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.
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…
The Algebra of Complex Numbers.
ERIC Educational Resources Information Center
LePage, Wilbur R.
This programed text is an introduction to the algebra of complex numbers for engineering students, particularly because of its relevance to important problems of applications in electrical engineering. It is designed for a person who is well experienced with the algebra of real numbers and calculus, but who has no experience with complex number…
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.
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.
Role of division algebra in seven-dimensional gauge theory
NASA Astrophysics Data System (ADS)
Kalauni, Pushpa; Barata, J. C. A.
2015-03-01
The algebra of octonions 𝕆 forms the largest normed division algebra over the real numbers ℝ, complex numbers ℂ and quaternions ℍ. The usual three-dimensional vector product is given by quaternions, while octonions produce seven-dimensional vector product. Thus, octonionic algebra is closely related to the seven-dimensional algebra, therefore one can extend generalization of rotations in three dimensions to seven dimensions using octonions. An explicit algebraic description of octonions has been given to describe rotational transformation in seven-dimensional space. We have also constructed a gauge theory based on non-associative algebra to discuss Yang-Mills theory and field equation in seven-dimensional space.
On the binary expansions of algebraic numbers
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}.
Algebraic K-theory, K-regularity, and -duality of -stable C ∗-algebras
NASA Astrophysics Data System (ADS)
Mahanta, Snigdhayan
2015-12-01
We develop an algebraic formalism for topological -duality. More precisely, we show that topological -duality actually induces an isomorphism between noncommutative motives that in turn implements the well-known isomorphism between twisted K-theories (up to a shift). In order to establish this result we model topological K-theory by algebraic K-theory. We also construct an E ∞ -operad starting from any strongly self-absorbing C ∗-algebra . Then we show that there is a functorial topological K-theory symmetric spectrum construction on the category of separable C ∗-algebras, such that is an algebra over this operad; moreover, is a module over this algebra. Along the way we obtain a new symmetric spectra valued functorial model for the (connective) topological K-theory of C ∗-algebras. We also show that -stable C ∗-algebras are K-regular providing evidence for a conjecture of Rosenberg. We conclude with an explicit description of the algebraic K-theory of a x+ b-semigroup C ∗-algebras coming from number theory and that of -stabilized noncommutative tori.
Computer algebra and transport theory.
Warsa, J. S.
2004-01-01
Modern symbolic algebra computer software augments and complements more traditional approaches to transport theory applications in several ways. The first area is in the development and enhancement of numerical solution methods for solving the Boltzmann transport equation. Typically, special purpose computer codes are designed and written to solve specific transport problems in particular ways. Different aspects of the code are often written from scratch and the pitfalls of developing complex computer codes are numerous and well known. Software such as MAPLE and MATLAB can be used to prototype, analyze, verify and determine the suitability of numerical solution methods before a full-scale transport application is written. Once it is written, the relevant pieces of the full-scale code can be verified using the same tools I that were developed for prototyping. Another area is in the analysis of numerical solution methods or the calculation of theoretical results that might otherwise be difficult or intractable. Algebraic manipulations are done easily and without error and the software also provides a framework for any additional numerical calculations that might be needed to complete the analysis. We will discuss several applications in which we have extensively used MAPLE and MATLAB in our work. All of them involve numerical solutions of the S{sub N} transport equation. These applications encompass both of the two main areas in which we have found computer algebra software essential.
Class Numbers and Groups of Algebraic Groups
NASA Astrophysics Data System (ADS)
Platonov, V. P.; Bondarenko, A. A.; Rapinčuk, A. S.
1980-06-01
The class number of an algebraic group G defined over a global field is the number of double cosets of the adele group GA with respect to the subgroups of integral and principal adeles. In most cases the set of double cosets has the natural structure of an abelian group, called the class group of G. In this article the class number of a semisimple group G is computed, and it is proved that any finite abelian group can be realized as a class group.Bibliography: 24 titles.
Fusion rule algebras from graph theory
NASA Astrophysics Data System (ADS)
Caselle, M.; Ponzano, G.
1989-06-01
We describe a new class of fusion algebras related to graph theory which bear intriguing connections with group algebras. The structure constants and the matrix S, which diagonalizes the fusion rules, are explicitly computed in terms of SU(2) coupling coefficients.
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.
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)
Vertex operator algebras and conformal field theory
Huang, Y.Z. )
1992-04-20
This paper discusses conformal field theory, an important physical theory, describing both two-dimensional critical phenomena in condensed matter physics and classical motions of strings in string theory. The study of conformal field theory will deepen the understanding of these theories and will help to understand string theory conceptually. Besides its importance in physics, the beautiful and rich mathematical structure of conformal field theory has interested many mathematicians. New relations between different branches of mathematics, such as representations of infinite-dimensional Lie algebras and Lie groups, Riemann surfaces and algebraic curves, the Monster sporadic group, modular functions and modular forms, elliptic genera and elliptic cohomology, Calabi-Yau manifolds, tensor categories, and knot theory, are revealed in the study of conformal field theory. It is therefore believed that the study of the mathematics involved in conformal field theory will ultimately lead to new mathematical structures which would be important to both mathematics and physics.
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.
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.
The arithmetic theory of algebraic groups
NASA Astrophysics Data System (ADS)
Platonov, V. P.
1982-06-01
CONTENTS Introduction § 1. Arithmetic groups § 2. Adèle groups § 3. Tamagawa numbers § 4. Approximations in algebraic groups § 5. Class numbers and class groups of algebraic groups § 6. The genus problem in arithmetic groups § 7. Classification of maximal arithmetic subgroups § 8. The congruence problem § 9. Groups of rational points over global fields § 10. Galois cohomology and the Hasse principle § 11. Cohomology of arithmetic groups References
Excision in algebraic K-theory and Karoubi's conjecture.
Suslin, A A; Wodzicki, M
1990-12-15
We prove that the property of excision in algebraic K-theory is for a Q-algebra A equivalent to the H-unitality of the latter. Our excision theorem, in particular, implies Karoubi's conjecture on the equality of algebraic and topological K-theory groups of stable C*-algebras. It also allows us to identify the algebraic K-theory of the symbol map in the theory of pseudodifferential operators. PMID:11607130
Excision in algebraic K-theory and Karoubi's conjecture.
Suslin, A A; Wodzicki, M
1990-01-01
We prove that the property of excision in algebraic K-theory is for a Q-algebra A equivalent to the H-unitality of the latter. Our excision theorem, in particular, implies Karoubi's conjecture on the equality of algebraic and topological K-theory groups of stable C*-algebras. It also allows us to identify the algebraic K-theory of the symbol map in the theory of pseudodifferential operators. PMID:11607130
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.
Algebraic theory of recombination spaces.
Stadler, P F; Wagner, G P
1997-01-01
A new mathematical representation is proposed for the configuration space structure induced by recombination, which we call "P-structure." It consists of a mapping of pairs of objects to the power set of all objects in the search space. The mapping assigns to each pair of parental "genotypes" the set of all recombinant genotypes obtainable from the parental ones. It is shown that this construction allows a Fourier decomposition of fitness landscapes into a superposition of "elementary landscapes." This decomposition is analogous to the Fourier decomposition of fitness landscapes on mutation spaces. The elementary landscapes are obtained as eigenfunctions of a Laplacian operator defined for P-structures. For binary string recombination, the elementary landscapes are exactly the p-spin functions (Walsh functions), that is, the same as the elementary landscapes of the string point mutation spaces (i.e., the hypercube). This supports the notion of a strong homomorphism between string mutation and recombination spaces. However, the effective nearest neighbor correlations on these elementary landscapes differ between mutation and recombination and among different recombination operators. On average, the nearest neighbor correlation is higher for one-point recombination than for uniform recombination. For one-point recombination, the correlations are higher for elementary landscapes with fewer interacting sites as well as for sites that have closer linkage, confirming the qualitative predictions of the Schema Theorem. We conclude that the algebraic approach to fitness landscape analysis can be extended to recombination spaces and provides an effective way to analyze the relative hardness of a landscape for a given recombination operator. PMID:10021760
Algebraic Theories and (Infinity,1)-Categories
NASA Astrophysics Data System (ADS)
Cranch, James
2010-11-01
We adapt the classical framework of algebraic theories to work in the setting of (infinity,1)-categories developed by Joyal and Lurie. This gives a suitable approach for describing highly structured objects from homotopy theory. A central example, treated at length, is the theory of E_infinity spaces: this has a tidy combinatorial description in terms of span diagrams of finite sets. We introduce a theory of distributive laws, allowing us to describe objects with two distributing E_infinity stuctures. From this we produce a theory of E_infinity ring spaces. We also study grouplike objects, and produce theories modelling infinite loop spaces (or connective spectra), and infinite loop spaces with coherent multiplicative structure (or connective ring spectra). We use this to construct the units of a grouplike E_infinity ring space in a natural manner. Lastly we provide a speculative pleasant description of the K-theory of monoidal quasicategories and quasicategories with ring-like structures.
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.
Quantum field theories on algebraic curves. I. Additive bosons
NASA Astrophysics Data System (ADS)
Takhtajan, Leon A.
2013-04-01
Using Serre's adelic interpretation of cohomology, we develop a `differential and integral calculus' on an algebraic curve X over an algebraically closed field k of constants of characteristic zero, define algebraic analogues of additive multi-valued functions on X and prove the corresponding generalized residue theorem. Using the representation theory of the global Heisenberg algebra and lattice Lie algebra, we formulate quantum field theories of additive and charged bosons on an algebraic curve X. These theories are naturally connected with the algebraic de Rham theorem. We prove that an extension of global symmetries (Witten's additive Ward identities) from the k-vector space of rational functions on X to the vector space of additive multi-valued functions uniquely determines these quantum theories of additive and charged bosons.
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…
Do malaria parasites follow the algebra of sex ratio theory?
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
Topics in Number Theory: The Number Game.
ERIC Educational Resources Information Center
Batra, Laj, Ed.; And Others
This teacher's guide contains nine topics in number theory. Suggested questions for the teacher, short investigations, and possible exercises for the student are included. Chapter 1 is an introduction to sequences and series using geoboard activities involving triangular numbers, square numbers, rectangular numbers, and pentagonal numbers. The…
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…
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…
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.
Imaginary numbers are not real—The geometric algebra of spacetime
NASA Astrophysics Data System (ADS)
Gull, Stephen; Lasenby, Anthony; Doran, Chris
1993-09-01
This paper contains a tutorial introduction to the ideas of geometric algebra, concentrating on its physical applications. We show how the definition of a “geometric product” of vectors in 2-and 3-dimensional space provides precise geometrical interpretations of the imaginary numbers often used in conventional methods. Reflections and rotations are analyzed in terms of bilinear spinor transformations, and are then related to the theory of analytic functions and their natural extension in more than two dimensions (monogenics), Physics is greatly facilitated by the use of Hestenes' spacetime algebra, which automatically incorporates the geometric structure of spacetime. This is demonstrated by examples from electromagnetism. In the course of this purely classical exposition many surprising results are obtained—results which are usually thought to belong to the preserve of quantum theory. We conclude that geometric algebra is the most powerful and general language available for the development of mathematical physics.
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.
Topological insulators and C*-algebras: Theory and numerical practice
Hastings, Matthew B.; Loring, Terry A.
2011-07-15
Research Highlights: > We classify topological insulators using C* algebras. > We present new K-theory invariants. > We develop efficient numerical algorithms based on this technique. > We observe unexpected quantum phase transitions using our algorithm. - Abstract: We apply ideas from C*-algebra to the study of disordered topological insulators. We extract certain almost commuting matrices from the free Fermi Hamiltonian, describing band projected coordinate matrices. By considering topological obstructions to approximating these matrices by exactly commuting matrices, we are able to compute invariants quantifying different topological phases. We generalize previous two dimensional results to higher dimensions; we give a general expression for the topological invariants for arbitrary dimension and several symmetry classes, including chiral symmetry classes, and we present a detailed K-theory treatment of this expression for time reversal invariant three dimensional systems. We can use these results to show non-existence of localized Wannier functions for these systems. We use this approach to calculate the index for time-reversal invariant systems with spin-orbit scattering in three dimensions, on sizes up to 12{sup 3}, averaging over a large number of samples. The results show an interesting separation between the localization transition and the point at which the average index (which can be viewed as an 'order parameter' for the topological insulator) begins to fluctuate from sample to sample, implying the existence of an unsuspected quantum phase transition separating two different delocalized phases in this system. One of the particular advantages of the C*-algebraic technique that we present is that it is significantly faster in practice than other methods of computing the index, allowing the study of larger systems. In this paper, we present a detailed discussion of numerical implementation of our method.
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.
From string theory to algebraic geometry and back
Brinzanescu, Vasile
2011-02-10
We describe some facts in physics which go up to the modern string theory and the related concepts in algebraic geometry. Then we present some recent results on moduli-spaces of vector bundles on non-Kaehler Calabi-Yau 3-folds and their consequences for heterotic string theory.
A Cohomology Theory of Grading-Restricted Vertex Algebras
NASA Astrophysics Data System (ADS)
Huang, Yi-Zhi
2014-04-01
We introduce a cohomology theory of grading-restricted vertex algebras. To construct the correct cohomologies, we consider linear maps from tensor powers of a grading-restricted vertex algebra to "rational functions valued in the algebraic completion of a module for the algebra," instead of linear maps from tensor powers of the algebra to a module for the algebra. One subtle complication arising from such functions is that we have to carefully address the issue of convergence when we compose these linear maps with vertex operators. In particular, for each , we have an inverse system of nth cohomologies and an additional nth cohomology of a grading-restricted vertex algebra V with coefficients in a V-module W such that is isomorphic to the inverse limit of the inverse system . In the case of n = 2, there is an additional second cohomology denoted by which will be shown in a sequel to the present paper to correspond to what we call square-zero extensions of V and to first order deformations of V when W = V.
Some Ideas About Number Theory.
ERIC Educational Resources Information Center
Barnett, I. A.
The material in this booklet is designed for non-professional mathematicians who have an interest in the theory of numbers. The author presents some elementary results of number theory without involving detailed proofs. Much of the material has direct application for secondary school mathematics teachers. A brief account of the nature of number…
Algebraic isomorphism in two-dimensional anomalous gauge theories
Carvalhaes, C.G.; Natividade, C.P.
1997-08-01
The operator solution of the anomalous chiral Schwinger model is discussed on the basis of the general principles of Wightman field theory. Some basic structural properties of the model are analyzed taking a careful control on the Hilbert space associated with the Wightman functions. The isomorphism between gauge noninvariant and gauge invariant descriptions of the anomalous theory is established in terms of the corresponding field algebras. We show that (i) the {Theta}-vacuum representation and (ii) the suggested equivalence of vector Schwinger model and chiral Schwinger model cannot be established in terms of the intrinsic field algebra. {copyright} 1997 Academic Press, Inc.
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).
Category of trees in representation theory of quantum algebras
Moskaliuk, N. M.; Moskaliuk, S. S.
2013-10-15
New applications of categorical methods are connected with new additional structures on categories. One of such structures in representation theory of quantum algebras, the category of Kuznetsov-Smorodinsky-Vilenkin-Smirnov (KSVS) trees, is constructed, whose objects are finite rooted KSVS trees and morphisms generated by the transition from a KSVS tree to another one.
Algebraic K-theory of discrete subgroups of Lie groups.
Farrell, F T; Jones, L E
1987-05-01
Let G be a Lie group (with finitely many connected components) and Gamma be a discrete, cocompact, torsion-free subgroup of G. We rationally calculate the algebraic K-theory of the integral group ring ZGamma in terms of the homology of Gamma with trivial rational coefficients. PMID:16593834
Algebraic K-theory of discrete subgroups of Lie groups
Farrell, F. T.; Jones, L. E.
1987-01-01
Let G be a Lie group (with finitely many connected components) and Γ be a discrete, cocompact, torsion-free subgroup of G. We rationally calculate the algebraic K-theory of the integral group ring ZΓ in terms of the homology of Γ with trivial rational coefficients. PMID:16593834
Forms and algebras in (half-)maximal supergravity theories
NASA Astrophysics Data System (ADS)
Howe, Paul; Palmkvist, Jakob
2015-05-01
The forms in D-dimensional (half-)maximal supergravity theories are discussed for 3 ≤ D ≤ 11. Superspace methods are used to derive consistent sets of Bianchi identities for all the forms for all degrees, and to show that they are soluble and fully compatible with supersymmetry. The Bianchi identities determine Lie superalgebras that can be extended to Borcherds superalgebras of a special type. It is shown that any Borcherds superalgebra of this type gives the same form spectrum, up to an arbitrary degree, as an associated Kac-Moody algebra. For maximal supergravity up to D-form potentials, this is the very extended Kac-Moody algebra E 11. It is also shown how gauging can be carried out in a simple fashion by deforming the Bianchi identities by means of a new algebraic element related to the embedding tensor. In this case the appropriate extension of the form algebra is a truncated version of the so-called tensor hierarchy algebra.
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
Pattern vectors from algebraic graph theory.
Wilson, Richard C; Hancock, Edwin R; Luo, Bin
2005-07-01
Graph structures have proven computationally cumbersome for pattern analysis. The reason for this is that, before graphs can be converted to pattern vectors, correspondences must be established between the nodes of structures which are potentially of different size. To overcome this problem, in this paper, we turn to the spectral decomposition of the Laplacian matrix. We show how the elements of the spectral matrix for the Laplacian can be used to construct symmetric polynomials that are permutation invariants. The coefficients of these polynomials can be used as graph features which can be encoded in a vectorial manner. We extend this representation to graphs in which there are unary attributes on the nodes and binary attributes on the edges by using the spectral decomposition of a Hermitian property matrix that can be viewed as a complex analogue of the Laplacian. To embed the graphs in a pattern space, we explore whether the vectors of invariants can be embedded in a low-dimensional space using a number of alternative strategies, including principal components analysis (PCA), multidimensional scaling (MDS), and locality preserving projection (LPP). Experimentally, we demonstrate that the embeddings result in well-defined graph clusters. Our experiments with the spectral representation involve both synthetic and real-world data. The experiments with synthetic data demonstrate that the distances between spectral feature vectors can be used to discriminate between graphs on the basis of their structure. The real-world experiments show that the method can be used to locate clusters of graphs. PMID:16013758
NASA Astrophysics Data System (ADS)
Hoppe, Jens
Over the past years, associative algebras have come to play a major role in several areas of theoretical physics. Firstly, it has been realized that Yang Baxter algebras [1] constitute the relevant structure underlying 1+1 dimensional integrable models; in addition, their relation to braid groups, the theory of knots and links, and the exchange algebras of 1+1 dimensional conformal field theories [2] has been quite well understood by now. Secondly, deformations of Poisson structures that appeared in 2+1 dimensional field theories as infinite dimensional symmetry algebras possess underlying associative structures, which have also been studied in some detail (concerning higher spin theories see, e.g., [3, 4] and references therein, concerning the enveloping algebra of sl(2, C) see, e.g., [5], concerning deformations of diffAT2 — the Lie algebra of infinitesimal area preserving diffeomorphisms of the Torus — see [6, 7, 8, 9]). Ideas on how both investigations could eventually converge (i.e., a relation between 2+1 and 1+1 dimensions) have, e.g., been expressed in [10]. As indicated by the two subtitles there will be two parts to my paper: the first one presents a view on something I met long ago [11], and recently got interested in again [5, 7, 9, 12], while the second part introduces some algebraic structures that seem to be interesting, and possibly new.
Non-simply laced Lie algebras via F theory strings
NASA Astrophysics Data System (ADS)
Bonora, L.; Savelli, R.
2010-11-01
In order to describe the appearance in F theory of the non-simply-laced Lie algebras, we use the representation of symmetry enhancements by means of string junctions. After an introduction to the techniques used to describe symmetry enhancement, that is algebraic geometry, BPS states analysis and string junctions, we concentrate on the latter. We give an explicit description of the folding of D 2n to B n , of the folding of E 6 to F 4 and that of D 4 to G 2 in terms of junctions and Jordan strings. We also discuss the case of C n , but we are unable in this case to provide a string interpretation.
ERIC Educational Resources Information Center
Stanford Univ., CA. School Mathematics Study Group.
Not all of mathematics can be taught in formal textbooks. Just as an English course can be enlivened by selections from literature, a mathematics course can gain depth and interest from special readings. This volume can be read in conjunction with the SMSG First Course in Algebra or Intermediate Mathematics. It introduces the subject of number…
From symmetries to number theory
Tempesta, P.
2009-05-15
It is shown that the finite-operator calculus provides a simple formalism useful for constructing symmetry-preserving discretizations of quantum-mechanical integrable models. A related algebraic approach can also be used to define a class of Appell polynomials and of L series.
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).
On the algebraic K-theory of the complex K-theory spectrum
NASA Astrophysics Data System (ADS)
Ausoni, Christian
2010-03-01
Let p>3 be a prime, let ku be the connective complex K-theory spectrum, and let K(ku) be the algebraic K-theory spectrum of ku. We study the p-primary homotopy type of the spectrum K(ku) by computing its mod (p,v_1) homotopy groups. We show that up to a finite summand, these groups form a finitely generated free module over a polynomial algebra F_p[b], where b is a class of degree 2p+2 defined as a higher Bott element.
Calculation of exchange energies using algebraic perturbation theory
Burrows, B. L.; Dalgarno, A.; Cohen, M.
2010-04-15
An algebraic perturbation theory is presented for efficient calculations of localized states and hence of exchange energies, which are the differences between low-lying states of the valence electron of a molecule, formed by the collision of an ion Y{sup +} with an atom X. For the case of a homonuclear molecule these are the gerade and ungerade states and the exchange energy is an exponentially decreasing function of the internuclear distance. For such homonuclear systems the theory is used in conjunction with the Herring-Holstein technique to give accurate exchange energies for a range of intermolecular separations R. Since the perturbation parameter is essentially 1/R, this method is suitable for large R. In particular, exchange energies are calculated for X{sub 2}{sup +} systems, where X is H, Li, Na, K, Rb, or Cs.
COMMENT: Comment on `Dirac theory in spacetime algebra'
NASA Astrophysics Data System (ADS)
Baylis, William E.
2002-06-01
In contrast to formulations of the Dirac theory by Hestenes and by the present author, the formulation recently presented by Joyce (Joyce W P 2001 J. Phys. A: Math. Gen. 34 1991-2005) is equivalent to the usual Dirac equation only in the case of vanishing mass. For nonzero mass, solutions to Joyce's equation can be solutions either of the Dirac equation in the Hestenes form or of the same equation with the sign of the mass reversed, and in general they are mixtures of the two possibilities. Because of this relationship, Joyce obtains twice as many linearly independent plane-wave solutions for a given momentum eigenvalue as exist in the conventional theory. A misconception about the symmetry of the Hestenes equation and the geometric significance of the algebraic spinors is also briefly discussed.
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.
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.
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.
Solutions in bosonic string field theory and higher spin algebras in AdS
NASA Astrophysics Data System (ADS)
Polyakov, Dimitri
2015-11-01
We find a class of analytic solutions in open bosonic string field theory, parametrized by the chiral copy of higher spin algebra in AdS3. The solutions are expressed in terms of the generating function for the products of Bell polynomials in derivatives of bosonic space-time coordinates Xm(z ) of the open string, the form of which is determined in this work. The products of these polynomials form a natural operator algebra realizations of w∞ (area-preserving diffeomorphisms), enveloping algebra of SU(2) and higher spin algebra in AdS3. The class of string field theory solutions found can, in turn, be interpreted as the "enveloping of enveloping," or the enveloping of AdS3 higher spin algebra. We also discuss the extensions of this class of solutions to superstring theory and their relations to higher spin algebras in higher space-time dimensions.
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.
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.
Equivariant algebraic vector bundles over representations of reductive groups: theory.
Masuda, M; Petrie, T
1991-01-01
Let G be a reductive algebraic group and let B be an affine variety with an algebraic action of G. Everything is defined over the field C of complex numbers. Consider the trivial G-vector bundle B x S = S over B where S is a G-module. From the endomorphism ring R of the G-vector bundle S a construction of G-vector bundles over B is given. The bundles constructed this way have the property that when added to S they are isomorphic to F + S for a fixed G-module F. For such a bundle E an invariant rho(E) is defined that lies in a quotient of R. This invariant allows us to distinguish nonisomorphic G-vector bundles. This is applied to the case where B is a G-module and, in that case, an invariant of the underlying equivariant variety is given too. These constructions and invariants are used to produce families of inequivalent G-vector bundles over G-modules and families of inequivalent G actions on affine spaces for some finite and some connected semisimple groups. PMID:11607220
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).
Geary, David C; Hoard, Mary K; Nugent, Lara; Rouder, Jeffrey N
2015-12-01
The relation between performance on measures of algebraic cognition and acuity of the approximate number system (ANS) and memory for addition facts was assessed for 171 ninth graders (92 girls) while controlling for parental education, sex, reading achievement, speed of numeral processing, fluency of symbolic number processing, intelligence, and the central executive component of working memory. The algebraic tasks assessed accuracy in placing x,y pairs in the coordinate plane, speed and accuracy of expression evaluation, and schema memory for algebra equations. ANS acuity was related to accuracy of placements in the coordinate plane and expression evaluation but not to schema memory. Frequency of fact retrieval errors was related to schema memory but not to coordinate plane or expression evaluation accuracy. The results suggest that the ANS may contribute to or be influenced by spatial-numerical and numerical-only quantity judgments in algebraic contexts, whereas difficulties in committing addition facts to long-term memory may presage slow formation of memories for the basic structure of algebra equations. More generally, the results suggest that different brain and cognitive systems are engaged during the learning of different components of algebraic competence while controlling for demographic and domain general abilities. PMID:26255604
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…
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…
Quantum Computing and Number Theory
NASA Astrophysics Data System (ADS)
Sasaki, Yoshitaka
2013-09-01
The prime factorization can be efficiently solved on a quantum computer. This result was given by Shor in 1994. In the first half of this article, a review of Shor's algorithm with mathematical setups is given. In the second half of this article, the prime number theorem which is an essential tool to understand the distribution of prime numbers is given.
Clifford Algebra Cℓ 3(ℂ) for Applications to Field Theories
NASA Astrophysics Data System (ADS)
Panicaud, B.
2011-10-01
The multivectorial algebras present yet both an academic and a technological interest. Difficulties can occur for their use. Indeed, in all applications care is taken to distinguish between polar and axial vectors and between scalars and pseudo scalars. Then a total of eight elements are often considered even if they are not given the correct name of multivectors. Eventually because of their simplicity, only the vectorial algebra or the quaternions algebra are explicitly used for physical applications. Nevertheless, it should be more convenient to use directly more complex algebras in order to have a wider range of application. The aim of this paper is to inquire into one particular Clifford algebra which could solve this problem. The present study is both didactic concerning its construction and pragmatic because of the introduced applications. The construction method is not an original one. But this latter allows to build up the associated real algebra as well as a peculiar formalism that enables a formal analogy with the classical vectorial algebra. Finally several fields of the theoretical physics will be described thanks to this algebra, as well as a more applied case in general relativity emphasizing simultaneously its relative validity in this particular domain and the easiness of modeling some physical problems.
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…
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.)
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)
Perturbative quantization of Yang-Mills theory with classical double as gauge algebra
NASA Astrophysics Data System (ADS)
Ruiz Ruiz, F.
2016-02-01
Perturbative quantization of Yang-Mills theory with a gauge algebra given by the classical double of a semisimple Lie algebra is considered. The classical double of a real Lie algebra is a nonsemisimple real Lie algebra that admits a nonpositive definite invariant metric, the indefiniteness of the metric suggesting an apparent lack of unitarity. It is shown that the theory is UV divergent at one loop and that there are no radiative corrections at higher loops. One-loop UV divergences are removed through renormalization of the coupling constant, thus introducing a renormalization scale. The terms in the classical action that would spoil unitarity are proved to be cohomologically trivial with respect to the Slavnov-Taylor operator that controls gauge invariance for the quantum theory. Hence they do not contribute gauge invariant radiative corrections to the quantum effective action and the theory is unitary.
Algebraic methods for diagonalization of a quaternion matrix in quaternionic quantum theory
Jiang Tongsong
2005-05-01
By means of complex representation and real representation of a quaternion matrix, this paper studies the problem of diagonalization of a quaternion matrix, gives two algebraic methods for diagonalization of quaternion matrices in quaternionic quantum theory.
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.
Relativistic theory of tidal Love numbers
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.
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)
Adèlic formulas for string amplitudes in fields of algebraic numbers
NASA Astrophysics Data System (ADS)
Vladimirov, V. S.; Sapuzhak, T. M.
1996-06-01
On the basis of the analysis on adèle groups (Tate's formula) for any field of algebraic numbers, a regularization of infinite adèlic products of gamma and beta functions of local fields is proposed. The formulas obtained are applied to representations of the four-point crossing symmetric Veneziano and Virasoro-Shapiro amplitudes through regularized adèlic products of the corresponding string (open and closed, resp.) amplitudes.
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.
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.
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].
Number Theory in the Elementary School.
ERIC Educational Resources Information Center
Beougher, Elton E.
The paper presents reasons for teaching topics from number theory to elementary school students: (1) it can help reveal why numbers "act" in a certain way when added, multiplied, etc., (2) it offers drill material in new areas of mathematics, (3) it can develop interest - as mathematical enrichment, (4) it offers opportunities for students to…
Laboratory Mathematics. Curriculum Booklet 6 - Number Theory.
ERIC Educational Resources Information Center
Rogers, Sandra
This booklet is one of a set of five booklets which comprise the basic curriculum for "Mathematics Laboratories for Disadvantaged Students," a nationally validated Title III ESEA project. This publication provides evaluation materials and student materials related to number theory. Topics included in this booklet are prime and composite numbers,…
Cluster Algebras from Dualities of 2d = (2, 2) Quiver Gauge Theories
NASA Astrophysics Data System (ADS)
Benini, Francesco; Park, Daniel S.; Zhao, Peng
2015-11-01
We interpret certain Seiberg-like dualities of two-dimensional = (2,2) quiver gauge theories with unitary groups as cluster mutations in cluster algebras, originally formulated by Fomin and Zelevinsky. In particular, we show how the complexified Fayet-Iliopoulos parameters of the gauge group factors transform under those dualities and observe that they are in fact related to the dual cluster variables of cluster algebras. This implies that there is an underlying cluster algebra structure in the quantum Kähler moduli space of manifolds constructed from the corresponding Kähler quotients. We study the S 2 partition function of the gauge theories, showing that it is invariant under dualities/mutations, up to an overall normalization factor, whose physical origin and consequences we spell out in detail. We also present similar dualities in = (2,2)* quiver gauge theories, which are related to dualities of quantum integrable spin chains.
Classification of operator algebraic conformal field theories in dimensions one and two
NASA Astrophysics Data System (ADS)
Kawahigashi, Yasuyuki
2006-03-01
We formulate conformal field theory in the setting of algebraic quantum field theory as Haag-Kastler nets of local observable algebras with diffeomorphism covariance on the two-dimensional Minkowski space. We then obtain a decomposition of a two-dimensional theory into two chiral theories. We give the first classification result of such chiral theories with representation theoretic invariants. That is, we use the central charge as the first invariant, and if it is less than 1, we obtain a complete classification. Our classification list contains a new net which does not seem to arise from the known constructions such as the coset or orbifold constructions. We also present a classification of full two-dimensional conformal theories. These are joint works with Roberto Longo.
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…
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.
Regular perturbation theory of relativistic corrections: II. Algebraic approximation
NASA Astrophysics Data System (ADS)
Rutkowski, A.; Kozłowski, R.; Rutkowska, D.
2001-01-01
A four-component equivalent of the Schrödinger equation, describing both the nonrelativistic electron and the nonrelativistic positron, is introduced. The difference between this equation and the Dirac equation is treated as a perturbation. The relevant perturbation equations and formulas for corrections to the energy are derived. Owing to the semibounded character of the Schrödinger Hamiltonian of the unperturbed equation the variational perturbation method is formulated. The Hylleraas functionals become then either upper or lower bounds to the respective exact corrections to the energy. In order to demonstrate the usefulness of this approach to the problem of the variational optimization of nonlinear parameters, the perturbation corrections to wave functions for the of hydrogenlike atoms have been approximated in terms of exponential basis functions. The Dirac equation in this algebraic approximation is solved iteratively starting with the solution of the Schrödinger equation.
Towards a loop representation of connection theories defined over a super Lie algebra
Urrutia, L.F. |
1996-02-01
The purpose of this contribution is to review some aspects of the loop space formulation of pure gauge theories having the connection defined over a Lie algebra. The emphasis is focused on the discussion of the Mandelstam identities, which provide the basic constraints upon both the classical and the quantum degrees of freedom of the theory. In the case where the connection is extended to be valued on a super Lie algebra, some new results are presented which can be considered as first steps towards the construction of the Mandelstam identities in this situation, which encompasses such interesting cases as supergravity in 3+1 dimensions together with 2+1 super Chern-Simons theories, for example. Also, these ideas could be useful in the loop space formulation of fully supersymmetric theories. {copyright} {ital 1996 American Institute of Physics.}
Towards a loop representation of connection theories defined over a super Lie algebra
Urrutia, Luis F.
1996-02-20
The purpose of this contribution is to review some aspects of the loop space formulation of pure gauge theories having the connection defined over a Lie algebra. The emphasis is focused on the discussion of the Mandelstam identities, which provide the basic constraints upon both the classical and the quantum degrees of freedom of the theory. In the case where the connection is extended to be valued on a super Lie algebra, some new results are presented which can be considered as first steps towards the construction of the Mandelstam identities in this situation, which encompasses such interesting cases as supergravity in 3+1 dimensions together with 2+1 super Chern-Simons theories, for example. Also, these ideas could be useful in the loop space formulation of fully supersymmetric theories.
An algebraic PT-symmetric quantum theory with a maximal mass
NASA Astrophysics Data System (ADS)
Rodionov, V. N.; Kravtsova, G. A.
2016-03-01
In this paper, we draw attention to the fact that the studies by V.G. Kadyshevsky devoted to the creation of the geometric quantum field theory with a fundamental mass have had great development recently, as regards a non-Hermitian algebraic approach to construction of the quantum theory. The central idea of such theories is to construct a new scalar product in which the average values of non-Hermitian Hamiltonians are real. Many studies in this field include both purely mathematical ones and those containing the discussion of experimental results. We consider the development of an algebraic relativistic pseudo-Hermitian quantum theory with a maximal mass and discuss its experimentally important corollaries.
Small numbers in supersymmetric theories of nature
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
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.
K-Theory of Crossed Products of Tiling C*-Algebras by Rotation Groups
NASA Astrophysics Data System (ADS)
Starling, Charles
2015-02-01
Let Ω be a tiling space and let G be the maximal group of rotations which fixes Ω. Then the cohomology of Ω and Ω/ G are both invariants which give useful geometric information about the tilings in Ω. The noncommutative analog of the cohomology of Ω is the K-theory of a C*-algebra associated to Ω, and for translationally finite tilings of dimension 2 or less, the K-theory is isomorphic to the direct sum of cohomology groups. In this paper we give a prescription for calculating the noncommutative analog of the cohomology of Ω/ G, that is, the K-theory of the crossed product of the tiling C*-algebra by G. We also provide a table with some calculated K-groups for many common examples, including the Penrose and pinwheel tilings.
Algebraic Characterization of the Vacuum in Light-Front Field Theory
NASA Astrophysics Data System (ADS)
Herrmann, Marc; Polyzou, Wayne
2016-03-01
In the light-front formulation of quantum field theory, the vacuum vector of an interacting field theory has a relatively simple relationship to the vacuum of a free field theory. This is a benefit over the usual equal-time formulation where the interacting vacuum vector has infinite norm with respect to the Hilbert space of the free field theory. By describing the vacuum as a positive linear functional on an operator algebra constructed from free fields with two distinct masses, it can be demonstrated that the complications associated with adding dynamics to the vacuum of a free theory are not present in the construction of the light-front vacuum. Instead, the complications are moved into defining a subalgebra of the light-front algebra which corresponds to the physically relevant algebra of local fields. These results can then be applied to interacting fields by first describing them in terms of asymptotic in or out fields. However, in order to treat local operators products, the vacuum functional may need to be modified to include states with zero eigenvalue of the generator of translations in the direction along the light front, x- =1/√(2) >x0-x3. This work supported by DOE contract No. DE-FG02-86ER40286.
LieART-A Mathematica application for Lie algebras and representation theory
NASA Astrophysics Data System (ADS)
Feger, Robert; Kephart, Thomas W.
2015-07-01
We present the Mathematica application "LieART" (Lie Algebras and Representation Theory) for computations frequently encountered in Lie algebras and representation theory, such as tensor product decomposition and subalgebra branching of irreducible representations. LieART can handle all classical and exceptional Lie algebras. It computes root systems of Lie algebras, weight systems and several other properties of irreducible representations. LieART's user interface has been created with a strong focus on usability and thus allows the input of irreducible representations via their dimensional name, while the output is in the textbook style used in most particle-physics publications. The unique Dynkin labels of irreducible representations are used internally and can also be used for input and output. LieART exploits the Weyl reflection group for most of the calculations, resulting in fast computations and a low memory consumption. Extensive tables of properties, tensor products and branching rules of irreducible representations are included as online supplementary material (see Appendix A).
Entanglement distillation protocols and number theory
Bombin, H.; Martin-Delgado, M.A.
2005-09-15
We show that the analysis of entanglement distillation protocols for qudits of arbitrary dimension D benefits from applying basic concepts from number theory, since the set Z{sub D}{sup n} associated with Bell diagonal states is a module rather than a vector space. We find that a partition of Z{sub D}{sup n} into divisor classes characterizes the invariant properties of mixed Bell diagonal states under local permutations. We construct a very general class of recursion protocols by means of unitary operations implementing these local permutations. We study these distillation protocols depending on whether we use twirling operations in the intermediate steps or not, and we study them both analytically and numerically with Monte Carlo methods. In the absence of twirling operations, we construct extensions of the quantum privacy algorithms valid for secure communications with qudits of any dimension D. When D is a prime number, we show that distillation protocols are optimal both qualitatively and quantitatively.
Scale-adaptive tensor algebra for local many-body methods of electronic structure theory
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).
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.
On the Algebraic K Theory of the Massive D8 and M9-Branes
NASA Astrophysics Data System (ADS)
Vancea, Ion V.
In this paper we review some basic relations of algebraic K theory and we formulate them in the language of D-branes. Then we study the relation between the D8-branes wrapped on an orientable compact manifold W in a massive Type IIA supergravity background and the M9-branes wrapped on a compact manifold Z in a massive d=11 supergravity background from the K-theoretic point of view. By interpreting the D8-brane charges as elements of K0(C(W)) and the (inequivalent classes of) spaces of gauge fields on the M9-branes as the elements of K0(C(Z)x{¯ {k}*}G) where G is a one-dimensional compact group, a connection between charges and gauge fields is argued to exists. This connection could be realized as a composition map between the corresponding algebraic K theory groups.
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.
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…
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.
Extending the Number Line to Make Connections with Number Theory.
ERIC Educational Resources Information Center
Graviss, Tom; Greaver, Joanne
1992-01-01
Shares a coded version of the number line to provide concrete experiences for learning abstract concepts. Using the fundamental theorem of arithmetic, appropriate coded symbols are determined for the prime factorization of each natural number and used to study the concepts of greatest common divisor, least common multiple, square roots, and…
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.
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.
Modified large number theory with constant G
Recami, E.
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 R-bar/r-bar of the cosmos typical length R-bar to the hadron typical length r-bar is constant in time (for instance, if both cosmos and hadrons undergo an expansion/contraction cycle: according to the ''cyclical big-bang'' hypothesis: then R-bar and r-bar 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. Pavsic.
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.
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.
The Many Pearls of Number Theory
ERIC Educational Resources Information Center
Prielipp, Robert W.
1970-01-01
Presents information about prime numbers which the elementary or junior high school teacher might introduce to students. Gives five definition statements followed by at least one theorem and its proof. Introduces some historical aspects to prime numbers. (RR)
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…
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…
Algebra I, Package 03-03, Addition and Multiplication of Real Numbers.
ERIC Educational Resources Information Center
Thompson, Russ; Fuller, Albert
This teacher guide is part of the materials prepared for an individualized program for ninth-grade algebra and basic mathematics students. Materials written for the program are to be used with audiovisual lessons recorded on tape cassettes. For an evaluation of the program, see ED 086 545. In this guide, the teacher is provided with objectives for…
Figueroa-O'Farrill, Jose Miguel
2009-11-15
We phrase deformations of n-Leibniz algebras in terms of the cohomology theory of the associated Leibniz algebra. We do the same for n-Lie algebras and for the metric versions of n-Leibniz and n-Lie algebras. We place particular emphasis on the case of n=3 and explore the deformations of 3-algebras of relevance to three-dimensional superconformal Chern-Simons theories with matter.
Algebraic vs physical N = 6 3-algebras
Cantarini, Nicoletta; Kac, Victor G.
2014-01-15
In our previous paper, we classified linearly compact algebraic simple N = 6 3-algebras. In the present paper, we classify their “physical” counterparts, which actually appear in the N = 6 supersymmetric 3-dimensional Chern-Simons theories.
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.…
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.
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…
The algebras of large N matrix mechanics
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.
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).
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…
NASA Astrophysics Data System (ADS)
Putnam, Ian F.
2010-03-01
We investigate the C*-algebras associated to aperiodic structures called model sets obtained by the cut-and-project method. These C*-algebras are Morita equivalent to crossed product C*-algebras obtained from dynamics on a disconnected version of the internal space. This construction may be made from more general data, which we call a hyperplane system. From a hyperplane system, others may be constructed by a process of reduction and we show how the C*-algebras involved are related to each other. In particular, there are natural elements in the Kasparov KK-groups for the C*-algebra of a hyperplane system and that of its reduction. The induced map on K-theory fits in a six-term exact sequence. This provides a new method of the computation of the K-theory of such C*-algebras which is done completely in the setting of non-commutative geometry.
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…
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…
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…
Fermi-Dirac statistics and the number theory
NASA Astrophysics Data System (ADS)
Kubasiak, Anna; Korbicz, Jaroslaw K.; Zakrzewski, Jakub; Lewenstein, Maciej
2005-11-01
We relate the Fermi-Dirac statistics of an ideal Fermi gas in a harmonic trap to partitions of given integers into distinct parts, studied in number theory. Using methods of quantum statistical physics we derive analytic expressions for cumulants of the probability distribution of the number of different partitions.
Algebraic K-theory of spaces stratified fibered over hyperbolic orbifolds.
Farrell, F T; Jones, L E
1986-08-01
Among other results, we rationally calculate the algebraic K-theory of any discrete cocompact subgroup of a Lie group G, where G is either O(n, 1), U(n, 1), Sp(n, 1), or F(4), in terms of the homology of the double coset space Gamma\\G/K, where K is a maximal cocompact subgroup of G. We obtain the formula K(n)(ZGamma) [unk] [unk] congruent with [unk](i=0) (infinity)H(i)(Gamma\\G/K; [unk](n-i)), where [unk](j) is a stratified system of Q vector spaces over Gamma\\G/K and the vector space [unk](j)(GammagK) corresponding to the double coset GammagK is isomorphic to K(J)(Z(Gamma [unk] gKg(-1))) [unk] Q. Note Gamma [unk] gKg(-1) is a finite subgroup of Gamma. Earlier, a similar formula for discrete cocompact subgroups Gamma of the group of rigid motions of Euclidean space was conjectured by F. T. Farrell and W. C. Hsiang and proven by F. Quinn. PMID:16593733
Algebraic K-theory of spaces stratified fibered over hyperbolic orbifolds
Farrell, F. T.; Jones, L. E.
1986-01-01
Among other results, we rationally calculate the algebraic K-theory of any discrete cocompact subgroup of a Lie group G, where G is either O(n, 1), U(n, 1), Sp(n, 1), or F4, in terms of the homology of the double coset space Γ\\G/K, where K is a maximal cocompact subgroup of G. We obtain the formula Kn(ZΓ) [unk] [unk] ≅ [unk]i=0∞Hi(Γ\\G/K; [unk]n-i), where [unk]j is a stratified system of Q vector spaces over Γ\\G/K and the vector space [unk]j(ΓgK) corresponding to the double coset ΓgK is isomorphic to KJ(Z(Γ [unk] gKg-1)) [unk] Q. Note Γ [unk] gKg-1 is a finite subgroup of Γ. Earlier, a similar formula for discrete cocompact subgroups Γ of the group of rigid motions of Euclidean space was conjectured by F. T. Farrell and W. C. Hsiang and proven by F. Quinn. PMID:16593733
Hard-core lattice bosons: new insights from algebraic graph theory
NASA Astrophysics Data System (ADS)
Squires, Randall W.; Feder, David L.
2014-03-01
Determining the characteristics of hard-core lattice bosons is a problem of long-standing interest in condensed matter physics. While in one-dimensional systems the ground state can be formally obtained via a mapping to free fermions, various properties (such as correlation functions) are often difficult to calculate. In this work we discuss the application of techniques from algebraic graph theory to hard-core lattice bosons in one dimension. Graphs are natural representations of many-body Hamiltonians, with vertices representing Fock basis states and edges representing matrix elements. We prove that the graphs for hard-core bosons and non-interacting bosons have identical connectivity; the only difference is the existence of edge weights. A formal mapping between the two is therefore possible by manipulating the graph incidence matrices. We explore the implications of these insights, in particular the intriguing possibility that ground-state properties of hard-core bosons can be calculated directly from those of non-interacting bosons.
S-duality and the prepotential of N={2}^{star } theories (II): the non-simply laced algebras
NASA Astrophysics Data System (ADS)
Billó, M.; Frau, M.; Fucito, F.; Lerda, A.; Morales, J. F.
2015-11-01
We derive a modular anomaly equation satisfied by the prepotential of the N={2}^{star } supersymmetric theories with non-simply laced gauge algebras, including the classical B r and C r infinite series and the exceptional F 4 and G 2 cases. This equation determines the exact prepotential recursively in an expansion for small mass in terms of quasi-modular forms of the S-duality group. We also discuss the behaviour of these theories under S-duality and show that the prepotential of the SO(2 r + 1) theory is mapped to that of the Sp(2 r) theory and viceversa, while the exceptional F 4 and G 2 theories are mapped into themselves (up to a rotation of the roots) in analogy with what happens for the N=4 supersymmetric theories. These results extend the analysis for the simply laced groups presented in a companion paper.
Stochastic-Conceptual Models Applied to Number Theory
NASA Astrophysics Data System (ADS)
Sotiropoulos, Megaklis Th.
2011-09-01
Concepts are defined as couples (O, A) of sets O and A: the object O (a set of none, or one or more elements) is assigned to the set A of these elements' (common) attributes. The objects change according to the sequence of attributes. Only couples of objects and attributes, that is concepts, are adequate for our world. The connections and links we need in databases and multimedia are expressed, naturally, by concepts, since concepts have been proved to dispose the order of a lattice (more complex and rich than linear and hierarchical ones). The lattice can be created by two algebraic operations: "intersection" as the multiplication and "symmetric-difference (!)" as the addition (!). There are, also, two other operations: the "union" and the "complement of a concept". Intersection and union (which cannot play the role neither of the addition nor of the multiplication) express similarities, while the other two operations express dissimilarities. The operation "complement of a concept" expresses the different, the uncommon, the variety. The symmetric-difference of two concepts has been proved to be a "distance" between them (in the mathematical sense!). We must not always see the natural numbers with their linear order (1,2,…,n,n+1,…), but is rather better to give them a more complex structure: the structure of a lattice. We need three operations: "union" of two numbers, "intersection" of two numbers and "complement" of a number. Research conclusion: the conceptual distance of (O1, A1) and (O2,A2) and is always a prime number! (conceptual means, from the point of view of the characteristic "divisibility" we are examining now and not the Euclidean or any other distance). This is the unique way the prime numbers are generated: not by unions and intersections(which express similarities), but by distances(differences)! …
Orientation in operator algebras
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
Twisted vertex algebras, bicharacter construction and boson-fermion correspondences
Anguelova, Iana I.
2013-12-15
The boson-fermion correspondences are an important phenomena on the intersection of several areas in mathematical physics: representation theory, vertex algebras and conformal field theory, integrable systems, number theory, cohomology. Two such correspondences are well known: the types A and B (and their super extensions). As a main result of this paper we present a new boson-fermion correspondence of type D-A. Further, we define a new concept of twisted vertex algebra of order N, which generalizes super vertex algebra. We develop the bicharacter construction which we use for constructing classes of examples of twisted vertex algebras, as well as for deriving formulas for the operator product expansions, analytic continuations, and normal ordered products. By using the underlying Hopf algebra structure we prove general bicharacter formulas for the vacuum expectation values for two important groups of examples. We show that the correspondences of types B, C, and D-A are isomorphisms of twisted vertex algebras.
NASA Astrophysics Data System (ADS)
Khudaverdian, H. M.
2014-03-01
We consider differential operators acting on densities of arbitrary weights on manifold M identifying pencils of such operators with operators on algebra of densities of all weights. This algebra can be identified with the special subalgebra of functions on extended manifold . On one hand there is a canonical lift of projective structures on M to affine structures on extended manifold . On the other hand the restriction of algebra of all functions on extended manifold to this special subalgebra of functions implies the canonical scalar product. This leads in particular to classification of second order operators with use of Kaluza-Klein-like mechanisms.
Prime Numbers, Quantum Field Theory and the Goldbach Conjecture
NASA Astrophysics Data System (ADS)
Sanchis-Lozano, Miguel-Angel; Barbero G., J. Fernando; Navarro-Salas, José
2012-09-01
Motivated by the Goldbach conjecture in number theory and the Abelian bosonization mechanism on a cylindrical two-dimensional space-time, we study the reconstruction of a real scalar field as a product of two real fermion (so-called prime) fields whose Fourier expansion exclusively contains prime modes. We undertake the canonical quantization of such prime fields and construct the corresponding Fock space by introducing creation operators bp\\dag — labeled by prime numbers p — acting on the vacuum. The analysis of our model, based on the standard rules of quantum field theory and the assumption of the Riemann hypothesis, allows us to prove that the theory is not renormalizable. We also comment on the potential consequences of this result concerning the validity or breakdown of the Goldbach conjecture for large integer numbers.
NASA Astrophysics Data System (ADS)
Vladimirov, Vasilii S.
1996-02-01
On the basis of analysis on the adele group (the Tate formula) of an algebraic number field, a regularization is constructed for the divergent adelic products of the gamma and beta functions of all (non-isomorphic) completions of this field. The formulae obtained are applied to representations of the four-point crossing-symmetric Veneziano amplitudes and Virasoro-Shapiro amplitudes in terms of regularized adelic products of string amplitudes (for open or closed strings) corresponding to all non-Archimedean completions of the algebraic number field under consideration.
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.
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.
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.
Learning Activity Package, Pre-Algebra.
ERIC Educational Resources Information Center
Evans, Diane
A set of ten teacher-prepared Learning Activity Packages (LAPs) for individualized instruction in topics in pre-algebra, the units cover the decimal numeration system; number theory; fractions and decimals; ratio, proportion, and percent; sets; properties of operations; rational numbers; real numbers; open expressions; and open rational…
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…
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.
Quantum cluster algebras and quantum nilpotent algebras
Goodearl, Kenneth R.; Yakimov, Milen T.
2014-01-01
A major direction in the theory of cluster algebras is to construct (quantum) cluster algebra structures on the (quantized) coordinate rings of various families of varieties arising in Lie theory. We prove that all algebras in a very large axiomatically defined class of noncommutative algebras possess canonical quantum cluster algebra structures. Furthermore, they coincide with the corresponding upper quantum cluster algebras. We also establish analogs of these results for a large class of Poisson nilpotent algebras. Many important families of coordinate rings are subsumed in the class we are covering, which leads to a broad range of applications of the general results to the above-mentioned types of problems. As a consequence, we prove the Berenstein–Zelevinsky conjecture [Berenstein A, Zelevinsky A (2005) Adv Math 195:405–455] for the quantized coordinate rings of double Bruhat cells and construct quantum cluster algebra structures on all quantum unipotent groups, extending the theorem of Geiß et al. [Geiß C, et al. (2013) Selecta Math 19:337–397] for the case of symmetric Kac–Moody groups. Moreover, we prove that the upper cluster algebras of Berenstein et al. [Berenstein A, et al. (2005) Duke Math J 126:1–52] associated with double Bruhat cells coincide with the corresponding cluster algebras. PMID:24982197
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…
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,…
Realizations of Galilei algebras
NASA Astrophysics Data System (ADS)
Nesterenko, Maryna; Pošta, Severin; Vaneeva, Olena
2016-03-01
All inequivalent realizations of the Galilei algebras of dimensions not greater than five are constructed using the algebraic approach proposed by Shirokov. The varieties of the deformed Galilei algebras are discussed and families of one-parametric deformations are presented in explicit form. It is also shown that a number of well-known and physically interesting equations and systems are invariant with respect to the considered Galilei algebras or their deformations.
Lepton number violation in theories with a large number of standard model copies
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.
The Algebra of Lexical Semantics
NASA Astrophysics Data System (ADS)
Kornai, András
The current generative theory of the lexicon relies primarily on tools from formal language theory and mathematical logic. Here we describe how a different formal apparatus, taken from algebra and automata theory, resolves many of the known problems with the generative lexicon. We develop a finite state theory of word meaning based on machines in the sense of Eilenberg [11], a formalism capable of describing discrepancies between syntactic type (lexical category) and semantic type (number of arguments). This mechanism is compared both to the standard linguistic approaches and to the formalisms developed in AI/KR.
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.
Pseudorational Impulse Responses — Algebraic System Theory for Distributed Parameter Systems
NASA Astrophysics Data System (ADS)
Yamamoto, Yutaka
This paper gives a comprehensive account on a class of distributed parameter systems, whose impulse response is called pseudorational. This notion was introduced by the author in 1980's, and is particularly amenable for the study of systems with bounded-time memory. We emphasize algebraic structures induced by this class of systems. Some recent results on coprimeness issues and H∞ control are discussed and illustrated.
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…
Transportation Optimization with Fuzzy Trapezoidal Numbers Based on Possibility Theory
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. PMID:25137239
Number theory, periodic orbits, and superconductivity in nanocubes
NASA Astrophysics Data System (ADS)
Mayoh, James; García-García, Antonio M.
2014-07-01
We study superconductivity in isolated superconducting nanocubes and nanosquares of size L in the limit of negligible disorder δ /Δ0≪1 and kFL≫1 for which mean-field theory and semiclassical techniques are applicable, with kF the Fermi wave vector, δ the mean level spacing, and Δ0 the bulk gap. By using periodic orbit theory and number theory we find explicit analytical expressions for the size dependence of the superconducting order parameter. Our formalism takes into account contributions from both the spectral density and the interaction matrix elements in a basis of one-body eigenstates. The leading size dependence of the energy gap in three dimensions seems to be universal as it agrees with the result for chaotic grains. In the region of parameters corresponding to conventional metallic superconductors, and for sizes L ≳10 nm, the contribution to the superconducting gap from the matrix elements is substantial (˜20%). Deviations from the bulk limit are still clearly observed even for comparatively large grains L ˜50 nm. These analytical results are in excellent agreement with the numerical solution of the mean-field gap equation.
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.
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.
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.
A decoding problem in dynamics and in number theory.
Siegel, Ralph M.; Tresser, Charles; Zettler, George
1992-10-01
Given a homeomorphism f of the circle, any splitting of this circle in two semiopen arcs induces a coding process for the orbits of f, which can be determined by recording the successive arcs visited by the orbit. The problem of describing these codes has a two hundred year history (that we briefly recall) in the particular case when the arcs are limited by a point and its image; in modern language, it is the kneading theory of such maps, and as such is relevant for our understanding of dynamical problems involving oscillations. This paper deals with questions attached to the general case, a problem considered by many mathematicians in the 50's and 60's in the case where f is a rotation, and which has recently found some applications in physiology. We show that, except for trivial cases, any code determines the rotation number, up to the orientation, of the homeomorphism which generates it. In the case the code is periodic, we can also determine whether or not it can be generated in this way. An equivalent problem in arithmetic consists of finding +/-p, knowing a collection of classes in Z/qZ of the form {m,m+p,.,m+(k-1)p}, where 2number theoretic context. PMID:12779997
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)
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.
Thermodynamic limit in number theory: Riemann-Beurling gases
NASA Astrophysics Data System (ADS)
Julia, B. L.
1994-03-01
We study the grand canonical version of a solved statistical model, the Riemann gas: a collection of bosonic oscillators with energies the logarithms of the prime numbers. The introduction of a chemical potential μ amounts to multiply each prime by e -μ, the corresponding gases could be called Beurling gases because they are defined by the choice of appropriate generalized primes when considered as canonical ensembles; one finds generalized Hagedorn singularities in the temperature. The discrete spectrum can be treated as continuous in its high energy region; this approximation allows us to study the high energy level density and is applied to Beurling gases. It is expected to be accurate for the high temperature behaviour. One model (the logarithmic gases) will be studied in more detail, it corresponds to the choice of all the integers strictly larger than one as Beurling primes; we give an explicit formula for its grand canonical thermodynamic potential F - μ N in terms of a hypergeometric function and check the approximation on the Hagedorn phenomenon. Related physical situations include string theories and quark deconfinement where one needs a better understanding of the nature of the Hagedorn transitions.
Quasifinite highest weight modules over the Lie algebra of differential operators on the circle
NASA Astrophysics Data System (ADS)
Kac, Victor; Radul, Andrey
1993-11-01
We classify positive energy representations with finite degeneracies of the Lie algebra W 1+∞ and construct them in terms of representation theory of the Lie algebrahat gl(infty ,R_m ) of infinites matrices with finite number of non-zero diagonals over the algebra R m =ℂ[ t]/( t m+1). The unitary ones are classified as well. Similar results are obtained for the sin-algebras.
Generator algebra of the asymptotic Poincare group in the general theory of relativity
Solovev, V.O.
1986-06-01
This paper obtains the Poisson brackets of the generators of the Hamiltonian formalism for general relativity with allowance for surface terms of aritrary form. For Minkowski space, there exists the asymptotic Poincare group, which is the semi-direct product of the Poincare group and an infinite subgroup for which the algebra of generators with surface terms closes. A criterion invariant with respect to the choice of the coordinate system on the hypersurfaces is obtained for realization of the Poincare group in asymptotically flat space-time. The ''background'' flat metric on the hypersurfaces and Poincare group that preserve it are determined nonuniquely; however, the numerical values of the generators do not depend on the freedom of this choice on solutions of the constraint equations. For an asymptotically Galilean metric, the widely used boundary cnoditins are determined more accurately. A prescription is given for application of the Arnowitt-Deser-Misner decomposition in the case of a slowly decreasing contribution from coordinate and time transformations.
Constraint algebra in bigravity
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.
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
Refactorable Numbers - A Machine Invention
NASA Astrophysics Data System (ADS)
Colton, Simon
1999-02-01
The HR (or Hardy-Ramanujan) program invents and analyses definitions in areas of pure mathematics, including finite algebras, graph theory and number theory. While working in number theory, HR recently invented a new integer sequence, the refactorable numbers, which are defined and developed here. A discussion of how HR works, along with details of well known sequences reinvented by HR and other new sequences invented by HR is also given.
C∗-algebras of Penrose hyperbolic tilings
NASA Astrophysics Data System (ADS)
Oyono-Oyono, Hervé; Petite, Samuel
2011-02-01
Penrose hyperbolic tilings are tilings of the hyperbolic plane which admit, up to affine transformations a finite number of prototiles. In this paper, we give a complete description of the C∗-algebras and of the K-theory for such tilings. Since the continuous hull of these tilings have no transversally invariant measure, these C∗-algebras are traceless. Nevertheless, harmonic currents give rise to 3-cyclic cocycles and we discuss in this setting a higher-order version of the gap-labeling.
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…
Microcomputer-Assisted Mathematics: Exploring Number Theory with a Microcomputer.
ERIC Educational Resources Information Center
Fischer, Frederic
1986-01-01
Presents three problems, each with a computer solution in BASIC programming language: (1) finding sum equals product for three-digit numbers; (2) finding cases of improper cancellation that give a correct answer; and (3) finding perfect numbers. Some suggestions for further investigation are included. (JN)
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…
Higher-order M-theory corrections and the Kac Moody algebra E10
NASA Astrophysics Data System (ADS)
Damour, Thibault; Nicolai, Hermann
2005-07-01
It has been conjectured that the classical dynamics of M-theory is equivalent to a null geodesic motion in the infinite-dimensional coset space E10/K(E10), where K(E10) is the maximal compact subgroup of the hyperbolic Kac Moody group E10. We here provide further evidence for this conjecture by showing that the leading higher-order corrections, quartic in the curvature and related 3-form-dependent terms, correspond to negative imaginary roots of E10. The conjecture entails certain predictions for which higher-order corrections are allowed: in particular corrections of type RM(DF)N are compatible with E10 only for M + N = 3k + 1. Furthermore, the leading parts of the R4, R7, ... terms are predicted to be associated with singlets under the {\\mathfrak{sl}}_{10} decomposition of E10. Although singlets are extremely rare among the 4400 752 653 representations of {\\mathfrak{sl}}_{10} appearing in E10 up to level ell <= 28, there are indeed singlets at levels ell = 10 and ell = 20 which do match with the R4 and the expected R7 corrections. Our analysis indicates a far more complicated behaviour of the theory near the cosmological singularity than suggested by the standard homogeneous ansätze.
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.
3D Winding Number: Theory and Application to Medical Imaging
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
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…
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.
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.
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.
Number crunching vs. number theory: computers and FLT, from Kummer to SWAC (1850-1960), and beyond
NASA Astrophysics Data System (ADS)
Corry, Leo
2008-07-01
The article discusses the computational tools (both conceptual and material) used in various attempts to deal with individual cases of FLT [Fermat's Last Theorem], as well as the changing historical contexts in which these tools were developed and used, and affected research. It also explores the changing conceptions about the role of computations within the overall disciplinary picture of number theory, how they influenced research on the theorem, and the kinds of general insights thus achieved. After an overview of Kummer's contributions and its immediate influence, the author presents work that favored intensive computations of particular cases of FLT as a legitimate, fruitful, and worth-pursuing number-theoretical endeavor, and that were part of a coherent and active, but essentially low-profile tradition within nineteenth century number theory. This work was related to table making activity that was encouraged by institutions and individuals whose motivations came mainly from applied mathematics, astronomy, and engineering, and seldom from number theory proper. A main section of the article is devoted to the fruitful collaboration between Harry S. Vandiver and Emma and Dick Lehmer. The author shows how their early work led to the hesitant introduction of electronic computers for research related with FLT. Their joint work became a milestone for computer-assisted activity in number theory at large.
The Weyl realizations of Lie algebras, and left-right duality
NASA Astrophysics Data System (ADS)
Meljanac, Stjepan; Krešić-Jurić, Saša; Martinić, Tea
2016-05-01
We investigate dual realizations of non-commutative spaces of Lie algebra type in terms of formal power series in the Weyl algebra. To each realization of a Lie algebra 𝔤 we associate a star-product on the symmetric algebra S(𝔤) and an ordering on the enveloping algebra U(𝔤). Dual realizations of 𝔤 are defined in terms of left-right duality of the star-products on S(𝔤). It is shown that the dual realizations are related to an extension problem for 𝔤 by shift operators whose action on U(𝔤) describes left and right shift of the generators of U(𝔤) in a given monomial. Using properties of the extended algebra, in the Weyl symmetric ordering we derive closed form expressions for the dual realizations of 𝔤 in terms of two generating functions for the Bernoulli numbers. The theory is illustrated by considering the κ-deformed space.
Quantitative K-Theory Related to Spin Chern Numbers
NASA Astrophysics Data System (ADS)
Loring, Terry A.
2014-07-01
We examine the various indices defined on pairs of almost commuting unitary matrices that can detect pairs that are far from commuting pairs. We do this in two symmetry classes, that of general unitary matrices and that of self-dual matrices, with an emphasis on quantitative results. We determine which values of the norm of the commutator guarantee that the indices are defined, where they are equal, and what quantitative results on the distance to a pair with a different index are possible. We validate a method of computing spin Chern numbers that was developed with Hastings and only conjectured to be correct. Specifically, the Pfaffian-Bott index can be computed by the ''log method'' for commutator norms up to a specific constant.
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.
Permutation centralizer algebras and multimatrix invariants
NASA Astrophysics Data System (ADS)
Mattioli, Paolo; Ramgoolam, Sanjaye
2016-03-01
We introduce a class of permutation centralizer algebras which underly the combinatorics of multimatrix gauge-invariant observables. One family of such noncommutative algebras is parametrized by two integers. Its Wedderburn-Artin decomposition explains the counting of restricted Schur operators, which were introduced in the physics literature to describe open strings attached to giant gravitons and were subsequently used to diagonalize the Gaussian inner product for gauge invariants of two-matrix models. The structure of the algebra, notably its dimension, its center and its maximally commuting subalgebra, is related to Littlewood-Richardson numbers for composing Young diagrams. It gives a precise characterization of the minimal set of charges needed to distinguish arbitrary matrix gauge invariants, which are related to enhanced symmetries in gauge theory. The algebra also gives a star product for matrix invariants. The center of the algebra allows efficient computation of a sector of multimatrix correlators. These generate the counting of a certain class of bicoloured ribbon graphs with arbitrary genus.
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.…
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
Béguinot, Jean
2014-01-01
Anne Chao proposed a very popular, nonparametric estimator of the species richness of a community, on the basis of a limited size sampling of this community. This expression was originally derived on a statistical basis as a lower-bound estimate of the number of missing species in the sample and provides accordingly a minimal threshold for the estimation of the total species richness of the community. Hereafter, we propose an alternative, algebraic derivation of Chao's estimator, demonstrating thereby that Chao's formulation may also provide centered estimates (and not only a lower bound threshold), provided that the sampled communities satisfy a specific type of SAD (species abundance distribution). This particular SAD corresponds to the case when the number of unrecorded species in the sample tends to decrease exponentially with increasing sampling size. It turns out that the shape of this “ideal” SAD often conforms approximately to the usually recorded types in nature, such as “log-normal” or “broken-stick.”. Accordingly, this may explain why Chao's formulation is generally recognized as a particularly satisfying nonparametric estimator.
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…
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%…
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.
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…
Secondary School Mathematics, Chapter 5, Number Theory, Chapter 6, The Integers. Student's Text.
ERIC Educational Resources Information Center
Stanford Univ., CA. School Mathematics Study Group.
The third student text in this SMSG series of 14 covers the following topics from number theory: the division algorithm, divisibility, prime numbers, prime factorization, common divisors and common multiples, and properties of the whole number system. A second chapter discusses properties and operations with integers. For a special edition of this…
ERIC Educational Resources Information Center
Boiteau, Denise; Stansfield, David
This document describes mathematical programs on the basic concepts of algebra produced by Louisiana Public Broadcasting. Programs included are: (1) "Inverse Operations"; (2) "The Order of Operations"; (3) "Basic Properties" (addition and multiplication of numbers and variables); (4) "The Positive and Negative Numbers"; and (5) "Using Positive…
ERIC Educational Resources Information Center
Kennedy, John
This text provides information and exercises on arithmetic topics which should be mastered before a student enrolls in an Elementary Algebra course. Section I describes the fundamental properties and relationships of whole numbers, focusing on basic operations, divisibility tests, exponents, order of operations, prime numbers, greatest common…
Vertex Algebras, Kac-Moody Algebras, and the Monster
NASA Astrophysics Data System (ADS)
Borcherds, Richard E.
1986-05-01
It is known that the adjoint representation of any Kac-Moody algebra A can be identified with a subquotient of a certain Fock space representation constructed from the root lattice of A. I define a product on the whole of the Fock space that restricts to the Lie algebra product on this subquotient. This product (together with a infinite number of other products) is constructed using a generalization of vertex operators. I also construct an integral form for the universal enveloping algebra of any Kac-Moody algebra that can be used to define Kac-Moody groups over finite fields, some new irreducible integrable representations, and a sort of affinization of any Kac-Moody algebra. The ``Moonshine'' representation of the Monster constructed by Frenkel and others also has products like the ones constructed for Kac-Moody algebras, one of which extends the Griess product on the 196884-dimensional piece to the whole representation.
Semigroups and computer algebra in algebraic structures
NASA Astrophysics Data System (ADS)
Bijev, G.
2012-11-01
Some concepts in semigroup theory can be interpreted in several algebraic structures. A generalization fA,B,fA,B(X) = A(X')B of the complement operator (') on Boolean matrices is made, where A and B denote any rectangular Boolean matrices. While (') is an isomorphism between Boolean semilattices, the generalized complement operator is homomorphism in the general case. The map fA,B and its general inverse (fA,B)+ have quite similar properties to those in the linear algebra and are useful for solving linear equations in Boolean matrix algebras. For binary relations on a finite set, necessary and sufficient conditions for the equation αξβ = γ to have a solution ξ are proved. A generalization of Green's equivalence relations in semigroups for rectangular matrices is proposed. Relationships between them and the Moore-Penrose inverses are investigated. It is shown how any generalized Green's H-class could be constructed by given its corresponding linear subspaces and converted into a group isomorphic to a linear group. Some information about using computer algebra methods concerning this paper is given.
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!!!
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.
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.
Realizations of conformal current-type Lie algebras
Pei Yufeng; Bai Chengming
2010-05-15
In this paper we obtain the realizations of some infinite-dimensional Lie algebras, named 'conformal current-type Lie algebras', in terms of a two-dimensional Novikov algebra and its deformations. Furthermore, Ovsienko and Roger's loop cotangent Virasoro algebra, which can be regarded as a nice generalization of the Virasoro algebra with two space variables, is naturally realized as an affinization of the tensor product of a deformation of the two-dimensional Novikov algebra and the Laurent polynomial algebra. These realizations shed new light on various aspects of the structure and representation theory of the corresponding infinite-dimensional Lie algebras.
Quantum computation using geometric algebra
NASA Astrophysics Data System (ADS)
Matzke, Douglas James
This dissertation reports that arbitrary Boolean logic equations and operators can be represented in geometric algebra as linear equations composed entirely of orthonormal vectors using only addition and multiplication Geometric algebra is a topologically based algebraic system that naturally incorporates the inner and anticommutative outer products into a real valued geometric product, yet does not rely on complex numbers or matrices. A series of custom tools was designed and built to simplify geometric algebra expressions into a standard sum of products form, and automate the anticommutative geometric product and operations. Using this infrastructure, quantum bits (qubits), quantum registers and EPR-bits (ebits) are expressed symmetrically as geometric algebra expressions. Many known quantum computing gates, measurement operators, and especially the Bell/magic operators are also expressed as geometric products. These results demonstrate that geometric algebra can naturally and faithfully represent the central concepts, objects, and operators necessary for quantum computing, and can facilitate the design and construction of quantum computing tools.
Higher level twisted Zhu algebras
Ekeren, Jethro van
2011-05-15
The study of twisted representations of graded vertex algebras is important for understanding orbifold models in conformal field theory. In this paper, we consider the general setup of a vertex algebra V, graded by {Gamma}/Z for some subgroup {Gamma} of R containing Z, and with a Hamiltonian operator H having real (but not necessarily integer) eigenvalues. We construct the directed system of twisted level p Zhu algebras Zhu{sub p,{Gamma}}(V), and we prove the following theorems: For each p, there is a bijection between the irreducible Zhu{sub p,{Gamma}}(V)-modules and the irreducible {Gamma}-twisted positive energy V-modules, and V is ({Gamma}, H)-rational if and only if all its Zhu algebras Zhu{sub p,{Gamma}}(V) are finite dimensional and semisimple. The main novelty is the removal of the assumption of integer eigenvalues for H. We provide an explicit description of the level p Zhu algebras of a universal enveloping vertex algebra, in particular of the Virasoro vertex algebra Vir{sup c} and the universal affine Kac-Moody vertex algebra V{sup k}(g) at non-critical level. We also compute the inverse limits of these directed systems of algebras.
Dual numbers and supersymmetric mechanics
NASA Astrophysics Data System (ADS)
Frydryszak, Andrzej M.
2005-11-01
We show that dual numbers, apart from the known practical applications to the description of a rigid body movements in three dimensional space and natural presence in abstract differential algebra, play a role in field theory and are related to supersymmetry as well. Relevant models are considered.
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…
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…
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…
On N = 2 compactifications of M-theory to AdS{sub 3} using geometric algebra techniques
Babalic, E. M.; Coman, I. A.; Condeescu, C.; Micu, A.; Lazaroiu, C. I.
2013-11-13
We investigate the most general warped compactification of eleven-dimensional supergravity on eight-dimensional manifolds to AdS{sub 3} spaces (in the presence of non-vanishing four-form flux) which preserves N = 2 supersymmetry in three dimensions. Without imposing any restrictions on the chirality of the internal part of the supersymmetry generators, we use geometric algebra techniques to study some implications of the supersymmetry constraints. In particular, we discuss the Lie bracket of certain vector fields constructed as pinor bilinears on the compactification manifold.
Filiform Lie algebras of order 3
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.
Invariants of triangular Lie algebras
NASA Astrophysics Data System (ADS)
Boyko, Vyacheslav; Patera, Jiri; Popovych, Roman
2007-07-01
Triangular Lie algebras are the Lie algebras which can be faithfully represented by triangular matrices of any finite size over the real/complex number field. In the paper invariants ('generalized Casimir operators') are found for three classes of Lie algebras, namely those which are either strictly or non-strictly triangular, and for so-called special upper triangular Lie algebras. Algebraic algorithm of Boyko et al (2006 J. Phys. A: Math. Gen.39 5749 (Preprint math-ph/0602046)), developed further in Boyko et al (2007 J. Phys. A: Math. Theor.40 113 (Preprint math-ph/0606045)), is used to determine the invariants. A conjecture of Tremblay and Winternitz (2001 J. Phys. A: Math. Gen.34 9085), concerning the number of independent invariants and their form, is corroborated.
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.
ERIC Educational Resources Information Center
Ketterlin-Geller, Leanne R.; Jungjohann, Kathleen; Chard, David J.; Baker, Scott
2007-01-01
Much of the difficulty that students encounter in the transition from arithmetic to algebra stems from their early learning and understanding of arithmetic. Too often, students learn about the whole number system and the operations that govern that system as a set of procedures to solve addition, subtraction, multiplication, and division problems.…
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.
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.
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.
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!!!
Rank of quantized universal enveloping algebras and modular functions
NASA Astrophysics Data System (ADS)
Majid, Shahn; Soibelman, Ya. S.
1991-04-01
We compute an intrinsic rank invariant for quasitriangular Hopf algebras in the case of general quantum groups U q (g). As a function of q the rank has remarkable number theoretic properties connected with modular covariance and Galois theory. A number of examples are treated in detail, including rank ( U q (su(3))) and rank ( U q ( e 8)). We briefly indicate a physical interpretation as relating Chern-Simons theory with the theory of a quantum particle confined to an alcove of g.
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
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.
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…
Application of abelian holonomy formalism to the elementary theory of numbers
NASA Astrophysics Data System (ADS)
Abe, Yasuhiro
2012-05-01
We consider an abelian holonomy operator in two-dimensional conformal field theory with zero-mode contributions. The analysis is made possible by use of a geometric-quantization scheme for abelian Chern-Simons theory on S1 × S1 × R. We find that a purely zero-mode part of the holonomy operator can be expressed in terms of Riemann's zeta function. We also show that a generalization of linking numbers can be obtained in terms of the vacuum expectation values of the zero-mode holonomy operators. Inspired by mathematical analogies between linking numbers and Legendre symbols, we then apply these results to a space of Fp = Z/pZ, where p is an odd prime number. This enables us to calculate "scattering amplitudes" of identical odd primes in the holonomy formalism. In this framework, the Riemann hypothesis can be interpreted by means of a physically obvious fact, i.e., there is no notion of "scattering" for a single-particle system. Abelian gauge theories described by the zero-mode holonomy operators will be useful for studies on quantum aspects of topology and number theory.
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…
ERIC Educational Resources Information Center
Campbell, Stephen R.
This discussion focuses upon potential implications of research conducted on preservice teachers' understanding of introductory topics from elementary number theory. Comments interlace three levels of consideration and recapitulate what is striking as some of the most interesting and important findings raised in the presentations, and flesh out…
NASA Astrophysics Data System (ADS)
Lannes, A.; Teunissen, P. J. G.
2011-05-01
The first objective of this paper is to show that some basic concepts used in global navigation satellite systems (GNSS) are similar to those introduced in Fourier synthesis for handling some phase calibration problems. In experimental astronomy, the latter are at the heart of what is called `phase closure imaging.' In both cases, the analysis of the related structures appeals to the algebraic graph theory and the algebraic number theory. For example, the estimable functions of carrier-phase ambiguities, which were introduced in GNSS to correct some rank defects of the undifferenced equations, prove to be `closure-phase ambiguities:' the so-called `closure-delay' (CD) ambiguities. The notion of closure delay thus generalizes that of double difference (DD). The other estimable functional variables involved in the phase and code undifferenced equations are the receiver and satellite pseudo-clock biases. A related application, which corresponds to the second objective of this paper, concerns the definition of the clock information to be broadcasted to the network users for their precise point positioning (PPP). It is shown that this positioning can be achieved by simply having access to the satellite pseudo-clock biases. For simplicity, the study is restricted to relatively small networks. Concerning the phase for example, these biases then include five components: a frequency-dependent satellite-clock error, a tropospheric satellite delay, an ionospheric satellite delay, an initial satellite phase, and an integer satellite ambiguity. The form of the PPP equations to be solved by the network user is then similar to that of the traditional PPP equations. As soon as the CD ambiguities are fixed and validated, an operation which can be performed in real time via appropriate decorrelation techniques, estimates of these float biases can be immediately obtained. No other ambiguity is to be fixed. The satellite pseudo-clock biases can thus be obtained in real time. This is
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…
Results of Using Algebra Tiles as Meaningful Representations of Algebra Concepts.
ERIC Educational Resources Information Center
Sharp, Janet M.
Mathematical meanings can be developed when individuals construct translations between algebra symbol systems and physical systems that represent one another. Previous research studies indicated (1) few high school students connect whole number manipulations to algebraic manipulations and (2) students who encounter algebraic ideas through…
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!!!
A note on derivations of Murray–von Neumann algebras
Kadison, Richard V.; Liu, Zhe
2014-01-01
A Murray–von Neumann algebra is the algebra of operators affiliated with a finite von Neumann algebra. In this article, we first present a brief introduction to the theory of derivations of operator algebras from both the physical and mathematical points of view. We then describe our recent work on derivations of Murray–von Neumann algebras. We show that the “extended derivations” of a Murray–von Neumann algebra, those that map the associated finite von Neumann algebra into itself, are inner. In particular, we prove that the only derivation that maps a Murray–von Neumann algebra associated with a factor of type II1 into that factor is 0. Those results are extensions of Singer’s seminal result answering a question of Kaplansky, as applied to von Neumann algebras: The algebra may be noncommutative and may even contain unbounded elements. PMID:24469831
Hom-Lie algebras with symmetric invariant nondegenerate bilinear forms
NASA Astrophysics Data System (ADS)
Benayadi, Saïd; Makhlouf, Abdenacer
2014-02-01
The aim of this paper is to introduce and study quadratic Hom-Lie algebras, which are Hom-Lie algebras equipped with symmetric invariant nondegenerate bilinear forms. We provide several constructions leading to examples and extend the Double Extension Theory to this class of nonassociative algebras. Elements of Representation Theory for Hom-Lie algebras, including adjoint and coadjoint representations, are supplied with application to quadratic Hom-Lie algebras. Centerless involutive quadratic Hom-Lie algebras are characterized. We reduce the case where the twist map is invertible to the study of involutive quadratic Lie algebras. Also, we establish a correspondence between the class of involutive quadratic Hom-Lie algebras and quadratic simple Lie algebras with symmetric involution.
Structural chemistry and number theory amalgamized: crystal structure of Na11Hg52.
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. PMID:26634733
Highest-weight representations of Brocherd`s algebras
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.
Theory of exciton annihilation in complexes of a finite number of molecular sites.
Gülen, D
1990-11-01
A theory of the kinematics of singlet exciton annihilation in complexes of a finite number of molecular sites is developed. The theory is based on a specific scheme suggested earlier by Gülen, Wittmershaus, and Knox [Biophys J. 49:469-477 (1986)]. It is adequate to address the excitation kinetics and dynamics in such systems, especially under high excitation intensities. A Pauli master equation is formulated and is solved to give explicit expressions for observables such as quantum yield and fluorescence intensity. The excitation intensity dependence of the observables is taken into account by introducing Poisson statistics. Details relevant to its application to the annihilation of excitons in photosynthetic systems and its connection to earlier theories are presented. PMID:2134489
Representations of Super Yang-Mills Algebras
NASA Astrophysics Data System (ADS)
Herscovich, Estanislao
2013-06-01
We study in this article the representation theory of a family of super algebras, called the super Yang-Mills algebras, by exploiting the Kirillov orbit method à la Dixmier for nilpotent super Lie algebras. These super algebras are an extension of the so-called Yang-Mills algebras, introduced by A. Connes and M. Dubois-Violette in (Lett Math Phys 61(2):149-158, 2002), and in fact they appear as a "background independent" formulation of supersymmetric gauge theory considered in physics, in a similar way as Yang-Mills algebras do the same for the usual gauge theory. Our main result states that, under certain hypotheses, all Clifford-Weyl super algebras {{Cliff}q(k) ⊗ Ap(k)}, for p ≥ 3, or p = 2 and q ≥ 2, appear as a quotient of all super Yang-Mills algebras, for n ≥ 3 and s ≥ 1. This provides thus a family of representations of the super Yang-Mills algebras.
On the Design of Lifting Airfoils with High Critical Mach Number Using Full Potential Theory
NASA Astrophysics Data System (ADS)
Kropinski, M. C. A.
We wish to construct airfoils that have the highest free-stream Mach number for a given set of geometric constraints for which the flow is nowhere supersonic. Nonlifting airfoils that maximize the critical Mach number for a given cross-sectional area are known to possess long sonic segments at their critical speed. To construct lifting airfoils, we proceed under the conjecture that an airfoil with a high value of has the longest possible arc length of sonic velocity over its upper and lower surface. In Kropinski etal. (1995) the lifting problem was tackled in transonic small-disturbance theory. In this paper we numerically construct lifting airfoils with high using the full potential theory and we show that these airfoils have significantly higher than some standard airfoils. We also construct airfoils with higher values of the lift coefficient, by relaxing the speed constraint on the lower surface of the airfoil to have a value less than sonic.
Phase transitions in number theory: from the birthday problem to Sidon sets.
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. PMID:24329226
Number-conserving master equation theory for a dilute Bose-Einstein condensate
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.
Twisted Logarithmic Modules of Vertex Algebras
NASA Astrophysics Data System (ADS)
Bakalov, Bojko
2016-07-01
Motivated by logarithmic conformal field theory and Gromov-Witten theory, we introduce a notion of a twisted module of a vertex algebra under an arbitrary (not necessarily semisimple) automorphism. Its main feature is that the twisted fields involve the logarithm of the formal variable. We develop the theory of such twisted modules and, in particular, derive a Borcherds identity and commutator formula for them. We investigate in detail the examples of affine and Heisenberg vertex algebras.
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. PMID:27080590
NASA Astrophysics Data System (ADS)
Kimura, Yusuke
2015-07-01
It has been understood that correlation functions of multi-trace operators in SYM can be neatly computed using the group algebra of symmetric groups or walled Brauer algebras. On the other hand, such algebras have been known to construct 2D topological field theories (TFTs). After reviewing the construction of 2D TFTs based on symmetric groups, we construct 2D TFTs based on walled Brauer algebras. In the construction, the introduction of a dual basis manifests a similarity between the two theories. We next construct a class of 2D field theories whose physical operators have the same symmetry as multi-trace operators constructed from some matrices. Such field theories correspond to non-commutative Frobenius algebras. A matrix structure arises as a consequence of the noncommutativity. Correlation functions of the Gaussian complex multi-matrix models can be translated into correlation functions of the two-dimensional field theories.
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"…
Twisted Quantum Toroidal Algebras
NASA Astrophysics Data System (ADS)
Jing, Naihuan; Liu, Rongjia
2014-09-01
We construct a principally graded quantum loop algebra for the Kac-Moody algebra. As a special case a twisted analog of the quantum toroidal algebra is obtained together with the quantum Serre relations.
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.
Type-Decomposition of an Effect Algebra
NASA Astrophysics Data System (ADS)
Foulis, David J.; Pulmannová, Sylvia
2010-10-01
Effect algebras (EAs), play a significant role in quantum logic, are featured in the theory of partially ordered Abelian groups, and generalize orthoalgebras, MV-algebras, orthomodular posets, orthomodular lattices, modular ortholattices, and boolean algebras. We study centrally orthocomplete effect algebras (COEAs), i.e., EAs satisfying the condition that every family of elements that is dominated by an orthogonal family of central elements has a supremum. For COEAs, we introduce a general notion of decomposition into types; prove that a COEA factors uniquely as a direct sum of types I, II, and III; and obtain a generalization for COEAs of Ramsay’s fourfold decomposition of a complete orthomodular lattice.
C*-Algebras Associated with Endomorphisms and Polymorphisms of Compact Abelian Groups
NASA Astrophysics Data System (ADS)
Cuntz, Joachim; Vershik, Anatoly
2013-07-01
A surjective endomorphism or, more generally, a polymorphism in the sense of Schmidt and Vershik [Erg Th Dyn Sys 28(2):633-642, 2008], of a compact abelian group H induces a transformation of L 2( H). We study the C*-algebra generated by this operator together with the algebra of continuous functions C( H) which acts as multiplication operators on L 2( H). Under a natural condition on the endo- or polymorphism, this algebra is simple and can be described by generators and relations. In the case of an endomorphism it is always purely infinite, while for a polymorphism in the class we consider, it is either purely infinite or has a unique trace. We prove a formula allowing to determine the K-theory of these algebras and use it to compute the K-groups in a number of interesting examples.
Algebraic Lattices in QFT Renormalization
NASA Astrophysics Data System (ADS)
Borinsky, Michael
2016-04-01
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.
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.
Classification of central extensions of Lax operator algebras
Schlichenmaier, Martin
2008-11-18
Lax operator algebras were introduced by Krichever and Sheinman as further developments of Krichever's theory of Lax operators on algebraic curves. They are infinite dimensional Lie algebras of current type with meromorphic objects on compact Riemann surfaces (resp. algebraic curves) as elements. Here we report on joint work with Oleg Sheinman on the classification of their almost-graded central extensions. It turns out that in case that the finite-dimensional Lie algebra on which the Lax operator algebra is based on is simple there is a unique almost-graded central extension up to equivalence and rescaling of the central element.
Banach Algebras Associated to Lax Pairs
NASA Astrophysics Data System (ADS)
Glazebrook, James F.
2015-04-01
Lax pairs featuring in the theory of integrable systems are known to be constructed from a commutative algebra of formal pseudodifferential operators known as the Burchnall- Chaundy algebra. Such pairs induce the well known KP flows on a restricted infinite-dimensional Grassmannian. The latter can be exhibited as a Banach homogeneous space constructed from a Banach *-algebra. It is shown that this commutative algebra of operators generating Lax pairs can be associated with a commutative C*-subalgebra in the C*-norm completion of the *-algebra. In relationship to the Bose-Fermi correspondence and the theory of vertex operators, this C*-algebra has an association with the CAR algebra of operators as represented on Fermionic Fock space by the Gelfand-Naimark-Segal construction. Instrumental is the Plücker embedding of the restricted Grassmannian into the projective space of the associated Hilbert space. The related Baker and tau-functions provide a connection between these two C*-algebras, following which their respective state spaces and Jordan-Lie-Banach algebras structures can be compared.
Lax operator algebras and integrable systems
NASA Astrophysics Data System (ADS)
Sheinman, O. K.
2016-02-01
A new class of infinite-dimensional Lie algebras, called Lax operator algebras, is presented, along with a related unifying approach to finite-dimensional integrable systems with a spectral parameter on a Riemann surface such as the Calogero-Moser and Hitchin systems. In particular, the approach includes (non-twisted) Kac-Moody algebras and integrable systems with a rational spectral parameter. The presentation is based on quite simple ideas about the use of gradings of semisimple Lie algebras and their interaction with the Riemann-Roch theorem. The basic properties of Lax operator algebras and the basic facts about the theory of the integrable systems in question are treated (and proved) from this general point of view. In particular, the existence of commutative hierarchies and their Hamiltonian properties are considered. The paper concludes with an application of Lax operator algebras to prequantization of finite-dimensional integrable systems. Bibliography: 51 titles.
NASA Astrophysics Data System (ADS)
Palmkvist, Jakob
2014-01-01
We introduce an infinite-dimensional Lie superalgebra which is an extension of the U-duality Lie algebra of maximal supergravity in D dimensions, for 3 ⩽ D ⩽ 7. The level decomposition with respect to the U-duality Lie algebra gives exactly the tensor hierarchy of representations that arises in gauge deformations of the theory described by an embedding tensor, for all positive levels p. We prove that these representations are always contained in those coming from the associated Borcherds-Kac-Moody superalgebra, and we explain why some of the latter representations are not included in the tensor hierarchy. The most remarkable feature of our Lie superalgebra is that it does not admit a triangular decomposition like a (Borcherds-)Kac-Moody (super)algebra. Instead the Hodge duality relations between level p and D - 2 - p extend to negative p, relating the representations at the first two negative levels to the supersymmetry and closure constraints of the embedding tensor.
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.
Palmkvist, Jakob
2014-01-15
We introduce an infinite-dimensional Lie superalgebra which is an extension of the U-duality Lie algebra of maximal supergravity in D dimensions, for 3 ⩽ D ⩽ 7. The level decomposition with respect to the U-duality Lie algebra gives exactly the tensor hierarchy of representations that arises in gauge deformations of the theory described by an embedding tensor, for all positive levels p. We prove that these representations are always contained in those coming from the associated Borcherds-Kac-Moody superalgebra, and we explain why some of the latter representations are not included in the tensor hierarchy. The most remarkable feature of our Lie superalgebra is that it does not admit a triangular decomposition like a (Borcherds-)Kac-Moody (super)algebra. Instead the Hodge duality relations between level p and D − 2 − p extend to negative p, relating the representations at the first two negative levels to the supersymmetry and closure constraints of the embedding tensor.
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.
C∗-completions and the DFR-algebra
NASA Astrophysics Data System (ADS)
Forger, Michael; Paulino, Daniel V.
2016-02-01
The aim of this paper is to present the construction of a general family of C∗-algebras which includes, as a special case, the "quantum spacetime algebra" introduced by Doplicher, Fredenhagen, and Roberts. It is based on an extension of the notion of C∗-completion from algebras to bundles of algebras, compatible with the usual C∗-completion of the appropriate algebras of sections, combined with a novel definition for the algebra of the canonical commutation relations using Rieffel's theory of strict deformation quantization. Taking the C∗-algebra of continuous sections vanishing at infinity, we arrive at a functor associating a C∗-algebra to any Poisson vector bundle and recover the original DFR-algebra as a particular example.
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.
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.
Baryon and lepton number violation in the electroweak theory at TeV energies
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.
Nonlinear theory of classical cylindrical Richtmyer-Meshkov instability for arbitrary Atwood numbers
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.
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.
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.
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.
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.
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…
Algebras with convergent star products and their representations in Hilbert spaces
Soloviev, M. A.
2013-07-15
We study star product algebras of analytic functions for which the power series defining the products converge absolutely. Such algebras arise naturally in deformation quantization theory and in noncommutative quantum field theory. We consider different star products in a unifying way and present results on the structure and basic properties of these algebras, which are useful for applications. Special attention is given to the Hilbert space representation of the algebras and to the exact description of their corresponding operator algebras.
Riemann Zeta Zeros and Prime Number Spectra in Quantum Field Theory
NASA Astrophysics Data System (ADS)
Menezes, G.; Svaiter, B. F.; Svaiter, N. F.
2013-10-01
The Riemann hypothesis states that all nontrivial zeros of the zeta function lie in the critical line Re(s) = 1/2. Hilbert and Pólya suggested that one possible way to prove the Riemann hypothesis is to interpret the nontrivial zeros in the light of spectral theory. Using the construction of the so-called super-zeta functions or secondary zeta functions built over the Riemann nontrivial zeros and the regularity property of one of this function at the origin, we show that it is possible to extend the Hilbert-Pólya conjecture to systems with countably infinite number of degrees of freedom. The sequence of the nontrivial zeros of the Riemann zeta function can be interpreted as the spectrum of a self-adjoint operator of some hypothetical system described by the functional approach to quantum field theory. However, if one considers the same situation with numerical sequences whose asymptotic distributions are not "far away" from the asymptotic distribution of prime numbers, the associated functional integral cannot be constructed. Finally, we discuss possible relations between the asymptotic behavior of a sequence and the analytic domain of the associated zeta function.
ERIC Educational Resources Information Center
National Council of Teachers of Mathematics, Inc., Reston, VA.
This is a reprint of the historical capsules dealing with algebra from the 31st Yearbook of NCTM,"Historical Topics for the Mathematics Classroom." Included are such themes as the change from a geometric to an algebraic solution of problems, the development of algebraic symbolism, the algebraic contributions of different countries, the origin and…
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.
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.
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.
Knots, BPS States, and Algebraic Curves
NASA Astrophysics Data System (ADS)
Garoufalidis, Stavros; Kucharski, Piotr; Sułkowski, Piotr
2016-07-01
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.
Algebraic Thinking: A Problem Solving Approach
ERIC Educational Resources Information Center
Windsor, Will
2010-01-01
Algebraic thinking is a crucial and fundamental element of mathematical thinking and reasoning. It initially involves recognising patterns and general mathematical relationships among numbers, objects and geometric shapes. This paper will highlight how the ability to think algebraically might support a deeper and more useful knowledge, not only of…
Extending Birthday Paradox Theory to Estimate the Number of Tags in RFID Systems
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
Extending birthday paradox theory to estimate the number of tags in RFID systems.
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
Six-dimensional (1,0) superconformal models and higher gauge theory
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.
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.
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.
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
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…
Law of Large Numbers: the Theory, Applications and Technology-based Education
Dinov, Ivo D.; Christou, Nicolas; Gould, Robert
2011-01-01
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). PMID:21603584
The Universal C*-Algebra of the Electromagnetic Field
NASA Astrophysics Data System (ADS)
Buchholz, Detlev; Ciolli, Fabio; Ruzzi, Giuseppe; Vasselli, Ezio
2016-02-01
A universal C*-algebra of the electromagnetic field is constructed. It is represented in any quantum field theory which incorporates electromagnetism and expresses basic features of the field such as Maxwell's equations, Poincaré covariance and Einstein causality. Moreover, topological properties of the field resulting from Maxwell's equations are encoded in the algebra, leading to commutation relations with values in its center. The representation theory of the algebra is discussed with focus on vacuum representations, fixing the dynamics of the field.
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.
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…
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…
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…
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…
Three-algebra for supermembrane and two-algebra for superstring
NASA Astrophysics Data System (ADS)
Lee, Kanghoon; Park, Jeong-Hyuck
2009-04-01
While string or Yang-Mills theories are based on Lie algebra or two-algebra structure, recent studies indicate that Script M-theory may require a one higher, three-algebra structure. Here we construct a covariant action for a supermembrane in eleven dimensions, which is invariant under global supersymmetry, local fermionic symmetry and worldvolume diffeomorphism. Our action is classically on-shell equivalent to the celebrated Bergshoeff-Sezgin-Townsend action. However, the novelty is that we spell the action genuinely in terms of Nambu three-brackets: All the derivatives appear through Nambu brackets and hence it manifests the three-algebra structure. Further the double dimensional reduction of our action gives straightforwardly to a type IIA string action featuring two-algebra. Applying the same method, we also construct a covariant action for type IIB superstring, leading directly to the IKKT matrix model.
Clifford algebras and Hestenes spinors
NASA Astrophysics Data System (ADS)
Lounesto, Pertti
1993-09-01
This article reviews Hestenes' work on the Dirac theory, where his main achievement is a real formulation of the theory within the real Clifford algebra Cl 1,3 ≃ M2 (H). Hestenes invented first in 1966 his ideal spinorsφ in Cl_{1,3 _2}^1 (1 - γ _{03} ) and later 1967/75 he recognized the importance of his operator spinors ψ ∈ Cl{/1,3 + } ≃ M2 (C). This article starts from the conventional Dirac equation as presented with matrices by Bjorken-Drell. Explicit mappings are given for a passage between Hestenes' operator spinors and Dirac's column spinors. Hestenes' operator spinors are seen to be multiples of even parts of real parts of Dirac spinors (real part in the decomposition C ⊗ Cl 1,3 and not in C ⊗ M4 (R)=M4 (C)). It will become apparent that the standard matrix formulation contains superfluous parts, which ought to be cut out by Occam's razor. Fierz identities of bilinear covariants are known to be sufficient to study the non-null case but are seen to be insufficient for the null case ψ†γ0ψ=0, ψ†γ0γ0123ψ=0. The null case is thoroughly scrutinized for the first time with a new concept called boomerang. This permits a new intrinsically geometric classification of spinors. This in turn reveals a new class of spinors which has not been discussed before. This class supplements the spinors of Dirac, Weyl, and Majorana; it describes neither the electron nor the neutron; it is awaiting a physical interpretation and a possible observation. Projection operators P±, Σ± are resettled among their new relatives in End(Cl 1,3 ). Finally, a new mapping, called tilt, is introduced to enable a transition from Cl 1,3 to the (graded) opposite algebra Cl 3,1 without resorting to complex numbers, that is, not using a replacement γμ → iγμ.
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…
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.
Phase Boundaries in Algebraic Conformal QFT
NASA Astrophysics Data System (ADS)
Bischoff, Marcel; Kawahigashi, Yasuyuki; Longo, Roberto; Rehren, Karl-Henning
2016-02-01
We study the structure of local algebras in relativistic conformal quantum field theory with phase boundaries. Phase boundaries are instances of a more general notion of boundaries that give rise to a variety of algebraic structures. These can be formulated in a common framework originating in Algebraic QFT, with the principle of Einstein Causality playing a prominent role. We classify the phase boundary conditions by the centre of a certain universal construction, which produces a reducible representation in which all possible boundary conditions are realized. For a large class of models, the classification reproduces results obtained in a different approach by Fuchs et al. before.
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.
Theory of cylindrical and spherical Langmuir probes in the limit of vanishing Debye number
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/.
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…
Applied Algebra Curriculum Modules.
ERIC Educational Resources Information Center
Texas State Technical Coll., Marshall.
This collection of 11 applied algebra curriculum modules can be used independently as supplemental modules for an existing algebra curriculum. They represent diverse curriculum styles that should stimulate the teacher's creativity to adapt them to other algebra concepts. The selected topics have been determined to be those most needed by students…
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…
Ternary Virasoro - Witt algebra.
Zachos, C.; Curtright, T.; Fairlie, D.; High Energy Physics; Univ. of Miami; Univ. of Durham
2008-01-01
A 3-bracket variant of the Virasoro-Witt algebra is constructed through the use of su(1,1) enveloping algebra techniques. The Leibniz rules for 3-brackets acting on other 3-brackets in the algebra are discussed and verified in various situations.
Edge covers and independence: Algebraic approach
NASA Astrophysics Data System (ADS)
Kalinina, E. A.; Khitrov, G. M.; Pogozhev, S. V.
2016-06-01
In this paper, linear algebra methods are applied to solve some problems of graph theory. For ordinary connected graphs, edge coverings and independent sets are considered. Some results concerning minimum edge covers and maximum matchings are proved with the help of linear algebraic approach. The problem of finding a maximum matching of a graph is fundamental both practically and theoretically, and has numerous applications, e.g., in computational chemistry and mathematical chemistry.
Vague Congruences and Quotient Lattice Implication Algebras
Qin, Xiaoyan; Xu, Yang
2014-01-01
The aim of this paper is to further develop the congruence theory on lattice implication algebras. Firstly, we introduce the notions of vague similarity relations based on vague relations and vague congruence relations. Secondly, the equivalent characterizations of vague congruence relations are investigated. Thirdly, the relation between the set of vague filters and the set of vague congruences is studied. Finally, we construct a new lattice implication algebra induced by a vague congruence, and the homomorphism theorem is given. PMID:25133207
Algebraic Apect of Helicities in Hadron Physics
NASA Astrophysics Data System (ADS)
An, Murat; Ji, Chueng
2015-04-01
We examined the relation of polarization vectors and spinors of (1 , 0) ⊕(0 , 1) representation of Lorentz group in Clifford algebra Cl1 , 3 , their relation with standard algebra, and properties of these spinors. Cl1 , 3 consists of different grades:e.g. the first and the second grades represent (1 / 2 , 1 / 2) and (1 , 0) ⊕(0 , 1) representation of spin groups respectively with 4 and 6 components. However, these Clifford numbers are not the helicity eigenstates and thus we transform them into combinations of helicity eigenstates by expressing them as spherical harmonics. We relate the spin-one polarization vectors and (1 , 0) ⊕(0 , 1) spinors under one simple transformation with the spin operators. We also link our work with Winnberg's work of a superfield of a spinors of Clifford algebra by giving a physical meaning to Grassmann variables and discuss how Grassman algebra is linked with Clifford algebra.
Nilpotent orbits in classical Lie algebras over F2n and the Springer correspondence
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
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…
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.
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.
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…
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…
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.
Dark matter, lepton and baryon number, and left-right symmetric theories
NASA Astrophysics Data System (ADS)
Patra, Sudhanwa
2016-05-01
A lepto-baryonic left-right symmetric theory is considered along with pointing out stable dark matter candidates whose stability is ensured automatically where leptons and baryons are defined as local gauge symmetries. These theories are generally anomalous, and the possible gauge anomaly free solutions for these theories are presented. It is found that the neutral component of fermion triplets can be a viable dark matter candidate originally introduced for gauge anomaly cancellation. The other dark matter possibilities within this lepto-baryonic left-right symmetric theory are also presented.
Lie algebras and linear differential equations.
NASA Technical Reports Server (NTRS)
Brockett, R. W.; Rahimi, A.
1972-01-01
Certain symmetry properties possessed by the solutions of linear differential equations are examined. For this purpose, some basic ideas from the theory of finite dimensional linear systems are used together with the work of Wei and Norman on the use of Lie algebraic methods in differential equation theory.
Modules as Learning Tools in Linear Algebra
ERIC Educational Resources Information Center
Cooley, Laurel; Vidakovic, Draga; Martin, William O.; Dexter, Scott; Suzuki, Jeff; Loch, Sergio
2014-01-01
This paper reports on the experience of STEM and mathematics faculty at four different institutions working collaboratively to integrate learning theory with curriculum development in a core undergraduate linear algebra context. The faculty formed a Professional Learning Community (PLC) with a focus on learning theories in mathematics and…
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…
Introduction to Algebra (Part 2). Preliminary Edition.
ERIC Educational Resources Information Center
Haag, V. H.; And Others
This is part two of a two-part SMSG algebra text for ninth-grade students. The text was written for those students whose mathematical talent is underdeveloped. Chapter topics include the real numbers, addition of real numbers, multiplication of real numbers, properties of order, and subtraction and division for real numbers. (MP)
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 determination of prime numbers vs. composite numbers from a "negative" approach. This approach applies a certain "revolving chamber" to generate the set of composite numbers not having 2, 3 or 5 as factor, and by this indirectly to also deduce the complementary set of prime numbers larger than 5 as an exact, complete and non-trivial pattern. Since this approach is anchored in a certain geometric positioning of natural numbers, the paper will consider some 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.
C-Graded vertex algebras and conformal flow
Laber, Rob; Mason, Geoffrey
2014-01-15
We consider C-graded vertex algebras, which are vertex algebras V with a C-grading such that V is an admissible V-module generated by “lowest weight vectors.” We show that such vertex algebras have a “good” representation theory in the sense that there is a Zhu algebra A(V) and a bijection between simple admissible V-modules and simple A(V)-modules. We also consider pseudo vertex operator algebras (PVOAs), which are C-graded vertex algebras with a conformal vector such that the homogeneous subspaces of V are generalized eigenspaces for L(0); essentially, these are VOAs that lack any semisimplicity or integrality assumptions on L(0). As a motivating example, we show that deformation of the conformal structure (conformal flow) of a strongly regular VOA (e.g., a lattice theory, or Wess-Zumino-Witten model) is a path in a space whose points are PVOAs.
Toward robust scalable algebraic multigrid solvers.
Waisman, Haim; Schroder, Jacob; Olson, Luke; Hiriyur, Badri; Gaidamour, Jeremie; Siefert, Christopher; Hu, Jonathan Joseph; Tuminaro, Raymond Stephen
2010-10-01
This talk highlights some multigrid challenges that arise from several application areas including structural dynamics, fluid flow, and electromagnetics. A general framework is presented to help introduce and understand algebraic multigrid methods based on energy minimization concepts. Connections between algebraic multigrid prolongators and finite element basis functions are made to explored. It is shown how the general algebraic multigrid framework allows one to adapt multigrid ideas to a number of different situations. Examples are given corresponding to linear elasticity and specifically in the solution of linear systems associated with extended finite elements for fracture problems.
Some functional metrics in algebraic and combinatorial coding
NASA Astrophysics Data System (ADS)
Choen, G.
1980-06-01
Three approaches to coding problems can be systematically distinguished: probabilistic (essentially existential), algebraic, and combinatorial. This last approach searches for optimal configurations and relegates to the second order, the problems of complexity related to decoding. Enumeration, graphs, designs, and the extreme theory of groups are used. The optimization of a functional metric was used with the combinatorial approach in order to define the space considered and the distance. The codes then become particular groups of the metric space, which is defined by parameters such as length, number of words, and capacity for correction. Some of these parameters are imposed.
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.
Supersymmetric extension of Galilean conformal algebras
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.
Semigroups And Computer Algebra In Discrete Structures
NASA Astrophysics Data System (ADS)
Bijev, G.
2010-10-01
Some concepts in semigroup theory are interpreted in discrete structures such as finite lattices, binary relations, and finite semilattices. An algebraic approach to the pseudoinverse generalization problem in Boolean vector spaces is used. By analogy with the linear spaces in the linear algebra semilattice homomorphisms, isomorphisms, projections on Boolean vector spaces are defined and some properties of them are investigated in detail. Maps, corresponding to them in the linear algebra, are connected with matrices and their pseudouinverse. Important properties of these maps, which are essential for solving linear systems, remain the same in the Boolean vector spaces. Stochastic experiments using the maps defined and computer algebra methods have been made for solving linear equations Ax = b. The Hamming distance between b and the projection p(b) = Ax of b is equal or close to the least possible one, if the system has no solutions.
Atomic Theory and Multiple Combining Proportions: The Search for Whole Number Ratios.
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
Fuzzy-algebra uncertainty assessment
Cooper, J.A.; Cooper, D.K.
1994-12-01
A significant number of analytical problems (for example, abnormal-environment safety analysis) depend on data that are partly or mostly subjective. Since fuzzy algebra depends on subjective operands, we have been investigating its applicability to these forms of assessment, particularly for portraying uncertainty in the results of PRA (probabilistic risk analysis) and in risk-analysis-aided decision-making. Since analysis results can be a major contributor to a safety-measure decision process, risk management depends on relating uncertainty to only known (not assumed) information. The uncertainties due to abnormal environments are even more challenging than those in normal-environment safety assessments; and therefore require an even more judicious approach. Fuzzy algebra matches these requirements well. One of the most useful aspects of this work is that we have shown the potential for significant differences (especially in perceived margin relative to a decision threshold) between fuzzy assessment and probabilistic assessment based on subtle factors inherent in the choice of probability distribution models. We have also shown the relation of fuzzy algebra assessment to ``bounds`` analysis, as well as a description of how analyses can migrate from bounds analysis to fuzzy-algebra analysis, and to probabilistic analysis as information about the process to be analyzed is obtained. Instructive examples are used to illustrate the points.
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…
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
Algorithmic Questions for Linear Algebraic Groups. Ii
NASA Astrophysics Data System (ADS)
Sarkisjan, R. A.
1982-04-01
It is proved that, given a linear algebraic group defined over an algebraic number field and satisfying certain conditions, there exists an algorithm which determines whether or not two double cosets of a special type coincide in its adele group, and which enumerates all such double cosets. This result is applied to the isomorphism problem for finitely generated nilpotent groups, and also to other problems.Bibliography: 18 titles.
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)
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.
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.
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…
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…
East-West migration in Europe: can migration theories help estimate the numbers?
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
ERIC Educational Resources Information Center
Turner, Colin M.
This working paper examines the theories of Fred Fiedler concerning the effectiveness of different leadership styles in different situations. Discussed first are the distinctions that Fiedler makes between what he terms relationship-motivated leaders and task-motivated leaders. The next section comprises an explanation of Fiedler's view of…
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…
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…
Modular operads and the quantum open-closed homotopy algebra
NASA Astrophysics Data System (ADS)
Doubek, Martin; Jurčo, Branislav; Münster, Korbinian
2015-12-01
We verify that certain algebras appearing in string field theory are algebras over Feynman transform of modular operads which we describe explicitly. Equivalent description in terms of solutions of generalized BV master equations are explained from the operadic point of view.
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.
Infinitesimal deformations of filiform Lie algebras of order 3
NASA Astrophysics Data System (ADS)
Navarro, R. M.
2015-12-01
The Lie algebras of order F have important applications for the fractional supersymmetry, and on the other hand the filiform Lie (super)algebras have very important properties into the Lie Theory. Thus, the aim of this work is to study filiform Lie algebras of order F which were introduced in Navarro (2014). In this work we obtain new families of filiform Lie algebras of order 3, in which the complexity of the problem rises considerably respecting to the cases considered in Navarro (2014).
Simulations on the number of entanglements of a polymer network using knot theory.
Michalke, W; Lang, M; Kreitmeier, S; Göritz, D
2001-07-01
Polymer networks, created on the computer using the Bond-Fluctuation-Algorithm, offer the possibility to count the number of entanglements. We generated networks consisting of 5000 chains that were cross linked at their end groups via tetra-functional cross linkers. The analysis of the topology was performed by computing the Homfly polynomial of the entanglements offering a much more precise determination of the knot and entanglement type than the Gaussian linking number. It also allows us to determine the influence of Brunnian links. Results concerning the connection between the chain length and the number of entanglements are shown. PMID:11461310
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.
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.
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.
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.
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.
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.
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…
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…
NASA Technical Reports Server (NTRS)
Lawson, C. L.; Krogh, F. T.; Gold, S. S.; Kincaid, D. R.; Sullivan, J.; Williams, E.; Hanson, R. J.; Haskell, K.; Dongarra, J.; Moler, C. B.
1982-01-01
The Basic Linear Algebra Subprograms (BLAS) library is a collection of 38 FORTRAN-callable routines for performing basic operations of numerical linear algebra. BLAS library is portable and efficient source of basic operations for designers of programs involving linear algebriac computations. BLAS library is supplied in portable FORTRAN and Assembler code versions for IBM 370, UNIVAC 1100 and CDC 6000 series computers.
ERIC Educational Resources Information Center
Cavanagh, Sean
2008-01-01
A popular humorist and avowed mathphobe once declared that in real life, there's no such thing as algebra. Kathie Wilson knows better. Most of the students in her 8th grade class will be thrust into algebra, the definitive course that heralds the beginning of high school mathematics, next school year. The problem: Many of them are about three…
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
Degenerations of generalized Krichever-Novikov algebras on tori
NASA Astrophysics Data System (ADS)
Schlichenmaier, Martin
1993-08-01
Degenerations of Lie algebras of meromorphic vector fields on elliptic curves (i.e., complex tori) which are holomorphic outside a certain set of points (markings) are studied. By an algebraic geometric degeneration process certain subalgebras of Lie algebras of meromorphic vector fields on P1, the Riemann sphere, are obtained. In case of some natural choices of the markings these subalgebras are explicitly determined. It is shown that the number of markings can change.
Affine Vertex Operator Algebras and Modular Linear Differential Equations
NASA Astrophysics Data System (ADS)
Arike, Yusuke; Kaneko, Masanobu; Nagatomo, Kiyokazu; Sakai, Yuichi
2016-05-01
In this paper, we list all affine vertex operator algebras of positive integral levels whose dimensions of spaces of characters are at most 5 and show that a basis of the space of characters of each affine vertex operator algebra in the list gives a fundamental system of solutions of a modular linear differential equation. Further, we determine the dimensions of the spaces of characters of affine vertex operator algebras whose numbers of inequivalent simple modules are not exceeding 20.
Contractions of the irreducible representations of the quantum algebras suq(2) and soq(3)
NASA Astrophysics Data System (ADS)
Gromov, N. A.; Man'ko, V. I.
1992-04-01
The contractions of the irreducible representations of the unitary quantum algebra suq(2) and the orthogonal quantum algebra soq(3) in the Gel'fand-Tsetlin basis are regarded in detail with the help of the dual numbers.
Supersymmetry algebra cohomology. III. Primitive elements in four and five dimensions
NASA Astrophysics Data System (ADS)
Brandt, Friedemann
2011-05-01
The primitive elements of the supersymmetry algebra cohomology as defined in a previous paper are computed for standard supersymmetry algebras in four and five dimensions, for all signatures of the metric and any number of supersymmetries.
Supersymmetry algebra cohomology. III. Primitive elements in four and five dimensions
Brandt, Friedemann
2011-05-15
The primitive elements of the supersymmetry algebra cohomology as defined in a previous paper are computed for standard supersymmetry algebras in four and five dimensions, for all signatures of the metric and any number of supersymmetries.
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…
The geometric semantics of algebraic quantum mechanics.
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. PMID:26124252
Using geometric algebra to study optical aberrations
Hanlon, J.; Ziock, H.
1997-05-01
This paper uses Geometric Algebra (GA) to study vector aberrations in optical systems with square and round pupils. GA is a new way to produce the classical optical aberration spot diagrams on the Gaussian image plane and surfaces near the Gaussian image plane. Spot diagrams of the third, fifth and seventh order aberrations for square and round pupils are developed to illustrate the theory.
A Short Sheffer axiom for Boolean algebra.
Veroff, R.; McCune, W.
2000-06-30
A short Sheffer stroke identity is shown to be a single axiom for Boolean algebra. The axiom has length 15 and 3 variables. The proof shows that it is equivalent to Sheffer's original 3-basis for the theory. Automated deduction techniques were used to find the proof. The shortest single axiom previously known to us has length 105 and 6 variables.
Short single axioms for boolean algebra.
McCune, W.; Veroff, R.; Fitelson, B.; Harris, K.; Feist, A.; Wos, L.; Mathematics and Computer Science; Univ. of New Mexico; Univ. of Wisconsin at Madison; Duke Univ.
2002-01-01
We present short single equational axioms for Boolean algebra in terms of disjunction and negation and in terms of the Sheffer stroke. Previously known single axioms for these theories are much longer than the ones we present. We show that there is no shorter axiom in terms of the Sheffer stroke. Automated deduction techniques were used in several parts of the work.
An Application of Boolean Algebra to Biology
ERIC Educational Resources Information Center
McConnell, John W.
1971-01-01
Examines the model of interacting nerve systems based on a switching theory, which uses a mathematical structure familiar to many high school students and requires little knowledge of biology. Reviews the basic operation of nerves, and demonstrates how Boolean algebraic statements are applied to synaptic interactions. (PR)
One-loop gauge theory amplitudes with an arbitrary number of external legs
Bern, Z.; Dunbar, D.C.; Dixon, L.; Kosower, D.A.
1994-05-01
The authors review recent progress in calculations of one-loop QCD amplitudes. By imposing the consistency requirements of unitarity and correct behavior as the momenta of two legs become collinear, they construct ansaetze for one-loop amplitudes with an arbitrary number of external legs. For supersymmetric amplitudes, which can be thought of as components of QCD amplitudes, the cuts uniquely specify the amplitude.
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.
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.
Finite-particle-number approach to physics
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.
Chern-Simons number diffusion in (1+1)-dimensional Higgs theory
Forcrand, P.d.; Krasnitz, A. ); Potting, R. )
1994-11-15
We study the Chern-Simons number diffusion rate in the (1+1)-dimensional lattice Abelian Higgs model at temperatures much higher than, as well as comparable to, the sphaleron energy. It is found that in the high-temperature limit the rate is likely to grow as a power of 2/3 of the temperature. In the intermediate-temperature regime, our numerical simulations show that the very weak temperature dependence of the rate, found in previous work, persists at smaller lattice spacings. We discuss possibilities of relating the observed behavior of the rate to static finite-temperature properties of the model.
Coreflections in Algebraic Quantum Logic
NASA Astrophysics Data System (ADS)
Jacobs, Bart; Mandemaker, Jorik
2012-07-01
Various generalizations of Boolean algebras are being studied in algebraic quantum logic, including orthomodular lattices, orthomodular po-sets, orthoalgebras and effect algebras. This paper contains a systematic study of the structure in and between categories of such algebras. It does so via a combination of totalization (of partially defined operations) and transfer of structure via coreflections.
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.
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.
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.
Algebraic models of flexible manufacturing systems
NASA Astrophysics Data System (ADS)
Leskin, Aleksei Alekseevich
Various aspects of the use of mathematical methods in the development of flexible manufacturing systems are examined. Attention is given to dynamical and structural models of flexible manufacturing systems developed by using methods of algebraic and differential geometry, topology, polynomial algebra, and extreme value problem theory. The principles of model integration are discussed, and approaches are proposed for solving problems related to the selection of flexible manufacturing equipment, real-time modeling of the manufacturing process, and optimization of local automation systems. The discussion is illustrated by examples.
The early history of current Algebra
NASA Astrophysics Data System (ADS)
Pietschmann, Herbert
2011-07-01
The history of Current Algebra is reviewed up to the appearance of the Adler-Weisberger sum rule. Particular emphasis is given to the role of current algebra in the historical struggle in strong interaction physics of elementary particles between field theory and the S-matrix approach based on dispersion relations. The question as to whether some particles are truly fundamental or all hadrons are bound or resonant states of one another played an important role in this struggle and is thus also regarded.
NASA Astrophysics Data System (ADS)
Kuzmin, Dmitri; Möller, Matthias; Gurris, Marcel
Flux limiting for hyperbolic systems requires a careful generalization of the design principles and algorithms introduced in the context of scalar conservation laws. In this chapter, we develop FCT-like algebraic flux correction schemes for the Euler equations of gas dynamics. In particular, we discuss the construction of artificial viscosity operators, the choice of variables to be limited, and the transformation of antidiffusive fluxes. An a posteriori control mechanism is implemented to make the limiter failsafe. The numerical treatment of initial and boundary conditions is discussed in some detail. The initialization is performed using an FCT-constrained L 2 projection. The characteristic boundary conditions are imposed in a weak sense, and an approximate Riemann solver is used to evaluate the fluxes on the boundary. We also present an unconditionally stable semi-implicit time-stepping scheme and an iterative solver for the fully discrete problem. The results of a numerical study indicate that the nonlinearity and non-differentiability of the flux limiter do not inhibit steady state convergence even in the case of strongly varying Mach numbers. Moreover, the convergence rates improve as the pseudo-time step is increased.
High Reynolds number droplet de-pinning on textured surfaces: theory and experiments
NASA Astrophysics Data System (ADS)
Lee, Sungyon; Wilcox, Benjamin; Xu, Feng; White, Edward
2014-11-01
The stability of drops on surfaces subject to forcing by wind and gravity is relevant to heat exchangers, fuel cells, and aircraft icing, and it lacks understanding in a high Reynolds number regime. To experimentally investigate this phenomenon, water drops are placed on the rough aluminum floor of a tiltable wind tunnel and brought to critical conditions for varying drop sizes, inclination angles, and flow speeds. In particular, the evolving 3D droplet shapes under flow are reconstructed based on a laser-speckle interface measurement tool, while the critical flow rates of droplet depinning are also noted. By accounting for the contact angle hysteresis and the pressure build-up in a nearly turbulent boundary layer, the critical depinning flow rate is theoretically predicted and is compared to the experimental results. We also observe and explain the transition of the drop depinning behavior from inertia-dominated to gravity-dominated regimes at non-zero inclination angles.
Developing Algebraic Thinking.
ERIC Educational Resources Information Center
Alejandre, Suzanne
2002-01-01
Presents a teaching experience that resulted in students getting to a point of full understanding of the kinesthetic activity and the algebra behind it. Includes a lesson plan for a traffic jam activity. (KHR)
Algebraic integrability: a survey.
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
Algebra. Instructor Edition. Post-Secondary Vocational-Technical Level.
ERIC Educational Resources Information Center
Eads, Freeman D.; Hinton, Barbara E.
This instructor's guide consists of materials for use in teaching a course in algebra designed for students enrolled in postsecondary vocational and technical education programs. Addressed in the eight individual units of the guide are the following topics: signed numbers, algebraic notation, equations and inequalities, solving algebraic…
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.
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.
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.
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.
DG Poisson algebra and its universal enveloping algebra
NASA Astrophysics Data System (ADS)
Lü, JiaFeng; Wang, XingTing; Zhuang, GuangBin
2016-05-01
In this paper, we introduce the notions of differential graded (DG) Poisson algebra and DG Poisson module. Let $A$ be any DG Poisson algebra. We construct the universal enveloping algebra of $A$ explicitly, which is denoted by $A^{ue}$. We show that $A^{ue}$ has a natural DG algebra structure and it satisfies certain universal property. As a consequence of the universal property, it is proved that the category of DG Poisson modules over $A$ is isomorphic to the category of DG modules over $A^{ue}$. Furthermore, we prove that the notion of universal enveloping algebra $A^{ue}$ is well-behaved under opposite algebra and tensor product of DG Poisson algebras. Practical examples of DG Poisson algebras are given throughout the paper including those arising from differential geometry and homological algebra.
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.
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.
The kinematic algebras from the scattering equations
NASA Astrophysics Data System (ADS)
Monteiro, Ricardo; O'Connell, Donal
2014-03-01
We study kinematic algebras associated to the recently proposed scattering equations, which arise in the description of the scattering of massless particles. In particular, we describe the role that these algebras play in the BCJ duality between colour and kinematics in gauge theory, and its relation to gravity. We find that the scattering equations are a consistency condition for a self-dual-type vertex which is associated to each solution of those equations. We also identify an extension of the anti-self-dual vertex, such that the two vertices are not conjugate in general. Both vertices correspond to the structure constants of Lie algebras. We give a prescription for the use of the generators of these Lie algebras in trivalent graphs that leads to a natural set of BCJ numerators. In particular, we write BCJ numerators for each contribution to the amplitude associated to a solution of the scattering equations. This leads to a decomposition of the determinant of a certain kinematic matrix, which appears naturally in the amplitudes, in terms of trivalent graphs. We also present the kinematic analogues of colour traces, according to these algebras, and the associated decomposition of that determinant.
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.
Open-closed homotopy algebra in mathematical physics
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)
Walker, A.D.M. )
1987-09-01
A new hydromagnetic theory is developed for describing compressional pulsations with azimuthal wave number. It is assumed that there are two plasma, one hot, in which pressure effects are important, and the other cold. The equations are derived in a general set of magnetic coordinates which allow realistic calculations including geometrical effects in the magnetosphere. The equations describe the three hydromagnetic modes which are coupled by the geometry. When the azimuthal wave number is large, the fast mode is strongly evanescent. This allows an expansion in 1/m in order to decouple the fast wave. The remaining equations describe the coupled transverse Alfven and magnetosonic modes. Some of the puzzling features of the observations of polarization are discussed.
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
Robbins algebra : conditions that make a near-Boolean algebra Boolean.
Winker, S.; Mathematics and Computer Science
1990-01-01
Some problems posed years ago remain challenging today. In particular, the Robbins problem, which is still open and which is the focus of attention in this paper, offers interesting challenges for attack with the assistance of an automated reasoning program; for the study presented here, we used the program OTTER. For example, when one submits this problem, which asks for a proof that every Robbins algebra is a Boolean algebra, a large number of deduced clauses results. One must, therefore, consider the possibility that there exists a Robbins algebra that is not Boolean; such an algebra would have to be infinite. One can instead search for properties that, if adjoined to those of a Robbins algebra, guarantee that the algebra is Boolean. Here we present a number of such properties, and we show how an automated reasoning program was used to obtain the corresponding proofs. Additional properties have been identified, and we include here examples of using such a program to check that the corresponding hand-proofs are correct. We present the appropriate input for many of the examples and also include the resulting proofs in clause notation.
Federal Register 2010, 2011, 2012, 2013, 2014
2013-04-25
... Agency Information Collection Activities; Comment Request; Promoting Student Success in Algebra I Project... notice will be considered public records. Title of Collection: Promoting Student Success in Algebra I... Estimated Number of Annual Burden Hours: 208. Abstract: The Promoting Student Success in Algebra I...
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.
Knot theory and statistical mechanics
Jones, V.F.R. )
1990-11-01
Certain algebraic relations used to solve models in statistical mechanics were key to describing a mathematical property of knots known as a polynomial invariant. This connection, tenuous at first, has since developed into a significant flow of ideas. The appearance of such common ground is not atypical of recent developments in mathematics and physics--ideas from different fields interact and produce unexpected results. Indeed, the discovery of the connection between knots and statistical mechanics passed through a theory intimately related to the mathematical structure of quantum physics. This theory, called von Neumann algebras, is distinguished by the idea of continuous dimensionality. Spaces typically have dimensions that are natural numbers, such as 2, 3 or 11, but in von Neumann algebras dimensions such as 2 or {pi} are equally possible. This possibility for continuous dimension played a key role in joining knot theory and statistical mechanics. In another direction, the knot invariants were soon found to occur in quantum field theory. Indeed, Edward Witten of the Institute for Advanced Study in Princeton, N.J., has shown that topological quantum field theory provides a natural way of expressing the new ideas about knots. This advance, in turn, has allowed a beautiful generalization about the invariants of knots in more complicated three-dimensional spaces known as three-manifolds, in which space itself may contain holes and loops.
Algebra [Student's Individualized Career Source Package].
ERIC Educational Resources Information Center
Tingle, H. Burton
This is a volume of teacher-developed units to supplement the textbook in a first-year algebra course. The units consist mainly of statements of objectives and student worksheets with some examples and references to the textbook given as aids. Major topics covered are basic operations with signed rational numbers and with polynomials,…
Learning Activity Package, Algebra-Trigonometry.
ERIC Educational Resources Information Center
Holland, Bill
A series of ten teacher-prepared Learning Activity Packages (LAPs) in advanced algebra and trigonometry, the units cover logic; absolute value, inequalities, exponents, and complex numbers; functions; higher degree equations and the derivative; the trigonometric function; graphs and applications of the trigonometric functions; sequences and…
Algebra 1Q, Mathematics: 5215.12.
ERIC Educational Resources Information Center
Hirigoyen, Hector
This is the second of the six guidebooks on minimum course content for first-year algebra; it includes the ordered field properties of the real number system, solution of linear equations and inequalities, verbal problems, exponents and operations with polynomials. Overall goals for the course are stated; performance objectives for each unit, a…
Algebra 1p, Mathematics: 5215.11.
ERIC Educational Resources Information Center
Strachan, Florence; Hirigoyen, Hector
This is the first of six guidebooks on minimum course content for first-year algebra; it introduces the language of sets, the fundamental operations and properties of the real number system, the use of variables, and the solution of simple linear equations and inequalities. Overall goals for the course are stated; then performance objectives, a…
Pre-Algebra Groups. Concepts & Applications.
ERIC Educational Resources Information Center
Montgomery County Public Schools, Rockville, MD.
Discussion material and exercises related to pre-algebra groups are provided in this five chapter manual. Chapter 1 (mappings) focuses on restricted domains, order of operations (parentheses and exponents), rules of assignment, and computer extensions. Chapter 2 considers finite number systems, including binary operations, clock arithmetic,…
Piecewise lexsegment ideals in exterior algebras
NASA Astrophysics Data System (ADS)
Shakin, D. A.
2005-02-01
The problem of describing the Hilbert functions of homogeneous ideals of an exterior algebra over a field containing a fixed monomial ideal I is considered. For this purpose the notion of a piecewise lexsegment ideal in an exterior algebra is introduced generalizing the notion of a lexsegment ideal. It is proved that if I is a piecewise lexsegment ideal, then it is possible to describe the Hilbert functions of the homogeneous ideals containing I in a way similar to that suggested by Kruskal and Katona for the situation I=0. Moreover, a generalization of the extremal properties of lexsegment ideals is obtained (the inequality for the Betti numbers).
The Dirac equation and Hestenes' geometric algebra
NASA Astrophysics Data System (ADS)
Hamilton, J. Dwayne
1984-06-01
Hestenes' geometric algebra and Dirac spinors are reviewed and united into a common mathematical formalism, a unification that establishes the Dirac equation as being manifestly covariant under the Lorentz group, and one that needs no matrix representation of the Dirac algebra. New and simple methods of amplitude or ``trace'' calculations are then described. A number of problems are then considered within the context of the new approach, such as relativistic spin projections, new and covariant C and T-transformations and spinors for massless and Majorana fields.
Algebraic Approach to the Computation of the Defining Polynomial of the Algebraic Riccati Equation
NASA Astrophysics Data System (ADS)
Kitamoto, Takuya
The algebraic Riccati equation, which we denote by ’ARE’ in the rest of the paper, is one of the most important equations of the post modern control theory. It plays important role for solving H 2 and H ∞ optimal control problems.
NASA Astrophysics Data System (ADS)
Roitman, Michael
2008-08-01
In this paper we prove that for any commutative (but in general non-associative) algebra A with an invariant symmetric non-degenerate bilinear form there is a graded vertex algebra V = V0 Å V2 Å V3 Å ¼, such that dim V0 = 1 and V2 contains A. We can choose V so that if A has a unit e, then 2e is the Virasoro element of V, and if G is a finite group of automorphisms of A, then G acts on V as well. In addition, the algebra V can be chosen with a non-degenerate invariant bilinear form, in which case it is simple.
Adaptive Algebraic Multigrid Methods
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.
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.
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…
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.
NASA Technical Reports Server (NTRS)
Shahshahani, M.
1991-01-01
The performance characteristics are discussed of certain algebraic geometric codes. Algebraic geometric codes have good minimum distance properties. On many channels they outperform other comparable block codes; therefore, one would expect them eventually to replace some of the block codes used in communications systems. It is suggested that it is unlikely that they will become useful substitutes for the Reed-Solomon codes used by the Deep Space Network in the near future. However, they may be applicable to systems where the signal to noise ratio is sufficiently high so that block codes would be more suitable than convolutional or concatenated codes.
NASA Astrophysics Data System (ADS)
Bouwknegt, Peter
1988-06-01
We investigate extensions of the Virasoro algebra by a single primary field of integer or halfinteger conformal dimension Δ. We argue that for vanishing structure constant CΔΔΔ, the extended conformal algebra can only be associative for a generic c-value if Δ=1/2, 1, 3/2, 2 or 3. For the other Δ<=5 we compute the finite set of allowed c-values and identify the rational solutions. The case CΔΔΔ≠0 is also briefly discussed. I would like to thank Kareljan Schoutens for discussions and Sander Bais for a careful reading of the manuscript.
NASA Astrophysics Data System (ADS)
Dobrev, V. K.
2013-02-01
In the present paper we continue the project of systematic construction of invariant differential operators for non-compact semisimple Lie groups. Our starting points is the class of algebras, which we call 'conformal Lie algebras' (CLA), which have very similar properties to the conformal algebras of Minkowski space-time, though our aim is to go beyond this class in a natural way. For this we introduce the new notion of parabolic relation between two non-compact semisimple Lie algebras G and G ' that have the same complexification and possess maximal parabolic subalgebras with the same complexification. Thus, we consider the exceptional algebra E 7(7) which is parabolically related to the CLA E 7(-25) , the parabolic subalgebras including E 6(6) and E 6(-26). Other interesting examples are the orthogonal algebras so(p, q) all of which are parabolically related to the conformal algebra so( n, 2) with p + q = n + 2, the parabolic subalgebras including the Lorentz subalgebra so( n - 1, 1) and its analogs so( p - 1, q - 1). We consider also E6(6) and E6(2) which are parabolically related to the hermitian symmetric case E6(-14) , the parabolic subalgebras including real forms of sl(6). We also give a formula for the number of representations in the main multiplets valid for CLAs and all algebras that are parabolically related to them. In all considered cases we give the main multiplets of indecomposable elementary representations including the necessary data for all relevant invariant differential operators. In the case of so( p, q) we give also the reduced multiplets. We should stress that the multiplets are given in the most economic way in pairs of shadow fields. Furthermore we should stress that the classification of all invariant differential operators includes as special cases all possible conservation laws and conserved currents, unitary or not.
Taylor, J.P.H.; Walker, A.D.M. )
1987-09-01
When the azimuthal wave number is large, the equations describing standing hydromagnetic waves in the magnetosphere can be written as a set of coupled equations describing the couples magnetosonic and Alfven waves. These equations are decoupled when the filed lines are straight. The eigenfrequencies of the decoupled oscillations are computed. For typical conditions in the outer magnetosphere these give periods in the Pc 4-5 band or above. The longitudinal magnetosonic wave consists of oscillations in the plasma pressure, the longitudinal plasma drift velocity and the compressional magnetic field. Higher harmonics of the standing waves have nodes quite near the equator. These higher harmonics have larger fractional pressure perturbations at high latitudes. The compressional magnetic field for all modes, however, is substantially attenuated at higher latitudes, and the theory predicts that compressional oscillations of B are only likely to be seen near the equator. Conditions can be favorable for resonance to occur between the magnetosonic mode and the transverse Alfven mode. The computed results show periods of the right order of magnitude to explain observations of compressional pulsations. The theory has the potential to explain the polarization when coupling is fully taken into account.
The three-dimensional origin of the classifying algebra
NASA Astrophysics Data System (ADS)
Fuchs, Jürgen; Schweigert, Christoph; Stigner, Carl
2010-01-01
It is known that reflection coefficients for bulk fields of a rational conformal field theory in the presence of an elementary boundary condition can be obtained as representation matrices of irreducible representations of the classifying algebra, a semisimple commutative associative complex algebra. We show how this algebra arises naturally from the three-dimensional geometry of factorization of correlators of bulk fields on the disk. This allows us to derive explicit expressions for the structure constants of the classifying algebra as invariants of ribbon graphs in the three-manifold S×S. Our result unravels a precise relation between intertwiners of the action of the mapping class group on spaces of conformal blocks and boundary conditions in rational conformal field theories.
Tensor Algebra Library for NVidia Graphics Processing Units
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).
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
Not each sequential effect algebra is sharply dominating
NASA Astrophysics Data System (ADS)
Shen, Jun; Wu, Junde
2009-04-01
Let E be an effect algebra and E be the set of all sharp elements of E. E is said to be sharply dominating if for each a∈E there exists a smallest element aˆ∈E such that a⩽aˆ. In 2002, Professors Gudder and Greechie proved that each σ-sequential effect algebra is sharply dominating. In 2005, Professor Gudder presented 25 open problems in [S. Gudder, Int. J. Theory Phys. 44 (2005) 2219], the 3rd problem asked: Is each sequential effect algebra sharply dominating? Now, we construct an example to answer the problem negatively.
The Hochschild cohomology problem for von Neumann algebras
Sinclair, Allan M.; Smith, Roger R.
1998-01-01
In 1967, when Kadison and Ringrose began the development of continuous cohomology theory for operator algebras, they conjectured that the cohomology groups Hn(M, M), n ≥ 1, for a von Neumann algebra M, should all be zero. This conjecture, which has important structural implications for von Neumann algebras, has been solved affirmatively in the type I, II∞, and III cases, leaving open only the type II1 case. In this paper, we describe a positive solution when M is type II1 and has a Cartan subalgebra and a separable predual. PMID:9520373
The hochschild cohomology problem for von neumann algebras.
Sinclair, A M; Smith, R R
1998-03-31
In 1967, when Kadison and Ringrose began the development of continuous cohomology theory for operator algebras, they conjectured that the cohomology groups Hn(M, M), n >/= 1, for a von Neumann algebra M, should all be zero. This conjecture, which has important structural implications for von Neumann algebras, has been solved affirmatively in the type I, IIinfinity, and III cases, leaving open only the type II1 case. In this paper, we describe a positive solution when M is type II1 and has a Cartan subalgebra and a separable predual. PMID:9520373
Some "Big Ideas" of Algebra in the Middle Grades.
ERIC Educational Resources Information Center
Edwards, Thomas G.
2000-01-01
Identifies big ideas for the informal study of algebra-related topics in the middle grades including notation, variable, function, and properties of number. Suggests meaningful ways to accomplish them. (YDS)
NASA Astrophysics Data System (ADS)
Leukhin, Anatolii N.
2005-08-01
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.
Combinatorial quantization of the Hamiltonian Chern-Simons theory II
NASA Astrophysics Data System (ADS)
Alekseev, Anton Yu.; Grosse, Harald; Schomerus, Volker
1996-01-01
This paper further develops the combinatorial approach to quantization of the Hamiltonian Chern Simons theory advertised in [1]. Using the theory of quantum Wilson lines, we show how the Verlinde algebra appears within the context of quantum group gauge theory. This allows to discuss flatness of quantum connections so that we can give a mathematically rigorous definition of the algebra of observables A CS of the Chern Simons model. It is a *-algebra of “functions on the quantum moduli space of flat connections” and comes equipped with a positive functional ω (“integration”). We prove that this data does not depend on the particular choices which have been made in the construction. Following ideas of Fock and Rosly [2], the algebra A CS provides a deformation quantization of the algebra of functions on the moduli space along the natural Poisson bracket induced by the Chern Simons action. We evaluate a volume of the quantized moduli space and prove that it coincides with the Verlinde number. This answer is also interpreted as a partition partition function of the lattice Yang-Mills theory corresponding to a quantum gauge group.
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
Teaching Arithmetic and Algebraic Expressions
ERIC Educational Resources Information Center
Subramaniam, K.; Banerjee, Rakhi
2004-01-01
A teaching intervention study was conducted with sixth grade students to explore the interconnections between students' growing understanding of arithmetic expressions and beginning algebra. Three groups of students were chosen, with two groups receiving instruction in arithmetic and algebra, and one group in algebra without arithmetic. Students…
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…
Spinors in the hyperbolic algebra
NASA Astrophysics Data System (ADS)
Ulrych, S.
2006-01-01
The three-dimensional universal complex Clifford algebra Cbar3,0 is used to represent relativistic vectors in terms of paravectors. In analogy to the Hestenes spacetime approach spinors are introduced in an algebraic form. This removes the dependance on an explicit matrix representation of the algebra.
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.
Pawlak Algebra and Approximate Structure on Fuzzy Lattice
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
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)
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…
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,…
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…
Dual algebraic formulation of differential GPS
NASA Astrophysics Data System (ADS)
Lannes, A.; Dur, S.
2003-05-01
A new approach to differential GPS is presented. The corresponding theoretical framework calls on elementary concepts of algebraic graph theory. The notion of double difference, which is related to that of closure in the sense of Kirchhoff, is revisited in this context. The Moore-Penrose pseudo-inverse of the closure operator plays a key role in the corresponding dual formulation. This approach, which is very attractive from a conceptual point of view, sheds a new light on the Teunissen formulation.
Teichmüller Theory of Bordered Surfaces
NASA Astrophysics Data System (ADS)
Chekhov, Leonid O.
2007-05-01
We propose the graph description of Teichmüller theory of surfaces with marked points on boundary components (bordered surfaces). Introducing new parameters, we formulate this theory in terms of hyperbolic geometry. We can then describe both classical and quantum theories having the proper number of Thurston variables (foliation-shear coordinates), mapping-class group invariance (both classical and quantum), Poisson and quantum algebra of geodesic functions, and classical and quantum braid-group relations. These new algebras can be defined on the double of the corresponding graph related (in a novel way) to a double of the Riemann surface (which is a Riemann surface with holes, not a smooth Riemann surface). We enlarge the mapping class group allowing transformations relating different Teichmüller spaces of bordered surfaces of the same genus, same number of boundary components, and same total number of marked points but with arbitrary distributions of marked po! ints among the boundary components. We describe the classical and quantum algebras and braid group relations for particular sets of geodesic functions corresponding to An and Dn algebras and discuss briefly the relation to the Thurston theory.
NASA Astrophysics Data System (ADS)
Zhang, Ming; Yao, JingTao
2004-04-01
The XML is a new standard for data representation and exchange on the Internet. There are studies on XML query languages as well as XML algebras in literature. However, attention has not been paid to research on XML algebras for data mining due to partially the fact that there is no widely accepted definition of XML mining tasks. This paper tries to examine the XML mining tasks and provide guidelines to design XML algebras for data mining. Some summarization and comparison have been done to existing XML algebras. We argue that by adding additional operators for mining tasks, XML algebras may work well for data mining with XML documents.
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…
Yang-Mills connections valued on the octonionic algebra
NASA Astrophysics Data System (ADS)
Restuccia, A.; Veiro, J. P.
2016-05-01
We consider a formulation of Yang-Mills theory where the gauge field is valued on an octonionic algebra and the gauge transformation is the group of automorphisms of it. We show, under mild assumptions, that the only possible gauge formulations are the usual su(2) or u(1) Yang-Mills theories.
On Dunkl angular momenta algebra
NASA Astrophysics Data System (ADS)
Feigin, Misha; Hakobyan, Tigran
2015-11-01
We consider the quantum angular momentum generators, deformed by means of the Dunkl operators. Together with the reflection operators they generate a subalgebra in the rational Cherednik algebra associated with a finite real reflection group. We find all the defining relations of the algebra, which appear to be quadratic, and we show that the algebra is of Poincaré-Birkhoff-Witt (PBW) type. We show that this algebra contains the angular part of the Calogero-Moser Hamiltonian and that together with constants it generates the centre of the algebra. We also consider the gl( N ) version of the subalge-bra of the rational Cherednik algebra and show that it is a non-homogeneous quadratic algebra of PBW type as well. In this case the central generator can be identified with the usual Calogero-Moser Hamiltonian associated with the Coxeter group in the harmonic confinement.
Algebraic connectivity and graph robustness.
Feddema, John Todd; Byrne, Raymond Harry; Abdallah, Chaouki T.
2009-07-01
Recent papers have used Fiedler's definition of algebraic connectivity to show that network robustness, as measured by node-connectivity and edge-connectivity, can be increased by increasing the algebraic connectivity of the network. By the definition of algebraic connectivity, the second smallest eigenvalue of the graph Laplacian is a lower bound on the node-connectivity. In this paper we show that for circular random lattice graphs and mesh graphs algebraic connectivity is a conservative lower bound, and that increases in algebraic connectivity actually correspond to a decrease in node-connectivity. This means that the networks are actually less robust with respect to node-connectivity as the algebraic connectivity increases. However, an increase in algebraic connectivity seems to correlate well with a decrease in the characteristic path length of these networks - which would result in quicker communication through the network. Applications of these results are then discussed for perimeter security.
Brigner, W L
1983-12-01
Prime numbers are used to code various dimensions of an input matrix (receptor surface), i.e., prime numbers code the position of each cell in the matrix, the position of each column and row constituted by the cells of the matrix, and the orientation of each such column or row. The coding permits any pattern or stimulus configuration to be changed into a single, unique number, viz., the serial product of the prime numbers which code the relevant dimensions of the pattern. Storage of a pattern is effected by storage of the serial product. By factoring the serial product, the pattern or stimulus configuration is analyzed into the dimensions (features) specified by the code. The factorization is also utilized in abstracting a schema, in approximating a feature count model, and in presenting a strategy for holistic vs sequential pattern processing. Finally, the relationship between the serial product representing a pattern and the phenomenal experience of a pattern is explored. PMID:6664792
Marquette, Ian
2013-07-15
We introduce the most general quartic Poisson algebra generated by a second and a fourth order integral of motion of a 2D superintegrable classical system. We obtain the corresponding quartic (associative) algebra for the quantum analog, extend Daskaloyannis construction obtained in context of quadratic algebras, and also obtain the realizations as deformed oscillator algebras for this quartic algebra. We obtain the Casimir operator and discuss how these realizations allow to obtain the finite-dimensional unitary irreducible representations of quartic algebras and obtain algebraically the degenerate energy spectrum of superintegrable systems. We apply the construction and the formula obtained for the structure function on a superintegrable system related to type I Laguerre exceptional orthogonal polynomials introduced recently.
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.
NASA Astrophysics Data System (ADS)
Norton, Michael; Ross, Frances; Bau, Haim
2015-11-01
Using a hermetically sealed liquid cell, we observed the growth and migration of bubbles (tens to hundreds of nanometers in diameter) in a tapered conduit and supersaturated solution with a transmission electron microscope. To better understand bubble shape and migration dynamics, we developed simple 2D and 3D models valid in the limit of zero capillary and Bond numbers. The 3D model is restricted to small taper slope, weakly non-circular contact line geometries and large bubble aspect ratio (high confinement), and was solved using a pseudo-spectral decomposition. Both models utilize the Blake-Haynes mechanism to relate dynamic contact angle to local contact line velocity The influence of pinning of a portion of the contact line on bubble geometry is also considered. Contact line dissipation controls curvature and regulates growth rate. Our 2D and 3D models predict growth rates in agreement with experimental observations, but several orders of magnitude lower than predicted by the classical Epstein - Plesset theory. The work was supported, in part, by NSF CBET grant 1066573.
Dual spaces of differential Lie algebras
Kupershmidt, B.A.
1982-01-01
We present a mathematical scheme which serves as an infinite-dimensional generalization of Poisson structures on dual spaces of finite-dimensional Lie algebras, which are well known and widely used in classical mechanics. These structures have recently appeared in the theory of Lax equations, long waves in hydrodynamics, and various other physical models: compressible hydrodynamics, magnetohydrodynamics, multifluid plasmas, elasticity, superfluid /sup 4/He and /sup 3/He-A, Ginzburg-Landau theory of superconductors, and classical chromohydrodynamics (the generalization of plasma physics to Yang-Mills interactions).
An algebraic approach to BCJ numerators
NASA Astrophysics Data System (ADS)
Fu, Chih-Hao; Du, Yi-Jian; Feng, Bo
2013-03-01
One important discovery in recent years is that the total amplitude of gauge theory can be written as BCJ form where kinematic numerators satisfy Jacobi identity. Although the existence of such kinematic numerators is no doubt, the simple and explicit construction is still an important problem. As a small step, in this note we provide an algebraic approach to construct these kinematic numerators. Under our Feynman-diagram-like construction, the Jacobi identity is manifestly satisfied. The corresponding color ordered amplitudes satisfy off-shell KK-relation and off-shell BCJ relation similar to the color ordered scalar theory. Using our construction, the dual DDM form is also established.
Heptagonal Numbers: A Neglected Set.
ERIC Educational Resources Information Center
O'Donnell, William J.
1979-01-01
An investigation of these numbers is shown to give students an opportunity to make use of such mathematical skills as algebraic substitution, modular arithmetic, counting arguments, and mathematical induction. (MP)
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®,…
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.
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
Generalizing the Connes Moscovici Hopf algebra to contain all rooted trees
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.
Generalizing the Connes Moscovici Hopf algebra to contain all rooted trees
NASA Astrophysics Data System (ADS)
Agarwala, Susama; Delaney, Colleen
2015-04-01
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.
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.
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.
Compactly Generated de Morgan Lattices, Basic Algebras and Effect Algebras
NASA Astrophysics Data System (ADS)
Paseka, Jan; Riečanová, Zdenka
2010-12-01
We prove that a de Morgan lattice is compactly generated if and only if its order topology is compatible with a uniformity on L generated by some separating function family on L. Moreover, if L is complete then L is (o)-topological. Further, if a basic algebra L (hence lattice with sectional antitone involutions) is compactly generated then L is atomic. Thus all non-atomic Boolean algebras as well as non-atomic lattice effect algebras (including non-atomic MV-algebras and orthomodular lattices) are not compactly generated.
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;…
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;…
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…
Dynamics of gelling liquids: algebraic relaxation.
Srivastava, Sunita; Kumar, C N; Tankeshwar, K
2009-08-19
The sol-gel system which is known, experimentally, to exhibit a power law decay of stress autocorrelation function has been studied theoretically. A second-order nonlinear differential equation obtained from Mori's integro-differential equation is derived which provides the algebraic decay of a time correlation function. Involved parameters in the expression obtained are related to exact properties of the corresponding correlation function. The algebraic model has been applied to Lennard-Jones and sol-gel systems. The model shows the behaviour of viscosity as has been observed in computer simulation and theoretical studies. The expression obtained for the viscosity predicts a logarithmic divergence at a critical value of the parameter in agreement with the prediction of other theories. PMID:21828600
Locally finite dimensional Lie algebras
NASA Astrophysics Data System (ADS)
Hennig, Johanna
We prove that in a locally finite dimensional Lie algebra L, any maximal, locally solvable subalgebra is the stabilizer of a maximal, generalized flag in an integrable, faithful module over L. Then we prove two structure theorems for simple, locally finite dimensional Lie algebras over an algebraically closed field of characteristic p which give sufficient conditions for the algebras to be of the form [K(R, *), K( R, *)] / (Z(R) ∩ [ K(R, *), K(R, *)]) for a simple, locally finite dimensional associative algebra R with involution *. Lastly, we explore the noncommutative geometry of locally simple representations of the diagonal locally finite Lie algebras sl(ninfinity), o( ninfinity), and sp(n infinity).
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
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.
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…
On the Generalized Unitary Parasupersymmetry Algebra of Beckers-Debergh
NASA Astrophysics Data System (ADS)
Chenaghlou, A.; Fakhri, H.
An appropriate generalization of the unitary parasupersymmetry algebra of Beckers-Debergh to arbitrary order is presented in this paper. A special representation for realizing the even arbitrary order unitary parasupersymmetry algebra of Beckers-Debergh is analyzed by one-dimensional shape invariance solvable models, 2D and 3D quantum solvable models obtained from the shape invariance theory as well. In particular, in the special representation, it is shown that the isospectrum Hamiltonians consist of the two partner Hamiltonians of the shape invariance theory.
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.
Quantum Phase Space from Schwinger's Measurement Algebra
NASA Astrophysics Data System (ADS)
Watson, P.; Bracken, A. J.
2014-07-01
Schwinger's algebra of microscopic measurement, with the associated complex field of transformation functions, is shown to provide the foundation for a discrete quantum phase space of known type, equipped with a Wigner function and a star product. Discrete position and momentum variables label points in the phase space, each taking distinct values, where is any chosen prime number. Because of the direct physical interpretation of the measurement symbols, the phase space structure is thereby related to definite experimental configurations.
Pure Spinors in AdS and Lie Algebra Cohomology
NASA Astrophysics Data System (ADS)
Mikhailov, Andrei
2014-10-01
We show that the BRST cohomology of the massless sector of the Type IIB superstring on AdS5 × S 5 can be described as the relative cohomology of an infinite-dimensional Lie superalgebra. We explain how the vertex operators of ghost number 1, which correspond to conserved currents, are described in this language. We also give some algebraic description of the ghost number 2 vertices, which appears to be new. We use this algebraic description to clarify the structure of the zero mode sector of the ghost number two states in flat space, and initiate the study of the vertices of the higher ghost number.
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…
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.
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.
NASA Astrophysics Data System (ADS)
Restrepo, Doriam; Gómez, Juan David; Jaramillo, Juan Diego
2014-09-01
We present a closed-form frequency-wave number ( ω - k) Green's function for a layered, elastic half-space under SH wave propagation. It is shown that for every ( ω - k) pair, the fundamental solution exhibits two distinctive features: (1) the original layered system can be reduced to a system composed by the uppermost superficial layer over an equivalent half-space; (2) the fundamental solution can be partitioned into three different fundamental solutions, each one carrying out a different physical interpretation, i.e., an equivalent half-space, source image impact, and dispersive wave effect, respectively. Such an interpretation allows the proper use of analytical and numerical integration schemes, and ensures the correct assessment of Cauchy principal value integrals. Our method is based upon a stiffness-matrix scheme, and as a first approach we assume that observation points and the impulsive SH line-source are spatially located within the uppermost superficial layer. We use a discrete wave number boundary element strategy to test the benefits of our fundamental solution. We benchmark our results against reported solutions for an infinitely long circular canyon subjected to oblique incident SH waves within a homogeneous half-space. Our results show an almost exact agreement with previous studies. We further shed light on the impact of horizontal strata by examining the dynamic response of the circular canyon to oblique incident SH waves under different layered half-space configurations and incident angles. Our results show that modifications in the layering structure manifest by larger peak ground responses, and stronger spatial variability due to interactions of the canyon geometry with trapped Love waves in combination with impedance contrast effects.
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…
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…
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…
Contraction-based classification of supersymmetric extensions of kinematical lie algebras
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.
Philip, Bobby; Chartier, Dr Timothy
2012-01-01
methods based on Local Sensitivity Analysis (LSA). The method can be used in the context of geometric and algebraic multigrid methods for constructing smoothers, and in the context of Krylov methods for constructing block preconditioners. It is suitable for both constant and variable coecient problems. Furthermore, the method can be applied to systems arising from both scalar and coupled system partial differential equations (PDEs), as well as linear systems that do not arise from PDEs. The simplicity of the method will allow it to be easily incorporated into existing multigrid and Krylov solvers while providing a powerful tool for adaptively constructing methods tuned to a problem.
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
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
Twisted fermionic oscillator algebra in κ-minkowski space-time
NASA Astrophysics Data System (ADS)
Verma, Ravikant
2015-04-01
In this paper, we investigate the twisted algebra of the fermionic oscillators associated with Dirac field defined in κ-Minkowski space-time. Starting from κ-deformed Dirac theory, which is invariant under the undeformed κ-Poincaré algebra, using the twisted flip operator, we derive the deformed algebra of the creation and annihilation operators corresponding to the Dirac field quanta in κ-Minkowski space-time. In the limit a → 0, the deformed algebra reduces to the commutative result.
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.
A Renormalisation Group Method. I. Gaussian Integration and Normed Algebras
NASA Astrophysics Data System (ADS)
Brydges, David C.; Slade, Gordon
2015-05-01
This paper is the first in a series devoted to the development of a rigorous renormalisation group method for lattice field theories involving boson fields, fermion fields, or both. Our immediate motivation is a specific model, involving both boson and fermion fields, which arises as a representation of the continuous-time weakly self-avoiding walk. In this paper, we define normed algebras suitable for a renormalisation group analysis, and develop methods for performing analysis on these algebras. We also develop the theory of Gaussian integration on these normed algebras, and prove estimates for Gaussian integrals. The concepts and results developed here provide a foundation for the continuation of the method presented in subsequent papers in the series.
Algebraic Squares: Complete and Incomplete.
ERIC Educational Resources Information Center
Gardella, Francis J.
2000-01-01
Illustrates ways of using algebra tiles to give students a visual model of competing squares that appear in algebra as well as in higher mathematics. Such visual representations give substance to the symbolic manipulation and give students who do not learn symbolically a way of understanding the underlying concepts of completing the square. (KHR)
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…
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.)
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…
Algebraic Thinking in Adult Education
ERIC Educational Resources Information Center
Manly, Myrna; Ginsburg, Lynda
2010-01-01
In adult education, algebraic thinking can be a sense-making tool that introduces coherence among mathematical concepts for those who previously have had trouble learning math. Further, a modeling approach to algebra connects mathematics and the real world, demonstrating the usefulness of math to those who have seen it as just an academic…
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.)
Exploring Algebraic Patterns through Literature.
ERIC Educational Resources Information Center
Austin, Richard A.; Thompson, Denisse R.
1997-01-01
Presents methods for using literature to develop algebraic thinking in an environment that connects algebra to various situations. Activities are based on the book "Anno's Magic Seeds" with additional resources listed. Students express a constant function, exponential function, and a recursive function in their own words as well as writing about…
Learning Algebra from Worked Examples
ERIC Educational Resources Information Center
Lange, Karin E.; Booth, Julie L.; Newton, Kristie J.
2014-01-01
For students to be successful in algebra, they must have a truly conceptual understanding of key algebraic features as well as the procedural skills to complete a problem. One strategy to correct students' misconceptions combines the use of worked example problems in the classroom with student self-explanation. "Self-explanation" is…
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.
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.
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
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.
Multifractal vector fields and stochastic Clifford algebra
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.
Multifractal vector fields and stochastic Clifford algebra.
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
Heisenberg uncertainty in reduced power algebras
NASA Astrophysics Data System (ADS)
Rosinger, Elemér E.
2012-12-01
The Heisenberg uncertainty relation is known to be obtainable by a purely mathematical argument. Based on that fact, here it is shown that the Heisenberg uncertainty relation remains valid when Quantum Mechanics is re-formulated within far wider frameworks of scalars, namely, within one or the other of the infinitely many reduced power algebras which can replace the usual real numbers R, or complex numbers C. Three possible major advantages in Physics of such a reformulation are: 1) the disappearance of the well known and hard to deal with problem of the so called "infinities in Physics", 2) the possibilitiy to have infinitely many "levels of precision" instead of the only one existing at present, 3) the possibility to model "hierarchies of Planck constants", [2]. Last and not least, the scalars given by reduced power algebras contain as a particular case those obtained by Nonstandard Analysis, yet they are far more simple and easy to deal with, being in fact on the level of a first course in Algebra. A detailed version of this paper can be found in arxiv:0901.4825.
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.
Supersymmetry algebra cohomology. IV. Primitive elements in all dimensions from D = 4 to D = 11
NASA Astrophysics Data System (ADS)
Brandt, Friedemann
2013-05-01
The primitive elements of the supersymmetry algebra cohomology as defined in previous work are derived for standard supersymmetry algebras in dimensions D = 5, …, 11 for all signatures of the related Clifford algebras of gamma matrices and all numbers of supersymmetries. The results are presented in a uniform notation along with results of previous work for D = 4, and derived by means of dimensional extension from D = 4 up to D = 11.
On Sasakian hypersurfaces in 6-dimensional Hermitian submanifolds of the Cayley algebra
Banaru, M B
2003-08-31
A criterion for the minimality of a Sasakian hypersurface in a 6-dimensional Hermitian submanifold of the octave algebra is found. It is proved that the type number of a Sasakian hypersurface in a 6-dimensional Hermitian submanifold of the octave algebra is four or five. It is also proved that a Sasakian hypersurface in a 6-dimensional Hermitian submanifold of the Cayley algebra is minimal if and only if it is ruled.
Supersymmetry algebra cohomology. IV. Primitive elements in all dimensions from D= 4 to D= 11
Brandt, Friedemann
2013-05-15
The primitive elements of the supersymmetry algebra cohomology as defined in previous work are derived for standard supersymmetry algebras in dimensions D= 5, Horizontal-Ellipsis , 11 for all signatures of the related Clifford algebras of gamma matrices and all numbers of supersymmetries. The results are presented in a uniform notation along with results of previous work for D= 4, and derived by means of dimensional extension from D= 4 up to D= 11.
NASA Astrophysics Data System (ADS)
Uribe Peláez, Simon
2010-04-01
Identifying coverage holes makes an important topic for optimization of quality service for wireless sensor network hosts. This paper introduces a new way to identify and describe how is the network's structure, its number of holes and its components, assuming there's a sensor covering an area where a network communication exists. The simplicial complex method and algebraic graph theory will be applied. Betti numbers and Euler characteristics will be used for a sensor network represented by a simplicial complex, and the Tutte polynomial will be used for describing visual graphs algebraically, for a complete identification.
Algebraic geometry realization of quantum Hall soliton
NASA Astrophysics Data System (ADS)
Abounasr, R.; Ait Ben Haddou, M.; El Rhalami, A.; Saidi, E. H.
2005-02-01
Using the Iqbal-Netzike-Vafa dictionary giving the correspondence between the H2 homology of del Pezzo surfaces and p-branes, we develop a way to approach the system of brane bounds in M-theory on S1. We first review the structure of 10-dimensional quantum Hall soliton (QHS) from the view of M-theory on S1. Then, we show how the D0 dissolution in D2-brane is realized in M-theory language and derive the p-brane constraint equations used to define appropriately the QHS. Finally, we build an algebraic geometry realization of the QHS in type IIA superstring and show how to get its type IIB dual. Other aspects are also discussed.
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.
The conceptual basis of mathematics in cardiology: (I) algebra, functions and graphs.
Bates, Jason H T; Sobel, Burton E
2003-02-01
This is the first in a series of four articles developed for the readers of. Without language ideas cannot be articulated. What may not be so immediately obvious is that they cannot be formulated either. One of the essential languages of cardiology is mathematics. Unfortunately, medical education does not emphasize, and in fact, often neglects empowering physicians to think mathematically. Reference to statistics, conditional probability, multicompartmental modeling, algebra, calculus and transforms is common but often without provision of genuine conceptual understanding. At the University of Vermont College of Medicine, Professor Bates developed a course designed to address these deficiencies. The course covered mathematical principles pertinent to clinical cardiovascular and pulmonary medicine and research. It focused on fundamental concepts to facilitate formulation and grasp of ideas. This series of four articles was developed to make the material available for a wider audience. The articles will be published sequentially in Coronary Artery Disease. Beginning with fundamental axioms and basic algebraic manipulations they address algebra, function and graph theory, real and complex numbers, calculus and differential equations, mathematical modeling, linear system theory and integral transforms and statistical theory. The principles and concepts they address provide the foundation needed for in-depth study of any of these topics. Perhaps of even more importance, they should empower cardiologists and cardiovascular researchers to utilize the language of mathematics in assessing the phenomena of immediate pertinence to diagnosis, pathophysiology and therapeutics. The presentations are interposed with queries (by Coronary Artery Disease, abbreviated as CAD) simulating the nature of interactions that occurred during the course itself. Each article concludes with one or more examples illustrating application of the concepts covered to cardiovascular medicine and
NASA Astrophysics Data System (ADS)
Fan, Hong-Yi; Lou, Sen-Yue
2015-07-01
Based on the operator Hermite polynomials method (OHPM), we study Stirling numbers in the context of quantum mechanics, i.e., we present operator realization of generating function formulas of Stirling numbers with some applications. As a by-product, we derive a summation formula involving both Stirling number and Hermite polynomials. Project supported by the National Natural Science Foundation of China (Grant No. 11175113).
Realization Of Algebraic Processor For XML Documents Processing
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.
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.
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…
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…
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…
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.…
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)
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…
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…