -system to prove that the generalized spectral determinants satisfy the Bethe Ansatz equations of the quantum g -KdV model. We also consider generalized Airy functions for twisted Kac-Moody algebras and we construct new explicit solutions to the Bethe Ansatz equations. The paper is a continuation of our previous work on the ODE/IM correspondence for simply-laced Lie algebras.

  11. Example of a quantum field theory based on a nonlinear Lie algebra

    SciTech Connect

    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.

  12. Example of a quantum field theory based on a nonlinear Lie algebra

    SciTech Connect

    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.

  13. Escape from intermittent repellers: periodic orbit theory for crossover from exponential to algebraic decay.

    PubMed

    Dahlqvist, P

    1999-12-01

    We apply periodic orbit theory to study the asymptotic distribution of escape times from an intermittent map. The dynamical zeta function exhibits a branch point which is associated with an asymptotic power law escape. By an analytic continuation technique we compute a pair of complex conjugate zeroes beyond the branch point, associated with a preasymptotic exponential decay. The crossover time from an exponential to a power law is also predicted. The theoretical predictions are confirmed by numerical simulation. Applications to conductance fluctuations in quantum dots are discussed.

  14. Twining characters and orbit Lie algebras

    SciTech Connect

    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.

  15. Theory of the Decoherence Effect in Finite and Infinite Open Quantum Systems Using the Algebraic Approach

    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.

  16. Similarity Theory and Dimensionless Numbers in Heat Transfer

    ERIC Educational Resources Information Center

    Marin, E.; Calderon, A.; Delgado-Vasallo, O.

    2009-01-01

    We present basic concepts underlying the so-called similarity theory that in our opinion should be explained in basic undergraduate general physics courses when dealing with heat transport problems, in particular with those involving natural or free convection. A simple example is described that can be useful in showing a criterion for neglecting…

  17. The Logical Syntax of Number Words: Theory, Acquisition and Processing

    ERIC Educational Resources Information Center

    Musolino, Julien

    2009-01-01

    Recent work on the acquisition of number words has emphasized the importance of integrating linguistic and developmental perspectives [Musolino, J. (2004). The semantics and acquisition of number words: Integrating linguistic and developmental perspectives. "Cognition 93", 1-41; Papafragou, A., Musolino, J. (2003). Scalar implicatures: Scalar…

  18. 3D Winding Number: Theory and Application to Medical Imaging

    PubMed Central

    Becciu, Alessandro; Fuster, Andrea; Pottek, Mark; van den Heuvel, Bart; ter Haar Romeny, Bart; van Assen, Hans

    2011-01-01

    We develop a new formulation, mathematically elegant, to detect critical points of 3D scalar images. It is based on a topological number, which is the generalization to three dimensions of the 2D winding number. We illustrate our method by considering three different biomedical applications, namely, detection and counting of ovarian follicles and neuronal cells and estimation of cardiac motion from tagged MR images. Qualitative and quantitative evaluation emphasizes the reliability of the results. PMID:21317978

  19. Lorentz-diffeomorphism quasi-local conserved charges and Virasoro algebra in Chern-Simons-like theories of gravity

    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.

  20. 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.

  1. Applying Insights from Research on Learning: Teaching Number Theory for Preservice Elementary Teachers.

    ERIC Educational Resources Information Center

    Brown, Anne E.

    Elementary number theory is a standard topic in the mathematical preparation of preservice elementary teachers. To understand elementary number theory, a student must be comfortable with the representation of natural numbers as the product of primes. This paper discusses methods for accomplishing this goal in a mathematics course. It also…

  2. Algebra: A Challenge at the Crossroads of Policy and Practice

    ERIC Educational Resources Information Center

    Stein, Mary Kay; Kaufman, Julia Heath; Sherman, Milan; Hillen, Amy F.

    2011-01-01

    The authors review what is known about early and universal algebra, including who is getting access to algebra and student outcomes associated with algebra course taking in general and specifically with universal algebra policies. The findings indicate that increasing numbers of students, some of whom are underprepared, are taking algebra earlier.…

  3. The logical syntax of number words: theory, acquisition and processing.

    PubMed

    Musolino, Julien

    2009-04-01

    Recent work on the acquisition of number words has emphasized the importance of integrating linguistic and developmental perspectives [Musolino, J. (2004). The semantics and acquisition of number words: Integrating linguistic and developmental perspectives. Cognition93, 1-41; Papafragou, A., Musolino, J. (2003). Scalar implicatures: Scalar implicatures: Experiments at the semantics-pragmatics interface. Cognition, 86, 253-282; Hurewitz, F., Papafragou, A., Gleitman, L., Gelman, R. (2006). Asymmetries in the acquisition of numbers and quantifiers. Language Learning and Development, 2, 76-97; Huang, Y. T., Snedeker, J., Spelke, L. (submitted for publication). What exactly do numbers mean?]. Specifically, these studies have shown that data from experimental investigations of child language can be used to illuminate core theoretical issues in the semantic and pragmatic analysis of number terms. In this article, I extend this approach to the logico-syntactic properties of number words, focusing on the way numerals interact with each other (e.g. Three boys are holding two balloons) as well as with other quantified expressions (e.g. Three boys are holding each balloon). On the basis of their intuitions, linguists have claimed that such sentences give rise to at least four different interpretations, reflecting the complexity of the linguistic structure and syntactic operations involved. Using psycholinguistic experimentation with preschoolers (n=32) and adult speakers of English (n=32), I show that (a) for adults, the intuitions of linguists can be verified experimentally, (b) by the age of 5, children have knowledge of the core aspects of the logical syntax of number words, (c) in spite of this knowledge, children nevertheless differ from adults in systematic ways, (d) the differences observed between children and adults can be accounted for on the basis of an independently motivated, linguistically-based processing model [Geurts, B. (2003). Quantifying kids. Language

  4. Positive approach: Implications for the relation between number theory and geometry, including connection to Santilli mathematics, from Fibonacci reconstitution of natural numbers and of prime numbers

    NASA Astrophysics Data System (ADS)

    Johansen, Stein E.

    2014-12-01

    The paper recapitulates some key elements in previously published results concerning exact and complete reconstitution of the field of natural numbers, both as ordinal and as cardinal numbers, from systematic unfoldment of the Fibonacci algorithm. By this natural numbers emerge as Fibonacci "atoms" and "molecules" consistent with the notion of Zeckendorf sums. Here, the sub-set of prime numbers appears not as the primary numbers, but as an epistructure from a deeper Fibonacci constitution, and is thus targeted from a "positive approach". In the Fibonacci reconstitution of number theory natural numbers show a double geometrical aspect: partly as extension in space and partly as position in a successive structuring of space. More specifically, the natural numbers are shown to be distributed by a concise 5:3 code structured from the Fibonacci algorithm via Pascal's triangle. The paper discusses possible implications for the more general relation between number theory and geometry, as well as more specifically in relation to hadronic mathematics, initiated by R.M. Santilli, and also briefly to some other recent science linking number theory more directly to geometry and natural systems.

  5. Positive approach: Implications for the relation between number theory and geometry, including connection to Santilli mathematics, from Fibonacci reconstitution of natural numbers and of prime numbers

    SciTech Connect

    Johansen, Stein E.

    2014-12-10

    The paper recapitulates some key elements in previously published results concerning exact and complete reconstitution of the field of natural numbers, both as ordinal and as cardinal numbers, from systematic unfoldment of the Fibonacci algorithm. By this natural numbers emerge as Fibonacci 'atoms' and 'molecules' consistent with the notion of Zeckendorf sums. Here, the sub-set of prime numbers appears not as the primary numbers, but as an epistructure from a deeper Fibonacci constitution, and is thus targeted from a 'positive approach'. In the Fibonacci reconstitution of number theory natural numbers show a double geometrical aspect: partly as extension in space and partly as position in a successive structuring of space. More specifically, the natural numbers are shown to be distributed by a concise 5:3 code structured from the Fibonacci algorithm via Pascal's triangle. The paper discusses possible implications for the more general relation between number theory and geometry, as well as more specifically in relation to hadronic mathematics, initiated by R.M. Santilli, and also briefly to some other recent science linking number theory more directly to geometry and natural systems.

  6. On the direct numerical simulation of moderate-Stokes-number turbulent particulate flows using algebraic-closure-based and kinetic-based moments methods

    NASA Astrophysics Data System (ADS)

    Vie, Aymeric; Masi, Enrica; Simonin, Olivier; Massot, Marc; EM2C/Ecole Centrale Paris Team; IMFT Team

    2012-11-01

    To simulate particulate flows, a convenient formalism for HPC is to use Eulerian moment methods, which describe the evolution of velocity moments instead of tracking directly the number density function (NDF) of the droplets. By using a conditional PDF approach, the Mesoscopic Eulerian Formalism (MEF) of Février et al. 2005 offers a solution for the direct numerical simulation of turbulent particulate flows, even at relatively high Stokes number. Here, we propose to compare to existing approaches used to solved for this formalism: the Algebraic-Closure-Based Moment method (Kaufmann et al. 2008, Masi et al. 2011), and the Kinetic-Based Moment Method (Yuan et al. 2010, Chalons et al. 2010, Vié et al. 2012). Therefore, the goal of the current work is to evaluate both strategies in turbulent test cases. For the ACBMM, viscosity-type and non-linear closures are envisaged, whereas for the KBMM, isotropic and anisotropic closures are investigated. A main aspect of the current methodology for the comparison is that the same numerical methods are used for both approaches. Results show that the new non-linear closure and the Anisotropic Gaussian closures are both accurate in shear flows, whereas viscosity-type and isotropic closures lead to wrong results.

  7. Earth Algebra.

    ERIC Educational Resources Information Center

    Schaufele, Christopher; Zumoff, Nancy

    Earth Algebra is an entry level college algebra course that incorporates the spirit of the National Council of Teachers of Mathematics (NCTM) Curriculum and Evaluation Standards for School Mathematics at the college level. The context of the course places mathematics at the center of one of the major current concerns of the world. Through…

  8. New family of Maxwell like algebras

    NASA Astrophysics Data System (ADS)

    Concha, P. K.; Durka, R.; Merino, N.; Rodríguez, E. K.

    2016-08-01

    We introduce an alternative way of closing Maxwell like algebras. We show, through a suitable change of basis, that resulting algebras are given by the direct sums of the AdS and the Maxwell algebras already known in the literature. Casting the result into the S-expansion method framework ensures the straightaway construction of the gravity theories based on a found enlargement.

  9. A Balancing Act: Making Sense of Algebra

    ERIC Educational Resources Information Center

    Gavin, M. Katherine; Sheffield, Linda Jensen

    2015-01-01

    For most students, algebra seems like a totally different subject than the number topics they studied in elementary school. In reality, the procedures followed in arithmetic are actually based on the properties and laws of algebra. Algebra should be a logical next step for students in extending the proficiencies they developed with number topics…

  10. [Feature extraction for breast cancer data based on geometric algebra theory and feature selection using differential evolution].

    PubMed

    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. PMID:25868233

  11. [Feature extraction for breast cancer data based on geometric algebra theory and feature selection using differential evolution].

    PubMed

    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.

  12. Discrimination in a General Algebraic Setting.

    PubMed

    Fine, Benjamin; Gaglione, Anthony; Lipschutz, Seymour; Spellman, Dennis

    2015-01-01

    Discriminating groups were introduced by G. Baumslag, A. Myasnikov, and V. Remeslennikov as an outgrowth of their theory of algebraic geometry over groups. Algebraic geometry over groups became the main method of attack on the solution of the celebrated Tarski conjectures. In this paper we explore the notion of discrimination in a general universal algebra context. As an application we provide a different proof of a theorem of Malcev on axiomatic classes of Ω-algebras.

  13. Discrimination in a General Algebraic Setting

    PubMed Central

    Fine, Benjamin; Gaglione, Anthony; Lipschutz, Seymour; Spellman, Dennis

    2015-01-01

    Discriminating groups were introduced by G. Baumslag, A. Myasnikov, and V. Remeslennikov as an outgrowth of their theory of algebraic geometry over groups. Algebraic geometry over groups became the main method of attack on the solution of the celebrated Tarski conjectures. In this paper we explore the notion of discrimination in a general universal algebra context. As an application we provide a different proof of a theorem of Malcev on axiomatic classes of Ω-algebras. PMID:26171421

  14. Redesigning College Algebra: Combining Educational Theory and Web-Based Learning to Improve Student Attitudes and Performance

    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%…

  15. College Algebra II.

    ERIC Educational Resources Information Center

    Benjamin, Carl; And Others

    Presented are student performance objectives, a student progress chart, and assignment sheets with objective and diagnostic measures for the stated performance objectives in College Algebra II. Topics covered include: differencing and complements; real numbers; factoring; fractions; linear equations; exponents and radicals; complex numbers,…

  16. Algebra: Grades 8-12.

    ERIC Educational Resources Information Center

    Instructional Objectives Exchange, Los Angeles, CA.

    A complete set of behavioral objectives for first-year algebra taught in any of grades 8 through 12 is presented. Three to six sample test items and answers are provided for each objective. Objectives were determined by surveying the most used secondary school algebra textbooks. Fourteen major categories are included: (1) whole numbers--operations…

  17. 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.

  18. Novel Linear Algebraic Theory and One-Hundred-Million-Atom Electronic Structure Calculation on The K Computer

    NASA Astrophysics Data System (ADS)

    Hoshi, Takeo; Yamazaki, Keita; Akiyama, Yohei

    A novel linear-algebraic algorithm, multiple Arnoldi method, was developed in an interdisciplinary study between physics and applied mathematics and realized one-hundred-million-atom (100-nm-scale) electronic state calculations on the K computer. The algorithms are Krylov-subspace solvers for generalized shifted linear equations and were implemented in our order-N calculation code ELSES (http://www.elses.jp/). Moreover, a method for calculating eigen states is presented as a theoretical extension.

  19. Bethe Ansatz and the Spectral Theory of Affine Lie Algebra-Valued Connections I. The simply-laced Case

    NASA Astrophysics Data System (ADS)

    Masoero, Davide; Raimondo, Andrea; Valeri, Daniele

    2016-06-01

    We study the ODE/IM correspondence for ODE associated to {widehat{mathfrak{g}}}-valued connections, for a simply-laced Lie algebra {mathfrak{g}}. We prove that subdominant solutions to the ODE defined in different fundamental representations satisfy a set of quadratic equations called {Ψ}-system. This allows us to show that the generalized spectral determinants satisfy the Bethe Ansatz equations.

  20. First time combination of frozen density embedding theory with the algebraic diagrammatic construction scheme for the polarization propagator of second order

    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.

  1. Filiform Lie algebras of order 3

    SciTech Connect

    Navarro, R. M.

    2014-04-15

    The aim of this work is to generalize a very important type of Lie algebras and superalgebras, i.e., filiform Lie (super)algebras, into the theory of Lie algebras of order F. Thus, the concept of filiform Lie algebras of order F is obtained. In particular, for F = 3 it has been proved that by using infinitesimal deformations of the associated model elementary Lie algebra it can be obtained families of filiform elementary lie algebras of order 3, analogously as that occurs into the theory of Lie algebras [M. Vergne, “Cohomologie des algèbres de Lie nilpotentes. Application à l’étude de la variété des algèbres de Lie nilpotentes,” Bull. Soc. Math. France 98, 81–116 (1970)]. Also we give the dimension, using an adaptation of the sl(2,C)-module Method, and a basis of such infinitesimal deformations in some generic cases.

  2. Number-Theory in Nuclear-Physics in Number-Theory: Non-Primality Factorization As Fission VS. Primality As Fusion; Composites' Islands of INstability: Feshbach-Resonances?

    NASA Astrophysics Data System (ADS)

    Smith, A.; Siegel, Edward Carl-Ludwig

    2011-03-01

    Numbers: primality/indivisibility/non-factorization versus compositeness/divisibility/ factorization, often in tandem but not always, provocatively close analogy to nuclear-physics: (2 + 1)=(fusion)=3; (3+1)=(fission)=4[=2 x 2]; (4+1)=(fusion)=5; (5 +1)=(fission)=6[=2 x 3]; (6 + 1)=(fusion)=7; (7+1)=(fission)=8[= 2 x 4 = 2 x 2 x 2]; (8 + 1) =(non: fission nor fusion)= 9[=3 x 3]; then ONLY composites' Islands of fusion-INstability: 8, 9, 10; then 14, 15, 16, ... Could inter-digit Feshbach-resonances exist??? Possible applications to: quantum-information/ computing non-Shore factorization, millennium-problem Riemann-hypotheses proof as Goodkin BEC intersection with graph-theory "short-cut" method: Rayleigh(1870)-Polya(1922)-"Anderson"(1958)-localization, Goldbach-conjecture, financial auditing/accounting as quantum-statistical-physics; ...abound!!! Watkins [www.secamlocal.ex.ac.uk/people/staff/mrwatkin/] "Number-Theory in Physics" many interconnections: "pure"-maths number-theory to physics including Siegel [AMS Joint Mtg.(2002)-Abs.# 973-60-124] inversion of statistics on-average digits' Newcomb(1881)-Weyl(14-16)-Benford(38)-law to reveal both the quantum and BEQS (digits = bosons = digits:"spinEless-boZos"). 1881 1885 1901 1905 1925 < 1927, altering quantum-theory history!!!

  3. Coverings of topological semi-abelian algebras

    NASA Astrophysics Data System (ADS)

    Mucuk, Osman; Demir, Serap

    2016-08-01

    In this work, we study on a category of topological semi-abelian algebras which are topological models of given an algebraic theory T whose category of models is semi-abelian; and investigate some results on the coverings of topological models of such theories yielding semi-abelian categories. We also consider the internal groupoid structure in the semi-abelian category of T-algebras, and give a criteria for the lifting of internal groupoid structure to the covering groupoids.

  4. From the Law of Large Numbers to Large Deviation Theory in Statistical Physics: An Introduction

    NASA Astrophysics Data System (ADS)

    Cecconi, Fabio; Cencini, Massimo; Puglisi, Andrea; Vergni, Davide; Vulpiani, Angelo

    This contribution aims at introducing the topics of this book. We start with a brief historical excursion on the developments from the law of large numbers to the central limit theorem and large deviations theory. The same topics are then presented using the language of probability theory. Finally, some applications of large deviations theory in physics are briefly discussed through examples taken from statistical mechanics, dynamical and disordered systems.

  5. [Experimental Course in Elementary Number Theory, Cambridge Conference on School Mathematics Feasibility Study No. 35.

    ERIC Educational Resources Information Center

    Hatch, Mary Jacqueline

    In the winter of 1965, an experimental course in Elementary Number Theory was presented to a 6th grade class in the Hosmer School, Watertown, Massachusetts. Prior to the introduction of the present material, students had been exposed in class to such topics from the University of Illinois Arithmetic Project as lattices, number lines, frame…

  6. Describing Pre-Service Teachers' Developing Understanding of Elementary Number Theory Topics

    ERIC Educational Resources Information Center

    Feldman, Ziv

    2012-01-01

    Although elementary number theory topics are closely linked to foundational topics in number and operations and are prevalent in elementary and middle grades mathematics curricula, little is currently known about how students and teachers make sense of them. This study investigated pre-service elementary teachers' developing understanding of…

  7. An Algebra-Integrated Physics and Chemistry Workshop for Teachers as a Model for Increasing the Number of Minority Students in Science and Engineering

    NASA Astrophysics Data System (ADS)

    Obot, V.; Brown, B.; Wu, T.; Wunsch, G.; Miles, A.; Morris, P.; Lindstrom, M.; Allen, J.

    The need to increase minority representation in science and engineering disciplines is well documented. Many strategies for achieving this goal have evolved over the years; yet, minority representation is still minimal. It appears that while students are naturally curious about the universe, once mention is made of mathematics as a pre-requisite to the study of science and engineering, interest seems to wane. Perhaps a possible way to get around this phobia is to incorporate the mathematics into the science courses and the science into the mathematics courses at the secondary level. This will require mathematics and science teachers to work together, re-enforcing each other so that lessons can be truly interdisciplinary. For the past two summers, we have conducted workshops for secondary school mathematics and science teachers in a large urban school district. The workshops are called "Algebra-Integrated Physics and Chemistry". These workshops are designed to introduce the teachers to mathematical modeling of physical and chemical phenomenon. T chnology (graphic calculators) is used to dis covere functions that model a particular process. We have modeled linear functions by looking at the Celsius and Fahrenheit scales. A simple experiment is heating water, measuring the temperature in both Celsius and Fahrenheit scales, plotting Celsius versus Fahrenheit temperatures, and determining their mathematical relationship. At this point, the science teacher can also go into a discussion of the meaning of temperature. In some cases readily available data can be analyzed. The ellipse and Kepler's third law is ideal when studying conic sections. In this case, available data can be used, and by plotting appropriately, cubic functions can be studied and motions of planets in their orbits near and far from the sun can be discussed. This new approach to mathematics and science will take the student to a certain comfort level so that statements such as either " I like science

  8. 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.

  9. Prospective Teachers' Views on the Use of Calculators with Computer Algebra System in Algebra Instruction

    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…

  10. Simple left-right theory: Lepton number violation at the LHC

    NASA Astrophysics Data System (ADS)

    Fileviez Perez, Pavel; Murgui, Clara; Ohmer, Sebastian

    2016-09-01

    We propose a simple left-right symmetric theory where the neutrino masses are generated at the quantum level. In this context the neutrinos are Majorana fermions and the model has the minimal degrees of freedom in the scalar sector needed for symmetry breaking and mass generation. We discuss the lepton number violating signatures with two charged leptons of different flavor and missing energy at the Large Hadron Collider in order to understand the testability of the theory.

  11. Assessment of a transitional boundary layer theory at low hypersonic Mach numbers

    NASA Technical Reports Server (NTRS)

    Shamroth, S. J.; Mcdonald, H.

    1972-01-01

    An investigation was carried out to assess the accuracy of a transitional boundary layer theory in the low hypersonic Mach number regime. The theory is based upon the simultaneous numerical solution of the boundary layer partial differential equations for the mean motion and an integral form of the turbulence kinetic energy equation which controls the magnitude and development of the Reynolds stress. Comparisions with experimental data show the theory is capable of accurately predicting heat transfer and velocity profiles through the transitional regime and correctly predicts the effects of Mach number and wall cooling on transition Reynolds number. The procedure shows promise of predicting the initiation of transition for given free stream disturbance levels. The effects on transition predictions of the pressure dilitation term and of direct absorption of acoustic energy by the boundary layer were evaluated.

  12. A note on derivations of Murray–von Neumann algebras

    PubMed Central

    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

  13. Analytic theory for the selection of 2-D needle crystal at arbitrary Peclet number

    NASA Technical Reports Server (NTRS)

    Tanveer, Saleh

    1989-01-01

    An accurate analytic theory is presented for the velocity selection of a two-dimensional needle crystal for arbitrary Peclet number for small values of the surface tension parameter. The velocity selection is caused by the effect of transcendentally small terms which are determined by analytic continuation to the complex plane and analysis of nonlinear equations. The work supports the general conclusion of previous small Peclet number analytical results of other investigators, though there are some discrepancies in details. It also addresses questions raised on the validity of selection theory owing to assumptions made on shape corrections at large distances from the tip.

  14. Analytic theory for the selection of a two-dimensional needle crystal at arbitrary Peclet number

    NASA Technical Reports Server (NTRS)

    Tanveer, S.

    1989-01-01

    An accurate analytic theory is presented for the velocity selection of a two-dimensional needle crystal for arbitrary Peclet number for small values of the surface tension parameter. The velocity selection is caused by the effect of transcendentally small terms which are determined by analytic continuation to the complex plane and analysis of nonlinear equations. The work supports the general conclusion of previous small Peclet number analytical results of other investigators, though there are some discrepancies in details. It also addresses questions raised on the validity of selection theory owing to assumptions made on shape corrections at large distances from the tip.

  15. Number-Theory in Nuclear-Physics in Number-Theory: Non-Primality Factorization As Fission VS. Primality As Fusion; Composites' Islands of INstability: Feshbach-Resonances?

    NASA Astrophysics Data System (ADS)

    Siegel, Edward

    2011-04-01

    Numbers: primality/indivisibility/non-factorization versus compositeness/divisibility /factor-ization, often in tandem but not always, provocatively close analogy to nuclear-physics: (2 + 1)=(fusion)=3; (3+1)=(fission)=4[=2 x 2]; (4+1)=(fusion)=5; (5+1)=(fission)=6[=2 x 3]; (6 + 1)=(fusion)=7; (7+1)=(fission)=8[= 2 x 4 = 2 x 2 x 2]; (8 + 1) =(non: fission nor fusion)= 9[=3 x 3]; then ONLY composites' Islands of fusion-INstability: 8, 9, 10; then 14, 15, 16,... Could inter-digit Feshbach-resonances exist??? Applications to: quantum-information and computing non-Shore factorization, millennium-problem Riemann-hypotheses physics-proof as numbers/digits Goodkin Bose-Einstein Condensation intersection with graph-theory ``short-cut'' method: Rayleigh(1870)-Polya(1922)-``Anderson'' (1958)-localization, Goldbach-conjecture, financial auditing/accounting as quantum-statistical-physics;... abound!!!

  16. Highest-weight representations of Brocherd`s algebras

    SciTech Connect

    Slansky, R.

    1997-01-01

    General features of highest-weight representations of Borcherd`s algebras are described. to show their typical features, several representations of Borcherd`s extensions of finite-dimensional algebras are analyzed. Then the example of the extension of affine- su(2) to a Borcherd`s algebra is examined. These algebras provide a natural way to extend a Kac-Moody algebra to include the hamiltonian and number-changing operators in a generalized symmetry structure.

  17. Modelling Problem-Solving Situations into Number Theory Tasks: The Route towards Generalisation

    ERIC Educational Resources Information Center

    Papadopoulos, Ioannis; Iatridou, Maria

    2010-01-01

    This paper examines the way two 10th graders cope with a non-standard generalisation problem that involves elementary concepts of number theory (more specifically linear Diophantine equations) in the geometrical context of a rectangle's area. Emphasis is given on how the students' past experience of problem solving (expressed through interplay…

  18. An Instructional Model for Teaching Proof Writing in the Number Theory Classroom

    ERIC Educational Resources Information Center

    Schabel, Carmen

    2005-01-01

    I discuss an instructional model that I have used in my number theory classes. Facets of the model include using small group work and whole class discussion, having students generate examples and counterexamples, and giving students the opportunity to write proofs and make conjectures in class. The model is designed to actively engage students in…

  19. Mean-field theory of spin-glasses with finite coordination number

    NASA Technical Reports Server (NTRS)

    Kanter, I.; Sompolinsky, H.

    1987-01-01

    The mean-field theory of dilute spin-glasses is studied in the limit where the average coordination number is finite. The zero-temperature phase diagram is calculated and the relationship between the spin-glass phase and the percolation transition is discussed. The present formalism is applicable also to graph optimization problems.

  20. The operator algebra approach to quantum groups

    PubMed Central

    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

  1. 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 solve"…

  2. Number-Theory in Nuclear-Physics in Number-Theory: Non-Primality Factorization As Fission VS. Primality As Fusion; Composites' Islands of INstability: Feshbach-Resonances?

    NASA Astrophysics Data System (ADS)

    Siegel, Edward

    2011-10-01

    Numbers: primality/indivisibility/non-factorization versus compositeness/divisibility /factor-ization, often in tandem but not always, provocatively close analogy to nuclear-physics: (2 + 1)=(fusion)=3; (3+1)=(fission)=4[=2 × 2]; (4+1)=(fusion)=5; (5 +1)=(fission)=6[=2 × 3]; (6 + 1)=(fusion)=7; (7+1)=(fission)=8[= 2 × 4 = 2 × 2 × 2]; (8 + 1) =(non: fission nor fusion)= 9[=3 × 3]; then ONLY composites' Islands of fusion-INstability: 8, 9, 10; then 14, 15, 16,... Could inter-digit Feshbach-resonances exist??? Applications to: quantum-information/computing non-Shore factorization, millennium-problem Riemann-hypotheses proof as Goodkin BEC intersection with graph-theory ``short-cut'' method: Rayleigh(1870)-Polya(1922)-``Anderson'' (1958)-localization, Goldbach-conjecture, financial auditing/accounting as quantum-statistical-physics;... abound!!!

  3. Multi-instanton-induced baryon- and lepton-number violation in the electroweak theory

    SciTech Connect

    Aoyama, H.; Kikuchi, H. )

    1991-03-15

    The {Delta}{ital B}={Delta}{ital L}{ne}0 process in the standard electroweak theory is investigated by using the instanton formalism. Fermion zero modes yield a two-body interaction between an instanton and anti-instanton. The expansion in terms of numbers of instantons is proven to be equivalent (at all orders) to the perturbation of a field theory whose vertices are induced by an instanton or an anti-instanton. The resulting cross section satisfies the unitarity bound.

  4. Structural chemistry and number theory amalgamized: crystal structure of Na11Hg52.

    PubMed

    Hornfeck, Wolfgang; Hoch, Constantin

    2015-12-01

    The recently elucidated crystal structure of the technologically important amalgam Na11Hg52 is described by means of a method employing some fundamental concept of number theory, namely modular arithmetical (congruence) relations observed between a slightly idealized set of atomic coordinates. In combination with well known ideas from group theory, regarding lattice-sublattice transformations, these allow for a deeper mutual understanding of both and provide the structural chemist with a slightly different kind of spectacles, thus enabling a distinct viw on complex crystal structures in general.

  5. Classification of central extensions of Lax operator algebras

    SciTech Connect

    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.

  6. Algebraic Systems and Pushdown Automata

    NASA Astrophysics Data System (ADS)

    Petre, Ion; Salomaa, Arto

    We concentrate in this chapter on the core aspects of algebraic series, pushdown automata, and their relation to formal languages. We choose to follow here a presentation of their theory based on the concept of properness. We introduce in Sect. 2 some auxiliary notions and results needed throughout the chapter, in particular the notions of discrete convergence in semirings and C-cycle free infinite matrices. In Sect. 3 we introduce the algebraic power series in terms of algebraic systems of equations. We focus on interconnections with context-free grammars and on normal forms. We then conclude the section with a presentation of the theorems of Shamir and Chomsky-Schützenberger. We discuss in Sect. 4 the algebraic and the regulated rational transductions, as well as some representation results related to them. Section 5 is dedicated to pushdown automata and focuses on the interconnections with classical (non-weighted) pushdown automata and on the interconnections with algebraic systems. We then conclude the chapter with a brief discussion of some of the other topics related to algebraic systems and pushdown automata.

  7. A Metric Conceptual Space Algebra

    NASA Astrophysics Data System (ADS)

    Adams, Benjamin; Raubal, Martin

    The modeling of concepts from a cognitive perspective is important for designing spatial information systems that interoperate with human users. Concept representations that are built using geometric and topological conceptual space structures are well suited for semantic similarity and concept combination operations. In addition, concepts that are more closely grounded in the physical world, such as many spatial concepts, have a natural fit with the geometric structure of conceptual spaces. Despite these apparent advantages, conceptual spaces are underutilized because existing formalizations of conceptual space theory have focused on individual aspects of the theory rather than the creation of a comprehensive algebra. In this paper we present a metric conceptual space algebra that is designed to facilitate the creation of conceptual space knowledge bases and inferencing systems. Conceptual regions are represented as convex polytopes and context is built in as a fundamental element. We demonstrate the applicability of the algebra to spatial information systems with a proof-of-concept application.

  8. 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.

  9. Priority in Process Algebras

    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.

  10. Critical Analysis of the Mathematical Formalism of Theoretical Physics. V. Foundations of the Theory of Negative Numbers

    NASA Astrophysics Data System (ADS)

    Kalanov, Temur Z.

    2015-04-01

    Analysis of the foundations of the theory of negative numbers is proposed. The unity of formal logic and of rational dialectics is methodological basis of the analysis. Statement of the problem is as follows. As is known, point O in the Cartesian coordinate system XOY determines the position of zero on the scale. The number ``zero'' belongs to both the scale of positive numbers and the scale of negative numbers. In this case, the following formallogical contradiction arises: the number 0 is both positive number and negative number; or, equivalently, the number 0 is neither positive number nor negative number, i.e. number 0 has no sign. Then the following question arises: Do negative numbers exist in science and practice? A detailed analysis of the problem shows that negative numbers do not exist because the foundations of the theory of negative numbers contrary to the formal-logical laws. It is proved that: (a) all numbers have no signs; (b) the concepts ``negative number'' and ``negative sign of number'' represent a formallogical error; (c) signs ``plus'' and ``minus'' are only symbols of mathematical operations. The logical errors determine the essence of the theory of negative numbers: the theory of negative number is a false theory.

  11. Phase transitions in number theory: from the birthday problem to Sidon sets.

    PubMed

    Luque, Bartolo; Torre, Iván G; Lacasa, Lucas

    2013-11-01

    In this work, we show how number theoretical problems can be fruitfully approached with the tools of statistical physics. We focus on g-Sidon sets, which describe sequences of integers whose pairwise sums are different, and propose a random decision problem which addresses the probability of a random set of k integers to be g-Sidon. First, we provide numerical evidence showing that there is a crossover between satisfiable and unsatisfiable phases which converts to an abrupt phase transition in a properly defined thermodynamic limit. Initially assuming independence, we then develop a mean-field theory for the g-Sidon decision problem. We further improve the mean-field theory, which is only qualitatively correct, by incorporating deviations from independence, yielding results in good quantitative agreement with the numerics for both finite systems and in the thermodynamic limit. Connections between the generalized birthday problem in probability theory, the number theory of Sidon sets and the properties of q-Potts models in condensed matter physics are briefly discussed.

  12. Algebraic solution of the synthesis problem for coded sequences

    SciTech Connect

    Leukhin, Anatolii N

    2005-08-31

    The algebraic solution of a 'complex' problem of synthesis of phase-coded (PC) sequences with the zero level of side lobes of the cyclic autocorrelation function (ACF) is proposed. It is shown that the solution of the synthesis problem is connected with the existence of difference sets for a given code dimension. The problem of estimating the number of possible code combinations for a given code dimension is solved. It is pointed out that the problem of synthesis of PC sequences is related to the fundamental problems of discrete mathematics and, first of all, to a number of combinatorial problems, which can be solved, as the number factorisation problem, by algebraic methods by using the theory of Galois fields and groups. (fourth seminar to the memory of d.n. klyshko)

  13. Number-conserving master equation theory for a dilute Bose-Einstein condensate

    SciTech Connect

    Schelle, Alexej; Wellens, Thomas; Buchleitner, Andreas; Delande, Dominique

    2011-01-15

    We describe the transition of N weakly interacting atoms into a Bose-Einstein condensate within a number-conserving quantum master equation theory. Based on the separation of time scales for condensate formation and noncondensate thermalization, we derive a master equation for the condensate subsystem in the presence of the noncondensate environment under the inclusion of all two-body interaction processes. We numerically monitor the condensate particle number distribution during condensate formation, and derive a condition under which the unique equilibrium steady state of a dilute, weakly interacting Bose-Einstein condensate is given by a Gibbs-Boltzmann thermal state of N noninteracting atoms.

  14. Approach of the Azimuthal and Magnetic Quantum Numbers l and m to Representation of a Cubic Deformation of su(2) Algebra

    NASA Astrophysics Data System (ADS)

    Fakhri, H.; Hashemzadeh, R.

    It is shown that the space of spherical harmonics Ylm(θ ,φ ) whose 2l - m = p - 1 is given, represent irreducibly a cubic deformation of su(2) algebra, the so-called suΦp(2), with deformation function as Φ p(x) = (27)/(2)x2 + 3(7-3p2)x. The irreducible representation spaces are classified in three different bunches, depending on one of values 3k - 2, 3k - 1 and 3k, with k as a positive integer, to be chosen for p. So, three different methods for generating the spectrum of spherical harmonics are presented by using the cubic deformation of su(2). Moreover, it is shown that p plays the role of deformation parameter.

  15. Quadratically convergent algorithm for fractional occupation numbers in density functional theory

    NASA Astrophysics Data System (ADS)

    Cancès, Eric; Kudin, Konstantin N.; Scuseria, Gustavo E.; Turinici, Gabriel

    2003-03-01

    The numerical solution of the electronic structure problem in Kohn-Sham density functional theory may in certain cases yield fractional occupancy of the single-particle orbitals. In this paper, we propose a quadratically convergent approach for simultaneous optimization of orbitals and occupancies in systems with fractional occupation numbers (FONs). The starting guess for orbitals and FONs is obtained via the relaxed constraint algorithm. Numerical results are presented for benchmark cases.

  16. Number density distribution of solvent molecules on a substrate: a transform theory for atomic force microscopy.

    PubMed

    Amano, Ken-Ichi; Liang, Yunfeng; Miyazawa, Keisuke; Kobayashi, Kazuya; Hashimoto, Kota; Fukami, Kazuhiro; Nishi, Naoya; Sakka, Tetsuo; Onishi, Hiroshi; Fukuma, Takeshi

    2016-06-21

    Atomic force microscopy (AFM) in liquids can measure a force curve between a probe and a buried substrate. The shape of the measured force curve is related to hydration structure on the substrate. However, until now, there has been no practical theory that can transform the force curve into the hydration structure, because treatment of the liquid confined between the probe and the substrate is a difficult problem. Here, we propose a robust and practical transform theory, which can generate the number density distribution of solvent molecules on a substrate from the force curve. As an example, we analyzed a force curve measured by using our high-resolution AFM with a newly fabricated ultrashort cantilever. It is demonstrated that the hydration structure on muscovite mica (001) surface can be reproduced from the force curve by using the transform theory. The transform theory will enhance AFM's ability and support structural analyses of solid/liquid interfaces. By using the transform theory, the effective diameter of a real probe apex is also obtained. This result will be important for designing a model probe of molecular scale simulations.

  17. 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.

  18. An Algebraic Route to Pi

    ERIC Educational Resources Information Center

    Deakin, Michael A. B.

    1974-01-01

    Euler's famous formula, e to the (i, pi) power equals -1, is developed by a purely algebraic method that avoids the use of both trigonometry and calculus. A heuristic outline is given followed by the rigorous theory. Pedagogical considerations for classroom presentation are suggested. (LS)

  19. Algebra from Chips and Chopsticks

    ERIC Educational Resources Information Center

    Yun, Jeong Oak; Flores, Alfinio

    2012-01-01

    Students can use geometric representations of numbers as a way to explore algebraic ideas. With the help of these representations, students can think about the relations among the numbers, express them using their own words, and represent them with letters. The activities discussed here can stimulate students to try to find various ways of solving…

  20. Representations of filtered solvable Lie algebras

    SciTech Connect

    Panov, Alexander N

    2012-01-31

    The representation theory of filtered solvable Lie algebras is constructed. In this framework a classification of irreducible representations is obtained and spectra of some reducible representations are found. Bibliography: 9 titles.

  1. Principal fiber bundle description of number scaling for scalars and vectors: application to gauge theory

    NASA Astrophysics Data System (ADS)

    Benioff, Paul

    2015-05-01

    The purpose of this paper is to put the description of number scaling and its effects on physics and geometry on a firmer foundation, and to make it more understandable. A main point is that two different concepts, number and number value are combined in the usual representations of number structures. This is valid as long as just one structure of each number type is being considered. It is not valid when different structures of each number type are being considered. Elements of base sets of number structures, considered by themselves, have no meaning. They acquire meaning or value as elements of a number structure. Fiber bundles over a space or space time manifold, M, are described. The fiber consists of a collection of many real or complex number structures and vector space structures. The structures are parameterized by a real or complex scaling factor, s. A vector space at a fiber level, s, has, as scalars, real or complex number structures at the same level. Connections are described that relate scalar and vector space structures at both neighbor M locations and at neighbor scaling levels. Scalar and vector structure valued fields are described and covariant derivatives of these fields are obtained. Two complex vector fields, each with one real and one imaginary field, appear, with one complex field associated with positions in M and the other with position dependent scaling factors. A derivation of the covariant derivative for scalar and vector valued fields gives the same vector fields. The derivation shows that the complex vector field associated with scaling fiber levels is the gradient of a complex scalar field. Use of these results in gauge theory shows that the imaginary part of the vector field associated with M positions acts like the electromagnetic field. The physical relevance of the other three fields, if any, is not known.

  2. An Algebraic Model for the mathfrak{su}(2|2) Light-Cone String Field Theory

    NASA Astrophysics Data System (ADS)

    Moriyama, S.

    We first revisit the light-cone string field theory on the flat andpp-wave background. By our systematic analysis, we find that some unsatisfactions in the previous construction can be overcome. After that, we head for the construction of the LCSFT on the bubbling geometry with an isometry [mathfrak{psu}(2|2)]^2 ltimes {mathbb R}. We clarify the structure of expansion and propose toy models for it. This proceeding is based on the collaboration with Kishimoto [I. Kishimoto and S. Moriyama, J. High Energy Phys. textbf{08} (2010), 013, arXiv:1005.4719 (Ref. 1)].

  3. Literal algebra for satellite dynamics. [perturbation analysis

    NASA Technical Reports Server (NTRS)

    Gaposchkin, E. M.

    1975-01-01

    A description of the rather general class of operations available is given and the operations are related to problems in satellite dynamics. The implementation of an algebra processor is discussed. The four main categories of symbol processors are related to list processing, string manipulation, symbol manipulation, and formula manipulation. Fundamental required operations for an algebra processor are considered. It is pointed out that algebra programs have been used for a number of problems in celestial mechanics with great success. The advantage of computer algebra is its accuracy and speed.

  4. The Middle Number World: A View of Complexity Theory and Methods in Ecology

    NASA Astrophysics Data System (ADS)

    Bradshaw, G.; Bradshaw, G.

    2001-12-01

    Ecosystems, like the porridge and chair that Goldilocks found in the Three Bear's house, are characterized by numbers neither too large nor too small; they belong instead to the class of middle number systems. As such, complexity theory and methods complement the web of structures and interactions which make up landscapes and ecosystems and concern the inception of "life itself" (Rosen, 1991). As a field integral to critical socio-ecological issues confronting the globe today, and one concerned with intricate scale relationships between observer (ecologist) and observed (ecosystem), ecology brings an intriguing perspective to complex systems analysis. We discuss these new findings from complexity theory within ecological research. In this overview, we describe a systematics of ecosystem dynamics (emergence, unfolding, embedding, and operational closure) which is evolving for ecological phenomena and is common to other complex adaptive systems. Further, we discuss future research directions which are emerging with the integration of complexity and social sciences theories as they develop into a new post-modern epistemology.

  5. Baryon and lepton number violation in the electroweak theory at TeV energies

    SciTech Connect

    Mottola, E.

    1990-01-01

    In the standard Weinberg-Salam electroweak theory baryon and lepton number (B and L) are NOT exactly conserved. The nonconservation of B and L can be traced to the existence of parity violation in the electroweak theory, together with the chiral current anomaly. This subtle effect gives negligibly small amplitudes for B and L violation at energies and temperatures significantly smaller than M{sub w} sin{sup 2} {theta}{sub w}/{alpha} {approximately} 10 TeV. However, recent theoretical work shows that the rate for B and L nonconservation is unsuppressed at higher energies. The consequences of this for cosmology and the baryon asymmetry of the universe, as well as the prospects for direct verification at the SSC are discussed. 13 refs., 3 figs.

  6. Theory of viscous transonic flow over airfoils at high Reynolds number

    NASA Technical Reports Server (NTRS)

    Melnik, R. E.; Chow, R.; Mead, H. R.

    1977-01-01

    This paper considers viscous flows with unseparated turbulent boundary layers over two-dimensional airfoils at transonic speeds. Conventional theoretical methods are based on boundary layer formulations which do not account for the effect of the curved wake and static pressure variations across the boundary layer in the trailing edge region. In this investigation an extended viscous theory is developed that accounts for both effects. The theory is based on a rational analysis of the strong turbulent interaction at airfoil trailing edges. The method of matched asymptotic expansions is employed to develop formal series solutions of the full Reynolds equations in the limit of Reynolds numbers tending to infinity. Procedures are developed for combining the local trailing edge solution with numerical methods for solving the full potential flow and boundary layer equations. Theoretical results indicate that conventional boundary layer methods account for only about 50% of the viscous effect on lift, the remaining contribution arising from wake curvature and normal pressure gradient effects.

  7. The Mach number of the cosmic flow - A critical test for current theories

    NASA Technical Reports Server (NTRS)

    Ostriker, Jeremiah P.; Suto, Yusushi

    1990-01-01

    A new cosmological, self-contained test using the ratio of mean velocity and the velocity dispersion in the mean flow frame of a group of test objects is presented. To allow comparison with linear theory, the velocity field must first be smoothed on a suitable scale. In the context of linear perturbation theory, the Mach number M(R) which measures the ratio of power on scales larger than to scales smaller than the patch size R, is independent of the perturbation amplitude and also of bias. An apparent inconsistency is found for standard values of power-law index n = 1 and cosmological density parameter Omega = 1, when comparing values of M(R) predicted by popular models with tentative available observations. Nonstandard models based on adiabatic perturbations with either negative n or small Omega value also fail, due to creation of unacceptably large microwave background fluctuations.

  8. How To Prepare Students for Algebra.

    ERIC Educational Resources Information Center

    Wu, H.

    2001-01-01

    Suggests that no matter how much algebraic thinking is introduced in the early grades, and no matter how worthwhile this might be, the failure rate in algebra will continue unless the teaching of fractions and decimals is radically revamped. The proper study of fractions provides a ramp that leads students gently from whole number arithmetic up to…

  9. Nonlinear theory of classical cylindrical Richtmyer-Meshkov instability for arbitrary Atwood numbers

    SciTech Connect

    Liu, Wan Hai; Ping Yu, Chang; Hua Ye, Wen; Feng Wang, Li; Tu He, Xian

    2014-06-15

    A nonlinear theory is developed to describe the cylindrical Richtmyer-Meshkov instability (RMI) of an impulsively accelerated interface between incompressible fluids, which is based on both a technique of Padé approximation and an approach of perturbation expansion directly on the perturbed interface rather than the unperturbed interface. When cylindrical effect vanishes (i.e., in the large initial radius of the interface), our explicit results reproduce those [Q. Zhang and S.-I. Sohn, Phys. Fluids 9, 1106 (1996)] related to the planar RMI. The present prediction in agreement with previous simulations [C. Matsuoka and K. Nishihara, Phys. Rev. E 73, 055304(R) (2006)] leads us to better understand the cylindrical RMI at arbitrary Atwood numbers for the whole nonlinear regime. The asymptotic growth rate of the cylindrical interface finger (bubble or spike) tends to its initial value or zero, depending upon mode number of the initial cylindrical interface and Atwood number. The explicit conditions, directly affecting asymptotic behavior of the cylindrical interface finger, are investigated in this paper. This theory allows a straightforward extension to other nonlinear problems related closely to an instable interface.

  10. Effective Lagrangians and Current Algebra in Three Dimensions

    NASA Astrophysics Data System (ADS)

    Ferretti, Gabriele

    In this thesis we study three dimensional field theories that arise as effective Lagrangians of quantum chromodynamics in Minkowski space with signature (2,1) (QCD3). In the first chapter, we explain the method of effective Langrangians and the relevance of current algebra techniques to field theory. We also provide the physical motivations for the study of QCD3 as a toy model for confinement and as a theory of quantum antiferromagnets (QAF). In chapter two, we derive the relevant effective Lagrangian by studying the low energy behavior of QCD3, paying particular attention to how the global symmetries are realized at the quantum level. In chapter three, we show how baryons arise as topological solitons of the effective Lagrangian and also show that their statistics depends on the number of colors as predicted by the quark model. We calculate mass splitting and magnetic moments of the soliton and find logarithmic corrections to the naive quark model predictions. In chapter four, we drive the current algebra of the theory. We find that the current algebra is a co -homologically non-trivial generalization of Kac-Moody algebras to three dimensions. This fact may provide a new, non -perturbative way to quantize the theory. In chapter five, we discuss the renormalizability of the model in the large-N expansion. We prove the validity of the non-renormalization theorem and compute the critical exponents in a specific limiting case, the CP^ {N-1} model with a Chern-Simons term. Finally, chapter six contains some brief concluding remarks.

  11. Knots, BPS States, and Algebraic Curves

    NASA Astrophysics Data System (ADS)

    Garoufalidis, Stavros; Kucharski, Piotr; Sułkowski, Piotr

    2016-08-01

    We analyze relations between BPS degeneracies related to Labastida-Mariño-Ooguri-Vafa (LMOV) invariants and algebraic curves associated to knots. We introduce a new class of such curves, which we call extremal A-polynomials, discuss their special properties, and determine exact and asymptotic formulas for the corresponding (extremal) BPS degeneracies. These formulas lead to nontrivial integrality statements in number theory, as well as to an improved integrality conjecture, which is stronger than the known M-theory integrality predictions. Furthermore, we determine the BPS degeneracies encoded in augmentation polynomials and show their consistency with known colored HOMFLY polynomials. Finally, we consider refined BPS degeneracies for knots, determine them from the knowledge of super-A-polynomials, and verify their integrality. We illustrate our results with twist knots, torus knots, and various other knots with up to 10 crossings.

  12. On the instabilities of supersonic mixing layers - A high-Mach-number asymptotic theory

    NASA Technical Reports Server (NTRS)

    Balsa, Thomas F.; Goldstein, M. E.

    1990-01-01

    The stability of a family of tanh mixing layers is studied at large Mach numbers using perturbation methods. It is found that the eigenfunction develops a multilayered structure, and the eigenvalue is obtained by solving a simplified version of the Rayleigh equation (with homogeneous boundary conditions) in one of these layers which lies in either of the external streams. This analysis leads to a simple hypersonic similarity law which explains how spatial and temporal phase speeds and growth rates scale with Mach number and temperature ratio. Comparisons are made with numerical results, and it is found that this similarity law provides a good qualitative guide for the behavior of the instability at high Mach numbers. In addition to this asymptotic theory, some fully numerical results are also presented (with no limitation on the Mach number) in order to explain the origin of the hypersonic modes (through mode splitting) and to discuss the role of oblique modes over a very wide range of Mach number and temperature ratio.

  13. Using dynamo theory to predict the sunspot number during solar cycle 21

    NASA Technical Reports Server (NTRS)

    Schatten, K. H.; Scherrer, P. H.; Svalgaard, L.; Wilcox, J. M.

    1978-01-01

    On physical grounds it is suggested that the polar field strength of the sun near a solar minimum is closely related to the solar activity of the following cycle. Four methods of estimating the polar magnetic field strength of the sun near solar minimum are employed to provide an estimate of the yearly mean sunspot number of cycle 21 at solar maximum of 140 + or - 20. This estimate may be considered a first-order attempt to predict the cycle activity using one parameter of physical importance based upon dynamo theory.

  14. Profile of a low-Mach-number shock in two-fluid plasma theory

    NASA Astrophysics Data System (ADS)

    Gedalin, M.; Kushinsky, Y.; Balikhin, M.

    2015-08-01

    Magnetic profiles of low-Mach-number collisionless shocks in space plasmas are studied within the two-fluid plasma theory. Particular attention is given to the upstream magnetic oscillations generated at the ramp. By including weak resistive dissipation in the equations of motion for electrons and protons, the dependence of the upstream wave train features on the ratio of the dispersion length to the dissipative length is established quantitatively. The dependence of the oscillation amplitude and spatial damping scale on the shock normal angle θ is found.

  15. Numerical algebraic geometry and algebraic kinematics

    NASA Astrophysics Data System (ADS)

    Wampler, Charles W.; Sommese, Andrew J.

    In this article, the basic constructs of algebraic kinematics (links, joints, and mechanism spaces) are introduced. This provides a common schema for many kinds of problems that are of interest in kinematic studies. Once the problems are cast in this algebraic framework, they can be attacked by tools from algebraic geometry. In particular, we review the techniques of numerical algebraic geometry, which are primarily based on homotopy methods. We include a review of the main developments of recent years and outline some of the frontiers where further research is occurring. While numerical algebraic geometry applies broadly to any system of polynomial equations, algebraic kinematics provides a body of interesting examples for testing algorithms and for inspiring new avenues of work.

  16. 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.

  17. Algebra for All: The Effect of Algebra Coursework and Classroom Peer Academic Composition on Low-Achieving Students

    ERIC Educational Resources Information Center

    Nomi, Takako; Raudenbush, Stephen W.

    2014-01-01

    Algebra is often considered as a gateway for later achievement. A recent report by the Mathematics Advisory Panel (2008) underscores the importance of improving algebra learning in secondary school. Today, a growing number of states and districts require algebra for all students in ninth grade or earlier. Chicago is at the forefront of this…

  18. Six-dimensional (1,0) superconformal models and higher gauge theory

    SciTech Connect

    Palmer, Sam; Sämann, Christian

    2013-11-15

    We analyze the gauge structure of a recently proposed superconformal field theory in six dimensions. We find that this structure amounts to a weak Courant-Dorfman algebra, which, in turn, can be interpreted as a strong homotopy Lie algebra. This suggests that the superconformal field theory is closely related to higher gauge theory, describing the parallel transport of extended objects. Indeed we find that, under certain restrictions, the field content and gauge transformations reduce to those of higher gauge theory. We also present a number of interesting examples of admissible gauge structures such as the structure Lie 2-algebra of an abelian gerbe, differential crossed modules, the 3-algebras of M2-brane models, and string Lie 2-algebras.

  19. ON THE MAXIMAL DIMENSION OF IRREDUCIBLE REPRESENTATIONS OF SIMPLE LIE p-ALGEBRAS OF THE CARTAN SERIES S AND H

    NASA Astrophysics Data System (ADS)

    Krylyuk, Ya S.

    1985-02-01

    The maximal dimension is computed for irreducible representations of the Hamiltonian Lie p-algebra and the special Lie p-algebra of an even number of variables over an algebraically closed field of characteristic p>3.Bibliography: 11 titles.

  20. 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…

  1. Putting the Modern in Algebra

    ERIC Educational Resources Information Center

    Bosse, Michael J.; Ries, Heather; Chandler, Kayla

    2012-01-01

    Secondary school mathematics teachers often need to answer the "Why do we do that?" question in such a way that avoids confusion and evokes student interest. Understanding the properties of number systems can provide an avenue to better grasp algebraic structures, which in turn builds students' conceptual knowledge of secondary mathematics. This…

  2. Algebra for All. Research Brief

    ERIC Educational Resources Information Center

    Bleyaert, Barbara

    2009-01-01

    The call for "algebra for all" is not a recent phenomenon. Concerns about the inadequacy of math (and science) preparation in America's high schools have been a steady drumbeat since the 1957 launch of Sputnik; a call for raising standards and the number of math (and science) courses required for graduation has been a part of countless national…

  3. Math Sense: Algebra and Geometry.

    ERIC Educational Resources Information Center

    Howett, Jerry

    This book is designed to help students gain the range of math skills they need to succeed in life, work, and on standardized tests; overcome math anxiety; discover math as interesting and purposeful; and develop good number sense. Topics covered in this book include algebra and geometry. Lessons are organized around four strands: (1) skill lessons…

  4. The Impact of the Number of School Transitions and Self-Efficacy about High School on Algebra I End-of-Course Test Scores

    ERIC Educational Resources Information Center

    Hamer, LaJuana Maurice

    2012-01-01

    The purpose of this study was to examine the impact of the number of transitions by students from school to school on their mathematics achievement. Transition is defined as the number of times a student has changed schools from Kindergarten to the current school year. This study also looked at the relationship between the students' perceived…

  5. Detecting prime numbers via roots of polynomials

    NASA Astrophysics Data System (ADS)

    Dobbs, David E.

    2012-04-01

    It is proved that an integer n ≥ 2 is a prime (resp., composite) number if and only if there exists exactly one (resp., more than one) nth-degree monic polynomial f with coefficients in Z n , the ring of integers modulo n, such that each element of Z n is a root of f. This classroom note could find use in any introductory course on abstract algebra or elementary number theory.

  6. Extending Birthday Paradox Theory to Estimate the Number of Tags in RFID Systems

    PubMed Central

    Shakiba, Masoud; Singh, Mandeep Jit; Sundararajan, Elankovan; Zavvari, Azam; Islam, Mohammad Tariqul

    2014-01-01

    The main objective of Radio Frequency Identification systems is to provide fast identification for tagged objects. However, there is always a chance of collision, when tags transmit their data to the reader simultaneously. Collision is a time-consuming event that reduces the performance of RFID systems. Consequently, several anti-collision algorithms have been proposed in the literature. Dynamic Framed Slotted ALOHA (DFSA) is one of the most popular of these algorithms. DFSA dynamically modifies the frame size based on the number of tags. Since the real number of tags is unknown, it needs to be estimated. Therefore, an accurate tag estimation method has an important role in increasing the efficiency and overall performance of the tag identification process. In this paper, we propose a novel estimation technique for DFSA anti-collision algorithms that applies birthday paradox theory to estimate the number of tags accurately. The analytical discussion and simulation results prove that the proposed method increases the accuracy of tag estimation and, consequently, outperforms previous schemes. PMID:24752285

  7. Extending birthday paradox theory to estimate the number of tags in RFID systems.

    PubMed

    Shakiba, Masoud; Singh, Mandeep Jit; Sundararajan, Elankovan; Zavvari, Azam; Islam, Mohammad Tariqul

    2014-01-01

    The main objective of Radio Frequency Identification systems is to provide fast identification for tagged objects. However, there is always a chance of collision, when tags transmit their data to the reader simultaneously. Collision is a time-consuming event that reduces the performance of RFID systems. Consequently, several anti-collision algorithms have been proposed in the literature. Dynamic Framed Slotted ALOHA (DFSA) is one of the most popular of these algorithms. DFSA dynamically modifies the frame size based on the number of tags. Since the real number of tags is unknown, it needs to be estimated. Therefore, an accurate tag estimation method has an important role in increasing the efficiency and overall performance of the tag identification process. In this paper, we propose a novel estimation technique for DFSA anti-collision algorithms that applies birthday paradox theory to estimate the number of tags accurately. The analytical discussion and simulation results prove that the proposed method increases the accuracy of tag estimation and, consequently, outperforms previous schemes.

  8. A new algebra core for the minimal form' problem

    SciTech Connect

    Purtill, M.R. . Center for Communications Research); Oliveira, J.S.; Cook, G.O. Jr. )

    1991-12-20

    The demands of large-scale algebraic computation have led to the development of many new algorithms for manipulating algebraic objects in computer algebra systems. For instance, parallel versions of many important algorithms have been discovered. Simultaneously, more effective symbolic representations of algebraic objects have been sought. Also, while some clever techniques have been found for improving the speed of the algebraic simplification process, little attention has been given to the issue of restructuring expressions, or transforming them into minimal forms.'' By minimal form,'' we mean that form of an expression that involves a minimum number of operations. In a companion paper, we introduce some new algorithms that are very effective at finding minimal forms of expressions. These algorithms require algebraic and combinatorial machinery that is not readily available in most algebra systems. In this paper we describe a new algebra core that begins to provide the necessary capabilities.

  9. Quantization of Algebraic Reduction

    SciTech Connect

    Sniatycki, Jeodrzej

    2007-11-14

    For a Poisson algebra obtained by algebraic reduction of symmetries of a quantizable system we develop an analogue of geometric quantization based on the quantization structure of the original system.

  10. Correlation of theory to wind-tunnel data at Reynolds numbers below 500,000

    NASA Technical Reports Server (NTRS)

    Evangelista, Raquel; Mcghee, Robert J.; Walker, Betty S.

    1989-01-01

    This paper presents results obtained from two airfoil analysis methods compared with previously published wind tunnel test data at chord Reynolds numbers below 500,000. The analysis methods are from the Eppler-Somers airfoil design/analysis code and from ISES, the Drela-Giles Airfoil design/analysis code. The experimental data are from recent tests of the Eppler 387 airfoil in the NASA Langley Low Turbulence Pressure Tunnel. For R not less than 200,000, lift and pitching moment predictions from both theories compare well with experiment. Drag predictions from both theories also agree with experiment, although to different degrees. However, most of the drag predictions from the Eppler-Somers code are accompanied with separation bubble warnings which indicate that the drag predictions are too low. With the Drela-Giles code, there is a large discrepancy between the computed and experimental pressure distributions in cases with laminar separation bubbles, although the drag polar predictions are similar in trend to experiment.

  11. Learning Algebra in a Computer Algebra Environment

    ERIC Educational Resources Information Center

    Drijvers, Paul

    2004-01-01

    This article summarises a doctoral thesis entitled "Learning algebra in a computer algebra environment, design research on the understanding of the concept of parameter" (Drijvers, 2003). It describes the research questions, the theoretical framework, the methodology and the results of the study. The focus of the study is on the understanding of…

  12. 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…

  13. 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…

  14. Analytic MHD Theory for Earth's Bow Shock at Low Mach Numbers

    NASA Technical Reports Server (NTRS)

    Grabbe, Crockett L.; Cairns, Iver H.

    1995-01-01

    A previous MHD theory for the density jump at the Earth's bow shock, which assumed the Alfven M(A) and sonic M(s) Mach numbers are both much greater than 1, is reanalyzed and generalized. It is shown that the MHD jump equation can be analytically solved much more directly using perturbation theory, with the ordering determined by M(A) and M(s), and that the first-order perturbation solution is identical to the solution found in the earlier theory. The second-order perturbation solution is calculated, whereas the earlier approach cannot be used to obtain it. The second-order terms generally are important over most of the range of M(A) and M(s) in the solar wind when the angle theta between the normal to the bow shock and magnetic field is not close to 0 deg or 180 deg (the solutions are symmetric about 90 deg). This new perturbation solution is generally accurate under most solar wind conditions at 1 AU, with the exception of low Mach numbers when theta is close to 90 deg. In this exceptional case the new solution does not improve on the first-order solutions obtained earlier, and the predicted density ratio can vary by 10-20% from the exact numerical MHD solutions. For theta approx. = 90 deg another perturbation solution is derived that predicts the density ratio much more accurately. This second solution is typically accurate for quasi-perpendicular conditions. Taken together, these two analytical solutions are generally accurate for the Earth's bow shock, except in the rare circumstance that M(A) is less than or = 2. MHD and gasdynamic simulations have produced empirical models in which the shock's standoff distance a(s) is linearly related to the density jump ratio X at the subsolar point. Using an empirical relationship between a(s) and X obtained from MHD simulations, a(s) values predicted using the MHD solutions for X are compared with the predictions of phenomenological models commonly used for modeling observational data, and with the predictions of a

  15. Nilpotent orbits in classical Lie algebras over F2n and the Springer correspondence

    PubMed Central

    Xue, Ting

    2008-01-01

    We give the number of nilpotent orbits in the Lie algebras of orthogonal groups under the adjoint action of the groups over F2n. Let G be an adjoint algebraic group of type B, C, or D defined over an algebraically closed field of characteristic 2. We construct the Springer correspondence for the nilpotent variety in the Lie algebra of G. PMID:18202179

  16. Vague Congruences and Quotient Lattice Implication Algebras

    PubMed Central

    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

  17. Law of Large Numbers: the Theory, Applications and Technology-based Education.

    PubMed

    Dinov, Ivo D; Christou, Nicolas; Gould, Robert

    2009-03-01

    Modern approaches for technology-based blended education utilize a variety of recently developed novel pedagogical, computational and network resources. Such attempts employ technology to deliver integrated, dynamically-linked, interactive-content and heterogeneous learning environments, which may improve student comprehension and information retention. In this paper, we describe one such innovative effort of using technological tools to expose students in probability and statistics courses to the theory, practice and usability of the Law of Large Numbers (LLN). We base our approach on integrating pedagogical instruments with the computational libraries developed by the Statistics Online Computational Resource (www.SOCR.ucla.edu). To achieve this merger we designed a new interactive Java applet and a corresponding demonstration activity that illustrate the concept and the applications of the LLN. The LLN applet and activity have common goals - to provide graphical representation of the LLN principle, build lasting student intuition and present the common misconceptions about the law of large numbers. Both the SOCR LLN applet and activity are freely available online to the community to test, validate and extend (Applet: http://socr.ucla.edu/htmls/exp/Coin_Toss_LLN_Experiment.html, and Activity: http://wiki.stat.ucla.edu/socr/index.php/SOCR_EduMaterials_Activities_LLN).

  18. Algebra Is a Civil Right: Increasing Achievement for African American Males in Algebra through Collaborative Inquiry

    ERIC Educational Resources Information Center

    Davies Gomez, Lisa

    2012-01-01

    Algebra is the gatekeeper of access to higher-level math and science courses, higher education and future earning opportunities. Unequal numbers of African-American males drop out of Algebra and mathematics courses and underperform on tests of mathematical competency and are thus denied both essential skills and a particularly important pathway to…

  19. Generalization of Patterns: The Tension between Algebraic Thinking and Algebraic Notation.

    ERIC Educational Resources Information Center

    Zazkis, Rina; Liljedahl, Peter

    2002-01-01

    Explores the attempts of a group of preservice elementary school teachers to generalize a repeating visual number pattern. Discusses students' emergent algebraic thinking. Indicates that students' ability to express generalities verbally was not accompanied by algebraic notation, but participants often perceived complete and accurate solutions…

  20. FAST TRACK COMMUNICATION: Kac Moody algebras and controlled chaos

    NASA Astrophysics Data System (ADS)

    Wesley, Daniel H.

    2007-02-01

    Compactification can control chaotic Mixmaster behaviour in gravitational systems with p-form matter: we consider this in light of the connection between supergravity models and Kac Moody algebras. We show that different compactifications define 'mutations' of the algebras associated with the noncompact theories. We list the algebras obtained in this way, and find novel examples of wall systems determined by Lorentzian (but not hyperbolic) algebras. Cosmological models with a smooth pre-big bang phase require that chaos is absent: we show that compactification alone cannot eliminate chaos in the simplest compactifications of the heterotic string on a Calabi Yau, or M theory on a manifold of G2 holonomy.

  1. 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.

  2. Gauge Theories of Vector Particles

    DOE R&D Accomplishments Database

    Glashow, S. L.; Gell-Mann, M.

    1961-04-24

    The possibility of generalizing the Yang-Mills trick is examined. Thus we seek theories of vector bosons invariant under continuous groups of coordinate-dependent linear transformations. All such theories may be expressed as superpositions of certain "simple" theories; we show that each "simple theory is associated with a simple Lie algebra. We may introduce mass terms for the vector bosons at the price of destroying the gauge-invariance for coordinate-dependent gauge functions. The theories corresponding to three particular simple Lie algebras - those which admit precisely two commuting quantum numbers - are examined in some detail as examples. One of them might play a role in the physics of the strong interactions if there is an underlying super-symmetry, transcending charge independence, that is badly broken. The intermediate vector boson theory of weak interactions is discussed also. The so-called "schizon" model cannot be made to conform to the requirements of partial gauge-invariance.

  3. Algebraic Trigonometry

    ERIC Educational Resources Information Center

    Vaninsky, Alexander

    2011-01-01

    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 sin[superscript 2] + cos[superscript 2] = 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…

  4. The Bell states in noncommutative algebraic geometry

    NASA Astrophysics Data System (ADS)

    Beil, Charlie

    2014-10-01

    We introduce new mathematical aspects of the Bell states using matrix factorizations, non-noetherian singularities, and noncommutative blowups. A matrix factorization of a polynomial p consists of two matrices ϕ1, ϕ2 such that ϕ1ϕ2 = ϕ2ϕ1 = p id. Using this notion, we show how the Bell states emerge from the separable product of two mixtures, by defining pure states over complex matrices rather than just the complex numbers. We then show in an idealized algebraic setting that pure states are supported on non-noetherian singularities. Moreover, we find that the collapse of a Bell state is intimately related to the representation theory of the noncommutative blowup along its singular support. This presents an exchange in geometry: the nonlocal commutative spacetime of the entangled state emerges from an underlying local noncommutative spacetime.

  5. 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…

  6. 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.

  7. Fundamental Theorems of Algebra for the Perplexes

    ERIC Educational Resources Information Center

    Poodiak, Robert; LeClair, Kevin

    2009-01-01

    The fundamental theorem of algebra for the complex numbers states that a polynomial of degree n has n roots, counting multiplicity. This paper explores the "perplex number system" (also called the "hyperbolic number system" and the "spacetime number system") In this system (which has extra roots of +1 besides the usual [plus or minus]1 of the…

  8. Superconformal algebras on the boundary of AdS3

    NASA Astrophysics Data System (ADS)

    Rasmussen, Jørgen

    1999-07-01

    Motivated by recent progress on the correspondence between string theory on nti-de Sitter space and conformal field theory, we provide an explicit construction of an infinite dimensional class of superconformal algebras on the boundary of AdS3. These space-time algebras are N extended superconformal algebras of the kind obtainable by hamiltonian reduction of affine SL(2|N/2) current superalgebras for N even, and are induced by the same current superalgebras residing on the world sheet. Thus, such an extended superconformal algebra is generated by N supercurrents and an SL(N/2) current algebra in addition to a U(1) current algebra. The results are obtained within the framework of free field realizations.

  9. The problem-solving approach in the teaching of number theory

    NASA Astrophysics Data System (ADS)

    Toh, Pee Choon; Hoong Leong, Yew; Toh, Tin Lam; Dindyal, Jaguthsing; Quek, Khiok Seng; Guan Tay, Eng; Him Ho, Foo

    2014-02-01

    Mathematical problem solving is the mainstay of the mathematics curriculum for Singapore schools. In the preparation of prospective mathematics teachers, the authors, who are mathematics teacher educators, deem it important that pre-service mathematics teachers experience non-routine problem solving and acquire an attitude that predisposes them to adopt a Pólya-style approach in learning mathematics. The Practical Worksheet is an instructional scaffold we adopted to help our pre-service mathematics teachers develop problem-solving dispositions alongside the learning of the subject matter. The Worksheet was initially used in a design experiment aimed at teaching problem solving in a secondary school. In this paper, we describe an application and adaptation of the MProSE (Mathematical Problem Solving for Everyone) design experiment to a university level number theory course for pre-service mathematics teachers. The goal of the enterprise was to help the pre-service mathematics teachers develop problem-solving dispositions alongside the learning of the subject matter. Our analysis of the pre-service mathematics teachers' work shows that the MProSE design holds promise for mathematics courses at the tertiary level.

  10. Kaluza-Klein theory in the limit of large number of extra dimensions

    SciTech Connect

    Canfora, Fabrizio; Giacomini, Alex; Zerwekh, Alfonso R.

    2009-10-15

    The Kaluza-Klein compactification in the limit of a large number of extra dimensions is studied. The starting point is the Einstein-Hilbert action plus cosmological constant in 4+D dimensions. It is shown that in the large D limit the effective four-dimensional cosmological constant is of order 1/D, whereas the size of the extra dimensions remains finite. A 't Hooft-like large D expansion of the effective Lagrangian for the Kaluza-Klein scalar and gauge fields arising from the dimensional reduction is considered. It is shown that the propagator of the scalar field associated to the determinant of the metric of the extra dimensions is strongly suppressed. This is an interesting result as in standard Kaluza-Klein theory this scalar degree of freedom is responsible for the constraint on the gauge fields which makes it impossible to recover the usual Yang-Mills equations. Moreover in the large D limit it turns out that the ultraviolet divergences due to the interactions between gauge and scalar fields are softened.

  11. Theory of cylindrical and spherical Langmuir probes in the limit of vanishing Debye number

    SciTech Connect

    Parrot, M.J.M.; Storey, L.R.O.; Parker, L.W.; Laframboise, J.G.

    1982-12-01

    A theory has been developed for cylindrical and spherical probes and other collectors in collisionless plasmas, in the limit where the ratio of Debye length to probe radius (the Debye number lambda/sub D/) vanishes. Results are presented for the case of equal electron and ion temperatures. On the scale of the probe radius, the distributions of potential and density in the presheath appear to have infinite slope at the probe surface. The dimensionless current--voltage characteristic is the same for the cylinder as for the sphere, within the limits of error of the numerical results, although no physical reason for this is evident. As the magnitude of probe potential (relative to space) increases, the current does not saturate abruptly but only asymptotically; its limiting value is about 45% larger than at space potential. Probe currents for small nonzero lambda/sub D/ approach those for zero lambda/sub D/ only very slowly, showing power-law behavior as function of lambda/sub D/ in the limit as lambda/sub D/ ..-->.. 0, with power-law exponents less than unity, resulting in infinite limiting derivatives with respect to lambda/sub D/.

  12. Algebraic independence properties related to certain infinite products

    NASA Astrophysics Data System (ADS)

    Tanaka, Taka-aki

    2011-09-01

    In this paper we establish algebraic independence of the values of a certain infinite product as well as its all successive derivatives at algebraic points other than its zeroes, using the fact that the logarithmic derivative of an infinite product gives a partial fraction expansion. Such an infinite product is generated by a linear recurrence. The method used for proving the algebraic independence is based on the theory of Mahler functions of several variables.

  13. Supersymmetric extension of Galilean conformal algebras

    SciTech Connect

    Bagchi, Arjun; Mandal, Ipsita

    2009-10-15

    The Galilean conformal algebra has recently been realized in the study of the nonrelativistic limit of the AdS/CFT conjecture. This was obtained by a systematic parametric group contraction of the parent relativistic conformal field theory. In this paper, we extend the analysis to include supersymmetry. We work at the level of the coordinates in superspace to construct the N=1 super-Galilean conformal algebra. One of the interesting outcomes of the analysis is that one is able to naturally extend the finite algebra to an infinite one. This looks structurally similar to the N=1 superconformal algebra in two dimensions, but is different. We also comment on the extension of our construction to cases of higher N.

  14. Motivation Theories and Motivation Systems. The Gulf between Theory and Practice. Coombe Lodge Working Paper. Information Bank Number 1937.

    ERIC Educational Resources Information Center

    Turner, Colin

    Managers have an obvious interest in motivation, yet there are few connections between the needs of the manager and research on motivation theory and system building. Motivation can be defined as the degree to which an individual wants and chooses to engage in certain specified behaviors. This definition assumes that motivation is an individual…

  15. 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)

  16. The kinematic algebra from the self-dual sector

    NASA Astrophysics Data System (ADS)

    Monteiro, Ricardo; O'Connell, Donal

    2011-07-01

    We identify a diffeomorphism Lie algebra in the self-dual sector of Yang-Mills theory, and show that it determines the kinematic numerators of tree-level MHV amplitudes in the full theory. These amplitudes can be computed off-shell from Feynman diagrams with only cubic vertices, which are dressed with the structure constants of both the Yang-Mills colour algebra and the diffeomorphism algebra. Therefore, the latter algebra is the dual of the colour algebra, in the sense suggested by the work of Bern, Carrasco and Johansson. We further study perturbative gravity, both in the self-dual and in the MHV sectors, finding that the kinematic numerators of the theory are the BCJ squares of the Yang-Mills numerators.

  17. Contribution from the interaction Hamiltonian to the expectation value of particle number with the non-equilibrium quantum field theory

    SciTech Connect

    Hotta, Ryuuichi; Morozumi, Takuya; Takata, Hiroyuki

    2012-07-27

    We develop the method analyzing particle number non-conserving phenomena with non-equilibrium quantum field-theory. In this study, we consider a CP violating model with interaction Hamiltonian that breaks particle number conservation. To derive the quantum Boltzmann equation for the particle number, we solve Schwinger-Dyson equation, which are obtained from two particle irreducible closed-time-path (2PI CTP) effective action. In this calculation, we show the contribution from interaction Hamiltonian to the time evolution of expectation value of particle number.

  18. Enhancing Undergraduate Mathematics Curriculum via Coding Theory and Cryptography

    ERIC Educational Resources Information Center

    Aydin, Nuh

    2009-01-01

    The theory of error-correcting codes and cryptography are two relatively recent applications of mathematics to information and communication systems. The mathematical tools used in these fields generally come from algebra, elementary number theory, and combinatorics, including concepts from computational complexity. It is possible to introduce the…

  19. 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…

  20. Max-plus Algebraic Tools for Discrete Event Systems, Static Analysis, and Zero-Sum Games

    NASA Astrophysics Data System (ADS)

    Gaubert, Stéphane

    The max-plus algebraic approach of timed discrete event systems emerged in the eighties, after the discovery that synchronization phenomena can be modeled in a linear way in the max-plus setting. This led to a number of results, like the determination of long term characteristics (throughput, stationary regime) by spectral theory methods or the representation of the input-output behavior by rational series.

  1. The algebra of Grassmann canonical anticommutation relations and its applications to fermionic systems

    SciTech Connect

    Keyl, Michael; Schlingemann, Dirk-M.

    2010-02-15

    We present an approach to a noncommutativelike phase space which allows to analyze quasifree states on the algebra of canonical anti-commutation relations (CAR) in analogy to quasifree states on the algebra of canonical commutation relations (CCR). The used mathematical tools are based on a new algebraic structure the 'Grassmann algebra of canonical anticommutation relations' (GAR algebra) which is given by the twisted tensor product of a Grassmann and a CAR algebra. As a new application, the corresponding theory provides an elegant tool for calculating the fidelity of two quasifree fermionic states which is needed for the study of entanglement distillation within fermionic systems.

  2. Atomic Theory and Multiple Combining Proportions: The Search for Whole Number Ratios.

    PubMed

    Usselman, Melvyn C; Brown, Todd A

    2015-04-01

    John Dalton's atomic theory, with its postulate of compound formation through atom-to-atom combination, brought a new perspective to weight relationships in chemical reactions. A presumed one-to-one combination of atoms A and B to form a simple compound AB allowed Dalton to construct his first table of relative atomic weights from literature analyses of appropriate binary compounds. For such simple binary compounds, the atomic theory had little advantages over affinity theory as an explanation of fixed proportions by weight. For ternary compounds of the form AB2, however, atomic theory made quantitative predictions that were not deducible from affinity theory. Atomic theory required that the weight of B in the compound AB2 be exactly twice that in the compound AB. Dalton, Thomas Thomson and William Hyde Wollaston all published within a few years of each other experimental data that claimed to give the predicted results with the required accuracy. There are nonetheless several experimental barriers to obtaining the desired integral multiple proportions. In this paper I will discuss replication experiments which demonstrate that only Wollaston's results are experimentally reliable. It is likely that such replicability explains why Wollaston's experiments were so influential.

  3. Atomic Theory and Multiple Combining Proportions: The Search for Whole Number Ratios.

    PubMed

    Usselman, Melvyn C; Brown, Todd A

    2015-04-01

    John Dalton's atomic theory, with its postulate of compound formation through atom-to-atom combination, brought a new perspective to weight relationships in chemical reactions. A presumed one-to-one combination of atoms A and B to form a simple compound AB allowed Dalton to construct his first table of relative atomic weights from literature analyses of appropriate binary compounds. For such simple binary compounds, the atomic theory had little advantages over affinity theory as an explanation of fixed proportions by weight. For ternary compounds of the form AB2, however, atomic theory made quantitative predictions that were not deducible from affinity theory. Atomic theory required that the weight of B in the compound AB2 be exactly twice that in the compound AB. Dalton, Thomas Thomson and William Hyde Wollaston all published within a few years of each other experimental data that claimed to give the predicted results with the required accuracy. There are nonetheless several experimental barriers to obtaining the desired integral multiple proportions. In this paper I will discuss replication experiments which demonstrate that only Wollaston's results are experimentally reliable. It is likely that such replicability explains why Wollaston's experiments were so influential. PMID:26104162

  4. Operator product expansion algebra

    SciTech Connect

    Holland, Jan; Hollands, Stefan

    2013-07-15

    We establish conceptually important properties of the operator product expansion (OPE) in the context of perturbative, Euclidean φ{sup 4}-quantum field theory. First, we demonstrate, generalizing earlier results and techniques of hep-th/1105.3375, that the 3-point OPE, =Σ{sub C}C{sub A{sub 1A{sub 2A{sub 3}{sup C}}}}, usually interpreted only as an asymptotic short distance expansion, actually converges at finite, and even large, distances. We further show that the factorization identity C{sub A{sub 1A{sub 2A{sub 3}{sup B}}}}=Σ{sub C}C{sub A{sub 1A{sub 2}{sup C}}}C{sub CA{sub 3}{sup B}} is satisfied for suitable configurations of the spacetime arguments. Again, the infinite sum is shown to be convergent. Our proofs rely on explicit bounds on the remainders of these expansions, obtained using refined versions, mostly due to Kopper et al., of the renormalization group flow equation method. These bounds also establish that each OPE coefficient is a real analytic function in the spacetime arguments for non-coinciding points. Our results hold for arbitrary but finite loop orders. They lend support to proposals for a general axiomatic framework of quantum field theory, based on such “consistency conditions” and akin to vertex operator algebras, wherein the OPE is promoted to the defining structure of the theory.

  5. 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.

  6. Linear-Algebra Programs

    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.

  7. Ready, Set, Algebra?

    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…

  8. 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…

  9. Teaching Structure in Algebra

    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…

  10. Extended conformal field theories

    NASA Astrophysics Data System (ADS)

    Taormina, Anne

    1990-08-01

    Some extended conformal field theories are briefly reviewed. They illustrate how non minimal models of the Virasoro algebra (c≥1) can become minimal with respect to a larger algebra. The accent is put on N-extended superconformal algebras, which are relevant in superstring compactification.

  11. ALGEBRA IIVer 1.22

    SciTech Connect

    2003-06-03

    The ALGEBRA II program allows the user to manipulate data from a finite element analysis before it is plotted by evaluating algebraic expressions. The equation variables are dependent on the input database variable names. The finite element output data is in the form of variable values (e.g., stress, strain, and velocity components) in an EXODUS II database which can be read by plot programs. Code is written in a portable form as possible. Fortran code is written in ANSI Standard FORTRAN-77. Machine-specific routines are limited in number and are grouped together to minimize the time required to adapt them to a new system. SEACAS codes has been ported to several Unix systems.

  12. ALGEBRA IIVer 1.22

    2003-06-03

    The ALGEBRA II program allows the user to manipulate data from a finite element analysis before it is plotted by evaluating algebraic expressions. The equation variables are dependent on the input database variable names. The finite element output data is in the form of variable values (e.g., stress, strain, and velocity components) in an EXODUS II database which can be read by plot programs. Code is written in a portable form as possible. Fortran codemore » is written in ANSI Standard FORTRAN-77. Machine-specific routines are limited in number and are grouped together to minimize the time required to adapt them to a new system. SEACAS codes has been ported to several Unix systems.« less

  13. The general theory of convolutional codes

    NASA Technical Reports Server (NTRS)

    Mceliece, R. J.; Stanley, R. P.

    1993-01-01

    This article presents a self-contained introduction to the algebraic theory of convolutional codes. This introduction is partly a tutorial, but at the same time contains a number of new results which will prove useful for designers of advanced telecommunication systems. Among the new concepts introduced here are the Hilbert series for a convolutional code and the class of compact codes.

  14. East-West migration in Europe: can migration theories help estimate the numbers?

    PubMed

    Oberg, S; Wils, A B

    1992-01-01

    "In this article, we discuss the types of scientific knowledge that could be used to estimate migration flows." Theories from the disciplines of economics, geography, geopolitics, sociology, demography, econometrics, and history are reviewed. The authors find that "each field provides a partial explanation of...migration flows." The geographical focus is on Europe. PMID:12286068

  15. From Number Agreement to the Subjunctive: Evidence for Processability Theory in L2 Spanish

    ERIC Educational Resources Information Center

    Bonilla, Carrie L.

    2015-01-01

    This article contributes to typological plausibility of Processability Theory (PT) (Pienemann, 1998, 2005) by providing empirical data that show that the stages predicted by PT are followed in the second language (L2) acquisition of Spanish syntax and morphology. In the present article, the PT stages for L2 Spanish morphology and syntax are first…

  16. East-West migration in Europe: can migration theories help estimate the numbers?

    PubMed

    Oberg, S; Wils, A B

    1992-01-01

    "In this article, we discuss the types of scientific knowledge that could be used to estimate migration flows." Theories from the disciplines of economics, geography, geopolitics, sociology, demography, econometrics, and history are reviewed. The authors find that "each field provides a partial explanation of...migration flows." The geographical focus is on Europe.

  17. Teaching Public Policy: Theory, Research, and Practice. Contributions in Political Science, Number 268.

    ERIC Educational Resources Information Center

    Bergerson, Peter J., Ed.

    The 16 chapters of this book offer innovative instructional techniques used to train public managers. It presents public management concepts along with such subtopics as organizational theory and ethics, research skills, program evaluation, financial management, computers and communication skills in public administration, comparative public…

  18. Learning Study Guided by Variation Theory: Exemplified by Children Learning to Halve and Double Whole Numbers

    ERIC Educational Resources Information Center

    Holmqvist Olander, Mona; Nyberg, Eva

    2014-01-01

    This study aims to describe how the learning study model can be used to improve lesson design and children's learning outcomes by enabling them to perceive and define the critical aspects of the object of learning, guided by variation theory. Three lesson designs were used with three groups of children (A = 24, B = 13, C = 14) from two…

  19. The geometric semantics of algebraic quantum mechanics.

    PubMed

    Cruz Morales, John Alexander; Zilber, Boris

    2015-08-01

    In this paper, we will present an ongoing project that aims to use model theory as a suitable mathematical setting for studying the formalism of quantum mechanics. We argue that this approach provides a geometric semantics for such a formalism by means of establishing a (non-commutative) duality between certain algebraic and geometric objects.

  20. Using Group Explorer in Teaching Abstract Algebra

    ERIC Educational Resources Information Center

    Schubert, Claus; Gfeller, Mary; Donohue, Christopher

    2013-01-01

    This study explores the use of Group Explorer in an undergraduate mathematics course in abstract algebra. The visual nature of Group Explorer in representing concepts in group theory is an attractive incentive to use this software in the classroom. However, little is known about students' perceptions on this technology in learning concepts in…

  1. Leibniz algebras associated with representations of filiform Lie algebras

    NASA Astrophysics Data System (ADS)

    Ayupov, Sh. A.; Camacho, L. M.; Khudoyberdiyev, A. Kh.; Omirov, B. A.

    2015-12-01

    In this paper we investigate Leibniz algebras whose quotient Lie algebra is a naturally graded filiform Lie algebra nn,1. We introduce a Fock module for the algebra nn,1 and provide classification of Leibniz algebras L whose corresponding Lie algebra L / I is the algebra nn,1 with condition that the ideal I is a Fock nn,1-module, where I is the ideal generated by squares of elements from L. We also consider Leibniz algebras with corresponding Lie algebra nn,1 and such that the action I ×nn,1 → I gives rise to a minimal faithful representation of nn,1. The classification up to isomorphism of such Leibniz algebras is given for the case of n = 4.

  2. The algebra of diffeomorphisms from the world sheet

    NASA Astrophysics Data System (ADS)

    Schulgin, Waldemar; Troost, Jan

    2014-09-01

    The quantum theory of a massless spin two particle is strongly constrained by diffeomorphism invariance, which is in turn implied by unitarity. We explicitly exhibit the space-time diffeomorphism algebra of string theory, realizing it in terms of world sheet vertex operators. Viewing diffeomorphisms as field redefinitions in the two-dimensional conformal field theory renders the calculation of their algebra straightforward. Next, we generalize the analysis to combinations of space-time anti-symmetric tensor gauge transformations and diffeomorphisms. We also point out a left-right split of the algebra combined with a twist that reproduces the C-bracket of double field theory. We further compare our derivation to an analysis in terms of marginal deformations as well as vertex operator algebras.

  3. Confluences of the Painlevé equations, Cherednik algebras and q-Askey scheme

    NASA Astrophysics Data System (ADS)

    Mazzocco, Marta

    2016-09-01

    In this paper we produce seven new algebras as confluences of the Cherednik algebra of type \\check {{{{C}1}}} {{C}1} and we characterise their spherical-sub-algebras. The limit of the spherical sub-algebra of the Cherednik algebra of type \\check {{{{C}1}}} {{C}1} is the monodromy manifold of the Painlevé VI equation (Oblomkov 2004 Int. Math. Res. Not. 2004 877–912). Here we prove that by considering the limits of the spherical sub-algebras of our new confluent algebras, one obtains the monodromy manifolds of all other Painlevé differential equations. Moreover, we introduce confluent versions of the Zhedanov algebra and prove that each of them (quotiented by their Casimir) is isomorphic to the corresponding spherical sub-algebra of our new confluent Cherednik algebras. We show that in the basic representation our confluent Zhedanov algebras act as symmetries of certain elements of the q-Askey scheme, thus setting a stepping stone towards the solution of the open problem of finding the corresponding quantum algebra for each element of the q-Askey scheme. These results establish a new link between the theory of the Painlevé equations and the theory of the q-Askey scheme making a step towards the construction of a representation theoretic approach for the Painlevé theory.

  4. Confluences of the Painlevé equations, Cherednik algebras and q-Askey scheme

    NASA Astrophysics Data System (ADS)

    Mazzocco, Marta

    2016-09-01

    In this paper we produce seven new algebras as confluences of the Cherednik algebra of type \\check {{{{C}1}}} {{C}1} and we characterise their spherical-sub-algebras. The limit of the spherical sub-algebra of the Cherednik algebra of type \\check {{{{C}1}}} {{C}1} is the monodromy manifold of the Painlevé VI equation (Oblomkov 2004 Int. Math. Res. Not. 2004 877-912). Here we prove that by considering the limits of the spherical sub-algebras of our new confluent algebras, one obtains the monodromy manifolds of all other Painlevé differential equations. Moreover, we introduce confluent versions of the Zhedanov algebra and prove that each of them (quotiented by their Casimir) is isomorphic to the corresponding spherical sub-algebra of our new confluent Cherednik algebras. We show that in the basic representation our confluent Zhedanov algebras act as symmetries of certain elements of the q-Askey scheme, thus setting a stepping stone towards the solution of the open problem of finding the corresponding quantum algebra for each element of the q-Askey scheme. These results establish a new link between the theory of the Painlevé equations and the theory of the q-Askey scheme making a step towards the construction of a representation theoretic approach for the Painlevé theory.

  5. Application of a transitional boundary-layer theory in the low hypersonic Mach number regime

    NASA Technical Reports Server (NTRS)

    Shamroth, S. J.; Mcdonald, H.

    1975-01-01

    An investigation is made to assess the capability of a finite-difference boundary-layer procedure to predict the mean profile development across a transition from laminar to turbulent flow in the low hypersonic Mach-number regime. The boundary-layer procedure uses an integral form of the turbulence kinetic-energy equation to govern the development of the Reynolds apparent shear stress. The present investigation shows the ability of this procedure to predict Stanton number, velocity profiles, and density profiles through the transition region and, in addition, to predict the effect of wall cooling and Mach number on transition Reynolds number. The contribution of the pressure-dilatation term to the energy balance is examined and it is suggested that transition can be initiated by the direct absorption of acoustic energy even if only a small amount (1 per cent) of the incident acoustic energy is absorbed.

  6. Cosmological baryon number domain structure from symmetry-breaking in grand unified field theories

    NASA Technical Reports Server (NTRS)

    Brown, R. W.; Stecker, F. W.

    1979-01-01

    It is suggested that grand unified field theories with spontaneous symmetry breaking in the very early big-bang can lead more naturally to a baryon symmetric cosmology with a domain structure than to a totally baryon asymmetric cosmology. The symmetry is broken in a randomized manner in causally independent domains, favoring neither a baryon nor an antibaryon excess on a universal scale. Arguments in favor of this cosmology and observational tests are discussed.

  7. Cosmological baryon-number domain structure from symmetry breaking in grand unified field theories

    NASA Technical Reports Server (NTRS)

    Brown, R. W.; Stecker, F. W.

    1979-01-01

    It is suggested that grand unified field theories with spontaneous symmetry breaking in the very early big bang can lead more naturally to a baryon-symmetric cosmology with a domain structure than to a totally baryon-asymmetric cosmology. The symmetry is broken in a randomized manner in causally independent domains, favoring neither a baryon nor an antibaryon excess on a universal scale. Arguments in favor of this cosmology and observational tests are discussed.

  8. Degenerate Sklyanin algebras

    NASA Astrophysics Data System (ADS)

    Smirnov, Andrey

    2010-08-01

    New trigonometric and rational solutions of the quantum Yang-Baxter equation (QYBE) are obtained by applying some singular gauge transformations to the known Belavin-Drinfeld elliptic R-matrix for sl(2;?). These solutions are shown to be related to the standard ones by the quasi-Hopf twist. We demonstrate that the quantum algebras arising from these new R-matrices can be obtained as special limits of the Sklyanin algebra. A representation for these algebras by the difference operators is found. The sl( N;?)-case is discussed.

  9. Degenerate Sklyanin algebras

    NASA Astrophysics Data System (ADS)

    Smirnov, Andrey

    2010-08-01

    New trigonometric and rational solutions of the quantum Yang-Baxter equation (QYBE) are obtained by applying some singular gauge transformations to the known Belavin-Drinfeld elliptic R-matrix for sl(2;?). These solutions are shown to be related to the standard ones by the quasi-Hopf twist. We demonstrate that the quantum algebras arising from these new R-matrices can be obtained as special limits of the Sklyanin algebra. A representation for these algebras by the difference operators is found. The sl(N;?)-case is discussed.

  10. Finite-particle-number approach to physics

    SciTech Connect

    Noyes, H.P.

    1982-10-01

    Starting from a discrete, self-generating and self-organizing, recursive model and self-consistent interpretive rules we construct: the scale constants of physics (3,10,137,1.7x10/sup 38/); 3+1 Minkowski space with a discrete metric and the algebraic bound ..delta.. is an element of ..delta.. tau is greater than or equal to 1; the Einstein-deBroglie relation; algebraic double slit interference; a single-time momentum-space scattering theory connected to laboratory experience; an approximation to wave functions; local phase severance and hence both distant correlations and separability; baryon number, lepton number, charge and helicity; m/sub p//m/sub e/; a cosmology not in disagreement with current observations.

  11. BPS preons and the AdS-M-algebra

    NASA Astrophysics Data System (ADS)

    Bandos, Igor A.; de Azcárraga, José A.

    2008-04-01

    We present here the AdS generalization of BPS preons, which were introduced as the hypothetical constituents of M-theory preserving all but one supersymmetries. Our construction, suggested by the relation of `lower dimensional preons' with higher spin theories, can be considered as a deformation of the M-algebraic description of the single supersymmetry broken by a preon, and provides another reason to identify the AdS generalization of the M-algebra, which we call the AdS-M-algebra, with osp(1|32).

  12. Shapes and stability of algebraic nuclear models

    NASA Technical Reports Server (NTRS)

    Lopez-Moreno, Enrique; Castanos, Octavio

    1995-01-01

    A generalization of the procedure to study shapes and stability of algebraic nuclear models introduced by Gilmore is presented. One calculates the expectation value of the Hamiltonian with respect to the coherent states of the algebraic structure of the system. Then equilibrium configurations of the resulting energy surface, which depends in general on state variables and a set of parameters, are classified through the Catastrophe theory. For one- and two-body interactions in the Hamiltonian of the interacting Boson model-1, the critical points are organized through the Cusp catastrophe. As an example, we apply this Separatrix to describe the energy surfaces associated to the Rutenium and Samarium isotopes.

  13. Towards a cladistics of double Yangians and elliptic algebras*

    NASA Astrophysics Data System (ADS)

    Arnaudon, D.; Avan, J.; Frappat, L.; Ragoucy, E.; Rossi, M.

    2000-09-01

    A self-contained description of algebraic structures, obtained by combinations of various limit procedures applied to vertex and face sl(2) elliptic quantum affine algebras, is given. New double Yangian structures of dynamical type are defined. Connections between these structures are established. A number of them take the form of twist-like actions. These are conjectured to be evaluations of universal twists.

  14. Algebraic integrability: a survey.

    PubMed

    Vanhaecke, Pol

    2008-03-28

    We give a concise introduction to the notion of algebraic integrability. Our exposition is based on examples and phenomena, rather than on detailed proofs of abstract theorems. We mainly focus on algebraic integrability in the sense of Adler-van Moerbeke, where the fibres of the momentum map are affine parts of Abelian varieties; as it turns out, most examples from classical mechanics are of this form. Two criteria are given for such systems (Kowalevski-Painlevé and Lyapunov) and each is illustrated in one example. We show in the case of a relatively simple example how one proves algebraic integrability, starting from the differential equations for the integrable vector field. For Hamiltonian systems that are algebraically integrable in the generalized sense, two examples are given, which illustrate the non-compact analogues of Abelian varieties which typically appear in such systems. PMID:17588863

  15. Covariant deformed oscillator algebras

    NASA Technical Reports Server (NTRS)

    Quesne, Christiane

    1995-01-01

    The general form and associativity conditions of deformed oscillator algebras are reviewed. It is shown how the latter can be fulfilled in terms of a solution of the Yang-Baxter equation when this solution has three distinct eigenvalues and satisfies a Birman-Wenzl-Murakami condition. As an example, an SU(sub q)(n) x SU(sub q)(m)-covariant q-bosonic algebra is discussed in some detail.

  16. Aprepro - Algebraic Preprocessor

    2005-08-01

    Aprepro is an algebraic preprocessor that reads a file containing both general text and algebraic, string, or conditional expressions. It interprets the expressions and outputs them to the output file along witht the general text. Aprepro contains several mathematical functions, string functions, and flow control constructs. In addition, functions are included that, with some additional files, implement a units conversion system and a material database lookup system.

  17. The Algebraic Way

    NASA Astrophysics Data System (ADS)

    Hiley, B. J.

    In this chapter, we examine in detail the non-commutative symplectic algebra underlying quantum dynamics. By using this algebra, we show that it contains both the Weyl-von Neumann and the Moyal quantum algebras. The latter contains the Wigner distribution as the kernel of the density matrix. The underlying non-commutative geometry can be projected into either of two Abelian spaces, so-called `shadow phase spaces'. One of these is the phase space of Bohmian mechanics, showing that it is a fragment of the basic underlying algebra. The algebraic approach is much richer, giving rise to two fundamental dynamical time development equations which reduce to the Liouville equation and the Hamilton-Jacobi equation in the classical limit. They also include the Schrödinger equation and its wave-function, showing that these features are a partial aspect of the more general non-commutative structure. We discuss briefly the properties of this more general mathematical background from which the non-commutative symplectic algebra emerges.

  18. A process algebra model of QED

    NASA Astrophysics Data System (ADS)

    Sulis, William

    2016-03-01

    The process algebra approach to quantum mechanics posits a finite, discrete, determinate ontology of primitive events which are generated by processes (in the sense of Whitehead). In this ontology, primitive events serve as elements of an emergent space-time and of emergent fundamental particles and fields. Each process generates a set of primitive elements, using only local information, causally propagated as a discrete wave, forming a causal space termed a causal tapestry. Each causal tapestry forms a discrete and finite sampling of an emergent causal manifold (space-time) M and emergent wave function. Interactions between processes are described by a process algebra which possesses 8 commutative operations (sums and products) together with a non-commutative concatenation operator (transitions). The process algebra possesses a representation via nondeterministic combinatorial games. The process algebra connects to quantum mechanics through the set valued process and configuration space covering maps, which associate each causal tapestry with sets of wave functions over M. Probabilities emerge from interactions between processes. The process algebra model has been shown to reproduce many features of the theory of non-relativistic scalar particles to a high degree of accuracy, without paradox or divergences. This paper extends the approach to a semi-classical form of quantum electrodynamics.

  19. Open-closed homotopy algebra in mathematical physics

    SciTech Connect

    Kajiura, Hiroshige; Stasheff, Jim

    2006-02-15

    In this paper we discuss various aspects of open-closed homotopy algebras (OCHAs) presented in our previous paper, inspired by Zwiebach's open-closed string field theory, but that first paper concentrated on the mathematical aspects. Here we show how an OCHA is obtained by extracting the tree part of Zwiebach's quantum open-closed string field theory. We clarify the explicit relation of an OCHA with Kontsevich's deformation quantization and with the B-models of homological mirror symmetry. An explicit form of the minimal model for an OCHA is given as well as its relation to the perturbative expansion of open-closed string field theory. We show that our open-closed homotopy algebra gives us a general scheme for deformation of open string structures (A{sub {infinity}} algebras) by closed strings (L{sub {infinity}} algebras)

  20. Local Algebras of Differential Operators

    NASA Astrophysics Data System (ADS)

    Church, P. T.; Timourian, J. G.

    2002-05-01

    There is an increasing literature devoted to the study of boundary value problems using singularity theory. The resulting differential operators are typically Fredholm with index 0, defined on infinite-dimensional spaces, and they have often led to folds, cusps, and even higher-order Morin singularities. In this paper we develop some of the local algebras of germs of such differential Fredholm operators, extending the theory of the finite-dimensional case. We apply this work to nonlinear elliptic boundary value problems: in particular, we make further progress on a question proposed and initially studied by Ruf [1999, J. Differential Equations 151, 111-133]. We also make comments on several problems raised by others.

  1. Locally Compact Quantum Groups. A von Neumann Algebra Approach

    NASA Astrophysics Data System (ADS)

    Van Daele, Alfons

    2014-08-01

    In this paper, we give an alternative approach to the theory of locally compact quantum groups, as developed by Kustermans and Vaes. We start with a von Neumann algebra and a comultiplication on this von Neumann algebra. We assume that there exist faithful left and right Haar weights. Then we develop the theory within this von Neumann algebra setting. In [Math. Scand. 92 (2003), 68-92] locally compact quantum groups are also studied in the von Neumann algebraic context. This approach is independent of the original C^*-algebraic approach in the sense that the earlier results are not used. However, this paper is not really independent because for many proofs, the reader is referred to the original paper where the C^*-version is developed. In this paper, we give a completely self-contained approach. Moreover, at various points, we do things differently. We have a different treatment of the antipode. It is similar to the original treatment in [Ann. Sci. & #201;cole Norm. Sup. (4) 33 (2000), 837-934]. But together with the fact that we work in the von Neumann algebra framework, it allows us to use an idea from [Rev. Roumaine Math. Pures Appl. 21 (1976), 1411-1449] to obtain the uniqueness of the Haar weights in an early stage. We take advantage of this fact when deriving the other main results in the theory. We also give a slightly different approach to duality. Finally, we collect, in a systematic way, several important formulas. In an appendix, we indicate very briefly how the C^*-approach and the von Neumann algebra approach eventually yield the same objects. The passage from the von Neumann algebra setting to the C^*-algebra setting is more or less standard. For the other direction, we use a new method. It is based on the observation that the Haar weights on the C^*-algebra extend to weights on the double dual with central support and that all these supports are the same. Of course, we get the von Neumann algebra by cutting down the double dual with this unique

  2. A Process Algebraic Framework for Modeling Resource Demand and Supply

    NASA Astrophysics Data System (ADS)

    Philippou, Anna; Lee, Insup; Sokolsky, Oleg; Choi, Jin-Young

    As real-time embedded systems become more complex, resource partitioning is increasingly used to guarantee real-time performance. Recently, several compositional frameworks of resource partitioning have been proposed using real-time scheduling theory with various notions of real-time tasks running under restricted resource supply environments. However, these real-time scheduling-based approaches are limited in their expressiveness in that, although capable of describing resource-demand tasks, they are unable to model resource supply. This paper describes a process algebraic framework for reasoning about resource demand and supply inspired by the timed process algebra ACSR. In ACSR, real-time tasks are specified by enunciating their consumption needs for resources. To also accommodate resource-supply processes we define PADS where, given a resource CPU, the complimented resource overline{CPU} denotes for availability of CPU for the corresponding demand process. Using PADS, we define a supply-demand relation where a pair (S, T) belongs to the relation if the demand process T can be scheduled under supply S. We develop a theory of compositional schedulability analysis as well as a technique for synthesizing an optimal supply process for a set of tasks. We illustrate our technique via a number of examples.

  3. On vertex algebra representations of the Schrödinger-Virasoro Lie algebra

    NASA Astrophysics Data System (ADS)

    Unterberger, Jérémie

    2009-12-01

    The Schrödinger-Virasoro Lie algebra sv is an extension of the Virasoro Lie algebra by a nilpotent Lie algebra formed with a bosonic current of weight 3/2 and a bosonic current of weight 1. It is also a natural infinite-dimensional extension of the Schrödinger Lie algebra, which — leaving aside the invariance under time-translation — has been proved to be a symmetry algebra for many statistical physics models undergoing a dynamics with dynamical exponent z=2. We define in this article general Schrödinger-Virasoro primary fields by analogy with conformal field theory, characterized by a 'spin' index and a (non-relativistic) mass, and construct vertex algebra representations of sv out of a charged symplectic boson and a free boson and its associated vertex operators. We also compute two- and three-point functions of still conjectural massive fields that are defined by an analytic continuation with respect to a formal parameter.

  4. Kinetic-theory predictions of clustering instabilities in granular flows: beyond the small-Knudsen-number regime

    SciTech Connect

    Mitrano, Peter P.; Zenk, John R.; Benyahia, Sofiane; Galvin, Janine E.; Dahl, Steven R.; Hrenya, Christine M.

    2013-12-04

    In this work we quantitatively assess, via instabilities, a Navier–Stokes-order (small- Knudsen-number) continuum model based on the kinetic theory analogy and applied to inelastic spheres in a homogeneous cooling system. Dissipative collisions are known to give rise to instabilities, namely velocity vortices and particle clusters, for sufficiently large domains. We compare predictions for the critical length scales required for particle clustering obtained from transient simulations using the continuum model with molecular dynamics (MD) simulations. The agreement between continuum simulations and MD simulations is excellent, particularly given the presence of well-developed velocity vortices at the onset of clustering. More specifically, spatial mapping of the local velocity-field Knudsen numbers (Knu) at the time of cluster detection reveals Knu » 1 due to the presence of large velocity gradients associated with vortices. Although kinetic-theory-based continuum models are based on a small- Kn (i.e. small-gradient) assumption, our findings suggest that, similar to molecular gases, Navier–Stokes-order (small-Kn) theories are surprisingly accurate outside their expected range of validity.

  5. Topological basis realization for BMW algebra and Heisenberg XXZ spin chain model

    NASA Astrophysics Data System (ADS)

    Liu, Bo; Xue, Kang; Wang, Gangcheng; Liu, Ying; Sun, Chunfang

    2015-04-01

    In this paper, we study three-dimensional (3D) reduced Birman-Murakami-Wenzl (BMW) algebra based on topological basis theory. Several examples of BMW algebra representations are reviewed. We also discuss a special solution of BMW algebra, which can be used to construct Heisenberg XXZ model. The theory of topological basis provides a useful method to solve quantum spin chain models. It is also shown that the ground state of XXZ spin chain is superposition state of topological basis.

  6. Algebraic description of external and internal attributes of fundamental fermions

    NASA Astrophysics Data System (ADS)

    Sogami, Ikuo S.

    2012-02-01

    To describe external and internal attributes of fundamental fermions, a theory of multi-spinor fields is developed on an algebra, a triplet algebra, which consists of all the triple-direct-products of Dirac γ-matrices. The triplet algebra is decomposed into the product of two subalgebras, an external algebra and an internal algebra, which are exclusively related with external and internal characteristic of the multi-spinor field named triplet fields. All elements of the external algebra which is isomorphic to the original Dirac algebra Aγ are invariant under the action of permutation group S3 which works to exchange the order of the Aγ elements in the triple-direct-product. The internal algebra is decomposed into the product of two 42 dimensional algebras, called the family and color algebras, which describe the family and color degrees of freedom. The family and color algebras have fine substructures with "trio plus solo" (3 + 1) conformations which are irreducible under the action of S3. The triplet field has trio plus solo family modes with ordinary tricolor quark and colorless solo lepton components. To incorporate the Weinberg-Salam mechanism, it is required to introduce two types of triplet fields, a left-handed doublet and right-handed singlets of electroweak iso-spin. It is possible to qualify the Yukawa interaction and to make a new interpretation of its coupling constants naturally in an intrinsic mechanism of the triplet field formalism. The ordinary Higgs mechanism leads to the Dirac mass matrices which can explain all data of quark sector within experimental accuracy.

  7. Mathematics: Algebra and Geometry. GED Scoreboost.

    ERIC Educational Resources Information Center

    Hoyt, Cathy

    GED "Scoreboost" materials target exactly the skills one needs to pass the General Educational Development (GED) tests. This book focuses on the GED Mathematics test. To prepare for the test, the test taker needs to learn skills in number and operation sense, data and statistics, geometry and measurement, and algebra. To pass the test, the test…

  8. A Visual Approach to Algebra Concepts.

    ERIC Educational Resources Information Center

    Morelli, Lynn

    1992-01-01

    Presents activities to visually explore the algebraic concepts of variable, constant, the distributive property, and combining like terms. Presents four transparencies that use visual models to understand exercises in students perform the same mental calculations on a number of their choice and obtain the same result. (MDH)

  9. A dynamo theory prediction for solar cycle 22: Sunspot number, radio flux, exospheric temperature, and total density at 400 km

    NASA Technical Reports Server (NTRS)

    Schatten, K. H.; Hedin, A. E.

    1986-01-01

    Using the dynamo theory method to predict solar activity, a value for the smoothed sunspot number of 109 + or - 20 is obtained for solar cycle 22. The predicted cycle is expected to peak near December, 1990 + or - 1 year. Concommitantly, F(10.7) radio flux is expected to reach a smoothed value of 158 + or - 18 flux units. Global mean exospheric temperature is expected to reach 1060 + or - 50 K and global total average total thermospheric density at 400 km is expected to reach 4.3 x 10 to the -15th gm/cu cm + or - 25 percent.

  10. A dynamo theory prediction for solar cycle 22 - Sunspot number, radio flux, exospheric temperature, and total density at 400 km

    NASA Technical Reports Server (NTRS)

    Schatten, K. H.; Hedin, A. E.

    1984-01-01

    Using the 'dynamo theory' method to predict solar activity, a value for the smoothed sunspot number of 109 + or - 20 is obtained for solar cycle 22. The predicted cycle is expected to peak near December, 1990 + or - 1 year. Concommitantly, F(10.7) radio flux is expected to reach a smoothed value of 158 + or - 18 flux units. Global mean exospheric temperature is expected to reach 1060 + or - 50 K and global total average total thermospheric density at 400 km is expected to reach 4.3 x 10 to the -15th gm/cu cm + or - 25 percent.

  11. Computational algebraic geometry of epidemic models

    NASA Astrophysics Data System (ADS)

    Rodríguez Vega, Martín.

    2014-06-01

    Computational Algebraic Geometry is applied to the analysis of various epidemic models for Schistosomiasis and Dengue, both, for the case without control measures and for the case where control measures are applied. The models were analyzed using the mathematical software Maple. Explicitly the analysis is performed using Groebner basis, Hilbert dimension and Hilbert polynomials. These computational tools are included automatically in Maple. Each of these models is represented by a system of ordinary differential equations, and for each model the basic reproductive number (R0) is calculated. The effects of the control measures are observed by the changes in the algebraic structure of R0, the changes in Groebner basis, the changes in Hilbert dimension, and the changes in Hilbert polynomials. It is hoped that the results obtained in this paper become of importance for designing control measures against the epidemic diseases described. For future researches it is proposed the use of algebraic epidemiology to analyze models for airborne and waterborne diseases.

  12. Abstract Algebra for Algebra Teaching: Influencing School Mathematics Instruction

    ERIC Educational Resources Information Center

    Wasserman, Nicholas H.

    2016-01-01

    This article explores the potential for aspects of abstract algebra to be influential for the teaching of school algebra (and early algebra). Using national standards for analysis, four primary areas common in school mathematics--and their progression across elementary, middle, and secondary mathematics--where teaching may be transformed by…

  13. Adaptive Algebraic Multigrid Methods

    SciTech Connect

    Brezina, M; Falgout, R; MacLachlan, S; Manteuffel, T; McCormick, S; Ruge, J

    2004-04-09

    Our ability to simulate physical processes numerically is constrained by our ability to solve the resulting linear systems, prompting substantial research into the development of multiscale iterative methods capable of solving these linear systems with an optimal amount of effort. Overcoming the limitations of geometric multigrid methods to simple geometries and differential equations, algebraic multigrid methods construct the multigrid hierarchy based only on the given matrix. While this allows for efficient black-box solution of the linear systems associated with discretizations of many elliptic differential equations, it also results in a lack of robustness due to assumptions made on the near-null spaces of these matrices. This paper introduces an extension to algebraic multigrid methods that removes the need to make such assumptions by utilizing an adaptive process. The principles which guide the adaptivity are highlighted, as well as their application to algebraic multigrid solution of certain symmetric positive-definite linear systems.

  14. Computer Program For Linear Algebra

    NASA Technical Reports Server (NTRS)

    Krogh, F. T.; Hanson, R. J.

    1987-01-01

    Collection of routines provided for basic vector operations. Basic Linear Algebra Subprogram (BLAS) library is collection from FORTRAN-callable routines for employing standard techniques to perform basic operations of numerical linear algebra.

  15. Algebra for Gifted Third Graders.

    ERIC Educational Resources Information Center

    Borenson, Henry

    1987-01-01

    Elementary school children who are exposed to a concrete, hands-on experience in algebraic linear equations will more readily develop a positive mind-set and expectation for success in later formal, algebraic studies. (CB)

  16. Pseudo Algebraically Closed Extensions

    NASA Astrophysics Data System (ADS)

    Bary-Soroker, Lior

    2009-07-01

    This PhD deals with the notion of pseudo algebraically closed (PAC) extensions of fields. It develops a group-theoretic machinery, based on a generalization of embedding problems, to study these extensions. Perhaps the main result is that although there are many PAC extensions, the Galois closure of a proper PAC extension is separably closed. The dissertation also contains the following subjects. The group theoretical counterpart of pseudo algebraically closed extensions, the so-called projective pairs. Applications to seemingly unrelated subjects, e.g., an analog of Dirichlet's theorem about primes in arithmetic progression for polynomial rings in one variable over infinite fields.

  17. Tensor Algebra Library for NVidia Graphics Processing Units

    SciTech Connect

    Liakh, Dmitry

    2015-03-16

    This is a general purpose math library implementing basic tensor algebra operations on NVidia GPU accelerators. This software is a tensor algebra library that can perform basic tensor algebra operations, including tensor contractions, tensor products, tensor additions, etc., on NVidia GPU accelerators, asynchronously with respect to the CPU host. It supports a simultaneous use of multiple NVidia GPUs. Each asynchronous API function returns a handle which can later be used for querying the completion of the corresponding tensor algebra operation on a specific GPU. The tensors participating in a particular tensor operation are assumed to be stored in local RAM of a node or GPU RAM. The main research area where this library can be utilized is the quantum many-body theory (e.g., in electronic structure theory).

  18. Tensor Algebra Library for NVidia Graphics Processing Units

    2015-03-16

    This is a general purpose math library implementing basic tensor algebra operations on NVidia GPU accelerators. This software is a tensor algebra library that can perform basic tensor algebra operations, including tensor contractions, tensor products, tensor additions, etc., on NVidia GPU accelerators, asynchronously with respect to the CPU host. It supports a simultaneous use of multiple NVidia GPUs. Each asynchronous API function returns a handle which can later be used for querying the completion ofmore » the corresponding tensor algebra operation on a specific GPU. The tensors participating in a particular tensor operation are assumed to be stored in local RAM of a node or GPU RAM. The main research area where this library can be utilized is the quantum many-body theory (e.g., in electronic structure theory).« less

  19. Stochastic theory of large-scale enzyme-reaction networks: finite copy number corrections to rate equation models.

    PubMed

    Thomas, Philipp; Straube, Arthur V; Grima, Ramon

    2010-11-21

    Chemical reactions inside cells occur in compartment volumes in the range of atto- to femtoliters. Physiological concentrations realized in such small volumes imply low copy numbers of interacting molecules with the consequence of considerable fluctuations in the concentrations. In contrast, rate equation models are based on the implicit assumption of infinitely large numbers of interacting molecules, or equivalently, that reactions occur in infinite volumes at constant macroscopic concentrations. In this article we compute the finite-volume corrections (or equivalently the finite copy number corrections) to the solutions of the rate equations for chemical reaction networks composed of arbitrarily large numbers of enzyme-catalyzed reactions which are confined inside a small subcellular compartment. This is achieved by applying a mesoscopic version of the quasisteady-state assumption to the exact Fokker-Planck equation associated with the Poisson representation of the chemical master equation. The procedure yields impressively simple and compact expressions for the finite-volume corrections. We prove that the predictions of the rate equations will always underestimate the actual steady-state substrate concentrations for an enzyme-reaction network confined in a small volume. In particular we show that the finite-volume corrections increase with decreasing subcellular volume, decreasing Michaelis-Menten constants, and increasing enzyme saturation. The magnitude of the corrections depends sensitively on the topology of the network. The predictions of the theory are shown to be in excellent agreement with stochastic simulations for two types of networks typically associated with protein methylation and metabolism.

  20. Stochastic theory of large-scale enzyme-reaction networks: Finite copy number corrections to rate equation models

    NASA Astrophysics Data System (ADS)

    Thomas, Philipp; Straube, Arthur V.; Grima, Ramon

    2010-11-01

    Chemical reactions inside cells occur in compartment volumes in the range of atto- to femtoliters. Physiological concentrations realized in such small volumes imply low copy numbers of interacting molecules with the consequence of considerable fluctuations in the concentrations. In contrast, rate equation models are based on the implicit assumption of infinitely large numbers of interacting molecules, or equivalently, that reactions occur in infinite volumes at constant macroscopic concentrations. In this article we compute the finite-volume corrections (or equivalently the finite copy number corrections) to the solutions of the rate equations for chemical reaction networks composed of arbitrarily large numbers of enzyme-catalyzed reactions which are confined inside a small subcellular compartment. This is achieved by applying a mesoscopic version of the quasisteady-state assumption to the exact Fokker-Planck equation associated with the Poisson representation of the chemical master equation. The procedure yields impressively simple and compact expressions for the finite-volume corrections. We prove that the predictions of the rate equations will always underestimate the actual steady-state substrate concentrations for an enzyme-reaction network confined in a small volume. In particular we show that the finite-volume corrections increase with decreasing subcellular volume, decreasing Michaelis-Menten constants, and increasing enzyme saturation. The magnitude of the corrections depends sensitively on the topology of the network. The predictions of the theory are shown to be in excellent agreement with stochastic simulations for two types of networks typically associated with protein methylation and metabolism.

  1. Assessing Elementary Algebra with STACK

    ERIC Educational Resources Information Center

    Sangwin, Christopher J.

    2007-01-01

    This paper concerns computer aided assessment (CAA) of mathematics in which a computer algebra system (CAS) is used to help assess students' responses to elementary algebra questions. Using a methodology of documentary analysis, we examine what is taught in elementary algebra. The STACK CAA system, http://www.stack.bham.ac.uk/, which uses the CAS…

  2. q-graded Heisenberg algebras and deformed supersymmetries

    SciTech Connect

    Ben Geloun, Joseph; Hounkonnou, Mahouton Norbert

    2010-02-15

    The notion of q-grading on the enveloping algebra generated by products of q-deformed Heisenberg algebras is introduced for q complex number in the unit disk. Within this formulation, we consider the extension of the notion of supersymmetry in the enveloping algebra. We recover the ordinary Z{sub 2} grading or Grassmann parity for associative superalgebra and a modified version of the usual supersymmetry. As a specific problem, we focus on the interesting limit q{yields}-1 for which the Arik and Coon deformation [J. Math. Phys. 17, 524 (1976)] of the Heisenberg algebra allows one to map fermionic modes to bosonic ones in a modified sense. Different algebraic consequences are discussed.

  3. Gene algebra from a genetic code algebraic structure.

    PubMed

    Sanchez, R; Morgado, E; Grau, R

    2005-10-01

    By considering two important factors involved in the codon-anticodon interactions, the hydrogen bond number and the chemical type of bases, a codon array of the genetic code table as an increasing code scale of interaction energies of amino acids in proteins was obtained. Next, in order to consecutively obtain all codons from the codon AAC, a sum operation has been introduced in the set of codons. The group obtained over the set of codons is isomorphic to the group (Z(64), +) of the integer module 64. On the Z(64)-algebra of the set of 64(N) codon sequences of length N, gene mutations are described by means of endomorphisms f:(Z(64))(N)-->(Z(64))(N). Endomorphisms and automorphisms helped us describe the gene mutation pathways. For instance, 77.7% mutations in 749 HIV protease gene sequences correspond to unique diagonal endomorphisms of the wild type strain HXB2. In particular, most of the reported mutations that confer drug resistance to the HIV protease gene correspond to diagonal automorphisms of the wild type. What is more, in the human beta-globin gene a similar situation appears where most of the single codon mutations correspond to automorphisms. Hence, in the analyses of molecular evolution process on the DNA sequence set of length N, the Z(64)-algebra will help us explain the quantitative relationships between genes.

  4. Pawlak Algebra and Approximate Structure on Fuzzy Lattice

    PubMed Central

    Zhuang, Ying; Liu, Wenqi; Wu, Chin-Chia; Li, Jinhai

    2014-01-01

    The aim of this paper is to investigate the general approximation structure, weak approximation operators, and Pawlak algebra in the framework of fuzzy lattice, lattice topology, and auxiliary ordering. First, we prove that the weak approximation operator space forms a complete distributive lattice. Then we study the properties of transitive closure of approximation operators and apply them to rough set theory. We also investigate molecule Pawlak algebra and obtain some related properties. PMID:25152922

  5. Pawlak algebra and approximate structure on fuzzy lattice.

    PubMed

    Zhuang, Ying; Liu, Wenqi; Wu, Chin-Chia; Li, Jinhai

    2014-01-01

    The aim of this paper is to investigate the general approximation structure, weak approximation operators, and Pawlak algebra in the framework of fuzzy lattice, lattice topology, and auxiliary ordering. First, we prove that the weak approximation operator space forms a complete distributive lattice. Then we study the properties of transitive closure of approximation operators and apply them to rough set theory. We also investigate molecule Pawlak algebra and obtain some related properties.

  6. Using Group Explorer in teaching abstract algebra

    NASA Astrophysics Data System (ADS)

    Schubert, Claus; Gfeller, Mary; Donohue, Christopher

    2013-04-01

    This study explores the use of Group Explorer in an undergraduate mathematics course in abstract algebra. The visual nature of Group Explorer in representing concepts in group theory is an attractive incentive to use this software in the classroom. However, little is known about students' perceptions on this technology in learning concepts in abstract algebra. A total of 26 participants in an undergraduate course studying group theory were surveyed regarding their experiences using Group Explorer. Findings indicate that all participants believed that the software was beneficial to their learning and described their attitudes regarding the software in terms of using the technology and its helpfulness in learning concepts. A multiple regression analysis reveals that representational fluency of concepts with the software correlated significantly with participants' understanding of group concepts yet, participants' attitudes about Group Explorer and technology in general were not significant factors.

  7. Algebraic curves of maximal cyclicity

    NASA Astrophysics Data System (ADS)

    Caubergh, Magdalena; Dumortier, Freddy

    2006-01-01

    The paper deals with analytic families of planar vector fields, studying methods to detect the cyclicity of a non-isolated closed orbit, i.e. the maximum number of limit cycles that can locally bifurcate from it. It is known that this multi-parameter problem can be reduced to a single-parameter one, in the sense that there exist analytic curves in parameter space along which the maximal cyclicity can be attained. In that case one speaks about a maximal cyclicity curve (mcc) in case only the number is considered and of a maximal multiplicity curve (mmc) in case the multiplicity is also taken into account. In view of obtaining efficient algorithms for detecting the cyclicity, we investigate whether such mcc or mmc can be algebraic or even linear depending on certain general properties of the families or of their associated Bautin ideal. In any case by well chosen examples we show that prudence is appropriate.

  8. Thinking Visually about Algebra

    ERIC Educational Resources Information Center

    Baroudi, Ziad

    2015-01-01

    Many introductions to algebra in high school begin with teaching students to generalise linear numerical patterns. This article argues that this approach needs to be changed so that students encounter variables in the context of modelling visual patterns so that the variables have a meaning. The article presents sample classroom activities,…

  9. Computer Algebra versus Manipulation

    ERIC Educational Resources Information Center

    Zand, Hossein; Crowe, David

    2004-01-01

    In the UK there is increasing concern about the lack of skill in algebraic manipulation that is evident in students entering mathematics courses at university level. In this note we discuss how the computer can be used to ameliorate some of the problems. We take as an example the calculations needed in three dimensional vector analysis in polar…

  10. Algebraic Artful Aids.

    ERIC Educational Resources Information Center

    Glick, David

    1995-01-01

    Presents a technique that helps students concentrate more on the science and less on the mechanics of algebra while dealing with introductory physics formulas. Allows the teacher to do complex problems at a lower level and not be too concerned about the mathematical abilities of the students. (JRH)

  11. Computers in Abstract Algebra

    ERIC Educational Resources Information Center

    Nwabueze, Kenneth K.

    2004-01-01

    The current emphasis on flexible modes of mathematics delivery involving new information and communication technology (ICT) at the university level is perhaps a reaction to the recent change in the objectives of education. Abstract algebra seems to be one area of mathematics virtually crying out for computer instructional support because of the…

  12. On Cohen-Macaulayness of Algebras Generated by Generalized Power Sums. With an appendix by Misha Feigin

    NASA Astrophysics Data System (ADS)

    Etingof, Pavel; Rains, Eric

    2016-10-01

    Generalized power sums are linear combinations of ith powers of coordinates. We consider subalgebras of the polynomial algebra generated by generalized power sums, and study when such algebras are Cohen-Macaulay. It turns out that the Cohen-Macaulay property of such algebras is rare, and tends to be related to quantum integrability and representation theory of Cherednik algebras. Using representation theoretic results and deformation theory, we establish Cohen-Macaulayness of the algebra of q, t-deformed power sums defined by Sergeev and Veselov, and of some generalizations of this algebra, proving a conjecture of Brookner, Corwin, Etingof, and Sam. We also apply representation-theoretic techniques to studying m-quasi-invariants of deformed Calogero-Moser systems. In an appendix to this paper, M. Feigin uses representation theory of Cherednik algebras to compute Hilbert series for such quasi-invariants, and show that in the case of one light particle, the ring of quasi-invariants is Gorenstein.

  13. Achievements and Problems in Diophantine Approximation Theory

    NASA Astrophysics Data System (ADS)

    Sprindzhuk, V. G.

    1980-08-01

    ContentsIntroduction I. Metrical theory of approximation on manifolds § 1. The basic problem § 2. Brief survey of results § 3. The principal conjecture II. Metrical theory of transcendental numbers § 1. Mahler's classification of numbers § 2. Metrical characterization of numbers with a given type of approximation § 3. Further problems III. Approximation of algebraic numbers by rationals § 1. Simultaneous approximations § 2. The inclusion of p-adic metrics § 3. Effective improvements of Liouville's inequality IV. Estimates of linear forms in logarithms of algebraic numbers § 1. The basic method § 2. Survey of results § 3. Estimates in the p-adic metric V. Diophantine equations § 1. Ternary exponential equations § 2. The Thue and Thue-Mahler equations § 3. Equations of hyperelliptic type § 4. Algebraic-exponential equations VI. The arithmetic structure of polynomials and the class number § 1. The greatest prime divisor of a polynomial in one variable § 2. The greatest prime divisor of a polynomial in two variables § 3. Square-free divisors of polynomials and the class number § 4. The general problem of the size of the class number Conclusion References

  14. Algebraic connectivity and graph robustness.

    SciTech Connect

    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.

  15. The Progressive Development of Early Embodied Algebraic Thinking

    ERIC Educational Resources Information Center

    Radford, Luis

    2014-01-01

    In this article I present some results from a 5-year longitudinal investigation with young students about the genesis of embodied, non-symbolic algebraic thinking and its progressive transition to culturally evolved forms of symbolic thinking. The investigation draws on a cultural-historical theory of teaching and learning--the theory of…

  16. Phase transitions for rotational states within an algebraic cluster model

    NASA Astrophysics Data System (ADS)

    López Moreno, E.; Morales Hernández, G. E.; Hess, P. O.; Yépez Martínez, H.

    2016-07-01

    The ground state and excited, rotational phase transitions are investigated within the Semimicroscopic Algebraic Cluster Model (SACM). The catastrophe theory is used to describe these phase transitions. Short introductions to the SACM and the catastrophe theory are given. We apply the formalism to the case of 16O+α→20Ne.

  17. A possible framework of the Lipkin model obeying the SU(n) algebra in arbitrary fermion number. II: Two subalgebras in the SU(n) Lipkin model and an approach to the construction of a linearly independent basis

    NASA Astrophysics Data System (ADS)

    Tsue, Yasuhiko; Providência, Constança; Providência, João da; Yamamura, Masatoshi

    2016-08-01

    Based on the results for the minimum weight states obtained in the previous paper (I), an idea of how to construct the linearly independent basis is proposed for the SU(n) Lipkin model. This idea starts in setting up m independent SU(2) subalgebras in the cases with n=2m and n=2m+1 (m=2,3,4,…). The original representation is re-formed in terms of the spherical tensors for the SU(n) generators built under the SU(2) subalgebras. Through this re-formation, the SU(m) subalgebra can be found. For constructing the linearly independent basis, not only the SU(2) algebras but also the SU(m) subalgebra play a central role. Some concrete results in the cases with n=2, 3, 4, and 5 are presented.

  18. Unified derivation of exact solutions to the relativistic Coulomb problem: Lie algebraic approach

    NASA Astrophysics Data System (ADS)

    Panahi, H.; Baradaran, M.; Savadi, A.

    2015-10-01

    Exact algebraic solutions of the D-dimensional Dirac and Klein-Gordon equations for the Coulomb potential are obtained in a unified treatment. It is shown that two cases are reducible to the same basic equation, which can be solved exactly. Using the Lie algebraic approach, the general exact solutions of the problem are obtained within the framework of representation theory of the sl(2) Lie algebra.

  19. Generalizing the Connes Moscovici Hopf algebra to contain all rooted trees

    SciTech Connect

    Agarwala, Susama; Delaney, Colleen

    2015-04-15

    This paper defines a generalization of the Connes-Moscovici Hopf algebra, H(1), that contains the entire Hopf algebra of rooted trees. A relationship between the former, a much studied object in non-commutative geometry, and the latter, a much studied object in perturbative quantum field theory, has been established by Connes and Kreimer. The results of this paper open the door to study the cohomology of the Hopf algebra of rooted trees.

  20. Remarkable algebraic independence property of certain series related to continued fractions

    NASA Astrophysics Data System (ADS)

    Tanaka, Taka-aki

    2008-01-01

    We prove, using Mahler's method, the following results: Theorem 1 asserts that the series Θ(x,a,q) are algebraically independent for any distinct triplets (x,a,q) of nonzero algebraic numbers, where Θ(x,a,q) has the property shown in Corollary 1 that Θ(a,a,q) is expressed as a continued fraction. Theorem 2 asserts, under the weaker condition than that of Theorem 1, that the values Θ(x,1,q) are algebraically independent for any distinct pairs (x,q) of nonzero algebraic numbers. Typical examples of these results are generated by Fibonacci numbers.

  1. The Algebra Artist

    ERIC Educational Resources Information Center

    Beigie, Darin

    2014-01-01

    Most people who are attracted to STEM-related fields are drawn not by a desire to take mathematics tests but to create things. The opportunity to create an algebra drawing gives students a sense of ownership and adventure that taps into the same sort of energy that leads a young person to get lost in reading a good book, building with Legos®,…

  2. Algebra of Majorana doubling.

    PubMed

    Lee, Jaehoon; Wilczek, Frank

    2013-11-27

    Motivated by the problem of identifying Majorana mode operators at junctions, we analyze a basic algebraic structure leading to a doubled spectrum. For general (nonlinear) interactions the emergent mode creation operator is highly nonlinear in the original effective mode operators, and therefore also in the underlying electron creation and destruction operators. This phenomenon could open up new possibilities for controlled dynamical manipulation of the modes. We briefly compare and contrast related issues in the Pfaffian quantum Hall state.

  3. Algebraic Multigrid Benchmark

    SciTech Connect

    2013-05-06

    AMG2013 is a parallel algebraic multigrid solver for linear systems arising from problems on unstructured grids. It has been derived directly from the Boomer AMG solver in the hypre library, a large linear solvers library that is being developed in the Center for Applied Scientific Computing (CASC) at LLNL. The driver provided in the benchmark can build various test problems. The default problem is a Laplace type problem on an unstructured domain with various jumps and an anisotropy in one part.

  4. PC Basic Linear Algebra Subroutines

    1992-03-09

    PC-BLAS is a highly optimized version of the Basic Linear Algebra Subprograms (BLAS), a standardized set of thirty-eight routines that perform low-level operations on vectors of numbers in single and double-precision real and complex arithmetic. Routines are included to find the index of the largest component of a vector, apply a Givens or modified Givens rotation, multiply a vector by a constant, determine the Euclidean length, perform a dot product, swap and copy vectors, andmore » find the norm of a vector. The BLAS have been carefully written to minimize numerical problems such as loss of precision and underflow and are designed so that the computation is independent of the interface with the calling program. This independence is achieved through judicious use of Assembly language macros. Interfaces are provided for Lahey Fortran 77, Microsoft Fortran 77, and Ryan-McFarland IBM Professional Fortran.« less

  5. Matematica Para La Escuela Secundaria, Primer Curso de Algebra (Parte 1), Comentario. Traduccion Preliminar de la Edicion en Ingles Revisada. (Mathematics for High School, First Course in Algebra, Part 1, Teacher's Commentary. Translation of the Revised English Edition).

    ERIC Educational Resources Information Center

    Allen, Frank B.; And Others

    This is the teacher's commentary for part one of a three-part SMSG algebra text for high school students. The principal objective of the text is to help the student develop an understanding and appreciation of some of the algebraic structure as a basis for the techniques of algebra. Chapter topics include congruence; numbers and variables;…

  6. Matematica Para La Escuela Secundaria, Primer Curso de Algebra (Parte 2). Traduccion Preliminar de la Edicion Inglesa Revisada. (Mathematics for High School, First Course in Algebra, Part 2. Preliminary Translation of the Revised English Edition).

    ERIC Educational Resources Information Center

    Allen, Frank B.; And Others

    This is part two of a three-part SMSG algebra text for high school students. The principal objective of the text is to help the student develop an understanding and appreciation of some of the algebraic structure as a basis for the techniques of algebra. Chapter topics include addition and multiplication of real numbers, subtraction and division…

  7. Matematica Para La Escuela Secundaria, Primer Curso de Algebra (Parte 1). Traduccion Preliminar de la Edicion Inglesa Revisada. (Mathematics for High School, First Course in Algebra, Part 1. Preliminary Translation of the Revised English Edition).

    ERIC Educational Resources Information Center

    Allen, Frank B.; And Others

    This is the student text for part one of a three-part SMSG algebra course for high school students. The principal objective of the text is to help the student develop an understanding and appreciation of some of the algebraic structure as a basis for the techniques of algebra. Chapter topics include congruence; numbers and variables; operations;…

  8. Algebraic Modeling of Information Retrieval in XML Documents

    NASA Astrophysics Data System (ADS)

    Georgiev, Bozhidar; Georgieva, Adriana

    2009-11-01

    This paper presents an information retrieval approach in XML documents using tools, based on the linear algebra. The well-known transformation languages as XSLT (XPath) are grounded on the features of higher-order logic for manipulating hierarchical trees. The presented conception is compared to existing higher-order logic formalisms, where the queries are realized by both languages XSLT and XPath. The possibilities of the proposed linear algebraic model combined with hierarchy data models permit more efficient solutions for searching, extracting and manipulating semi-structured data with hierarchical structures avoiding the global navigation over the XML tree components. The main purpose of this algebraic model representation, applied to the hierarchical relationships in the XML data structures, is to make the implementation of linear algebra tools possible for XML data manipulations and to eliminate existing problems, related to regular grammars theory and also to avoid the difficulties, connected with higher -order logic (first-order logic, monadic second- order logic etc.).

  9. D-algebra structure of topological insulators

    NASA Astrophysics Data System (ADS)

    Estienne, B.; Regnault, N.; Bernevig, B. A.

    2012-12-01

    In the quantum Hall effect, the density operators at different wave vectors generally do not commute and give rise to the Girvin-MacDonald-Plazmann (GMP) algebra, with important consequences such as ground-state center-of-mass degeneracy at fractional filling fraction, and W1+∞ symmetry of the filled Landau levels. We show that the natural generalization of the GMP algebra to higher-dimensional topological insulators involves the concept of a D commutator. For insulators in even-dimensional space, the D commutator is isotropic and closes, and its structure factors are proportional to the D/2 Chern number. In odd dimensions, the algebra is not isotropic, contains the weak topological insulator index (layers of the topological insulator in one fewer dimension), and does not contain the Chern-Simons θ form. This algebraic structure paves the way towards the identification of fractional topological insulators through the counting of their excitations. The possible relation to D-dimensional volume-preserving diffeomorphisms and parallel transport of extended objects is also discussed.

  10. On the cohomology of Leibniz conformal algebras

    NASA Astrophysics Data System (ADS)

    Zhang, Jiao

    2015-04-01

    We construct a new cohomology complex of Leibniz conformal algebras with coefficients in a representation instead of a module. The low-dimensional cohomology groups of this complex are computed. Meanwhile, we construct a Leibniz algebra from a Leibniz conformal algebra and prove that the category of Leibniz conformal algebras is equivalent to the category of equivalence classes of formal distribution Leibniz algebras.

  11. Assessing Algebraic Solving Ability: A Theoretical Framework

    ERIC Educational Resources Information Center

    Lian, Lim Hooi; Yew, Wun Thiam

    2012-01-01

    Algebraic solving ability had been discussed by many educators and researchers. There exists no definite definition for algebraic solving ability as it can be viewed from different perspectives. In this paper, the nature of algebraic solving ability in terms of algebraic processes that demonstrate the ability in solving algebraic problem is…

  12. Duncan F. Gregory, William Walton and the development of British algebra: 'algebraical geometry', 'geometrical algebra', abstraction.

    PubMed

    Verburgt, Lukas M

    2016-01-01

    This paper provides a detailed account of the period of the complex history of British algebra and geometry between the publication of George Peacock's Treatise on Algebra in 1830 and William Rowan Hamilton's paper on quaternions of 1843. During these years, Duncan Farquharson Gregory and William Walton published several contributions on 'algebraical geometry' and 'geometrical algebra' in the Cambridge Mathematical Journal. These contributions enabled them not only to generalize Peacock's symbolical algebra on the basis of geometrical considerations, but also to initiate the attempts to question the status of Euclidean space as the arbiter of valid geometrical interpretations. At the same time, Gregory and Walton were bound by the limits of symbolical algebra that they themselves made explicit; their work was not and could not be the 'abstract algebra' and 'abstract geometry' of figures such as Hamilton and Cayley. The central argument of the paper is that an understanding of the contributions to 'algebraical geometry' and 'geometrical algebra' of the second generation of 'scientific' symbolical algebraists is essential for a satisfactory explanation of the radical transition from symbolical to abstract algebra that took place in British mathematics in the 1830s-1840s. PMID:26806075

  13. Duncan F. Gregory, William Walton and the development of British algebra: 'algebraical geometry', 'geometrical algebra', abstraction.

    PubMed

    Verburgt, Lukas M

    2016-01-01

    This paper provides a detailed account of the period of the complex history of British algebra and geometry between the publication of George Peacock's Treatise on Algebra in 1830 and William Rowan Hamilton's paper on quaternions of 1843. During these years, Duncan Farquharson Gregory and William Walton published several contributions on 'algebraical geometry' and 'geometrical algebra' in the Cambridge Mathematical Journal. These contributions enabled them not only to generalize Peacock's symbolical algebra on the basis of geometrical considerations, but also to initiate the attempts to question the status of Euclidean space as the arbiter of valid geometrical interpretations. At the same time, Gregory and Walton were bound by the limits of symbolical algebra that they themselves made explicit; their work was not and could not be the 'abstract algebra' and 'abstract geometry' of figures such as Hamilton and Cayley. The central argument of the paper is that an understanding of the contributions to 'algebraical geometry' and 'geometrical algebra' of the second generation of 'scientific' symbolical algebraists is essential for a satisfactory explanation of the radical transition from symbolical to abstract algebra that took place in British mathematics in the 1830s-1840s.

  14. How Structure Sense for Algebraic Expressions or Equations Is Related to Structure Sense for Abstract Algebra

    ERIC Educational Resources Information Center

    Novotna, Jarmila; Hoch, Maureen

    2008-01-01

    Many students have difficulties with basic algebraic concepts at high school and at university. In this paper two levels of algebraic structure sense are defined: for high school algebra and for university algebra. We suggest that high school algebra structure sense components are sub-components of some university algebra structure sense…

  15. Renormalization group flows and continual Lie algebras

    NASA Astrophysics Data System (ADS)

    Bakas, Ioannis

    2003-08-01

    We study the renormalization group flows of two-dimensional metrics in sigma models using the one-loop beta functions, and demonstrate that they provide a continual analogue of the Toda field equations in conformally flat coordinates. In this algebraic setting, the logarithm of the world-sheet length scale, t, is interpreted as Dynkin parameter on the root system of a novel continual Lie algebra, denoted by Script G(d/dt;1), with anti-symmetric Cartan kernel K(t,t') = delta'(t-t'); as such, it coincides with the Cartan matrix of the superalgebra sl(N|N+1) in the large-N limit. The resulting Toda field equation is a non-linear generalization of the heat equation, which is integrable in target space and shares the same dissipative properties in time, t. We provide the general solution of the renormalization group flows in terms of free fields, via Bäcklund transformations, and present some simple examples that illustrate the validity of their formal power series expansion in terms of algebraic data. We study in detail the sausage model that arises as geometric deformation of the O(3) sigma model, and give a new interpretation to its ultra-violet limit by gluing together two copies of Witten's two-dimensional black hole in the asymptotic region. We also provide some new solutions that describe the renormalization group flow of negatively curved spaces in different patches, which look like a cane in the infra-red region. Finally, we revisit the transition of a flat cone C/Zn to the plane, as another special solution, and note that tachyon condensation in closed string theory exhibits a hidden relation to the infinite dimensional algebra Script G(d/dt;1) in the regime of gravity. Its exponential growth holds the key for the construction of conserved currents and their systematic interpretation in string theory, but they still remain unknown.

  16. Algebraic methods for the solution of some linear matrix equations

    NASA Technical Reports Server (NTRS)

    Djaferis, T. E.; Mitter, S. K.

    1979-01-01

    The characterization of polynomials whose zeros lie in certain algebraic domains (and the unification of the ideas of Hermite and Lyapunov) is the basis for developing finite algorithms for the solution of linear matrix equations. Particular attention is given to equations PA + A'P = Q (the Lyapunov equation) and P - A'PA = Q the (discrete Lyapunov equation). The Lyapunov equation appears in several areas of control theory such as stability theory, optimal control (evaluation of quadratic integrals), stochastic control (evaluation of covariance matrices) and in the solution of the algebraic Riccati equation using Newton's method.

  17. 2-Local derivations on matrix algebras over semi-prime Banach algebras and on AW*-algebras

    NASA Astrophysics Data System (ADS)

    Ayupov, Shavkat; Kudaybergenov, Karimbergen

    2016-03-01

    The paper is devoted to 2-local derivations on matrix algebras over unital semi-prime Banach algebras. For a unital semi-prime Banach algebra A with the inner derivation property we prove that any 2-local derivation on the algebra M 2n (A), n ≥ 2, is a derivation. We apply this result to AW*-algebras and show that any 2-local derivation on an arbitrary AW*-algebra is a derivation.

  18. Solving stochastic epidemiological models using computer algebra

    NASA Astrophysics Data System (ADS)

    Hincapie, Doracelly; Ospina, Juan

    2011-06-01

    Mathematical modeling in Epidemiology is an important tool to understand the ways under which the diseases are transmitted and controlled. The mathematical modeling can be implemented via deterministic or stochastic models. Deterministic models are based on short systems of non-linear ordinary differential equations and the stochastic models are based on very large systems of linear differential equations. Deterministic models admit complete, rigorous and automatic analysis of stability both local and global from which is possible to derive the algebraic expressions for the basic reproductive number and the corresponding epidemic thresholds using computer algebra software. Stochastic models are more difficult to treat and the analysis of their properties requires complicated considerations in statistical mathematics. In this work we propose to use computer algebra software with the aim to solve epidemic stochastic models such as the SIR model and the carrier-borne model. Specifically we use Maple to solve these stochastic models in the case of small groups and we obtain results that do not appear in standard textbooks or in the books updated on stochastic models in epidemiology. From our results we derive expressions which coincide with those obtained in the classical texts using advanced procedures in mathematical statistics. Our algorithms can be extended for other stochastic models in epidemiology and this shows the power of computer algebra software not only for analysis of deterministic models but also for the analysis of stochastic models. We also perform numerical simulations with our algebraic results and we made estimations for the basic parameters as the basic reproductive rate and the stochastic threshold theorem. We claim that our algorithms and results are important tools to control the diseases in a globalized world.

  19. Trigonometrical sums connected with the chiral Potts model, Verlinde dimension formula, two-dimensional resistor network, and number theory

    SciTech Connect

    Chair, Noureddine

    2014-02-15

    We have recently developed methods for obtaining exact two-point resistance of the complete graph minus N edges. We use these methods to obtain closed formulas of certain trigonometrical sums that arise in connection with one-dimensional lattice, in proving Scott’s conjecture on permanent of Cauchy matrix, and in the perturbative chiral Potts model. The generalized trigonometrical sums of the chiral Potts model are shown to satisfy recursion formulas that are transparent and direct, and differ from those of Gervois and Mehta. By making a change of variables in these recursion formulas, the dimension of the space of conformal blocks of SU(2) and SO(3) WZW models may be computed recursively. Our methods are then extended to compute the corner-to-corner resistance, and the Kirchhoff index of the first non-trivial two-dimensional resistor network, 2×N. Finally, we obtain new closed formulas for variant of trigonometrical sums, some of which appear in connection with number theory. -- Highlights: • Alternative derivation of certain trigonometrical sums of the chiral Potts model are given. • Generalization of these trigonometrical sums satisfy recursion formulas. • The dimension of the space of conformal blocks may be computed from these recursions. • Exact corner-to-corner resistance, the Kirchhoff index of 2×N are given.

  20. NPN fuzzy sets and NPN qualitative algebra: a computational framework for bipolar cognitive modeling and multiagent decision analysis.

    PubMed

    Zhang, W R

    1996-01-01

    An NPN (Negative-Positive-Neutral) fuzzy set theory and an NPN qualitative algebra (Q-algebra) are proposed which form a computational framework for bipolar cognitive modeling and multiagent decision analysis. First a 6-valued NPN logic is introduced which extends the usual 4-valued Q-algebra (S, approximately , plus sign in circle,multiply sign in circle) and S={+,-,0,?} by adding one more level of specification; and then a real-valued NPN fuzzy logic is introduced which extends the 6-valued model to the real space { for all(x,y)|(x,y)in[-1,0]x[0,1]} and adds infinite levels of specifications, As a generalization, a fuzzy set theory is presented that allows beta-level fuzzy number-based NPN variables (x,y) to be substituted into (S, approximately , plus sign in circle,multiply sign in circle) where multiply sign in circle stands for any NPN T-norm; plus sign in circle stands for disjunction (V) or union ( union or logical sum), and beta is the number of alpha-cuts.

  1. Optical linear algebra processors - Architectures and algorithms

    NASA Technical Reports Server (NTRS)

    Casasent, David

    1986-01-01

    Attention is given to the component design and optical configuration features of a generic optical linear algebra processor (OLAP) architecture, as well as the large number of OLAP architectures, number representations, algorithms and applications encountered in current literature. Number-representation issues associated with bipolar and complex-valued data representations, high-accuracy (including floating point) performance, and the base or radix to be employed, are discussed, together with case studies on a space-integrating frequency-multiplexed architecture and a hybrid space-integrating and time-integrating multichannel architecture.

  2. Contraction-based classification of supersymmetric extensions of kinematical lie algebras

    SciTech Connect

    Campoamor-Stursberg, R.; Rausch de Traubenberg, M.

    2010-02-15

    We study supersymmetric extensions of classical kinematical algebras from the point of view of contraction theory. It is shown that contracting the supersymmetric extension of the anti-de Sitter algebra leads to a hierarchy similar in structure to the classical Bacry-Levy-Leblond classification.

  3. On an approach for computing the generating functions of the characters of simple Lie algebras

    NASA Astrophysics Data System (ADS)

    Fernández Núñez, José; García Fuertes, Wifredo; Perelomov, Askold M.

    2014-04-01

    We describe a general approach to obtain the generating functions of the characters of simple Lie algebras which is based on the theory of the quantum trigonometric Calogero-Sutherland model. We show how the method works in practice by means of a few examples involving some low rank classical algebras.

  4. Algebra and Algebraic Thinking in School Math: 70th YB

    ERIC Educational Resources Information Center

    National Council of Teachers of Mathematics, 2008

    2008-01-01

    Algebra is no longer just for college-bound students. After a widespread push by the National Council of Teachers of Mathematics (NCTM) and teachers across the country, algebra is now a required part of most curricula. However, students' standardized test scores are not at the level they should be. NCTM's seventieth yearbook takes a look at the…

  5. Abstract Algebra to Secondary School Algebra: Building Bridges

    ERIC Educational Resources Information Center

    Christy, Donna; Sparks, Rebecca

    2015-01-01

    The authors have experience with secondary mathematics teacher candidates struggling to make connections between the theoretical abstract algebra course they take as college students and the algebra they will be teaching in secondary schools. As a mathematician and a mathematics educator, the authors collaborated to create and implement a…

  6. Handheld Computer Algebra Systems in the Pre-Algebra Classroom

    ERIC Educational Resources Information Center

    Gantz, Linda Ann Galofaro

    2010-01-01

    This mixed method analysis sought to investigate several aspects of student learning in pre-algebra through the use of computer algebra systems (CAS) as opposed to non-CAS learning. This research was broken into two main parts, one which compared results from both the experimental group (instruction using CAS, N = 18) and the control group…

  7. Renormalization in general theories with intergeneration mixing

    NASA Astrophysics Data System (ADS)

    Kniehl, Bernd A.; Sirlin, Alberto

    2012-02-01

    We derive general and explicit expressions for the unrenormalized and renormalized dressed propagators of fermions in parity-nonconserving theories with intergeneration mixing. The mass eigenvalues, the corresponding mass counterterms, and the effect of intergeneration mixing on their determination are discussed. Invoking the Aoki-Hioki-Kawabe-Konuma-Muta renormalization conditions and employing a number of very useful relations from matrix algebra, we show explicitly that the renormalized dressed propagators satisfy important physical properties.

  8. Statecharts Via Process Algebra

    NASA Technical Reports Server (NTRS)

    Luttgen, Gerald; vonderBeeck, Michael; Cleaveland, Rance

    1999-01-01

    Statecharts is a visual language for specifying the behavior of reactive systems. The Language extends finite-state machines with concepts of hierarchy, concurrency, and priority. Despite its popularity as a design notation for embedded system, precisely defining its semantics has proved extremely challenging. In this paper, a simple process algebra, called Statecharts Process Language (SPL), is presented, which is expressive enough for encoding Statecharts in a structure-preserving and semantic preserving manner. It is establish that the behavioral relation bisimulation, when applied to SPL, preserves Statecharts semantics

  9. Algebraic Multigrid Benchmark

    2013-05-06

    AMG2013 is a parallel algebraic multigrid solver for linear systems arising from problems on unstructured grids. It has been derived directly from the Boomer AMG solver in the hypre library, a large linear solvers library that is being developed in the Center for Applied Scientific Computing (CASC) at LLNL. The driver provided in the benchmark can build various test problems. The default problem is a Laplace type problem on an unstructured domain with various jumpsmore » and an anisotropy in one part.« less

  10. Enhanced gauge groups in N=4 topological amplitudes and Lorentzian Borcherds algebras

    NASA Astrophysics Data System (ADS)

    Hohenegger, Stefan; Persson, Daniel

    2011-11-01

    We continue our study of algebraic properties of N=4 topological amplitudes in heterotic string theory compactified on T2, initiated in arXiv:1102.1821. In this work we evaluate a particular one-loop amplitude for any enhanced gauge group h⊂e8⊕e8, i.e. for arbitrary choice of Wilson line moduli. We show that a certain analytic part of the result has an infinite product representation, where the product is taken over the positive roots of a Lorentzian Kac-Moody algebra g++. The latter is obtained through double extension of the complement g=(e8⊕e8)/h. The infinite product is automorphic with respect to a finite index subgroup of the full T-duality group SO(2,18;Z) and, through the philosophy of Borcherds-Gritsenko-Nikulin, this defines the denominator formula of a generalized Kac-Moody algebra G(g++), which is an ’automorphic correction’ of g++. We explicitly give the root multiplicities of G(g++) for a number of examples.

  11. What is conditional event algebra and why should you care?

    NASA Astrophysics Data System (ADS)

    Goodman, I. R.; Mahler, Ronald P. S.; Nguyen, H. T.

    1999-07-01

    Building practical intelligent-system algorithms requires appropriate tools for capturing the basic features of highly complex real-world environments. One of the most important of these tools, probability theory, is a calculus of events (e.g. EVENT = 'A fire-control radar of type A is detected' with Prob(EVENT) = 0.80). Conditional Event Algebra (CEA) is a relatively new inference calculus which rigorously extends standard probability theory to include events which are contingent--e.g. rules such as `If fire-control radar A is detected, then weapon B will be launched'; or conditionals such as `observation Z given target state X.' CEA allows one to (1) probabilistically model a contingent event; (2) assign a probability Prob(COND_EVENT) equals 0.50 to it; and (3) compute with such conditional events and probabilities using the same basic rules that govern ordinary events and probabilities. Since CEA is only about ten years old, it has achieved visibility primarily among specialists in expert-systems theory and mathematical logic. Recently, however, it has become clear that CEA has potentially radical implications for engineering practice as well. The purpose of this paper is to bring this promising new tool to the attention of the wider engineering community. We will give a tutorial introduction to CEA, based on simple motivational examples, and describe its potential applications in a number of practical engineering problems.

  12. An analogue of Wagner's theorem for decompositions of matrix algebras

    NASA Astrophysics Data System (ADS)

    Ivanov, D. N.

    2004-12-01

    Wagner's celebrated theorem states that a finite affine plane whose collineation group is transitive on lines is a translation plane. The notion of an orthogonal decomposition (OD) of a classically semisimple associative algebra introduced by the author allows one to draw an analogy between finite affine planes of order n and ODs of the matrix algebra M_n(\\mathbb C) into a sum of subalgebras conjugate to the diagonal subalgebra. These ODs are called WP-decompositions and are equivalent to the well-known ODs of simple Lie algebras of type A_{n-1} into a sum of Cartan subalgebras. In this paper we give a detailed and improved proof of the analogue of Wagner's theorem for WP-decompositions of the matrix algebra of odd non-square order an outline of which was earlier published in a short note in "Russian Math. Surveys" in 1994. In addition, in the framework of the theory of ODs of associative algebras, based on the method of idempotent bases, we obtain an elementary proof of the well-known Kostrikin-Tiep theorem on irreducible ODs of Lie algebras of type A_{n-1} in the case where n is a prime-power.

  13. Linear Algebra and Image Processing

    ERIC Educational Resources Information Center

    Allali, Mohamed

    2010-01-01

    We use the computing technology digital image processing (DIP) to enhance the teaching of linear algebra so as to make the course more visual and interesting. Certainly, this visual approach by using technology to link linear algebra to DIP is interesting and unexpected to both students as well as many faculty. (Contains 2 tables and 11 figures.)

  14. Linear algebra and image processing

    NASA Astrophysics Data System (ADS)

    Allali, Mohamed

    2010-09-01

    We use the computing technology digital image processing (DIP) to enhance the teaching of linear algebra so as to make the course more visual and interesting. Certainly, this visual approach by using technology to link linear algebra to DIP is interesting and unexpected to both students as well as many faculty.

  15. A Programmed Course in Algebra.

    ERIC Educational Resources Information Center

    Mewborn, Ancel C.; Hively, Wells II

    This programed textbook consists of short sections of text interspersed with questions designed to aid the student in understanding the material. The course is designed to increase the student's understanding of some of the basic ideas of algebra. Some general experience and manipulative skill with respect to high school algebra is assumed.…

  16. Astro Algebra [CD-ROM].

    ERIC Educational Resources Information Center

    1997

    Astro Algebra is one of six titles in the Mighty Math Series from Edmark, a comprehensive line of math software for students from kindergarten through ninth grade. Many of the activities in Astro Algebra contain a unique technology that uses the computer to help students make the connection between concrete and abstract mathematics. This software…

  17. Gamow functionals on operator algebras

    NASA Astrophysics Data System (ADS)

    Castagnino, M.; Gadella, M.; Betán, R. Id; Laura, R.

    2001-11-01

    We obtain the precise form of two Gamow functionals representing the exponentially decaying part of a quantum resonance and its mirror image that grows exponentially, as a linear, positive and continuous functional on an algebra containing observables. These functionals do not admit normalization and, with an appropriate choice of the algebra, are time reversal of each other.

  18. Online Algebraic Tools for Teaching

    ERIC Educational Resources Information Center

    Kurz, Terri L.

    2011-01-01

    Many free online tools exist to complement algebraic instruction at the middle school level. This article presents findings that analyzed the features of algebraic tools to support learning. The findings can help teachers select appropriate tools to facilitate specific topics. (Contains 1 table and 4 figures.)

  19. Patterns to Develop Algebraic Reasoning

    ERIC Educational Resources Information Center

    Stump, Sheryl L.

    2011-01-01

    What is the role of patterns in developing algebraic reasoning? This important question deserves thoughtful attention. In response, this article examines some differing views of algebraic reasoning, discusses a controversy regarding patterns, and describes how three types of patterns--in contextual problems, in growing geometric figures, and in…

  20. Elementary maps on nest algebras

    NASA Astrophysics Data System (ADS)

    Li, Pengtong

    2006-08-01

    Let , be algebras and let , be maps. An elementary map of is an ordered pair (M,M*) such that for all , . In this paper, the general form of surjective elementary maps on standard subalgebras of nest algebras is described. In particular, such maps are automatically additive.

  1. Condensing Algebra for Technical Mathematics.

    ERIC Educational Resources Information Center

    Greenfield, Donald R.

    Twenty Algebra-Packets (A-PAKS) were developed by the investigator for technical education students at the community college level. Each packet contained a statement of rationale, learning objectives, performance activities, performance test, and performance test answer key. The A-PAKS condensed the usual sixteen weeks of algebra into a six-week…

  2. The Algebra of the Arches

    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…

  3. Maximizing algebraic connectivity in air transportation networks

    NASA Astrophysics Data System (ADS)

    Wei, Peng

    In air transportation networks the robustness of a network regarding node and link failures is a key factor for its design. An experiment based on the real air transportation network is performed to show that the algebraic connectivity is a good measure for network robustness. Three optimization problems of algebraic connectivity maximization are then formulated in order to find the most robust network design under different constraints. The algebraic connectivity maximization problem with flight routes addition or deletion is first formulated. Three methods to optimize and analyze the network algebraic connectivity are proposed. The Modified Greedy Perturbation Algorithm (MGP) provides a sub-optimal solution in a fast iterative manner. The Weighted Tabu Search (WTS) is designed to offer a near optimal solution with longer running time. The relaxed semi-definite programming (SDP) is used to set a performance upper bound and three rounding techniques are discussed to find the feasible solution. The simulation results present the trade-off among the three methods. The case study on two air transportation networks of Virgin America and Southwest Airlines show that the developed methods can be applied in real world large scale networks. The algebraic connectivity maximization problem is extended by adding the leg number constraint, which considers the traveler's tolerance for the total connecting stops. The Binary Semi-Definite Programming (BSDP) with cutting plane method provides the optimal solution. The tabu search and 2-opt search heuristics can find the optimal solution in small scale networks and the near optimal solution in large scale networks. The third algebraic connectivity maximization problem with operating cost constraint is formulated. When the total operating cost budget is given, the number of the edges to be added is not fixed. Each edge weight needs to be calculated instead of being pre-determined. It is illustrated that the edge addition and the

  4. Using computer algebra and SMT solvers in algebraic biology

    NASA Astrophysics Data System (ADS)

    Pineda Osorio, Mateo

    2014-05-01

    Biologic processes are represented as Boolean networks, in a discrete time. The dynamics within these networks are approached with the help of SMT Solvers and the use of computer algebra. Software such as Maple and Z3 was used in this case. The number of stationary states for each network was calculated. The network studied here corresponds to the immune system under the effects of drastic mood changes. Mood is considered as a Boolean variable that affects the entire dynamics of the immune system, changing the Boolean satisfiability and the number of stationary states of the immune network. Results obtained show Z3's great potential as a SMT Solver. Some of these results were verified in Maple, even though it showed not to be as suitable for the problem approach. The solving code was constructed using Z3-Python and Z3-SMT-LiB. Results obtained are important in biology systems and are expected to help in the design of immune therapies. As a future line of research, more complex Boolean network representations of the immune system as well as the whole psychological apparatus are suggested.

  5. PREFACE: Infinite Dimensional Algebras and their Applications to Quantum Integrable Systems

    NASA Astrophysics Data System (ADS)

    Fring, Andreas; Kulish, Petr P.; Manojlović, Nenad; Nagy, Zoltán; Nunes da Costa, Joana; Samtleben, Henning

    2008-05-01

    This special issue is centred around the workshop Infinite Dimensional Algebras and Quantum Integrable Systems II—IDAQUIS 2007, held at the University of Algarve, Faro, Portugal in July 2007. It was the second workshop in the IDAQUIS series following a previous meeting at the same location in 2003. The latest workshop gathered around forty experts in the field reviewing recent developments in the theory and applications of integrable systems in the form of invited lectures and in a number of contributions from the participants. All contributions contain significant new results or provide a survey of the state of the art of the subject or a critical assessment of the present understanding of the topic and a discussion of open problems. Original contributions from non-participants are also included. The origins of the topic of this issue can be traced back a long way to the early investigations of completely integrable systems of classical mechanics in the fundamental papers by Euler, Lagrange, Jacobi, Liouville, Kowalevski and others. By the end of the nineteenth century all interesting examples seemed to have been exhausted. A revival in the study of integrable systems began with the development of the classical inverse scattering method, or the theory of solitons. Later developments led to the basic geometrical ideas of the theory, of which infinite dimensional algebras are a key ingredient. In a loose sense one may think that all integrable systems possess some hidden symmetry. In the quantum version of these systems the representation theory of these algebras may be exploited in the description of the structure of the Hilbert space of states. Modern examples of field theoretical systems such as conformal field theories, with the Liouville model being a prominent example, affine Toda field theories and the AdS/CFT correspondence are based on algebraic structures like quantum groups, modular doubles, global conformal invariance, Hecke algebras, Kac

  6. Thermodynamics. [algebraic structure

    NASA Technical Reports Server (NTRS)

    Zeleznik, F. J.

    1976-01-01

    The fundamental structure of thermodynamics is purely algebraic, in the sense of atopological, and it is also independent of partitions, composite systems, the zeroth law, and entropy. The algebraic structure requires the notion of heat, but not the first law. It contains a precise definition of entropy and identifies it as a purely mathematical concept. It also permits the construction of an entropy function from heat measurements alone when appropriate conditions are satisfied. Topology is required only for a discussion of the continuity of thermodynamic properties, and then the weak topology is the relevant topology. The integrability of the differential form of the first law can be examined independently of Caratheodory's theorem and his inaccessibility axiom. Criteria are established by which one can determine when an integrating factor can be made intensive and the pseudopotential extensive and also an entropy. Finally, a realization of the first law is constructed which is suitable for all systems whether they are solids or fluids, whether they do or do not exhibit chemical reactions, and whether electromagnetic fields are or are not present.

  7. Multifractal vector fields and stochastic Clifford algebra

    NASA Astrophysics Data System (ADS)

    Schertzer, Daniel; Tchiguirinskaia, Ioulia

    2015-12-01

    In the mid 1980s, the development of multifractal concepts and techniques was an important breakthrough for complex system analysis and simulation, in particular, in turbulence and hydrology. Multifractals indeed aimed to track and simulate the scaling singularities of the underlying equations instead of relying on numerical, scale truncated simulations or on simplified conceptual models. However, this development has been rather limited to deal with scalar fields, whereas most of the fields of interest are vector-valued or even manifold-valued. We show in this paper that the combination of stable Lévy processes with Clifford algebra is a good candidate to bridge up the present gap between theory and applications. We show that it indeed defines a convenient framework to generate multifractal vector fields, possibly multifractal manifold-valued fields, based on a few fundamental and complementary properties of Lévy processes and Clifford algebra. In particular, the vector structure of these algebra is much more tractable than the manifold structure of symmetry groups while the Lévy stability grants a given statistical universality.

  8. Multifractal vector fields and stochastic Clifford algebra

    SciTech Connect

    Schertzer, Daniel Tchiguirinskaia, Ioulia

    2015-12-15

    In the mid 1980s, the development of multifractal concepts and techniques was an important breakthrough for complex system analysis and simulation, in particular, in turbulence and hydrology. Multifractals indeed aimed to track and simulate the scaling singularities of the underlying equations instead of relying on numerical, scale truncated simulations or on simplified conceptual models. However, this development has been rather limited to deal with scalar fields, whereas most of the fields of interest are vector-valued or even manifold-valued. We show in this paper that the combination of stable Lévy processes with Clifford algebra is a good candidate to bridge up the present gap between theory and applications. We show that it indeed defines a convenient framework to generate multifractal vector fields, possibly multifractal manifold-valued fields, based on a few fundamental and complementary properties of Lévy processes and Clifford algebra. In particular, the vector structure of these algebra is much more tractable than the manifold structure of symmetry groups while the Lévy stability grants a given statistical universality.

  9. Multifractal vector fields and stochastic Clifford algebra.

    PubMed

    Schertzer, Daniel; Tchiguirinskaia, Ioulia

    2015-12-01

    In the mid 1980s, the development of multifractal concepts and techniques was an important breakthrough for complex system analysis and simulation, in particular, in turbulence and hydrology. Multifractals indeed aimed to track and simulate the scaling singularities of the underlying equations instead of relying on numerical, scale truncated simulations or on simplified conceptual models. However, this development has been rather limited to deal with scalar fields, whereas most of the fields of interest are vector-valued or even manifold-valued. We show in this paper that the combination of stable Lévy processes with Clifford algebra is a good candidate to bridge up the present gap between theory and applications. We show that it indeed defines a convenient framework to generate multifractal vector fields, possibly multifractal manifold-valued fields, based on a few fundamental and complementary properties of Lévy processes and Clifford algebra. In particular, the vector structure of these algebra is much more tractable than the manifold structure of symmetry groups while the Lévy stability grants a given statistical universality. PMID:26723166

  10. Multifractal vector fields and stochastic Clifford algebra.

    PubMed

    Schertzer, Daniel; Tchiguirinskaia, Ioulia

    2015-12-01

    In the mid 1980s, the development of multifractal concepts and techniques was an important breakthrough for complex system analysis and simulation, in particular, in turbulence and hydrology. Multifractals indeed aimed to track and simulate the scaling singularities of the underlying equations instead of relying on numerical, scale truncated simulations or on simplified conceptual models. However, this development has been rather limited to deal with scalar fields, whereas most of the fields of interest are vector-valued or even manifold-valued. We show in this paper that the combination of stable Lévy processes with Clifford algebra is a good candidate to bridge up the present gap between theory and applications. We show that it indeed defines a convenient framework to generate multifractal vector fields, possibly multifractal manifold-valued fields, based on a few fundamental and complementary properties of Lévy processes and Clifford algebra. In particular, the vector structure of these algebra is much more tractable than the manifold structure of symmetry groups while the Lévy stability grants a given statistical universality.

  11. Algebraic quantum gravity (AQG): I. Conceptual setup

    NASA Astrophysics Data System (ADS)

    Giesel, K.; Thiemann, T.

    2007-05-01

    We introduce a new top down approach to canonical quantum gravity, called algebraic quantum gravity (AQG). The quantum kinematics of AQG is determined by an abstract *-algebra generated by a countable set of elementary operators labelled by an algebraic graph. The quantum dynamics of AQG is governed by a single master constraint operator. While AQG is inspired by loop quantum gravity (LQG), it differs drastically from it because in AQG there is fundamentally no topology or differential structure. A natural Hilbert space representation acquires the structure of an infinite tensor product (ITP) whose separable strong equivalence class Hilbert subspaces (sectors) are left invariant by the quantum dynamics. The missing information about the topology and differential structure of the spacetime manifold as well as about the background metric to be approximated is supplied by coherent states. Given such data, the corresponding coherent state defines a sector in the ITP which can be identified with a usual QFT on the given manifold and background. Thus, AQG contains QFT on all curved spacetimes at once, possibly has something to say about topology change and provides the contact with the familiar low energy physics. In particular, in two companion papers we develop semiclassical perturbation theory for AQG and LQG and thereby show that the theory admits a semiclassical limit whose infinitesimal gauge symmetry agrees with that of general relativity. In AQG everything is computable with sufficient precision and no UV divergences arise due to the background independence of the fundamental combinatorial structure. Hence, in contrast to lattice gauge theory on a background metric, no continuum limit has to be taken. There simply is no lattice regulator that must be sent to zero.

  12. Realization Of Algebraic Processor For XML Documents Processing

    NASA Astrophysics Data System (ADS)

    Georgiev, Bozhidar; Georgieva, Adriana

    2010-10-01

    In this paper, are presented some possibilities concerning the implementation of an algebraic method for XML hierarchical data processing which makes faster the XML search mechanism. Here is offered a different point of view for creation of advanced algebraic processor (with all necessary software tools and programming modules respectively). Therefore, this nontraditional approach for fast XML navigation with the presented algebraic processor may help to build an easier user-friendly interface provided XML transformations, which can avoid the difficulties in the complicated language constructions of XSL, XSLT and XPath. This approach allows comparatively simple search of XML hierarchical data by means of the following types of functions: specification functions and so named build-in functions. The choice of programming language Java may appear strange at first, but it isn't when you consider that the applications can run on different kinds of computers. The specific search mechanism based on the linear algebra theory is faster in comparison with MSXML parsers (on the basis of the developed examples with about 30%). Actually, there exists the possibility for creating new software tools based on the linear algebra theory, which cover the whole navigation and search techniques characterizing XSLT/XPath. The proposed method is able to replace more complicated operations in other SOA components.

  13. Realization Of Algebraic Processor For XML Documents Processing

    SciTech Connect

    Georgiev, Bozhidar; Georgieva, Adriana

    2010-10-25

    In this paper, are presented some possibilities concerning the implementation of an algebraic method for XML hierarchical data processing which makes faster the XML search mechanism. Here is offered a different point of view for creation of advanced algebraic processor (with all necessary software tools and programming modules respectively). Therefore, this nontraditional approach for fast XML navigation with the presented algebraic processor may help to build an easier user-friendly interface provided XML transformations, which can avoid the difficulties in the complicated language constructions of XSL, XSLT and XPath. This approach allows comparatively simple search of XML hierarchical data by means of the following types of functions: specification functions and so named build-in functions. The choice of programming language Java may appear strange at first, but it isn't when you consider that the applications can run on different kinds of computers. The specific search mechanism based on the linear algebra theory is faster in comparison with MSXML parsers (on the basis of the developed examples with about 30%). Actually, there exists the possibility for creating new software tools based on the linear algebra theory, which cover the whole navigation and search techniques characterizing XSLT/XPath. The proposed method is able to replace more complicated operations in other SOA components.

  14. Using Linear Algebra to Introduce Computer Algebra, Numerical Analysis, Data Structures and Algorithms (and To Teach Linear Algebra, Too).

    ERIC Educational Resources Information Center

    Gonzalez-Vega, Laureano

    1999-01-01

    Using a Computer Algebra System (CAS) to help with the teaching of an elementary course in linear algebra can be one way to introduce computer algebra, numerical analysis, data structures, and algorithms. Highlights the advantages and disadvantages of this approach to the teaching of linear algebra. (Author/MM)

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

    PubMed

    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

  16. Algebraic description of intrinsic modes in nuclei

    SciTech Connect

    Leviatan, A.

    1989-01-01

    We present a procedure for extracting normal modes in algebraic number-conserving systems of interacting bosons relevant for collective states in even-even nuclei. The Hamiltonian is resolved into intrinsic (bandhead related) and collective (in-band related) parts. Shape parameters are introduced through non-spherical boson bases. Intrinsic modes decoupled from the spurious modes are obtained from the intinsic part of the Hamiltonian in the limit of large number of bosons. Intrinsic states are constructed and serve to evaluate electromagnetic transition rates. The method is illustrated for systems with one type of boson as well as with proton-neutron bosons. 28 refs., 1 fig.

  17. Infinite-dimensional spin-2 symmetries in Kaluza-Klein theories

    NASA Astrophysics Data System (ADS)

    Hohm, Olaf

    2006-02-01

    We consider the couplings of an infinite number of spin-2 fields to gravity appearing in Kaluza-Klein theories. They are constructed as the broken phase of a massless theory possessing an infinite-dimensional spin-2 symmetry. Focusing on a circle compactification of four-dimensional gravity we show that the resulting gravity/spin-2 system in D=3 has in its unbroken phase an interpretation as a Chern-Simons theory of the Kac-Moody algebra iso(1,2)^ associated to the Poincaré group and also fits into the geometrical framework of algebra-valued differential geometry developed by Wald. Assigning all degrees of freedom to scalar fields, the matter couplings in the unbroken phase are determined, and it is shown that their global symmetry algebra contains the Virasoro algebra together with an enhancement of the Ehlers group SL(2,R) to its affine extension. The broken phase is then constructed by gauging a subgroup of the global symmetries. It is shown that metric, spin-2 fields and Kaluza-Klein vectors combine into a Chern-Simons theory for an extended algebra, in which the affine Poincaré subalgebra acquires a central extension.

  18. MV-Algebra for Cultural Rules

    NASA Astrophysics Data System (ADS)

    Ballonoff, Paul

    2008-01-01

    This paper reports preliminary results on a new area of application of quantum structures, motivated by a reading of the 2004 monograph Reasoning in Quantum Theory. Ethnographers often describe a particular culture by describing rules of social relations that they assert characterize that culture. Viable cultures exist over periods of time, that is, over sequences of “generations”. To embody this, we define a suitable set of objects and relations, and a structure on which cultural rules act as “operators” on a set of “configurations” on generations. This yields an MV-algebra of those operators. This implies that culture theory might be studied as an example of the theory of quantum structures.

  19. Inverse Modelling Problems in Linear Algebra Undergraduate Courses

    ERIC Educational Resources Information Center

    Martinez-Luaces, Victor E.

    2013-01-01

    This paper will offer an analysis from a theoretical point of view of mathematical modelling, applications and inverse problems of both causation and specification types. Inverse modelling problems give the opportunity to establish connections between theory and practice and to show this fact, a simple linear algebra example in two different…

  20. A Framework for Mathematical Thinking: The Case of Linear Algebra

    ERIC Educational Resources Information Center

    Stewart, Sepideh; Thomas, Michael O. J.

    2009-01-01

    Linear algebra is one of the unavoidable advanced courses that many mathematics students encounter at university level. The research reported here was part of the first author's recent PhD study, where she created and applied a theoretical framework combining the strengths of two major mathematics education theories in order to investigate the…

  1. Perceiving the General: The Multisemiotic Dimension of Students' Algebraic Activity

    ERIC Educational Resources Information Center

    Radford, Luis; Bardino, Caroline; Sabena, Cristina

    2007-01-01

    In this article, we deal with students' algebraic generalizations set in the context of elementary geometric-numeric patterns. Drawing from Vygotsky's psychology, Leont'ev's Activity Theory, and Husserl's phenomenology, we focus on the various semiotic resources mobilized by students in their passage from the particular to the general. Two small…

  2. Enumerating Small Sudoku Puzzles in a First Abstract Algebra Course

    ERIC Educational Resources Information Center

    Lorch, Crystal; Lorch, John

    2008-01-01

    Two methods are presented for counting small "essentially different" sudoku puzzles using elementary group theory: one method (due to Jarvis and Russell) uses Burnside's counting formula, while the other employs an invariant property of sudoku puzzles. Ideas are included for incorporating this material into an introductory abstract algebra course.…

  3. Algebra 33: Introduction through Unit V-B.

    ERIC Educational Resources Information Center

    Nederland Independent School District, TX.

    GRADES OR AGES: No mention. SUBJECT MATTER: Algebra. ORGANIZATION AND PHYSICAL APPEARANCE: The guide is divided into seven separately bound units. Unit headings are introduction, sets of numbers and axioms, open sentences in one variable, systems of linear open sentences, polynomials and factoring, rational numbers and expressions (A), and…

  4. Quantum algebra of N superspace

    SciTech Connect

    Hatcher, Nicolas; Restuccia, A.; Stephany, J.

    2007-08-15

    We identify the quantum algebra of position and momentum operators for a quantum system bearing an irreducible representation of the super Poincare algebra in the N>1 and D=4 superspace, both in the case where there are no central charges in the algebra, and when they are present. This algebra is noncommutative for the position operators. We use the properties of superprojectors acting on the superfields to construct explicit position and momentum operators satisfying the algebra. They act on the projected wave functions associated to the various supermultiplets with defined superspin present in the representation. We show that the quantum algebra associated to the massive superparticle appears in our construction and is described by a supermultiplet of superspin 0. This result generalizes the construction for D=4, N=1 reported recently. For the case N=2 with central charges, we present the equivalent results when the central charge and the mass are different. For the {kappa}-symmetric case when these quantities are equal, we discuss the reduction to the physical degrees of freedom of the corresponding superparticle and the construction of the associated quantum algebra.

  5. Algebraic distance on graphs.

    SciTech Connect

    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.

  6. SD-CAS: Spin Dynamics by Computer Algebra System.

    PubMed

    Filip, Xenia; Filip, Claudiu

    2010-11-01

    A computer algebra tool for describing the Liouville-space quantum evolution of nuclear 1/2-spins is introduced and implemented within a computational framework named Spin Dynamics by Computer Algebra System (SD-CAS). A distinctive feature compared with numerical and previous computer algebra approaches to solving spin dynamics problems results from the fact that no matrix representation for spin operators is used in SD-CAS, which determines a full symbolic character to the performed computations. Spin correlations are stored in SD-CAS as four-entry nested lists of which size increases linearly with the number of spins into the system and are easily mapped into analytical expressions in terms of spin operator products. For the so defined SD-CAS spin correlations a set of specialized functions and procedures is introduced that are essential for implementing basic spin algebra operations, such as the spin operator products, commutators, and scalar products. They provide results in an abstract algebraic form: specific procedures to quantitatively evaluate such symbolic expressions with respect to the involved spin interaction parameters and experimental conditions are also discussed. Although the main focus in the present work is on laying the foundation for spin dynamics symbolic computation in NMR based on a non-matrix formalism, practical aspects are also considered throughout the theoretical development process. In particular, specific SD-CAS routines have been implemented using the YACAS computer algebra package (http://yacas.sourceforge.net), and their functionality was demonstrated on a few illustrative examples.

  7. SD-CAS: Spin Dynamics by Computer Algebra System

    NASA Astrophysics Data System (ADS)

    Filip, Xenia; Filip, Claudiu

    2010-11-01

    A computer algebra tool for describing the Liouville-space quantum evolution of nuclear 1/2-spins is introduced and implemented within a computational framework named Spin Dynamics by Computer Algebra System (SD-CAS). A distinctive feature compared with numerical and previous computer algebra approaches to solving spin dynamics problems results from the fact that no matrix representation for spin operators is used in SD-CAS, which determines a full symbolic character to the performed computations. Spin correlations are stored in SD-CAS as four-entry nested lists of which size increases linearly with the number of spins into the system and are easily mapped into analytical expressions in terms of spin operator products. For the so defined SD-CAS spin correlations a set of specialized functions and procedures is introduced that are essential for implementing basic spin algebra operations, such as the spin operator products, commutators, and scalar products. They provide results in an abstract algebraic form: specific procedures to quantitatively evaluate such symbolic expressions with respect to the involved spin interaction parameters and experimental conditions are also discussed. Although the main focus in the present work is on laying the foundation for spin dynamics symbolic computation in NMR based on a non-matrix formalism, practical aspects are also considered throughout the theoretical development process. In particular, specific SD-CAS routines have been implemented using the YACAS computer algebra package (http://yacas.sourceforge.net), and their functionality was demonstrated on a few illustrative examples.

  8. Readiness and Preparation for Beginning Algebra.

    ERIC Educational Resources Information Center

    Rotman, Jack W.

    Drawing from experience at Lansing Community College (LCC), this paper discusses how to best prepare students for success in a beginning algebra course. First, an overview is presented of LCC's developmental math sequence, which includes Basic Arithmetic (MTH 008), Pre-Algebra (MTH 009), Beginning Algebra (MTH 012), and Intermediate Algebra (MTH…

  9. Hopf algebras and Dyson-Schwinger equations

    NASA Astrophysics Data System (ADS)

    Weinzierl, Stefan

    2016-06-01

    In this paper I discuss Hopf algebras and Dyson-Schwinger equations. This paper starts with an introduction to Hopf algebras, followed by a review of the contribution and application of Hopf algebras to particle physics. The final part of the paper is devoted to the relation between Hopf algebras and Dyson-Schwinger equations.

  10. Two-parameter twisted quantum affine algebras

    NASA Astrophysics Data System (ADS)

    Jing, Naihuan; Zhang, Honglian

    2016-09-01

    We establish Drinfeld realization for the two-parameter twisted quantum affine algebras using a new method. The Hopf algebra structure for Drinfeld generators is given for both untwisted and twisted two-parameter quantum affine algebras, which include the quantum affine algebras as special cases.

  11. Application of supersonic linear theory and hypersonic impact methods to three nonslender hypersonic airplane concepts at Mach numbers from 1.10 to 2.86

    NASA Technical Reports Server (NTRS)

    Pittman, J. L.

    1979-01-01

    Aerodynamic predictions from supersonic linear theory and hypersonic impact theory were compared with experimental data for three hypersonic research airplane concepts over a Mach number range from 1.10 to 2.86. The linear theory gave good lift prediction and fair to good pitching-moment prediction over the Mach number (M) range. The tangent-cone theory predictions were good for lift and fair to good for pitching moment for M more than or equal to 2.0. The combined tangent-cone theory predictions were good for lift and fair to good for pitching moment for M more than or equal to 2.0. The combined tangent-cone/tangent-wedge method gave the least accurate prediction of lift and pitching moment. The zero-lift drag was overestimated, especially for M less than 2.0. The linear theory drag prediction was generally poor, with areas of good agreement only for M less than or equal to 1.2. For M more than or equal to 2.), the tangent-cone method predicted the zero-lift drag most accurately.

  12. Development of abstract mathematical reasoning: the case of algebra

    PubMed Central

    Susac, Ana; Bubic, Andreja; Vrbanc, Andrija; Planinic, Maja

    2014-01-01

    Algebra typically represents the students’ first encounter with abstract mathematical reasoning and it therefore causes significant difficulties for students who still reason concretely. The aim of the present study was to investigate the developmental trajectory of the students’ ability to solve simple algebraic equations. 311 participants between the ages of 13 and 17 were given a computerized test of equation rearrangement. Equations consisted of an unknown and two other elements (numbers or letters), and the operations of multiplication/division. The obtained results showed that younger participants are less accurate and slower in solving equations with letters (symbols) than those with numbers. This difference disappeared for older participants (16–17 years), suggesting that they had reached an abstract reasoning level, at least for this simple task. A corresponding conclusion arises from the analysis of their strategies which suggests that younger participants mostly used concrete strategies such as inserting numbers, while older participants typically used more abstract, rule-based strategies. These results indicate that the development of algebraic thinking is a process which unfolds over a long period of time. In agreement with previous research, we can conclude that, on average, children at the age of 15–16 transition from using concrete to abstract strategies while solving the algebra problems addressed within the present study. A better understanding of the timing and speed of students’ transition from concrete arithmetic reasoning to abstract algebraic reasoning might help in designing better curricula and teaching materials that would ease that transition. PMID:25228874

  13. Development of abstract mathematical reasoning: the case of algebra.

    PubMed

    Susac, Ana; Bubic, Andreja; Vrbanc, Andrija; Planinic, Maja

    2014-01-01

    Algebra typically represents the students' first encounter with abstract mathematical reasoning and it therefore causes significant difficulties for students who still reason concretely. The aim of the present study was to investigate the developmental trajectory of the students' ability to solve simple algebraic equations. 311 participants between the ages of 13 and 17 were given a computerized test of equation rearrangement. Equations consisted of an unknown and two other elements (numbers or letters), and the operations of multiplication/division. The obtained results showed that younger participants are less accurate and slower in solving equations with letters (symbols) than those with numbers. This difference disappeared for older participants (16-17 years), suggesting that they had reached an abstract reasoning level, at least for this simple task. A corresponding conclusion arises from the analysis of their strategies which suggests that younger participants mostly used concrete strategies such as inserting numbers, while older participants typically used more abstract, rule-based strategies. These results indicate that the development of algebraic thinking is a process which unfolds over a long period of time. In agreement with previous research, we can conclude that, on average, children at the age of 15-16 transition from using concrete to abstract strategies while solving the algebra problems addressed within the present study. A better understanding of the timing and speed of students' transition from concrete arithmetic reasoning to abstract algebraic reasoning might help in designing better curricula and teaching materials that would ease that transition.

  14. Minimal dynamics and the classification of C*-algebras.

    PubMed

    Toms, Andrew S; Winter, Wilhelm

    2009-10-01

    Let X be an infinite, compact, metrizable space of finite covering dimension and alpha: X --> X a minimal homeomorphism. We prove that the crossed product C(X) times sign, right closed(alpha) Z absorbs the Jiang-Su algebra tensorially and has finite nuclear dimension. As a consequence, these algebras are determined up to isomorphism by their graded ordered K-theory under the necessary condition that their projections separate traces. This result applies, in particular, to those crossed products arising from uniquely ergodic homeomorphisms.

  15. Cartooning in Algebra and Calculus

    ERIC Educational Resources Information Center

    Moseley, L. Jeneva

    2014-01-01

    This article discusses how teachers can create cartoons for undergraduate math classes, such as college algebra and basic calculus. The practice of cartooning for teaching can be helpful for communication with students and for students' conceptual understanding.

  16. Ada Linear-Algebra Program

    NASA Technical Reports Server (NTRS)

    Klumpp, A. R.; Lawson, C. L.

    1988-01-01

    Routines provided for common scalar, vector, matrix, and quaternion operations. Computer program extends Ada programming language to include linear-algebra capabilities similar to HAS/S programming language. Designed for such avionics applications as software for Space Station.

  17. GCD, LCM, and Boolean Algebra?

    ERIC Educational Resources Information Center

    Cohen, Martin P.; Juraschek, William A.

    1976-01-01

    This article investigates the algebraic structure formed when the process of finding the greatest common divisor and the least common multiple are considered as binary operations on selected subsets of positive integers. (DT)

  18. Saturation of the magnetorotational instability at large Elsasser number

    NASA Astrophysics Data System (ADS)

    Jamroz, B.; Julien, K.; Knobloch, E.

    2008-09-01

    The magnetorotational instability is investigated within the shearing box approximation in the large Elsasser number regime. In this regime, which is of fundamental importance to astrophysical accretion disk theory, shear is the dominant source of energy, but the instability itself requires the presence of a weaker vertical magnetic field. Dissipative effects are weaker still but not negligible. The regime explored retains the condition that (viscous and ohmic) dissipative forces do not play a role in the leading order linear instability mechanism. However, they are sufficiently large to permit a nonlinear feedback mechanism whereby the turbulent stresses generated by the MRI act on and modify the local background shear in the angular velocity profile. To date this response has been omitted in shearing box simulations and is captured by a reduced pde model derived here from the global MHD fluid equations using multiscale asymptotic perturbation theory. Results from numerical simulations of the reduced pde model indicate a linear phase of exponential growth followed by a nonlinear adjustment to algebraic growth and decay in the fluctuating quantities. Remarkably, the velocity and magnetic field correlations associated with these algebraic growth and decay laws conspire to achieve saturation of the angular momentum transport. The inclusion of subdominant ohmic dissipation arrests the algebraic growth of the fluctuations on a longer, dissipative time scale.

  19. ALGEBRA v.1.27

    SciTech Connect

    Sjaardema, G.; Gilkey, A.; Smith, M.; Forsythe, C.

    2005-04-11

    The ALGEBRA program allows the user to manipulate data from a finite element analysis before it is plotted. The finite element output data is in the form of variable values (e.g., stress, strain, and velocity components) in an EXODUS II database. The ALGEBRA program evaluates user-supplied functions of the data and writes the results to an output EXODUS II database that can be read by plot programs.

  20. Modular properties of doubly extended N = 4 superconformal algebras and their connection to rational torus models (I)

    NASA Astrophysics Data System (ADS)

    Petersen, Jens Lyng; Taormina, Anne

    1991-05-01

    The doubly extended N = 4 superconformal algebra, which contains all conventional extended superconformal algebras, is analyzed when one of the central extensions is set to 1. The modular transformations of the characters are derived, the relation between the characters and those of the N = 2 minimal is clarified, and in the process it is shown how rather simple extensions of the algebra based on rational torus theories, give rise to finite dimensional representations of the modular group.

  1. Quantized Nambu-Poisson manifolds and n-Lie algebras

    NASA Astrophysics Data System (ADS)

    DeBellis, Joshua; Sämann, Christian; Szabo, Richard J.

    2010-12-01

    We investigate the geometric interpretation of quantized Nambu-Poisson structures in terms of noncommutative geometries. We describe an extension of the usual axioms of quantization in which classical Nambu-Poisson structures are translated to n-Lie algebras at quantum level. We demonstrate that this generalized procedure matches an extension of Berezin-Toeplitz quantization yielding quantized spheres, hyperboloids, and superspheres. The extended Berezin quantization of spheres is closely related to a deformation quantization of n-Lie algebras as well as the approach based on harmonic analysis. We find an interpretation of Nambu-Heisenberg n-Lie algebras in terms of foliations of {{R}}^n by fuzzy spheres, fuzzy hyperboloids, and noncommutative hyperplanes. Some applications to the quantum geometry of branes in M-theory are also briefly discussed.

  2. Quantized Nambu-Poisson manifolds and n-Lie algebras

    SciTech Connect

    DeBellis, Joshua; Saemann, Christian; Szabo, Richard J.

    2010-12-15

    We investigate the geometric interpretation of quantized Nambu-Poisson structures in terms of noncommutative geometries. We describe an extension of the usual axioms of quantization in which classical Nambu-Poisson structures are translated to n-Lie algebras at quantum level. We demonstrate that this generalized procedure matches an extension of Berezin-Toeplitz quantization yielding quantized spheres, hyperboloids, and superspheres. The extended Berezin quantization of spheres is closely related to a deformation quantization of n-Lie algebras as well as the approach based on harmonic analysis. We find an interpretation of Nambu-Heisenberg n-Lie algebras in terms of foliations of R{sup n} by fuzzy spheres, fuzzy hyperboloids, and noncommutative hyperplanes. Some applications to the quantum geometry of branes in M-theory are also briefly discussed.

  3. Increasing the Number of Replications in Item Response Theory Simulations: Automation through SAS and Disk Operating System

    ERIC Educational Resources Information Center

    Gagne, Phill; Furlow, Carolyn; Ross, Terris

    2009-01-01

    In item response theory (IRT) simulation research, it is often necessary to use one software package for data generation and a second software package to conduct the IRT analysis. Because this can substantially slow down the simulation process, it is sometimes offered as a justification for using very few replications. This article provides…

  4. Predicting the Number of Public Computer Terminals Needed for an On-Line Catalog: A Queuing Theory Approach.

    ERIC Educational Resources Information Center

    Knox, A. Whitney; Miller, Bruce A.

    1980-01-01

    Describes a method for estimating the number of cathode ray tube terminals needed for public use of an online library catalog. Authors claim method could also be used to estimate needed numbers of microform readers for a computer output microform (COM) catalog. Formulae are included. (Author/JD)

  5. Is the brain a Clifford algebra quantum computer?

    NASA Astrophysics Data System (ADS)

    Labunets, Valeri G.; Labunets-Rundblad, Ekaterina V.; Astola, Jaakko T.

    2001-11-01

    We propose a novel method to calculate invariants of colour and multicolour images. It employs an idea of classical and quantum hypercomplex numbers and combines it with the idea of classical and quantum number theoretical transforms over hypercomplex algebras, which reduce the computational complexity of the global recognition algorithm for nD k-multispectral images from O(knNn+1)to O(kNn log N) and to O(kn log N), respectively. Our hypotheses are 1) the brain of primates calculates hypercomplex-valued invariants of an image during recognizing, 2) visual systems of animals with different evolutionary history use different hypercomplex algebras. The main goal of the paper is to show that quantum Clifford algebras can be used to solve pattern recognition in multispectral environment in a natural and effective manner.

  6. Compatible Relaxation and Coarsening in Algebraic Multigrid

    SciTech Connect

    Brannick, J J; Falgout, R D

    2009-09-22

    We introduce a coarsening algorithm for algebraic multigrid (AMG) based on the concept of compatible relaxation (CR). The algorithm is significantly different from standard methods, most notably because it does not rely on any notion of strength of connection. We study its behavior on a number of model problems, and evaluate the performance of an AMG algorithm that incorporates the coarsening approach. Lastly, we introduce a variant of CR that provides a sharper metric of coarse-grid quality and demonstrate its potential with two simple examples.

  7. Invertible linear transformations and the Lie algebras

    NASA Astrophysics Data System (ADS)

    Zhang, Yufeng; Tam, Honwah; Guo, Fukui

    2008-07-01

    With the help of invertible linear transformations and the known Lie algebras, a way to generate new Lie algebras is given. These Lie algebras obtained have a common feature, i.e. integrable couplings of solitary hierarchies could be obtained by using them, specially, the Hamiltonian structures of them could be worked out. Some ways to construct the loop algebras of the Lie algebras are presented. It follows that some various loop algebras are given. In addition, a few new Lie algebras are explicitly constructed in terms of the classification of Lie algebras proposed by Ma Wen-Xiu, which are bases for obtaining new Lie algebras by using invertible linear transformations. Finally, some solutions of a (2 + 1)-dimensional partial-differential equation hierarchy are obtained, whose Hamiltonian form-expressions are manifested by using the quadratic-form identity.

  8. Super-Poincarè algebras, space-times, and supergravities. II

    NASA Astrophysics Data System (ADS)

    Santi, A.; Spiro, A.

    2012-03-01

    The presentation of supergravity theories of our previous paper "Super-Poincarè algebras, space-times, and supergravities. I" is re-formulated in the language of Berezin-Leites-Kostant theory of supermanifolds. It is also shown that the equations of Cremmer, Julia, and Scherk's theory of 11D-supergravity are equivalent to manifestly covariant equations on a supermanifold.

  9. Sixth SIAM conference on applied linear algebra: Final program and abstracts. Final technical report

    SciTech Connect

    1997-12-31

    Linear algebra plays a central role in mathematics and applications. The analysis and solution of problems from an amazingly wide variety of disciplines depend on the theory and computational techniques of linear algebra. In turn, the diversity of disciplines depending on linear algebra also serves to focus and shape its development. Some problems have special properties (numerical, structural) that can be exploited. Some are simply so large that conventional approaches are impractical. New computer architectures motivate new algorithms, and fresh ways to look at old ones. The pervasive nature of linear algebra in analyzing and solving problems means that people from a wide spectrum--universities, industrial and government laboratories, financial institutions, and many others--share an interest in current developments in linear algebra. This conference aims to bring them together for their mutual benefit. Abstracts of papers presented are included.

  10. A modification to linearized theory for prediction of pressure loadings on lifting surfaces at high supersonic Mach numbers and large angles of attack

    NASA Technical Reports Server (NTRS)

    Carlson, H. W.

    1979-01-01

    A new linearized-theory pressure-coefficient formulation was studied. The new formulation is intended to provide more accurate estimates of detailed pressure loadings for improved stability analysis and for analysis of critical structural design conditions. The approach is based on the use of oblique-shock and Prandtl-Meyer expansion relationships for accurate representation of the variation of pressures with surface slopes in two-dimensional flow and linearized-theory perturbation velocities for evaluation of local three-dimensional aerodynamic interference effects. The applicability and limitations of the modification to linearized theory are illustrated through comparisons with experimental pressure distributions for delta wings covering a Mach number range from 1.45 to 4.60 and angles of attack from 0 to 25 degrees.

  11. BRST charges for finite nonlinear algebras

    NASA Astrophysics Data System (ADS)

    Isaev, A. P.; Krivonos, S. O.; Ogievetsky, O. V.

    2010-07-01

    Some ingredients of the BRST construction for quantum Lie algebras are applied to a wider class of quadratic algebras of constraints. We build the BRST charge for a quantum Lie algebra with three generators and ghost-anti-ghosts commuting with constraints. We consider a one-parametric family of quadratic algebras with three generators and show that the BRST charge acquires the conventional form after a redefinition of ghosts. The modified ghosts form a quadratic algebra. The family possesses a nonlinear involution, which implies the existence of two independent BRST charges for each algebra in the family. These BRST charges anticommute and form a double BRST complex.

  12. Some Remarks on Kite Pseudo Effect Algebras

    NASA Astrophysics Data System (ADS)

    Dvurečenskij, Anatolij; Holland, W. Charles

    2014-05-01

    Recently a new family of pseudo effect algebras, called kite pseudo effect algebras, was introduced. Such an algebra starts with a po-group G, a set I and with two bijections λ, ρ: I→ I. Using a clever construction on the ordinal sum of ( G +) I and ( G -) I , we can define a pseudo effect algebra which can be non-commutative even if G is an Abelian po-group. In the paper we give a characterization of subdirect product of subdirectly irreducible kite pseudo effect algebras, and we show that every kite pseudo effect algebra is an interval in a unital po-loop.

  13. Time-dependent occupation numbers in reduced-density-matrix-functional theory: Application to an interacting Landau-Zener model

    SciTech Connect

    Requist, Ryan; Pankratov, Oleg

    2011-05-15

    We prove that if the two-body terms in the equation of motion for the one-body reduced density matrix are approximated by ground-state functionals, the eigenvalues of the one-body reduced density matrix (occupation numbers) remain constant in time. This deficiency is related to the inability of such an approximation to account for relative phases in the two-body reduced density matrix. We derive an exact differential equation giving the functional dependence of these phases in an interacting Landau-Zener model and study their behavior in short- and long-time regimes. The phases undergo resonances whenever the occupation numbers approach the boundaries of the interval [0,1]. In the long-time regime, the occupation numbers display correlation-induced oscillations and the memory dependence of the functionals assumes a simple form.

  14. The Structure of the Kac-Wang-Yan Algebra

    NASA Astrophysics Data System (ADS)

    Linshaw, Andrew R.

    2016-07-01

    The Lie algebra {mathcal{D}} of regular differential operators on the circle has a universal central extension {hat{mathcal{D}}}. The invariant subalgebra {hat{mathcal{D}}^+} under an involution preserving the principal gradation was introduced by Kac, Wang, and Yan. The vacuum {hat{mathcal{D}}^+}-module with central charge {c in mathbb{C}}, and its irreducible quotient {mathcal{V}_c}, possess vertex algebra structures, and {mathcal{V}_c} has a nontrivial structure if and only if {c in 1/2mathbb{Z}}. We show that for each integer {n > 0}, {mathcal{V}_{n/2}} and {mathcal{V}_{-n}} are {mathcal{W}}-algebras of types {mathcal{W}(2, 4,dots,2n)} and {mathcal{W}(2, 4,dots, 2n^2 + 4n)}, respectively. These results are formal consequences of Weyl's first and second fundamental theorems of invariant theory for the orthogonal group {O(n)} and the symplectic group {Sp(2n)}, respectively. Based on Sergeev's theorems on the invariant theory of {Osp(1, 2n)} we conjecture that {mathcal{V}_{-n+1/2}} is of type {mathcal{W}(2, 4,dots, 4n^2 + 8n + 2)}, and we prove this for {n = 1}. As an application, we show that invariant subalgebras of {βγ}-systems and free fermion algebras under arbitrary reductive group actions are strongly finitely generated.

  15. Strongly-local reductions and the complexity/efficient approximability of algebra and optimization on abstract algebraic structures

    SciTech Connect

    Hunt, H. B.; Marathe, M. V.; Stearns, R. E.

    2001-01-01

    We demonstrate how the concepts of algebraic representability and strongly-local reductions developed here and in [HSM00] can be used to characterize the computational complexity/efficient approximability of a number of basic problems and their variants, on various abstract algebraic structures F. These problems include the following: (1) A1gebra:Determine the solvability, unique solvability, number of solutions, etc., of a system of equations on F. Determine the equivalence of two formulas or straight-line programs on F. 2. 0ptimization:Let {epsilon} > 0. (a) Determine the maximum number of simultaneously satisfiable equations in a system of equations on F; or approximate this number within a multiplicative factor of n{sup {epsilon}}. (b) Determine the maximum value of an objective function subject to satisfiable algebraically expressed constraints on F; or approximate this maximum value within a multiplicative factor of n{sup {epsilon}}. (c) Given a formula or straight-line program, find a minimum size equivalent formula or straightline program; or find an equivalent formula or straight-line program of size {le} f (minimum). Both finite and infinite algebraic structures are considered. These finite structures include all finite nondegenerate lattices and all finite rings or semi-rings with a nonzero element idempotent under multiplication (e.g. all non-degenerate finite unitary rings or semi-rings); and these infinite structures include the natural numbers, integers, real numbers, various algebras on these structures, all ordered rings, many cancellative semi-rings, and all infinite lattices with two elements a,b such that a is covered by b. Our results significantly extend a number of results by Ladner [La89], Condon, et. al. [CF+93], Khanna, et.al [KSW97], Cr951 and Zuckerman [Zu93] on the complexity and approximbaility of combinatorial problems.

  16. Early Algebra to Reach the Range of Learners

    ERIC Educational Resources Information Center

    Schifter, Deborah; Russell, Susan Jo; Bastable, Virginia

    2009-01-01

    Since 2001, the authors have been working with groups of teachers to investigate students' early algebraic thinking--learning representations, connections, and generalizations in the elementary school grades. They began paying attention to students' explicit remarks about regularities in the number system or what students imply by their…

  17. Algebraic criteria for positive realness relative to the unit circle.

    NASA Technical Reports Server (NTRS)

    Siljak, D. D.

    1973-01-01

    A purely algebraic algorithm is developed for testing positive real character of real rational functions and matrices relative to the unit circle in the complex plane. Since the algorithm is entirely recursive and is performed in finite number of steps, it is suitable for machine computations.

  18. Math Sense: Algebra and Geometry. Teacher's Resource Guide.

    ERIC Educational Resources Information Center

    Phillips, Jan; Osmus, Kathy

    This book is a teacher's resource guide designed to help students gain the range of math skills they need to succeed in life, work, and on standardized tests; overcome math anxiety; discover math as interesting and purposeful; and develop good number sense. Topics covered in this book include algebra and geometry. Lessons are organized around four…

  19. 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…

  20. Constructing a parasupersymmetric Virasoro algebra

    NASA Astrophysics Data System (ADS)

    Kuwata, S.

    2011-03-01

    We construct a para SUSY Virasoro algebra by generalizing the ordinary fermion in SUSY Virasoro algebra (Ramond or Neveu-Schwarz algebra) to the parafermion. First, we obtain a polynomial relation (PR) between different-mode parafermion fi's by generalizing the corresponding single-mode PR to such that is invariant under the unitary transformation of fi (Green's condition). Differently from a usual context, where the Green's condition is imposed only on the defining relation of fi (degree three with respect to fi and fi†), we impose it on any degree of PR. For the case of order-two parafermion (the simplest case of para SUSY), we calculate a PR between the parasupercharge G0, the bosonic hamiltonian LB0 and parafermionic one LF0, although it is difficult to obtain a PR between G0 and the total hamiltonian L0 (= LB0 + LF0). Finally, we construct a para SUSY Virasoro algebra by generalizing L0 to the Ln's such that form a Virasoro algebra.

  1. Reducing Communication in Algebraic Multigrid Using Additive Variants

    SciTech Connect

    Vassilevski, Panayot S.; Yang, Ulrike Meier

    2014-02-12

    Algebraic multigrid (AMG) has proven to be an effective scalable solver on many high performance computers. However, its increasing communication complexity on coarser levels has shown to seriously impact its performance on computers with high communication cost. Moreover, additive AMG variants provide not only increased parallelism as well as decreased numbers of messages per cycle but also generally exhibit slower convergence. Here we present various new additive variants with convergence rates that are significantly improved compared to the classical additive algebraic multigrid method and investigate their potential for decreased communication, and improved communication-computation overlap, features that are essential for good performance on future exascale architectures.

  2. Algebraic Dynamic Programming over general data structures

    PubMed Central

    2015-01-01

    Background Dynamic programming algorithms provide exact solutions to many problems in computational biology, such as sequence alignment, RNA folding, hidden Markov models (HMMs), and scoring of phylogenetic trees. Structurally analogous algorithms compute optimal solutions, evaluate score distributions, and perform stochastic sampling. This is explained in the theory of Algebraic Dynamic Programming (ADP) by a strict separation of state space traversal (usually represented by a context free grammar), scoring (encoded as an algebra), and choice rule. A key ingredient in this theory is the use of yield parsers that operate on the ordered input data structure, usually strings or ordered trees. The computation of ensemble properties, such as a posteriori probabilities of HMMs or partition functions in RNA folding, requires the combination of two distinct, but intimately related algorithms, known as the inside and the outside recursion. Only the inside recursions are covered by the classical ADP theory. Results The ideas of ADP are generalized to a much wider scope of data structures by relaxing the concept of parsing. This allows us to formalize the conceptual complementarity of inside and outside variables in a natural way. We demonstrate that outside recursions are generically derivable from inside decomposition schemes. In addition to rephrasing the well-known algorithms for HMMs, pairwise sequence alignment, and RNA folding we show how the TSP and the shortest Hamiltonian path problem can be implemented efficiently in the extended ADP framework. As a showcase application we investigate the ancient evolution of HOX gene clusters in terms of shortest Hamiltonian paths. Conclusions The generalized ADP framework presented here greatly facilitates the development and implementation of dynamic programming algorithms for a wide spectrum of applications. PMID:26695390

  3. A brief review of E theory

    NASA Astrophysics Data System (ADS)

    West, Peter

    2016-09-01

    I begin with some memories of Abdus Salam who was my PhD supervisor. After reviewing the theory of nonlinear realisations and Kac-Moody algebras, I explain how to construct the nonlinear realisation based on the Kac-Moody algebra E11 and its vector representation. I explain how this field theory leads to dynamical equations which contain an infinite number of fields defined on a space-time with an infinite number of coordinates. I then show that these unique dynamical equations, when truncated to low level fields and the usual coordinates of space-time, lead to precisely the equations of motion of 11-dimensional supergravity theory. By taking different group decompositions of E11 we find all the maximal supergravity theories, including the gauged maximal supergravities, and as a result the nonlinear realisation should be thought of as a unified theory that is the low energy effective action for type II strings and branes. These results essentially confirm the E11 conjecture given many years ago.

  4. The application of cryogenics to high Reynolds number testing in wind tunnels. I - Evolution, theory, and advantages

    NASA Technical Reports Server (NTRS)

    Kilgore, R. A.; Dress, D. A.

    1984-01-01

    During the time which has passed since the construction of the first wind tunnel in 1870, wind tunnels have been developed to a high degree of sophistication. However, their development has consistently failed to keep pace with the demands placed on them. One of the more serious problems to be found with existing transonic wind tunnels is their inability to test subscale aircraft models at Reynolds numbers sufficiently near full-scale values to ensure the validity of using the wind tunnel data to predict flight characteristics. The Reynolds number capability of a wind tunnel may be increased by a number of different approaches. However, the best solution in terms of model, balance, and model support loads, as well as in terms of capital and operating cost appears to be related to the reduction of the temperature of the test gas to cryogenic temperatures. The present paper has the objective to review the evolution of the cryogenic wind tunnel concept and to describe its more important advantages.

  5. Moving frames and prolongation algebras

    NASA Technical Reports Server (NTRS)

    Estabrook, F. B.

    1982-01-01

    Differential ideals generated by sets of 2-forms which can be written with constant coefficients in a canonical basis of 1-forms are considered. By setting up a Cartan-Ehresmann connection, in a fiber bundle over a base space in which the 2-forms live, one finds an incomplete Lie algebra of vector fields in the fields in the fibers. Conversely, given this algebra (a prolongation algebra), one can derive the differential ideal. The two constructs are thus dual, and analysis of either derives properties of both. Such systems arise in the classical differential geometry of moving frames. Examples of this are discussed, together with examples arising more recently: the Korteweg-de Vries and Harrison-Ernst systems.

  6. Algebra and statistics of the solar wind

    NASA Astrophysics Data System (ADS)

    Veselovsky, I. S.; Dmitriev, A. V.; Suvorova, A. V.

    2010-04-01

    Statistical studies of properties of the solar wind and interplanetary magnetic field, based on an extended database for the period 1963-2007 including four solar cycles, show that the Gaussian approximation well suites for some parameters as the probability distribution of their numerical values, while for others the lognormal law is preferred. This paper gives an interpretation of these results as associated with predominance of linear or nonlinear processes in composition and interaction of various disturbances and irregularities propagating and originating in the interior of the Sun and its atmosphere, including the solar corona and the solar wind running away from it. Summation of independent random components of disturbances leads, according to the central limit theorem of the probability theory, to the normal (Gaussian) distributions of quantities proper, while their multiplication leads to the normal distributions of logarithms. Thus, one can discuss the algebra of events and associate observed statistical distinctions with one or another process of formation of irregularities in the solar wind. Among them there are impossible events (having null probability) and reliable events (occurring with 100% probability). For better understanding of the relationship between algebra and statistics of events in the solar wind further investigations are necessary.

  7. Algebraic approach to electronic spectroscopy and dynamics.

    PubMed

    Toutounji, Mohamad

    2008-04-28

    Lie algebra, Zassenhaus, and parameter differentiation techniques are utilized to break up the exponential of a bilinear Hamiltonian operator into a product of noncommuting exponential operators by the virtue of the theory of Wei and Norman [J. Math. Phys. 4, 575 (1963); Proc. Am. Math. Soc., 15, 327 (1964)]. There are about three different ways to find the Zassenhaus exponents, namely, binomial expansion, Suzuki formula, and q-exponential transformation. A fourth, and most reliable method, is provided. Since linearly displaced and distorted (curvature change upon excitation/emission) Hamiltonian and spin-boson Hamiltonian may be classified as bilinear Hamiltonians, the presented algebraic algorithm (exponential operator disentanglement exploiting six-dimensional Lie algebra case) should be useful in spin-boson problems. The linearly displaced and distorted Hamiltonian exponential is only treated here. While the spin-boson model is used here only as a demonstration of the idea, the herein approach is more general and powerful than the specific example treated. The optical linear dipole moment correlation function is algebraically derived using the above mentioned methods and coherent states. Coherent states are eigenvectors of the bosonic lowering operator a and not of the raising operator a(+). While exp(a(+)) translates coherent states, exp(a(+)a(+)) operation on coherent states has always been a challenge, as a(+) has no eigenvectors. Three approaches, and the results, of that operation are provided. Linear absorption spectra are derived, calculated, and discussed. The linear dipole moment correlation function for the pure quadratic coupling case is expressed in terms of Legendre polynomials to better show the even vibronic transitions in the absorption spectrum. Comparison of the present line shapes to those calculated by other methods is provided. Franck-Condon factors for both linear and quadratic couplings are exactly accounted for by the herein calculated

  8. Algebraic approach to electronic spectroscopy and dynamics

    NASA Astrophysics Data System (ADS)

    Toutounji, Mohamad

    2008-04-01

    Lie algebra, Zassenhaus, and parameter differentiation techniques are utilized to break up the exponential of a bilinear Hamiltonian operator into a product of noncommuting exponential operators by the virtue of the theory of Wei and Norman [J. Math. Phys. 4, 575 (1963); Proc. Am. Math. Soc., 15, 327 (1964)]. There are about three different ways to find the Zassenhaus exponents, namely, binomial expansion, Suzuki formula, and q-exponential transformation. A fourth, and most reliable method, is provided. Since linearly displaced and distorted (curvature change upon excitation/emission) Hamiltonian and spin-boson Hamiltonian may be classified as bilinear Hamiltonians, the presented algebraic algorithm (exponential operator disentanglement exploiting six-dimensional Lie algebra case) should be useful in spin-boson problems. The linearly displaced and distorted Hamiltonian exponential is only treated here. While the spin-boson model is used here only as a demonstration of the idea, the herein approach is more general and powerful than the specific example treated. The optical linear dipole moment correlation function is algebraically derived using the above mentioned methods and coherent states. Coherent states are eigenvectors of the bosonic lowering operator a and not of the raising operator a+. While exp(a +) translates coherent states, exp(a +a+) operation on coherent states has always been a challenge, as a+ has no eigenvectors. Three approaches, and the results, of that operation are provided. Linear absorption spectra are derived, calculated, and discussed. The linear dipole moment correlation function for the pure quadratic coupling case is expressed in terms of Legendre polynomials to better show the even vibronic transitions in the absorption spectrum. Comparison of the present line shapes to those calculated by other methods is provided. Franck-Condon factors for both linear and quadratic couplings are exactly accounted for by the herein calculated linear

  9. Colored Quantum Algebra and Its Bethe State

    NASA Astrophysics Data System (ADS)

    Wang, Jin-Zheng; Jia, Xiao-Yu; Wang, Shi-Kun

    2014-12-01

    We investigate the colored Yang—Baxter equation. Based on a trigonometric solution of colored Yang—Baxter equation, we construct a colored quantum algebra. Moreover we discuss its algebraic Bethe ansatz state and highest wight representation.

  10. Generalized Galilean algebras and Newtonian gravity

    NASA Astrophysics Data System (ADS)

    González, N.; Rubio, G.; Salgado, P.; Salgado, S.

    2016-04-01

    The non-relativistic versions of the generalized Poincaré algebras and generalized AdS-Lorentz algebras are obtained. These non-relativistic algebras are called, generalized Galilean algebras of type I and type II and denoted by GBn and GLn respectively. Using a generalized Inönü-Wigner contraction procedure we find that the generalized Galilean algebras of type I can be obtained from the generalized Galilean algebras type II. The S-expansion procedure allows us to find the GB5 algebra from the Newton Hooke algebra with central extension. The procedure developed in Ref. [1] allows us to show that the nonrelativistic limit of the five dimensional Einstein-Chern-Simons gravity is given by a modified version of the Poisson equation. The modification could be compatible with the effects of Dark Matter, which leads us to think that Dark Matter can be interpreted as a non-relativistic limit of Dark Energy.

  11. Motivating Activities that Lead to Algebra

    ERIC Educational Resources Information Center

    Menon, Ramakrishnan

    2004-01-01

    Four activities consisting of puzzles are introduced, which help students to recognize the strength of algebraic generalizations. They also assist them to comprehend algebraic concepts, and enable them to develop their individual puzzles and games.

  12. Higher spin approaches to quantum field theory and (psuedo)-Riemannian geometries

    NASA Astrophysics Data System (ADS)

    Hallowell, Karl Evan

    In this thesis, we study a number of higher spin quantum field theories and some of their algebraic and geometric consequences. These theories apply mostly either over constant curvature or more generally symmetric pseudo-Riemannian manifolds. The first part of this dissertation covers a superalgebra coming from a family of particle models over symmetric spaces. These theories are novel in that the symmetries of the (super)algebra osp( Q|2p) are larger and more elaborate than traditional symmetries. We construct useful (super)algebras related to and generalizing old work by Lichnerowicz and describe their role in developing the geometry of massless models with osp(Q|2 p) symmetry. The result is two practical applications of these (super)algebras: (1) a lunch more concise description of a family of higher spin quantum field theories; and (2) an interesting algebraic probe of underlying background geometries. We also consider massive models over constant curvature spaces. We use a radial dimensional reduction process which converts massless models into massive ones over a lower dimensional space. In our case, we take from the family of theories above the particular free, massless model over flat space associated with sp(2, R ) and derive a massive model. In the process, we develop a novel associative algebra, which is a deformation of the original differential operator algebra associated with the sp(2, R ) model. This algebra is interesting in its own right since its operators realize the representation structure of the sp(2, R ) group. The massive model also has implications for a sequence of unusual, "partially massless" theories. The derivation illuminates how reduced degrees of freedom become manifest in these particular models. Finally, we study a Yang-Mills model using an on-shell Poincare Yang-Mills twist of the Maxwell complex along with a non-minimal coupling. This is a special, higher spin case of a quantum field theory called a Yang-Mills detour complex

  13. Lack of Set Theory Relevant Prerequisite Knowledge

    ERIC Educational Resources Information Center

    Dogan-Dunlap, Hamide

    2006-01-01

    Many students struggle with college mathematics topics due to a lack of mastery of prerequisite knowledge. Set theory language is one such prerequisite for linear algebra courses. Many students' mistakes on linear algebra questions reveal a lack of mastery of set theory knowledge. This paper reports the findings of a qualitative analysis of a…

  14. Scalable Parallel Algebraic Multigrid Solvers

    SciTech Connect

    Bank, R; Lu, S; Tong, C; Vassilevski, P

    2005-03-23

    The authors propose a parallel algebraic multilevel algorithm (AMG), which has the novel feature that the subproblem residing in each processor is defined over the entire partition domain, although the vast majority of unknowns for each subproblem are associated with the partition owned by the corresponding processor. This feature ensures that a global coarse description of the problem is contained within each of the subproblems. The advantages of this approach are that interprocessor communication is minimized in the solution process while an optimal order of convergence rate is preserved; and the speed of local subproblem solvers can be maximized using the best existing sequential algebraic solvers.

  15. Spatial-Operator Algebra For Robotic Manipulators

    NASA Technical Reports Server (NTRS)

    Rodriguez, Guillermo; Kreutz, Kenneth K.; Milman, Mark H.

    1991-01-01

    Report discusses spatial-operator algebra developed in recent studies of mathematical modeling, control, and design of trajectories of robotic manipulators. Provides succinct representation of mathematically complicated interactions among multiple joints and links of manipulator, thereby relieving analyst of most of tedium of detailed algebraic manipulations. Presents analytical formulation of spatial-operator algebra, describes some specific applications, summarizes current research, and discusses implementation of spatial-operator algebra in the Ada programming language.

  16. The symmetry of M-theories

    NASA Astrophysics Data System (ADS)

    Englert, François; Houart, Laurent; Taormina, Anne; West, Peter

    2003-09-01

    We consider the Cartan subalgebra of any very extended algebra Script G+++ where Script G is a simple Lie algebra and let the parameters be space-time fields. These are identified with diagonal metrics and dilatons. Using the properties of the algebra, we find that for all very extensions Script G+++ of simple Lie algebras there are theories of gravity and matter, which admit classical solutions carrying representations of the Weyl group of Script G+++. We also identify the T and S-dualities of superstrings and of the bosonic string with Weyl reflections and outer automorphisms of well-chosen very extended algebras and we exhibit specific features of the very extensions. We take these results as indication that very extended algebras underlie symmetries of any consistent theory of gravity and matter, and might encode basic information for the construction of such theory.

  17. Unifying the Algebra for All Movement

    ERIC Educational Resources Information Center

    Eddy, Colleen M.; Quebec Fuentes, Sarah; Ward, Elizabeth K.; Parker, Yolanda A.; Cooper, Sandi; Jasper, William A.; Mallam, Winifred A.; Sorto, M. Alejandra; Wilkerson, Trena L.

    2015-01-01

    There exists an increased focus on school mathematics, especially first-year algebra, due to recent efforts for all students to be college and career ready. In addition, there are calls, policies, and legislation advocating for all students to study algebra epitomized by four rationales of the "Algebra for All" movement. In light of this…

  18. Build an Early Foundation for Algebra Success

    ERIC Educational Resources Information Center

    Knuth, Eric; Stephens, Ana; Blanton, Maria; Gardiner, Angela

    2016-01-01

    Research tells us that success in algebra is a factor in many other important student outcomes. Emerging research also suggests that students who are started on an algebra curriculum in the earlier grades may have greater success in the subject in secondary school. What's needed is a consistent, algebra-infused mathematics curriculum all…

  19. Difficulties in Initial Algebra Learning in Indonesia

    ERIC Educational Resources Information Center

    Jupri, Al; Drijvers, Paul; van den Heuvel-Panhuizen, Marja

    2014-01-01

    Within mathematics curricula, algebra has been widely recognized as one of the most difficult topics, which leads to learning difficulties worldwide. In Indonesia, algebra performance is an important issue. In the Trends in International Mathematics and Science Study (TIMSS) 2007, Indonesian students' achievement in the algebra domain was…

  20. Algebra? A Gate! A Barrier! A Mystery!

    ERIC Educational Resources Information Center

    Mathematics Educatio Dialogues, 2000

    2000-01-01

    This issue of Mathematics Education Dialogues focuses on the nature and the role of algebra in the K-14 curriculum. Articles on this theme include: (1) "Algebra For All? Why?" (Nel Noddings); (2) "Algebra For All: It's a Matter of Equity, Expectations, and Effectiveness" (Dorothy S. Strong and Nell B. Cobb); (3) "Don't Delay: Build and Talk about…