Generalized two-dimensional chiral QED: Anomaly and exotic statistics

DOE Office of Scientific and Technical Information (OSTI.GOV)

Saradzhev, F.M.

1997-07-01

We study the influence of the anomaly on the physical quantum picture of the generalized chiral Schwinger model defined on S{sup 1}. We show that the anomaly (i) results in the background linearly rising electric field and (ii) makes the spectrum of the physical Hamiltonian nonrelativistic without a massive boson. The physical matter fields acquire exotic statistics. We construct explicitly the algebra of the Poincar{acute e} generators and show that it differs from the Poincar{acute e} one. We exhibit the role of the vacuum Berry phase in the failure of the Poincar{acute e} algebra to close. We prove that, inmore » spite of the background electric field, such phenomenon as the total screening of external charges characteristic for the standard Schwinger model takes place in the generalized chiral Schwinger model, too. {copyright} {ital 1997} {ital The American Physical Society}« less

Identities of Finitely Generated Algebras Over AN Infinite Field

NASA Astrophysics Data System (ADS)

Kemer, A. R.

1991-02-01

It is proved that for each finitely generated associative PI-algebra U over an infinite field F, there is a finite-dimensional F-algebra C such that the ideals of identities of the algebras U and C coincide. This yields a positive solution to the local problem of Specht for algebras over an infinite field: A finitely generated free associative algebra satisfies the maximum condition for T-ideals.

The hopf algebra of vector fields on complex quantum groups

NASA Astrophysics Data System (ADS)

Drabant, Bernhard; Jurčo, Branislav; Schlieker, Michael; Weich, Wolfgang; Zumino, Bruno

1992-10-01

We derive the equivalence of the complex quantum enveloping algebra and the algebra of complex quantum vector fields for the Lie algebra types A n , B n , C n , and D n by factorizing the vector fields uniquely into a triangular and a unitary part and identifying them with the corresponding elements of the algebra of regular functionals.

A calculus based on a q-deformed Heisenberg algebra

DOE PAGES

Cerchiai, B. L.; Hinterding, R.; Madore, J.; ...

1999-04-27

We show how one can construct a differential calculus over an algebra where position variables $x$ and momentum variables p have be defined. As the simplest example we consider the one-dimensional q-deformed Heisenberg algebra. This algebra has a subalgebra generated by cursive Greek chi and its inverse which we call the coordinate algebra. A physical field is considered to be an element of the completion of this algebra. We can construct a derivative which leaves invariant the coordinate algebra and so takes physical fields into physical fields. A generalized Leibniz rule for this algebra can be found. Based on thismore » derivative differential forms and an exterior differential calculus can be constructed.« less

Gapless bosonic excitation without symmetry breaking: An algebraic spin liquid with soft gravitons

NASA Astrophysics Data System (ADS)

Xu, Cenke

2006-12-01

A quantum ground state of matter is realized in a bosonic model on a three-dimensional fcc lattice with emergent low energy excitations. The phase obtained is a stable gapless boson liquid phase, with algebraic boson density correlations. The stability of this phase is protected against the instanton effect and superfluidity by self-duality and large gauge symmetries on both sides of the duality. The gapless collective excitations of this phase closely resemble the graviton, although they have a soft ω˜k2 dispersion relation. There are three branches of gapless excitations in this phase, one of which is gapless scalar trace mode, the other two have the same polarization and gauge symmetries as the gravitons. The dynamics of this phase is described by a set of Maxwell’s equations. The defects carrying gauge charges can drive the system into the superfluid order when the defects are condensed; also the topological defects are coupled to the dual gauge field in the same manner as the charge defects couple to the original gauge field, after the condensation of the topological defects, the system is driven into the Mott insulator phase. In the two-dimensional case, the gapless soft graviton as well as the algebraic liquid phase are destroyed by the vertex operators in the dual theory, and the stripe order is most likely to take place close to the two-dimensional quantum critical point at which the vertex operators are tuned to zero.

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.

Canonical formulation and conserved charges of double field theory

DOE PAGES

Naseer, Usman

2015-10-26

We provide the canonical formulation of double field theory. It is shown that this dynamics is subject to primary and secondary constraints. The Poisson bracket algebra of secondary constraints is shown to close on-shell according to the C-bracket. We also give a systematic way of writing boundary integrals in doubled geometry. Finally, by including appropriate boundary terms in the double field theory Hamiltonian, expressions for conserved energy and momentum of an asymptotically flat doubled space-time are obtained and applied to a number of solutions.

Algebraic Structure of tt * Equations for Calabi-Yau Sigma Models

NASA Astrophysics Data System (ADS)

Alim, Murad

2017-08-01

The tt * equations define a flat connection on the moduli spaces of {2d, \\mathcal{N}=2} quantum field theories. For conformal theories with c = 3 d, which can be realized as nonlinear sigma models into Calabi-Yau d-folds, this flat connection is equivalent to special geometry for threefolds and to its analogs in other dimensions. We show that the non-holomorphic content of the tt * equations, restricted to the conformal directions, in the cases d = 1, 2, 3 is captured in terms of finitely many generators of special functions, which close under derivatives. The generators are understood as coordinates on a larger moduli space. This space parameterizes a freedom in choosing representatives of the chiral ring while preserving a constant topological metric. Geometrically, the freedom corresponds to a choice of forms on the target space respecting the Hodge filtration and having a constant pairing. Linear combinations of vector fields on that space are identified with the generators of a Lie algebra. This Lie algebra replaces the non-holomorphic derivatives of tt * and provides these with a finer and algebraic meaning. For sigma models into lattice polarized K3 manifolds, the differential ring of special functions on the moduli space is constructed, extending known structures for d = 1 and 3. The generators of the differential rings of special functions are given by quasi-modular forms for d = 1 and their generalizations in d = 2, 3. Some explicit examples are worked out including the case of the mirror of the quartic in {\\mathbbm{P}^3}, where due to further algebraic constraints, the differential ring coincides with quasi modular forms.

On the Hamiltonian formalism of the tetrad-gravity with fermions

NASA Astrophysics Data System (ADS)

Lagraa, M. H.; Lagraa, M.

2018-06-01

We extend the analysis of the Hamiltonian formalism of the d-dimensional tetrad-connection gravity to the fermionic field by fixing the non-dynamic part of the spatial connection to zero (Lagraa et al. in Class Quantum Gravity 34:115010, 2017). Although the reduced phase space is equipped with complicated Dirac brackets, the first-class constraints which generate the diffeomorphisms and the Lorentz transformations satisfy a closed algebra with structural constants analogous to that of the pure gravity. We also show the existence of a canonical transformation leading to a new reduced phase space equipped with Dirac brackets having a canonical form leading to the same algebra of the first-class constraints.

Topics in elementary particle physics

NASA Astrophysics Data System (ADS)

Jin, Xiang

The author of this thesis discusses two topics in elementary particle physics: n-ary algebras and their applications to M-theory (Part I), and functional evolution and Renormalization Group flows (Part II). In part I, Lie algebra is extended to four different n-ary algebraic structure: generalized Lie algebra, Filippov algebra, Nambu algebra and Nambu-Poisson tensor; though there are still many other n-ary algebras. A natural property of Generalized Lie algebras — the Bremner identity, is studied, and proved with a totally different method from its original version. We extend Bremner identity to n-bracket cases, where n is an arbitrary odd integer. Filippov algebras do not focus on associativity, and are defined by the Fundamental identity. We add associativity to Filippov algebras, and give examples of how to construct Filippov algebras from su(2), bosonic oscillator, Virasoro algebra. We try to include fermionic charges into the ternary Virasoro-Witt algebra, but the attempt fails because fermionic charges keep generating new charges that make the algebra not closed. We also study the Bremner identity restriction on Nambu algebras and Nambu-Poisson tensors. So far, the only example 3-algebra being used in physics is the BLG model with 3-algebra A4, describing two M2-branes interactions. Its extension with Nambu algebra, BLG-NB model, is believed to describe infinite M2-branes condensation. Also, there is another propose for M2-brane interactions, the ABJM model, which is constructed by ordinary Lie algebra. We compare the symmetry properties between them, and discuss the possible approaches to include these three models into a grand unification theory. In Part II, we give an approximate solution for Schroeder's equations, based on series and conjugation methods. We use the logistic map as an example, and demonstrate that this approximate solution converges to known analytical solutions around the fixed point, around which the approximate solution is constructed. Although the closed-form solutions for Schroeder's equations can not always be approached analytically, by fitting the approximation solutions, one can still obtain closed-form solutions sometimes. Based on Schroeder's theory, approximate solutions for trajectories, velocities and potentials can also be constructed. The approximate solution is significantly useful to calculate the beta function in renormalization group trajectory. By "wrapping" the series solutions with the conjugations from different inverse functions, we generate different branches of the trajectory, and construct a counterexample for a folk theorem about limited cycles.

BRST formulation of 4-monopoles

NASA Astrophysics Data System (ADS)

Gianvittorio, R.; Martin, I.; Restuccia, A.

1996-11-01

A supersymmetric gauge-invariant action is constructed over any four-dimensional Riemannian manifold describing Witten's theory of 4-monopoles. The topological supersymmetric algebra closes off-shell. The multiplets include the auxiliary fields and the Wess - Zumino fields in an unusual way, arising naturally from BRST gauge fixing. A new canonical approach over Riemann manifolds is followed, using a Morse function as a Euclidean time and taking into account the BRST boundary conditions that come from the BFV formulation. This allows a construction of the effective action starting from gauge principles.

BFV-BRST quantization of two-dimensional supergravity

NASA Astrophysics Data System (ADS)

Fujiwara, T.; Igarashi, Y.; Kuriki, R.; Tabei, T.

1996-01-01

Two-dimensional supergravity theory is quantized as an anomalous gauge theory. In the Batalin-Fradkin (BF) formalism, the anomaly-canceling super-Liouville fields are introduced to identify the original second-class constrained system with a gauge-fixed version of a first-class system. The BFV-BRST quantization applies to formulate the theory in the most general class of gauges. A local effective action constructed in the configuration space contains two super-Liouville actions; one is a noncovariant but local functional written only in terms of two-dimensional supergravity fields, and the other contains the super-Liouville fields canceling the super-Weyl anomaly. Auxiliary fields for the Liouville and the gravity supermultiplets are introduced to make the BRST algebra close off-shell. Inclusion of them turns out to be essentially important especially in the super-light-cone gauge fixing, where the supercurvature equations (∂3-g++=∂2-χ++=0) are obtained as a result of BRST invariance of the theory. Our approach reveals the origin of the OSp(1,2) current algebra symmetry in a transparent manner.

Double field theory at order α'

NASA Astrophysics Data System (ADS)

Hohm, Olaf; Zwiebach, Barton

2014-11-01

We investigate α' corrections of bosonic strings in the framework of double field theory. The previously introduced "doubled α'-geometry" gives α'-deformed gauge transformations arising in the Green-Schwarz anomaly cancellation mechanism but does not apply to bosonic strings. These require a different deformation of the duality-covariantized Courant bracket which governs the gauge structure. This is revealed by examining the α' corrections in the gauge algebra of closed string field theory. We construct a four-derivative cubic double field theory action invariant under the deformed gauge transformations, giving a first glimpse of the gauge principle underlying bosonic string α' corrections. The usual metric and b-field are related to the duality covariant fields by non-covariant field redefinitions.

Lie algebra of conformal Killing-Yano forms

NASA Astrophysics Data System (ADS)

Ertem, Ümit

2016-06-01

We provide a generalization of the Lie algebra of conformal Killing vector fields to conformal Killing-Yano forms. A new Lie bracket for conformal Killing-Yano forms that corresponds to slightly modified Schouten-Nijenhuis bracket of differential forms is proposed. We show that conformal Killing-Yano forms satisfy a graded Lie algebra in constant curvature manifolds. It is also proven that normal conformal Killing-Yano forms in Einstein manifolds also satisfy a graded Lie algebra. The constructed graded Lie algebras reduce to the graded Lie algebra of Killing-Yano forms and the Lie algebras of conformal Killing and Killing vector fields in special cases.

Renormalization group flows and continual Lie algebras

NASA Astrophysics Data System (ADS)

Bakas, Ioannis

2003-08-01

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

A Domain-Specific Language for Discrete Mathematics

NASA Astrophysics Data System (ADS)

Jha, Rohit; Samuel, Alfy; Pawar, Ashmee; Kiruthika, M.

2013-05-01

This paper discusses a Domain Specific Language (DSL) that has been developed to enable implementation of concepts of discrete mathematics. A library of data types and functions provides functionality which is frequently required by users. Covering the areas of Mathematical Logic, Set Theory, Functions, Graph Theory, Number Theory, Linear Algebra and Combinatorics, the language's syntax is close to the actual notation used in the specific fields.

Twist for Snyder space

NASA Astrophysics Data System (ADS)

Meljanac, Daniel; Meljanac, Stjepan; Mignemi, Salvatore; Pikutić, Danijel; Štrajn, Rina

2018-03-01

We construct the twist operator for the Snyder space. Our starting point is a non-associative star product related to a Hermitian realisation of the noncommutative coordinates originally introduced by Snyder. The corresponding coproduct of momenta is non-coassociative. The twist is constructed using a general definition of the star product in terms of a bi-differential operator in the Hopf algebroid approach. The result is given by a closed analytical expression. We prove that this twist reproduces the correct coproducts of the momenta and the Lorentz generators. The twisted Poincaré symmetry is described by a non-associative Hopf algebra, while the twisted Lorentz symmetry is described by the undeformed Hopf algebra. This new twist might be important in the construction of different types of field theories on Snyder space.

Generalized Quantum Field Theory Based on a Nonlinear Deformed Heisenberg Algebra

NASA Astrophysics Data System (ADS)

Ribeiro-Silva, C. I.; Oliveira-Neto, N. M.

We consider a quantum field theory based on a nonlinear Heisenberg algebra which describes phenomenologically a composite particle. Perturbative computation, considering the λϕ4 interaction was done and we also performed some comparison with a quantum field theory based on the q-oscillator algebra.

Enveloping algebra-valued gauge transformations for non-abelian gauge groups on non-commutative spaces

NASA Astrophysics Data System (ADS)

Jurco, B.; Schraml, S.; Schupp, P.; Wess, J.

2000-11-01

An enveloping algebra-valued gauge field is constructed, its components are functions of the Lie algebra-valued gauge field and can be constructed with the Seiberg-Witten map. This allows the formulation of a dynamics for a finite number of gauge field components on non-commutative spaces.

Systems with outer constraints. Gupta-Bleuler electromagnetism as an algebraic field theory

NASA Astrophysics Data System (ADS)

Grundling, Hendrik

1988-03-01

Since there are some important systems which have constraints not contained in their field algebras, we develop here in a C*-context the algebraic structures of these. The constraints are defined as a group G acting as outer automorphisms on the field algebra ℱ, α: G ↦ Aut ℱ, α G ⊄ Inn ℱ, and we find that the selection of G-invariant states on ℱ is the same as the selection of states ω on M( G M(Gmathop × limits_α F) ℱ) by ω( U g)=1∨ g∈ G, where U g ∈ M ( G M(Gmathop × limits_α F) ℱ)/ℱ are the canonical elements implementing α g . These states are taken as the physical states, and this specifies the resulting algebraic structure of the physics in M( G M(Gmathop × limits_α F) ℱ), and in particular the maximal constraint free physical algebra ℛ. A nontriviality condition is given for ℛ to exist, and we extend the notion of a crossed product to deal with a situation where G is not locally compact. This is necessary to deal with the field theoretical aspect of the constraints. Next the C*-algebra of the CCR is employed to define the abstract algebraic structure of Gupta-Bleuler electromagnetism in the present framework. The indefinite inner product representation structure is obtained, and this puts Gupta-Bleuler electromagnetism on a rigorous footing. Finally, as a bonus, we find that the algebraic structures just set up, provide a blueprint for constructive quadratic algebraic field theory.

Variations on a theme of Heisenberg, Pauli and Weyl

NASA Astrophysics Data System (ADS)

Kibler, Maurice R.

2008-09-01

The parentage between Weyl pairs, the generalized Pauli group and the unitary group is investigated in detail. We start from an abstract definition of the Heisenberg-Weyl group on the field {\\bb R} and then switch to the discrete Heisenberg-Weyl group or generalized Pauli group on a finite ring {\\bb Z}_d . The main characteristics of the latter group, an abstract group of order d3 noted Pd, are given (conjugacy classes and irreducible representation classes or equivalently Lie algebra of dimension d3 associated with Pd). Leaving the abstract sector, a set of Weyl pairs in dimension d is derived from a polar decomposition of SU(2) closely connected to angular momentum theory. Then, a realization of the generalized Pauli group Pd and the construction of generalized Pauli matrices in dimension d are revisited in terms of Weyl pairs. Finally, the Lie algebra of the unitary group U(d) is obtained as a subalgebra of the Lie algebra associated with Pd. This leads to a development of the Lie algebra of U(d) in a basis consisting of d2 generalized Pauli matrices. In the case where d is a power of a prime integer, the Lie algebra of SU(d) can be decomposed into d - 1 Cartan subalgebras. Dedicated to the memory of my teacher and friend Moshé Flato on the occasion of the tenth anniversary of his death.

Coherent States for the Two-Dimensional Dirac-Moshinsky Oscillator Coupled to an External Magnetic Field

NASA Astrophysics Data System (ADS)

Ojeda-Guillén, D.; Mota, R. D.; Granados, V. D.

2015-03-01

We show that the (2+1)-dimensional Dirac-Moshinsky oscillator coupled to an external magnetic field can be treated algebraically with the SU(1,1) group theory and its group basis. We use the su(1,1) irreducible representation theory to find the energy spectrum and the eigenfunctions. Also, with the su(1,1) group basis we construct the relativistic coherent states in a closed form for this problem. Supported by SNI-México, COFAA-IPN, EDI-IPN, EDD-IPN, SIP-IPN project number 20140598

Leveling the Playing Field: Using a One-to-One Laptop Initiative to Close the Achievement Gap

ERIC Educational Resources Information Center

Smith, Luke Andrew

2012-01-01

The purpose of this study was to examine how the one-to-one laptop initiative affected student achievement gaps for students at a single high school in Mooresville, NC. The variable in this study was the preexisting End of Course exams for Algebra I and English I for the two school years prior to the one-to-one laptop implementation year…

BFV-BRST quantization of two-dimensional supergravity

DOE Office of Scientific and Technical Information (OSTI.GOV)

Fujiwara, T.; Igarashi, Y.; Kuriki, R.

1996-01-01

Two-dimensional supergravity theory is quantized as an anomalous gauge theory. In the Batalin-Fradkin (BF) formalism, the anomaly-canceling super-Liouville fields are introduced to identify the original second-class constrained system with a gauge-fixed version of a first-class system. The BFV-BRST quantization applies to formulate the theory in the most general class of gauges. A local effective action constructed in the configuration space contains two super-Liouville actions; one is a noncovariant but local functional written only in terms of two-dimensional supergravity fields, and the other contains the super-Liouville fields canceling the super-Weyl anomaly. Auxiliary fields for the Liouville and the gravity supermultiplets aremore » introduced to make the BRST algebra close off-shell. Inclusion of them turns out to be essentially important especially in the super-light-cone gauge fixing, where the supercurvature equations ({partial_derivative}{sup 3}{sub {minus}}{ital g}{sub +}{sub +}={partial_derivative}{sup 2}{sub {minus}}{chi}{sub +}{sub +}=0) are obtained as a result of BRST invariance of the theory. Our approach reveals the origin of the OSp(1,2) current algebra symmetry in a transparent manner. {copyright} {ital 1996 The American Physical Society.}« less

Solitons and the energy-momentum tensor for affine Toda theory

NASA Astrophysics Data System (ADS)

Olive, D. I.; Turok, N.; Underwood, J. W. R.

1993-07-01

Following Leznov and Saveliev, we present the general solution to Toda field theories of conformal, affine or conformal affine type, associated with a simple Lie algebra g. These depend on a free massless field and on a group element. By putting the former to zero, soliton solutions to the affine Toda theories with imaginary coupling constant result with the soliton data encoded in the group element. As this requires a reformulation of the affine Kac-Moody algebra closely related to that already used to formulate the physical properties of the particle excitations, including their scattering matrices, a unified treatment of particles and solitons emerges. The physical energy—momentum tensor for a general solution is broken into a total derivative plus a part dependent only on the derivatives of the free field. Despite the non-linearity of the field equations and their complex nature the energy and momentum of the N-soliton solution is shown to be real, equalling the sum of contributions from the individual solitons. There are rank-g species of soliton, with masses given by a generalisation of a formula due to Hollowood, being proportional to the components of the left Perron-Frobenius eigenvector of the Cartan matrix of g.

On the quantum symmetry of the chiral Ising model

NASA Astrophysics Data System (ADS)

Vecsernyés, Peter

1994-03-01

We introduce the notion of rational Hopf algebras that we think are able to describe the superselection symmetries of rational quantum field theories. As an example we show that a six-dimensional rational Hopf algebra H can reproduce the fusion rules, the conformal weights, the quantum dimensions and the representation of the modular group of the chiral Ising model. H plays the role of the global symmetry algebra of the chiral Ising model in the following sense: (1) a simple field algebra F and a representation π on Hπ of it is given, which contains the c = {1}/{2} unitary representations of the Virasoro algebra as subrepresentations; (2) the embedding U: H → B( Hπ) is such that the observable algebra π( A) - is the invariant subalgebra of B( Hπ) with respect to the left adjoint action of H and U(H) is the commutant of π( A); (3) there exist H-covariant primary fields in B( Hπ), which obey generalized Cuntz algebra properties and intertwine between the inequivalent sectors of the observables.

Boundaries for algebras of holomorphic functions on Marcinkiewicz sequence spaces

NASA Astrophysics Data System (ADS)

Choi, Yun Sung; Han, Kwang Hee

2006-11-01

Let be the Banach algebra of all complex-valued bounded continuous functions on the closed unit ball BE of a complex Banach space E and holomorphic in the interior of BE and let be the closed subalgebra of those functions which are uniformly continuous on BE. For the case whose bidual is a Marcinkiewicz sequence space Mw, we describe some sufficient conditions for a set to be a boundary of either or . Moreover, we consider some analogous problems on to those which were studied on the Gowers space Gp of characteristic p by Grados and Moraes [L.R. Grados, L.A. Moraes, Boundaries for algebras of holomorphic functions, J. Math. Anal. Appl. 281 (2003) 575-586; L.R. Grados, L.A. Moraes, Boundaries for an algebra of bounded holomorphic functions, J. Korean Math. Soc. 41 (1) (2004) 231-242].

Analysis of algebraic reasoning ability of cognitive style perspectives on field dependent field independent and gender

NASA Astrophysics Data System (ADS)

Rosita, N. T.

2018-03-01

The purpose of this study is to analyse algebraic reasoning ability using the SOLO model as a theoretical framework to assess students’ algebraic reasoning abilities of Field Dependent cognitive (FD), Field Independent (FI) and Gender perspectives. The method of this study is a qualitative research. The instrument of this study is the researcher himself assisted with algebraic reasoning tests, the problems have been designed based on NCTM indicators and algebraic reasoning according to SOLO model. While the cognitive style of students is determined using Group Embedded Figure Test (GEFT), as well as interviews on the subject as triangulation. The subjects are 15 female and 15 males of the sixth semester students of mathematics education, STKIP Sebelas April. The results of the qualitative data analysis is that most subjects are at the level of unistructural and multi-structural, subjects at the relational level have difficulty in forming a new linear pattern. While the subjects at the extended abstract level are able to meet all the indicators of algebraic reasoning ability even though some of the answers are not perfect yet. Subjects of FI tend to have higher algebraic reasoning abilities than of the subject of FD.

Massless conformal fields, AdS (d+1)/CFT d higher spin algebras and their deformations

DOE PAGES

Fernando, Sudarshan; Gunaydin, Murat

2016-02-04

Here, we extend our earlier work on the minimal unitary representation of SO(d, 2)and its deformations for d=4, 5and 6to arbitrary dimensions d. We show that there is a one-to-one correspondence between the minrep of SO(d, 2)and its deformations and massless conformal fields in Minkowskian spacetimes in ddimensions. The minrep describes a massless conformal scalar field, and its deformations describe massless conformal fields of higher spin. The generators of Joseph ideal vanish identically as operators for the quasiconformal realization of the minrep, and its enveloping algebra yields directly the standard bosonic AdS (d+1)/CFT d higher spin algebra. For deformed minrepsmore » the generators of certain deformations of Joseph ideal vanish as operators and their enveloping algebras lead to deformations of the standard bosonic higher spin algebra. In odd dimensions there is a unique deformation of the higher spin algebra corresponding to the spinor singleton. In even dimensions one finds infinitely many deformations of the higher spin algebra labelled by the eigenvalues of Casimir operator of the little group SO(d–2)for massless representations.« less

q-Derivatives, quantization methods and q-algebras

DOE Office of Scientific and Technical Information (OSTI.GOV)

Twarock, Reidun

1998-12-15

Using the example of Borel quantization on S{sup 1}, we discuss the relation between quantization methods and q-algebras. In particular, it is shown that a q-deformation of the Witt algebra with generators labeled by Z is realized by q-difference operators. This leads to a discrete quantum mechanics. Because of Z, the discretization is equidistant. As an approach to a non-equidistant discretization of quantum mechanics one can change the Witt algebra using not the number field Z as labels but a quadratic extension of Z characterized by an irrational number {tau}. This extension is denoted as quasi-crystal Lie algebra, because thismore » is a relation to one-dimensional quasicrystals. The q-deformation of this quasicrystal Lie algebra is discussed. It is pointed out that quasicrystal Lie algebras can be considered also as a 'deformed' Witt algebra with a 'deformation' of the labeling number field. Their application to the theory is discussed.« less

Three-dimensional fractional-spin gravity

NASA Astrophysics Data System (ADS)

Boulanger, Nicolas; Sundell, Per; Valenzuela, Mauricio

2014-02-01

Using Wigner-deformed Heisenberg oscillators, we construct 3D Chern-Simons models consisting of fractional-spin fields coupled to higher-spin gravity and internal nonabelian gauge fields. The gauge algebras consist of Lorentz-tensorial Blencowe-Vasiliev higher-spin algebras and compact internal algebras intertwined by infinite-dimensional generators in lowest-weight representations of the Lorentz algebra with fractional spin. In integer or half-integer non-unitary cases, there exist truncations to gl(ℓ , ℓ ± 1) or gl(ℓ|ℓ ± 1) models. In all non-unitary cases, the internal gauge fields can be set to zero. At the semi-classical level, the fractional-spin fields are either Grassmann even or odd. The action requires the enveloping-algebra representation of the deformed oscillators, while their Fock-space representation suffices on-shell. The project was funded in part by F.R.S.-FNRS " Ulysse" Incentive Grant for Mobility in Scientific Research.

Weak completions, bornologies and rigid cohomology

NASA Astrophysics Data System (ADS)

Cortiñas, Guillermo; Cuntz, Joachim; Meyer, Ralf; Tamme, Georg

2018-07-01

Let V be a complete discrete valuation ring with residue field k of positive characteristic and with fraction field K of characteristic 0. We clarify the analysis behind the Monsky-Washnitzer completion of a commutative V-algebra using completions of bornological V-algebras. This leads us to a functorial chain complex for a finitely generated commutative algebra over the residue field k that computes its rigid cohomology in the sense of Berthelot.

A non-commutative *-algebra of Borel functions

NASA Astrophysics Data System (ADS)

Hart, Robert

To the pair (E, sigma), where E is a countable Borel equivalence relation on a standard Borel space ( X, A ) and sigma a normalized Borel T -valued 2-cocycle on E, we associate a sequentially weakly closed Borel *-algebra B*r (E, sigma), contained in the bounded linear operators on ℓ2(E). Associated to B*r (E, sigma) is a natural (Borel) Cartan subalgebra (Definition 6.4.10) L( Bo (X)) isomorphic to the bounded Borel functions on X. Then L( Bo (X)) and its normalizer (the set of the unitaries u ∈ B*r (E, sigma) such that u* fu ∈ L( Bo (X)), f ∈ L( Bo (X))) countably generates the Borel *-algebra B*r (E, sigma). In this thesis, we study B*r (E, sigma) and in particular prove that: i) If E is smooth, then B*r (E, sigma) is a type I Borel *-algebra (Definition 6.3.10). ii) If E is a hyperfinite, then B*r (E, sigma) is a Borel AF-algebra (Definition 7.5.1). iii) Generalizing Kumjian's definition, we define a Borel twist Gamma over E and its associated sequentially closed Borel *-algebra B*r (Gamma). iv) Let a Borel Cartan pair ( B,B0 ) denote a sequentially closed Borel *-algebra B with a Borel Cartan subalgebra B0 , where B is countably B0 -generated. Generalizing Feldman-Moore's result, we prove that any pair ( B,B0 ) can be realized uniquely as a pair ( B*r (E, sigma), L( Bo (X))). Moreover, we show that the pair ( B*r (E), L( Bo (X))) is a complete invariant of the countable Borel equivalence relation E. v) We prove a Krieger type theorem, by showing that two aperiodic hyperfinite countable equivalence relations are isomorphic if and only if their associated Borel *-algebras B*r (E1) and B*r (E2) are isomorphic.

Modified non-Abelian Toda field equations and twisted quasigraded Lie algebras

DOE Office of Scientific and Technical Information (OSTI.GOV)

Skrypnyk, T.

We construct a new family of quasigraded Lie algebras that admit the Kostant-Adler scheme. They coincide with special quasigraded deformations of twisted subalgebras of the loop algebras. Using them we obtain new hierarchies of integrable equations in partial derivatives which we call 'modified' non-Abelian Toda field hierarchies.

Remarks towards the spectrum of the Heisenberg spin chain type models

NASA Astrophysics Data System (ADS)

Burdík, Č.; Fuksa, J.; Isaev, A. P.; Krivonos, S. O.; Navrátil, O.

2015-05-01

The integrable close and open chain models can be formulated in terms of generators of the Hecke algebras. In this review paper, we describe in detail the Bethe ansatz for the XXX and the XXZ integrable close chain models. We find the Bethe vectors for two-component and inhomogeneous models. We also find the Bethe vectors for the fermionic realization of the integrable XXX and XXZ close chain models by means of the algebraic and coordinate Bethe ansatz. Special modification of the XXZ closed spin chain model ("small polaron model") is considered. Finally, we discuss some questions relating to the general open Hecke chain models.

Math Remediation for the College Bound: How Teachers Can Close the Gap, from the Basics through Algebra

ERIC Educational Resources Information Center

Khatri, Daryao

2011-01-01

Algebra is the language that must be mastered for any course that uses math because it is the gateway for entry into any science, technology, engineering, and mathematics (STEM) discipline. This book fosters mastery of critical math and algebraic concepts and skills essential to all of the STEM disciplines and some of the social sciences. This…

Continuum analogues of contragredient Lie algebras (Lie algebras with a Cartan operator and nonlinear dynamical systems)

NASA Astrophysics Data System (ADS)

Saveliev, M. V.; Vershik, A. M.

1989-12-01

We present an axiomatic formulation of a new class of infinitedimensional Lie algebras-the generalizations of Z-graded Lie algebras with, generally speaking, an infinite-dimensional Cartan subalgebra and a contiguous set of roots. We call such algebras “continuum Lie algebras.” The simple Lie algebras of constant growth are encapsulated in our formulation. We pay particular attention to the case when the local algebra is parametrized by a commutative algebra while the Cartan operator (the generalization of the Cartan matrix) is a linear operator. Special examples of these algebras are the Kac-Moody algebras, algebras of Poisson brackets, algebras of vector fields on a manifold, current algebras, and algebras with differential or integro-differential cartan operator. The nonlinear dynamical systems associated with the continuum contragredient Lie algebras are also considered.

Conformal field algebras with quantum symmetry from the theory of superselection sectors

NASA Astrophysics Data System (ADS)

Mack, Gerhard; Schomerus, Volker

1990-11-01

According to the theory of superselection sectors of Doplicher, Haag, and Roberts, field operators which make transitions between different superselection sectors—i.e. different irreducible representations of the observable algebra—are to be constructed by adjoining localized endomorphisms to the algebra of local observables. We find the relevant endomorphisms of the chiral algebra of observables in the minimal conformal model with central charge c=1/2 (Ising model). We show by explicit and elementary construction how they determine a representation of the braid group B ∞ which is associated with a Temperley-Lieb-Jones algebra. We recover fusion rules, and compute the quantum dimensions of the superselection sectors. We exhibit a field algebra which is quantum group covariant and acts in the Hilbert space of physical states. It obeys local braid relations in an appropriate weak sense.

Toward the classification of differential calculi on κ-Minkowski space and related field theories

NASA Astrophysics Data System (ADS)

Jurić, Tajron; Meljanac, Stjepan; Pikutić, Danijel; Štrajn, Rina

2015-07-01

Classification of differential forms on κ-Minkowski space, particularly, the classification of all bicovariant differential calculi of classical dimension is presented. By imposing super-Jacobi identities we derive all possible differential algebras compatible with the κ-Minkowski algebra for time-like, space-like and light-like deformations. Embedding into the super-Heisenberg algebra is constructed using non-commutative (NC) coordinates and one-forms. Particularly, a class of differential calculi with an undeformed exterior derivative and one-forms is considered. Corresponding NC differential calculi are elaborated. Related class of new Drinfeld twists is proposed. It contains twist leading to κ-Poincaré Hopf algebra for light-like deformation. Corresponding super-algebra and deformed super-Hopf algebras, as well as the symmetries of differential algebras are presented and elaborated. Using the NC differential calculus, we analyze NC field theory, modified dispersion relations, and discuss further physical applications.

Symmetric tops in combined electric fields: Conditional quasisolvability via the quantum Hamilton-Jacobi theory

NASA Astrophysics Data System (ADS)

Schatz, Konrad; Friedrich, Bretislav; Becker, Simon; Schmidt, Burkhard

2018-05-01

We make use of the quantum Hamilton-Jacobi (QHJ) theory to investigate conditional quasisolvability of the quantum symmetric top subject to combined electric fields (symmetric top pendulum). We derive the conditions of quasisolvability of the time-independent Schrödinger equation as well as the corresponding finite sets of exact analytic solutions. We do so for this prototypical trigonometric system as well as for its anti-isospectral hyperbolic counterpart. An examination of the algebraic and numerical spectra of these two systems reveals mutually closely related patterns. The QHJ approach allows us to retrieve the closed-form solutions for the spherical and planar pendula and the Razavy system that had been obtained in our earlier work via supersymmetric quantum mechanics as well as to find a cornucopia of additional exact analytic solutions.

Diffeomorphism invariance and black hole entropy

NASA Astrophysics Data System (ADS)

Huang, Chao-Guang; Guo, Han-Ying; Wu, Xiaoning

2003-11-01

The Noether-charge and the Hamiltonian realizations for the diff(M) algebra in diffeomorphism-invariant gravitational theories without a cosmological constant in any dimension are studied in a covariant formalism. We analyze how the Hamiltonian functionals form the diff(M) algebra under the Poisson brackets and show how the Noether charges with respect to the diffeomorphism generated by the vector fields and their variations in n-dimensional general relativity form this algebra. The asymptotic behaviors of vector fields generating diffeomorphism of the manifold with boundaries are discussed. It is shown that the “central extension” for a large class of vector fields is always zero on the Killing horizon. We also check whether choosing the vector fields near the horizon may pick up the Virasoro algebra. The conclusion is unfortunately negative in any dimension.

Superspace geometrical realization of the N-extended super Virasoro algebra and its dual

NASA Astrophysics Data System (ADS)

Curto, C.; Gates, S. J., Jr.; Rodgers, V. G. J.

2000-05-01

We derive properties of N-extended /GR super Virasoro algebras. These include adding central extensions, identification of all primary fields and the action of the adjoint representation on its dual. The final result suggest identification with the spectrum of fields in supergravity theories and superstring/M-theory constructed from NSR N-extended supersymmetric /GR Virasoro algebras.

Closed form of the Baker-Campbell-Hausdorff formula for the generators of semisimple complex Lie algebras

NASA Astrophysics Data System (ADS)

Matone, Marco

2016-11-01

Recently it has been introduced an algorithm for the Baker-Campbell-Hausdorff (BCH) formula, which extends the Van-Brunt and Visser recent results, leading to new closed forms of BCH formula. More recently, it has been shown that there are 13 types of such commutator algebras. We show, by providing the explicit solutions, that these include the generators of the semisimple complex Lie algebras. More precisely, for any pair, X, Y of the Cartan-Weyl basis, we find W, linear combination of X, Y, such that exp (X) exp (Y)=exp (W). The derivation of such closed forms follows, in part, by using the above mentioned recent results. The complete derivation is provided by considering the structure of the root system. Furthermore, if X, Y, and Z are three generators of the Cartan-Weyl basis, we find, for a wide class of cases, W, a linear combination of X, Y and Z, such that exp (X) exp (Y) exp (Z)=exp (W). It turns out that the relevant commutator algebras are type 1c-i, type 4 and type 5. A key result concerns an iterative application of the algorithm leading to relevant extensions of the cases admitting closed forms of the BCH formula. Here we provide the main steps of such an iteration that will be developed in a forthcoming paper.

Strings on complex multiplication tori and rational conformal field theory with matrix level

NASA Astrophysics Data System (ADS)

Nassar, Ali

Conformal invariance in two dimensions is a powerful symmetry. Two-dimensional quantum field theories which enjoy conformal invariance, i.e., conformal field theories (CFTs) are of great interest in both physics and mathematics. CFTs describe the dynamics of the world sheet in string theory where conformal symmetry arises as a remnant of reparametrization invariance of the world-sheet coordinates. In statistical mechanics, CFTs describe the critical points of second order phase transitions. On the mathematics side, conformal symmetry gives rise to infinite dimensional chiral algebras like the Virasoro algebra or extensions thereof. This gave rise to the study of vertex operator algebras (VOAs) which is an interesting branch of mathematics. Rational conformal theories are a simple class of CFTs characterized by a finite number of representations of an underlying chiral algebra. The chiral algebra leads to a set of Ward identities which gives a complete non-perturbative solution of the RCFT. Identifying the chiral algebra of an RCFT is a very important step in solving it. Particularly interesting RCFTs are the ones which arise from the compactification of string theory as sigma-models on a target manifold M. At generic values of the geometric moduli of M, the corresponding CFT is not rational. Rationality can arise at particular values of the moduli of M. At these special values of the moduli, the chiral algebra is extended. This interplay between the geometric picture and the algebraic description encoded in the chiral algebra makes CFTs/RCFTs a perfect link between physics and mathematics. It is always useful to find a geometric interpretation of a chiral algebra in terms of a sigma-model on some target manifold M. Then the next step is to figure out the conditions on the geometric moduli of M which gives a RCFT. In this thesis, we limit ourselves to the simplest class of string compactifications, i.e., strings on tori. As Gukov and Vafa proved, rationality selects the complex-multiplication tori. On the other hand, the study of the matrix-level affine algebra Um,K is motivated by conformal field theory and the fractional quantum Hall effect. Gannon completed the classification of U m,K modular-invariant partition functions. Here we connect the algebra U2,K to strings on 2-tori describable by rational conformal field theories. We point out that the rational conformal field theories describing strings on complex-multiplication tori have characters and partition functions identical to those of the matrix-level algebra Um,K. This connection makes obvious that the rational theories are dense in the moduli space of strings on Tm, and may prove useful in other ways.

Enlarged symmetry algebras of spin chains, loop models, and S-matrices

NASA Astrophysics Data System (ADS)

Read, N.; Saleur, H.

2007-08-01

The symmetry algebras of certain families of quantum spin chains are considered in detail. The simplest examples possess m states per site ( m⩾2), with nearest-neighbor interactions with U(m) symmetry, under which the sites transform alternately along the chain in the fundamental m and its conjugate representation m¯. We find that these spin chains, even with arbitrary coefficients of these interactions, have a symmetry algebra A much larger than U(m), which implies that the energy eigenstates fall into sectors that for open chains (i.e., free boundary conditions) can be labeled by j=0,1,…,L, for the 2 L-site chain such that the degeneracies of all eigenvalues in the jth sector are generically the same and increase rapidly with j. For large j, these degeneracies are much larger than those that would be expected from the U(m) symmetry alone. The enlarged symmetry algebra A(2L) consists of operators that commute in this space of states with the Temperley-Lieb algebra that is generated by the set of nearest-neighbor interaction terms; A(2L) is not a Yangian. There are similar results for supersymmetric chains with gl(m+n|n) symmetry of nearest-neighbor interactions, and a richer representation structure for closed chains (i.e., periodic boundary conditions). The symmetries also apply to the loop models that can be obtained from the spin chains in a spacetime or transfer matrix picture. In the loop language, the symmetries arise because the loops cannot cross. We further define tensor products of representations (for the open chains) by joining chains end to end. The fusion rules for decomposing the tensor product of representations labeled j and j take the same form as the Clebsch-Gordan series for SU(2). This and other structures turn the symmetry algebra A into a ribbon Hopf algebra, and we show that this is "Morita equivalent" to the quantum group U(sl) for m=q+q. The open-chain results are extended to the cases |m|<2 for which the algebras are no longer semisimple; these possess continuum limits that are critical (conformal) field theories, or massive perturbations thereof. Such models, for open and closed boundary conditions, arise in connection with disordered fermions, percolation, and polymers (self-avoiding walks), and certain non-linear sigma models, all in two dimensions. A product operation is defined in a related way for the Temperley-Lieb representations also, and the fusion rules for this are related to those for A or U(sl) representations; this is useful for the continuum limits also, as we discuss in a companion paper.

On Correspondence of BRST-BFV, Dirac, and Refined Algebraic Quantizations of Constrained Systems

NASA Astrophysics Data System (ADS)

Shvedov, O. Yu.

2002-11-01

The correspondence between BRST-BFV, Dirac, and refined algebraic (group averaging, projection operator) approaches to quantizing constrained systems is analyzed. For the closed-algebra case, it is shown that the component of the BFV wave function corresponding to maximal (minimal) value of number of ghosts and antighosts in the Schrodinger representation may be viewed as a wave function in the refined algebraic (Dirac) quantization approach. The Giulini-Marolf group averaging formula for the inner product in the refined algebraic quantization approach is obtained from the Batalin-Marnelius prescription for the BRST-BFV inner product, which should be generally modified due to topological problems. The considered prescription for the correspondence of states is observed to be applicable to the open-algebra case. The refined algebraic quantization approach is generalized then to the case of nontrivial structure functions. A simple example is discussed. The correspondence of observables for different quantization methods is also investigated.

Some elements of a theory of multidimensional complex variables. I - General theory. II - Expansions of analytic functions and application to fluid flows

NASA Technical Reports Server (NTRS)

Martin, E. Dale

1989-01-01

The paper introduces a new theory of N-dimensional complex variables and analytic functions which, for N greater than 2, is both a direct generalization and a close analog of the theory of ordinary complex variables. The algebra in the present theory is a commutative ring, not a field. Functions of a three-dimensional variable were defined and the definition of the derivative then led to analytic functions.

The Pontryagin class for pre-Courant algebroids

NASA Astrophysics Data System (ADS)

Liu, Zhangju; Sheng, Yunhe; Xu, Xiaomeng

2016-06-01

In this paper, we show that the Jacobiator J of a pre-Courant algebroid is closed naturally. The corresponding equivalence class [J♭ ] is defined as the Pontryagin class, which is the obstruction of a pre-Courant algebroid to be deformed into a Courant algebroid. We construct a Leibniz 2-algebra and a Lie 2-algebra associated to a pre-Courant algebroid and prove that these algebraic structures are isomorphic under deformations. Finally, we introduce the twisted action of a Lie algebra on a manifold to give more examples of pre-Courant algebroids, which include the Cartan geometry.

Dimensional regularization in position space and a Forest Formula for Epstein-Glaser renormalization

NASA Astrophysics Data System (ADS)

Dütsch, Michael; Fredenhagen, Klaus; Keller, Kai Johannes; Rejzner, Katarzyna

2014-12-01

We reformulate dimensional regularization as a regularization method in position space and show that it can be used to give a closed expression for the renormalized time-ordered products as solutions to the induction scheme of Epstein-Glaser. This closed expression, which we call the Epstein-Glaser Forest Formula, is analogous to Zimmermann's Forest Formula for BPH renormalization. For scalar fields, the resulting renormalization method is always applicable, we compute several examples. We also analyze the Hopf algebraic aspects of the combinatorics. Our starting point is the Main Theorem of Renormalization of Stora and Popineau and the arising renormalization group as originally defined by Stückelberg and Petermann.

DOE Office of Scientific and Technical Information (OSTI.GOV)

Fernando, Sudarshan; Gunaydin, Murat

Here, we extend our earlier work on the minimal unitary representation of SO(d, 2)and its deformations for d=4, 5and 6to arbitrary dimensions d. We show that there is a one-to-one correspondence between the minrep of SO(d, 2)and its deformations and massless conformal fields in Minkowskian spacetimes in ddimensions. The minrep describes a massless conformal scalar field, and its deformations describe massless conformal fields of higher spin. The generators of Joseph ideal vanish identically as operators for the quasiconformal realization of the minrep, and its enveloping algebra yields directly the standard bosonic AdS (d+1)/CFT d higher spin algebra. For deformed minrepsmore » the generators of certain deformations of Joseph ideal vanish as operators and their enveloping algebras lead to deformations of the standard bosonic higher spin algebra. In odd dimensions there is a unique deformation of the higher spin algebra corresponding to the spinor singleton. In even dimensions one finds infinitely many deformations of the higher spin algebra labelled by the eigenvalues of Casimir operator of the little group SO(d–2)for massless representations.« less

Hopf algebras of rooted forests, cocyles, and free Rota-Baxter algebras

NASA Astrophysics Data System (ADS)

Zhang, Tianjie; Gao, Xing; Guo, Li

2016-10-01

The Hopf algebra and the Rota-Baxter algebra are the two algebraic structures underlying the algebraic approach of Connes and Kreimer to renormalization of perturbative quantum field theory. In particular, the Hopf algebra of rooted trees serves as the "baby model" of Feynman graphs in their approach and can be characterized by certain universal properties involving a Hochschild 1-cocycle. Decorated rooted trees have also been applied to study Feynman graphs. We will continue the study of universal properties of various spaces of decorated rooted trees with such a 1-cocycle, leading to the concept of a cocycle Hopf algebra. We further apply the universal properties to equip a free Rota-Baxter algebra with the structure of a cocycle Hopf algebra.

Bilinear covariants and spinor fields duality in quantum Clifford algebras

DOE Office of Scientific and Technical Information (OSTI.GOV)

Abłamowicz, Rafał, E-mail: rablamowicz@tntech.edu; Gonçalves, Icaro, E-mail: icaro.goncalves@ufabc.edu.br; Centro de Matemática, Computação e Cognição, Universidade Federal do ABC, 09210-170 Santo André, SP

Classification of quantum spinor fields according to quantum bilinear covariants is introduced in a context of quantum Clifford algebras on Minkowski spacetime. Once the bilinear covariants are expressed in terms of algebraic spinor fields, the duality between spinor and quantum spinor fields can be discussed. Thus, by endowing the underlying spacetime with an arbitrary bilinear form with an antisymmetric part in addition to a symmetric spacetime metric, quantum algebraic spinor fields and deformed bilinear covariants can be constructed. They are thus compared to the classical (non quantum) ones. Classes of quantum spinor fields classes are introduced and compared with Lounesto'smore » spinor field classification. A physical interpretation of the deformed parts and the underlying Z-grading is proposed. The existence of an arbitrary bilinear form endowing the spacetime already has been explored in the literature in the context of quantum gravity [S. W. Hawking, “The unpredictability of quantum gravity,” Commun. Math. Phys. 87, 395 (1982)]. Here, it is shown further to play a prominent role in the structure of Dirac, Weyl, and Majorana spinor fields, besides the most general flagpoles and flag-dipoles. We introduce a new duality between the standard and the quantum spinor fields, by showing that when Clifford algebras over vector spaces endowed with an arbitrary bilinear form are taken into account, a mixture among the classes does occur. Consequently, novel features regarding the spinor fields can be derived.« less

Serre duality, Abel's theorem, and Jacobi inversion for supercurves over a thick superpoint

NASA Astrophysics Data System (ADS)

Rothstein, Mitchell J.; Rabin, Jeffrey M.

2015-04-01

The principal aim of this paper is to extend Abel's theorem to the setting of complex supermanifolds of dimension 1 | q over a finite-dimensional local supercommutative C-algebra. The theorem is proved by establishing a compatibility of Serre duality for the supercurve with Poincaré duality on the reduced curve. We include an elementary algebraic proof of the requisite form of Serre duality, closely based on the account of the reduced case given by Serre in Algebraic groups and class fields, combined with an invariance result for the topology on the dual of the space of répartitions. Our Abel map, taking Cartier divisors of degree zero to the dual of the space of sections of the Berezinian sheaf, modulo periods, is defined via Penkov's characterization of the Berezinian sheaf as the cohomology of the de Rham complex of the sheaf D of differential operators. We discuss the Jacobi inversion problem for the Abel map and give an example demonstrating that if n is an integer sufficiently large that the generic divisor of degree n is linearly equivalent to an effective divisor, this need not be the case for all divisors of degree n.

Linear algebraic theory of partial coherence: discrete fields and measures of partial coherence.

PubMed

Ozaktas, Haldun M; Yüksel, Serdar; Kutay, M Alper

2002-08-01

A linear algebraic theory of partial coherence is presented that allows precise mathematical definitions of concepts such as coherence and incoherence. This not only provides new perspectives and insights but also allows us to employ the conceptual and algebraic tools of linear algebra in applications. We define several scalar measures of the degree of partial coherence of an optical field that are zero for full incoherence and unity for full coherence. The mathematical definitions are related to our physical understanding of the corresponding concepts by considering them in the context of Young's experiment.

Type II superstring field theory: geometric approach and operadic description

NASA Astrophysics Data System (ADS)

Jurčo, Branislav; Münster, Korbinian

2013-04-01

We outline the construction of type II superstring field theory leading to a geometric and algebraic BV master equation, analogous to Zwiebach's construction for the bosonic string. The construction uses the small Hilbert space. Elementary vertices of the non-polynomial action are described with the help of a properly formulated minimal area problem. They give rise to an infinite tower of superstring field products defining a {N} = 1 generalization of a loop homotopy Lie algebra, the genus zero part generalizing a homotopy Lie algebra. Finally, we give an operadic interpretation of the construction.

Using CAS to Solve Classical Mathematics Problems

ERIC Educational Resources Information Center

Burke, Maurice J.; Burroughs, Elizabeth A.

2009-01-01

Historically, calculus has displaced many algebraic methods for solving classical problems. This article illustrates an algebraic method for finding the zeros of polynomial functions that is closely related to Newton's method (devised in 1669, published in 1711), which is encountered in calculus. By exploring this problem, precalculus students…

Intuitionistic fuzzy n-fold KU-ideal of KU-algebra

NASA Astrophysics Data System (ADS)

Mostafa, Samy M.; Kareem, Fatema F.

2018-05-01

In this paper, we apply the notion of intuitionistic fuzzy n-fold KU-ideal of KU-algebra. Some types of ideals such as intuitionistic fuzzy KU-ideal, intuitionistic fuzzy closed ideal and intuitionistic fuzzy n-fold KU-ideal are studied. Also, the relations between intuitionistic fuzzy n-fold KU-ideal and intuitionistic fuzzy KU-ideal are discussed. Furthermore, a few results of intuitionistic fuzzy n-fold KU-ideals of a KU-algebra under homomorphism are discussed.

N = 2 S-duality revisited

NASA Astrophysics Data System (ADS)

Buican, Matthew; Laczko, Zoltan; Nishinaka, Takahiro

2017-09-01

Using the chiral algebra bootstrap, we revisit the simplest Argyres-Douglas (AD) generalization of Argyres-Seiberg S-duality. We argue that the exotic AD superconformal field theory (SCFT), T_{3,3/2} , emerging in this duality splits into a free piece and an interacting piece, T_X , even though this factorization seems invisible in the Seiberg-Witten (SW) curve derived from the corresponding M5-brane construction. Without a Lagrangian, an associated topological field theory, a BPS spectrum, or even an SW curve, we nonetheless obtain exact information about T_X by bootstrapping its chiral algebra, {}_X(T_X) , and finding the corresponding vacuum character in terms of Affine Kac-Moody characters. By a standard 4D/2D correspondence, this result gives us the Schur index for T_X and, by studying this quantity in the limit of small S 1, we make contact with a proposed S 1 reduction. Along the way, we discuss various properties of T_X : as an N = 1 theory, it has flavor symmetry SU(3) × SU(2) × U(1), the central charge of {}_X(T_X) matches the central charge of the bc ghosts in bosonic string theory, and its global SU(2) symmetry has a Witten anomaly. This anomaly does not prevent us from building conformal manifolds out of arbitrary numbers of T_X theories (giving us a surprisingly close AD relative of Gaiotto's T N theories), but it does lead to some open questions in the context of the chiral algebra/4D N =2SCFT correspondence.

Introducing Algebraic Structures through Solving Equations: Vertical Content Knowledge for K-12 Mathematics Teachers

ERIC Educational Resources Information Center

Wasserman, Nicholas H.

2014-01-01

Algebraic structures are a necessary aspect of algebraic thinking for K-12 students and teachers. An approach for introducing the algebraic structure of groups and fields through the arithmetic properties required for solving simple equations is summarized; the collective (not individual) importance of these axioms as a foundation for algebraic…

Mathematics of Quantization and Quantum Fields

NASA Astrophysics Data System (ADS)

Dereziński, Jan; Gérard, Christian

2013-03-01

Preface; 1. Vector spaces; 2. Operators in Hilbert spaces; 3. Tensor algebras; 4. Analysis in L2(Rd); 5. Measures; 6. Algebras; 7. Anti-symmetric calculus; 8. Canonical commutation relations; 9. CCR on Fock spaces; 10. Symplectic invariance of CCR in finite dimensions; 11. Symplectic invariance of the CCR on Fock spaces; 12. Canonical anti-commutation relations; 13. CAR on Fock spaces; 14. Orthogonal invariance of CAR algebras; 15. Clifford relations; 16. Orthogonal invariance of the CAR on Fock spaces; 17. Quasi-free states; 18. Dynamics of quantum fields; 19. Quantum fields on space-time; 20. Diagrammatics; 21. Euclidean approach for bosons; 22. Interacting bosonic fields; Subject index; Symbols index.

Multifractal vector fields and stochastic Clifford algebra.

PubMed

Schertzer, Daniel; Tchiguirinskaia, Ioulia

2015-12-01

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

D{sub {infinity}}-differential E{sub {infinity}}-algebras and spectral sequences of fibrations

DOE Office of Scientific and Technical Information (OSTI.GOV)

Lapin, Sergei V

2007-10-31

The notion of an E{sub {infinity}}-algebra with a filtration is introduced. The connections are established between E{sub {infinity}}-algebras with filtrations and the theory of D{sub {infinity}}-differential E{sub {infinity}}-algebras over fields. Based on the technique of D{sub {infinity}}-differential E{sub {infinity}}-algebras, the apparatus of spectral sequences is developed for E{sub {infinity}}-algebras with filtrations, and applications of this apparatus to the multiplicative cohomology spectral sequences of fibrations are given. Bibliography: 21 titles.

Frobenius manifolds and Frobenius algebra-valued integrable systems

NASA Astrophysics Data System (ADS)

Strachan, Ian A. B.; Zuo, Dafeng

2017-06-01

The notion of integrability will often extend from systems with scalar-valued fields to systems with algebra-valued fields. In such extensions the properties of, and structures on, the algebra play a central role in ensuring integrability is preserved. In this paper, a new theory of Frobenius algebra-valued integrable systems is developed. This is achieved for systems derived from Frobenius manifolds by utilizing the theory of tensor products for such manifolds, as developed by Kaufmann (Int Math Res Not 19:929-952, 1996), Kontsevich and Manin (Inv Math 124: 313-339, 1996). By specializing this construction, using a fixed Frobenius algebra A, one can arrive at such a theory. More generally, one can apply the same idea to construct an A-valued topological quantum field theory. The Hamiltonian properties of two classes of integrable evolution equations are then studied: dispersionless and dispersive evolution equations. Application of these ideas are discussed, and as an example, an A-valued modified Camassa-Holm equation is constructed.

Field-antifield and BFV formalisms for quadratic systems with open gauge algebras

DOE Office of Scientific and Technical Information (OSTI.GOV)

Nirov, K.S.; Razumov, A.V.

1992-09-20

In this paper the Lagrangian field-antifield (BV) and Hamiltonian (BFV) BRST formalisms for the general quadratic systems with open gauge algebra are considered. The equivalence between the Lagrangian and Hamiltonian formalisms is proven.

Deformed twistors and higher spin conformal (super-)algebras in four dimensions

DOE PAGES

Govil, Karan; Gunaydin, Murat

2015-03-05

Massless conformal scalar field in d = 4 corresponds to the minimal unitary representation (minrep) of the conformal group SU(2, 2) which admits a one-parameter family of deformations that describe massless fields of arbitrary helicity. The minrep and its deformations were obtained by quantization of the nonlinear realization of SU(2, 2) as a quasiconformal group in arXiv:0908.3624. We show that the generators of SU(2,2) for these unitary irreducible representations can be written as bilinears of deformed twistorial oscillators which transform nonlinearly under the Lorentz group and apply them to define and study higher spin algebras and superalgebras in AdS 5.more » The higher spin (HS) algebra of Fradkin-Vasiliev type in AdS 5 is simply the enveloping algebra of SU(2, 2) quotiented by a two-sided ideal (Joseph ideal) which annihilates the minrep. We show that the Joseph ideal vanishes identically for the quasiconformal realization of the minrep and its enveloping algebra leads directly to the HS algebra in AdS 5. Furthermore, the enveloping algebras of the deformations of the minrep define a one parameter family of HS algebras in AdS 5 for which certain 4d covariant deformations of the Joseph ideal vanish identically. These results extend to superconformal algebras SU(2, 2|N) and we find a one parameter family of HS superalgebras as enveloping algebras of the minimal unitary supermultiplet and its deformations. Our results suggest the existence of a family of (supersymmetric) HS theories in AdS 5 which are dual to free (super)conformal field theories (CFTs) or to interacting but integrable (supersymmetric) CFTs in 4d. We also discuss the corresponding picture in HS algebras in AdS 4 where the corresponding 3d conformal group Sp(4,R) admits only two massless representations (minreps), namely the scalar and spinor singletons.« less

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

ERIC Educational Resources Information Center

Yantz, Jennifer

2013-01-01

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

Unification of the general non-linear sigma model and the Virasoro master equation

DOE Office of Scientific and Technical Information (OSTI.GOV)

Boer, J. de; Halpern, M.B.

1997-06-01

The Virasoro master equation describes a large set of conformal field theories known as the affine-Virasoro constructions, in the operator algebra (affinie Lie algebra) of the WZW model, while the einstein equations of the general non-linear sigma model describe another large set of conformal field theories. This talk summarizes recent work which unifies these two sets of conformal field theories, together with a presumable large class of new conformal field theories. The basic idea is to consider spin-two operators of the form L{sub ij}{partial_derivative}x{sup i}{partial_derivative}x{sup j} in the background of a general sigma model. The requirement that these operators satisfymore » the Virasoro algebra leads to a set of equations called the unified Einstein-Virasoro master equation, in which the spin-two spacetime field L{sub ij} cuples to the usual spacetime fields of the sigma model. The one-loop form of this unified system is presented, and some of its algebraic and geometric properties are discussed.« less

Algebraic invariant curves of plane polynomial differential systems

NASA Astrophysics Data System (ADS)

Tsygvintsev, Alexei

2001-01-01

We consider a plane polynomial vector field P(x,y) dx + Q(x,y) dy of degree m>1. With each algebraic invariant curve of such a field we associate a compact Riemann surface with the meromorphic differential ω = dx/P = dy/Q. The asymptotic estimate of the degree of an arbitrary algebraic invariant curve is found. In the smooth case this estimate has already been found by Cerveau and Lins Neto in a different way.

Classification of commutator algebras leading to the new type of closed Baker-Campbell-Hausdorff formulas

NASA Astrophysics Data System (ADS)

Matone, Marco

2015-11-01

We show that there are {\\it 13 types} of commutator algebras leading to the new closed forms of the Baker-Campbell-Hausdorff (BCH) formula $$\\exp(X)\\exp(Y)\\exp(Z)=\\exp({AX+BZ+CY+DI}) \\ , $$ derived in arXiv:1502.06589, JHEP {\\bf 1505} (2015) 113. This includes, as a particular case, $\\exp(X) \\exp(Z)$, with $[X,Z]$ containing other elements in addition to $X$ and $Z$. The algorithm exploits the associativity of the BCH formula and is based on the decomposition $\\exp(X)\\exp(Y)\\exp(Z)=\\exp(X)\\exp({\\alpha Y}) \\exp({(1-\\alpha) Y}) \\exp(Z)$, with $\\alpha$ fixed in such a way that it reduces to $\\exp({\\tilde X})\\exp({\\tilde Y})$, with $\\tilde X$ and $\\tilde Y$ satisfying the Van-Brunt and Visser condition $[\\tilde X,\\tilde Y]=\\tilde u\\tilde X+\\tilde v\\tilde Y+\\tilde cI$. It turns out that $e^\\alpha$ satisfies, in the generic case, an algebraic equation whose exponents depend on the parameters defining the commutator algebra. In nine {\\it types} of commutator algebras, such an equation leads to rational solutions for $\\alpha$. We find all the equations that characterize the solution of the above decomposition problem by combining it with the Jacobi identity.

Global differential geometry: An introduction for control engineers

NASA Technical Reports Server (NTRS)

Doolin, B. F.; Martin, C. F.

1982-01-01

The basic concepts and terminology of modern global differential geometry are discussed as an introduction to the Lie theory of differential equations and to the role of Grassmannians in control systems analysis. To reach these topics, the fundamental notions of manifolds, tangent spaces, vector fields, and Lie algebras are discussed and exemplified. An appendix reviews such concepts needed for vector calculus as open and closed sets, compactness, continuity, and derivative. Although the content is mathematical, this is not a mathematical treatise but rather a text for engineers to understand geometric and nonlinear control.

Quiver W-algebras

NASA Astrophysics Data System (ADS)

Kimura, Taro; Pestun, Vasily

2018-06-01

For a quiver with weighted arrows, we define gauge-theory K-theoretic W-algebra generalizing the definition of Shiraishi et al. and Frenkel and Reshetikhin. In particular, we show that the qq-character construction of gauge theory presented by Nekrasov is isomorphic to the definition of the W-algebra in the operator formalism as a commutant of screening charges in the free field representation. Besides, we allow arbitrary quiver and expect interesting applications to representation theory of generalized Borcherds-Kac-Moody Lie algebras, their quantum affinizations and associated W-algebras.

Background Independence and Duality Invariance in String Theory.

PubMed

Hohm, Olaf

2017-03-31

Closed string theory exhibits an O(D,D) duality symmetry on tori, which in double field theory is manifest before compactification. I prove that to first order in α^{'} there is no manifestly background independent and duality invariant formulation of bosonic string theory in terms of a metric, b field, and dilaton. To this end I use O(D,D) invariant second order perturbation theory around flat space to show that the unique background independent candidate expression for the gauge algebra at order α^{'} is inconsistent with the Jacobi identity. A background independent formulation exists instead for frame variables subject to α^{'}-deformed frame transformations (generalized Green-Schwarz transformations). Potential applications for curved backgrounds, as in cosmology, are discussed.

Prescriptive unitarity

DOE PAGES

Bourjaily, Jacob L.; Herrmann, Enrico; Trnka, Jaroslav

2017-06-12

We introduce a prescriptive approach to generalized unitarity, resulting in a strictly-diagonal basis of loop integrands with coefficients given by specifically-tailored residues in field theory. We illustrate the power of this strategy in the case of planar, maximally supersymmetric Yang-Mills theory (SYM), where we construct closed-form representations of all (n-point N k MHV) scattering amplitudes through three loops. The prescriptive approach contrasts with the ordinary description of unitarity-based methods by avoiding any need for linear algebra to determine integrand coefficients. We describe this approach in general terms as it should have applications to many quantum field theories, including those withoutmore » planarity, supersymmetry, or massless spectra defined in any number of dimensions.« less

Wakimoto realization of drinfeld current for the elliptic quantum algebra U{sub q,p}( widehat(sl{sub 3}) )

DOE Office of Scientific and Technical Information (OSTI.GOV)

Kojima, T., E-mail: kojima@math.cst.nihon-u.ac.j

2010-02-15

We study a free field realization of the elliptic quantum algebra U{sub q,p}( widehat(sl{sub 3}) ) for arbitrary level k. We give the free field realization of elliptic analog of Drinfeld current associated with U{sub q,p}( widehat(sl{sub 3}) ) for arbitrary level k. In the limit p {yields} 0, q {yields} 1 our realization reproduces Wakimoto realization for the affine Lie algebra ( widehat(sl{sub 3}) ) .

Topological defects in open string field theory

NASA Astrophysics Data System (ADS)

Kojita, Toshiko; Maccaferri, Carlo; Masuda, Toru; Schnabl, Martin

2018-04-01

We show how conformal field theory topological defects can relate solutions of open string field theory for different boundary conditions. To this end we generalize the results of Graham and Watts to include the action of defects on boundary condition changing fields. Special care is devoted to the general case when nontrivial multiplicities arise upon defect action. Surprisingly the fusion algebra of defects is realized on open string fields only up to a (star algebra) isomorphism.

Moving frames and prolongation algebras

NASA Technical Reports Server (NTRS)

Estabrook, F. B.

1982-01-01

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

A Cohomological Perspective on Algebraic Quantum Field Theory

NASA Astrophysics Data System (ADS)

Hawkins, Eli

2018-05-01

Algebraic quantum field theory is considered from the perspective of the Hochschild cohomology bicomplex. This is a framework for studying deformations and symmetries. Deformation is a possible approach to the fundamental challenge of constructing interacting QFT models. Symmetry is the primary tool for understanding the structure and properties of a QFT model. This perspective leads to a generalization of the algebraic quantum field theory framework, as well as a more general definition of symmetry. This means that some models may have symmetries that were not previously recognized or exploited. To first order, a deformation of a QFT model is described by a Hochschild cohomology class. A deformation could, for example, correspond to adding an interaction term to a Lagrangian. The cohomology class for such an interaction is computed here. However, the result is more general and does not require the undeformed model to be constructed from a Lagrangian. This computation leads to a more concrete version of the construction of perturbative algebraic quantum field theory.

Algebraic Thinking in Solving Linier Program at High School Level: Female Student’s Field Independent Cognitive Style

NASA Astrophysics Data System (ADS)

Hardiani, N.; Budayasa, I. K.; Juniati, D.

2018-01-01

The aim of this study was to describe algebraic thinking of high school female student’s field independent cognitive style in solving linier program problem by revealing deeply the female students’ responses. Subjects in this study were 7 female students having field independent cognitive style in class 11. The type of this research was descriptive qualitative. The method of data collection used was observation, documentation, and interview. Data analysis technique was by reduction, presentation, and conclusion. The results of this study showed that the female students with field independent cognitive style in solving the linier program problem had the ability to represent algebraic ideas from the narrative question that had been read by manipulating symbols and variables presented in tabular form, creating and building mathematical models in two variables linear inequality system which represented algebraic ideas, and interpreting the solutions as variables obtained from the point of intersection in the solution area to obtain maximum benefit.

DOE Office of Scientific and Technical Information (OSTI.GOV)

Govil, Karan; Gunaydin, Murat

Massless conformal scalar field in d = 4 corresponds to the minimal unitary representation (minrep) of the conformal group SU(2, 2) which admits a one-parameter family of deformations that describe massless fields of arbitrary helicity. The minrep and its deformations were obtained by quantization of the nonlinear realization of SU(2, 2) as a quasiconformal group in arXiv:0908.3624. We show that the generators of SU(2,2) for these unitary irreducible representations can be written as bilinears of deformed twistorial oscillators which transform nonlinearly under the Lorentz group and apply them to define and study higher spin algebras and superalgebras in AdS 5.more » The higher spin (HS) algebra of Fradkin-Vasiliev type in AdS 5 is simply the enveloping algebra of SU(2, 2) quotiented by a two-sided ideal (Joseph ideal) which annihilates the minrep. We show that the Joseph ideal vanishes identically for the quasiconformal realization of the minrep and its enveloping algebra leads directly to the HS algebra in AdS 5. Furthermore, the enveloping algebras of the deformations of the minrep define a one parameter family of HS algebras in AdS 5 for which certain 4d covariant deformations of the Joseph ideal vanish identically. These results extend to superconformal algebras SU(2, 2|N) and we find a one parameter family of HS superalgebras as enveloping algebras of the minimal unitary supermultiplet and its deformations. Our results suggest the existence of a family of (supersymmetric) HS theories in AdS 5 which are dual to free (super)conformal field theories (CFTs) or to interacting but integrable (supersymmetric) CFTs in 4d. We also discuss the corresponding picture in HS algebras in AdS 4 where the corresponding 3d conformal group Sp(4,R) admits only two massless representations (minreps), namely the scalar and spinor singletons.« less

A spatial operator algebra for manipulator modeling and control

NASA Technical Reports Server (NTRS)

Rodriguez, G.; Kreutz, K.; Milman, M.

1988-01-01

A powerful new spatial operator algebra for modeling, control, and trajectory design of manipulators is discussed along with its implementation in the Ada programming language. Applications of this algebra to robotics include an operator representation of the manipulator Jacobian matrix; the robot dynamical equations formulated in terms of the spatial algebra, showing the complete equivalence between the recursive Newton-Euler formulations to robot dynamics; the operator factorization and inversion of the manipulator mass matrix which immediately results in O(N) recursive forward dynamics algorithms; the joint accelerations of a manipulator due to a tip contact force; the recursive computation of the equivalent mass matrix as seen at the tip of a manipulator; and recursive forward dynamics of a closed chain system. Finally, additional applications and current research involving the use of the spatial operator algebra are discussed in general terms.

Brick walls and AdS/CFT

NASA Astrophysics Data System (ADS)

Kay, Bernard S.; Ortíz, L.

2014-05-01

We discuss the relationship between the bulk-boundary correspondence in Rehren's algebraic holography (and in other `fixed-background', QFT-based, approaches to holography) and in mainstream string-theoretic `Maldacena AdS/CFT'. Especially, we contrast the understanding of black-hole entropy from the point of view of QFT in curved spacetime—in the framework of 't Hooft's `brick wall' model—with the understanding based on Maldacena AdS/CFT. We show that the brick-wall modification of a Klein-Gordon field in the Hartle-Hawking-Israel state on dimensional Schwarzschild AdS has a well-defined boundary limit with the same temperature and entropy as the brick-wall-modified bulk theory. One of our main purposes is to point out a close connection, for general AdS/CFT situations, between the puzzle raised by Arnsdorf and Smolin regarding the relationship between Rehren's algebraic holography and mainstream AdS/CFT and the puzzle embodied in the `complementarity principle' proposed by Mukohyama and Israel in their work on the brick-wall approach to black hole entropy. Working on the assumption that similar results will hold for bulk QFT other than the Klein-Gordon field and for Schwarzschild AdS in other dimensions, and recalling the first author's proposed resolution to the Mukohyama-Israel puzzle based on his `matter-gravity entanglement hypothesis', we argue that, in Maldacena AdS/CFT, the algebra of the boundary CFT is isomorphic only to a proper subalgebra of the bulk algebra, albeit (at non-zero temperature) the (GNS) Hilbert spaces of bulk and boundary theories are still the `same'—the total bulk state being pure, while the boundary state is mixed (thermal). We also argue from the finiteness of its boundary (and hence, on our assumptions, also bulk) entropy at finite temperature, that the Rehren dual of the Maldacena boundary CFT cannot itself be a QFT and must, instead, presumably be something like a string theory.

The Hom-Yang-Baxter equation and Hom-Lie algebras

DOE Office of Scientific and Technical Information (OSTI.GOV)

Yau, Donald

2011-05-15

Motivated by recent work on Hom-Lie algebras, a twisted version of the Yang-Baxter equation, called the Hom-Yang-Baxter equation (HYBE), was introduced by Yau [J. Phys. A 42, 165202 (2009)]. In this paper, several more classes of solutions of the HYBE are constructed. Some of the solutions of the HYBE are closely related to the quantum enveloping algebra of sl(2), the Jones-Conway polynomial, and Yetter-Drinfel'd modules. Under some invertibility conditions, we construct a new infinite sequence of solutions of the HYBE from a given one.

Hilbert space structure in quantum gravity: an algebraic perspective

DOE PAGES

Giddings, Steven B.

2015-12-16

If quantum gravity respects the principles of quantum mechanics, suitably generalized, it may be that a more viable approach to the theory is through identifying the relevant quantum structures rather than by quantizing classical spacetime. Here, this viewpoint is supported by difficulties of such quantization, and by the apparent lack of a fundamental role for locality. In finite or discrete quantum systems, important structure is provided by tensor factorizations of the Hilbert space. However, even in local quantum field theory properties of the generic type III von Neumann algebras and of long range gauge fields indicate that factorization of themore » Hilbert space is problematic. Instead it is better to focus on the structure of the algebra of observables, and in particular on its subalgebras corresponding to regions. This paper suggests that study of analogous algebraic structure in gravity gives an important perspective on the nature of the quantum theory. Significant departures from the subalgebra structure of local quantum field theory are found, working in the correspondence limit of long-distances/low-energies. Particularly, there are obstacles to identifying commuting algebras of localized operators. In addition to suggesting important properties of the algebraic structure, this and related observations pose challenges to proposals of a fundamental role for entanglement.« less

Hilbert space structure in quantum gravity: an algebraic perspective

DOE Office of Scientific and Technical Information (OSTI.GOV)

Giddings, Steven B.

If quantum gravity respects the principles of quantum mechanics, suitably generalized, it may be that a more viable approach to the theory is through identifying the relevant quantum structures rather than by quantizing classical spacetime. Here, this viewpoint is supported by difficulties of such quantization, and by the apparent lack of a fundamental role for locality. In finite or discrete quantum systems, important structure is provided by tensor factorizations of the Hilbert space. However, even in local quantum field theory properties of the generic type III von Neumann algebras and of long range gauge fields indicate that factorization of themore » Hilbert space is problematic. Instead it is better to focus on the structure of the algebra of observables, and in particular on its subalgebras corresponding to regions. This paper suggests that study of analogous algebraic structure in gravity gives an important perspective on the nature of the quantum theory. Significant departures from the subalgebra structure of local quantum field theory are found, working in the correspondence limit of long-distances/low-energies. Particularly, there are obstacles to identifying commuting algebras of localized operators. In addition to suggesting important properties of the algebraic structure, this and related observations pose challenges to proposals of a fundamental role for entanglement.« less

Quantum group structure and local fields in the algebraic approach to 2D gravity

NASA Astrophysics Data System (ADS)

Schnittger, J.

1995-07-01

This review contains a summary of the work by J.-L. Gervais and the author on the operator approach to 2d gravity. Special emphasis is placed on the construction of local observables — the Liouville exponentials and the Liouville field itself — and the underlying algebra of chiral vertex operators. The double quantum group structure arising from the presence of two screening charges is discussed and the generalized algebra and field operators are derived. In the last part, we show that our construction gives rise to a natural definition of a quantum tau function, which is a noncommutative version of the classical group-theoretic representation of the Liouville fields by Leznov and Saveliev.

Differential Models for B-Type Open-Closed Topological Landau-Ginzburg Theories

NASA Astrophysics Data System (ADS)

Babalic, Elena Mirela; Doryn, Dmitry; Lazaroiu, Calin Iuliu; Tavakol, Mehdi

2018-05-01

We propose a family of differential models for B-type open-closed topological Landau-Ginzburg theories defined by a pair (X,W), where X is any non-compact Calabi-Yau manifold and W is any holomorphic complex-valued function defined on X whose critical set is compact. The models are constructed at cochain level using smooth data, including the twisted Dolbeault algebra of polyvector-valued forms and a twisted Dolbeault category of holomorphic factorizations of W. We give explicit proposals for cochain level versions of the bulk and boundary traces and for the bulk-boundary and boundary-bulk maps of the Landau-Ginzburg theory. We prove that most of the axioms of an open-closed TFT (topological field theory) are satisfied on cohomology and conjecture that the remaining two axioms (namely non-degeneracy of bulk and boundary traces and the topological Cardy constraint) are also satisfied.

Braided Categories of Endomorphisms as Invariants for Local Quantum Field Theories

NASA Astrophysics Data System (ADS)

Giorgetti, Luca; Rehren, Karl-Henning

2018-01-01

We want to establish the "braided action" (defined in the paper) of the DHR category on a universal environment algebra as a complete invariant for completely rational chiral conformal quantum field theories. The environment algebra can either be a single local algebra, or the quasilocal algebra, both of which are model-independent up to isomorphism. The DHR category as an abstract structure is captured by finitely many data (superselection sectors, fusion, and braiding), whereas its braided action encodes the full dynamical information that distinguishes models with isomorphic DHR categories. We show some geometric properties of the "duality pairing" between local algebras and the DHR category that are valid in general (completely rational) chiral CFTs. Under some additional assumptions whose status remains to be settled, the braided action of its DHR category completely classifies a (prime) CFT. The approach does not refer to the vacuum representation, or the knowledge of the vacuum state.

RANS modeling of scalar dispersion from localized sources within a simplified urban-area model

NASA Astrophysics Data System (ADS)

Rossi, Riccardo; Capra, Stefano; Iaccarino, Gianluca

2011-11-01

The dispersion of a passive scalar downstream a localized source within a simplified urban-like geometry is examined by means of RANS scalar flux models. The computations are conducted under conditions of neutral stability and for three different incoming wind directions (0°, 45°, 90°) at a roughness Reynolds number of Ret = 391. A Reynolds stress transport model is used to close the flow governing equations whereas both the standard eddy-diffusivity closure and algebraic flux models are employed to close the transport equation for the passive scalar. The comparison with a DNS database shows improved reliability from algebraic scalar flux models towards predicting both the mean concentration and the plume structure. Since algebraic flux models do not increase substantially the computational effort, the results indicate that the use of tensorial-diffusivity can be promising tool for dispersion simulations for the urban environment.

Quantum corrections to Bekenstein-Hawking black hole entropy and gravity partition functions

NASA Astrophysics Data System (ADS)

Bytsenko, A. A.; Tureanu, A.

2013-08-01

Algebraic aspects of the computation of partition functions for quantum gravity and black holes in AdS3 are discussed. We compute the sub-leading quantum corrections to the Bekenstein-Hawking entropy. It is shown that the quantum corrections to the classical result can be included systematically by making use of the comparison with conformal field theory partition functions, via the AdS3/CFT2 correspondence. This leads to a better understanding of the role of modular and spectral functions, from the point of view of the representation theory of infinite-dimensional Lie algebras. Besides, the sum of known quantum contributions to the partition function can be presented in a closed form, involving the Patterson-Selberg spectral function. These contributions can be reproduced in a holomorphically factorized theory whose partition functions are associated with the formal characters of the Virasoro modules. We propose a spectral function formulation for quantum corrections to the elliptic genus from supergravity states.

Z2×Z2 generalizations of 𝒩 =2 super Schrödinger algebras and their representations

NASA Astrophysics Data System (ADS)

Aizawa, N.; Segar, J.

2017-11-01

We generalize the real and chiral N =2 super Schrödinger algebras to Z2×Z2-graded Lie superalgebras. This is done by D-module presentation, and as a consequence, the D-module presentations of Z2×Z2-graded superalgebras are identical to the ones of super Schrödinger algebras. We then generalize the calculus over the Grassmann number to Z2×Z2 setting. Using it and the standard technique of Lie theory, we obtain a vector field realization of Z2×Z2-graded superalgebras. A vector field realization of the Z2×Z2 generalization of N =1 super Schrödinger algebra is also presented.

The Resolvent Algebra of Non-relativistic Bose Fields: Observables, Dynamics and States

NASA Astrophysics Data System (ADS)

Buchholz, Detlev

2018-05-01

The structure of the gauge invariant (particle number preserving) C*-algebra generated by the resolvents of a non-relativistic Bose field is analyzed. It is shown to form a dense subalgebra of the bounded inverse limit of a directed system of approximately finite dimensional C*-algebras. Based on this observation, it is proven that the closure of the gauge invariant algebra is stable under the dynamics induced by Hamiltonians involving pair potentials. These facts allow to proceed to a description of interacting Bosons in terms of C*-dynamical systems. It is outlined how the present approach leads to simplifications in the construction of infinite bosonic states and sheds new light on topics in many body theory.

Some Applications Of Semigroups And Computer Algebra In Discrete Structures

NASA Astrophysics Data System (ADS)

Bijev, G.

2009-11-01

An algebraic approach to the pseudoinverse generalization problem in Boolean vector spaces is used. A map (p) is defined, which is similar to an orthogonal projection in linear vector spaces. Some other important maps with properties similar to those of the generalized inverses (pseudoinverses) of linear transformations and matrices corresponding to them are also defined and investigated. Let Ax = b be an equation with matrix A and vectors x and b Boolean. Stochastic experiments for solving the equation, which involves the maps defined and use computer algebra methods, have been made. As a result, the Hamming distance between vectors Ax = p(b) and b is equal or close to the least possible. We also share our experience in using computer algebra systems for teaching discrete mathematics and linear algebra and research. Some examples for computations with binary relations using Maple are given.

Aspects of QCD current algebra on a null plane

NASA Astrophysics Data System (ADS)

Beane, S. R.; Hobbs, T. J.

2016-09-01

Consequences of QCD current algebra formulated on a light-like hyperplane are derived for the forward scattering of vector and axial-vector currents on an arbitrary hadronic target. It is shown that current algebra gives rise to a special class of sum rules that are direct consequences of the independent chiral symmetry that exists at every point on the two-dimensional transverse plane orthogonal to the lightlike direction. These sum rules are obtained by exploiting the closed, infinite-dimensional algebra satisfied by the transverse moments of null-plane axial-vector and vector charge distributions. In the special case of a nucleon target, this procedure leads to the Adler-Weisberger, Gerasimov-Drell-Hearn, Cabibbo-Radicati and Fubini-Furlan-Rossetti sum rules. Matching to the dispersion-theoretic language which is usually invoked in deriving these sum rules, the moment sum rules are shown to be equivalent to algebraic constraints on forward S-matrix elements in the Regge limit.

Minimal unitary representation of 5d superconformal algebra F(4) and AdS 6/CFT 5 higher spin (super)-algebras

DOE PAGES

Fernando, Sudarshan; Günaydin, Murat

2014-11-28

We study the minimal unitary representation (minrep) of SO(5, 2), obtained by quantization of its geometric quasiconformal action, its deformations and supersymmetric extensions. The minrep of SO(5, 2) describes a massless conformal scalar field in five dimensions and admits a unique “deformation” which describes a massless conformal spinor. Scalar and spinor minreps of SO(5, 2) are the 5d analogs of Dirac’s singletons of SO(3, 2). We then construct the minimal unitary representation of the unique 5d supercon-formal algebra F(4) with the even subalgebra SO(5, 2) ×SU(2). The minrep of F(4) describes a massless conformal supermultiplet consisting of two scalar andmore » one spinor fields. We then extend our results to the construction of higher spin AdS 6/CFT 5 (super)-algebras. The Joseph ideal of the minrep of SO(5, 2) vanishes identically as operators and hence its enveloping algebra yields the AdS 6/CFT 5 bosonic higher spin algebra directly. The enveloping algebra of the spinor minrep defines a “deformed” higher spin algebra for which a deformed Joseph ideal vanishes identically as operators. These results are then extended to the construction of the unique higher spin AdS 6/CFT 5 superalgebra as the enveloping algebra of the minimal unitary realization of F(4) obtained by the quasiconformal methods.« less

Algebraic models of local period maps and Yukawa algebras

NASA Astrophysics Data System (ADS)

Bandiera, Ruggero; Manetti, Marco

2018-02-01

We describe some L_{∞} model for the local period map of a compact Kähler manifold. Applications include the study of deformations with associated variation of Hodge structure constrained by certain closed strata of the Grassmannian of the de Rham cohomology. As a by-product, we obtain an interpretation in the framework of deformation theory of the Yukawa coupling.

Labeled trees and the efficient computation of derivations

NASA Technical Reports Server (NTRS)

Grossman, Robert; Larson, Richard G.

1989-01-01

The effective parallel symbolic computation of operators under composition is discussed. Examples include differential operators under composition and vector fields under the Lie bracket. Data structures consisting of formal linear combinations of rooted labeled trees are discussed. A multiplication on rooted labeled trees is defined, thereby making the set of these data structures into an associative algebra. An algebra homomorphism is defined from the original algebra of operators into this algebra of trees. An algebra homomorphism from the algebra of trees into the algebra of differential operators is then described. The cancellation which occurs when noncommuting operators are expressed in terms of commuting ones occurs naturally when the operators are represented using this data structure. This leads to an algorithm which, for operators which are derivations, speeds up the computation exponentially in the degree of the operator. It is shown that the algebra of trees leads naturally to a parallel version of the algorithm.

Interaction of non-Abelian tensor gauge fields

NASA Astrophysics Data System (ADS)

Savvidy, George

2018-01-01

The non-Abelian tensor gauge fields take value in extended Poincaré algebra. In order to define the invariant Lagrangian we introduce a vector variable in two alternative ways: through the transversal representation of the extended Poincaré algebra and through the path integral over the auxiliary vector field with the U(1) Abelian action. We demonstrate that this allows to fix the unitary gauge and derive scattering amplitudes in spinor representation.

From Quantum Fields to Local Von Neumann Algebras

NASA Astrophysics Data System (ADS)

Borchers, H. J.; Yngvason, Jakob

The subject of the paper is an old problem of the general theory of quantized fields: When can the unbounded operators of a Wightman field theory be associated with local algebras of bounded operators in the sense of Haag? The paper reviews and extends previous work on this question, stressing its connections with a noncommutive generalization of the classical Hamburger moment problem. Necessary and sufficient conditions for the existence of a local net of von Neumann algebras corresponding to a given Wightman field are formulated in terms of strengthened versions of the usual positivity property of Wightman functionals. The possibility that the local net has to be defined in an enlarged Hilbert space cannot be ruled out in general. Under additional hypotheses, e.g., if the field operators obey certain energy bounds, such an extension of the Hilbert space is not necessary, however. In these cases a fairly simple condition for the existence of a local net can be given involving the concept of “central positivity” introduced by Powers. The analysis presented here applies to translationally covariant fields with an arbitrary number of components, whereas Lorentz covariance is not needed. The paper contains also a brief discussion of an approach to noncommutative moment problems due to Dubois-Violette, and concludes with some remarks on modular theory for algebras of unbounded operators.

Asymptotic symmetries of colored gravity in three dimensions

NASA Astrophysics Data System (ADS)

Joung, Euihun; Kim, Jaewon; Kim, Jihun; Rey, Soo-Jong

2018-03-01

Three-dimensional colored gravity refers to nonabelian isospin extension of Einstein gravity. We investigate the asymptotic symmetry algebra of the SU( N)-colored gravity in (2+1)-dimensional anti-de Sitter spacetime. Formulated by the Chern-Simons theory with SU( N, N) × SU( N, N) gauge group, the theory contains graviton, SU( N) Chern-Simons gauge fields and massless spin-two multiplets in the SU( N) adjoint representation, thus extending diffeomorphism to colored, nonabelian counterpart. We identify the asymptotic symmetry as Poisson algebra of generators associated with the residual global symmetries of the nonabelian diffeomorphism set by appropriately chosen boundary conditions. The resulting asymptotic symmetry algebra is a nonlinear extension of \\widehat{su(N)} Kac-Moody algebra, supplemented by additional generators corresponding to the massless spin-two adjoint matter fields.

DOE Office of Scientific and Technical Information (OSTI.GOV)

Fernando, Sudarshan; Günaydin, Murat

We study the minimal unitary representation (minrep) of SO(5, 2), obtained by quantization of its geometric quasiconformal action, its deformations and supersymmetric extensions. The minrep of SO(5, 2) describes a massless conformal scalar field in five dimensions and admits a unique “deformation” which describes a massless conformal spinor. Scalar and spinor minreps of SO(5, 2) are the 5d analogs of Dirac’s singletons of SO(3, 2). We then construct the minimal unitary representation of the unique 5d supercon-formal algebra F(4) with the even subalgebra SO(5, 2) ×SU(2). The minrep of F(4) describes a massless conformal supermultiplet consisting of two scalar andmore » one spinor fields. We then extend our results to the construction of higher spin AdS 6/CFT 5 (super)-algebras. The Joseph ideal of the minrep of SO(5, 2) vanishes identically as operators and hence its enveloping algebra yields the AdS 6/CFT 5 bosonic higher spin algebra directly. The enveloping algebra of the spinor minrep defines a “deformed” higher spin algebra for which a deformed Joseph ideal vanishes identically as operators. These results are then extended to the construction of the unique higher spin AdS 6/CFT 5 superalgebra as the enveloping algebra of the minimal unitary realization of F(4) obtained by the quasiconformal methods.« less

Closedness of orbits in a space with SU(2) Poisson structure

NASA Astrophysics Data System (ADS)

Fatollahi, Amir H.; Shariati, Ahmad; Khorrami, Mohammad

2014-06-01

The closedness of orbits of central forces is addressed in a three-dimensional space in which the Poisson bracket among the coordinates is that of the SU(2) Lie algebra. In particular it is shown that among problems with spherically symmetric potential energies, it is only the Kepler problem for which all bounded orbits are closed. In analogy with the case of the ordinary space, a conserved vector (apart from the angular momentum) is explicitly constructed, which is responsible for the orbits being closed. This is the analog of the Laplace-Runge-Lenz vector. The algebra of the constants of the motion is also worked out.

Spatial operator algebra framework for multibody system dynamics

NASA Technical Reports Server (NTRS)

Rodriguez, G.; Jain, Abhinandan; Kreutz, K.

1989-01-01

The Spatial Operator Algebra framework for the dynamics of general multibody systems is described. The use of a spatial operator-based methodology permits the formulation of the dynamical equations of motion of multibody systems in a concise and systematic way. The dynamical equations of progressively more complex grid multibody systems are developed in an evolutionary manner beginning with a serial chain system, followed by a tree topology system and finally, systems with arbitrary closed loops. Operator factorizations and identities are used to develop novel recursive algorithms for the forward dynamics of systems with closed loops. Extensions required to deal with flexible elements are also discussed.

Spatial Operator Algebra for multibody system dynamics

NASA Technical Reports Server (NTRS)

Rodriguez, G.; Jain, A.; Kreutz-Delgado, K.

1992-01-01

The Spatial Operator Algebra framework for the dynamics of general multibody systems is described. The use of a spatial operator-based methodology permits the formulation of the dynamical equations of motion of multibody systems in a concise and systematic way. The dynamical equations of progressively more complex grid multibody systems are developed in an evolutionary manner beginning with a serial chain system, followed by a tree topology system and finally, systems with arbitrary closed loops. Operator factorizations and identities are used to develop novel recursive algorithms for the forward dynamics of systems with closed loops. Extensions required to deal with flexible elements are also discussed.

Chiral higher spin theories and self-duality

NASA Astrophysics Data System (ADS)

Ponomarev, Dmitry

2017-12-01

We study recently proposed chiral higher spin theories — cubic theories of interacting massless higher spin fields in four-dimensional flat space. We show that they are naturally associated with gauge algebras, which manifest themselves in several related ways. Firstly, the chiral higher spin equations of motion can be reformulated as the self-dual Yang-Mills equations with the associated gauge algebras instead of the usual colour gauge algebra. We also demonstrate that the chiral higher spin field equations, similarly to the self-dual Yang-Mills equations, feature an infinite algebra of hidden symmetries, which ensures their integrability. Secondly, we show that off-shell amplitudes in chiral higher spin theories satisfy the generalised BCJ relations with the usual colour structure constants replaced by the structure constants of higher spin gauge algebras. We also propose generalised double copy procedures featuring higher spin theory amplitudes. Finally, using the light-cone deformation procedure we prove that the structure of the Lagrangian that leads to all these properties is universal and follows from Lorentz invariance.

Flatspace chiral supergravity

NASA Astrophysics Data System (ADS)

Bagchi, Arjun; Basu, Rudranil; Detournary, Stéphane; Parekh, Pulastya

2018-05-01

We propose a holographic duality between a 2 dimensional (2d) chiral superconformal field theory and a certain theory of supergravity in 3d with flatspace boundary conditions that is obtained as a double scaling limit of a parity breaking theory of supergravity. We show how the asymptotic symmetries of the bulk theory reduce from the "despotic" super Bondi-Metzner-Sachs algebra (or equivalently the inhomogeneous super Galilean conformal algebra) to a single copy of the super-Virasoro algebra in this limit and also reproduce the same reduction from a study of null vectors in the putative 2d dual field theory.

Combinatorial operad actions on cochains

NASA Astrophysics Data System (ADS)

Berger, Clemens; Fresse, Benoit

2004-07-01

A classical E-infinity operad is formed by the bar construction of the symmetric groups. Such an operad has been introduced by M. Barratt and P. Eccles in the context of simplicial sets in order to have an analogue of the Milnor FK-construction for infinite loop spaces. The purpose of this paper is to prove that the associative algebra structure on the normalized cochain complex of a simplicial set extends to the structure of an algebra over the Barratt-Eccles operad. We also prove that differential graded algebras over the Barratt-Eccles operad form a closed model category. Similar results hold for the normalized Hochschild cochain complex of an associative algebra. More precisely, the Hochschild cochain complex is acted on by a suboperad of the Barratt-Eccles operad which is equivalent to the classical little squares operad.

LETTER TO THE EDITOR: Landau levels on the hyperbolic plane

NASA Astrophysics Data System (ADS)

Fakhri, H.; Shariati, M.

2004-11-01

The quantum states of a spinless charged particle on a hyperbolic plane in the presence of a uniform magnetic field with a generalized quantization condition are proved to be the bases of the irreducible Hilbert representation spaces of the Lie algebra u(1, 1). The dynamical symmetry group U(1, 1) with the explicit form of the Lie algebra generators is extracted. It is also shown that the energy has an infinite-fold degeneracy in each of the representation spaces which are allocated to the different values of the magnetic field strength. Based on the simultaneous shift of two parameters, it is also noted that the quantum states realize the representations of Lie algebra u(2) by shifting the magnetic field strength.

Coherent population transfer in multilevel systems with magnetic sublevels. II. Algebraic analysis

NASA Astrophysics Data System (ADS)

Martin, J.; Shore, B. W.; Bergmann, K.

1995-07-01

We extend previous theoretical work on coherent population transfer by stimulated Raman adiabatic passage for states involving nonzero angular momentum. The pump and Stokes fields are either copropagating or counterpropagating with the corresponding linearly polarized electric-field vectors lying in a common plane with the magnetic-field direction. Zeeman splitting lifts the magnetic sublevel degeneracy. We present an algebraic analysis of dressed-state properties to explain the behavior noted in numerical studies. In particular, we discuss conditions which are likely to lead to a failure of complete population transfer. The applied strategy, based on simple methods of linear algebra, will also be successful for other types of discrete multilevel systems, provided the rotating-wave and adiabatic approximation are valid.

Applications: Using Algebra in an Accounting Practice.

ERIC Educational Resources Information Center

Eisner, Gail A.

1994-01-01

Presents examples of algebra from the field of accounting including proportional ownership of stock, separation of a loan payment into principal and interest portions, depreciation methods, and salary withholdings computations. (MKR)

Hidden symmetries and Lie algebra structures from geometric and supergravity Killing spinors

NASA Astrophysics Data System (ADS)

Açık, Özgür; Ertem, Ümit

2016-08-01

We consider geometric and supergravity Killing spinors and the spinor bilinears constructed out of them. The spinor bilinears of geometric Killing spinors correspond to the antisymmetric generalizations of Killing vector fields which are called Killing-Yano forms. They constitute a Lie superalgebra structure in constant curvature spacetimes. We show that the Dirac currents of geometric Killing spinors satisfy a Lie algebra structure up to a condition on 2-form spinor bilinears. We propose that the spinor bilinears of supergravity Killing spinors give way to different generalizations of Killing vector fields to higher degree forms. It is also shown that those supergravity Killing forms constitute a Lie algebra structure in six- and ten-dimensional cases. For five- and eleven-dimensional cases, the Lie algebra structure depends on an extra condition on supergravity Killing forms.

Wave scattering of complex local site in a layered half-space by using a multidomain IBEM: incident plane SH waves

NASA Astrophysics Data System (ADS)

Ba, Zhenning; Yin, Xiao

2016-06-01

A multidomain indirect boundary element method (IBEM) is proposed to study the wave scattering of plane SH waves by complex local site in a layered half-space. The new method, using both the full-space and layered half-space Green's functions as its fundamental solutions can also be regarded as a coupled method of the full-space IBEM and half-space IBEM. First, the whole model is decomposed into independent closed regions and an opened layered half-space region with all of the irregular interfaces; then, fictitious uniformly distributed loads are applied separately on the boundaries of each region, and scattered fields of the closed regions and the opened layered half-space region are constructed by calculating the full-space and layered half-space Green's functions, respectively; finally, all of the regions are assembled to establish the linear algebraic system that arises from discretization. The densities of the distributed loads are determined directly by solving the algebraic system. The accuracy and capability of the new approach are verified extensively by comparing its results with those of published approaches for a class of hills, valleys and embedded inclusions. And the capability of the new method is further displayed when it is used to investigate a hill-triple layered valley-hill coupled topography in a multilayered half-space. All of the numerical calculations presented in this paper demonstrate that the new method is very suitable for solving multidomain coupled multilayered wave scattering problems with the merits of high accuracy and representing the scattered fields in different kinds of regions more reasonably and flexibly.

The Effects of Cognitive Style and Piagetian Logical Reasoning on Solving a Propositional Relation Algebra Word Problem.

ERIC Educational Resources Information Center

Nasser, Ramzi; Carifio, James

The purpose of this study was to find out whether students perform differently on algebra word problems that have certain key context features and entail proportional reasoning, relative to their level of logical reasoning and their degree of field dependence/independence. Field-independent students tend to restructure and break stimuli into parts…

Properties of coupled-cluster equations originating in excitation sub-algebras

NASA Astrophysics Data System (ADS)

Kowalski, Karol

2018-03-01

In this paper, we discuss properties of single-reference coupled cluster (CC) equations associated with the existence of sub-algebras of excitations that allow one to represent CC equations in a hybrid fashion where the cluster amplitudes associated with these sub-algebras can be obtained by solving the corresponding eigenvalue problem. For closed-shell formulations analyzed in this paper, the hybrid representation of CC equations provides a natural way for extending active-space and seniority number concepts to provide an accurate description of electron correlation effects. Moreover, a new representation can be utilized to re-define iterative algorithms used to solve CC equations, especially for tough cases defined by the presence of strong static and dynamical correlation effects. We will also explore invariance properties associated with excitation sub-algebras to define a new class of CC approximations referred to in this paper as the sub-algebra-flow-based CC methods. We illustrate the performance of these methods on the example of ground- and excited-state calculations for commonly used small benchmark systems.

Algebraic Riccati equations in zero-sum differential games

NASA Technical Reports Server (NTRS)

Johnson, T. L.; Chao, A.

1974-01-01

The procedure for finding the closed-loop Nash equilibrium solution of two-player zero-sum linear time-invariant differential games with quadratic performance criteria and classical information pattern may be reduced in most cases to the solution of an algebraic Riccati equation. Based on the results obtained by Willems, necessary and sufficient conditions for existence of solutions to these equations are derived, and explicit conditions for a scalar example are given.

Spacetime algebra as a powerful tool for electromagnetism

NASA Astrophysics Data System (ADS)

Dressel, Justin; Bliokh, Konstantin Y.; Nori, Franco

2015-08-01

We present a comprehensive introduction to spacetime algebra that emphasizes its practicality and power as a tool for the study of electromagnetism. We carefully develop this natural (Clifford) algebra of the Minkowski spacetime geometry, with a particular focus on its intrinsic (and often overlooked) complex structure. Notably, the scalar imaginary that appears throughout the electromagnetic theory properly corresponds to the unit 4-volume of spacetime itself, and thus has physical meaning. The electric and magnetic fields are combined into a single complex and frame-independent bivector field, which generalizes the Riemann-Silberstein complex vector that has recently resurfaced in studies of the single photon wavefunction. The complex structure of spacetime also underpins the emergence of electromagnetic waves, circular polarizations, the normal variables for canonical quantization, the distinction between electric and magnetic charge, complex spinor representations of Lorentz transformations, and the dual (electric-magnetic field exchange) symmetry that produces helicity conservation in vacuum fields. This latter symmetry manifests as an arbitrary global phase of the complex field, motivating the use of a complex vector potential, along with an associated transverse and gauge-invariant bivector potential, as well as complex (bivector and scalar) Hertz potentials. Our detailed treatment aims to encourage the use of spacetime algebra as a readily available and mature extension to existing vector calculus and tensor methods that can greatly simplify the analysis of fundamentally relativistic objects like the electromagnetic field.

Algebra and topology for applications to physics

NASA Technical Reports Server (NTRS)

Rozhkov, S. S.

1987-01-01

The principal concepts of algebra and topology are examined with emphasis on applications to physics. In particular, attention is given to sets and mapping; topological spaces and continuous mapping; manifolds; and topological groups and Lie groups. The discussion also covers the tangential spaces of the differential manifolds, including Lie algebras, vector fields, and differential forms, properties of differential forms, mapping of tangential spaces, and integration of differential forms.

DOE Office of Scientific and Technical Information (OSTI.GOV)

Agarwala, Susama; Delaney, Colleen

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.

Beauty and the beast: Superconformal symmetry in a monster module

NASA Astrophysics Data System (ADS)

Dixon, L.; Ginsparg, P.; Harvey, J.

1988-06-01

Frenkel, Lepowsky, and Meurman have constructed a representation of the largest sporadic simple finite group, the Fischer-Griess monster, as the automorphism group of the operator product algebra of a conformal field theory with central charge c=24. In string terminology, their construction corresponds to compactification on a Z 2 asymmetric orbifold constructed from the torus R 24/∧, where ∧ is the Leech lattice. In this note we point out that their construction naturally embodies as well a larger algebraic structure, namely a super-Virasoro algebra with central charge ĉ=16, with the supersymmetry generator constructed in terms of bosonic twist fields.

Representations of spacetime diffeomorphisms. I. Canonical parametrized field theories

DOE Office of Scientific and Technical Information (OSTI.GOV)

Isham, C.J.; Kuchar, K.V.

The super-Hamiltonian and supermomentum in canonical geometrodynamics or in a parametried field theory on a given Riemannian background have Poisson brackets which obey the Dirac relations. By smearing the supermomentum with vector fields VepsilonL Diff..sigma.. on the space manifold ..sigma.., the Lie algebra L Diff ..sigma.. of the spatial diffeomorphism group Diff ..sigma.. can be mapped antihomomorphically into the Poisson bracket algebra on the phase space of the system. The explicit dependence of the Poisson brackets between two super-Hamiltonians on canonical coordinates (spatial metrics in geometrodynamics and embedding variables in parametrized theories) is usually regarded as an indication that themore » Dirac relations cannot be connected with a representation of the complete Lie algebra L Diff M of spacetime diffeomorphisms.« less

Elliptic biquaternion algebra

NASA Astrophysics Data System (ADS)

Özen, Kahraman Esen; Tosun, Murat

2018-01-01

In this study, we define the elliptic biquaternions and construct the algebra of elliptic biquaternions over the elliptic number field. Also we give basic properties of elliptic biquaternions. An elliptic biquaternion is in the form A0 + A1i + A2j + A3k which is a linear combination of {1, i, j, k} where the four components A0, A1, A2 and A3 are elliptic numbers. Here, 1, i, j, k are the quaternion basis of the elliptic biquaternion algebra and satisfy the same multiplication rules which are satisfied in both real quaternion algebra and complex quaternion algebra. In addition, we discuss the terms; conjugate, inner product, semi-norm, modulus and inverse for elliptic biquaternions.

Topological String Theory and Enumerative Geometry

DOE Office of Scientific and Technical Information (OSTI.GOV)

Song, Y. S

In this thesis we investigate several problems which have their roots in both topological string theory and enumerative geometry. In the former case, underlying theories are topological field theories, whereas the latter case is concerned with intersection theories on moduli spaces. A permeating theme in this thesis is to examine the close interplay between these two complementary fields of study. The main problems addressed are as follows: In considering the Hurwitz enumeration problem of branched covers of compact connected Riemann surfaces, we completely solve the problem in the case of simple Hurwitz numbers. In addition, utilizing the connection between Hurwitzmore » numbers and Hodge integrals, we derive a generating function for the latter on the moduli space {bar M}{sub g,2} of 2-pointed, genus-g Deligne-Mumford stable curves. We also investigate Givental's recent conjecture regarding semisimple Frobenius structures and Gromov-Witten invariants, both of which are closely related to topological field theories; we consider the case of a complex projective line P{sup 1} as a specific example and verify his conjecture at low genera. In the last chapter, we demonstrate that certain topological open string amplitudes can be computed via relative stable morphisms in the algebraic category.« less

Tropospheric wet refractivity tomography using multiplicative algebraic reconstruction technique

NASA Astrophysics Data System (ADS)

Xiaoying, Wang; Ziqiang, Dai; Enhong, Zhang; Fuyang, K. E.; Yunchang, Cao; Lianchun, Song

2014-01-01

Algebraic reconstruction techniques (ART) have been successfully used to reconstruct the total electron content (TEC) of the ionosphere and in recent years be tentatively used in tropospheric wet refractivity and water vapor tomography in the ground-based GNSS technology. The previous research on ART used in tropospheric water vapor tomography focused on the convergence and relaxation parameters for various algebraic reconstruction techniques and rarely discussed the impact of Gaussian constraints and initial field on the iteration results. The existing accuracy evaluation parameters calculated from slant wet delay can only evaluate the resultant precision of the voxels penetrated by slant paths and cannot evaluate that of the voxels not penetrated by any slant path. The paper proposes two new statistical parameters Bias and RMS, calculated from wet refractivity of the total voxels, to improve the deficiencies of existing evaluation parameters and then discusses the effect of the Gaussian constraints and initial field on the convergence and tomography results in multiplicative algebraic reconstruction technique (MART) to reconstruct the 4D tropospheric wet refractivity field using simulation method.

Nonlinear response of a harmonic diatomic molecule: Algebraic nonperturbative calculation

DOE Office of Scientific and Technical Information (OSTI.GOV)

Recamier, Jose; Mochan, W. Luis; Maytorena, Jesus A.

2005-08-15

Even harmonic molecules display a nonlinear behavior when driven by an inhomogeneous field. We calculate the response of single harmonic molecules to a monochromatic time and space dependent electric field E(r,t) of frequency {omega} employing exact algebraic methods. We evaluate the responses at the fundamental frequency {omega} and at successive harmonics 2{omega}, 3{omega}, etc., as a function of the intensity and of the frequency of the field and compare the results with those of first and second order perturbation theory.

Geometric interpretation of vertex operator algebras.

PubMed Central

Huang, Y Z

1991-01-01

In this paper, Vafa's approach to the formulation of conformal field theories is combined with the formal calculus developed in Frenkel, Lepowsky, and Meurman's work on the vertex operator construction of the Monster to give a geometric definition of vertex operator algebras. The main result announced is the equivalence between this definition and the algebraic one in the sense that the categories determined by these definitions are isomorphic. PMID:11607240

Quantum Superalgebras at Roots of Unity and Topological Invariants of Three-manifolds

NASA Astrophysics Data System (ADS)

Blumen, Sacha C.

2006-01-01

The general method of Reshetikhin and Turaev is followed to develop topological invariants of closed, connected, orientable 3-manifolds from a new class of algebras called pseudo-modular Hopf algebras. Pseudo-modular Hopf algebras are a class of Z_2-graded ribbon Hopf algebras that generalise the concept of a modular Hopf algebra. The quantum superalgebra U_q(osp(1|2n)) over C is considered with q a primitive N^th root of unity for all integers N >= 3. For such a q, a certain left ideal I of U_q(osp(1|2n)) is also a two-sided Hopf ideal, and the quotient algebra U_q^(N)(osp(1|2n)) = U_q(osp(1|2n)) / I is a Z_2-graded ribbon Hopf algebra. For all n and all N >= 3, a finite collection of finite dimensional representations of U_q^(N)(osp(1|2n)) is defined. Each such representation of U_q^(N)(osp(1|2n)) is labelled by an integral dominant weight belonging to the truncated dominant Weyl chamber. Properties of these representations are considered: the quantum superdimension of each representation is calculated, each representation is shown to be self-dual, and more importantly, the decomposition of the tensor product of an arbitrary number of such representations is obtained for even N. It is proved that the quotient algebra U_q^(N)(osp(1|2n)), together with the set of finite dimensional representations discussed above, form a pseudo-modular Hopf algebra when N >= 6 is twice an odd number. Using this pseudo-modular Hopf algebra, we construct a topological invariant of 3-manifolds. This invariant is shown to be different to the topological invariants of 3-manifolds arising from quantum so(2n+1) at roots of unity.

Differential Geometry and Lie Groups for Physicists

NASA Astrophysics Data System (ADS)

Fecko, Marián.

2006-10-01

Introduction; 1. The concept of a manifold; 2. Vector and tensor fields; 3. Mappings of tensors induced by mappings of manifolds; 4. Lie derivative; 5. Exterior algebra; 6. Differential calculus of forms; 7. Integral calculus of forms; 8. Particular cases and applications of Stoke's Theorem; 9. Poincaré Lemma and cohomologies; 10. Lie Groups - basic facts; 11. Differential geometry of Lie Groups; 12. Representations of Lie Groups and Lie Algebras; 13. Actions of Lie Groups and Lie Algebras on manifolds; 14. Hamiltonian mechanics and symplectic manifolds; 15. Parallel transport and linear connection on M; 16. Field theory and the language of forms; 17. Differential geometry on TM and T*M; 18. Hamiltonian and Lagrangian equations; 19. Linear connection and the frame bundle; 20. Connection on a principal G-bundle; 21. Gauge theories and connections; 22. Spinor fields and Dirac operator; Appendices; Bibliography; Index.

Differential Geometry and Lie Groups for Physicists

NASA Astrophysics Data System (ADS)

Fecko, Marián.

2011-03-01

Introduction; 1. The concept of a manifold; 2. Vector and tensor fields; 3. Mappings of tensors induced by mappings of manifolds; 4. Lie derivative; 5. Exterior algebra; 6. Differential calculus of forms; 7. Integral calculus of forms; 8. Particular cases and applications of Stoke's Theorem; 9. Poincaré Lemma and cohomologies; 10. Lie Groups - basic facts; 11. Differential geometry of Lie Groups; 12. Representations of Lie Groups and Lie Algebras; 13. Actions of Lie Groups and Lie Algebras on manifolds; 14. Hamiltonian mechanics and symplectic manifolds; 15. Parallel transport and linear connection on M; 16. Field theory and the language of forms; 17. Differential geometry on TM and T*M; 18. Hamiltonian and Lagrangian equations; 19. Linear connection and the frame bundle; 20. Connection on a principal G-bundle; 21. Gauge theories and connections; 22. Spinor fields and Dirac operator; Appendices; Bibliography; Index.

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.

Extended gauge theory and gauged free differential algebras

NASA Astrophysics Data System (ADS)

Salgado, P.; Salgado, S.

2018-01-01

Recently, Antoniadis, Konitopoulos and Savvidy introduced, in the context of the so-called extended gauge theory, a procedure to construct background-free gauge invariants, using non-abelian gauge potentials described by higher degree forms. In this article it is shown that the extended invariants found by Antoniadis, Konitopoulos and Savvidy can be constructed from an algebraic structure known as free differential algebra. In other words, we show that the above mentioned non-abelian gauge theory, where the gauge fields are described by p-forms with p ≥ 2, can be obtained by gauging free differential algebras.

Matrix De Rham Complex and Quantum A-infinity algebras

NASA Astrophysics Data System (ADS)

Barannikov, S.

2014-04-01

I establish the relation of the non-commutative BV-formalism with super-invariant matrix integration. In particular, the non-commutative BV-equation, defining the quantum A ∞-algebras, introduced in Barannikov (Modular operads and non-commutative Batalin-Vilkovisky geometry. IMRN, vol. 2007, rnm075. Max Planck Institute for Mathematics 2006-48, 2007), is represented via de Rham differential acting on the supermatrix spaces related with Bernstein-Leites simple associative algebras with odd trace q( N), and gl( N| N). I also show that the matrix Lagrangians from Barannikov (Noncommutative Batalin-Vilkovisky geometry and matrix integrals. Isaac Newton Institute for Mathematical Sciences, Cambridge University, 2006) are represented by equivariantly closed differential forms.

Formality Theorem for Hochschild Cochains via Transfer

NASA Astrophysics Data System (ADS)

Dolgushev, Vasily

2011-08-01

We construct a 2-colored operad Ger ∞ which, on the one hand, extends the operad Ger ∞ governing homotopy Gerstenhaber algebras and, on the other hand, extends the 2-colored operad governing open-closed homotopy algebras. We show that Tamarkin's Ger ∞-structure on the Hochschild cochain complex C •( A, A) of an A ∞-algebra A extends naturally to a {{Ger}^+_{infty}}-structure on the pair ( C •( A, A), A). We show that a formality quasi-isomorphism for the Hochschild cochains of the polynomial algebra can be obtained via transfer of this {{Ger}^+_{infty}}-structure to the cohomology of the pair ( C •( A, A), A). We show that {{Ger}^+_{infty}} is a sub DG operad of the first sheet E 1(SC) of the homology spectral sequence for the Fulton-MacPherson version SC of Voronov's Swiss Cheese operad. Finally, we prove that the DG operads {{Ger}^+_{infty}} and E 1(SC) are non-formal.

Classification of digital affine noncommutative geometries

NASA Astrophysics Data System (ADS)

Majid, Shahn; Pachoł, Anna

2018-03-01

It is known that connected translation invariant n-dimensional noncommutative differentials dxi on the algebra k[x1, …, xn] of polynomials in n-variables over a field k are classified by commutative algebras V on the vector space spanned by the coordinates. These data also apply to construct differentials on the Heisenberg algebra "spacetime" with relations [xμ, xν] = λΘμν, where Θ is an antisymmetric matrix, as well as to Lie algebras with pre-Lie algebra structures. We specialise the general theory to the field k =F2 of two elements, in which case translation invariant metrics (i.e., with constant coefficients) are equivalent to making V a Frobenius algebra. We classify all of these and their quantum Levi-Civita bimodule connections for n = 2, 3, with partial results for n = 4. For n = 2, we find 3 inequivalent differential structures admitting 1, 2, and 3 invariant metrics, respectively. For n = 3, we find 6 differential structures admitting 0, 1, 2, 3, 4, 7 invariant metrics, respectively. We give some examples for n = 4 and general n. Surprisingly, not all our geometries for n ≥ 2 have zero quantum Riemann curvature. Quantum gravity is normally seen as a weighted "sum" over all possible metrics but our results are a step towards a deeper approach in which we must also "sum" over differential structures. Over F2 we construct some of our algebras and associated structures by digital gates, opening up the possibility of "digital geometry."

Lambda: A Mathematica package for operator product expansions in vertex algebras

NASA Astrophysics Data System (ADS)

Ekstrand, Joel

2011-02-01

We give an introduction to the Mathematica package Lambda, designed for calculating λ-brackets in both vertex algebras, and in SUSY vertex algebras. This is equivalent to calculating operator product expansions in two-dimensional conformal field theory. The syntax of λ-brackets is reviewed, and some simple examples are shown, both in component notation, and in N=1 superfield notation. Program summaryProgram title: Lambda Catalogue identifier: AEHF_v1_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEHF_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: GNU General Public License No. of lines in distributed program, including test data, etc.: 18 087 No. of bytes in distributed program, including test data, etc.: 131 812 Distribution format: tar.gz Programming language: Mathematica Computer: See specifications for running Mathematica V7 or above. Operating system: See specifications for running Mathematica V7 or above. RAM: Varies greatly depending on calculation to be performed. Classification: 4.2, 5, 11.1. Nature of problem: Calculate operator product expansions (OPEs) of composite fields in 2d conformal field theory. Solution method: Implementation of the algebraic formulation of OPEs given by vertex algebras, and especially by λ-brackets. Running time: Varies greatly depending on calculation requested. The example notebook provided takes about 3 s to run.

Line defect Schur indices, Verlinde algebras and U(1) r fixed points

NASA Astrophysics Data System (ADS)

Neitzke, Andrew; Yan, Fei

2017-11-01

Given an N=2 superconformal field theory, we reconsider the Schur index ℐ L ( q) in the presence of a half line defect L. Recently Cordova-Gaiotto-Shao found that ℐ L ( q) admits an expansion in terms of characters of the chiral algebra A introduced by Beem et al., with simple coefficients υ L, β ( q). We report a puzzling new feature of this expansion: the q → 1 limit of the coefficients υ L, β ( q) is linearly related to the vacuum expectation values 〈 L〉 in U(1) r -invariant vacua of the theory compactified on S 1. This relation can be expressed algebraically as a commutative diagram involving three algebras: the algebra generated by line defects, the algebra of functions on U(1) r -invariant vacua, and a Verlindelike algebra associated to A . Our evidence is experimental, by direct computation in the Argyres-Douglas theories of type ( A 1, A 2), ( A 1, A 4), ( A 1, A 6), ( A 1, D 3) and ( A 1, D 5). In the latter two theories, which have flavor symmetries, the Verlinde-like algebra which appears is a new deformation of algebras previously considered.

Geometric low-energy effective action in a doubled spacetime

NASA Astrophysics Data System (ADS)

Ma, Chen-Te; Pezzella, Franco

2018-05-01

The ten-dimensional supergravity theory is a geometric low-energy effective theory and the equations of motion for its fields can be obtained from string theory by computing β functions. With d compact dimensions, an O (d , d ; Z) geometric structure can be added to it giving the supergravity theory with T-duality manifest. In this paper, this is constructed through the use of a suitable star product whose role is the one to implement the weak constraint on the fields and the gauge parameters in order to have a closed gauge symmetry algebra. The consistency of the action here proposed is based on the orthogonality of the momenta associated with fields in their triple star products in the cubic terms defined for d ≥ 1. This orthogonality holds also for an arbitrary number of star products of fields for d = 1. Finally, we extend our analysis to the double sigma model, non-commutative geometry and open string theory.

Situating the Debate on "Geometrical Algebra" within the Framework of Premodern Algebra.

PubMed

Sialaros, Michalis; Christianidis, Jean

2016-06-01

Argument The aim of this paper is to employ the newly contextualized historiographical category of "premodern algebra" in order to revisit the arguably most controversial topic of the last decades in the field of Greek mathematics, namely the debate on "geometrical algebra." Within this framework, we shift focus from the discrepancy among the views expressed in the debate to some of the historiographical assumptions and methodological approaches that the opposing sides shared. Moreover, by using a series of propositions related to Elem. II.5 as a case study, we discuss Euclid's geometrical proofs, the so-called "semi-algebraic" alternative demonstrations attributed to Heron of Alexandria, as well as the solutions given by Diophantus, al-Sulamī, and al-Khwārizmī to the corresponding numerical problem. This comparative analysis offers a new reading of Heron's practice, highlights the significance of contextualizing "premodern algebra," and indicates that the origins of algebraic reasoning should be sought in the problem-solving practice, rather than in the theorem-proving tradition.

Umbral Calculus and Holonomic Modules in Positive Characteristic

NASA Astrophysics Data System (ADS)

Kochubei, Anatoly N.

2006-03-01

In the framework of analysis over local fields of positive characteristic, we develop algebraic tools for introducing and investigating various polynomial systems. In this survey paper we describe a function field version of umbral calculus developed on the basis of a relation of binomial type satisfied by the Carlitz polynomials. We consider modules over the Weyl-Carlitz ring, a function field counterpart of the Weyl algebra. It is shown that some basic objects of function field arithmetic, like the Carlitz module, Thakur's hypergeometric polynomials, and analogs of binomial coefficients arising in the positive characteristic version of umbral calculus, generate holonomic modules.

Remarks on a one-parameter family of singular matrices

NASA Astrophysics Data System (ADS)

Sharma, Ramesh; Pariso, Chris; Duda, Michelle

2015-01-01

This short article will present to the reader a family of matrices that form an algebra over the reals. This presentation provides both current and former students of modern abstract algebra a better illustration of the concepts of rings, fields, and algebra itself. In addition, this article relates eigenspaces of 3×3 matrices with the arithmetic-geometric mean equality, an attribute that teachers might enjoy utilizing as a teaching tool in their classes.

New phases of D ge 2 current and diffeomorphism algebras in particle physics

DOE Office of Scientific and Technical Information (OSTI.GOV)

Tze, Chia-Hsiung.

We survey some global results and open issues of current algebras and their canonical field theoretical realization in D {ge} 2 dimensional spacetime. We assess the status of the representation theory of their generalized Kac-Moody and diffeomorphism algebras. Particular emphasis is put on higher dimensional analogs of fermi-bose correspondence, complex analyticity and the phase entanglements of anyonic solitons with exotic spin and statistics. 101 refs.

Homological Order in Three and Four dimensions: Wilson Algebra, Entanglement Entropy and Twist Defects

NASA Astrophysics Data System (ADS)

Roy, Abhishek; Chen, Xiao; Teo, Jeffrey

2013-03-01

We investigate homological orders in two, three and four dimensions by studying Zk toric code models on simplicial, cellular or in general differential complexes. The ground state degeneracy is obtained from Wilson loop and surface operators, and the homological intersection form. We compute these for a series of closed 3 and 4 dimensional manifolds and study the projective representations of mapping class groups (modular transformations). Braiding statistics between point and string excitations in (3+1)-dimensions or between dual string excitations in (4+1)-dimensions are topologically determined by the higher dimensional linking number, and can be understood by an effective topological field theory. An algorithm for calculating entanglemnent entropy of any bipartition of closed manifolds is presented, and its topological signature is completely characterized homologically. Extrinsic twist defects (or disclinations) are studied in 2,3 and 4 dimensions and are shown to carry exotic fusion and braiding properties. Simons Fellowship

Initial Orbit Determination Based on Propagation of Admissible Regions with Differential Algebra

DTIC Science & Technology

2017-01-19

Asteroid close encounter characterization using differential algebra: the case of aphophis. Celestial Mechanics and Dynamical Astronomy , 107(4), 2010...Mechanics and Dynamical Astronomy , 112 (3):331–352, 2012. ISSN 09232958. doi: 10.1007/s10569-012-9400-8. Roberto Armellin, Pierluigi Di Lizia, and Renato... Astronomy , 90(1-2):59–87, 2004. ISSN 09232958. doi: 10.1007/s10569-004-6593-5. 50 DISTRIBUTION A. Approved for public release: distribution unlimited

Noncommutative Common Cause Principles in algebraic quantum field theory

DOE Office of Scientific and Technical Information (OSTI.GOV)

Hofer-Szabo, Gabor; Vecsernyes, Peter

2013-04-15

States in algebraic quantum field theory 'typically' establish correlation between spacelike separated events. Reichenbach's Common Cause Principle, generalized to the quantum field theoretical setting, offers an apt tool to causally account for these superluminal correlations. In the paper we motivate first why commutativity between the common cause and the correlating events should be abandoned in the definition of the common cause. Then we show that the Noncommutative Weak Common Cause Principle holds in algebraic quantum field theory with locally finite degrees of freedom. Namely, for any pair of projections A, B supported in spacelike separated regions V{sub A} and V{submore » B}, respectively, there is a local projection C not necessarily commuting with A and B such that C is supported within the union of the backward light cones of V{sub A} and V{sub B} and the set {l_brace}C, C{sup Up-Tack }{r_brace} screens off the correlation between A and B.« less

Lie-Hamilton systems on the plane: Properties, classification and applications

NASA Astrophysics Data System (ADS)

Ballesteros, A.; Blasco, A.; Herranz, F. J.; de Lucas, J.; Sardón, C.

2015-04-01

We study Lie-Hamilton systems on the plane, i.e. systems of first-order differential equations describing the integral curves of a t-dependent vector field taking values in a finite-dimensional real Lie algebra of planar Hamiltonian vector fields with respect to a Poisson structure. We start with the local classification of finite-dimensional real Lie algebras of vector fields on the plane obtained in González-López, Kamran, and Olver (1992) [23] and we interpret their results as a local classification of Lie systems. By determining which of these real Lie algebras consist of Hamiltonian vector fields relative to a Poisson structure, we provide the complete local classification of Lie-Hamilton systems on the plane. We present and study through our results new Lie-Hamilton systems of interest which are used to investigate relevant non-autonomous differential equations, e.g. we get explicit local diffeomorphisms between such systems. We also analyse biomathematical models, the Milne-Pinney equations, second-order Kummer-Schwarz equations, complex Riccati equations and Buchdahl equations.

DOE Office of Scientific and Technical Information (OSTI.GOV)

Schertzer, Daniel, E-mail: Daniel.Schertzer@enpc.fr; Tchiguirinskaia, Ioulia, E-mail: Ioulia.Tchiguirinskaia@enpc.fr

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 upmore » 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.« less

Algebraic classification of Weyl anomalies in arbitrary dimensions.

PubMed

Boulanger, Nicolas

2007-06-29

Conformally invariant systems involving only dimensionless parameters are known to describe particle physics at very high energy. In the presence of an external gravitational field, the conformal symmetry may generalize to the Weyl invariance of classical massless field systems in interaction with gravity. In the quantum theory, the latter symmetry no longer survives: A Weyl anomaly appears. Anomalies are a cornerstone of quantum field theory, and, for the first time, a general, purely algebraic understanding of the universal structure of the Weyl anomalies is obtained, in arbitrary dimensions and independently of any regularization scheme.

Knotted optical vortices in exact solutions to Maxwell's equations

NASA Astrophysics Data System (ADS)

de Klerk, Albertus J. J. M.; van der Veen, Roland I.; Dalhuisen, Jan Willem; Bouwmeester, Dirk

2017-05-01

We construct a family of exact solutions to Maxwell's equations in which the points of zero intensity form knotted lines topologically equivalent to a given but arbitrary algebraic link. These lines of zero intensity, more commonly referred to as optical vortices, and their topology are preserved as time evolves and the fields have finite energy. To derive explicit expressions for these new electromagnetic fields that satisfy the nullness property, we make use of the Bateman variables for the Hopf field as well as complex polynomials in two variables whose zero sets give rise to algebraic links. The class of algebraic links includes not only all torus knots and links thereof, but also more intricate cable knots. While the unknot has been considered before, the solutions presented here show that more general knotted structures can also arise as optical vortices in exact solutions to Maxwell's equations.

Combinatorial Formulas for Characteristic Classes, and Localization of Secondary Topological Invariants.

NASA Astrophysics Data System (ADS)

Smirnov, Mikhail

1995-01-01

The problems solved in this thesis originated from combinatorial formulas for characteristic classes. This thesis deals with Chern-Simons classes, their generalizations and related algebraic and analytic problems. (1) In this thesis, I describe a new class of algebras whose elements contain Chern and generalized Chern -Simons classes. There is a Poisson bracket in these algebras, similar to the bracket in Kontsevich's noncommutative symplectic geometry (Kon). I prove that the Poisson bracket gives rise to a graded Lie algebra containing differential forms representing Chern and Chern-Simons classes. This is a new result. I describe algebraic analogs of the dilogarithm and higher polylogarithms in the algebra corresponding to Chern-Simons classes. (2) I study the properties of this bracket. It is possible to write the exterior differential and other operations in the algebra using this bracket. The bracket of any two Chern classes is zero and the bracket of a Chern class and a Chern-Simons class is d-closed. The construction developed here easily gives explicit formulas for known secondary classes and makes it possible to construct new ones. (3) I develop an algebraic model for the action of the gauge group and describe how elements of algebra corresponding to the secondary characteristic classes change under this action (see theorem 3 page xi). (4) It is possible give new explicit formulas for cocycles on a gauge group of a bundle and for the corresponding cocycles on the Lie algebra of the gauge group. I use formulas for secondary characteristic classes and an algebraic approach developed in chapter 1. I also use the work of Faddeev, Reiman and Semyonov-Tian-Shanskii (FRS) on cocycles as quantum anomalies. (5) I apply the methods of differential geometry of formal power series to construct universal characteristic and secondary characteristic classes. Given a pair of gauge equivalent connections using local formulas I obtain dilogarithmic and trilogarithmic analogs of Chern-Simons classes.

DOE Office of Scientific and Technical Information (OSTI.GOV)

Xin, Qiaoling, E-mail: xinqiaoling0923@163.com; Jiang, Lining, E-mail: jianglining@bit.edu.cn

Let G be a finite group and H a normal subgroup. D(H; G) is the crossed product of C(H) and CG which is only a subalgebra of D(G), the double algebra of G. One can construct a C*-subalgebra F{sub H} of the field algebra F of G-spin models, so that F{sub H} is a D(H; G)-module algebra, whereas F is not. Then the observable algebra A{sub (H,G)} is obtained as the D(H; G)-invariant subalgebra of F{sub H}, and there exists a unique C*-representation of D(H; G) such that D(H; G) and A{sub (H,G)} are commutants with each other.

Geometric model of topological insulators from the Maxwell algebra

NASA Astrophysics Data System (ADS)

Palumbo, Giandomenico

2017-11-01

We propose a novel geometric model of time-reversal-invariant topological insulators in three dimensions in presence of an external electromagnetic field. Their gapped boundary supports relativistic quantum Hall states and is described by a Chern-Simons theory, where the gauge connection takes values in the Maxwell algebra. This represents a non-central extension of the Poincaré algebra and takes into account both the Lorentz and magnetic-translation symmetries of the surface states. In this way, we derive a relativistic version of the Wen-Zee term and we show that the non-minimal coupling between the background geometry and the electromagnetic field in the model is in agreement with the main properties of the relativistic quantum Hall states in the flat space.

On the Solutions of Two-Extended Principal Conformal Toda Theory

NASA Astrophysics Data System (ADS)

Chao, L.; Hou, B. Y.

1994-02-01

The solutions of the two-extended principal conformal Toda theory (2-EPCT theory, also called bosonic superconformal Toda theory) are constructed in two different ways: (1) Leznov-Saveliev algebraic analysis and (2) the associated chiral embedding surface. The first approach gives rise to the general solution in terms of appropriate matrix elements in different fundamental representations of the underlying Lie algebra, whilst the second one leads to a special solution in the form of Wronski determinants and their co-minors, and it gives an explicit geometrical interpretation of the WZNW → 2-EPCT reduction. The key points of both approaches are the chiral vectors derived recently by the authors, which constitute a closed exchange algebra of the theory.

Continuous-spin mixed-symmetry fields in AdS(5)

NASA Astrophysics Data System (ADS)

Metsaev, R. R.

2018-05-01

Free mixed-symmetry continuous-spin fields propagating in AdS(5) space and flat R(4,1) space are studied. In the framework of a light-cone gauge formulation of relativistic dynamics, we build simple actions for such fields. The realization of relativistic symmetries on the space of light-cone gauge mixed-symmetry continuous-spin fields is also found. Interrelations between constant parameters entering the light-cone gauge actions and eigenvalues of the Casimir operators of space-time symmetry algebras are obtained. Using these interrelations and requiring that the field dynamics in AdS(5) be irreducible and classically unitary, we derive restrictions on the constant parameters and eigenvalues of the second-order Casimir operator of the algebra.

Super-Lie n-algebra extensions, higher WZW models and super-p-branes with tensor multiplet fields

NASA Astrophysics Data System (ADS)

Fiorenza, Domenico; Sati, Hisham; Schreiber, Urs

2015-12-01

We formalize higher-dimensional and higher gauge WZW-type sigma-model local prequantum field theory, and discuss its rationalized/perturbative description in (super-)Lie n-algebra homotopy theory (the true home of the "FDA"-language used in the supergravity literature). We show generally how the intersection laws for such higher WZW-type σ-model branes (open brane ending on background brane) are encoded precisely in (super-)L∞-extension theory and how the resulting "extended (super-)space-times" formalize spacetimes containing σ-model brane condensates. As an application we prove in Lie n-algebra homotopy theory that the complete super-p-brane spectrum of superstring/M-theory is realized this way, including the pure σ-model branes (the "old brane scan") but also the branes with tensor multiplet worldvolume fields, notably the D-branes and the M5-brane. For instance the degree-0 piece of the higher symmetry algebra of 11-dimensional (11D) spacetime with an M2-brane condensate turns out to be the "M-theory super-Lie algebra". We also observe that in this formulation there is a simple formal proof of the fact that type IIA spacetime with a D0-brane condensate is the 11D sugra/M-theory spacetime, and of (prequantum) S-duality for type IIB string theory. Finally we give the non-perturbative description of all this by higher WZW-type σ-models on higher super-orbispaces with higher WZW terms in stacky differential cohomology.

Quantum Physics, Fields and Closed Timelike Curves: The D-CTC Condition in Quantum Field Theory

NASA Astrophysics Data System (ADS)

Tolksdorf, Jürgen; Verch, Rainer

2018-01-01

The D-CTC condition has originally been proposed by David Deutsch as a condition on states of a quantum communication network that contains "backward time-steps" in some of its branches. It has been argued that this is an analogue for quantum processes in the presence of closed timelike curves (CTCs). The unusual properties of states of quantum communication networks that fulfill the D-CTC condition have been discussed extensively in recent literature. In this work, the D-CTC condition is investigated in the framework of quantum field theory in the local, operator-algebraic approach due to Haag and Kastler. It is shown that the D-CTC condition cannot be fulfilled in states that are analytic in the energy, or satisfy the Reeh-Schlieder property, for a certain class of processes and initial conditions. On the other hand, if a quantum field theory admits sufficiently many uncorrelated states across acausally related spacetime regions (as implied by the split property), then the D-CTC condition can always be fulfilled approximately to arbitrary precision. As this result pertains to quantum field theory on globally hyperbolic spacetimes where CTCs are absent, one may conclude that interpreting the D-CTC condition as characteristic for quantum processes in the presence of CTCs could be misleading, and should be regarded with caution. Furthermore, a construction of the quantized massless Klein-Gordon field on the Politzer spacetime, often viewed as spacetime analogue for quantum communication networks with backward time-steps, is proposed in this work.

Generalized Cartan Calculus in general dimension

DOE PAGES

Wang, Yi -Nan

2015-07-22

We develop the generalized Cartan Calculus for the groups G = SL(2,R) × R +, SL(5,R) and SO(5,5). They are the underlying algebraic structures of d=9,7,6 exceptional field theory, respectively. These algebraic identities are needed for the "tensor hierarchy" structure in exceptional field theory. The validity of Poincar\\'e lemmas in this new differential geometry is also discussed. Lastly, we explore some possible extension of the generalized Cartan calculus beyond the exceptional series.

Using trees to compute approximate solutions to ordinary differential equations exactly

NASA Technical Reports Server (NTRS)

Grossman, Robert

1991-01-01

Some recent work is reviewed which relates families of trees to symbolic algorithms for the exact computation of series which approximate solutions of ordinary differential equations. It turns out that the vector space whose basis is the set of finite, rooted trees carries a natural multiplication related to the composition of differential operators, making the space of trees an algebra. This algebraic structure can be exploited to yield a variety of algorithms for manipulating vector fields and the series and algebras they generate.

Galilei group with multiple central extension, vorticity, and entropy generation: Exotic fluid in 3 +1 dimensions

NASA Astrophysics Data System (ADS)

Das, Praloy; Ghosh, Subir

2017-12-01

A noncommutative extension of an ideal (Hamiltonian) fluid model in 3 +1 dimensions is proposed. The model enjoys several interesting features: it allows a multiparameter central extension in Galilean boost algebra (which is significant being contrary to the existing belief that a similar feature can appear only in 2 +1 -dimensions); noncommutativity generates vorticity in a canonically irrotational fluid; it induces a nonbarotropic pressure leading to a nonisentropic system. (Barotropic fluids are entropy preserving as the pressure depends only on the matter density.) Our fluid model is termed "exotic" since it has a close resemblance with the extensively studied planar (2 +1 dimensions) exotic models and exotic (noncommutative) field theories.

Equivalent magnetic vector potential model for low-frequency magnetic exposure assessment

NASA Astrophysics Data System (ADS)

Diao, Y. L.; Sun, W. N.; He, Y. Q.; Leung, S. W.; Siu, Y. M.

2017-10-01

In this paper, a novel source model based on a magnetic vector potential for the assessment of induced electric field strength in a human body exposed to the low-frequency (LF) magnetic field of an electrical appliance is presented. The construction of the vector potential model requires only a single-component magnetic field to be measured close to the appliance under test, hence relieving considerable practical measurement effort—the radial basis functions (RBFs) are adopted for the interpolation of discrete measurements; the magnetic vector potential model can then be directly constructed by summing a set of simple algebraic functions of RBF parameters. The vector potentials are then incorporated into numerical calculations as the equivalent source for evaluations of the induced electric field in the human body model. The accuracy and effectiveness of the proposed model are demonstrated by comparing the induced electric field in a human model to that of the full-wave simulation. This study presents a simple and effective approach for modelling the LF magnetic source. The result of this study could simplify the compliance test procedure for assessing an electrical appliance regarding LF magnetic exposure.

Equivalent magnetic vector potential model for low-frequency magnetic exposure assessment.

PubMed

Diao, Y L; Sun, W N; He, Y Q; Leung, S W; Siu, Y M

2017-09-21

In this paper, a novel source model based on a magnetic vector potential for the assessment of induced electric field strength in a human body exposed to the low-frequency (LF) magnetic field of an electrical appliance is presented. The construction of the vector potential model requires only a single-component magnetic field to be measured close to the appliance under test, hence relieving considerable practical measurement effort-the radial basis functions (RBFs) are adopted for the interpolation of discrete measurements; the magnetic vector potential model can then be directly constructed by summing a set of simple algebraic functions of RBF parameters. The vector potentials are then incorporated into numerical calculations as the equivalent source for evaluations of the induced electric field in the human body model. The accuracy and effectiveness of the proposed model are demonstrated by comparing the induced electric field in a human model to that of the full-wave simulation. This study presents a simple and effective approach for modelling the LF magnetic source. The result of this study could simplify the compliance test procedure for assessing an electrical appliance regarding LF magnetic exposure.

Algebraic integrability: a survey.

PubMed

Vanhaecke, Pol

2008-03-28

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

N=2 gauge theories and degenerate fields of Toda theory

DOE Office of Scientific and Technical Information (OSTI.GOV)

Kanno, Shoichi; Matsuo, Yutaka; Shiba, Shotaro

We discuss the correspondence between degenerate fields of the W{sub N} algebra and punctures of Gaiotto's description of the Seiberg-Witten curve of N=2 superconformal gauge theories. Namely, we find that the type of degenerate fields of the W{sub N} algebra, with null states at level one, is classified by Young diagrams with N boxes, and that the singular behavior of the Seiberg-Witten curve near the puncture agrees with that of W{sub N} generators. We also find how to translate mass parameters of the gauge theory to the momenta of the Toda theory.

A Guided Reinvention of Ring, Integral Domain, and Field

ERIC Educational Resources Information Center

Cook, John Paul

2012-01-01

Abstract algebra enjoys a prestigious position in mathematics and the undergraduate mathematics curriculum. A typical abstract algebra course aims to provide students with a glimpse into the elegance of mathematics by exposing them to structures that form its foundation--it arguably approximates the actual practice of mathematics better than any…

Symmetries and stochastic symmetry breaking in multifractal geophysics: analysis and simulation with the help of the Lévy-Clifford algebra of cascade generators..

NASA Astrophysics Data System (ADS)

Schertzer, D. J. M.; Tchiguirinskaia, I.

2016-12-01

Multifractal fields, whose definition is rather independent of their domain dimension, have opened a new approach of geophysics enabling to explore its spatial extension that is of prime importance as underlined by the expression "spatial chaos". However multifractals have been until recently restricted to be scalar valued, i.e. to one-dimensional codomains. This has prevented to deal with the key question of complex component interactions and their non trivial symmetries. We first emphasize that the Lie algebra of stochastic generators of cascade processes enables us to generalize multifractals to arbitrarily large codomains, e.g. flows of vector fields on large dimensional manifolds. In particular, we have recently investigated the neat example of stable Levy generators on Clifford algebra that have a number of seductive properties, e.g. universal statistical and robust algebra properties, both defining the basic symmetries of the corresponding fields (Schertzer and Tchiguirinskaia, 2015). These properties provide a convenient multifractal framework to study both the symmetries of the fields and how they stochastically break the symmetries of the underlying equations due to boundary conditions, large scale rotations and forcings. These developments should help us to answer to challenging questions such as the climatology of (exo-) planets based on first principles (Pierrehumbert, 2013), to fully address the question of the limitations of quasi- geostrophic turbulence (Schertzer et al., 2012) and to explore the peculiar phenomenology of turbulent dynamics of the atmosphere or oceans that is neither two- or three-dimensional. Pierrehumbert, R.T., 2013. Strange news from other stars. Nature Geoscience, 6(2), pp.8183. Schertzer, D. et al., 2012. Quasi-geostrophic turbulence and generalized scale invariance, a theoretical reply. Atmos. Chem. Phys., 12, pp.327336. Schertzer, D. & Tchiguirinskaia, I., 2015. Multifractal vector fields and stochastic Clifford algebra. Chaos: An Interdisciplinary Journal of Nonlinear Science, 25(12), p.123127

On Fock-space representations of quantized enveloping algebras related to noncommutative differential geometry

NASA Astrophysics Data System (ADS)

Jurčo, B.; Schlieker, M.

1995-07-01

In this paper explicitly natural (from the geometrical point of view) Fock-space representations (contragradient Verma modules) of the quantized enveloping algebras are constructed. In order to do so, one starts from the Gauss decomposition of the quantum group and introduces the differential operators on the corresponding q-deformed flag manifold (assumed as a left comodule for the quantum group) by a projection to it of the right action of the quantized enveloping algebra on the quantum group. Finally, the representatives of the elements of the quantized enveloping algebra corresponding to the left-invariant vector fields on the quantum group are expressed as first-order differential operators on the q-deformed flag manifold.

Intrinsic non-commutativity of closed string theory

DOE PAGES

Freidel, Laurent; Leigh, Robert G.; Minic, Djordje

2017-09-14

We show that the proper interpretation of the cocycle operators appearing in the physical vertex operators of compactified strings is that the closed string target is noncommutative. We track down the appearance of this non-commutativity to the Polyakov action of the at closed string in the presence of translational monodromies (i.e., windings). Here, in view of the unexpected nature of this result, we present detailed calculations from a variety of points of view, including a careful understanding of the consequences of mutual locality in the vertex operator algebra, as well as a detailed analysis of the symplectic structure of themore » Polyakov string. Finally, we also underscore why this non-commutativity was not emphasized previously in the existing literature. This non-commutativity can be thought of as a central extension of the zero-mode operator algebra, an effect set by the string length scale $-$ it is present even in trivial backgrounds. Clearly, this result indicates that the α'→0 limit is more subtle than usually assumed.« less

Intrinsic non-commutativity of closed string theory

DOE Office of Scientific and Technical Information (OSTI.GOV)

Freidel, Laurent; Leigh, Robert G.; Minic, Djordje

We show that the proper interpretation of the cocycle operators appearing in the physical vertex operators of compactified strings is that the closed string target is noncommutative. We track down the appearance of this non-commutativity to the Polyakov action of the at closed string in the presence of translational monodromies (i.e., windings). Here, in view of the unexpected nature of this result, we present detailed calculations from a variety of points of view, including a careful understanding of the consequences of mutual locality in the vertex operator algebra, as well as a detailed analysis of the symplectic structure of themore » Polyakov string. Finally, we also underscore why this non-commutativity was not emphasized previously in the existing literature. This non-commutativity can be thought of as a central extension of the zero-mode operator algebra, an effect set by the string length scale $-$ it is present even in trivial backgrounds. Clearly, this result indicates that the α'→0 limit is more subtle than usually assumed.« less

On irregular singularity wave functions and superconformal indices

NASA Astrophysics Data System (ADS)

Buican, Matthew; Nishinaka, Takahiro

2017-09-01

We generalize, in a manifestly Weyl-invariant way, our previous expressions for irregular singularity wave functions in two-dimensional SU(2) q-deformed Yang-Mills theory to SU( N). As an application, we give closed-form expressions for the Schur indices of all ( A N - 1 , A N ( n - 1)-1) Argyres-Douglas (AD) superconformal field theories (SCFTs), thus completing the computation of these quantities for the ( A N , A M ) SCFTs. With minimal effort, our wave functions also give new Schur indices of various infinite sets of "Type IV" AD theories. We explore the discrete symmetries of these indices and also show how highly intricate renormalization group (RG) flows from isolated theories and conformal manifolds in the ultraviolet to isolated theories and (products of) conformal manifolds in the infrared are encoded in these indices. We compare our flows with dimensionally reduced flows via a simple "monopole vev RG" formalism. Finally, since our expressions are given in terms of concise Lie algebra data, we speculate on extensions of our results that might be useful for probing the existence of hypothetical SCFTs based on other Lie algebras. We conclude with a discussion of some open problems.

Higher-Order Containers

NASA Astrophysics Data System (ADS)

Altenkirch, Thorsten; Levy, Paul; Staton, Sam

Containers are a semantic way to talk about strictly positive types. In previous work it was shown that containers are closed under various constructions including products, coproducts, initial algebras and terminal coalgebras. In the present paper we show that, surprisingly, the category of containers is cartesian closed, giving rise to a full cartesian closed subcategory of endofunctors. The result has interesting applications in generic programming and representation of higher order abstract syntax. We also show that the category of containers has finite limits, but it is not locally cartesian closed.

Hamiltonian structure of Dubrovin's equation of associativity in 2-d topological field theory

NASA Astrophysics Data System (ADS)

Galvão, C. A. P.; Nutku, Y.

1996-12-01

A third order Monge-Ampère type equation of associativity that Dubrovin has obtained in 2-d topological field theory is formulated in terms of a variational principle subject to second class constraints. Using Dirac's theory of constraints this degenerate Lagrangian system is cast into Hamiltonian form and the Hamiltonian operator is obtained from the Dirac bracket. There is a new type of Kac-Moody algebra that corresponds to this Hamiltonian operator. In particular, it is not a W-algebra.

Closed, analytic, boson realizations for Sp(4)

NASA Astrophysics Data System (ADS)

Klein, Abraham; Zhang, Qing-Ying

1986-08-01

The problem of determing a boson realization for an arbitrary irrep of the unitary simplectic algebra Sp(2d) [or of the corresponding discrete unitary irreps of the unbounded algebra Sp(2d,R)] has been solved completely in recent papers by Deenen and Quesne [J. Deenen and C. Quesne, J. Math. Phys. 23, 878, 2004 (1982); 25, 1638 (1984); 26, 2705 (1985)] and by Moshinsky and co-workers [O. Castaños, E. Chacón, M. Moshinsky, and C. Quesne, J. Math. Phys. 26, 2107 (1985); M. Moshinsky, ``Boson realization of symplectic algebras,'' to be published]. This solution is not known in closed analytic form except for d=1 and for special classes of irreps for d>1. A different method of obtaining a boson realization that solves the full problem for Sp(4) is described. The method utilizes the chain Sp(2d)⊇SU(2)×SU(2) ×ṡṡṡ×SU(2) (d times), which, for d≥4, does not provide a complete set of quantum numbers. Though a simple solution of the missing label problem can be given, this solution does not help in the construction of a mapping algorithm for general d.

Nonunitary Lagrangians and Unitary Non-Lagrangian Conformal Field Theories.

PubMed

Buican, Matthew; Laczko, Zoltan

2018-02-23

In various dimensions, we can sometimes compute observables of interacting conformal field theories (CFTs) that are connected to free theories via the renormalization group (RG) flow by computing protected quantities in the free theories. On the other hand, in two dimensions, it is often possible to algebraically construct observables of interacting CFTs using free fields without the need to explicitly construct an underlying RG flow. In this Letter, we begin to extend this idea to higher dimensions by showing that one can compute certain observables of an infinite set of unitary strongly interacting four-dimensional N=2 superconformal field theories (SCFTs) by performing simple calculations involving sets of nonunitary free four-dimensional hypermultiplets. These free fields are distant cousins of the Majorana fermion underlying the two-dimensional Ising model and are not obviously connected to our interacting theories via an RG flow. Rather surprisingly, this construction gives us Lagrangians for particular observables in certain subsectors of many "non-Lagrangian" SCFTs by sacrificing unitarity while preserving the full N=2 superconformal algebra. As a by-product, we find relations between characters in unitary and nonunitary affine Kac-Moody algebras. We conclude by commenting on possible generalizations of our construction.

Nonunitary Lagrangians and Unitary Non-Lagrangian Conformal Field Theories

NASA Astrophysics Data System (ADS)

Buican, Matthew; Laczko, Zoltan

2018-02-01

In various dimensions, we can sometimes compute observables of interacting conformal field theories (CFTs) that are connected to free theories via the renormalization group (RG) flow by computing protected quantities in the free theories. On the other hand, in two dimensions, it is often possible to algebraically construct observables of interacting CFTs using free fields without the need to explicitly construct an underlying RG flow. In this Letter, we begin to extend this idea to higher dimensions by showing that one can compute certain observables of an infinite set of unitary strongly interacting four-dimensional N =2 superconformal field theories (SCFTs) by performing simple calculations involving sets of nonunitary free four-dimensional hypermultiplets. These free fields are distant cousins of the Majorana fermion underlying the two-dimensional Ising model and are not obviously connected to our interacting theories via an RG flow. Rather surprisingly, this construction gives us Lagrangians for particular observables in certain subsectors of many "non-Lagrangian" SCFTs by sacrificing unitarity while preserving the full N =2 superconformal algebra. As a by-product, we find relations between characters in unitary and nonunitary affine Kac-Moody algebras. We conclude by commenting on possible generalizations of our construction.

Close Encounters of a Sparse Kind.

ERIC Educational Resources Information Center

Westerberg, Arthur W.

1980-01-01

By providing an example problem in solving sets of nonlinear algebraic equations, the advantages and disadvantages of two methods for its solution, the tearing approach v the Newton-Raphson approach, are elucidated. (CS)

Deformation Theory and Physics Model Building

NASA Astrophysics Data System (ADS)

Sternheimer, Daniel

2006-08-01

The mathematical theory of deformations has proved to be a powerful tool in modeling physical reality. We start with a short historical and philosophical review of the context and concentrate this rapid presentation on a few interrelated directions where deformation theory is essential in bringing a new framework - which has then to be developed using adapted tools, some of which come from the deformation aspect. Minkowskian space-time can be deformed into Anti de Sitter, where massless particles become composite (also dynamically): this opens new perspectives in particle physics, at least at the electroweak level, including prediction of new mesons. Nonlinear group representations and covariant field equations, coming from interactions, can be viewed as some deformation of their linear (free) part: recognizing this fact can provide a good framework for treating problems in this area, in particular global solutions. Last but not least, (algebras associated with) classical mechanics (and field theory) on a Poisson phase space can be deformed to (algebras associated with) quantum mechanics (and quantum field theory). That is now a frontier domain in mathematics and theoretical physics called deformation quantization, with multiple ramifications, avatars and connections in both mathematics and physics. These include representation theory, quantum groups (when considering Hopf algebras instead of associative or Lie algebras), noncommutative geometry and manifolds, algebraic geometry, number theory, and of course what is regrouped under the name of M-theory. We shall here look at these from the unifying point of view of deformation theory and refer to a limited number of papers as a starting point for further study.

A braided monoidal category for free super-bosons

DOE Office of Scientific and Technical Information (OSTI.GOV)

Runkel, Ingo, E-mail: ingo.runkel@uni-hamburg.de

The chiral conformal field theory of free super-bosons is generated by weight one currents whose mode algebra is the affinisation of an abelian Lie super-algebra h with non-degenerate super-symmetric pairing. The mode algebras of a single free boson and of a single pair of symplectic fermions arise for even|odd dimension 1|0 and 0|2 of h, respectively. In this paper, the representations of the untwisted mode algebra of free super-bosons are equipped with a tensor product, a braiding, and an associator. In the symplectic fermion case, i.e., if h is purely odd, the braided monoidal structure is extended to representations ofmore » the Z/2Z-twisted mode algebra. The tensor product is obtained by computing spaces of vertex operators. The braiding and associator are determined by explicit calculations from three- and four-point conformal blocks.« less

A comparison of three algebraic stress closures for combustor flow calculations

NASA Technical Reports Server (NTRS)

Nikjooy, M.; So, R. M. C.; Hwang, B. C.

1985-01-01

A comparison is made of the performance of two locally nonequilibrium and one equilibrium algebraic stress closures in calculating combustor flows. Effects of four different pressure-strain models on these closure models are also analyzed. The results show that the pressure-strain models have a much greater influence on the calculated mean velocity and turbulence field than the algebraic stress closures, and that the best mean strain model for the pressure-strain terms is that proposed by Launder, Reece and Rodi (1975). However, the equilibrium algebraic stress closure with the Rotta return-to-isotropy model (1951) for the pressure-strain terms gives as good a correlation with measurements as when the Launder et al. mean strain model is included in the pressure-strain model. Finally, comparison of the calculations with the standard k-epsilon closure results show that the algebraic stress closures are better suited for simple turbulent flow calculations.

Undergraduate Mathematics Students' Emotional Experiences in Linear Algebra Courses

ERIC Educational Resources Information Center

Martínez-Sierra, Gustavo; García-González, María del Socorro

2016-01-01

Little is known about students' emotions in the field of Mathematics Education that go beyond students' emotions in problem solving. To start filling this gap this qualitative research has the aim to identify emotional experiences of undergraduate mathematics students in Linear Algebra courses. In order to obtain data, retrospective focus group…

Algebraic grid adaptation method using non-uniform rational B-spline surface modeling

NASA Technical Reports Server (NTRS)

Yang, Jiann-Cherng; Soni, B. K.

1992-01-01

An algebraic adaptive grid system based on equidistribution law and utilized by the Non-Uniform Rational B-Spline (NURBS) surface for redistribution is presented. A weight function, utilizing a properly weighted boolean sum of various flow field characteristics is developed. Computational examples are presented to demonstrate the success of this technique.

Principal Component Analysis: Resources for an Essential Application of Linear Algebra

ERIC Educational Resources Information Center

Pankavich, Stephen; Swanson, Rebecca

2015-01-01

Principal Component Analysis (PCA) is a highly useful topic within an introductory Linear Algebra course, especially since it can be used to incorporate a number of applied projects. This method represents an essential application and extension of the Spectral Theorem and is commonly used within a variety of fields, including statistics,…

Student Connections of Linear Algebra Concepts: An Analysis of Concept Maps

ERIC Educational Resources Information Center

Lapp, Douglas A.; Nyman, Melvin A.; Berry, John S.

2010-01-01

This article examines the connections of linear algebra concepts in a first course at the undergraduate level. The theoretical underpinnings of this study are grounded in the constructivist perspective (including social constructivism), Vernaud's theory of conceptual fields and Pirie and Kieren's model for the growth of mathematical understanding.…

Hawking fluxes, fermionic currents, W{sub 1+{infinity}} algebra, and anomalies

DOE Office of Scientific and Technical Information (OSTI.GOV)

Bonora, L.; Cvitan, M.; Theoretical Physics Department, Faculty of Science, University of Zagreb Bijenicka cesta 32, HR-10002 Zagreb

2009-10-15

We complete the analysis carried out in previous papers by studying the Hawking radiation for a Kerr black hole carried to infinity by fermionic currents of any spin. We find agreement with the thermal spectrum of the Hawking radiation for fermionic degrees of freedom. We start by showing that the near-horizon physics for a Kerr black hole is approximated by an effective two-dimensional field theory of fermionic fields. Then, starting from two-dimensional currents of any spin that form a W{sub 1+{infinity}} algebra, we construct an infinite set of covariant currents, each of which carries the corresponding moment of the Hawkingmore » radiation. All together they agree with the thermal spectrum of the latter. We show that the predictive power of this method is based not on the anomalies of the higher-spin currents (which are trivial) but on the underlying W{sub 1+{infinity}} structure. Our results point toward the existence in the near-horizon geometry of a symmetry larger than the Virasoro algebra, which very likely takes the form of a W{sub {infinity}} algebra.« less

A note on powers in finite fields

NASA Astrophysics Data System (ADS)

Aabrandt, Andreas; Lundsgaard Hansen, Vagn

2016-08-01

The study of solutions to polynomial equations over finite fields has a long history in mathematics and is an interesting area of contemporary research. In recent years, the subject has found important applications in the modelling of problems from applied mathematical fields such as signal analysis, system theory, coding theory and cryptology. In this connection, it is of interest to know criteria for the existence of squares and other powers in arbitrary finite fields. Making good use of polynomial division in polynomial rings over finite fields, we have examined a classical criterion of Euler for squares in odd prime fields, giving it a formulation that is apt for generalization to arbitrary finite fields and powers. Our proof uses algebra rather than classical number theory, which makes it convenient when presenting basic methods of applied algebra in the classroom.

On B-type Open-Closed Landau-Ginzburg Theories Defined on Calabi-Yau Stein Manifolds

NASA Astrophysics Data System (ADS)

Babalic, Elena Mirela; Doryn, Dmitry; Lazaroiu, Calin Iuliu; Tavakol, Mehdi

2018-05-01

We consider the bulk algebra and topological D-brane category arising from the differential model of the open-closed B-type topological Landau-Ginzburg theory defined by a pair (X,W), where X is a non-compact Calabi-Yau manifold and W is a complex-valued holomorphic function. When X is a Stein manifold (but not restricted to be a domain of holomorphy), we extract equivalent descriptions of the bulk algebra and of the category of topological D-branes which are constructed using only the analytic space associated to X. In particular, we show that the D-brane category is described by projective factorizations defined over the ring of holomorphic functions of X. We also discuss simplifications of the analytic models which arise when X is holomorphically parallelizable and illustrate these in a few classes of examples.

Three-dimensional polarization algebra for all polarization sensitive optical systems.

PubMed

Li, Yahong; Fu, Yuegang; Liu, Zhiying; Zhou, Jianhong; Bryanston-Cross, P J; Li, Yan; He, Wenjun

2018-05-28

Using three-dimensional (3D) coherency vector (9 × 1), we develop a new 3D polarization algebra to calculate the polarization properties of all polarization sensitive optical systems, especially when the incident optical field is partially polarized or un-polarized. The polarization properties of a high numerical aperture (NA) microscope objective (NA = 1.25 immersed in oil) are analyzed based on the proposed 3D polarization algebra. Correspondingly, the polarization simulation of this high NA optical system is performed by the commercial software VirtualLAB Fusion. By comparing the theoretical calculations with polarization simulations, a perfect matching relation is obtained, which demonstrates that this 3D polarization algebra is valid to quantify the 3D polarization properties for all polarization sensitive optical systems.

Hearing Math: Algebra Supported eText for Students With Visual Impairments.

PubMed

Bouck, Emily C; Weng, Pei-Lin

2014-01-01

Supported eText for students with visual impairments in mathematics has a promising, emerging literature base, although little of the existing research focuses on implementation within a classroom setting. This qualitative study sought to understand the use of supported eText to deliver algebra to students with visual impairments enrolled in algebra mathematics courses. The study also sought to explore supported eText in contrast to students' traditional means of accessing an algebra text. The main results suggest supported eText holds potential in terms of delivering mathematics content; however, more research and more reflection on the field is needed regarding this approach as a sole means of presenting text. Implications for teacher professional development and implementation practices are discussed.

Higher symmetries of the Schrödinger operator in Newton-Cartan geometry

NASA Astrophysics Data System (ADS)

Gundry, James

2017-03-01

We establish several relationships between the non-relativistic conformal symmetries of Newton-Cartan geometry and the Schrödinger equation. In particular we discuss the algebra sch(d) of vector fields conformally-preserving a flat Newton-Cartan spacetime, and we prove that its curved generalisation generates the symmetry group of the covariant Schrödinger equation coupled to a Newtonian potential and generalised Coriolis force. We provide intrinsic Newton-Cartan definitions of Killing tensors and conformal Schrödinger-Killing tensors, and we discuss their respective links to conserved quantities and to the higher symmetries of the Schrödinger equation. Finally we consider the role of conformal symmetries in Newtonian twistor theory, where the infinite-dimensional algebra of holomorphic vector fields on twistor space corresponds to the symmetry algebra cnc(3) on the Newton-Cartan spacetime.

On an algebraic structure of dimensionally reduced magical supergravity theories

NASA Astrophysics Data System (ADS)

Fukuchi, Shin; Mizoguchi, Shun'ya

2018-06-01

We study an algebraic structure of magical supergravities in three dimensions. We show that if the commutation relations among the generators of the quasi-conformal group in the super-Ehlers decomposition are in a particular form, then one can always find a parameterization of the group element in terms of various 3d bosonic fields that reproduces the 3d reduced Lagrangian of the corresponding magical supergravity. This provides a unified treatment of all the magical supergravity theories in finding explicit relations between the 3d dimensionally reduced Lagrangians and particular coset nonlinear sigma models. We also verify that the commutation relations of E 6 (+ 2), the quasi-conformal group for A = C, indeed satisfy this property, allowing the algebraic interpretation of the structure constants and scalar field functions as was done in the F 4 (+ 4) magical supergravity.

On the central charge for the W algebras

NASA Astrophysics Data System (ADS)

Hornfeck, K.

1991-02-01

For the W algebra that is built besides by the stress-energy tensor T by one additional field W we translate the (WWW) Jacobi-identity (that restricts in general the allowed values for the central charge) into a set of conditions on the structure constants CWWΦ and CΦWW with quasi-primary fields Φ. For the case where the additional field has half-integer dimension >3/2 we give a restriction on the possible central charges that includes that in this case c must be less than one. Address after 1 November 1990: Department of Mathematics, King's College, Strand, London WC2R 2LS, UK.

Quantization of spinor fields. III. Fermions on coherent (Bose) domains

NASA Astrophysics Data System (ADS)

Garbaczewski, Piotr

1983-02-01

A formulation of the c-number classics-quanta correspondence rule for spinor systems requires all elements of the quantum field algebra to be expanded into power series with respect to the generators of the canonical commutation relation (CCR) algebra. On the other hand, the asymptotic completeness demand would result in the (Haag) expansions with respect to the canonical anticommutation relation (CAR) generators. We establish the conditions under which the above correspondence rule can be reconciled with the existence of Haag expansions in terms of asymptotic free Fermi fields. Then, the CAR become represented on the state space of the Bose (CCR) system.

Application of symbolic and algebraic manipulation software in solving applied mechanics problems

NASA Technical Reports Server (NTRS)

Tsai, Wen-Lang; Kikuchi, Noboru

1993-01-01

As its name implies, symbolic and algebraic manipulation is an operational tool which not only can retain symbols throughout computations but also can express results in terms of symbols. This report starts with a history of symbolic and algebraic manipulators and a review of the literatures. With the help of selected examples, the capabilities of symbolic and algebraic manipulators are demonstrated. These applications to problems of applied mechanics are then presented. They are the application of automatic formulation to applied mechanics problems, application to a materially nonlinear problem (rigid-plastic ring compression) by finite element method (FEM) and application to plate problems by FEM. The advantages and difficulties, contributions, education, and perspectives of symbolic and algebraic manipulation are discussed. It is well known that there exist some fundamental difficulties in symbolic and algebraic manipulation, such as internal swelling and mathematical limitation. A remedy for these difficulties is proposed, and the three applications mentioned are solved successfully. For example, the closed from solution of stiffness matrix of four-node isoparametrical quadrilateral element for 2-D elasticity problem was not available before. Due to the work presented, the automatic construction of it becomes feasible. In addition, a new advantage of the application of symbolic and algebraic manipulation found is believed to be crucial in improving the efficiency of program execution in the future. This will substantially shorten the response time of a system. It is very significant for certain systems, such as missile and high speed aircraft systems, in which time plays an important role.

Metric 3-Leibniz algebras and M2-branes

NASA Astrophysics Data System (ADS)

Méndez-Escobar, Elena

2010-08-01

This thesis is concerned with superconformal Chern-Simons theories with matter in 3 dimensions. The interest in these theories is two-fold. On the one hand, it is a new family of theories in which to test the AdS/CFT correspondence and on the other, they are important to study one of the main objects of M-theory (M2-branes). All these theories have something in common: they can be written in terms of 3-Leibniz algebras. Here we study the structure theory of such algebras, paying special attention to a subclass of them that gives rise to maximal supersymmetry and that was the first to appear in this context: 3-Lie algebras. In chapter 2, we review the structure theory of metric Lie algebras and their unitary representations. In chapter 3, we study metric 3-Leibniz algebras and show, by specialising a construction originally due to Faulkner, that they are in one to one correspondence with pairs of real metric Lie algebras and unitary representations of them. We also show a third characterisation for six extreme cases of 3-Leibniz algebras as graded Lie (super)algebras. In chapter 4, we study metric 3-Lie algebras in detail. We prove a structural result and also classify those with a maximally isotropic centre, which is the requirement that ensures unitarity of the corresponding conformal field theory. Finally, in chapter 5, we study the universal structure of superpotentials in this class of superconformal Chern-Simons theories with matter in three dimensions. We provide a uniform formulation for all these theories and establish the connection between the amount of supersymmetry preserved and the gauge Lie algebra and the appropriate unitary representation to be used to write down the Lagrangian. The conditions for supersymmetry enhancement are then expressed equivalently in the language of representation theory of Lie algebras or the language of 3-Leibniz algebras.

The structure and Gel'fand patterns inherent in the Heisenberg supergenerator algebra of dual Liouville-space ||kq{K-tilde.}(k1 - kn)[lambda- tilde],scriptn>> tensors and the Cayley algebra of scalar invariants over a (k1 - kn) field

NASA Astrophysics Data System (ADS)

Temme, F. P.

1991-06-01

For many-body spin cluster problems, dual-symmetry recoupled tensors over Liouville space provide suitable bases for a generalized torque formalism using the Sn-adapted density operator in which to discuss NMR and related techniques. The explicit structure of such tensors is considered in the context of the Cayley algebra of scalar invariants over a field, specified by the inner ki rank labels of the Tkq(kl-kn)s. The pertinence of both lexical combinatorial architectures over inner rank sets and SU2 propagative topologies in specifying the structure of dual recoupling tensors is considered in the context of the Sn partitional aspects of spin clusters. The form of Heisenberg superoperator generators whose algebra underlies the Gel'fand pattern algebra of SU(2) and SU(2)×Sn tensor bases over Liouville space is presented together with both the related s-boson algebras and a description of the associated {||2k 0>>} pattern sets of CF29H carrier space under the appropriate symmetry. These concepts are correlated with recent work on SU(2)×Sn induced symmetry hierarchies over Liouville spin space. The pertinence of this theoretical work to an understanding of multiquantum NMR in Liouville space formalisms is stressed in a discussion of the nature of pathways for intracluster J coupling, which also gives a valuable physical insight into the nature of coherence transfer in more general spin-1/2 systems.

Real-Time Algebraic Derivative Estimations Using a Novel Low-Cost Architecture Based on Reconfigurable Logic

PubMed Central

Morales, Rafael; Rincón, Fernando; Gazzano, Julio Dondo; López, Juan Carlos

2014-01-01

Time derivative estimation of signals plays a very important role in several fields, such as signal processing and control engineering, just to name a few of them. For that purpose, a non-asymptotic algebraic procedure for the approximate estimation of the system states is used in this work. The method is based on results from differential algebra and furnishes some general formulae for the time derivatives of a measurable signal in which two algebraic derivative estimators run simultaneously, but in an overlapping fashion. The algebraic derivative algorithm presented in this paper is computed online and in real-time, offering high robustness properties with regard to corrupting noises, versatility and ease of implementation. Besides, in this work, we introduce a novel architecture to accelerate this algebraic derivative estimator using reconfigurable logic. The core of the algorithm is implemented in an FPGA, improving the speed of the system and achieving real-time performance. Finally, this work proposes a low-cost platform for the integration of hardware in the loop in MATLAB. PMID:24859033

Bootstrapping non-commutative gauge theories from L∞ algebras

NASA Astrophysics Data System (ADS)

Blumenhagen, Ralph; Brunner, Ilka; Kupriyanov, Vladislav; Lüst, Dieter

2018-05-01

Non-commutative gauge theories with a non-constant NC-parameter are investigated. As a novel approach, we propose that such theories should admit an underlying L∞ algebra, that governs not only the action of the symmetries but also the dynamics of the theory. Our approach is well motivated from string theory. We recall that such field theories arise in the context of branes in WZW models and briefly comment on its appearance for integrable deformations of AdS5 sigma models. For the SU(2) WZW model, we show that the earlier proposed matrix valued gauge theory on the fuzzy 2-sphere can be bootstrapped via an L∞ algebra. We then apply this approach to the construction of non-commutative Chern-Simons and Yang-Mills theories on flat and curved backgrounds with non-constant NC-structure. More concretely, up to the second order, we demonstrate how derivative and curvature corrections to the equations of motion can be bootstrapped in an algebraic way from the L∞ algebra. The appearance of a non-trivial A∞ algebra is discussed, as well.

Nonlinear, nonbinary cyclic group codes

NASA Technical Reports Server (NTRS)

Solomon, G.

1992-01-01

New cyclic group codes of length 2(exp m) - 1 over (m - j)-bit symbols are introduced. These codes can be systematically encoded and decoded algebraically. The code rates are very close to Reed-Solomon (RS) codes and are much better than Bose-Chaudhuri-Hocquenghem (BCH) codes (a former alternative). The binary (m - j)-tuples are identified with a subgroup of the binary m-tuples which represents the field GF(2 exp m). Encoding is systematic and involves a two-stage procedure consisting of the usual linear feedback register (using the division or check polynomial) and a small table lookup. For low rates, a second shift-register encoding operation may be invoked. Decoding uses the RS error-correcting procedures for the m-tuple codes for m = 4, 5, and 6.

Poisson sigma models, reduction and nonlinear gauge theories

NASA Astrophysics Data System (ADS)

Signori, Daniele

This dissertation comprises two main lines of research. Firstly, we study non-linear gauge theories for principal bundles, where the structure group is replaced by a Lie groupoid. We follow the approach of Moerdijk-Mrcun and establish its relation with the existing physics literature. In particular, we derive a new formula for the gauge transformation which closely resembles and generalizes the classical formulas found in Yang Mills gauge theories. Secondly, we give a field theoretic interpretation of the of the BRST (Becchi-Rouet-Stora-Tyutin) and BFV (Batalin-Fradkin-Vilkovisky) methods for the reduction of coisotropic submanifolds of Poisson manifolds. The generalized Poisson sigma models that we define are related to the quantization deformation problems of coisotropic submanifolds using homotopical algebras.

Superbranes, D = 11 CJS Supergravity and Enlarged Superspace Coordinates/Fields Correspondence

DOE Office of Scientific and Technical Information (OSTI.GOV)

Azcarraga, J.A. de; IFIC - CSIC-UVEG, Facultad de Fisica, 46100-Burjassot, Valencia

2005-04-25

We discuss the role of enlarged superspaces in two seemingly different contexts, the structure of the p-brane actions and that of the Cremmer-Julia-Scherk eleven-dimensional supergravity. Both provide examples of a common principle: the existence of an enlarged superspaces coordinates/fields correspondence by which all the (worldvolume or spacetime) fields of the theory are associated to coordinates of enlarged superspaces. In the context of p-branes, enlarged superspaces may be used to construct manifestly supersymmetry-invariant Wess-Zumino terms and as a way of expressing the Born-Infeld worldvolume fields of D-branes and the worldvolume M5-brane two-form in terms of fields associated to the coordinates ofmore » these enlarged superspaces. This is tantamount to saying that the Born-Infeld fields have a superspace origin, as do the other worldvolume fields, and that they have a composite structure. In D=11 supergravity theory enlarged superspaces arise when its underlying gauge structure is investigated and, as a result, the composite nature of the A3 field is revealed: there is a full one-parametric family of enlarged superspace groups that solve the problem of expressing A3 in terms of spacetime fields associated to their coordinates. The corresponding enlarged supersymmetry algebras turn out to be deformations of an expansion of the osp(1 vertical bar 32) algebra. The unifying mathematical structure underlying all these facts is the cohomology of the supersymmetry algebras involved.« less

A Heisenberg Algebra Bundle of a Vector Field in Three-Space and its Weyl Quantization

NASA Astrophysics Data System (ADS)

Binz, Ernst; Pods, Sonja

2006-01-01

In these notes we associate a natural Heisenberg group bundle Ha with a singularity free smooth vector field X = (id,a) on a submanifold M in a Euclidean three-space. This bundle yields naturally an infinite dimensional Heisenberg group HX∞. A representation of the C*-group algebra of HX∞ is a quantization. It causes a natural Weyl-deformation quantization of X. The influence of the topological structure of M on this quantization is encoded in the Chern class of a canonical complex line bundle inside Ha.

Mastering algebra retrains the visual system to perceive hierarchical structure in equations.

PubMed

Marghetis, Tyler; Landy, David; Goldstone, Robert L

2016-01-01

Formal mathematics is a paragon of abstractness. It thus seems natural to assume that the mathematical expert should rely more on symbolic or conceptual processes, and less on perception and action. We argue instead that mathematical proficiency relies on perceptual systems that have been retrained to implement mathematical skills. Specifically, we investigated whether the visual system-in particular, object-based attention-is retrained so that parsing algebraic expressions and evaluating algebraic validity are accomplished by visual processing. Object-based attention occurs when the visual system organizes the world into discrete objects, which then guide the deployment of attention. One classic signature of object-based attention is better perceptual discrimination within, rather than between, visual objects. The current study reports that object-based attention occurs not only for simple shapes but also for symbolic mathematical elements within algebraic expressions-but only among individuals who have mastered the hierarchical syntax of algebra. Moreover, among these individuals, increased object-based attention within algebraic expressions is associated with a better ability to evaluate algebraic validity. These results suggest that, in mastering the rules of algebra, people retrain their visual system to represent and evaluate abstract mathematical structure. We thus argue that algebraic expertise involves the regimentation and reuse of evolutionarily ancient perceptual processes. Our findings implicate the visual system as central to learning and reasoning in mathematics, leading us to favor educational approaches to mathematics and related STEM fields that encourage students to adapt, not abandon, their use of perception.

Algebra of implicitly defined constraints for gravity as the general form of embedding theory

NASA Astrophysics Data System (ADS)

Paston, S. A.; Semenova, E. N.; Franke, V. A.; Sheykin, A. A.

2017-01-01

We consider the embedding theory, the approach to gravity proposed by Regge and Teitelboim, in which 4D space-time is treated as a surface in high-dimensional flat ambient space. In its general form, which does not contain artificially imposed constraints, this theory can be viewed as an extension of GR. In the present paper we study the canonical description of the embedding theory in this general form. In this case, one of the natural constraints cannot be written explicitly, in contrast to the case where additional Einsteinian constraints are imposed. Nevertheless, it is possible to calculate all Poisson brackets with this constraint. We prove that the algebra of four emerging constraints is closed, i.e., all of them are first-class constraints. The explicit form of this algebra is also obtained.

Lie symmetries of systems of second-order linear ordinary differential equations with constant coefficients.

PubMed

Boyko, Vyacheslav M; Popovych, Roman O; Shapoval, Nataliya M

2013-01-01

Lie symmetries of systems of second-order linear ordinary differential equations with constant coefficients are exhaustively described over both the complex and real fields. The exact lower and upper bounds for the dimensions of the maximal Lie invariance algebras possessed by such systems are obtained using an effective algebraic approach.

A Practical Approach to Inquiry-Based Learning in Linear Algebra

ERIC Educational Resources Information Center

Chang, J.-M.

2011-01-01

Linear algebra has become one of the most useful fields of mathematics since last decade, yet students still have trouble seeing the connection between some of the abstract concepts and real-world applications. In this article, we propose the use of thought-provoking questions in lesson designs to allow two-way communications between instructors…

Lie symmetries of systems of second-order linear ordinary differential equations with constant coefficients

PubMed Central

Boyko, Vyacheslav M.; Popovych, Roman O.; Shapoval, Nataliya M.

2013-01-01

Lie symmetries of systems of second-order linear ordinary differential equations with constant coefficients are exhaustively described over both the complex and real fields. The exact lower and upper bounds for the dimensions of the maximal Lie invariance algebras possessed by such systems are obtained using an effective algebraic approach. PMID:23564972

Moving beyond Solving for "x": Teaching Abstract Algebra in a Liberal Arts Mathematics Course

ERIC Educational Resources Information Center

Cook, John Paul

2015-01-01

This paper details an inquiry-based approach for teaching the basic notions of rings and fields to liberal arts mathematics students. The task sequence seeks to encourage students to identify and comprehend core concepts of introductory abstract algebra by thinking like mathematicians; that is, by investigating an open-ended mathematical context,…

The Algebra Artist

ERIC Educational Resources Information Center

Beigie, Darin

2014-01-01

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

Examples of Complete Solvability of 2D Classical Superintegrable Systems

NASA Astrophysics Data System (ADS)

Chen, Yuxuan; Kalnins, Ernie G.; Li, Qiushi; Miller, Willard, Jr.

2015-11-01

Classical (maximal) superintegrable systems in n dimensions are Hamiltonian systems with 2n-1 independent constants of the motion, globally defined, the maximum number possible. They are very special because they can be solved algebraically. In this paper we show explicitly, mostly through examples of 2nd order superintegrable systems in 2 dimensions, how the trajectories can be determined in detail using rather elementary algebraic, geometric and analytic methods applied to the closed quadratic algebra of symmetries of the system, without resorting to separation of variables techniques or trying to integrate Hamilton's equations. We treat a family of 2nd order degenerate systems: oscillator analogies on Darboux, nonzero constant curvature, and flat spaces, related to one another via contractions, and obeying Kepler's laws. Then we treat two 2nd order nondegenerate systems, an analogy of a caged Coulomb problem on the 2-sphere and its contraction to a Euclidean space caged Coulomb problem. In all cases the symmetry algebra structure provides detailed information about the trajectories, some of which are rather complicated. An interesting example is the occurrence of ''metronome orbits'', trajectories confined to an arc rather than a loop, which are indicated clearly from the structure equations but might be overlooked using more traditional methods. We also treat the Post-Winternitz system, an example of a classical 4th order superintegrable system that cannot be solved using separation of variables. Finally we treat a superintegrable system, related to the addition theorem for elliptic functions, whose constants of the motion are only rational in the momenta. It is a system of special interest because its constants of the motion generate a closed polynomial algebra. This paper contains many new results but we have tried to present most of the materials in a fashion that is easily accessible to nonexperts, in order to provide entrée to superintegrablity theory.

Yangians in Integrable Field Theories, Spin Chains and Gauge-String Dualities

NASA Astrophysics Data System (ADS)

Spill, Fabian

In the following paper, which is based on the author's PhD thesis submitted to Imperial College London, we explore the applicability of Yangian symmetry to various integrable models, in particular, in relation with S-matrices. One of the main themes in this work is that, after a careful study of the mathematics of the symmetry algebras one finds that in an integrable model, one can directly reconstruct S-matrices just from the algebra. It has been known for a long time that S-matrices in integrable models are fixed by symmetry. However, Lie algebra symmetry, the Yang-Baxter equation, crossing and unitarity, which constrain the S-matrix in integrable models, are often taken to be separate, independent properties of the S-matrix. Here, we construct scattering matrices purely from the Yangian, showing that the Yangian is the right algebraic object to unify all required symmetries of many integrable models. In particular, we reconstruct the S-matrix of the principal chiral field, and, up to a CDD factor, of other integrable field theories with 𝔰𝔲(n) symmetry. Furthermore, we study the AdS/CFT correspondence, which is also believed to be integrable in the planar limit. We reconstruct the S-matrices at weak and at strong coupling from the Yangian or its classical limit. We give a pedagogical introduction into the subject, presenting a unified perspective of Yangians and their applications in physics. This paper should hence be accessible to mathematicians who would like to explore the application of algebraic objects to physics as well as to physicists interested in a deeper understanding of the mathematical origin of physical quantities.

Weak Lie symmetry and extended Lie algebra

DOE Office of Scientific and Technical Information (OSTI.GOV)

Goenner, Hubert

2013-04-15

The concept of weak Lie motion (weak Lie symmetry) is introduced. Applications given exhibit a reduction of the usual symmetry, e.g., in the case of the rotation group. In this context, a particular generalization of Lie algebras is found ('extended Lie algebras') which turns out to be an involutive distribution or a simple example for a tangent Lie algebroid. Riemannian and Lorentz metrics can be introduced on such an algebroid through an extended Cartan-Killing form. Transformation groups from non-relativistic mechanics and quantum mechanics lead to such tangent Lie algebroids and to Lorentz geometries constructed on them (1-dimensional gravitational fields).

More on the hidden symmetries of 11D supergravity

NASA Astrophysics Data System (ADS)

Andrianopoli, Laura; D'Auria, Riccardo; Ravera, Lucrezia

2017-09-01

In this paper we clarify the relations occurring among the osp (1 | 32) algebra, the M-algebra and the hidden superalgebra underlying the Free Differential Algebra of D=11 supergravity (to which we will refer as DF-algebra) that was introduced in the literature by D'Auria and Frè in 1981 and is actually a (Lorentz valued) central extension of the M-algebra including a nilpotent spinor generator, Q‧. We focus in particular on the 4-form cohomology in 11D superspace of the supergravity theory, strictly related to the presence in the theory of a 3-form A (3). Once formulated in terms of its hidden superalgebra of 1-forms, we find that A (3) can be decomposed into the sum of two parts having different group-theoretical meaning: One of them allows to reproduce the FDA of the 11D Supergravity due to non-trivial contributions to the 4-form cohomology in superspace, while the second one does not contribute to the 4-form cohomology, being a closed 3-form in the vacuum, defining however a one parameter family of trilinear forms invariant under a symmetry algebra related to osp (1 | 32) by redefining the spin connection and adding a new Maurer-Cartan equation. We further discuss about the crucial role played by the 1-form spinor η (dual to the nilpotent generator Q‧) for the 4-form cohomology of the eleven dimensional theory on superspace.

Asymptotic symmetries in p-form theories

NASA Astrophysics Data System (ADS)

Afshar, Hamid; Esmaeili, Erfan; Sheikh-Jabbari, M. M.

2018-05-01

We consider ( p + 1)-form gauge fields in flat (2 p + 4)-dimensions for which radiation and Coulomb solutions have the same asymptotic fall-off behavior. Imposing appropriate fall-off behavior on fields and adopting a Maxwell-type action, we construct the boundary term which renders the action principle well-defined in the Lorenz gauge. We then compute conserved surface charges and the corresponding asymptotic charge algebra associated with nontrivial gauge transformations. We show that for p ≥ 1, there are three sets of conserved asymptotic charges associated with exact, coexact and zero-mode parts of the corresponding p-form gauge transformations on the asymptotic S 2 p+2. The coexact and zero-mode charges are higher form extensions of the four dimensional electrodynamics ( p = 0), and are commuting. Charges associated with exact gauge transformations have no counterparts in four dimensions and form infinite copies of Heisenberg algebras. We briefly discuss physical implications of these charges and their algebra.

Geometric Model of Topological Insulators from the Maxwell Algebra

NASA Astrophysics Data System (ADS)

Palumbo, Giandomenico

I propose a novel geometric model of time-reversal-invariant topological insulators in three dimensions in presence of an external electromagnetic field. Their gapped boundary supports relativistic quantum Hall states and is described by a Chern-Simons theory, where the gauge connection takes values in the Maxwell algebra. This represents a non-central extension of the Poincare' algebra and takes into account both the Lorentz and magnetic-translation symmetries of the surface states. In this way, I derive a relativistic version of the Wen-Zee term and I show that the non-minimal coupling between the background geometry and the electromagnetic field in the model is in agreement with the main properties of the relativistic quantum Hall states in the flat space. This work is part of the DITP consortium, a program of the Netherlands Organisation for Scientific Research (NWO) that is funded by the Dutch Ministry of Education, Culture and Science (OCW).

Partition functions for heterotic WZW conformal field theories

NASA Astrophysics Data System (ADS)

Gannon, Terry

1993-08-01

Thus far in the search for, and classification of, "physical" modular invariant partition functions ΣN LRχ Lχ R∗ the attention has been focused on the symmetric case where the holomorphic and anti-holomorphic sectors, and hence the characters χLand χR, are associated with the same Kac-Moody algebras ĝL = ĝR and levels κ L = κ R. In this paper we consider the more general possibility where ( ĝL, κ L) may not equal ( ĝR, κ R). We discuss which choices of algebras and levels may correspond to well-defined conformal field theories, we find the "smallest" such heterotic (i.e. asymmetric) partition functions, and we give a method, generalizing the Roberts-Terao-Warner lattice method, for explicitly constructing many other modular invariants. We conclude the paper by proving that this new lattice method will succeed in generating all the heterotic partition functions, for all choices of algebras and levels.

Flux-driven algebraic damping of m=2 diocotron mode

NASA Astrophysics Data System (ADS)

Chim, C. Y.; O'Neil, T. M.

2016-10-01

Recent experiments with pure electron plasmas in a Malmberg-Penning trap have observed the algebraic damping of m = 2 diocotron modes. Due to small field asymmetries a low density halo of electrons is transported radially outward from the plasma core, and the mode damping begins when the halo reaches the resonant radius rres, where f = mfE × B (rres) . The damping rate is proportional to the flux of halo particles through the resonant layer. The damping is related to, but distinct from the exponential spatial Landau damping in a linear wave-particle resonance. This poster uses analytic theory and simulations to explain the new flux-driven algebraic damping of the mode. As electrons are swept around the nonlinear ``cat's eye'' orbits of the resonant wave-particle interaction, they form a quadrupole (m = 2) density distribution, which sets up an electric field that acts back on the plasma core. The field causes an E × B drift motion that symmetrizes the core, i.e. damps the m = 2 mode. Supported by NSF Grant PHY-1414570, and DOE Grants DE-SC0002451.

Local phase space and edge modes for diffeomorphism-invariant theories

NASA Astrophysics Data System (ADS)

Speranza, Antony J.

2018-02-01

We discuss an approach to characterizing local degrees of freedom of a subregion in diffeomorphism-invariant theories using the extended phase space of Donnelly and Freidel [36]. Such a characterization is important for defining local observables and entanglement entropy in gravitational theories. Traditional phase space constructions for subregions are not invariant with respect to diffeomorphisms that act at the boundary. The extended phase space remedies this problem by introducing edge mode fields at the boundary whose transformations under diffeomorphisms render the extended symplectic structure fully gauge invariant. In this work, we present a general construction for the edge mode symplectic structure. We show that the new fields satisfy a surface symmetry algebra generated by the Noether charges associated with the edge mode fields. For surface-preserving symmetries, the algebra is universal for all diffeomorphism-invariant theories, comprised of diffeomorphisms of the boundary, SL(2, ℝ) transformations of the normal plane, and, in some cases, normal shearing transformations. We also show that if boundary conditions are chosen such that surface translations are symmetries, the algebra acquires a central extension.

Explorations in fuzzy physics and non-commutative geometry

NASA Astrophysics Data System (ADS)

Kurkcuoglu, Seckin

Fuzzy spaces arise as discrete approximations to continuum manifolds. They are usually obtained through quantizing coadjoint orbits of compact Lie groups and they can be described in terms of finite-dimensional matrix algebras, which for large matrix sizes approximate the algebra of functions of the limiting continuum manifold. Their ability to exactly preserve the symmetries of their parent manifolds is especially appealing for physical applications. Quantum Field Theories are built over them as finite-dimensional matrix models preserving almost all the symmetries of their respective continuum models. In this dissertation, we first focus our attention to the study of fuzzy supersymmetric spaces. In this regard, we obtain the fuzzy supersphere S2,2F through quantizing the supersphere, and demonstrate that it has exact supersymmetry. We derive a finite series formula for the *-product of functions over S2,2F and analyze the differential geometric information encoded in this formula. Subsequently, we show that quantum field theories on S2,2F are realized as finite-dimensional supermatrix models, and in particular we obtain the non-linear sigma model over the fuzzy supersphere by constructing the fuzzy supersymmetric extensions of a certain class of projectors. We show that this model too, is realized as a finite-dimensional supermatrix model with exact supersymmetry. Next, we show that fuzzy spaces have a generalized Hopf algebra structure. By focusing on the fuzzy sphere, we establish that there is a *-homomorphism from the group algebra SU(2)* of SU(2) to the fuzzy sphere. Using this and the canonical Hopf algebra structure of SU(2)* we show that both the fuzzy sphere and their direct sum are Hopf algebras. Using these results, we discuss processes in which a fuzzy sphere with angular momenta J splits into fuzzy spheres with angular momenta K and L. Finally, we study the formulation of Chern-Simons (CS) theory on an infinite strip of the non-commutative plane. We develop a finite-dimensional matrix model, whose large size limit approximates the CS theory on the infinite strip, and show that there are edge observables in this model obeying a finite-dimensional Lie algebra, that resembles the Kac-Moody algebra.

Relativistic Causality and Quasi-Orthomodular Algebras

NASA Astrophysics Data System (ADS)

Nobili, Renato

2006-05-01

The concept of fractionability or decomposability in parts of a physical system has its mathematical counterpart in the lattice--theoretic concept of orthomodularity. Systems with a finite number of degrees of freedom can be decomposed in different ways, corresponding to different groupings of the degrees of freedom. The orthomodular structure of these simple systems is trivially manifest. The problem then arises as to whether the same property is shared by physical systems with an infinite number of degrees of freedom, in particular by the quantum relativistic ones. The latter case was approached several years ago by Haag and Schroer (1962; Haag, 1992) who started from noting that the causally complete sets of Minkowski spacetime form an orthomodular lattice and posed the question of whether the subalgebras of local observables, with topological supports on such subsets, form themselves a corresponding orthomodular lattice. Were it so, the way would be paved to interpreting spacetime as an intrinsic property of a local quantum field algebra. Surprisingly enough, however, the hoped property does not hold for local algebras of free fields with superselection rules. The possibility seems to be instead open if the local currents that govern the superselection rules are driven by gauge fields. Thus, in the framework of local quantum physics, the request for algebraic orthomodularity seems to imply physical interactions! Despite its charm, however, such a request appears plagued by ambiguities and criticities that make of it an ill--posed problem. The proposers themselves, indeed, concluded that the orthomodular correspondence hypothesis is too strong for having a chance of being practicable. Thus, neither the idea was taken seriously by the proposers nor further investigated by others up to a reasonable degree of clarification. This paper is an attempt to re--formulate and well--pose the problem. It will be shown that the idea is viable provided that the algebra of local observables: (1) is considered all over the whole range of its irreducible representations; (2) is widened with the addition of the elements of a suitable intertwining group of automorphisms; (3) the orthomodular correspondence requirement is modified to an extent sufficient to impart a natural topological structure to the intertwined algebra of observables so obtained. A novel scenario then emerges in which local quantum physics appears to provide a general framework for non--perturbative quantum field dynamics.

Reflective thinking in solving an algebra problem: a case study of field independent-prospective teacher

NASA Astrophysics Data System (ADS)

Agustan, S.; Juniati, Dwi; Yuli Eko Siswono, Tatag

2017-10-01

Nowadays, reflective thinking is one of the important things which become a concern in learning mathematics, especially in solving a mathematical problem. The purpose of this paper is to describe how the student used reflective thinking when solved an algebra problem. The subject of this research is one female student who has field independent cognitive style. This research is a descriptive exploratory study with data analysis using qualitative approach to describe in depth reflective thinking of prospective teacher in solving an algebra problem. Four main categories are used to analyse the reflective thinking in solving an algebra problem: (1) formulation and synthesis of experience, (2) orderliness of experience, (3) evaluating the experience and (4) testing the selected solution based on the experience. The results showed that the subject described the problem by using another word and the subject also found the difficulties in making mathematical modelling. The subject analysed two concepts used in solving problem. For instance, geometry related to point and line while algebra is related to algebra arithmetic operation. The subject stated that solution must have four aspect to get effective solution, specifically the ability to (a) understand the meaning of every words; (b) make mathematical modelling; (c) calculate mathematically; (d) interpret solution obtained logically. To test the internal consistency or error in solution, the subject checked and looked back related procedures and operations used. Moreover, the subject tried to resolve the problem in a different way to compare the answers which had been obtained before. The findings supported the assertion that reflective thinking provides an opportunity for the students in improving their weakness in mathematical problem solving. It can make a grow accuracy and concentration in solving a mathematical problem. Consequently, the students will get the right and logic answer by reflective thinking.

Testing an algebraic model of self-reflexion.

PubMed

Grice, James W; McDaniel, Brenda L; Thompsen, Dana

2005-06-01

Self-reflexion is the conscious process of taking the position of an observer in relation to one's own thoughts, feelings, and experiences. Building on the work of Lefebvre, Lefebvre, and Adams-Webber, we used a formal algebraic model of self-reflexion to derive several predictions regarding the frequencies with which individuals would rate themselves and others positively on bipolar scales anchored by adjective terms. The current results from 108 participants (41 men, 67 women; M age= 20.2 yr.) confirmed two predictions derived from the model. Three other predictions, however, were not supported even though the observed frequencies were close to the predicted values. Although not as promising as results reported by Lefebvre, et al., these mixed findings were interpreted as encouraging support for the validity of Lefebvre's algebraic model of self-reflexion. Differences between the current methods and those from previous investigations were also examined, and methodological implications for further studies were discussed.

Generating a New Higher-Dimensional Coupled Integrable Dispersionless System: Algebraic Structures, Bäcklund Transformation and Hidden Structural Symmetries

NASA Astrophysics Data System (ADS)

Souleymanou, Abbagari; Thomas, B. Bouetou; Timoleon, C. Kofane

2013-08-01

The prolongation structure methodologies of Wahlquist—Estabrook [H.D. Wahlquist and F.B. Estabrook, J. Math. Phys. 16 (1975) 1] for nonlinear differential equations are applied to a more general set of coupled integrable dispersionless system. Based on the obtained prolongation structure, a Lie-Algebra valued connection of a closed ideal of exterior differential forms related to the above system is constructed. A Lie-Algebra representation of some hidden structural symmetries of the previous system, its Bäcklund transformation using the Riccati form of the linear eigenvalue problem and their general corresponding Lax-representation are derived. In the wake of the previous results, we extend the above prolongation scheme to higher-dimensional systems from which a new (2 + 1)-dimensional coupled integrable dispersionless system is unveiled along with its inverse scattering formulation, which applications are straightforward in nonlinear optics where additional propagating dimension deserves some attention.

Perturbative computation in a generalized quantum field theory

NASA Astrophysics Data System (ADS)

Bezerra, V. B.; Curado, E. M.; Rego-Monteiro, M. A.

2002-10-01

We consider a quantum field theory that creates at any point of the space-time particles described by a q-deformed Heisenberg algebra which is interpreted as a phenomenological quantum theory describing the scattering of spin-0 composed particles. We discuss the generalization of Wick's expansion for this case and we compute perturbatively the scattering 1+2-->1'+2' to second order in the coupling constant. The result we find shows that the structure of a composed particle, described here phenomenologically by the deformed algebraic structure, can modify in a simple but nontrivial way the perturbation expansion for the process under consideration.

On squares of representations of compact Lie algebras

DOE Office of Scientific and Technical Information (OSTI.GOV)

Zeier, Robert, E-mail: robert.zeier@ch.tum.de; Zimborás, Zoltán, E-mail: zimboras@gmail.com

We study how tensor products of representations decompose when restricted from a compact Lie algebra to one of its subalgebras. In particular, we are interested in tensor squares which are tensor products of a representation with itself. We show in a classification-free manner that the sum of multiplicities and the sum of squares of multiplicities in the corresponding decomposition of a tensor square into irreducible representations has to strictly grow when restricted from a compact semisimple Lie algebra to a proper subalgebra. For this purpose, relevant details on tensor products of representations are compiled from the literature. Since the summore » of squares of multiplicities is equal to the dimension of the commutant of the tensor-square representation, it can be determined by linear-algebra computations in a scenario where an a priori unknown Lie algebra is given by a set of generators which might not be a linear basis. Hence, our results offer a test to decide if a subalgebra of a compact semisimple Lie algebra is a proper one without calculating the relevant Lie closures, which can be naturally applied in the field of controlled quantum systems.« less

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.

Z n clock models and chains of so(n)2 non-Abelian anyons: symmetries, integrable points and low energy properties

NASA Astrophysics Data System (ADS)

Finch, Peter E.; Flohr, Michael; Frahm, Holger

2018-02-01

We study two families of quantum models which have been used previously to investigate the effect of topological symmetries in one-dimensional correlated matter. Various striking similarities are observed between certain {Z}n quantum clock models, spin chains generalizing the Ising model, and chains of non-Abelian anyons constructed from the so(n)2 fusion category for odd n, both subject to periodic boundary conditions. In spite of the differences between these two types of quantum chains, e.g. their Hilbert spaces being spanned by tensor products of local spin states or fusion paths of anyons, the symmetries of the lattice models are shown to be closely related. Furthermore, under a suitable mapping between the parameters describing the interaction between spins and anyons the respective Hamiltonians share part of their energy spectrum (although their degeneracies may differ). This spin-anyon correspondence can be extended by fine-tuning of the coupling constants leading to exactly solvable models. We show that the algebraic structures underlying the integrability of the clock models and the anyon chain are the same. For n  =  3,5,7 we perform an extensive finite size study—both numerical and based on the exact solution—of these models to map out their ground state phase diagram and to identify the effective field theories describing their low energy behaviour. We observe that the continuum limit at the integrable points can be described by rational conformal field theories with extended symmetry algebras which can be related to the discrete ones of the lattice models.

Gordan—Capelli series in superalgebras

PubMed Central

Brini, Andrea; Palareti, Aldopaolo; Teolis, Antonio G. B.

1988-01-01

We derive two Gordan—Capelli series for the supersymmetric algebra of the tensor product of two [unk]2-graded [unk]-vector spaces U and V, being [unk] a field of characteristic zero. These expansions yield complete decompositions of the supersymmetric algebra regarded as a pl(U)- and a pl(V)- module, where pl(U) and pl(V) are the general linear Lie superalgebras of U and V, respectively. PMID:16593911

A representation of solution of stochastic differential equations

NASA Astrophysics Data System (ADS)

Kim, Yoon Tae; Jeon, Jong Woo

2006-03-01

We prove that the logarithm of the formal power series, obtained from a stochastic differential equation, is an element in the closure of the Lie algebra generated by vector fields being coefficients of equations. By using this result, we obtain a representation of the solution of stochastic differential equations in terms of Lie brackets and iterated Stratonovich integrals in the algebra of formal power series.

Bv and Bfv Formulation of a Gauge Theory of Quadratic Lie Algebras in 2d and a Construction of W3 Topological Gravity

NASA Astrophysics Data System (ADS)

Dayi, Ömer F.

The recently proposed generalized field method for solving the master equation of Batalin and Vilkovisky is applied to a gauge theory of quadratic Lie algebras in two dimensions. The charge corresponding to BRST symmetry derived from this solution in terms of the phase space variables by using the Noether procedure, and the one found due to the BFV-method are compared and found to coincide. W3-algebra, formulated in terms of a continuous variable is exploit in the mentioned gauge theory to construct a W3 topological gravity. Moreover, its gauge fixing is briefly discussed.

Revisiting arithmetic solutions to the W =0 condition

NASA Astrophysics Data System (ADS)

Kanno, Keita; Watari, Taizan

2017-11-01

The gravitino mass is expected not to be much smaller than the Planck scale for a large fraction of vacua in flux compactifications. There is no continuous parameter to tune even by hand, and it seems that the gravitino mass can be small only as a result of accidental cancellation among period integrals weighted by integer-valued flux quanta. DeWolfe et al. [J. High Energy Phys. 02 (2005) 037, 10.1088/1126-6708/2005/02/037] proposed paying close attention to vacua where the Hodge decomposition is possible within a number field, so that the precise cancellation takes place as a result of algebra. We focus on a subclass of those vacua—those with complex multiplication—and explore more on the idea in this article. It turns out, in Type IIB compactifications, that those vacua admit nontrivial supersymmetric flux configurations if and only if the reflex field of the Weil intermediate Jacobian is isomorphic to the quadratic imaginary field generated by the axidilaton vacuum expectation value. We also found that flux statistics are highly enriched on such vacua, as F-term conditions become linearly dependent.

Direct localization of poles of a meromorphic function from measurements on an incomplete boundary

NASA Astrophysics Data System (ADS)

Nara, Takaaki; Ando, Shigeru

2010-01-01

This paper proposes an algebraic method to reconstruct the positions of multiple poles in a meromorphic function field from measurements on an arbitrary simple arc in it. A novel issue is the exactness of the algorithm depending on whether the arc is open or closed, and whether it encloses or does not enclose the poles. We first obtain a differential equation that can equivalently determine the meromorphic function field. From it, we derive linear equations that relate the elementary symmetric polynomials of the pole positions to weighted integrals of the field along the simple arc and end-point terms of the arc when it is an open one. Eliminating the end-point terms based on an appropriate choice of weighting functions and a combination of the linear equations, we obtain a simple system of linear equations for solving the elementary symmetric polynomials. We also show that our algorithm can be applied to a 2D electric impedance tomography problem. The effects of the proximity of the poles, the number of measurements and noise on the localization accuracy are numerically examined.

Quantum processes: A Whiteheadian interpretation of quantum field theory

NASA Astrophysics Data System (ADS)

Bain, Jonathan

Quantum processes: A Whiteheadian interpretation of quantum field theory is an ambitious and thought-provoking exercise in physics and metaphysics, combining an erudite study of the very complex metaphysics of A.N. Whitehead with a well-informed discussion of contemporary issues in the philosophy of algebraic quantum field theory. Hättich's overall goal is to construct an interpretation of quantum field theory. He does this by translating key concepts in Whitehead's metaphysics into the language of algebraic quantum field theory. In brief, this Hättich-Whitehead (H-W, hereafter) interpretation takes "actual occasions" as the fundamental ontological entities of quantum field theory. An actual occasion is the result of two types of processes: a "transition process" in which a set of initial possibly-possessed properties for the occasion (in the form of "eternal objects") is localized to a space-time region; and a "concrescence process" in which a subset of these initial possibly-possessed properties is selected and actualized to produce the occasion. Essential to these processes is the "underlying activity", which conditions the way in which properties are initially selected and subsequently actualized. In short, under the H-W interpretation of quantum field theory, an initial set of possibly-possessed eternal objects is represented by a Boolean sublattice of the lattice of projection operators determined by a von Neumann algebra R (O) associated with a region O of Minkowski space-time, and the underlying activity is represented by a state on R (O) obtained by conditionalizing off of the vacuum state. The details associated with the H-W interpretation involve imposing constraints on these representations motivated by principles found in Whitehead's metaphysics. These details are spelled out in the three sections of the book. The first section is a summary and critique of Whitehead's metaphysics, the second section introduces the formalism of algebraic quantum field theory, and the third section consists of a translation between the first two sections. This review will concentrate on the first and third sections, with an eye on making explicit the essential characteristics of the H-W interpretation.

Finding Coefficients of the Full Array of Motion-Independent N-Body Potentials of Metric Gravity from Gravity's Exterior and Interior Effacement Algebra

NASA Astrophysics Data System (ADS)

Nordtvedt, Kenneth

2018-01-01

In the author's previous publications, a recursive linear algebraic method was introduced for obtaining (without gravitational radiation) the full potential expansions for the gravitational metric field components and the Lagrangian for a general N-body system. Two apparent properties of gravity— Exterior Effacement and Interior Effacement—were defined and fully enforced to obtain the recursive algebra, especially for the motion-independent potential expansions of the general N-body situation. The linear algebraic equations of this method determine the potential coefficients at any order n of the expansions in terms of the lower-order coefficients. Then, enforcing Exterior and Interior Effacement on a selecedt few potential series of the full motion-independent potential expansions, the complete exterior metric field for a single, spherically-symmetric mass source was obtained, producing the Schwarzschild metric field of general relativity. In this fourth paper of this series, the complete spatial metric's motion-independent potentials for N bodies are obtained using enforcement of Interior Effacement and knowledge of the Schwarzschild potentials. From the full spatial metric, the complete set of temporal metric potentials and Lagrangian potentials in the motion-independent case can then be found by transfer equations among the coefficients κ( n, α) → λ( n, ɛ) → ξ( n, α) with κ( n, α), λ( n, ɛ), ξ( n, α) being the numerical coefficients in the spatial metric, the Lagrangian, and the temporal metric potential expansions, respectively.

Scalar field collapse in gauge theory gravity

NASA Astrophysics Data System (ADS)

Harke, Richard Eugene

A brief introduction to gravitational collapse in General Relativity is given. Then critical phenomena in the collapse of a massless scalar field as discovered by Choptuik are described. My own work in this area is described and some results are presented. Gauge Theory Gravity and its mathematical formalism, geometric algebra are introduced. Because geometric algebra is not widely known, a detailed and rigorous introduction to it is provided. The basic principles of Gauge Theory Gravity (GTG) are described and a derivation of the field equations is presented. An appropriate Lagrangian for the scalar field in GTG is introduced and the energy tensor is derived by the usual variational process. The equations of motion for the scalar field are derived for a spherically symmetric space. Finite difference approximations to these equations are constructed and simulations of gravitational collapse are run on a computer. Graphical results are presented. An unexpected phenomenon is found in which the passage of the scalar field leaves a persistent change in the gravitational gauge field.

Stiefel-Whitney classes of curve covers

NASA Astrophysics Data System (ADS)

Selander, Björn

2016-10-01

Let D be a Dedekind scheme with the characteristic of all residue fields not equal to 2. To every tame cover Cto D with only odd ramification we associate a second Stiefel-Whitney class in the second cohomology with mod 2 coefficients of a certain tame orbicurve [D] associated to D. This class is then related to the pull-back of the second Stiefel-Whitney class of the push-forward of the line bundle of half of the ramification divisor. This shows (indirectly) that our Stiefel-Whitney class is the pull-back of a sum of cohomology classes considered by Esnault, Kahn and Viehweg in `Coverings with odd ramification and Stiefel-Whitney classes'. Perhaps more importantly, in the case of a proper and smooth curve over an algebraically closed field, our Stiefel-Whitney class is shown to be the pull-back of an invariant considered by Serre in `Revêtements à ramification impaire et thêta-caractéristiques', and in this case our arguments give a new proof of the main result of that article.

Very special relativity as relativity of dark matter: the Elko connection

NASA Astrophysics Data System (ADS)

Ahluwalia, D. V.; Horvath, S. P.

2010-11-01

In the very special relativity (VSR) proposal by Cohen and Glashow, it was pointed out that invariance under HOM (2) is both necessary and sufficient to explain the null result of the Michelson-Morely experiment. It is the quantum field theoretic demand of locality, or the requirement of P, T, CP, or CT invariance, that makes invariance under the Lorentz group a necessity. Originally it was conjectured that VSR operates at the Planck scale; we propose that the natural arena for VSR is at energies similar to the standard model, but in the dark sector. To this end we provide an ab initio spinor representation invariant under the SIM (2) avatar of VSR and construct a mass dimension one fermionic quantum field of spin one half. This field turns out to be a very close sibling of Elko and it exhibits the same striking property of intrinsic darkness with respect to the standard model fields. In the new construct, the tension between Elko and Lorentz symmetries is fully resolved. We thus entertain the possibility that the symmetries underlying the standard model matter and gauge fields are those of Lorentz, while the event space underlying the dark matter and the dark gauge fields supports the algebraic structure underlying VSR.

Analysis on singular spaces: Lie manifolds and operator algebras

NASA Astrophysics Data System (ADS)

Nistor, Victor

2016-07-01

We discuss and develop some connections between analysis on singular spaces and operator algebras, as presented in my sequence of four lectures at the conference Noncommutative geometry and applications, Frascati, Italy, June 16-21, 2014. Therefore this paper is mostly a survey paper, but the presentation is new, and there are included some new results as well. In particular, Sections 3 and 4 provide a complete short introduction to analysis on noncompact manifolds that is geared towards a class of manifolds-called ;Lie manifolds; -that often appears in practice. Our interest in Lie manifolds is due to the fact that they provide the link between analysis on singular spaces and operator algebras. The groupoids integrating Lie manifolds play an important background role in establishing this link because they provide operator algebras whose structure is often well understood. The initial motivation for the work surveyed here-work that spans over close to two decades-was to develop the index theory of stratified singular spaces. Meanwhile, several other applications have emerged as well, including applications to Partial Differential Equations and Numerical Methods. These will be mentioned only briefly, however, due to the lack of space. Instead, we shall concentrate on the applications to Index theory.

Non-geometric fluxes, quasi-Hopf twist deformations, and nonassociative quantum mechanics

NASA Astrophysics Data System (ADS)

Mylonas, Dionysios; Schupp, Peter; Szabo, Richard J.

2014-12-01

We analyse the symmetries underlying nonassociative deformations of geometry in non-geometric R-flux compactifications which arise via T-duality from closed strings with constant geometric fluxes. Starting from the non-abelian Lie algebra of translations and Bopp shifts in phase space, together with a suitable cochain twist, we construct the quasi-Hopf algebra of symmetries that deforms the algebra of functions and the exterior differential calculus in the phase space description of nonassociative R-space. In this setting, nonassociativity is characterised by the associator 3-cocycle which controls non-coassociativity of the quasi-Hopf algebra. We use abelian 2-cocycle twists to construct maps between the dynamical nonassociative star product and a family of associative star products parametrized by constant momentum surfaces in phase space. We define a suitable integration on these nonassociative spaces and find that the usual cyclicity of associative noncommutative deformations is replaced by weaker notions of 2-cyclicity and 3-cyclicity. Using this star product quantization on phase space together with 3-cyclicity, we formulate a consistent version of nonassociative quantum mechanics, in which we calculate the expectation values of area and volume operators, and find coarse-graining of the string background due to the R-flux.

Assessment of an Explicit Algebraic Reynolds Stress Model

NASA Technical Reports Server (NTRS)

Carlson, Jan-Renee

2005-01-01

This study assesses an explicit algebraic Reynolds stress turbulence model in the in the three-dimensional Reynolds averaged Navier-Stokes (RANS) solver, ISAAC (Integrated Solution Algorithm for Arbitrary Con gurations). Additionally, it compares solutions for two select configurations between ISAAC and the RANS solver PAB3D. This study compares with either direct numerical simulation data, experimental data, or empirical models for several different geometries with compressible, separated, and high Reynolds number flows. In general, the turbulence model matched data or followed experimental trends well, and for the selected configurations, the computational results of ISAAC closely matched those of PAB3D using the same turbulence model.

Generalized Heisenberg Algebras, SUSYQM and Degeneracies: Infinite Well and Morse Potential

NASA Astrophysics Data System (ADS)

Hussin, Véronique; Marquette, Ian

2011-03-01

We consider classical and quantum one and two-dimensional systems with ladder operators that satisfy generalized Heisenberg algebras. In the classical case, this construction is related to the existence of closed trajectories. In particular, we apply these results to the infinite well and Morse potentials. We discuss how the degeneracies of the permutation symmetry of quantum two-dimensional systems can be explained using products of ladder operators. These products satisfy interesting commutation relations. The two-dimensional Morse quantum system is also related to a generalized two-dimensional Morse supersymmetric model. Arithmetical or accidental degeneracies of such system are shown to be associated to additional supersymmetry.

The Existence of the Solution to One Kind of Algebraic Riccati Equation

NASA Astrophysics Data System (ADS)

Liu, Jianming

2018-03-01

The matrix equation ATX + XA + XRX + Q = O is called algebraic Riccati equation, which is very important in the fields of automatic control and other engineering applications. Many researchers have studied the solutions to various algebraic Riccati equations and most of them mainly applied the matrix methods, while few used the functional analysis theories. This paper mainly studies the existence of the solution to the following kind of algebraic Riccati equation from the functional view point: ATX + XA + XRX ‑λX + Q = O Here, X, A, R, Q ∈ n×n , Q is a symmetric matrix, and R is a positive or negative semi-definite matrix, λ is arbitrary constants. This paper uses functional approach such as fixed point theorem and contraction mapping thinking so as to provide two sufficient conditions for the solvability about this kind of Riccati equation and to arrive at some relevant conclusions.

A rigorous approach to investigating common assumptions about disease transmission: Process algebra as an emerging modelling methodology for epidemiology.

PubMed

McCaig, Chris; Begon, Mike; Norman, Rachel; Shankland, Carron

2011-03-01

Changing scale, for example, the ability to move seamlessly from an individual-based model to a population-based model, is an important problem in many fields. In this paper, we introduce process algebra as a novel solution to this problem in the context of models of infectious disease spread. Process algebra allows us to describe a system in terms of the stochastic behaviour of individuals, and is a technique from computer science. We review the use of process algebra in biological systems, and the variety of quantitative and qualitative analysis techniques available. The analysis illustrated here solves the changing scale problem: from the individual behaviour we can rigorously derive equations to describe the mean behaviour of the system at the level of the population. The biological problem investigated is the transmission of infection, and how this relates to individual interactions.

Birth of the GUP and its effect on the entropy of the universe in Lie-N-algebra

NASA Astrophysics Data System (ADS)

Sepehri, Alireza; Pradhan, Anirudh; Pincak, Richard; Rahaman, Farook; Beesham, A.; Ghaffary, Tooraj

In this paper, the origin of the generalized uncertainty principle (GUP) in an M-dimensional theory with Lie-N-algebra is considered. This theory which we name Generalized Lie-N-Algebra (GLNA)-theory can be reduced to M-theory with M = 11 and N = 3. In this theory, at the beginning, two energies with positive and negative signs are created from nothing and produce two types of branes with opposite quantum numbers and different numbers of timing dimensions. Coincidence with the birth of these branes, various derivatives of bosonic fields emerge in the action of the system which produce the r GUP for bosons. These branes interact with each other, compact and various derivatives of spinor fields appear in the action of the system which leads to the creation of the GUP for fermions. The previous predicted entropy of branes in the GUP is corrected as due to the emergence of higher orders of derivatives and different number of timing dimensions.

Correlation functions of warped CFT

NASA Astrophysics Data System (ADS)

Song, Wei; Xu, Jianfei

2018-04-01

Warped conformal field theory (WCFT) is a two dimensional quantum field theory whose local symmetry algebra consists of a Virasoro algebra and a U(1) Kac-Moody algebra. In this paper, we study correlation functions for primary operators in WCFT. Similar to conformal symmetry, warped conformal symmetry is very constraining. The form of the two and three point functions are determined by the global warped conformal symmetry while the four point functions can be determined up to an arbitrary function of the cross ratio. The warped conformal bootstrap equation are constructed by formulating the notion of crossing symmetry. In the large central charge limit, four point functions can be decomposed into global warped conformal blocks, which can be solved exactly. Furthermore, we revisit the scattering problem in warped AdS spacetime (WAdS), and give a prescription on how to match the bulk result to a WCFT retarded Green's function. Our result is consistent with the conjectured holographic dualities between WCFT and WAdS.

Flux-driven algebraic damping of m = 1 diocotron mode

NASA Astrophysics Data System (ADS)

Chim, Chi Yung; O'Neil, Thomas

2015-11-01

Recent experiments with pure electron plasmas in a Malmberg-Penning trap have observed the algebraic damping of m = 1 diocotron modes. Transport due to small field asymmetries produce a low density halo of electrons moving radially outward from the plasma core, and the mode damping begins when the halo reaches the resonant radius rres, where f = mfE × B (rres) . The damping rate is proportional to the flux of halo particles through the resonant layer. The damping is related to, but distinct from spatial Landau damping, in which a linear wave-particle resonance produces exponential damping. This poster explains with analytic theory and simulations the new algebraic damping due to both mobility and diffusive fluxes. As electrons are swept around the ``cat's eye'' orbits of resonant wave-particle interaction, they form a dipole (m = 1) density distribution, and the electric field from this distribution produces an E × B drift of the core back to the axis, i.e. damps the m = 1 mode. Supported by National Science Foundation Grant PHY-1414570.

Thermal stress in high temperature cylindrical fasteners

NASA Technical Reports Server (NTRS)

Blosser, Max L.

1988-01-01

Uninsulated structures fabricated from carbon or silicon-based materials, which are allowed to become hot during flight, are attractive for the design of some components of hypersonic vehicles. They have the potential to reduce weight and increase vehicle efficiency. Because of manufacturing contraints, these structures will consist of parts which must be fastened together. The thermal expansion mismatch between conventional metal fasteners and carbon or silicon-based structural materials may make it difficult to design a structural joint which is tight over the operational temperature range without exceeding allowable stress limits. In this study, algebraic, closed-form solutions for calculating the thermal stresses resulting from radial thermal expansion mismatch around a cylindrical fastener are developed. These solutions permit a designer to quickly evaluate many combinations of materials for the fastener and the structure. Using the algebraic equations developed, material properties and joint geometry were varied to determine their effect on thermal stresses. Finite element analyses were used to verify that the closed-form solutions derived give the correct thermal stress distribution around a cylindrical fastener and to investigate the effect of some of the simplifying assumptions made in developing the closed-form solutions for thermal stresses.

Hypotrochoids in conformal restriction systems and Virasoro descendants

NASA Astrophysics Data System (ADS)

Doyon, Benjamin

2013-09-01

A conformal restriction system is a commutative, associative, unital algebra equipped with a representation of the groupoid of univalent conformal maps on connected open sets of the Riemann sphere, along with a family of linear functionals on subalgebras, satisfying a set of properties including conformal invariance and a type of restriction. This embodies some expected properties of expectation values in conformal loop ensembles CLEκ (at least for 8/3 < κ ≤ 4). In the context of conformal restriction systems, we study certain algebra elements associated with hypotrochoid simple curves (a family of curves including the ellipse). These have the CLE interpretation of being ‘renormalized random variables’ that are nonzero only if there is at least one loop of hypotrochoid shape. Each curve has a center w, a scale ɛ and a rotation angle θ, and we analyze the renormalized random variable as a function of u = ɛeiθ and w. We find that it has an expansion in positive powers of u and \\bar {u}, and that the coefficients of pure u (\\bar {u}) powers are holomorphic in w (\\bar {w}). We identify these coefficients (the ‘hypotrochoid fields’) with certain Virasoro descendants of the identity field in conformal field theory, thereby showing that they form part of a vertex operator algebraic structure. This largely generalizes works by the author (in CLE), and the author with his collaborators Riva and Cardy (in SLE8/3 and other restriction measures), where the case of the ellipse, at the order u2, led to the stress-energy tensor of CFT. The derivation uses in an essential way the Virasoro vertex operator algebra structure of conformal derivatives established recently by the author. The results suggest in particular the exact evaluation of CLE expectations of products of hypotrochoid fields as well as nontrivial relations amongst them through the vertex operator algebra, and further shed light onto the relationship between CLE and CFT.

Integrable systems with BMS3 Poisson structure and the dynamics of locally flat spacetimes

NASA Astrophysics Data System (ADS)

Fuentealba, Oscar; Matulich, Javier; Pérez, Alfredo; Pino, Miguel; Rodríguez, Pablo; Tempo, David; Troncoso, Ricardo

2018-01-01

We construct a hierarchy of integrable systems whose Poisson structure corresponds to the BMS3 algebra, and then discuss its description in terms of the Riemannian geometry of locally flat spacetimes in three dimensions. The analysis is performed in terms of two-dimensional gauge fields for isl(2,R) , being isomorphic to the Poincaré algebra in 3D. Although the algebra is not semisimple, the formulation can still be carried out à la Drinfeld-Sokolov because it admits a nondegenerate invariant bilinear metric. The hierarchy turns out to be bi-Hamiltonian, labeled by a nonnegative integer k, and defined through a suitable generalization of the Gelfand-Dikii polynomials. The symmetries of the hierarchy are explicitly found. For k ≥ 1, the corresponding conserved charges span an infinite-dimensional Abelian algebra without central extensions, so that they are in involution; while in the case of k = 0, they generate the BMS3 algebra. In the special case of k = 1, by virtue of a suitable field redefinition and time scaling, the field equations are shown to be equivalent to the ones of a specific type of the Hirota-Satsuma coupled KdV systems. For k ≥ 1, the hierarchy also includes the so-called perturbed KdV equations as a particular case. A wide class of analytic solutions is also explicitly constructed for a generic value of k. Remarkably, the dynamics can be fully geometrized so as to describe the evolution of spacelike surfaces embedded in locally flat spacetimes. Indeed, General Relativity in 3D can be endowed with a suitable set of boundary conditions, so that the Einstein equations precisely reduce to the ones of the hierarchy aforementioned. The symmetries of the integrable systems then arise as diffeomorphisms that preserve the asymptotic form of the spacetime metric, and therefore, they become Noetherian. The infinite set of conserved charges is then recovered from the corresponding surface integrals in the canonical approach.

Short Round Sub-Linear Zero-Knowledge Argument for Linear Algebraic Relations

NASA Astrophysics Data System (ADS)

Seo, Jae Hong

Zero-knowledge arguments allows one party to prove that a statement is true, without leaking any other information than the truth of the statement. In many applications such as verifiable shuffle (as a practical application) and circuit satisfiability (as a theoretical application), zero-knowledge arguments for mathematical statements related to linear algebra are essentially used. Groth proposed (at CRYPTO 2009) an elegant methodology for zero-knowledge arguments for linear algebraic relations over finite fields. He obtained zero-knowledge arguments of the sub-linear size for linear algebra using reductions from linear algebraic relations to equations of the form z = x *' y, where x, y ∈ Fnp are committed vectors, z ∈ Fp is a committed element, and *' : Fnp × Fnp → Fp is a bilinear map. These reductions impose additional rounds on zero-knowledge arguments of the sub-linear size. The round complexity of interactive zero-knowledge arguments is an important measure along with communication and computational complexities. We focus on minimizing the round complexity of sub-linear zero-knowledge arguments for linear algebra. To reduce round complexity, we propose a general transformation from a t-round zero-knowledge argument, satisfying mild conditions, to a (t - 2)-round zero-knowledge argument; this transformation is of independent interest.

On the correspondence between boundary and bulk lattice models and (logarithmic) conformal field theories

NASA Astrophysics Data System (ADS)

Belletête, J.; Gainutdinov, A. M.; Jacobsen, J. L.; Saleur, H.; Vasseur, R.

2017-12-01

The relationship between bulk and boundary properties is one of the founding features of (rational) conformal field theory (CFT). Our goal in this paper is to explore the possibility of having an equivalent relationship in the context of lattice models. We focus on models based on the Temperley-Lieb algebra, and use the concept of ‘braid translation’, which is a natural way, in physical terms, to ‘close’ an open spin chain by adding an interaction between the first and last spins using braiding to ‘bring’ them next to each other. The interaction thus obtained is in general non-local, but has the key feature that it is expressed solely in terms of the algebra for the open spin chain—the ‘ordinary’ Temperley-Lieb algebra and its blob algebra generalization. This is in contrast with the usual periodic spin chains which involve only local interactions, and are described by the periodic Temperley-Lieb algebra. We show that for the restricted solid-on-solid models, which are known to be described by minimal unitary CFTs (with central charge c<1 ) in the continuum limit, the braid translation in fact does provide the ordinary periodic model starting from the open model with fixed (identical) boundary conditions on the two sides of the strip. This statement has a precise mathematical formulation, which is a pull-back map between irreducible modules of, respectively, the blob algebra and the affine Temperley-Lieb algebra. We then turn to the same kind of analysis for two models whose continuum limits are logarithmic CFTs (LCFTs)—the alternating gl(1\\vert 1) and sl(2\\vert 1) spin chains. We find that the result for minimal models does not hold any longer: braid translation of the relevant (in that case, indecomposable but not irreducible) modules of the Temperley-Lieb algebra does not give rise to the modules known to be present in the periodic chains. In the gl(1\\vert 1) case, the content in terms of the irreducibles is the same, as well as the spectrum, but the detailed structure (like logarithmic coupling) is profoundly different. This carries over to the continuum limit. The situation is similar for the sl(2\\vert 1) case. The problem of relating bulk and boundary lattice models for LCFTs thus remains open.

Multifunctional, three-dimensional tomography for analysis of eletrectrohydrodynamic jetting

NASA Astrophysics Data System (ADS)

Nguyen, Xuan Hung; Gim, Yeonghyeon; Ko, Han Seo

2015-05-01

A three-dimensional optical tomography technique was developed to reconstruct three-dimensional objects using a set of two-dimensional shadowgraphic images and normal gray images. From three high-speed cameras, which were positioned at an offset angle of 45° between each other, number, size, and location of electrohydrodynamic jets with respect to the nozzle position were analyzed using shadowgraphic tomography employing multiplicative algebraic reconstruction technique (MART). Additionally, a flow field inside a cone-shaped liquid (Taylor cone) induced under an electric field was observed using a simultaneous multiplicative algebraic reconstruction technique (SMART), a tomographic method for reconstructing light intensities of particles, combined with three-dimensional cross-correlation. Various velocity fields of circulating flows inside the cone-shaped liquid caused by various physico-chemical properties of liquid were also investigated.

Electrokinetics Models for Micro and Nano Fluidic Impedance Sensors

DTIC Science & Technology

2010-11-01

primitive Differential-Algebraic Equations (DAEs), used to process and interpret the experimentally measured electrical impedance data (Sun and Morgan...field, and species respectively. A second-order scheme was used to calculate the ionic species distribution. The linearized algebraic equations were...is governed by the Poisson equation 2 0 0 r i i i F z cε ε φ∇ + =∑ where ε0 and εr are, respectively, the electrical permittivity in the vacuum

Linearized gravity in terms of differential forms

NASA Astrophysics Data System (ADS)

Baykal, Ahmet; Dereli, Tekin

2017-01-01

A technique to linearize gravitational field equations is developed in which the perturbation metric coefficients are treated as second rank, symmetric, 1-form fields belonging to the Minkowski background spacetime by using the exterior algebra of differential forms.

The octapolic ellipsoidal term in magnetoencephalography

NASA Astrophysics Data System (ADS)

Dassios, George; Hadjiloizi, Demetra; Kariotou, Fotini

2009-01-01

The forward problem of magnetoencephalography (MEG) in ellipsoidal geometry has been studied by Dassios and Kariotou ["Magnetoencephalography in ellipsoidal geometry," J. Math. Phys. 44, 220 (2003)] using the theory of ellipsoidal harmonics. In fact, the analytic solution of the quadrupolic term for the magnetic induction field has been calculated in the case of a dipolar neuronal current. Nevertheless, since the quadrupolic term is only the leading nonvanishing term in the multipole expansion of the magnetic field, it contains not enough information for the construction of an effective algorithm to solve the inverse MEG problem, i.e., to recover the position and the orientation of a dipole from measurements of the magnetic field outside the head. For this task, the next multipole of the magnetic field is also needed. The present work provides exactly this octapolic contribution of the dipolar current to the expansion of the magnetic induction field. The octapolic term is expressed in terms of the ellipsoidal harmonics of the third degree, and therefore it provides the highest order terms that can be expressed in closed form using long but reasonable analytic and algebraic manipulations. In principle, the knowledge of the quadrupolic and the octapolic terms is enough to solve the inverse problem of identifying a dipole inside an ellipsoid. Nevertheless, a simple inversion algorithm for this problem is not yet known.

Adult Reading, Rated G.

ERIC Educational Resources Information Center

Lewis, Anne C.

2003-01-01

Suggests books for summer reading for Department of Education officials involved in closing all 16 ERIC Clearinghouses or touting the superiority of a Christian education, standardized test-makers, planners of the Gates Foundation, and the creator of the Algebra Project. For example, suggests that standardized test-makers read William Bennett's…

Taiwanese Arithmetic and Algebra

ERIC Educational Resources Information Center

Lo, Jane-Jane; Tsai, Feng-Chiu

2011-01-01

Taiwanese students consistently rank near the top on international exams on mathematics and science. In 2007, Taiwan recorded the highest TIMSS math score for eighth grade. The central education agency in Taiwan publishes detailed mathematics curriculum guidelines, which textbooks and national exams follow closely. In May each year, all ninth…

Measurement theory in local quantum physics

DOE Office of Scientific and Technical Information (OSTI.GOV)

Okamura, Kazuya, E-mail: okamura@math.cm.is.nagoya-u.ac.jp; Ozawa, Masanao, E-mail: ozawa@is.nagoya-u.ac.jp

In this paper, we aim to establish foundations of measurement theory in local quantum physics. For this purpose, we discuss a representation theory of completely positive (CP) instruments on arbitrary von Neumann algebras. We introduce a condition called the normal extension property (NEP) and establish a one-to-one correspondence between CP instruments with the NEP and statistical equivalence classes of measuring processes. We show that every CP instrument on an atomic von Neumann algebra has the NEP, extending the well-known result for type I factors. Moreover, we show that every CP instrument on an injective von Neumann algebra is approximated bymore » CP instruments with the NEP. The concept of posterior states is also discussed to show that the NEP is equivalent to the existence of a strongly measurable family of posterior states for every normal state. Two examples of CP instruments without the NEP are obtained from this result. It is thus concluded that in local quantum physics not every CP instrument represents a measuring process, but in most of physically relevant cases every CP instrument can be realized by a measuring process within arbitrary error limits, as every approximately finite dimensional von Neumann algebra on a separable Hilbert space is injective. To conclude the paper, the concept of local measurement in algebraic quantum field theory is examined in our framework. In the setting of the Doplicher-Haag-Roberts and Doplicher-Roberts theory describing local excitations, we show that an instrument on a local algebra can be extended to a local instrument on the global algebra if and only if it is a CP instrument with the NEP, provided that the split property holds for the net of local algebras.« less

Conjunct rotation: Codman's paradox revisited.

PubMed

Wolf, Sebastian I; Fradet, Laetitia; Rettig, Oliver

2009-05-01

This contribution mathematically formalizes Codman's idea of conjunct rotation, a term he used in 1934 to describe a paradoxical phenomenon arising from a closed-loop arm movement. Real (axial) rotation is distinguished from conjunct rotation. For characterizing the latter, the idea of reference vector fields is developed to define the neutral axial position of the humerus for any given orientation of its long axis. This concept largely avoids typical coordinate singularities arising from decomposition of 3D joint motion and therefore can be used for postural (axial) assessment of the shoulder joint both clinically and in sports science in almost the complete accessible range of motion. The concept, even though algebraic rather complex, might help to get an easier and more intuitive understanding of axial rotation of the shoulder in complex movements present in daily life and in sports.

Algebraic Trigonometry

ERIC Educational Resources Information Center

Vaninsky, Alexander

2011-01-01

This article introduces a trigonometric field (TF) that extends the field of real numbers by adding two new elements: sin and cos--satisfying an axiom sin[superscript 2] + cos[superscript 2] = 1. It is shown that by assigning meaningful names to particular elements of the field, all known trigonometric identities may be introduced and proved. Two…

Exact analysis of the spectral properties of the anisotropic two-bosons Rabi model

NASA Astrophysics Data System (ADS)

Cui, Shuai; Cao, Jun-Peng; Fan, Heng; Amico, Luigi

2017-05-01

We introduce the anisotropic two-photon Rabi model in which the rotating and counter rotating terms enters the Hamiltonian with two different coupling constants. Eigenvalues and eigenvectors are studied with exact means. We employ a variation of the Braak method based on Bogolubov rotation of the underlying su(1, 1) Lie algebra. Accordingly, the spectrum is provided by the analytical properties of a suitable meromorphic function. Our formalism applies to the two-modes Rabi model as well, sharing the same algebraic structure of the two-photon model. Through the analysis of the spectrum, we discover that the model displays close analogies to many-body systems undergoing quantum phase transitions.

Symmetry operators of Killing spinors and superalgebras in AdS5

NASA Astrophysics Data System (ADS)

Ertem, Ümit

2016-04-01

We construct the first-order symmetry operators of Killing spinor equation in terms of odd Killing-Yano forms. By modifying the Schouten-Nijenhuis bracket of Killing-Yano forms, we show that the symmetry operators of Killing spinors close into an algebra in AdS5 spacetime. Since the symmetry operator algebra of Killing spinors corresponds to a Jacobi identity in extended Killing superalgebras, we investigate the possible extensions of Killing superalgebras to include higher-degree Killing-Yano forms. We found that there is a superalgebra extension but no Lie superalgebra extension of the Killing superalgebra constructed out of Killing spinors and odd Killing-Yano forms in AdS5 background.

FAST TRACK COMMUNICATION: \\ {P}\\ {T}-symmetry, Cartan decompositions, Lie triple systems and Krein space-related Clifford algebras

NASA Astrophysics Data System (ADS)

Günther, Uwe; Kuzhel, Sergii

2010-10-01

Gauged \\ {P}\\ {T} quantum mechanics (PTQM) and corresponding Krein space setups are studied. For models with constant non-Abelian gauge potentials and extended parity inversions compact and noncompact Lie group components are analyzed via Cartan decompositions. A Lie-triple structure is found and an interpretation as \\ {P}\\ {T}-symmetrically generalized Jaynes-Cummings model is possible with close relation to recently studied cavity QED setups with transmon states in multilevel artificial atoms. For models with Abelian gauge potentials a hidden Clifford algebra structure is found and used to obtain the fundamental symmetry of Krein space-related J-self-adjoint extensions for PTQM setups with ultra-localized potentials.

Almost commuting self-adjoint matrices: The real and self-dual cases

NASA Astrophysics Data System (ADS)

Loring, Terry A.; Sørensen, Adam P. W.

2016-08-01

We show that a pair of almost commuting self-adjoint, symmetric matrices is close to a pair of commuting self-adjoint, symmetric matrices (in a uniform way). Moreover, we prove that the same holds with self-dual in place of symmetric and also for paths of self-adjoint matrices. Since a symmetric, self-adjoint matrix is real, we get a real version of Huaxin Lin’s famous theorem on almost commuting matrices. Similarly, the self-dual case gives a version for matrices over the quaternions. To prove these results, we develop a theory of semiprojectivity for real C*-algebras and also examine various definitions of low-rank for real C*-algebras.

Infinite-Dimensional Symmetry Algebras as a Help Toward Solutions of the Self-Dual Field Equations with One Killing Vector

NASA Astrophysics Data System (ADS)

Finley, Daniel; McIver, John K.

2002-12-01

The sDiff(2) Toda equation determines all self-dual, vacuum solutions of the Einstein field equations with one rotational Killing vector. Some history of the searches for non-trivial solutions is given, including those that begin with the limit as n → ∞ of the An Toda lattice equations. That approach is applied here to the known prolongation structure for the Toda lattice, hoping to use Bäcklund transformations to generate new solutions. Although this attempt has not yet succeeded, new faithful (tangent-vector) realizations of A∞ are described, and a direct approach via the continuum Lie algebras of Saveliev and Leznov is given.

Geometric algebra description of polarization mode dispersion, polarization-dependent loss, and Stokes tensor transformations.

PubMed

Soliman, George; Yevick, David; Jessop, Paul

2014-09-01

This paper demonstrates that numerous calculations involving polarization transformations can be condensed by employing suitable geometric algebra formalism. For example, to describe polarization mode dispersion and polarization-dependent loss, both the material birefringence and differential loss enter as bivectors and can be combined into a single symmetric quantity. Their frequency and distance evolution, as well as that of the Stokes vector through an optical system, can then each be expressed as a single compact expression, in contrast to the corresponding Mueller matrix formulations. The intrinsic advantage of the geometric algebra framework is further demonstrated by presenting a simplified derivation of generalized Stokes parameters that include the electric field phase. This procedure simultaneously establishes the tensor transformation properties of these parameters.

Asymptotically spacelike warped anti-de Sitter spacetimes in generalized minimal massive gravity

NASA Astrophysics Data System (ADS)

Setare, M. R.; Adami, H.

2017-06-01

In this paper we show that warped AdS3 black hole spacetime is a solution of the generalized minimal massive gravity (GMMG) and introduce suitable boundary conditions for asymptotically warped AdS3 spacetimes. Then we find the Killing vector fields such that transformations generated by them preserve the considered boundary conditions. We calculate the conserved charges which correspond to the obtained Killing vector fields and show that the algebra of the asymptotic conserved charges is given as the semi direct product of the Virasoro algebra with U(1) current algebra. We use a particular Sugawara construction to reconstruct the conformal algebra. Thus, we are allowed to use the Cardy formula to calculate the entropy of the warped black hole. We demonstrate that the gravitational entropy of the warped black hole exactly coincides with what we obtain via Cardy’s formula. As we expect, the warped Cardy formula also gives us exactly the same result as we obtain from the usual Cardy’s formula. We calculate mass and angular momentum of the warped black hole and then check that obtained mass, angular momentum and entropy to satisfy the first law of the black hole mechanics. According to the results of this paper we believe that the dual theory of the warped AdS3 black hole solution of GMMG is a warped CFT.

Schwinger-Keldysh formalism. Part II: thermal equivariant cohomology

NASA Astrophysics Data System (ADS)

Haehl, Felix M.; Loganayagam, R.; Rangamani, Mukund

2017-06-01

Causally ordered correlation functions of local operators in near-thermal quantum systems computed using the Schwinger-Keldysh formalism obey a set of Ward identities. These can be understood rather simply as the consequence of a topological (BRST) algebra, called the universal Schwinger-Keldysh superalgebra, as explained in our compan-ion paper [1]. In the present paper we provide a mathematical discussion of this topological algebra. In particular, we argue that the structures can be understood in the language of extended equivariant cohomology. To keep the discussion self-contained, we provide a ba-sic review of the algebraic construction of equivariant cohomology and explain how it can be understood in familiar terms as a superspace gauge algebra. We demonstrate how the Schwinger-Keldysh construction can be succinctly encoded in terms a thermal equivariant cohomology algebra which naturally acts on the operator (super)-algebra of the quantum system. The main rationale behind this exploration is to extract symmetry statements which are robust under renormalization group flow and can hence be used to understand low-energy effective field theory of near-thermal physics. To illustrate the general prin-ciples, we focus on Langevin dynamics of a Brownian particle, rephrasing some known results in terms of thermal equivariant cohomology. As described elsewhere, the general framework enables construction of effective actions for dissipative hydrodynamics and could potentially illumine our understanding of black holes.

sl(1|2) Super-Toda Fields

NASA Astrophysics Data System (ADS)

Yang, Zhan-Ying; Xue, Pan-Pan; Zhao, Liu; Shi, Kang-Jie

2008-11-01

Explicit exact solution of supersymmetric Toda fields associated with the Lie superalgebra sl(2|1) is constructed. The approach used is a super extension of Leznov Saveliev algebraic analysis, which is based on a pair of chiral and antichiral Drienfeld Sokolov systems. Though such approach is well understood for Toda field theories associated with ordinary Lie algebras, its super analogue was only successful in the super Liouville case with the underlying Lie superalgebra osp(1|2). The problem lies in that a key step in the construction makes use of the tensor product decomposition of the highest weight representations of the underlying Lie superalgebra, which is not clear until recently. So our construction made in this paper presents a first explicit example of Leznov Saveliev analysis for super Toda systems associated with underlying Lie superalgebras of the rank higher than 1.

Lagrangian Formulation of a Magnetostatic Field in the Presence of a Minimal Length Scale Based on the Kempf Algebra

NASA Astrophysics Data System (ADS)

Moayedi, S. K.; Setare, M. R.; Khosropour, B.

2013-11-01

In the 1990s, Kempf and his collaborators Mangano and Mann introduced a D-dimensional (β, β‧)-two-parameter deformed Heisenberg algebra which leads to an isotropic minimal length (\\triangle Xi)\\min = \\hbar √ {Dβ +β '}, \\forall i\\in \\{1, 2, ..., D\\}. In this work, the Lagrangian formulation of a magnetostatic field in three spatial dimensions (D = 3) described by Kempf algebra is presented in the special case of β‧ = 2β up to the first-order over β. We show that at the classical level there is a similarity between magnetostatics in the presence of a minimal length scale (modified magnetostatics) and the magnetostatic sector of the Abelian Lee-Wick model in three spatial dimensions. The integral form of Ampere's law and the energy density of a magnetostatic field in the modified magnetostatics are obtained. Also, the Biot-Savart law in the modified magnetostatics is found. By studying the effect of minimal length corrections to the gyromagnetic moment of the muon, we conclude that the upper bound on the isotropic minimal length scale in three spatial dimensions is 4.42×10-19 m. The relationship between magnetostatics with a minimal length and the Gaete-Spallucci nonlocal magnetostatics [J. Phys. A: Math. Theor. 45, 065401 (2012)] is investigated.

Fusion of Positive Energy Representations of LSpin(2n)

NASA Astrophysics Data System (ADS)

Toledano-Laredo, V.

2004-09-01

Building upon the Jones-Wassermann program of studying Conformal Field Theory using operator algebraic tools, and the work of A. Wassermann on the loop group of LSU(n) (Invent. Math. 133 (1998), 467-538), we give a solution to the problem of fusion for the loop group of Spin(2n). Our approach relies on the use of A. Connes' tensor product of bimodules over a von Neumann algebra to define a multiplicative operation (Connes fusion) on the (integrable) positive energy representations of a given level. The notion of bimodules arises by restricting these representations to loops with support contained in an interval I of the circle or its complement. We study the corresponding Grothendieck ring and show that fusion with the vector representation is given by the Verlinde rules. The computation rests on 1) the solution of a 6-parameter family of Knizhnik-Zamolodchikhov equations and the determination of its monodromy, 2) the explicit construction of the primary fields of the theory, which allows to prove that they define operator-valued distributions and 3) the algebraic theory of superselection sectors developed by Doplicher-Haag-Roberts.

Moyal deformations of Clifford gauge theories of gravity

NASA Astrophysics Data System (ADS)

Castro, Carlos

2016-12-01

A Moyal deformation of a Clifford Cl(3, 1) Gauge Theory of (Conformal) Gravity is performed for canonical noncommutativity (constant Θμν parameters). In the very special case when one imposes certain constraints on the fields, there are no first-order contributions in the Θμν parameters to the Moyal deformations of Clifford gauge theories of gravity. However, when one does not impose constraints on the fields, there are first-order contributions in Θμν to the Moyal deformations in variance with the previous results obtained by other authors and based on different gauge groups. Despite that the generators of U(2, 2),SO(4, 2),SO(2, 3) can be expressed in terms of the Clifford algebra generators this does not imply that these algebras are isomorphic to the Clifford algebra. Therefore one should not expect identical results to those obtained by other authors. In particular, there are Moyal deformations of the Einstein-Hilbert gravitational action with a cosmological constant to first-order in Θμν. Finally, we provide a mechanism which furnishes a plausible cancellation of the huge vacuum energy density.

A Reduced Model for the Magnetorotational Instability

NASA Astrophysics Data System (ADS)

Jamroz, Ben; Julien, Keith; Knobloch, Edgar

2008-11-01

The magnetorotational instability is investigated within the shearing box approximation in the large Elsasser number regime. In this regime, which is of fundamental importance to astrophysical accretion disk theory, shear is the dominant source of energy, but the instability itself requires the presence of a weaker vertical magnetic field. Dissipative effects are weaker still. However, they are sufficiently large to permit a nonlinear feedback mechanism whereby the turbulent stresses generated by the MRI act on and modify the local background shear in the angular velocity profile. To date this response has been omitted in shearing box simulations and is captured by a reduced pde model derived here from the global MHD fluid equations using multiscale asymptotic perturbation theory. Results from numerical simulations of the reduced pde model indicate a linear phase of exponential growth followed by a nonlinear adjustment to algebraic growth and decay in the fluctuating quantities. Remarkably, the velocity and magnetic field correlations associated with these algebraic growth and decay laws conspire to achieve saturation of the angular momentum transport. The inclusion of subdominant ohmic dissipation arrests the algebraic growth of the fluctuations on a longer, dissipative time scale.

Flux-driven algebraic damping of m = 1 diocotron mode

NASA Astrophysics Data System (ADS)

Chim, Chi Yung; O'Neil, Thomas M.

2016-07-01

Recent experiments with pure electron plasmas in a Malmberg-Penning trap have observed the algebraic damping of m = 1 diocotron modes. Transport due to small field asymmetries produces a low density halo of electrons moving radially outward from the plasma core, and the mode damping begins when the halo reaches the resonant radius r = Rw at the wall of the trap. The damping rate is proportional to the flux of halo particles through the resonant layer. The damping is related to, but distinct from, spatial Landau damping, in which a linear wave-particle resonance produces exponential damping. This paper explains with analytic theory the new algebraic damping due to particle transport by both mobility and diffusion. As electrons are swept around the "cat's eye" orbits of the resonant wave-particle interaction, they form a dipole (m = 1) density distribution. From this distribution, the electric field component perpendicular to the core displacement produces E × B-drift of the core back to the axis, that is, damps the m = 1 mode. The parallel component produces drift in the azimuthal direction, that is, causes a shift in the mode frequency.

Solving Multi-variate Polynomial Equations in a Finite Field

DTIC Science & Technology

2013-06-01

Algebraic Background In this section, some algebraic definitions and basics are discussed as they pertain to this re- search. For a more detailed...definitions and basics are discussed as they pertain to this research. For a more detailed treatment, consult a graph theory text such as [10]. A graph G...graph if V(G) can be partitioned into k subsets V1,V2, ...,Vk such that uv is only an edge of G if u and v belong to different partite sets. If, in

Singular vectors for the WN algebras

NASA Astrophysics Data System (ADS)

Ridout, David; Siu, Steve; Wood, Simon

2018-03-01

In this paper, we use free field realisations of the A-type principal, or Casimir, WN algebras to derive explicit formulae for singular vectors in Fock modules. These singular vectors are constructed by applying screening operators to Fock module highest weight vectors. The action of the screening operators is then explicitly evaluated in terms of Jack symmetric functions and their skew analogues. The resulting formulae depend on sequences of pairs of integers that completely determine the Fock module as well as the Jack symmetric functions.

Bialgebra deformations and algebras of trees

NASA Technical Reports Server (NTRS)

Grossman, Robert; Radford, David

1991-01-01

Let A denote a bialgebra over a field k and let A sub t = A((t)) denote the ring of formal power series with coefficients in A. Assume that A is also isomorphic to a free, associative algebra over k. A simple construction is given which makes A sub t a bialgebra deformation of A. In typical applications, A sub t is neither commutative nor cocommutative. In the terminology of Drinfeld, (1987), A sub t is a quantum group. This construction yields quantum groups associated with families of trees.

Boolean Operations with Prism Algebraic Patches

PubMed Central

Bajaj, Chandrajit; Paoluzzi, Alberto; Portuesi, Simone; Lei, Na; Zhao, Wenqi

2009-01-01

In this paper we discuss a symbolic-numeric algorithm for Boolean operations, closed in the algebra of curved polyhedra whose boundary is triangulated with algebraic patches (A-patches). This approach uses a linear polyhedron as a first approximation of both the arguments and the result. On each triangle of a boundary representation of such linear approximation, a piecewise cubic algebraic interpolant is built, using a C1-continuous prism algebraic patch (prism A-patch) that interpolates the three triangle vertices, with given normal vectors. The boundary representation only stores the vertices of the initial triangulation and their external vertex normals. In order to represent also flat and/or sharp local features, the corresponding normal-per-face and/or normal-per-edge may be also given, respectively. The topology is described by storing, for each curved triangle, the two triples of pointers to incident vertices and to adjacent triangles. For each triangle, a scaffolding prism is built, produced by its extreme vertices and normals, which provides a containment volume for the curved interpolating A-patch. When looking for the result of a regularized Boolean operation, the 0-set of a tri-variate polynomial within each such prism is generated, and intersected with the analogous 0-sets of the other curved polyhedron, when two prisms have non-empty intersection. The intersection curves of the boundaries are traced and used to decompose each boundary into the 3 standard classes of subpatches, denoted in, out and on. While tracing the intersection curves, the locally refined triangulation of intersecting patches is produced, and added to the boundary representation. PMID:21516262

FOURTH SEMINAR TO THE MEMORY OF D.N. KLYSHKO: Algebraic solution of the synthesis problem for coded sequences

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.

Deformations of vector-scalar models

NASA Astrophysics Data System (ADS)

Barnich, Glenn; Boulanger, Nicolas; Henneaux, Marc; Julia, Bernard; Lekeu, Victor; Ranjbar, Arash

2018-02-01

Abelian vector fields non-minimally coupled to uncharged scalar fields arise in many contexts. We investigate here through algebraic methods their consistent deformations ("gaugings"), i.e., the deformations that preserve the number (but not necessarily the form or the algebra) of the gauge symmetries. Infinitesimal consistent deformations are given by the BRST cohomology classes at ghost number zero. We parametrize explicitly these classes in terms of various types of global symmetries and corresponding Noether currents through the characteristic cohomology related to antifields and equations of motion. The analysis applies to all ghost numbers and not just ghost number zero. We also provide a systematic discussion of the linear and quadratic constraints on these parameters that follow from higher-order consistency. Our work is relevant to the gaugings of extended supergravities.

A locally supersymmetric SO(10, 2) invariant action for D = 12 supergravity

NASA Astrophysics Data System (ADS)

Castellani, Leonardo

2017-06-01

We present an action for N = 1 supergravity in 10 + 2 dimensions, containing the gauge fields of the OSp(1|64) superalgebra, i.e. one-forms B ( n) with n=1,2,5,6,9,10 antisymmetric D=12 Lorentz indices and a Majorana gravitino ψ. The vielbein and spin connection correspond to B (1) and B (2) respectively. The action is not gauge invariant under the full OSp(1|64) superalgebra, but only under a subalgebra \\tilde{F} (containing the F algebra OSp(1|32)), whose gauge fields are B (2), B (6), B (10) and the Weyl projected Majorana gravitino 1/2(1+{Γ}_{13})ψ . Supersymmetry transformations are therefore generated by a Majorana-Weyl supercharge and, being part of a gauge superalgebra, close off-shell. The action is simply ∫ STr( R 6 Γ) where R is the OSp(1|64) curvature supermatrix two-form, and Γ is a constant supermatrix involving Γ13 and breaking OSp(1|64) to its \\tilde{F} subalgebra. The usual Einstein-Hilbert term is included in the action.

Trivial solutions of generalized supergravity vs non-abelian T-duality anomaly

NASA Astrophysics Data System (ADS)

Wulff, Linus

2018-06-01

The equations that follow from kappa symmetry of the type II Green-Schwarz string are a certain deformation, by a Killing vector field K, of the type II supergravity equations. We analyze under what conditions solutions of these 'generalized' supergravity equations are trivial in the sense that they solve also the standard supergravity equations. We argue that for this to happen K must be null and satisfy dK =iK H with H = dB the NSNS three-form field strength. Non-trivial examples are provided by symmetric pp-wave solutions. We then analyze the consequences for non-abelian T-duality and the closely related homogenous Yang-Baxter sigma models. When one performs non-abelian T-duality of a string sigma model on a non-unimodular (sub)algebra one generates a non-vanishing K proportional to the trace of the structure constants. This is expected to lead to an anomaly but we show that when K satisfies the same conditions the anomaly in fact goes away leading to more possibilities for non-anomalous non-abelian T-duality.

Three Interpretations of the Matrix Equation Ax = b

ERIC Educational Resources Information Center

Larson, Christine; Zandieh, Michelle

2013-01-01

Many of the central ideas in an introductory undergraduate linear algebra course are closely tied to a set of interpretations of the matrix equation Ax = b (A is a matrix, x and b are vectors): linear combination interpretations, systems interpretations, and transformation interpretations. We consider graphic and symbolic representations for each,…

The Routes of Unity

ERIC Educational Resources Information Center

McCartney, Mark; Gibson, Sharon

2006-01-01

A model for car following on a closed loop is defined. The stability of the solutions of the model is investigated by considering the evolution of the roots of the corresponding characteristic equation in the complex plane. The solution provides a motivation for investigating the behaviour of the roots of a simple class of algebraic equation.…

Using Disks as Models for Proofs of Series

ERIC Educational Resources Information Center

Somchaipeng, Tongta; Kruatong, Tussatrin; Panijpan, Bhinyo

2012-01-01

Exploring and deriving proofs of closed-form expressions for series can be fun for students. However, for some students, a physical representation of such problems is more meaningful. Various approaches have been designed to help students visualize squares of sums and sums of squares; these approaches may be arithmetic-algebraic or combinatorial…

Correlation Decay in Fermionic Lattice Systems with Power-Law Interactions at Nonzero Temperature

NASA Astrophysics Data System (ADS)

Hernández-Santana, Senaida; Gogolin, Christian; Cirac, J. Ignacio; Acín, Antonio

2017-09-01

We study correlations in fermionic lattice systems with long-range interactions in thermal equilibrium. We prove a bound on the correlation decay between anticommuting operators and generalize a long-range Lieb-Robinson-type bound. Our results show that in these systems of spatial dimension D with, not necessarily translation invariant, two-site interactions decaying algebraically with the distance with an exponent α ≥2 D , correlations between such operators decay at least algebraically to 0 with an exponent arbitrarily close to α at any nonzero temperature. Our bound is asymptotically tight, which we demonstrate by a high temperature expansion and by numerically analyzing density-density correlations in the one-dimensional quadratic (free, exactly solvable) Kitaev chain with long-range pairing.

Dirac delta representation by exact parametric equations.. Application to impulsive vibration systems

NASA Astrophysics Data System (ADS)

Chicurel-Uziel, Enrique

2007-08-01

A pair of closed parametric equations are proposed to represent the Heaviside unit step function. Differentiating the step equations results in two additional parametric equations, that are also hereby proposed, to represent the Dirac delta function. These equations are expressed in algebraic terms and are handled by means of elementary algebra and elementary calculus. The proposed delta representation complies exactly with the values of the definition. It complies also with the sifting property and the requisite unit area and its Laplace transform coincides with the most general form given in the tables. Furthermore, it leads to a very simple method of solution of impulsive vibrating systems either linear or belonging to a large class of nonlinear problems. Two example solutions are presented.

Analysis of eletrectrohydrodynamic jetting using multifunctional and three-dimensional tomography

NASA Astrophysics Data System (ADS)

Ko, Han Seo; Nguyen, Xuan Hung; Lee, Soo-Hong; Kim, Young Hyun

2013-11-01

Three-dimensional optical tomography technique was developed to reconstruct three-dimensional flow fields using a set of two-dimensional shadowgraphic images and normal gray images. From three high speed cameras, which were positioned at an offset angle of 45° relative to one another, number, size and location of electrohydrodynamic jets with respect to the nozzle position were analyzed using shadowgraphic tomography employing a multiplicative algebraic reconstruction technique (MART). Additionally, a flow field inside cone-shaped liquid (Taylor cone) which was induced under electric field was also observed using a simultaneous multiplicative algebraic reconstruction technique (SMART) for reconstructing intensities of particle light and combining with a three-dimensional cross correlation. Various velocity fields of a circulating flow inside the cone-shaped liquid due to different physico-chemical properties of liquid and applied voltages were also investigated. This work supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Korean government (MEST) (No. S-2011-0023457).

Vertical structure of medium-scale traveling ionospheric disturbances

NASA Astrophysics Data System (ADS)

Ssessanga, Nicholas; Kim, Yong Ha; Kim, Eunsol

2015-11-01

We develop an algorithm of computerized ionospheric tomography (CIT) to infer information on the vertical and horizontal structuring of electron density during nighttime medium-scale traveling ionospheric disturbances (MSTIDs). To facilitate digital CIT we have adopted total electron contents (TEC) from a dense Global Positioning System (GPS) receiver network, GEONET, which contains more than 1000 receivers. A multiplicative algebraic reconstruction technique was utilized with a calibrated IRI-2012 model as an initial solution. The reconstructed F2 peak layer varied in altitude with average peak-to-peak amplitude of ~52 km. In addition, the F2 peak layer anticorrelated with TEC variations. This feature supports a theory in which nighttime MSTID is composed of oscillating electric fields due to conductivity variations. Moreover, reconstructed TEC variations over two stations were reasonably close to variations directly derived from the measured TEC data set. Our tomographic analysis may thus help understand three-dimensional structure of MSTIDs in a quantitative way.

A Process Algebra Approach to Quantum Electrodynamics

NASA Astrophysics Data System (ADS)

Sulis, William

2017-12-01

The process algebra program is directed towards developing a realist model of quantum mechanics free of paradoxes, divergences and conceptual confusions. From this perspective, fundamental phenomena are viewed as emerging from primitive informational elements generated by processes. The process algebra has been shown to successfully reproduce scalar non-relativistic quantum mechanics (NRQM) without the usual paradoxes and dualities. NRQM appears as an effective theory which emerges under specific asymptotic limits. Space-time, scalar particle wave functions and the Born rule are all emergent in this framework. In this paper, the process algebra model is reviewed, extended to the relativistic setting, and then applied to the problem of electrodynamics. A semiclassical version is presented in which a Minkowski-like space-time emerges as well as a vector potential that is discrete and photon-like at small scales and near-continuous and wave-like at large scales. QED is viewed as an effective theory at small scales while Maxwell theory becomes an effective theory at large scales. The process algebra version of quantum electrodynamics is intuitive and realist, free from divergences and eliminates the distinction between particle, field and wave. Computations are carried out using the configuration space process covering map, although the connection to second quantization has not been fully explored.

Gauge symmetries of the free bosonic string field theory

NASA Astrophysics Data System (ADS)

Neveu, A.; Schwarz, J.; West, P. C.

1985-12-01

The gauge covariant local formulations of free bosonic string theories that contained a finite number of supplementary fields are extended to include an infinite number of supplementary fields. These new formulations allow the generators of the Virasoro algebra to appear on a more equal footing. Permanent address: King's College, Physics Department, London WC2R 2LS, UK.

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.

Quantum Field Theory in Two Dimensions: Light-front Versus Space-like Solutions

NASA Astrophysics Data System (ADS)

Martinovic̆, L'ubomír

2017-07-01

A few non-perturbative topics of quantum field theory in D=1+1 are studied in both the conventional (SL) and light-front (LF) versions. First, we give a concise review of the recently proposed quantization of the two-dimensional massless LF fields. The LF version of bosonization follows in a simple and natural way including the bosonized form of the Thirring model. As a further application, we demonstrate the closeness of the 2D massless LF quantum fields to conformal field theory (CFT). We calculate several correlation functions including those between the components of the LF energy-momentum tensor and derive the LF version of the Virasoro algebra. Using the Euclidean time variable, we can immediately transform calculated quantities to the (anti)holomorphic form. The results found are in agreement with those from CFT. Finally, we show that the proposed framework provides us with the elements needed for an independent LF study of exactly solvable models. We compute the non-perturbative correlation functions from the exact operator solution of the LF Thirring model and compare it to the analogous results in the SL theory. While the vacuum effects are automatically taken into account in the LF case, the non-trivial vacuum structure has to be incorported by an explicit diagonalization of the SL Hamiltonians, to obtain the equivalently complete solution.

Intermediate grouping on remotely sensed data using Gestalt algebra

NASA Astrophysics Data System (ADS)

Michaelsen, Eckart

2014-10-01

Human observers often achieve striking recognition performance on remotely sensed data unmatched by machine vision algorithms. This holds even for thermal images (IR) or synthetic aperture radar (SAR). Psychologists refer to these capabilities as Gestalt perceptive skills. Gestalt Algebra is a mathematical structure recently proposed for such laws of perceptual grouping. It gives operations for mirror symmetry, continuation in rows and rotational symmetric patterns. Each of these operations forms an aggregate-Gestalt of a tuple of part-Gestalten. Each Gestalt is attributed with a position, an orientation, a rotational frequency, a scale, and an assessment respectively. Any Gestalt can be combined with any other Gestalt using any of the three operations. Most often the assessment of the new aggregate-Gestalt will be close to zero. Only if the part-Gestalten perfectly fit into the desired pattern the new aggregate-Gestalt will be assessed with value one. The algebra is suitable in both directions: It may render an organized symmetric mandala using random numbers. Or it may recognize deep hidden visual relationships between meaningful parts of a picture. For the latter primitives must be obtained from the image by some key-point detector and a threshold. Intelligent search strategies are required for this search in the combinatorial space of possible Gestalt Algebra terms. Exemplarily, maximal assessed Gestalten found in selected aerial images as well as in IR and SAR images are presented.

Proper projective symmetry in LRS Bianchi type V spacetimes

NASA Astrophysics Data System (ADS)

Shabbir, Ghulam; Mahomed, K. S.; Mahomed, F. M.; Moitsheki, R. J.

2018-04-01

In this paper, we investigate proper projective vector fields of locally rotationally symmetric (LRS) Bianchi type V spacetimes using direct integration and algebraic techniques. Despite the non-degeneracy in the Riemann tensor eigenvalues, we classify proper Bianchi type V spacetimes and show that the above spacetimes do not admit proper projective vector fields. Here, in all the cases projective vector fields are Killing vector fields.

Two charges on a plane in a magnetic field: hidden algebra, (particular) integrability, polynomial eigenfunctions

NASA Astrophysics Data System (ADS)

Turbiner, A. V.; Escobar-Ruiz, M. A.

2013-07-01

The quantum mechanics of two Coulomb charges on a plane (e1, m1) and (e2, m2) subject to a constant magnetic field B perpendicular to the plane is considered. Four integrals of motion are explicitly indicated. It is shown that for two physically important particular cases, namely that of two particles of equal Larmor frequencies, {e_c} \\propto \\frac{e_1}{m_1}-\\frac{e_2}{m_2}=0 (e.g. two electrons) and one of a neutral system (e.g. the electron-positron pair, hydrogen atom) at rest (the center-of-mass momentum is zero) some outstanding properties occur. They are the most visible in double polar coordinates in CMS (R, ϕ) and relative (ρ, φ) coordinate systems: (i) eigenfunctions are factorizable, all factors except one with the explicit ρ-dependence are found analytically, they have definite relative angular momentum, (ii) dynamics in the ρ-direction is the same for both systems, it corresponds to a funnel-type potential and it has hidden sl(2) algebra, at some discrete values of dimensionless magnetic fields b ⩽ 1, (iii) particular integral(s) occur, (iv) the hidden sl(2) algebra emerges in finite-dimensional representation, thus, the system becomes quasi-exactly-solvable and (v) a finite number of polynomial eigenfunctions in ρ appear. Nine families of eigenfunctions are presented explicitly.

Reformulation of the symmetries of first-order general relativity

NASA Astrophysics Data System (ADS)

Montesinos, Merced; González, Diego; Celada, Mariano; Díaz, Bogar

2017-10-01

We report a new internal gauge symmetry of the n-dimensional Palatini action with cosmological term (n>3 ) that is the generalization of three-dimensional local translations. This symmetry is obtained through the direct application of the converse of Noether’s second theorem on the theory under consideration. We show that diffeomorphisms can be expressed as linear combinations of it and local Lorentz transformations with field-dependent parameters up to terms involving the variational derivatives of the action. As a result, the new internal symmetry together with local Lorentz transformations can be adopted as the fundamental gauge symmetries of general relativity. Although their gauge algebra is open in general, it allows us to recover, without resorting to the equations of motion, the very well-known Lie algebra satisfied by translations and Lorentz transformations in three dimensions. We also report the analog of the new gauge symmetry for the Holst action with cosmological term, finding that it explicitly depends on the Immirzi parameter. The same result concerning its relation to diffeomorphisms and the open character of the gauge algebra also hold in this case. Finally, we consider the non-minimal coupling of a scalar field to gravity in n dimensions and establish that the new gauge symmetry is affected by this matter field. Our results indicate that general relativity in dimension greater than three can be thought of as a gauge theory.

Observables and microscopic entropy of higher spin black holes

NASA Astrophysics Data System (ADS)

Compère, Geoffrey; Jottar, Juan I.; Song, Wei

2013-11-01

In the context of recently proposed holographic dualities between higher spin theories in AdS3 and (1 + 1)-dimensional CFTs with symmetry algebras, we revisit the definition of higher spin black hole thermodynamics and the dictionary between bulk fields and dual CFT operators. We build a canonical formalism based on three ingredients: a gauge-invariant definition of conserved charges and chemical potentials in the presence of higher spin black holes, a canonical definition of entropy in the bulk, and a bulk-to-boundary dictionary aligned with the asymptotic symmetry algebra. We show that our canonical formalism shares the same formal structure as the so-called holomorphic formalism, but differs in the definition of charges and chemical potentials and in the bulk-to-boundary dictionary. Most importantly, we show that it admits a consistent CFT interpretation. We discuss the spin-2 and spin-3 cases in detail and generalize our construction to theories based on the hs[ λ] algebra, and on the sl( N,[InlineMediaObject not available: see fulltext.]) algebra for any choice of sl(2 ,[InlineMediaObject not available: see fulltext.]) embedding.

The Motivic Cofiber of tau and Exotic Periodicities

NASA Astrophysics Data System (ADS)

Gheorghe, Bogdan

Consider the Tate twist tau epsilon H0,1 (S0,0) in the mod 2 cohomology of the motivic sphere. After 2-completion, the motivic Adams spectral sequence realizes this element as a map tau : S0,-1 → S0,0 . This thesis begins with the study of its cofiber, that we denote by Ctau. We first show that this motivic 2-cell complex can be endowed with a unique Einfinity ring structure. This promotes the known isomorphism pi *,* Ctau ≅ Ext *,* BP*BP (BP*, BP* ) to an isomorphism of rings which also preserves higher products. This structure allows us to consider its closed symmetric monoidal category of modules (Ctau Mod, - ∧Ctau -), which happens to live in the kernel of Betti realization. This category has surprising applications, and moreover contains many interesting motivic spectra. In particular, we construct exotic motivic fields K( wn), detecting motivic wn-periodicity. This theory of motivic wn-periodicity can be roughly seen as perpendicular to the vn-periodicity story, detected by the motivic Morava K-theories K( n). Finally, we also explain why the category Ctau Mod is so computable. The above isomor phism comes in a more structured version. In work that is joint with Zhouli Xu and Guozhen Wang, we show that there is an equivalence of infinity-categories Db ( MGL *,* MGL Comodev) ≅ → Ctau Cellcomp between an algebraic derived category, and the subcategory Ctau Cellcomp of cellular Ctau- modules that are complete with respect to a version of the algebraic cobordism spectrum MGL.

Close range fault tolerant noncontacting position sensor

DOEpatents

Bingham, D.N.; Anderson, A.A.

1996-02-20

A method and system are disclosed for locating the three dimensional coordinates of a moving or stationary object in real time. The three dimensional coordinates of an object in half space or full space are determined based upon the time of arrival or phase of the wave front measured by a plurality of receiver elements and an established vector magnitudes proportional to the measured time of arrival or phase at each receiver element. The coordinates of the object are calculated by solving a matrix equation or a set of closed form algebraic equations. 3 figs.

Some applications of mathematics in theoretical physics - A review

DOE Office of Scientific and Technical Information (OSTI.GOV)

Bora, Kalpana

2016-06-21

Mathematics is a very beautiful subject−very much an indispensible tool for Physics, more so for Theoretical Physics (by which we mean here mainly Field Theory and High Energy Physics). These branches of Physics are based on Quantum Mechanics and Special Theory of Relativity, and many mathematical concepts are used in them. In this work, we shall elucidate upon only some of them, like−differential geometry, infinite series, Mellin transforms, Fourier and integral transforms, special functions, calculus, complex algebra, topology, group theory, Riemannian geometry, functional analysis, linear algebra, operator algebra, etc. We shall also present, some physics issues, where these mathematical toolsmore » are used. It is not wrong to say that Mathematics is such a powerful tool, without which, there can not be any Physics theory!! A brief review on our research work is also presented.« less

Integrand-level reduction of loop amplitudes by computational algebraic geometry methods

NASA Astrophysics Data System (ADS)

Zhang, Yang

2012-09-01

We present an algorithm for the integrand-level reduction of multi-loop amplitudes of renormalizable field theories, based on computational algebraic geometry. This algorithm uses (1) the Gröbner basis method to determine the basis for integrand-level reduction, (2) the primary decomposition of an ideal to classify all inequivalent solutions of unitarity cuts. The resulting basis and cut solutions can be used to reconstruct the integrand from unitarity cuts, via polynomial fitting techniques. The basis determination part of the algorithm has been implemented in the Mathematica package, BasisDet. The primary decomposition part can be readily carried out by algebraic geometry softwares, with the output of the package BasisDet. The algorithm works in both D = 4 and D = 4 - 2 ɛ dimensions, and we present some two and three-loop examples of applications of this algorithm.

Some applications of mathematics in theoretical physics - A review

NASA Astrophysics Data System (ADS)

Bora, Kalpana

2016-06-01

Mathematics is a very beautiful subject-very much an indispensible tool for Physics, more so for Theoretical Physics (by which we mean here mainly Field Theory and High Energy Physics). These branches of Physics are based on Quantum Mechanics and Special Theory of Relativity, and many mathematical concepts are used in them. In this work, we shall elucidate upon only some of them, like-differential geometry, infinite series, Mellin transforms, Fourier and integral transforms, special functions, calculus, complex algebra, topology, group theory, Riemannian geometry, functional analysis, linear algebra, operator algebra, etc. We shall also present, some physics issues, where these mathematical tools are used. It is not wrong to say that Mathematics is such a powerful tool, without which, there can not be any Physics theory!! A brief review on our research work is also presented.

Pointless strings

DOE Office of Scientific and Technical Information (OSTI.GOV)

Periwal, V.

1988-01-01

The author proves that bosonic string perturbation theory diverges and is not Borel summable. This is an indication of a non-perturbative instability of the bosonic string vacuum. He formulates two-dimensional sigma models in terms of algebras of functions. He extends this formulation to general C* algebras. He illustrates the utility of these algebraic notions by calculating some determinants of interest in the study of string propagation in orbifold backgrounds. He studies the geometry of spaces of field theories and show that the vanishing of the curvature of the natural Gel'fand-Naimark-Segal metric on such spaces is exactly the strong associativity conditionmore » of the operator product expansion.He shows that string scattering amplitudes arise as invariants of renormalization, when he formulates renormalization in terms of rescalings of the metric on the string world-sheet.« less

Angular momentum and Zeeman effect in the presence of a minimal length based on the Kempf-Mann-Mangano algebra

NASA Astrophysics Data System (ADS)

Khosropour, B.

2016-07-01

In this work, we consider a D-dimensional ( β, β^' -two-parameters deformed Heisenberg algebra, which was introduced by Kempf et al. The angular-momentum operator in the presence of a minimal length scale based on the Kempf-Mann-Mangano algebra is obtained in the special case of β^' = 2β up to the first order over the deformation parameter β . It is shown that each of the components of the modified angular-momentum operator, commutes with the modified operator {L}2 . We find the magnetostatic field in the presence of a minimal length. The Zeeman effect in the deformed space is studied and also Lande's formula for the energy shift in the presence of a minimal length is obtained. We estimate an upper bound on the isotropic minimal length.

A lattice approach to the conformal OSp(2S+2|2S) supercoset sigma model. Part I: Algebraic structures in the spin chain. The Brauer algebra

NASA Astrophysics Data System (ADS)

Candu, Constantin; Saleur, Hubert

2009-02-01

We define and study a lattice model which we argue is in the universality class of the OSp(2S+2|2S) supercoset sigma model for a large range of values of the coupling constant gσ2. In this first paper, we analyze in details the symmetries of this lattice model, in particular the decomposition of the space of the quantum spin chain V as a bimodule over OSp(2S+2|2S) and its commutant, the Brauer algebra B(2). It turns out that V is a nonsemisimple module for both OSp(2S+2|2S) and B(2). The results are used in the companion paper to elucidate the structure of the (boundary) conformal field theory.

On the integration of a class of nonlinear systems of ordinary differential equations

NASA Astrophysics Data System (ADS)

Talyshev, Aleksandr A.

2017-11-01

For each associative, commutative, and unitary algebra over the field of real or complex numbers and an integrable nonlinear ordinary differential equation we can to construct integrable systems of ordinary differential equations and integrable systems of partial differential equations. In this paper we consider in some sense the inverse problem. Determine the conditions under which a given system of ordinary differential equations can be represented as a differential equation in some associative, commutative and unitary algebra. It is also shown that associativity is not a necessary condition.

Muzzle Flash Onset: An Algebraic Criterion and Further Validation of the Muzzle Exhaust Flow Field Model

DTIC Science & Technology

1983-03-01

Tic, equals to (NI/ Nic ) where Nic , defined as the net chemical production rate of i-th species, is in general the algebraic sum of terms which are...detailed analysis has shown that in preignition regions the chemical rates which make a significant contribution to any of the Nic are such that at least...Elkton Division Lab., Inc. ATTN. R. Biddle ATTN: M. Summeitield Tech Lib. 1041 US Hlighway One North P. 0. Box 241 Princeton, NJ 08540 Elkton, MD

Graviweak Unification, Invisible Universe and Dark Energy

NASA Astrophysics Data System (ADS)

Das, C. R.; Laperashvili, L. V.; Tureanu, A.

2013-07-01

We consider a graviweak unification model with the assumption of the existence of a hidden (invisible) sector of our Universe, parallel to the visible world. This Hidden World (HW) is assumed to be a Mirror World (MW) with broken mirror parity. We start with a diffeomorphism invariant theory of a gauge field valued in a Lie algebra g, which is broken spontaneously to the direct sum of the space-time Lorentz algebra and the Yang-Mills algebra: ˜ {g} = {{su}}(2) (grav)L ⊕ {{su}}(2)L — in the ordinary world, and ˜ {g}' = {{su}}(2){' (grav)}R ⊕ {{su}}(2)'R — in the hidden world. Using an extension of the Plebanski action for general relativity, we recover the actions for gravity, SU(2) Yang-Mills and Higgs fields in both (visible and invisible) sectors of the Universe, and also the total action. After symmetry breaking, all physical constants, including the Newton's constants, cosmological constants, Yang-Mills couplings, and other parameters, are determined by a single parameter g present in the initial action, and by the Higgs VEVs. The dark energy problem of this model predicts a too large supersymmetric breaking scale (MSUSY 1010GeV), which is not within the reach of the LHC experiments.

Operator bases, S-matrices, and their partition functions

NASA Astrophysics Data System (ADS)

Henning, Brian; Lu, Xiaochuan; Melia, Tom; Murayama, Hitoshi

2017-10-01

Relativistic quantum systems that admit scattering experiments are quantitatively described by effective field theories, where S-matrix kinematics and symmetry considerations are encoded in the operator spectrum of the EFT. In this paper we use the S-matrix to derive the structure of the EFT operator basis, providing complementary descriptions in (i) position space utilizing the conformal algebra and cohomology and (ii) momentum space via an algebraic formulation in terms of a ring of momenta with kinematics implemented as an ideal. These frameworks systematically handle redundancies associated with equations of motion (on-shell) and integration by parts (momentum conservation). We introduce a partition function, termed the Hilbert series, to enumerate the operator basis — correspondingly, the S-matrix — and derive a matrix integral expression to compute the Hilbert series. The expression is general, easily applied in any spacetime dimension, with arbitrary field content and (linearly realized) symmetries. In addition to counting, we discuss construction of the basis. Simple algorithms follow from the algebraic formulation in momentum space. We explicitly compute the basis for operators involving up to n = 5 scalar fields. This construction universally applies to fields with spin, since the operator basis for scalars encodes the momentum dependence of n-point amplitudes. We discuss in detail the operator basis for non-linearly realized symmetries. In the presence of massless particles, there is freedom to impose additional structure on the S- matrix in the form of soft limits. The most na¨ıve implementation for massless scalars leads to the operator basis for pions, which we confirm using the standard CCWZ formulation for non-linear realizations. Although primarily discussed in the language of EFT, some of our results — conceptual and quantitative — may be of broader use in studying conformal field theories as well as the AdS/CFT correspondence.

An Algebraic-Habits-of-Mind Perspective on Elementary School

ERIC Educational Resources Information Center

Goldenberg, E. Paul; Mark, June; Cuoco, Al

2010-01-01

More important than infusing courses and curricula with modern content is to give students the tools to use and understand mathematics--including some that does not yet exist. A curriculum organized around habits of mind tries to close the gap between what users and makers of mathematics do and what they say. These cross-journal articles are…

A Compact Formula for Rotations as Spin Matrix Polynomials

DOE PAGES

Curtright, Thomas L.; Fairlie, David B.; Zachos, Cosmas K.

2014-08-12

Group elements of SU(2) are expressed in closed form as finite polynomials of the Lie algebra generators, for all definite spin representations of the rotation group. Here, the simple explicit result exhibits connections between group theory, combinatorics, and Fourier analysis, especially in the large spin limit. Salient intuitive features of the formula are illustrated and discussed.

Revisiting Unemployment in Intermediate Macroeconomics: A New Approach for Teaching Diamond-Mortensen-Pissarides

ERIC Educational Resources Information Center

Bhattacharya, Arghya; Jackson, Paul; Jenkins, Brian C.

2018-01-01

The authors present a version of the Diamond-Mortensen-Pissarides model of unemployment that is accessible to undergraduates and preserve the dynamic structure of the original model. The model is solvable in closed form using basic algebra and admits a graphical representation useful for illustrating a variety of comparative statics. They show how…

Gupta-Bleuler Quantization of the Maxwell Field in Globally Hyperbolic Space-Times

NASA Astrophysics Data System (ADS)

Finster, Felix; Strohmaier, Alexander

2015-08-01

We give a complete framework for the Gupta-Bleuler quantization of the free electromagnetic field on globally hyperbolic space-times. We describe one-particle structures that give rise to states satisfying the microlocal spectrum condition. The field algebras in the so-called Gupta-Bleuler representations satisfy the time-slice axiom, and the corresponding vacuum states satisfy the microlocal spectrum condition. We also give an explicit construction of ground states on ultrastatic space-times. Unlike previous constructions, our method does not require a spectral gap or the absence of zero modes. The only requirement, the absence of zero-resonance states, is shown to be stable under compact perturbations of topology and metric. Usual deformation arguments based on the time-slice axiom then lead to a construction of Gupta-Bleuler representations on a large class of globally hyperbolic space-times. As usual, the field algebra is represented on an indefinite inner product space, in which the physical states form a positive semi-definite subspace. Gauge transformations are incorporated in such a way that the field can be coupled perturbatively to a Dirac field. Our approach does not require any topological restrictions on the underlying space-time.

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

NASA Astrophysics Data System (ADS)

Nazarov, Anton

2012-11-01

In this paper we present Affine.m-a program for computations in representation theory of finite-dimensional and affine Lie algebras and describe implemented algorithms. The algorithms are based on the properties of weights and Weyl symmetry. Computation of weight multiplicities in irreducible and Verma modules, branching of representations and tensor product decomposition are the most important problems for us. These problems have numerous applications in physics and we provide some examples of these applications. The program is implemented in the popular computer algebra system Mathematica and works with finite-dimensional and affine Lie algebras. Catalogue identifier: AENA_v1_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AENB_v1_0.html Program obtainable from: CPC Program Library, Queen’s University, Belfast, UK Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 24 844 No. of bytes in distributed program, including test data, etc.: 1 045 908 Distribution format: tar.gz Programming language: Mathematica. Computer: i386-i686, x86_64. Operating system: Linux, Windows, Mac OS, Solaris. RAM: 5-500 Mb Classification: 4.2, 5. Nature of problem: Representation theory of finite-dimensional Lie algebras has many applications in different branches of physics, including elementary particle physics, molecular physics, nuclear physics. Representations of affine Lie algebras appear in string theories and two-dimensional conformal field theory used for the description of critical phenomena in two-dimensional systems. Also Lie symmetries play a major role in a study of quantum integrable systems. Solution method: We work with weights and roots of finite-dimensional and affine Lie algebras and use Weyl symmetry extensively. Central problems which are the computations of weight multiplicities, branching and fusion coefficients are solved using one general recurrent algorithm based on generalization of Weyl character formula. We also offer alternative implementation based on the Freudenthal multiplicity formula which can be faster in some cases. Restrictions: Computational complexity grows fast with the rank of an algebra, so computations for algebras of ranks greater than 8 are not practical. Unusual features: We offer the possibility of using a traditional mathematical notation for the objects in representation theory of Lie algebras in computations if Affine.m is used in the Mathematica notebook interface. Running time: From seconds to days depending on the rank of the algebra and the complexity of the representation.

The algebra of complex 2 × 2 matrices and a general closed Baker-Campbell-Hausdorff formula

NASA Astrophysics Data System (ADS)

Foulis, D. L.

2017-07-01

We derive a closed formula for the Baker-Campbell-Hausdorff series expansion in the case of complex 2×2 matrices. For arbitrary matrices A and B, and a matrix Z such that \\exp Z = \\exp A \\exp B , our result expresses Z as a linear combination of A and B, their commutator [A, B] , and the identity matrix I. The coefficients in this linear combination are functions of the traces and determinants of A and B, and the trace of their product. The derivation proceeds purely via algebraic manipulations of the given matrices and their products, making use of relations developed here, based on the Cayley-Hamilton theorem, as well as a characterization of the consequences of [A, B] and/or its determinant being zero or otherwise. As a corollary of our main result we also derive a closed formula for the Zassenhaus expansion. We apply our results to several special cases, most notably the parametrization of the product of two SU(2) matrices and a verification of the recent result of Van-Brunt and Visser (2015 J. Phys. A: Math. Theor. 48 225207) for complex 2×2 matrices, in this latter case deriving also the related Zassenhaus formula which turns out to be quite simple. We then show that this simple formula should be valid for all matrices and operators.

Spin Number Coherent States and the Problem of Two Coupled Oscillators

NASA Astrophysics Data System (ADS)

Ojeda-Guillén, D.; Mota, R. D.; Granados, V. D.

2015-07-01

From the definition of the standard Perelomov coherent states we introduce the Perelomov number coherent states for any su(2) Lie algebra. With the displacement operator we apply a similarity transformation to the su(2) generators and construct a new set of operators which also close the su(2) Lie algebra, being the Perelomov number coherent states the new basis for its unitary irreducible representation. We apply our results to obtain the energy spectrum, the eigenstates and the partition function of two coupled oscillators. We show that the eigenstates of two coupled oscillators are the SU(2) Perelomov number coherent states of the two-dimensional harmonic oscillator with an appropriate choice of the coherent state parameters. Supported by SNI-México, COFAA-IPN, EDD-IPN, EDI-IPN, SIP-IPN Project No. 20150935

Adinkra (in)equivalence from Coxeter group representations: A case study

NASA Astrophysics Data System (ADS)

Chappell, Isaac; Gates, S. James; Hübsch, T.

2014-02-01

Using a MathematicaTM code, we present a straightforward numerical analysis of the 384-dimensional solution space of signed permutation 4×4 matrices, which in sets of four, provide representations of the 𝒢ℛ(4, 4) algebra, closely related to the 𝒩 = 1 (simple) supersymmetry algebra in four-dimensional space-time. Following after ideas discussed in previous papers about automorphisms and classification of adinkras and corresponding supermultiplets, we make a new and alternative proposal to use equivalence classes of the (unsigned) permutation group S4 to define distinct representations of higher-dimensional spin bundles within the context of adinkras. For this purpose, the definition of a dual operator akin to the well-known Hodge star is found to partition the space of these 𝒢ℛ(4, 4) representations into three suggestive classes.

Correlates of gender and achievement in introductory algebra based physics

NASA Astrophysics Data System (ADS)

Smith, Rachel Clara

The field of physics is heavily male dominated in America. Thus, half of the population of our country is underrepresented and underserved. The identification of factors that contribute to gender disparity in physics is necessary for educators to address the individual needs of students, and, in particular, the separate and specific needs of female students. In an effort to determine if any correlations could be established or strengthened between sex, gender identity, social network, algebra skill, scientific reasoning ability, and/or student attitude, a study was performed on a group of 82 students in an introductory algebra based physics course. The subjects each filled out a survey at the beginning of the semester of their first semester of algebra based physics. They filled out another survey at the end of that same semester. These surveys included physics content pretests and posttests, as well as questions about the students' habits, attitudes, and social networks. Correlates of posttest score were identified, in order of significance, as pretest score, emphasis on conceptual learning, preference for male friends, number of siblings (negatively correlated), motivation in physics, algebra score, and parents' combined education level. Number of siblings was also found to negatively correlate with, in order of significance, gender identity, preference for male friends, emphasis on conceptual learning, and motivation in physics. Preference for male friends was found to correlate with, in order of significance, emphasis on conceptual learning, gender identity, and algebra score. Also, gender identity was found to correlate with emphasis on conceptual learning, the strongest predictor of posttest score other than pretest score.

Questions Revisited: A Close Examination of Calculus of Inference and Inquiry

NASA Technical Reports Server (NTRS)

Knuth, Kevin H.; Koga, Dennis (Technical Monitor)

2003-01-01

In this paper I examine more closely the way in which probability theory, the calculus of inference, is derived from the Boolean lattice structure of logical assertions ordered by implication. I demonstrate how the duality between the logical conjunction and disjunction in Boolean algebra is lost when deriving the probability calculus. In addition, I look more closely at the other lattice identities to verify that they are satisfied by the probability calculus. Last, I look towards developing the calculus of inquiry demonstrating that there is a sum and product rule for the relevance measure as well as a Bayes theorem. Current difficulties in deriving the complete inquiry calculus will also be discussed.

Logarithmic conformal field theory: beyond an introduction

NASA Astrophysics Data System (ADS)

Creutzig, Thomas; Ridout, David

2013-12-01

This article aims to review a selection of central topics and examples in logarithmic conformal field theory. It begins with the remarkable observation of Cardy that the horizontal crossing probability of critical percolation may be computed analytically within the formalism of boundary conformal field theory. Cardy’s derivation relies on certain implicit assumptions which are shown to lead inexorably to indecomposable modules and logarithmic singularities in correlators. For this, a short introduction to the fusion algorithm of Nahm, Gaberdiel and Kausch is provided. While the percolation logarithmic conformal field theory is still not completely understood, there are several examples for which the formalism familiar from rational conformal field theory, including bulk partition functions, correlation functions, modular transformations, fusion rules and the Verlinde formula, has been successfully generalized. This is illustrated for three examples: the singlet model \\mathfrak {M} (1,2), related to the triplet model \\mathfrak {W} (1,2), symplectic fermions and the fermionic bc ghost system; the fractional level Wess-Zumino-Witten model based on \\widehat{\\mathfrak {sl}} \\left( 2 \\right) at k=-\\frac{1}{2}, related to the bosonic βγ ghost system; and the Wess-Zumino-Witten model for the Lie supergroup \\mathsf {GL} \\left( 1 {\\mid} 1 \\right), related to \\mathsf {SL} \\left( 2 {\\mid} 1 \\right) at k=-\\frac{1}{2} and 1, the Bershadsky-Polyakov algebra W_3^{(2)} and the Feigin-Semikhatov algebras W_n^{(2)}. These examples have been chosen because they represent the most accessible, and most useful, members of the three best-understood families of logarithmic conformal field theories. The logarithmic minimal models \\mathfrak {W} (q,p), the fractional level Wess-Zumino-Witten models, and the Wess-Zumino-Witten models on Lie supergroups (excluding \\mathsf {OSP} \\left( 1 {\\mid} 2n \\right)). In this review, the emphasis lies on the representation theory of the underlying chiral algebra and the modular data pertaining to the characters of the representations. Each of the archetypal logarithmic conformal field theories is studied here by first determining its irreducible spectrum, which turns out to be continuous, as well as a selection of natural reducible, but indecomposable, modules. This is followed by a detailed description of how to obtain character formulae for each irreducible, a derivation of the action of the modular group on the characters, and an application of the Verlinde formula to compute the Grothendieck fusion rules. In each case, the (genuine) fusion rules are known, so comparisons can be made and favourable conclusions drawn. In addition, each example admits an infinite set of simple currents, hence extended symmetry algebras may be constructed and a series of bulk modular invariants computed. The spectrum of such an extended theory is typically discrete and this is how the triplet model \\mathfrak {W} (1,2) arises, for example. Moreover, simple current technology admits a derivation of the extended algebra fusion rules from those of its continuous parent theory. Finally, each example is concluded by a brief description of the computation of some bulk correlators, a discussion of the structure of the bulk state space, and remarks concerning more advanced developments and generalizations. The final part gives a very short account of the theory of staggered modules, the (simplest class of) representations that are responsible for the logarithmic singularities that distinguish logarithmic theories from their rational cousins. These modules are discussed in a generality suitable to encompass all the examples met in this review and some of the very basic structure theory is proven. Then, the important quantities known as logarithmic couplings are reviewed for Virasoro staggered modules and their role as fundamentally important parameters, akin to the three-point constants of rational conformal field theory, is discussed. An appendix is also provided in order to introduce some of the necessary, but perhaps unfamiliar, language of homological algebra.

Gauge symmetries of the free supersymmetric string field theories

NASA Astrophysics Data System (ADS)

Neveu, A.; West, P. C.

1985-12-01

The gauge covariant local formulations of the free supersymmetric strings that contained a finite number of supplementary fields are extended so as to place all the generators of the Ramond-Neveu-Schwarz algebra on a more equal footing. Permanent address: King's College, Mathematics Department, London WC2R 2LS, UK.

What Does Algebra Look Like in Early Childhood?

ERIC Educational Resources Information Center

Lee, Joohi; Collins, Denise; Melton, Janet

2016-01-01

How can educators encourage and better prepare students to pursue science, technology, engineering, and mathematics (STEM)-based fields? To start, students are more likely to pursue these fields if they enjoy and perceive themselves to be good at them. This means introducing relevant concepts and skills at an early age and embedding them…

Existence and construction of Galilean invariant z ≠2 theories

NASA Astrophysics Data System (ADS)

Grinstein, Benjamín; Pal, Sridip

2018-06-01

We prove a no-go theorem for the construction of a Galilean boost invariant and z ≠2 anisotropic scale invariant field theory with a finite dimensional basis of fields. Two point correlators in such theories, we show, grow unboundedly with spatial separation. Correlators of theories with an infinite dimensional basis of fields, for example, labeled by a continuous parameter, do not necessarily exhibit this bad behavior. Hence, such theories behave effectively as if in one extra dimension. Embedding the symmetry algebra into the conformal algebra of one higher dimension also reveals the existence of an internal continuous parameter. Consideration of isometries shows that the nonrelativistic holographic picture assumes a canonical form, where the bulk gravitational theory lives in a space-time with one extra dimension. This can be contrasted with the original proposal by Balasubramanian and McGreevy, and by Son, where the metric of a (d +2 )-dimensional space-time is proposed to be dual of a d -dimensional field theory. We provide explicit examples of theories living at fixed point with anisotropic scaling exponent z =2/ℓ ℓ+1 , ℓ∈Z .

DOE Office of Scientific and Technical Information (OSTI.GOV)

Carpenter, J.A.

This report is a sequel to ORNL/CSD-106 in the ongoing supplements to Professor A.S. Householder's KWIC Index for Numerical Algebra. Beginning with the previous supplement, the subject has been restricted to Numerical Linear Algebra, roughly characterized by the American Mathematical Society's classification sections 15 and 65F but with little coverage of infinite matrices, matrices over fields of characteristics other than zero, operator theory, optimization and those parts of matrix theory primarily combinatorial in nature. Some consideration is given to the uses of graph theory in Numerical Linear Algebra, particularly with respect to algorithms for sparse matrix computations. The period coveredmore » by this report is roughly the calendar year 1982 as measured by the appearance of the articles in the American Mathematical Society's Contents of Mathematical Publications lagging actual appearance dates by up to nearly half a year. The review citations are limited to the Mathematical Reviews (MR).« less

Horizon fluffs: In the context of generalized minimal massive gravity

NASA Astrophysics Data System (ADS)

Setare, Mohammad Reza; Adami, Hamed

2018-02-01

We consider a metric which describes Bañados geometries and show that the considered metric is a solution of the generalized minimal massive gravity (GMMG) model. We consider the Killing vector field which preserves the form of the considered metric. Using the off-shell quasi-local approach we obtain the asymptotic conserved charges of the given solution. Similar to the Einstein gravity in the presence of negative cosmological constant, for the GMMG model, we also show that the algebra among the asymptotic conserved charges is isomorphic to two copies of the Virasoro algebra. Eventually, we find a relation between the algebra of the near-horizon and the asymptotic conserved charges. This relation shows that the main part of the horizon fluffs proposed by Afshar et al., Sheikh-Jabbari and Yavartanoo appear for generic black holes in the class of Bañados geometries in the context of the GMMG model.

Bi-orthogonal approach to non-Hermitian Hamiltonians with the oscillator spectrum: Generalized coherent states for nonlinear algebras

NASA Astrophysics Data System (ADS)

Rosas-Ortiz, Oscar; Zelaya, Kevin

2018-01-01

A set of Hamiltonians that are not self-adjoint but have the spectrum of the harmonic oscillator is studied. The eigenvectors of these operators and those of their Hermitian conjugates form a bi-orthogonal system that provides a mathematical procedure to satisfy the superposition principle. In this form the non-Hermitian oscillators can be studied in much the same way as in the Hermitian approaches. Two different nonlinear algebras generated by properly constructed ladder operators are found and the corresponding generalized coherent states are obtained. The non-Hermitian oscillators can be steered to the conventional one by the appropriate selection of parameters. In such limit, the generators of the nonlinear algebras converge to generalized ladder operators that would represent either intensity-dependent interactions or multi-photon processes if the oscillator is associated with single mode photon fields in nonlinear media.

Force-field parameters from the SAFT-γ equation of state for use in coarse-grained molecular simulations.

PubMed

Müller, Erich A; Jackson, George

2014-01-01

A description of fluid systems with molecular-based algebraic equations of state (EoSs) and by direct molecular simulation is common practice in chemical engineering and the physical sciences, but the two approaches are rarely closely coupled. The key for an integrated representation is through a well-defined force field and Hamiltonian at the molecular level. In developing coarse-grained intermolecular potential functions for the fluid state, one typically starts with a detailed, bottom-up quantum-mechanical or atomic-level description and then integrates out the unwanted degrees of freedom using a variety of techniques; an iterative heuristic simulation procedure is then used to refine the parameters of the model. By contrast, with a top-down technique, one can use an accurate EoS to link the macroscopic properties of the fluid and the force-field parameters. We discuss the latest developments in a top-down representation of fluids, with a particular focus on a group-contribution formulation of the statistical associating fluid theory (SAFT-γ). The accurate SAFT-γ EoS is used to estimate the parameters of the Mie force field, which can then be used with confidence in direct molecular simulations to obtain thermodynamic, structural, interfacial, and dynamical properties that are otherwise inaccessible from the EoS. This is exemplified for several prototypical fluids and mixtures, including carbon dioxide, hydrocarbons, perfluorohydrocarbons, and aqueous surfactants.

Control of nonlinear systems with applications to constrained robots and spacecraft attitude stabilization

NASA Technical Reports Server (NTRS)

Krishnan, Hariharan

1993-01-01

This thesis is organized in two parts. In Part 1, control systems described by a class of nonlinear differential and algebraic equations are introduced. A procedure for local stabilization based on a local state realization is developed. An alternative approach to local stabilization is developed based on a classical linearization of the nonlinear differential-algebraic equations. A theoretical framework is established for solving a tracking problem associated with the differential-algebraic system. First, a simple procedure is developed for the design of a feedback control law which ensures, at least locally, that the tracking error in the closed loop system lies within any given bound if the reference inputs are sufficiently slowly varying. Next, by imposing additional assumptions, a procedure is developed for the design of a feedback control law which ensures that the tracking error in the closed loop system approaches zero exponentially for reference inputs which are not necessarily slowly varying. The control design methodologies are used for simultaneous force and position control in constrained robot systems. The differential-algebraic equations are shown to characterize the slow dynamics of a certain nonlinear control system in nonstandard singularly perturbed form. In Part 2, the attitude stabilization (reorientation) of a rigid spacecraft using only two control torques is considered. First, the case of momentum wheel actuators is considered. The complete spacecraft dynamics are not controllable. However, the spacecraft dynamics are small time locally controllable in a reduced sense. The reduced spacecraft dynamics cannot be asymptotically stabilized using continuous feedback, but a discontinuous feedback control strategy is constructed. Next, the case of gas jet actuators is considered. If the uncontrolled principal axis is not an axis of symmetry, the complete spacecraft dynamics are small time locally controllable. However, the spacecraft attitude cannot be asymptotically stabilized using continuous feedback, but a discontinuous stabilizing feedback control strategy is constructed. If the uncontrolled principal axis is an axis of symmetry, the complete spacecraft dynamics cannot be stabilized. However, the spacecraft dynamics are small time locally controllable in a reduced sense. The reduced spacecraft dynamics cannot be asymptotically stabilized using continuous feedback, but again a discontinuous feedback control strategy is constructed.

Anomaly-corrected supersymmetry algebra and supersymmetric holographic renormalization

NASA Astrophysics Data System (ADS)

An, Ok Song

2017-12-01

We present a systematic approach to supersymmetric holographic renormalization for a generic 5D N=2 gauged supergravity theory with matter multiplets, including its fermionic sector, with all gauge fields consistently set to zero. We determine the complete set of supersymmetric local boundary counterterms, including the finite counterterms that parameterize the choice of supersymmetric renormalization scheme. This allows us to derive holographically the superconformal Ward identities of a 4D superconformal field theory on a generic background, including the Weyl and super-Weyl anomalies. Moreover, we show that these anomalies satisfy the Wess-Zumino consistency condition. The super-Weyl anomaly implies that the fermionic operators of the dual field theory, such as the supercurrent, do not transform as tensors under rigid supersymmetry on backgrounds that admit a conformal Killing spinor, and their anticommutator with the conserved supercharge contains anomalous terms. This property is explicitly checked for a toy model. Finally, using the anomalous transformation of the supercurrent, we obtain the anomaly-corrected supersymmetry algebra on curved backgrounds admitting a conformal Killing spinor.

A Unified Model of Phantom Energy and Dark Matter

NASA Astrophysics Data System (ADS)

Chaves, Max; Singleton, Douglas

2008-01-01

To explain the acceleration of the cosmological expansion researchers have considered an unusual form of mass-energy generically called dark energy. Dark energy has a ratio of pressure over mass density which obeys w = p/ρ < -1/3. This form of mass-energy leads to accelerated expansion. An extreme form of dark energy, called phantom energy, has been proposed which has w = p/ρ < -1. This possibility is favored by the observational data. The simplest model for phantom energy involves the introduction of a scalar field with a negative kinetic energy term. Here we show that theories based on graded Lie algebras naturally have such a negative kinetic energy and thus give a model for phantom energy in a less ad hoc manner. We find that the model also contains ordinary scalar fields and anti-commuting (Grassmann) vector fields which act as a form of two component dark matter. Thus from a gauge theory based o! n a graded algebra we naturally obtained both phantom energy and dark matter.

Gauge choices and entanglement entropy of two dimensional lattice gauge fields

NASA Astrophysics Data System (ADS)

Yang, Zhi; Hung, Ling-Yan

2018-03-01

In this paper, we explore the question of how different gauge choices in a gauge theory affect the tensor product structure of the Hilbert space in configuration space. In particular, we study the Coulomb gauge and observe that the naive gauge potential degrees of freedom cease to be local operators as soon as we impose the Dirac brackets. We construct new local set of operators and compute the entanglement entropy according to this algebra in 2 + 1 dimensions. We find that our proposal would lead to an entanglement entropy that behave very similar to a single scalar degree of freedom if we do not include further centers, but approaches that of a gauge field if we include non-trivial centers. We explore also the situation where the gauge field is Higgsed, and construct a local operator algebra that again requires some deformation. This should give us some insight into interpreting the entanglement entropy in generic gauge theories and perhaps also in gravitational theories.

BV Quantization of the Rozansky-Witten Model

NASA Astrophysics Data System (ADS)

Chan, Kwokwai; Leung, Naichung Conan; Li, Qin

2017-10-01

We investigate the perturbative aspects of Rozansky-Witten's 3d {σ}-model (Rozansky and Witten in Sel Math 3(3):401-458, 1997) using Costello's approach to the Batalin-Vilkovisky (BV) formalism (Costello in Renormalization and effective field theory, American Mathematical Society, Providence, 2011). We show that the BV quantization (in Costello's sense) of the model, which produces a perturbative quantum field theory, can be obtained via the configuration space method of regularization due to Kontsevich (First European congress of mathematics, Paris, 1992) and Axelrod-Singer (J Differ Geom 39(1):173-213, 1994). We also study the factorization algebra structure of quantum observables following Costello-Gwilliam (Factorization algebras in quantum field theory, Cambridge University Press, Cambridge 2017). In particular, we show that the cohomology of local quantum observables on a genus g handle body is given by {H^*(X, (\\wedge^*T_X)^{⊗ g})} (where X is the target manifold), and we prove that the partition function reproduces the Rozansky-Witten invariants.

Polarization ellipse and Stokes parameters in geometric algebra.

PubMed

Santos, Adler G; Sugon, Quirino M; McNamara, Daniel J

2012-01-01

In this paper, we use geometric algebra to describe the polarization ellipse and Stokes parameters. We show that a solution to Maxwell's equation is a product of a complex basis vector in Jackson and a linear combination of plane wave functions. We convert both the amplitudes and the wave function arguments from complex scalars to complex vectors. This conversion allows us to separate the electric field vector and the imaginary magnetic field vector, because exponentials of imaginary scalars convert vectors to imaginary vectors and vice versa, while exponentials of imaginary vectors only rotate the vector or imaginary vector they are multiplied to. We convert this expression for polarized light into two other representations: the Cartesian representation and the rotated ellipse representation. We compute the conversion relations among the representation parameters and their corresponding Stokes parameters. And finally, we propose a set of geometric relations between the electric and magnetic fields that satisfy an equation similar to the Poincaré sphere equation.

When Traditional Won't Do: Experiences from a "Lower-Level" Mathematics Classroom

ERIC Educational Resources Information Center

Hill, Crystal

2010-01-01

As the last bell rings, students scurry to their respective classrooms, doors begin to close, and the class period begins. Imagine that you are in the hallway of the school and you look into an advanced mathematics class, into an Algebra I, Part I, mathematics class (a course designed for students who have not found success in mathematics). What…

Teaching Algebra-Based Concepts to Students with Learning Disabilities: The Effects of Preteaching Using a Gradual Instructional Sequence

ERIC Educational Resources Information Center

Watt, Sarah Jean

2013-01-01

Research to identify validated instructional approaches to teach math to students with LD and those at-risk for failure in both core and supplemental instructional settings is necessary to assist teachers in closing the achievement gaps that exist across the country. The concrete-to-representational-to-abstract instructional sequence (CRA) has…

Geometrical Meaning of Arithmetic Series [Image Omitted], [Image Omitted] and [Image Omitted] in Terms of the Elementary Combinatorics

ERIC Educational Resources Information Center

Kobayashi, Yukio

2011-01-01

The formula [image omitted] is closely related to combinatorics through an elementary geometric exercise. This approach can be expanded to the formulas [image omitted], [image omitted] and [image omitted]. These formulas are also nice examples of showing two approaches, one algebraic and one combinatoric, to a problem of counting. (Contains 6…

Free vibration of functionally graded beams and frameworks using the dynamic stiffness method

NASA Astrophysics Data System (ADS)

Banerjee, J. R.; Ananthapuvirajah, A.

2018-05-01

The free vibration analysis of functionally graded beams (FGBs) and frameworks containing FGBs is carried out by applying the dynamic stiffness method and deriving the elements of the dynamic stiffness matrix in explicit algebraic form. The usually adopted rule that the material properties of the FGB vary continuously through the thickness according to a power law forms the fundamental basis of the governing differential equations of motion in free vibration. The differential equations are solved in closed analytical form when the free vibratory motion is harmonic. The dynamic stiffness matrix is then formulated by relating the amplitudes of forces to those of the displacements at the two ends of the beam. Next, the explicit algebraic expressions for the dynamic stiffness elements are derived with the help of symbolic computation. Finally the Wittrick-Williams algorithm is applied as solution technique to solve the free vibration problems of FGBs with uniform cross-section, stepped FGBs and frameworks consisting of FGBs. Some numerical results are validated against published results, but in the absence of published results for frameworks containing FGBs, consistency checks on the reliability of results are performed. The paper closes with discussion of results and conclusions.

Zooming in on AdS3/CFT2 near a BPS bound

NASA Astrophysics Data System (ADS)

Hartong, Jelle; Lei, Yang; Obers, Niels; Oling, Gerben

2018-05-01

Any ( d + 1)-dimensional CFT with a U(1) flavor symmetry, a BPS bound and an exactly marginal coupling admits a decoupling limit in which one zooms in on the spectrum close to the bound. This limit is an Inönü-Wigner contraction of so(2 , d+1)⊕ u(1) that leads to a relativistic algebra with a scaling generator but no conformal generators. In 2D CFTs, Lorentz boosts are abelian and by adding a second u(1) we find a contraction of two copies of sl(2, ℝ) ⊕ u(1) to two copies of P 2 c , the 2-dimensional centrally extended Poincaré algebra. We show that the bulk is described by a novel non-Lorentzian geometry that we refer to as pseudo-Newton-Cartan geometry. Both the Chern-Simons action on sl(2, ℝ) ⊕ u(1) and the entire phase space of asymptotically AdS3 spacetimes are well-behaved in the corresponding limit if we fix the radial component for the u(1) connection. With this choice, the resulting Newton-Cartan foliation structure is now associated not with time, but with the emerging holographic direction. Since the leaves of this foliation do not mix, the emergence of the holographic direction is much simpler than in AdS3 holography. Furthermore, we show that the asymptotic symmetry algebra of the limit theory consists of a left- and a right-moving warped Virasoro algebra.

Extended Riemannian geometry II: local heterotic double field theory

NASA Astrophysics Data System (ADS)

Deser, Andreas; Heller, Marc Andre; Sämann, Christian

2018-04-01

We continue our exploration of local Double Field Theory (DFT) in terms of symplectic graded manifolds carrying compatible derivations and study the case of heterotic DFT. We start by developing in detail the differential graded manifold that captures heterotic Generalized Geometry which leads to new observations on the generalized metric and its twists. We then give a symplectic pre-N Q-manifold that captures the symmetries and the geometry of local heterotic DFT. We derive a weakened form of the section condition, which arises algebraically from consistency of the symmetry Lie 2-algebra and its action on extended tensors. We also give appropriate notions of twists — which are required for global formulations — and of the torsion and Riemann tensors. Finally, we show how the observed α'-corrections are interpreted naturally in our framework.

Newton-Cartan gravity and torsion

NASA Astrophysics Data System (ADS)

Bergshoeff, Eric; Chatzistavrakidis, Athanasios; Romano, Luca; Rosseel, Jan

2017-10-01

We compare the gauging of the Bargmann algebra, for the case of arbitrary torsion, with the result that one obtains from a null-reduction of General Relativity. Whereas the two procedures lead to the same result for Newton-Cartan geometry with arbitrary torsion, the null-reduction of the Einstein equations necessarily leads to Newton-Cartan gravity with zero torsion. We show, for three space-time dimensions, how Newton-Cartan gravity with arbitrary torsion can be obtained by starting from a Schrödinger field theory with dynamical exponent z = 2 for a complex compensating scalar and next coupling this field theory to a z = 2 Schrödinger geometry with arbitrary torsion. The latter theory can be obtained from either a gauging of the Schrödinger algebra, for arbitrary torsion, or from a null-reduction of conformal gravity.

Modular constraints on conformal field theories with currents

NASA Astrophysics Data System (ADS)

Bae, Jin-Beom; Lee, Sungjay; Song, Jaewon

2017-12-01

We study constraints coming from the modular invariance of the partition function of two-dimensional conformal field theories. We constrain the spectrum of CFTs in the presence of holomorphic and anti-holomorphic currents using the semi-definite programming. In particular, we find the bounds on the twist gap for the non-current primaries depend dramatically on the presence of holomorphic currents, showing numerous kinks and peaks. Various rational CFTs are realized at the numerical boundary of the twist gap, saturating the upper limits on the degeneracies. Such theories include Wess-Zumino-Witten models for the Deligne's exceptional series, the Monster CFT and the Baby Monster CFT. We also study modular constraints imposed by W -algebras of various type and observe that the bounds on the gap depend on the choice of W -algebra in the small central charge region.

Development of an algebraic stress/two-layer model for calculating thrust chamber flow fields

NASA Technical Reports Server (NTRS)

Chen, C. P.; Shang, H. M.; Huang, J.

1993-01-01

Following the consensus of a workshop in Turbulence Modeling for Liquid Rocket Thrust Chambers, the current effort was undertaken to study the effects of second-order closure on the predictions of thermochemical flow fields. To reduce the instability and computational intensity of the full second-order Reynolds Stress Model, an Algebraic Stress Model (ASM) coupled with a two-layer near wall treatment was developed. Various test problems, including the compressible boundary layer with adiabatic and cooled walls, recirculating flows, swirling flows and the entire SSME nozzle flow were studied to assess the performance of the current model. Detailed calculations for the SSME exit wall flow around the nozzle manifold were executed. As to the overall flow predictions, the ASM removes another assumption for appropriate comparison with experimental data, to account for the non-isotropic turbulence effects.

Inverse Scattering and Local Observable Algebras in Integrable Quantum Field Theories

NASA Astrophysics Data System (ADS)

Alazzawi, Sabina; Lechner, Gandalf

2017-09-01

We present a solution method for the inverse scattering problem for integrable two-dimensional relativistic quantum field theories, specified in terms of a given massive single particle spectrum and a factorizing S-matrix. An arbitrary number of massive particles transforming under an arbitrary compact global gauge group is allowed, thereby generalizing previous constructions of scalar theories. The two-particle S-matrix S is assumed to be an analytic solution of the Yang-Baxter equation with standard properties, including unitarity, TCP invariance, and crossing symmetry. Using methods from operator algebras and complex analysis, we identify sufficient criteria on S that imply the solution of the inverse scattering problem. These conditions are shown to be satisfied in particular by so-called diagonal S-matrices, but presumably also in other cases such as the O( N)-invariant nonlinear {σ}-models.

Split Octonion Reformulation for Electromagnetic Chiral Media of Massive Dyons

NASA Astrophysics Data System (ADS)

Chanyal, B. C.

2017-12-01

In an explicit, unified, and covariant formulation of an octonion algebra, we study and generalize the electromagnetic chiral fields equations of massive dyons with the split octonionic representation. Starting with 2×2 Zorn’s vector matrix realization of split-octonion and its dual Euclidean spaces, we represent the unified structure of split octonionic electric and magnetic induction vectors for chiral media. As such, in present paper, we describe the chiral parameter and pairing constants in terms of split octonionic matrix representation of Drude-Born-Fedorov constitutive relations. We have expressed a split octonionic electromagnetic field vector for chiral media, which exhibits the unified field structure of electric and magnetic chiral fields of dyons. The beauty of split octonionic representation of Zorn vector matrix realization is that, the every scalar and vector components have its own meaning in the generalized chiral electromagnetism of dyons. Correspondingly, we obtained the alternative form of generalized Proca-Maxwell’s equations of massive dyons in chiral media. Furthermore, the continuity equations, Poynting theorem and wave propagation for generalized electromagnetic fields of chiral media of massive dyons are established by split octonionic form of Zorn vector matrix algebra.

Static Analysis of Large-Scale Multibody System Using Joint Coordinates and Spatial Algebra Operator

PubMed Central

Omar, Mohamed A.

2014-01-01

Initial transient oscillations inhibited in the dynamic simulations responses of multibody systems can lead to inaccurate results, unrealistic load prediction, or simulation failure. These transients could result from incompatible initial conditions, initial constraints violation, and inadequate kinematic assembly. Performing static equilibrium analysis before the dynamic simulation can eliminate these transients and lead to stable simulation. Most exiting multibody formulations determine the static equilibrium position by minimizing the system potential energy. This paper presents a new general purpose approach for solving the static equilibrium in large-scale articulated multibody. The proposed approach introduces an energy drainage mechanism based on Baumgarte constraint stabilization approach to determine the static equilibrium position. The spatial algebra operator is used to express the kinematic and dynamic equations of the closed-loop multibody system. The proposed multibody system formulation utilizes the joint coordinates and modal elastic coordinates as the system generalized coordinates. The recursive nonlinear equations of motion are formulated using the Cartesian coordinates and the joint coordinates to form an augmented set of differential algebraic equations. Then system connectivity matrix is derived from the system topological relations and used to project the Cartesian quantities into the joint subspace leading to minimum set of differential equations. PMID:25045732

A Procedure for Deriving Formulas to Convert Transition Rates to Probabilities for Multistate Markov Models.

PubMed

Jones, Edmund; Epstein, David; García-Mochón, Leticia

2017-10-01

For health-economic analyses that use multistate Markov models, it is often necessary to convert from transition rates to transition probabilities, and for probabilistic sensitivity analysis and other purposes it is useful to have explicit algebraic formulas for these conversions, to avoid having to resort to numerical methods. However, if there are four or more states then the formulas can be extremely complicated. These calculations can be made using packages such as R, but many analysts and other stakeholders still prefer to use spreadsheets for these decision models. We describe a procedure for deriving formulas that use intermediate variables so that each individual formula is reasonably simple. Once the formulas have been derived, the calculations can be performed in Excel or similar software. The procedure is illustrated by several examples and we discuss how to use a computer algebra system to assist with it. The procedure works in a wide variety of scenarios but cannot be employed when there are several backward transitions and the characteristic equation has no algebraic solution, or when the eigenvalues of the transition rate matrix are very close to each other.

Static analysis of large-scale multibody system using joint coordinates and spatial algebra operator.

PubMed

Omar, Mohamed A

2014-01-01

Initial transient oscillations inhibited in the dynamic simulations responses of multibody systems can lead to inaccurate results, unrealistic load prediction, or simulation failure. These transients could result from incompatible initial conditions, initial constraints violation, and inadequate kinematic assembly. Performing static equilibrium analysis before the dynamic simulation can eliminate these transients and lead to stable simulation. Most exiting multibody formulations determine the static equilibrium position by minimizing the system potential energy. This paper presents a new general purpose approach for solving the static equilibrium in large-scale articulated multibody. The proposed approach introduces an energy drainage mechanism based on Baumgarte constraint stabilization approach to determine the static equilibrium position. The spatial algebra operator is used to express the kinematic and dynamic equations of the closed-loop multibody system. The proposed multibody system formulation utilizes the joint coordinates and modal elastic coordinates as the system generalized coordinates. The recursive nonlinear equations of motion are formulated using the Cartesian coordinates and the joint coordinates to form an augmented set of differential algebraic equations. Then system connectivity matrix is derived from the system topological relations and used to project the Cartesian quantities into the joint subspace leading to minimum set of differential equations.

Solution of Inverse Kinematics for 6R Robot Manipulators With Offset Wrist Based on Geometric Algebra.

PubMed

Fu, Zhongtao; Yang, Wenyu; Yang, Zhen

2013-08-01

In this paper, we present an efficient method based on geometric algebra for computing the solutions to the inverse kinematics problem (IKP) of the 6R robot manipulators with offset wrist. Due to the fact that there exist some difficulties to solve the inverse kinematics problem when the kinematics equations are complex, highly nonlinear, coupled and multiple solutions in terms of these robot manipulators stated mathematically, we apply the theory of Geometric Algebra to the kinematic modeling of 6R robot manipulators simply and generate closed-form kinematics equations, reformulate the problem as a generalized eigenvalue problem with symbolic elimination technique, and then yield 16 solutions. Finally, a spray painting robot, which conforms to the type of robot manipulators, is used as an example of implementation for the effectiveness and real-time of this method. The experimental results show that this method has a large advantage over the classical methods on geometric intuition, computation and real-time, and can be directly extended to all serial robot manipulators and completely automatized, which provides a new tool on the analysis and application of general robot manipulators.

Dual number algebra method for Green's function derivatives in 3D magneto-electro-elasticity

NASA Astrophysics Data System (ADS)

Dziatkiewicz, Grzegorz

2018-01-01

The Green functions are the basic elements of the boundary element method. To obtain the boundary integral formulation the Green function and its derivative should be known for the considered differential operator. Today the interesting group of materials are electronic composites. The special case of the electronic composite is the magnetoelectroelastic continuum. The mentioned continuum is a model of the piezoelectric-piezomagnetic composites. The anisotropy of their physical properties makes the problem of Green's function determination very difficult. For that reason Green's functions for the magnetoelectroelastic continuum are not known in the closed form and numerical methods should be applied to determine such Green's functions. These means that the problem of the accurate and simply determination of Green's function derivatives is even harder. Therefore in the present work the dual number algebra method is applied to calculate numerically the derivatives of 3D Green's functions for the magnetoelectroelastic materials. The introduced method is independent on the step size and it can be treated as a special case of the automatic differentiation method. Therefore, the dual number algebra method can be applied as a tool for checking the accuracy of the well-known finite difference schemes.

Generalizations of the classical Yang-Baxter equation and O-operators

NASA Astrophysics Data System (ADS)

Bai, Chengming; Guo, Li; Ni, Xiang

2011-06-01

Tensor solutions (r-matrices) of the classical Yang-Baxter equation (CYBE) in a Lie algebra, obtained as the classical limit of the R-matrix solution of the quantum Yang-Baxter equation, is an important structure appearing in different areas such as integrable systems, symplectic geometry, quantum groups, and quantum field theory. Further study of CYBE led to its interpretation as certain operators, giving rise to the concept of {O}-operators. The O-operators were in turn interpreted as tensor solutions of CYBE by enlarging the Lie algebra [Bai, C., "A unified algebraic approach to the classical Yang-Baxter equation," J. Phys. A: Math. Theor. 40, 11073 (2007)], 10.1088/1751-8113/40/36/007. The purpose of this paper is to extend this study to a more general class of operators that were recently introduced [Bai, C., Guo, L., and Ni, X., "Nonabelian generalized Lax pairs, the classical Yang-Baxter equation and PostLie algebras," Commun. Math. Phys. 297, 553 (2010)], 10.1007/s00220-010-0998-7 in the study of Lax pairs in integrable systems. Relations between O-operators, relative differential operators, and Rota-Baxter operators are also discussed.

Learning to teach upper primary school algebra: changes to teachers' mathematical knowledge for teaching functional thinking

NASA Astrophysics Data System (ADS)

Wilkie, Karina J.

2016-06-01

A key aspect of learning algebra in the middle years of schooling is exploring the functional relationship between two variables: noticing and generalising the relationship, and expressing it mathematically. This article describes research on the professional learning of upper primary school teachers for developing their students' functional thinking through pattern generalisation. This aspect of algebra learning has been explicitly brought to the attention of upper primary teachers in the recently introduced Australian curriculum. Ten practising teachers participated over 1 year in a design-based research project involving a sequence of geometric pattern generalisation lessons with their classes. Initial and final survey responses and teachers' interactions in regular meetings and lessons were analysed from cognitive and situated perspectives on professional learning, using a theoretical model for the different types of knowledge needed for teaching mathematics. The teachers demonstrated an increase in certain aspects of their mathematical knowledge for teaching algebra as well as some residual issues. Implications for the professional learning of practising and pre-service teachers to develop their mathematics knowledge for teaching functional thinking, and challenges with operationalising knowledge categories for field-based research are presented.

Continuum limit and symmetries of the periodic gℓ(1|1) spin chain

NASA Astrophysics Data System (ADS)

Gainutdinov, A. M.; Read, N.; Saleur, H.

2013-06-01

This paper is the first in a series devoted to the study of logarithmic conformal field theories (LCFT) in the bulk. Building on earlier work in the boundary case, our general strategy consists in analyzing the algebraic properties of lattice regularizations (quantum spin chains) of these theories. In the boundary case, a crucial step was the identification of the space of states as a bimodule over the Temperley-Lieb (TL) algebra and the quantum group Uqsℓ(2). The extension of this analysis in the bulk case involves considerable difficulties, since the Uqsℓ(2) symmetry is partly lost, while the TL algebra is replaced by a much richer version (the Jones-Temperley-Lieb — JTL — algebra). Even the simplest case of the gℓ(1|1) spin chain — corresponding to the c=-2 symplectic fermions theory in the continuum limit — presents very rich aspects, which we will discuss in several papers. In this first work, we focus on the symmetries of the spin chain, that is, the centralizer of the JTL algebra in the alternating tensor product of the gℓ(1|1) fundamental representation and its dual. We prove that this centralizer is only a subalgebra of Uqsℓ(2) at q=i that we dub Uqoddsℓ(2). We then begin the analysis of the continuum limit of the JTL algebra: using general arguments about the regularization of the stress-energy tensor, we identify families of JTL elements going over to the Virasoro generators Ln,L in the continuum limit. We then discuss the sℓ(2) symmetry of the (continuum limit) symplectic fermions theory from the lattice and JTL point of view. The analysis of the spin chain as a bimodule over Uqoddsℓ(2) and JTLN is discussed in the second paper of this series.

The link between middle school mathematics course placement and achievement.

PubMed

Domina, Thurston

2014-01-01

The proportion of eighth graders in United States public schools enrolled in algebra or a more advanced mathematics course doubled between 1990 and 2011. This article uses Early Childhood Longitudinal Study's Kindergarten Cohort data to consider the selection process into advanced middle school mathematics courses and estimate the effects of advanced courses on students' mathematics achievement (n = 6,425; mean age at eighth grade = 13.7). Eighth-grade algebra and geometry course placements are academically selective, but considerable between-school variation exists in students' odds of taking these advanced courses. While analyses indicate that advanced middle school mathematics courses boost student achievement, these effects are most pronounced in content areas closely related to class content and may be contingent on student academic readiness. © 2014 The Author. Child Development © 2014 Society for Research in Child Development, Inc.

Group theoretical quantization of isotropic loop cosmology

NASA Astrophysics Data System (ADS)

Livine, Etera R.; Martín-Benito, Mercedes

2012-06-01

We achieve a group theoretical quantization of the flat Friedmann-Robertson-Walker model coupled to a massless scalar field adopting the improved dynamics of loop quantum cosmology. Deparemetrizing the system using the scalar field as internal time, we first identify a complete set of phase space observables whose Poisson algebra is isomorphic to the su(1,1) Lie algebra. It is generated by the volume observable and the Hamiltonian. These observables describe faithfully the regularized phase space underlying the loop quantization: they account for the polymerization of the variable conjugate to the volume and for the existence of a kinematical nonvanishing minimum volume. Since the Hamiltonian is an element in the su(1,1) Lie algebra, the dynamics is now implemented as SU(1, 1) transformations. At the quantum level, the system is quantized as a timelike irreducible representation of the group SU(1, 1). These representations are labeled by a half-integer spin, which gives the minimal volume. They provide superselection sectors without quantization anomalies and no factor ordering ambiguity arises when representing the Hamiltonian. We then explicitly construct SU(1, 1) coherent states to study the quantum evolution. They not only provide semiclassical states but truly dynamical coherent states. Their use further clarifies the nature of the bounce that resolves the big bang singularity.

Saturation of the magnetorotational instability at large Elsasser number

NASA Astrophysics Data System (ADS)

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

2008-09-01

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

Quantum Field Theory on Spacetimes with a Compactly Generated Cauchy Horizon

NASA Astrophysics Data System (ADS)

Kay, Bernard S.; Radzikowski, Marek J.; Wald, Robert M.

1997-02-01

We prove two theorems which concern difficulties in the formulation of the quantum theory of a linear scalar field on a spacetime, (M,g_{ab}), with a compactly generated Cauchy horizon. These theorems demonstrate the breakdown of the theory at certain base points of the Cauchy horizon, which are defined as 'past terminal accumulation points' of the horizon generators. Thus, the theorems may be interpreted as giving support to Hawking's 'Chronology Protection Conjecture', according to which the laws of physics prevent one from manufacturing a 'time machine'. Specifically, we prove: Theorem 1. There is no extension to (M,g_{ab}) of the usual field algebra on the initial globally hyperbolic region which satisfies the condition of F-locality at any base point. In other words, any extension of the field algebra must, in any globally hyperbolic neighbourhood of any base point, differ from the algebra one would define on that neighbourhood according to the rules for globally hyperbolic spacetimes. Theorem 2. The two-point distribution for any Hadamard state defined on the initial globally hyperbolic region must (when extended to a distributional bisolution of the covariant Klein-Gordon equation on the full spacetime) be singular at every base point x in the sense that the difference between this two point distribution and a local Hadamard distribution cannot be given by a bounded function in any neighbourhood (in M 2 M) of (x,x). In consequence of Theorem 2, quantities such as the renormalized expectation value of J2 or of the stress-energy tensor are necessarily ill-defined or singular at any base point. The proof of these theorems relies on the 'Propagation of Singularities' theorems of Duistermaat and Hörmander.

Quarks, Symmetries and Strings - a Symposium in Honor of Bunji Sakita's 60th Birthday

NASA Astrophysics Data System (ADS)

Kaku, M.; Jevicki, A.; Kikkawa, K.

1991-04-01

The Table of Contents for the full book PDF is as follows: * Preface * Evening Banquet Speech * I. Quarks and Phenomenology * From the SU(6) Model to Uniqueness in the Standard Model * A Model for Higgs Mechanism in the Standard Model * Quark Mass Generation in QCD * Neutrino Masses in the Standard Model * Solar Neutrino Puzzle, Horizontal Symmetry of Electroweak Interactions and Fermion Mass Hierarchies * State of Chiral Symmetry Breaking at High Temperatures * Approximate |ΔI| = 1/2 Rule from a Perspective of Light-Cone Frame Physics * Positronium (and Some Other Systems) in a Strong Magnetic Field * Bosonic Technicolor and the Flavor Problem * II. Strings * Supersymmetry in String Theory * Collective Field Theory and Schwinger-Dyson Equations in Matrix Models * Non-Perturbative String Theory * The Structure of Non-Perturbative Quantum Gravity in One and Two Dimensions * Noncritical Virasoro Algebra of d < 1 Matrix Model and Quantized String Field * Chaos in Matrix Models ? * On the Non-Commutative Symmetry of Quantum Gravity in Two Dimensions * Matrix Model Formulation of String Field Theory in One Dimension * Geometry of the N = 2 String Theory * Modular Invariance form Gauge Invariance in the Non-Polynomial String Field Theory * Stringy Symmetry and Off-Shell Ward Identities * q-Virasoro Algebra and q-Strings * Self-Tuning Fields and Resonant Correlations in 2d-Gravity * III. Field Theory Methods * Linear Momentum and Angular Momentum in Quaternionic Quantum Mechanics * Some Comments on Real Clifford Algebras * On the Quantum Group p-adics Connection * Gravitational Instantons Revisited * A Generalized BBGKY Hierarchy from the Classical Path-Integral * A Quantum Generated Symmetry: Group-Level Duality in Conformal and Topological Field Theory * Gauge Symmetries in Extended Objects * Hidden BRST Symmetry and Collective Coordinates * Towards Stochastically Quantizing Topological Actions * IV. Statistical Methods * A Brief Summary of the s-Channel Theory of Superconductivity * Neural Networks and Models for the Brain * Relativistic One-Body Equations for Planar Particles with Arbitrary Spin * Chiral Property of Quarks and Hadron Spectrum in Lattice QCD * Scalar Lattice QCD * Semi-Superconductivity of a Charged Anyon Gas * Two-Fermion Theory of Strongly Correlated Electrons and Charge-Spin Separation * Statistical Mechanics and Error-Correcting Codes * Quantum Statistics

Star products on graded manifolds and α′-corrections to Courant algebroids from string theory

DOE Office of Scientific and Technical Information (OSTI.GOV)

Deser, Andreas, E-mail: andreas.deser@itp.uni-hannover.de

2015-09-15

Courant algebroids, originally used to study integrability conditions for Dirac structures, have turned out to be of central importance to study the effective supergravity limit of string theory. The search for a geometric description of T-duality leads to Double Field Theory (DFT), whose gauge algebra is governed by the C-bracket, a generalization of the Courant bracket in the sense that it reduces to the latter by solving a specific constraint. Recently, in DFT deformations of the C-bracket and O(d, d)-invariant bilinear form to first order in the closed string sigma model coupling, α′ were derived by analyzing the transformation propertiesmore » of the Neveu-Schwarz B-field. By choosing a particular Poisson structure on the Drinfel’d double corresponding to the Courant algebroid structure of the generalized tangent bundle, we are able to interpret the C-bracket and bilinear form in terms of Poisson brackets. As a result, we reproduce the α′-deformations for a specific solution to the strong constraint of DFT as expansion of a graded version of the Moyal-Weyl star product.« less

Highly symmetric D-brane-anti-D-brane effective actions

NASA Astrophysics Data System (ADS)

Hatefi, Ehsan

2017-09-01

The entire S-matrix elements of four, five and six point functions of D-brane-anti D-brane system are explored. To deal with symmetries of string amplitudes as well as their all order α ' corrections we first address a four point function of one closed string Ramond-Ramond (RR) and two real tachyons on the world volume of brane-anti brane system. We then focus on symmetries of string theory as well as universal tachyon expansion to achieve both string and effective field theory of an RR and three tachyons where the complete algebraic analysis for the whole S-matrix < {V}_{C^{-1}}{V}_{T^{-1}}{V}_{T^0}{V}_{T^0}> was also revealed. Lastly, we employ all the conformal field theory techniques to < {V}_{C^{-1}}{V}_{T^{-1}}{V}_{T^0}{V}_{T^0}{V}_{T^0}> , working out with symmetries of theory and find out the expansion for the amplitude to be able to precisely discover all order singularity structures of D-brane-anti-D-brane effective actions of string theory. Various remarks about the so called generalized Veneziano amplitude and new string couplings are elaborated as well.

Topological charges in SL(2,R) covariant massive 11-dimensional and type IIB supergravity

DOE Office of Scientific and Technical Information (OSTI.GOV)

Callister, Andrew K.; Smith, Douglas J.

2009-12-15

In this paper we construct closed expressions that correspond to the topological charges of the various 1/2-BPS states of the maximal 10- and 11-dimensional supergravity theories. These expressions are related to the structure of the supersymmetry algebras in curved spacetimes. We mainly focus on IIB supergravity and 11-dimensional supergravity in a double M9-brane background, with an emphasis on the SL(2,R) multiplet structure of the charges and how these map between theories. This includes the charges corresponding to the multiplets of 7- and 9-branes in IIB. We find that examining the possible multiplet structures of the charges provides another tool formore » exploring the spectrum of BPS states that appear in these theories. As a prerequisite to constructing the charges we determine the field equations and multiplet structure of the 11-dimensional gauge potentials, extending previous results on the subject. The massive gauge transformations of the fields are also discussed. We also demonstrate how these massive gauge transformations are compatible with the construction of an SL(2,R) covariant kinetic term in the 11-dimensional Kaluza-Klein monopole worldvolume action.« less

Noncommutative Translations and *-PRODUCT Formalism

NASA Astrophysics Data System (ADS)

Daszkiewicz, Marcin; Lukierski, Jerzy; Woronowicz, Mariusz

2008-09-01

We consider the noncommutative space-times with Lie-algebraic noncommutativity (e.g. κ-deformed Minkowski space). In the framework with classical fields we extend the *-product in order to represent the noncommutative translations in terms of commutative ones. We show the translational invariance of noncommutative bilinear action with local product of noncommutative fields. The quadratic noncommutativity is also briefly discussed.

A Method of Computing Electric Field Parameters on Boundaries between Two Media

ERIC Educational Resources Information Center

Rizhov, Alexander

2010-01-01

Many problems of electric field strength on a boundary between two media require college-level mathematical analysis. However, when the boundary between media is represented by a sphere or a flat plane, these types of problems can be solved algebraically, placing them within reach of high school students. This article presents a solution analysis…

Closed-form solutions for linear regulator-design of mechanical systems including optimal weighting matrix selection

NASA Technical Reports Server (NTRS)

Hanks, Brantley R.; Skelton, Robert E.

1991-01-01

This paper addresses the restriction of Linear Quadratic Regulator (LQR) solutions to the algebraic Riccati Equation to design spaces which can be implemented as passive structural members and/or dampers. A general closed-form solution to the optimal free-decay control problem is presented which is tailored for structural-mechanical systems. The solution includes, as subsets, special cases such as the Rayleigh Dissipation Function and total energy. Weighting matrix selection is a constrained choice among several parameters to obtain desired physical relationships. The closed-form solution is also applicable to active control design for systems where perfect, collocated actuator-sensor pairs exist. Some examples of simple spring mass systems are shown to illustrate key points.

On the Poincare noninvariance of a recent alternative to the Dirac equation.

NASA Technical Reports Server (NTRS)

Madan, R. N.

1972-01-01

Explicit construction of the infinitesimal generators of the Poincare group, demonstrating that the algebra of commutators closes only for the case of zero mass. Hence, the so-called Stigma equation proposed by Biedenharn et al. (1971) as an alternative to the Dirac equation (1928) for spin one-half, finite mass particles is not Poincare invariant except when the leptonic mass is zero.

Solitary wave solutions of two-dimensional nonlinear Kadomtsev-Petviashvili dynamic equation in dust-acoustic plasmas

NASA Astrophysics Data System (ADS)

Seadawy, Aly R.

2017-09-01

Nonlinear two-dimensional Kadomtsev-Petviashvili (KP) equation governs the behaviour of nonlinear waves in dusty plasmas with variable dust charge and two temperature ions. By using the reductive perturbation method, the two-dimensional dust-acoustic solitary waves (DASWs) in unmagnetized cold plasma consisting of dust fluid, ions and electrons lead to a KP equation. We derived the solitary travelling wave solutions of the two-dimensional nonlinear KP equation by implementing sech-tanh, sinh-cosh, extended direct algebraic and fraction direct algebraic methods. We found the electrostatic field potential and electric field in the form travelling wave solutions for two-dimensional nonlinear KP equation. The solutions for the KP equation obtained by using these methods can be demonstrated precisely and efficiency. As an illustration, we used the readymade package of Mathematica program 10.1 to solve the original problem. These solutions are in good agreement with the analytical one.

String Theory: exact solutions, marginal deformations and hyperbolic spaces

NASA Astrophysics Data System (ADS)

Orlando, Domenico

2006-10-01

This thesis is almost entirely devoted to studying string theory backgrounds characterized by simple geometrical and integrability properties. The archetype of this type of system is given by Wess-Zumino-Witten models, describing string propagation in a group manifold or, equivalently, a class of conformal field theories with current algebras. We study the moduli space of such models by using truly marginal deformations. Particular emphasis is placed on asymmetric deformations that, together with the CFT description, enjoy a very nice spacetime interpretation in terms of the underlying Lie algebra. Then we take a slight detour so to deal with off-shell systems. Using a renormalization-group approach we describe the relaxation towards the symmetrical equilibrium situation. In he final chapter we consider backgrounds with Ramond-Ramond field and in particular we analyze direct products of constant-curvature spaces and find solutions with hyperbolic spaces.

Higgsing the stringy higher spin symmetry

DOE PAGES

Gaberdiel, Matthias R.; Peng, Cheng; Zadeh, Ida G.

2015-10-01

It has recently been argued that the symmetric orbifold theory of T 4 is dual to string theory on AdS 3 × S 3 × T 4 at the tensionless point. At this point in moduli space, the theory possesses a very large symmetry algebra that includes, in particular, a W ∞ algebra capturing the gauge fields of a dual higher spin theory. Using conformal perturbation theory, we study the behaviour of the symmetry generators of the symmetric orbifold theory under the deformation that corresponds to switching on the string tension. We show that the generators fall nicely into Reggemore » trajectories, with the higher spin fields corresponding to the leading Regge trajectory. We also estimate the form of the Regge trajectories for large spin, and find evidence for the familiar logarithmic behaviour, thereby suggesting that the symmetric orbifold theory is dual to an AdS background with pure RR flux.« less

Unconventional phases in quantum spin and pseudospin systems in two dimensional and three dimensional lattices

NASA Astrophysics Data System (ADS)

Xu, Cenke

Several examples of quantum spin systems and pseudo spin systems have been studied, and unconventional states of matters and phase transitions have been realized in all these systems under consideration. In the p +/- ip superconductor Josephson lattice and the p--band cold atomic system trapped in optical lattices, novel phases which behave similarly to 1+1 dimensional systems are realized, despite the fact that the real physical systems are in two or three dimensional spaces. For instance, by employing a spin-wave analysis together with a new duality transformation, we establish the existence and stability of a novel gapless "critical phase", which we refer to as a "bond algebraic liquid". This novel critical phase is analogous to the 1+1 dimensional algebraic boson liquid phase. The reason for the novel physics is that there is a quasilocal gauge symmetry in the effective low energy Hamiltonian. In a spin-1 system on the kagome lattice, and a hard-core boson system on the honeycomb lattice, the low energy physics is controlled by two components of compact U(1) gauge symmetries that emerge at low energy. Making use of the confinement nature of the 2+1 dimensional compact gauge theories and the powerful duality between gauge theories and height field theories, the crystalline phase diagrams are studied for both systems, and the transitions to other phases are also considered. These phase diagrams might be accessible in strongly correlated materials, or atomic systems in optical lattices. A novel quantum ground state of matter is realized in a bosonic model on three dimensional fcc lattice with emergent low energy excitations. The novel phase obtained is a stable gapless boson liquid phase, with algebraic boson density correlations. The stability of this phase is protected against the instanton effect and superfluidity by self-duality and large gauge symmetries on both sides of the duality. The gapless collective excitations of this phase closely resemble the graviton, although they have a soft w ˜ k2 dispersion relation. The dynamics of this novel phase is described by a new set of Maxwell's equations.

On choosing the start time of binary black hole ringdowns

NASA Astrophysics Data System (ADS)

Bhagwat, Swetha; Okounkova, Maria; Ballmer, Stefan W.; Brown, Duncan A.; Giesler, Matthew; Scheel, Mark A.; Teukolsky, Saul A.

2018-05-01

The final stage of a binary black hole merger is ringdown, in which the system is described by a Kerr black hole with quasinormal mode perturbations. It is far from straightforward to identify the time at which the ringdown begins. Yet determining this time is important for precision tests of the general theory of relativity that compare an observed signal with quasinormal mode descriptions of the ringdown, such as tests of the no-hair theorem. We present an algorithmic method to analyze the choice of ringdown start time in the observed waveform. This method is based on determining how close the strong field is to a Kerr black hole (Kerrness). Using numerical relativity simulations, we characterize the Kerrness of the strong-field region close to the black hole using a set of local, gauge-invariant geometric and algebraic conditions that measure local isometry to Kerr. We produce a map that associates each time in the gravitational waveform with a value of each of these Kerrness measures; this map is produced by following outgoing null characteristics from the strong and near-field regions to the wave zone. We perform this analysis on a numerical relativity simulation with parameters consistent with GW150914—the first gravitational-wave detection. We find that the choice of ringdown start time of 3 ms after merger used in the GW150914 study [B. P. Abbott et al. (Virgo Collaboration and LIGO Scientific Collaboration), Phys. Rev. Lett. 116, 221101 (2016)., 10.1103/PhysRevLett.116.221101] to test general relativity corresponds to a high dimensionless perturbation amplitude of ˜7.5 ×10-3 in the strong-field region. This suggests that in higher signal-to-noise detections, one would need to start analyzing the signal at a later time for studies that depend on the validity of black hole perturbation theory.

State-vector formalism and the Legendre polynomial solution for modelling guided waves in anisotropic plates

NASA Astrophysics Data System (ADS)

Zheng, Mingfang; He, Cunfu; Lu, Yan; Wu, Bin

2018-01-01

We presented a numerical method to solve phase dispersion curve in general anisotropic plates. This approach involves an exact solution to the problem in the form of the Legendre polynomial of multiple integrals, which we substituted into the state-vector formalism. In order to improve the efficiency of the proposed method, we made a special effort to demonstrate the analytical methodology. Furthermore, we analyzed the algebraic symmetries of the matrices in the state-vector formalism for anisotropic plates. The basic feature of the proposed method was the expansion of field quantities by Legendre polynomials. The Legendre polynomial method avoid to solve the transcendental dispersion equation, which can only be solved numerically. This state-vector formalism combined with Legendre polynomial expansion distinguished the adjacent dispersion mode clearly, even when the modes were very close. We then illustrated the theoretical solutions of the dispersion curves by this method for isotropic and anisotropic plates. Finally, we compared the proposed method with the global matrix method (GMM), which shows excellent agreement.

Mapping superintegrable quantum mechanics to resonant spacetimes

NASA Astrophysics Data System (ADS)

Evnin, Oleg; Demirchian, Hovhannes; Nersessian, Armen

2018-01-01

We describe a procedure naturally associating relativistic Klein-Gordon equations in static curved spacetimes to nonrelativistic quantum motion on curved spaces in the presence of a potential. Our procedure is particularly attractive in application to (typically, superintegrable) problems whose energy spectrum is given by a quadratic function of the energy level number, since for such systems the spacetimes one obtains possess evenly spaced, resonant spectra of frequencies for scalar fields of a certain mass. This construction emerges as a generalization of the previously studied correspondence between the Higgs oscillator and anti-de Sitter spacetime, which has been useful for both understanding weakly nonlinear dynamics in anti-de Sitter spacetime and algebras of conserved quantities of the Higgs oscillator. Our conversion procedure ("Klein-Gordonization") reduces to a nonlinear elliptic equation closely reminiscent of the one emerging in relation to the celebrated Yamabe problem of differential geometry. As an illustration, we explicitly demonstrate how to apply this procedure to superintegrable Rosochatius systems, resulting in a large family of spacetimes with resonant spectra for massless wave equations.

A Lie based 4-dimensional higher Chern-Simons theory

NASA Astrophysics Data System (ADS)

Zucchini, Roberto

2016-05-01

We present and study a model of 4-dimensional higher Chern-Simons theory, special Chern-Simons (SCS) theory, instances of which have appeared in the string literature, whose symmetry is encoded in a skeletal semistrict Lie 2-algebra constructed from a compact Lie group with non discrete center. The field content of SCS theory consists of a Lie valued 2-connection coupled to a background closed 3-form. SCS theory enjoys a large gauge and gauge for gauge symmetry organized in an infinite dimensional strict Lie 2-group. The partition function of SCS theory is simply related to that of a topological gauge theory localizing on flat connections with degree 3 second characteristic class determined by the background 3-form. Finally, SCS theory is related to a 3-dimensional special gauge theory whose 2-connection space has a natural symplectic structure with respect to which the 1-gauge transformation action is Hamiltonian, the 2-curvature map acting as moment map.

Schwinger-Keldysh superspace in quantum mechanics

NASA Astrophysics Data System (ADS)

Geracie, Michael; Haehl, Felix M.; Loganayagam, R.; Narayan, Prithvi; Ramirez, David M.; Rangamani, Mukund

2018-05-01

We examine, in a quantum mechanical setting, the Hilbert space representation of the Becchi, Rouet, Stora, and Tyutin (BRST) symmetry associated with Schwinger-Keldysh path integrals. This structure had been postulated to encode important constraints on influence functionals in coarse-grained systems with dissipation, or in open quantum systems. Operationally, this entails uplifting the standard Schwinger-Keldysh two-copy formalism into superspace by appending BRST ghost degrees of freedom. These statements were previously argued at the level of the correlation functions. We provide herein a complementary perspective by working out the Hilbert space structure explicitly. Our analysis clarifies two crucial issues not evident in earlier works: first, certain background ghost insertions necessary to reproduce the correct Schwinger-Keldysh correlators arise naturally, and, second, the Schwinger-Keldysh difference operators are systematically dressed by the ghost bilinears, which turn out to be necessary to give rise to a consistent operator algebra. We also elaborate on the structure of the final state (which is BRST closed) and the future boundary condition of the ghost fields.

Higher-n triangular dilatonic black holes

NASA Astrophysics Data System (ADS)

Zadora, Anton; Gal'tsov, Dmitri V.; Chen, Chiang-Mei

2018-04-01

Dilaton gravity with the form fields is known to possess dyon solutions with two horizons for the discrete "triangular" values of the dilaton coupling constant a =√{ n (n + 1) / 2 }. This sequence first obtained numerically and then explained analytically as consequence of the regularity of the dilaton, should have some higher-dimensional and/or group theoretical origin. Meanwhile, this origin was explained earlier only for n = 1 , 2 in which cases the solutions were known analytically. We extend this explanation to n = 3 , 5 presenting analytical triangular solutions for the theory with different dilaton couplings a , b in electric and magnetic sectors in which case the quantization condition reads ab = n (n + 1) / 2. The solutions are derived via the Toda chains for B2 and G2 Lie algebras. They are found in the closed form in general D space-time dimensions. Solutions satisfy the entropy product rules indicating on the microscopic origin of their entropy and have negative binding energy in the extremal case.

Operator bases, S-matrices, and their partition functions

DOE Office of Scientific and Technical Information (OSTI.GOV)

Henning, Brian; Lu, Xiaochuan; Melia, Tom

Relativistic quantum systems that admit scattering experiments are quantitatively described by effective field theories, where S-matrix kinematics and symmetry considerations are encoded in the operator spectrum of the EFT. Here in this paper we use the S-matrix to derive the structure of the EFT operator basis, providing complementary descriptions in (i) position space utilizing the conformal algebra and cohomology and (ii) momentum space via an algebraic formulation in terms of a ring of momenta with kinematics implemented as an ideal. These frameworks systematically handle redundancies associated with equations of motion (on-shell) and integration by parts (momentum conservation). We introduce amore » partition function, termed the Hilbert series, to enumerate the operator basis — correspondingly, the S-matrix — and derive a matrix integral expression to compute the Hilbert series. The expression is general, easily applied in any spacetime dimension, with arbitrary field content and (linearly realized) symmetries. In addition to counting, we discuss construction of the basis. Simple algorithms follow from the algebraic formulation in momentum space. We explicitly compute the basis for operators involving up to n = 5 scalar fields. This construction universally applies to fields with spin, since the operator basis for scalars encodes the momentum dependence of n-point amplitudes. We discuss in detail the operator basis for non-linearly realized symmetries. In the presence of massless particles, there is freedom to impose additional structure on the S- matrix in the form of soft limits. The most naÏve implementation for massless scalars leads to the operator basis for pions, which we confirm using the standard CCWZ formulation for non-linear realizations. Finally, although primarily discussed in the language of EFT, some of our results — conceptual and quantitative — may be of broader use in studying conformal field theories as well as the AdS/CFT correspondence.« less

Operator bases, S-matrices, and their partition functions

DOE PAGES

Henning, Brian; Lu, Xiaochuan; Melia, Tom; ...

2017-10-27

Relativistic quantum systems that admit scattering experiments are quantitatively described by effective field theories, where S-matrix kinematics and symmetry considerations are encoded in the operator spectrum of the EFT. Here in this paper we use the S-matrix to derive the structure of the EFT operator basis, providing complementary descriptions in (i) position space utilizing the conformal algebra and cohomology and (ii) momentum space via an algebraic formulation in terms of a ring of momenta with kinematics implemented as an ideal. These frameworks systematically handle redundancies associated with equations of motion (on-shell) and integration by parts (momentum conservation). We introduce amore » partition function, termed the Hilbert series, to enumerate the operator basis — correspondingly, the S-matrix — and derive a matrix integral expression to compute the Hilbert series. The expression is general, easily applied in any spacetime dimension, with arbitrary field content and (linearly realized) symmetries. In addition to counting, we discuss construction of the basis. Simple algorithms follow from the algebraic formulation in momentum space. We explicitly compute the basis for operators involving up to n = 5 scalar fields. This construction universally applies to fields with spin, since the operator basis for scalars encodes the momentum dependence of n-point amplitudes. We discuss in detail the operator basis for non-linearly realized symmetries. In the presence of massless particles, there is freedom to impose additional structure on the S- matrix in the form of soft limits. The most naÏve implementation for massless scalars leads to the operator basis for pions, which we confirm using the standard CCWZ formulation for non-linear realizations. Finally, although primarily discussed in the language of EFT, some of our results — conceptual and quantitative — may be of broader use in studying conformal field theories as well as the AdS/CFT correspondence.« less

A Note on Equivalence Among Various Scalar Field Models of Dark Energies

NASA Astrophysics Data System (ADS)

Mandal, Jyotirmay Das; Debnath, Ujjal

2017-08-01

In this work, we have tried to find out similarities between various available models of scalar field dark energies (e.g., quintessence, k-essence, tachyon, phantom, quintom, dilatonic dark energy, etc). We have defined an equivalence relation from elementary set theory between scalar field models of dark energies and used fundamental ideas from linear algebra to set up our model. Consequently, we have obtained mutually disjoint subsets of scalar field dark energies with similar properties and discussed our observation.

Quantum field theory and coalgebraic logic in theoretical computer science.

PubMed

Basti, Gianfranco; Capolupo, Antonio; Vitiello, Giuseppe

2017-11-01

We suggest that in the framework of the Category Theory it is possible to demonstrate the mathematical and logical dual equivalence between the category of the q-deformed Hopf Coalgebras and the category of the q-deformed Hopf Algebras in quantum field theory (QFT), interpreted as a thermal field theory. Each pair algebra-coalgebra characterizes a QFT system and its mirroring thermal bath, respectively, so to model dissipative quantum systems in far-from-equilibrium conditions, with an evident significance also for biological sciences. Our study is in fact inspired by applications to neuroscience where the brain memory capacity, for instance, has been modeled by using the QFT unitarily inequivalent representations. The q-deformed Hopf Coalgebras and the q-deformed Hopf Algebras constitute two dual categories because characterized by the same functor T, related with the Bogoliubov transform, and by its contravariant application T op , respectively. The q-deformation parameter is related to the Bogoliubov angle, and it is effectively a thermal parameter. Therefore, the different values of q identify univocally, and label the vacua appearing in the foliation process of the quantum vacuum. This means that, in the framework of Universal Coalgebra, as general theory of dynamic and computing systems ("labelled state-transition systems"), the so labelled infinitely many quantum vacua can be interpreted as the Final Coalgebra of an "Infinite State Black-Box Machine". All this opens the way to the possibility of designing a new class of universal quantum computing architectures based on this coalgebraic QFT formulation, as its ability of naturally generating a Fibonacci progression demonstrates. Copyright © 2017 Elsevier Ltd. All rights reserved.

Functors of White Noise Associated to Characters of the Infinite Symmetric Group

NASA Astrophysics Data System (ADS)

Bożejko, Marek; Guţă, Mădălin

The characters of the infinite symmetric group are extended to multiplicative positive definite functions on pair partitions by using an explicit representation due to Veršik and Kerov. The von Neumann algebra generated by the fields with f in an infinite dimensional real Hilbert space is infinite and the vacuum vector is not separating. For a family depending on an integer N< - 1 an ``exclusion principle'' is found allowing at most ``identical particles'' on the same state: The algebras are type factors. Functors of white noise are constructed and proved to be non-equivalent for different values of N.

On the elliptic genera of manifolds of Spin(7) holonomy

DOE PAGES

Benjamin, Nathan; Harrison, Sarah M.; Kachru, Shamit; ...

2015-12-16

Superstring compactification on a manifold of Spin(7) holonomy gives rise to a 2d worldsheet conformal field theory with an extended supersymmetry algebra. The N=1 superconformal algebra is extended by additional generators of spins 2 and 5/2, and instead of just superconformal symmetry one has a c = 12 realization of the symmetry group SW(3/2,2). In this paper, we compute the characters of this supergroup and decompose the elliptic genus of a general Spin(7) compactification in terms of these characters. Here, we find suggestive relations to various sporadic groups, which are made more precise in a companion paper.

Method of orbit sums in the theory of modular vector invariants

NASA Astrophysics Data System (ADS)

Stepanov, S. A.

2006-12-01

Let F be a field, V a finite-dimensional F-vector space, G\\leqslant \\operatorname{GL}_F(V) a finite group, and V^m=V\\oplus\\dots\\oplus V the m-fold direct sum with the diagonal action of G. The group G acts naturally on the symmetric graded algebra A_m=F \\lbrack V^m \\rbrack as a group of non-degenerate linear transformations of the variables. Let A_m^G be the subalgebra of invariants of the polynomial algebra A_m with respect to G. A classical result of Noether [1] says that if \\operatorname{char}F=0, then A_m^G is generated as an F-algebra by homogeneous polynomials of degree at most \\vert G\\vert, no matter how large m can be. On the other hand, it was proved by Richman [2], [3] that this result does not hold when the characteristic of F is positive and divides the order \\vert G\\vert of G. Let p, p>2, be a prime number, F=F_p a finite field of p elements, V a linear F_p-vector space of dimension n, and H\\leqslant \\operatorname{GL}_{F_p}(V) a cyclic group of order p generated by a matrix \\gamma of a certain special form. In this paper we describe explicitly (Theorem 1) one complete set of generators of A_m^H. After that, for an arbitrary complete set of generators of this algebra we find a lower bound for the highest degree of the generating elements of this algebra. This is a significant extension of the corresponding result of Campbell and Hughes [4] for the particular case of n=2. As a consequence we show (Theorem 3) that if m>n and G\\ge H is an arbitrary finite group, then each complete set of generators of A_m^G contains an element of degree at least 2(m-n+2r)(p-1)/r, where r=r(H) is a positive integer dependent on the structure of the generating matrix \\gamma of the group H. This result refines considerably the earlier lower bound obtained by Richman [3].

Quantum correlations are weaved by the spinors of the Euclidean primitives

PubMed Central

2018-01-01

The exceptional Lie group E8 plays a prominent role in both mathematics and theoretical physics. It is the largest symmetry group associated with the most general possible normed division algebra, namely, that of the non-associative real octonions, which—thanks to their non-associativity—form the only possible closed set of spinors (or rotors) that can parallelize the 7-sphere. By contrast, here we show how a similar 7-sphere also arises naturally from the algebraic interplay of the graded Euclidean primitives, such as points, lines, planes and volumes, which characterize the three-dimensional conformal geometry of the ambient physical space, set within its eight-dimensional Clifford-algebraic representation. Remarkably, the resulting algebra remains associative, and allows us to understand the origins and strengths of all quantum correlations locally, in terms of the geometry of the compactified physical space, namely, that of a quaternionic 3-sphere, S3, with S7 being its algebraic representation space. Every quantum correlation can thus be understood as a correlation among a set of points of this S7, computed using manifestly local spinors within S3, thereby extending the stringent bounds of ±2 set by Bell inequalities to the bounds of ±22 on the strengths of all possible strong correlations, in the same quantitatively precise manner as that predicted within quantum mechanics. The resulting geometrical framework thus overcomes Bell’s theorem by producing a strictly deterministic and realistic framework that allows a locally causal understanding of all quantum correlations, without requiring either remote contextuality or backward causation. We demonstrate this by first proving a general theorem concerning the geometrical origins of the correlations predicted by arbitrarily entangled quantum states, and then reproducing the correlations predicted by the EPR-Bohm and the GHZ states. The raison d’être of strong correlations turns out to be the Möbius-like twists in the Hopf bundles of S3 and S7. PMID:29893385

Algebraically special space-time in relativity, black holes, and pulsar models

NASA Technical Reports Server (NTRS)

Adler, R. J.; Sheffield, C.

1973-01-01

The entire field of astronomy is in very rapid flux, and at the center of interest are problems relating to the very dense, rotating, neutron stars observed as pulsars. the hypothesized collapsed remains of stars known as black holes, and quasars. Degenerate metric form, or Kerr-Schild metric form, was used to study several problems related to intense gravitational fields.

On curve veering and flutter of rotating blades

NASA Technical Reports Server (NTRS)

Afolabi, Dare; Mehmed, Oral

1993-01-01

The eigenvalues of rotating blades usually change with rotation speed according to the Stodola-Southwell criterion. Under certain circumstances, the loci of eigenvalues belonging to two distinct modes of vibration approach each other very closely, and it may appear as if the loci cross each other. However, our study indicates that the observable frequency loci of an undamped rotating blade do not cross, but must either repel each other (leading to 'curve veering'), or attract each other (leading to 'frequency coalescence'). Our results are reached by using standard arguments from algebraic geometry--the theory of algebraic curves and catastrophe theory. We conclude that it is important to resolve an apparent crossing of eigenvalue loci into either a frequency coalescence or a curve veering, because frequency coalescence is dangerous since it leads to flutter, whereas curve veering does not precipitate flutter and is, therefore, harmless with respect to elastic stability.

Investigation of triaxiality in 54122-128Xe isotopes in the framework of sdg-IBM

NASA Astrophysics Data System (ADS)

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

In this paper, a transitional interacting boson model (IBM) Hamiltonian in both sd-(IBM) and sdg-IBM versions based on affine SU(1, 1) Lie algebra is employed to describe deviations from the gamma-unstable nature of Hamiltonian along the chain of Xe isotopes. sdg-IBM Hamiltonian proposed a better interpretation of this deviation which cannot be explained in the sd-boson models. The nuclei studied have well-known γ bands close to the γ-unstable limit. The energy levels, B(E2) transition rates and signature splitting of the γ -vibrational band are calculated via the affine SU(1,1) Lie algebra. An acceptable degree of agreement was achieved based on this procedure. It is shown that in these isotopes the signature splitting is better reproduced by the inclusion of sdg-IBM. In none of them, any evidence for a stable, triaxial ground state shape is found.

Mixed-state fidelity susceptibility through iterated commutator series expansion

NASA Astrophysics Data System (ADS)

Tonchev, N. S.

2014-11-01

We present a perturbative approach to the problem of computation of mixed-state fidelity susceptibility (MFS) for thermal states. The mathematical techniques used provide an analytical expression for the MFS as a formal expansion in terms of the thermodynamic mean values of successively higher commutators of the Hamiltonian with the operator involved through the control parameter. That expression is naturally divided into two parts: the usual isothermal susceptibility and a constituent in the form of an infinite series of thermodynamic mean values which encodes the noncommutativity in the problem. If the symmetry properties of the Hamiltonian are given in terms of the generators of some (finite-dimensional) algebra, the obtained expansion may be evaluated in a closed form. This issue is tested on several popular models, for which it is shown that the calculations are much simpler if they are based on the properties from the representation theory of the Heisenberg or SU(1, 1) Lie algebra.

Excitation basis for (3+1)d topological phases

NASA Astrophysics Data System (ADS)

Delcamp, Clement

2017-12-01

We consider an exactly solvable model in 3+1 dimensions, based on a finite group, which is a natural generalization of Kitaev's quantum double model. The corresponding lattice Hamiltonian yields excitations located at torus-boundaries. By cutting open the three-torus, we obtain a manifold bounded by two tori which supports states satisfying a higher-dimensional version of Ocneanu's tube algebra. This defines an algebraic structure extending the Drinfel'd double. Its irreducible representations, labeled by two fluxes and one charge, characterize the torus-excitations. The tensor product of such representations is introduced in order to construct a basis for (3+1)d gauge models which relies upon the fusion of the defect excitations. This basis is defined on manifolds of the form Σ × S_1 , with Σ a two-dimensional Riemann surface. As such, our construction is closely related to dimensional reduction from (3+1)d to (2+1)d topological orders.

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.

Introduction to quantized LIE groups and algebras

DOE Office of Scientific and Technical Information (OSTI.GOV)

Tjin, T.

1992-10-10

In this paper, the authors give a self-contained introduction to the theory of quantum groups according to Drinfeld, highlighting the formal aspects as well as the applications to the Yang-Baxter equation and representation theory. Introductions to Hopf algebras, Poisson structures and deformation quantization are also provided. After defining Poisson Lie groups the authors study their relation to Lie bialgebras and the classical Yang-Baxter equation. Then the authors explain in detail the concept of quantization for them. As an example the quantization of sl[sub 2] is explicitly carried out. Next, the authors show how quantum groups are related to the Yang-Baxtermore » equation and how they can be used to solve it. Using the quantum double construction, the authors explicitly construct the universal R matrix for the quantum sl[sub 2] algebra. In the last section, the authors deduce all finite-dimensional irreducible representations for q a root of unity. The authors also give their tensor product decomposition (fusion rules), which is relevant to conformal field theory.« less

On the origin of dual Lax pairs and their r-matrix structure

NASA Astrophysics Data System (ADS)

Avan, Jean; Caudrelier, Vincent

2017-10-01

We establish the algebraic origin of the following observations made previously by the authors and coworkers: (i) A given integrable PDE in 1 + 1 dimensions within the Zakharov-Shabat scheme related to a Lax pair can be cast in two distinct, dual Hamiltonian formulations; (ii) Associated to each formulation is a Poisson bracket and a phase space (which are not compatible in the sense of Magri); (iii) Each matrix in the Lax pair satisfies a linear Poisson algebra a la Sklyanin characterized by the same classical r matrix. We develop the general concept of dual Lax pairs and dual Hamiltonian formulation of an integrable field theory. We elucidate the origin of the common r-matrix structure by tracing it back to a single Lie-Poisson bracket on a suitable coadjoint orbit of the loop algebra sl(2 , C) ⊗ C(λ ,λ-1) . The results are illustrated with the examples of the nonlinear Schrödinger and Gerdjikov-Ivanov hierarchies.

Quantum trilogy: discrete Toda, Y-system and chaos

NASA Astrophysics Data System (ADS)

Yamazaki, Masahito

2018-02-01

We discuss a discretization of the quantum Toda field theory associated with a semisimple finite-dimensional Lie algebra or a tamely-laced infinite-dimensional Kac-Moody algebra G, generalizing the previous construction of discrete quantum Liouville theory for the case G  =  A 1. The model is defined on a discrete two-dimensional lattice, whose spatial direction is of length L. In addition we also find a ‘discretized extra dimension’ whose width is given by the rank r of G, which decompactifies in the large r limit. For the case of G  =  A N or AN-1(1) , we find a symmetry exchanging L and N under appropriate spatial boundary conditions. The dynamical time evolution rule of the model is quantizations of the so-called Y-system, and the theory can be well described by the quantum cluster algebra. We discuss possible implications for recent discussions of quantum chaos, and comment on the relation with the quantum higher Teichmüller theory of type A N .

BRST Formalism in Self-Dual Chern-Simons Theory with Matter Fields

NASA Astrophysics Data System (ADS)

Dai, Jialiang; Fan, Engui

2018-04-01

We apply BRST method to the self-dual Chern-Simons gauge theory with matter fields and the generators of symmetries of the system from an elegant Lie algebra structure under the operation of Poisson bracket. We discuss four different cases: abelian, nonabelian, relativistic, and nonrelativistic situations and extend the system to the whole phase space including ghost fields. In addition, we obtain the BRST charge of the field system and check its nilpotence of the BRST transformation which plays an important role such as in topological quantum field theory and string theory.

On deformation of complex continuum immersed in a plane space

NASA Astrophysics Data System (ADS)

Kovalev, V. A.; Murashkin, E. V.; Radayev, Y. N.

2018-05-01

The present paper is devoted to mathematical modelling of complex continua deformations considered as immersed in an external plane space. The complex continuum is defined as a differential manifold supplied with metrics induced by the external space. A systematic derivation of strain tensors by notion of isometric immersion of the complex continuum into a plane space of a higher dimension is proposed. Problem of establishing complete systems of irreducible objective strain and extrastrain tensors for complex continuum immersed in an external plane space is resolved. The solution to the problem is obtained by methods of the field theory and the theory of rational algebraic invariants. Strain tensors of the complex continuum are derived as irreducible algebraic invariants of contravariant vectors of the external space emerging as functional arguments in the complex continuum action density. Present analysis is restricted to rational algebraic invariants. Completeness of the considered systems of rational algebraic invariants is established for micropolar elastic continua. Rational syzygies for non-quadratic invariants are discussed. Objective strain tensors (indifferent to frame rotations in the external plane space) for micropolar continuum are alternatively obtained by properly combining multipliers of polar decompositions of deformation and extra-deformation gradients. The latter is realized only for continua immersed in a plane space of the equal mathematical dimension.

Survey of Turbulence Models for the Computation of Turbulent Jet Flow and Noise

NASA Technical Reports Server (NTRS)

Nallasamy, N.

1999-01-01

The report presents an overview of jet noise computation utilizing the computational fluid dynamic solution of the turbulent jet flow field. The jet flow solution obtained with an appropriate turbulence model provides the turbulence characteristics needed for the computation of jet mixing noise. A brief account of turbulence models that are relevant for the jet noise computation is presented. The jet flow solutions that have been directly used to calculate jet noise are first reviewed. Then, the turbulent jet flow studies that compute the turbulence characteristics that may be used for noise calculations are summarized. In particular, flow solutions obtained with the k-e model, algebraic Reynolds stress model, and Reynolds stress transport equation model are reviewed. Since, the small scale jet mixing noise predictions can be improved by utilizing anisotropic turbulence characteristics, turbulence models that can provide the Reynolds stress components must now be considered for jet flow computations. In this regard, algebraic stress models and Reynolds stress transport models are good candidates. Reynolds stress transport models involve more modeling and computational effort and time compared to algebraic stress models. Hence, it is recommended that an algebraic Reynolds stress model (ASM) be implemented in flow solvers to compute the Reynolds stress components.

On Special Functions in the Context of Clifford Analysis

NASA Astrophysics Data System (ADS)

Malonek, H. R.; Falcão, M. I.

2010-09-01

Considering the foundation of Quaternionic Analysis by R. Fueter and his collaborators in the beginning of the 1930s as starting point of Clifford Analysis, we can look back to 80 years of work in this field. However the interest in multivariate analysis using Clifford algebras only started to grow significantly in the 70s. Since then a great amount of papers on Clifford Analysis referring different classes of Special Functions have appeared. This situation may have been triggered by a more systematic treatment of monogenic functions by their multiple series development derived from Gegenbauer or associated Legendre polynomials (and not only by their integral representation). Also approaches to Special Functions by means of algebraic methods, either Lie algebras or through Lie groups and symmetric spaces gained by that time importance and influenced their treatment in Clifford Analysis. In our talk we will rely on the generalization of the classical approach to Special Functions through differential equations with respect to the hypercomplex derivative, which is a more recently developed tool in Clifford Analysis. In this context special attention will be payed to the role of Special Functions as intermediator between continuous and discrete mathematics. This corresponds to a more recent trend in combinatorics, since it has been revealed that many algebraic structures have hidden combinatorial underpinnings.

Black holes in higher spin supergravity

NASA Astrophysics Data System (ADS)

Datta, Shouvik; David, Justin R.

2013-07-01

We study black hole solutions in Chern-Simons higher spin supergravity based on the superalgebra sl(3|2). These black hole solutions have a U(1) gauge field and a spin 2 hair in addition to the spin 3 hair. These additional fields correspond to the R-symmetry charges of the supergroup sl(3|2). Using the relation between the bulk field equations and the Ward identities of a CFT with {N} = 2 super- {{{W}}_3} symmetry, we identify the bulk charges and chemical potentials with those of the boundary CFT. From these identifications we see that a suitable set of variables to study this black hole is in terms of the charges present in three decoupled bosonic sub-algebras of the {N} = 2 super- {{{W}}_3} algebra. The entropy and the partition function of these R-charged black holes are then evaluated in terms of the charges of the bulk theory as well as in terms of its chemical potentials. We then compute the partition function in the dual CFT and find exact agreement with the bulk partition function.

A premier analysis of supersymmetric closed string tachyon cosmology

NASA Astrophysics Data System (ADS)

Vázquez-Báez, V.; Ramírez, C.

2018-04-01

From a previously found worldline supersymmetric formulation for the effective action of the closed string tachyon in a FRW background, the Hamiltonian of the theory is constructed, by means of the Dirac procedure, and written in a quantum version. Using the supersymmetry algebra we are able to find solutions to the Wheeler-DeWitt equation via a more simple set of first order differential equations. Finally, for the k = 0 case, we compute the expectation value of the scale factor with a suitably potential also favored in the present literature. We give some interpretations of the results and state future work lines on this matter.

Linearizing feedforward/feedback attitude control

NASA Technical Reports Server (NTRS)

Paielli, Russell A.; Bach, Ralph E.

1991-01-01

An approach to attitude control theory is introduced in which a linear form is postulated for the closed-loop rotation error dynamics, then the exact control law required to realize it is derived. The nonminimal (four-component) quaternion form is used to attitude because it is globally nonsingular, but the minimal (three-component) quaternion form is used for attitude error because it has no nonlinear constraints to prevent the rotational error dynamics from being linearized, and the definition of the attitude error is based on quaternion algebra. This approach produces an attitude control law that linearizes the closed-loop rotational error dynamics exactly, without any attitude singularities, even if the control errors become large.

Analytical expressions for the correlation function of a hard sphere dimer fluid

NASA Astrophysics Data System (ADS)

Kim, Soonho; Chang, Jaeeon; Kim, Hwayong

A closed form expression is given for the correlation function of a hard sphere dimer fluid. A set of integral equations is obtained from Wertheim's multidensity Ornstein-Zernike integral equation theory with Percus-Yevick approximation. Applying the Laplace transformation method to the integral equations and then solving the resulting equations algebraically, the Laplace transforms of the individual correlation functions are obtained. By the inverse Laplace transformation, the radial distribution function (RDF) is obtained in closed form out to 3D (D is the segment diameter). The analytical expression for the RDF of the hard dimer should be useful in developing the perturbation theory of dimer fluids.

Analytical expression for the correlation function of a hard sphere chain fluid

NASA Astrophysics Data System (ADS)

Chang, Jaeeon; Kim, Hwayong

A closed form expression is given for the correlation function of flexible hard sphere chain fluid. A set of integral equations obtained from Wertheim's multidensity Ornstein-Zernike integral equation theory with the polymer Percus-Yevick ideal chain approximation is considered. Applying the Laplace transformation method to the integral equations and then solving the resulting equations algebraically, the Laplace transforms of individual correlation functions are obtained. By inverse Laplace transformation the inter- and intramolecular radial distribution functions (RDFs) are obtained in closed forms up to 3D(D is segment diameter). These analytical expressions for the RDFs would be useful in developing the perturbation theory of chain fluids.

A Novel Hypercomplex Solution to Kepler's Problem

NASA Astrophysics Data System (ADS)

Condurache, C.; Martinuşi, V.

2007-05-01

By using a Sundman like regularization, we offer a unified solution to Kepler's problem by using hypercomplex numbers. The fundamental role in this paper is played by the Laplace-Runge-Lenz prime integral and by the hypercomplex numbers algebra. The procedure unifies and generalizes the regularizations offered by Levi-Civita and Kustaanheimo-Stiefel. Closed form hypercomplex expressions for the law of motion and velocity are deduced, together with inedite hypercomplex prime integrals.

On Alternative Formulations for Linearised Miss Distance Analysis

DTIC Science & Technology

2013-05-01

is traditionally employed by analysts as part of the solution process . To gain further insight into the nature of the missile-target engagement...a constant. Thus, following this process , the revised block diagram model for the linearised equations is presented in Figure 13. This model is... process is known as reducing the block to its fundamental closed loop form and has been achieved here using standard block diagram algebra. This

Supertranslations and Superrotations at the Black Hole Horizon.

PubMed

Donnay, Laura; Giribet, Gaston; González, Hernán A; Pino, Miguel

2016-03-04

We show that the asymptotic symmetries close to nonextremal black hole horizons are generated by an extension of supertranslations. This group is generated by a semidirect sum of Virasoro and Abelian currents. The charges associated with the asymptotic Killing symmetries satisfy the same algebra. When considering the special case of a stationary black hole, the zero mode charges correspond to the angular momentum and the entropy at the horizon.

Modern Quantum Field Theory II - Proceeeings of the International Colloquium

NASA Astrophysics Data System (ADS)

Das, S. R.; Mandal, G.; Mukhi, S.; Wadia, S. R.

1995-08-01

The Table of Contents for the book is as follows: * Foreword * 1. Black Holes and Quantum Gravity * Quantum Black Holes and the Problem of Time * Black Hole Entropy and the Semiclassical Approximation * Entropy and Information Loss in Two Dimensions * Strings on a Cone and Black Hole Entropy (Abstract) * Boundary Dynamics, Black Holes and Spacetime Fluctuations in Dilation Gravity (Abstract) * Pair Creation of Black Holes (Abstract) * A Brief View of 2-Dim. String Theory and Black Holes (Abstract) * 2. String Theory * Non-Abelian Duality in WZW Models * Operators and Correlation Functions in c ≤ 1 String Theory * New Symmetries in String Theory * A Look at the Discretized Superstring Using Random Matrices * The Nested BRST Structure of Wn-Symmetries * Landau-Ginzburg Model for a Critical Topological String (Abstract) * On the Geometry of Wn Gravity (Abstract) * O(d, d) Tranformations, Marginal Deformations and the Coset Construction in WZNW Models (Abstract) * Nonperturbative Effects and Multicritical Behaviour of c = 1 Matrix Model (Abstract) * Singular Limits and String Solutions (Abstract) * BV Algebra on the Moduli Spaces of Riemann Surfaces and String Field Theory (Abstract) * 3. Condensed Matter and Statistical Mechanics * Stochastic Dynamics in a Deposition-Evaporation Model on a Line * Models with Inverse-Square Interactions: Conjectured Dynamical Correlation Functions of the Calogero-Sutherland Model at Rational Couplings * Turbulence and Generic Scale Invariance * Singular Perturbation Approach to Phase Ordering Dynamics * Kinetics of Diffusion-Controlled and Ballistically-Controlled Reactions * Field Theory of a Frustrated Heisenberg Spin Chain * FQHE Physics in Relativistic Field Theories * Importance of Initial Conditions in Determining the Dynamical Class of Cellular Automata (Abstract) * Do Hard-Core Bosons Exhibit Quantum Hall Effect? (Abstract) * Hysteresis in Ferromagnets * 4. Fundamental Aspects of Quantum Mechanics and Quantum Field Theory * Finite Quantum Physics and Noncommutative Geometry * Higgs as Gauge Field and the Standard Model * Canonical Quantisation of an Off-Conformal Theory * Deterministic Quantum Mechanics in One Dimension * Spin-Statistics Relations for Topological Geons in 2+1 Quantum Gravity * Generalized Fock Spaces * Geometrical Expression for Short Distance Singularities in Field Theory * 5. Mathematics and Quantum Field Theory * Knot Invariants from Quantum Field Theories * Infinite Grassmannians and Moduli Spaces of G-Bundles * A Review of an Algebraic Geometry Approach to a Model Quantum Field Theory on a Curve (Abstract) * 6. Integrable Models * Spectral Representation of Correlation Functions in Two-Dimensional Quantum Field Theories * On Various Avatars of the Pasquier Algebra * Supersymmetric Integrable Field Theories and Eight Vertex Free Fermion Models (Abstract) * 7. Lattice Field Theory * From Kondo Model and Strong Coupling Lattice QCD to the Isgur-Wise Function * Effective Confinement from a Logarithmically Running Coupling (Abstract)

Vertex operator algebras of Argyres-Douglas theories from M5-branes

NASA Astrophysics Data System (ADS)

Song, Jaewon; Xie, Dan; Yan, Wenbin

2017-12-01

We study aspects of the vertex operator algebra (VOA) corresponding to Argyres-Douglas (AD) theories engineered using the 6d N=(2, 0) theory of type J on a punctured sphere. We denote the AD theories as ( J b [ k], Y), where J b [ k] and Y represent an irregular and a regular singularity respectively. We restrict to the `minimal' case where J b [ k] has no associated mass parameters, and the theory does not admit any exactly marginal deformations. The VOA corresponding to the AD theory is conjectured to be the W-algebra W^{k_{2d}}(J, Y ) , where {k}_{2d}=-h+b/b+k with h being the dual Coxeter number of J. We verify this conjecture by showing that the Schur index of the AD theory is identical to the vacuum character of the corresponding VOA, and the Hall-Littlewood index computes the Hilbert series of the Higgs branch. We also find that the Schur and Hall-Littlewood index for the AD theory can be written in a simple closed form for b = h. We also test the conjecture that the associated variety of such VOA is identical to the Higgs branch. The M5-brane construction of these theories and the corresponding TQFT structure of the index play a crucial role in our computations.

On the tensionless limit of gauged WZW models

NASA Astrophysics Data System (ADS)

Bakas, I.; Sourdis, C.

2004-06-01

The tensionless limit of gauged WZW models arises when the level of the underlying Kac-Moody algebra assumes its critical value, equal to the dual Coxeter number, in which case the central charge of the Virasoro algebra becomes infinite. We examine this limit from the world-sheet and target space viewpoint and show that gravity decouples naturally from the spectrum. Using the two-dimensional black-hole coset SL(2,Bbb R)k/U(1) as illustrative example, we find for k = 2 that the world-sheet symmetry is described by a truncated version of Winfty generated by chiral fields with integer spin s geq 3, whereas the Virasoro algebra becomes abelian and it can be consistently factored out. The geometry of target space looks like an infinitely curved hyperboloid, which invalidates the effective field theory description and conformal invariance can no longer be used to yield reliable space-time interpretation. We also compare our results with the null gauging of WZW models, which correspond to infinite boost in target space and they describe the Liouville mode that decouples in the tensionless limit. A formal BRST analysis of the world-sheet symmetry suggests that the central charge of all higher spin generators should be fixed to a critical value, which is not seen by the contracted Virasoro symmetry. Generalizations to higher dimensional coset models are also briefly discussed in the tensionless limit, where similar observations are made.

Numerical Methods for Forward and Inverse Problems in Discontinuous Media

DOE Office of Scientific and Technical Information (OSTI.GOV)

Chartier, Timothy P.

The research emphasis under this grant's funding is in the area of algebraic multigrid methods. The research has two main branches: 1) exploring interdisciplinary applications in which algebraic multigrid can make an impact and 2) extending the scope of algebraic multigrid methods with algorithmic improvements that are based in strong analysis.The work in interdisciplinary applications falls primarily in the field of biomedical imaging. Work under this grant demonstrated the effectiveness and robustness of multigrid for solving linear systems that result from highly heterogeneous finite element method models of the human head. The results in this work also give promise tomore » medical advances possible with software that may be developed. Research to extend the scope of algebraic multigrid has been focused in several areas. In collaboration with researchers at the University of Colorado, Lawrence Livermore National Laboratory, and Los Alamos National Laboratory, the PI developed an adaptive multigrid with subcycling via complementary grids. This method has very cheap computing costs per iterate and is showing promise as a preconditioner for conjugate gradient. Recent work with Los Alamos National Laboratory concentrates on developing algorithms that take advantage of the recent advances in adaptive multigrid research. The results of the various efforts in this research could ultimately have direct use and impact to researchers for a wide variety of applications, including, astrophysics, neuroscience, contaminant transport in porous media, bi-domain heart modeling, modeling of tumor growth, and flow in heterogeneous porous media. This work has already led to basic advances in computational mathematics and numerical linear algebra and will continue to do so into the future.« less

Successive phase transitions and kink solutions in Φ⁸, Φ¹⁰, and Φ¹² field theories

DOE PAGES

Khare, Avinash; Christov, Ivan C.; Saxena, Avadh

2014-08-27

We obtain exact solutions for kinks in Φ⁸, Φ¹⁰, and Φ¹² field theories with degenerate minima, which can describe a second-order phase transition followed by a first-order one, a succession of two first-order phase transitions and a second-order phase transition followed by two first-order phase transitions, respectively. Such phase transitions are known to occur in ferroelastic and ferroelectric crystals and in meson physics. In particular, we find that the higher-order field theories have kink solutions with algebraically-decaying tails and also asymmetric cases with mixed exponential-algebraic tail decay, unlike the lower-order Φ⁴ and Φ⁶ theories. Additionally, we construct distinct kinks withmore » equal energies in all three field theories considered, and we show the co-existence of up to three distinct kinks (for a Φ¹² potential with six degenerate minima). We also summarize phonon dispersion relations for these systems, showing that the higher-order field theories have specific cases in which only nonlinear phonons are allowed. For the Φ¹⁰ field theory, which is a quasi-exactly solvable (QES) model akin to Φ⁶, we are also able to obtain three analytical solutions for the classical free energy as well as the probability distribution function in the thermodynamic limit.« less

Virasoro algebra in the KN algebra; Bosonic string with fermionic ghosts on Riemann surfaces

DOE Office of Scientific and Technical Information (OSTI.GOV)

Koibuchi, H.

1991-10-10

In this paper the bosonic string model with fermionic ghosts is considered in the framework of the KN algebra. The authors' attentions are paid to representations of KN algebra and a Clifford algebra of the ghosts. The authors show that a Virasoro-like algebra is obtained from KN algebra when KN algebra has certain antilinear anti-involution, and that it is isomorphic to the usual Virasoro algebra. The authors show that there is an expected relation between a central charge of this Virasoro-like algebra and an anomaly of the combined system.

Mathematical Modeling for Inherited Diseases.

PubMed

Anis, Saima; Khan, Madad; Khan, Saqib

2017-01-01

We introduced a new nonassociative algebra, namely, left almost algebra, and discussed some of its genetic properties. We discussed the relation of this algebra with flexible algebra, Jordan algebra, and generalized Jordan algebra.

Lubin-Tate extensions, an elementary approach

NASA Astrophysics Data System (ADS)

Ershov, Yu L.

2007-12-01

We give an elementary proof of the assertion that the Lubin-Tate extension L\\ge K is an Abelian extension whose Galois group is isomorphic to U_K/N_{L/K}(U_L) for arbitrary fields K that have Henselian discrete valuation rings with finite residue fields. The term `elementary' only means that the proofs are algebraic (that is, no transcedental methods are used [1], pp. 327, 332).

Mathematical Modeling for Inherited Diseases

PubMed Central

Khan, Saqib

2017-01-01

We introduced a new nonassociative algebra, namely, left almost algebra, and discussed some of its genetic properties. We discussed the relation of this algebra with flexible algebra, Jordan algebra, and generalized Jordan algebra. PMID:28781606

Projective limits of state spaces IV. Fractal label sets

NASA Astrophysics Data System (ADS)

Lanéry, Suzanne; Thiemann, Thomas

2018-01-01

Instead of formulating the state space of a quantum field theory over one big Hilbert space, it has been proposed by Kijowski (1977) to represent quantum states as projective families of density matrices over a collection of smaller, simpler Hilbert spaces (see Lanéry (2016) [1] for a concise introduction to this formalism). One can thus bypass the need to select a vacuum state for the theory, and still be provided with an explicit and constructive description of the quantum state space, at least as long as the label set indexing the projective structure is countable. Because uncountable label sets are much less practical in this context, we develop in the present article a general procedure to trim an originally uncountable label set down to countable cardinality. In particular, we investigate how to perform this tightening of the label set in a way that preserves both the physical content of the algebra of observables and its symmetries. This work is notably motivated by applications to the holonomy-flux algebra underlying Loop Quantum Gravity. Building on earlier work by Okołów (2013), a projective state space was introduced for this algebra in Lanéry and Thiemann (2016). However, the non-trivial structure of the holonomy-flux algebra prevents the construction of satisfactory semi-classical states (Lanéry and Thiemann, 2017). Implementing the general procedure just mentioned in the case of a one-dimensional version of this algebra, we show how a discrete subalgebra can be extracted without destroying universality nor diffeomorphism invariance. On this subalgebra, quantum states can then be constructed which are more regular than was possible on the original algebra. In particular, this allows the design of semi-classical states whose semi-classicality is enforced step by step, starting from collective, macroscopic degrees of freedom and going down progressively toward smaller and smaller scales.

New infinite-dimensional hidden symmetries for heterotic string theory

DOE Office of Scientific and Technical Information (OSTI.GOV)

Gao Yajun

The symmetry structures of two-dimensional heterotic string theory are studied further. A (2d+n)x(2d+n) matrix complex H-potential is constructed and the field equations are extended into a complex matrix formulation. A pair of Hauser-Ernst-type linear systems are established. Based on these linear systems, explicit formulations of new hidden symmetry transformations for the considered theory are given and then these symmetry transformations are verified to constitute infinite-dimensional Lie algebras: the semidirect product of the Kac-Moody o(d,d+n-circumflex) and Virasoro algebras (without center charges). These results demonstrate that the heterotic string theory under consideration possesses more and richer symmetry structures than previously expected.

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

ERIC Educational Resources Information Center

Gonzalez-Vega, Laureano

1999-01-01

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

Quantum mechanical probability current as electromagnetic 4-current from topological EM fields

NASA Astrophysics Data System (ADS)

van der Mark, Martin B.

2015-09-01

Starting from a complex 4-potential A = αdβ we show that the 4-current density in electromagnetism and the probability current density in relativistic quantum mechanics are of identical form. With the Dirac-Clifford algebra Cl1,3 as mathematical basis, the given 4-potential allows topological solutions of the fields, quite similar to Bateman's construction, but with a double field solution that was overlooked previously. A more general nullvector condition is found and wave-functions of charged and neutral particles appear as topological configurations of the electromagnetic fields.

DOE Office of Scientific and Technical Information (OSTI.GOV)

Osipov, D V

We prove noncommutative reciprocity laws on an algebraic surface defined over a perfect field. These reciprocity laws establish that some central extensions of globally constructed groups split over certain subgroups constructed by points or projective curves on a surface. For a two-dimensional local field with a last finite residue field, the local central extension which is constructed is isomorphic to the central extension which comes from the case of tame ramification of the Abelian two-dimensional local Langlands correspondence suggested by Kapranov. Bibliography: 9 titles.

Localization of effective actions in open superstring field theory

NASA Astrophysics Data System (ADS)

Maccaferri, Carlo; Merlano, Alberto

2018-03-01

We consider the construction of the algebraic part of D-branes tree-level effective action from Berkovits open superstring field theory. Applying this construction to the quartic potential of massless fields carrying a specific worldsheet charge, we show that the full contribution to the potential localizes at the boundary of moduli space, reducing to elementary two-point functions. As examples of this general mechanism, we show how the Yang-Mills quartic potential and the instanton effective action of a Dp/D( p - 4) system are reproduced.

An elementary proof of a criterion for linear disjointness

NASA Astrophysics Data System (ADS)

Dobbs, David E.

2013-06-01

An elementary proof using matrix theory is given for the following criterion: if F/K and L/K are field extensions, with F and L both contained in a common extension field, then F and L are linearly disjoint over K if (and only if) some K-vector space basis of F is linearly independent over L. The material in this note could serve as enrichment material for the unit on fields in a first course on abstract algebra.

A minimal approach to the scattering of physical massless bosons

NASA Astrophysics Data System (ADS)

Boels, Rutger H.; Luo, Hui

2018-05-01

Tree and loop level scattering amplitudes which involve physical massless bosons are derived directly from physical constraints such as locality, symmetry and unitarity, bypassing path integral constructions. Amplitudes can be projected onto a minimal basis of kinematic factors through linear algebra, by employing four dimensional spinor helicity methods or at its most general using projection techniques. The linear algebra analysis is closely related to amplitude relations, especially the Bern-Carrasco-Johansson relations for gluon amplitudes and the Kawai-Lewellen-Tye relations between gluons and graviton amplitudes. Projection techniques are known to reduce the computation of loop amplitudes with spinning particles to scalar integrals. Unitarity, locality and integration-by-parts identities can then be used to fix complete tree and loop amplitudes efficiently. The loop amplitudes follow algorithmically from the trees. A number of proof-of-concept examples are presented. These include the planar four point two-loop amplitude in pure Yang-Mills theory as well as a range of one loop amplitudes with internal and external scalars, gluons and gravitons. Several interesting features of the results are highlighted, such as the vanishing of certain basis coefficients for gluon and graviton amplitudes. Effective field theories are naturally and efficiently included into the framework. Dimensional regularisation is employed throughout; different regularisation schemes are worked out explicitly. The presented methods appear most powerful in non-supersymmetric theories in cases with relatively few legs, but with potentially many loops. For instance, in the introduced approach iterated unitarity cuts of four point amplitudes for non-supersymmetric gauge and gravity theories can be computed by matrix multiplication, generalising the so-called rung-rule of maximally supersymmetric theories. The philosophy of the approach to kinematics also leads to a technique to control colour quantum numbers of scattering amplitudes with matter, especially efficient in the adjoint and fundamental representations.

Passive acoustic measurement of bedload grain size distribution using self-generated noise

NASA Astrophysics Data System (ADS)

Petrut, Teodor; Geay, Thomas; Gervaise, Cédric; Belleudy, Philippe; Zanker, Sebastien

2018-01-01

Monitoring sediment transport processes in rivers is of particular interest to engineers and scientists to assess the stability of rivers and hydraulic structures. Various methods for sediment transport process description were proposed using conventional or surrogate measurement techniques. This paper addresses the topic of the passive acoustic monitoring of bedload transport in rivers and especially the estimation of the bedload grain size distribution from self-generated noise. It discusses the feasibility of linking the acoustic signal spectrum shape to bedload grain sizes involved in elastic impacts with the river bed treated as a massive slab. Bedload grain size distribution is estimated by a regularized algebraic inversion scheme fed with the power spectrum density of river noise estimated from one hydrophone. The inversion methodology relies upon a physical model that predicts the acoustic field generated by the collision between rigid bodies. Here we proposed an analytic model of the acoustic energy spectrum generated by the impacts between a sphere and a slab. The proposed model computes the power spectral density of bedload noise using a linear system of analytic energy spectra weighted by the grain size distribution. The algebraic system of equations is then solved by least square optimization and solution regularization methods. The result of inversion leads directly to the estimation of the bedload grain size distribution. The inversion method was applied to real acoustic data from passive acoustics experiments realized on the Isère River, in France. The inversion of in situ measured spectra reveals good estimations of grain size distribution, fairly close to what was estimated by physical sampling instruments. These results illustrate the potential of the hydrophone technique to be used as a standalone method that could ensure high spatial and temporal resolution measurements for sediment transport in rivers.

A note on derivations of Murray-von Neumann algebras.

PubMed

Kadison, Richard V; Liu, Zhe

2014-02-11

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.

Banach Synaptic Algebras

NASA Astrophysics Data System (ADS)

Foulis, David J.; Pulmannov, Sylvia

2018-04-01

Using a representation theorem of Erik Alfsen, Frederic Schultz, and Erling Størmer for special JB-algebras, we prove that a synaptic algebra is norm complete (i.e., Banach) if and only if it is isomorphic to the self-adjoint part of a Rickart C∗-algebra. Also, we give conditions on a Banach synaptic algebra that are equivalent to the condition that it is isomorphic to the self-adjoint part of an AW∗-algebra. Moreover, we study some relationships between synaptic algebras and so-called generalized Hermitian algebras.

Asymptotic structure of N=2 supergravity in 3D: extended super-BMS3 and nonlinear energy bounds

NASA Astrophysics Data System (ADS)

Fuentealba, Oscar; Matulich, Javier; Troncoso, Ricardo

2017-09-01

The asymptotically flat structure of N=(2,0) supergravity in three spacetime dimensions is explored. The asymptotic symmetries are found to be spanned by an extension of the super-BMS3 algebra, endowed with two independent affine û(1) currents of electric and magnetic type. These currents are associated to U(1) fields being even and odd under parity, respectively. Remarkably, although the U(1) fields do not generate a backreaction on the metric, they provide nontrivial Sugawara-like contributions to the BMS3 generators, and hence to the energy and the angular momentum. Consequently, the entropy of flat cosmological spacetimes endowed with U(1) fields acquires a nontrivial dependence on the zero modes of the û(1) charges. If the spin structure is odd, the ground state corresponds to Minkowski spacetime, and although the anticommutator of the canonical supercharges is linear in the energy and in the electric-like û(1) charge, the energy becomes bounded from below by the energy of the ground state shifted by the square of the electric-like û(1) charge. If the spin structure is even, the same bound for the energy generically holds, unless the absolute value of the electric-like charge is less than minus the mass of Minkowski spacetime in vacuum, so that the energy has to be nonnegative. The explicit form of the global and asymptotic Killing spinors is found for a wide class of configurations that fulfills our boundary conditions, and they exist precisely when the corresponding bounds are saturated. It is also shown that the spectra with periodic or antiperiodic boundary conditions for the fermionic fields are related by spectral flow, in a similar way as it occurs for the N=2 super-Virasoro algebra. Indeed, our supersymmetric extension of BMS3 can be recovered from the Inönü-Wigner contraction of the superconformal algebra with N=(2,2) , once the fermionic generators of the right copy are truncated.

The Unitality of Quantum B-algebras

NASA Astrophysics Data System (ADS)

Han, Shengwei; Xu, Xiaoting; Qin, Feng

2018-02-01

Quantum B-algebras as a generalization of quantales were introduced by Rump and Yang, which cover the majority of implicational algebras and provide a unified semantic for a wide class of substructural logics. Unital quantum B-algebras play an important role in the classification of implicational algebras. The main purpose of this paper is to construct unital quantum B-algebras from non-unital quantum B-algebras.

Towards an M5-brane model I: A 6d superconformal field theory

NASA Astrophysics Data System (ADS)

Sämann, Christian; Schmidt, Lennart

2018-04-01

We present an action for a six-dimensional superconformal field theory containing a non-abelian tensor multiplet. All of the ingredients of this action have been available in the literature. We bring these pieces together by choosing the string Lie 2-algebra as a gauge structure, which we motivated in previous work. The kinematical data contains a connection on a categorified principal bundle, which is the appropriate mathematical description of the parallel transport of self-dual strings. Our action can be written down for each of the simply laced Dynkin diagrams, and each case reduces to a four-dimensional supersymmetric Yang-Mills theory with corresponding gauge Lie algebra. Our action also reduces nicely to an M2-brane model which is a deformation of the Aharony-Bergman-Jafferis-Maldacena (ABJM) model. While this action is certainly not the desired M5-brane model, we regard it as a key stepping stone towards a potential construction of the (2, 0)-theory.

Beyond E 11

NASA Astrophysics Data System (ADS)

Bossard, Guillaume; Kleinschmidt, Axel; Palmkvist, Jakob; Pope, Christopher N.; Sezgin, Ergin

2017-05-01

We study the non-linear realisation of E 11 originally proposed by West with particular emphasis on the issue of linearised gauge invariance. Our analysis shows even at low levels that the conjectured equations can only be invariant under local gauge transformations if a certain section condition that has appeared in a different context in the E 11 literature is satisfied. This section condition also generalises the one known from exceptional field theory. Even with the section condition, the E 11 duality equation for gravity is known to miss the trace component of the spin connection. We propose an extended scheme based on an infinite-dimensional Lie superalgebra, called the tensor hierarchy algebra, that incorporates the section condition and resolves the above issue. The tensor hierarchy algebra defines a generalised differential complex, which provides a systematic description of gauge invariance and Bianchi identities. It furthermore provides an E 11 representation for the field strengths, for which we define a twisted first order self-duality equation underlying the dynamics.

A novel encryption scheme for high-contrast image data in the Fresnelet domain

PubMed Central

Bibi, Nargis; Farwa, Shabieh; Jahngir, Adnan; Usman, Muhammad

2018-01-01

In this paper, a unique and more distinctive encryption algorithm is proposed. This is based on the complexity of highly nonlinear S box in Flesnelet domain. The nonlinear pattern is transformed further to enhance the confusion in the dummy data using Fresnelet technique. The security level of the encrypted image boosts using the algebra of Galois field in Fresnelet domain. At first level, the Fresnelet transform is used to propagate the given information with desired wavelength at specified distance. It decomposes given secret data into four complex subbands. These complex sub-bands are separated into two components of real subband data and imaginary subband data. At second level, the net subband data, produced at the first level, is deteriorated to non-linear diffused pattern using the unique S-box defined on the Galois field F28. In the diffusion process, the permuted image is substituted via dynamic algebraic S-box substitution. We prove through various analysis techniques that the proposed scheme enhances the cipher security level, extensively. PMID:29608609

A three-dimensional algebraic grid generation scheme for gas turbine combustors with inclined slots

NASA Technical Reports Server (NTRS)

Yang, S. L.; Cline, M. C.; Chen, R.; Chang, Y. L.

1993-01-01

A 3D algebraic grid generation scheme is presented for generating the grid points inside gas turbine combustors with inclined slots. The scheme is based on the 2D transfinite interpolation method. Since the scheme is a 2D approach, it is very efficient and can easily be extended to gas turbine combustors with either dilution hole or slot configurations. To demonstrate the feasibility and the usefulness of the technique, a numerical study of the quick-quench/lean-combustion (QQ/LC) zones of a staged turbine combustor is given. Preliminary results illustrate some of the major features of the flow and temperature fields in the QQ/LC zones. Formation of co- and counter-rotating bulk flow and shape temperature fields can be observed clearly, and the resulting patterns are consistent with experimental observations typical of the confined slanted jet-in-cross flow. Numerical solutions show the method to be an efficient and reliable tool for generating computational grids for analyzing gas turbine combustors with slanted slots.

Modular Hamiltonians on the null plane and the Markov property of the vacuum state

NASA Astrophysics Data System (ADS)

Casini, Horacio; Testé, Eduardo; Torroba, Gonzalo

2017-09-01

We compute the modular Hamiltonians of regions having the future horizon lying on a null plane. For a CFT this is equivalent to regions with a boundary of arbitrary shape lying on the null cone. These Hamiltonians have a local expression on the horizon formed by integrals of the stress tensor. We prove this result in two different ways, and show that the modular Hamiltonians of these regions form an infinite dimensional Lie algebra. The corresponding group of unitary transformations moves the fields on the null surface locally along the null generators with arbitrary null line dependent velocities, but act non-locally outside the null plane. We regain this result in greater generality using more abstract tools on the algebraic quantum field theory. Finally, we show that modular Hamiltonians on the null surface satisfy a Markov property that leads to the saturation of the strong sub-additive inequality for the entropies and to the strong super-additivity of the relative entropy.

Canonical gravity, diffeomorphisms and objective histories

NASA Astrophysics Data System (ADS)

Samuel, Joseph

2000-11-01

This paper discusses the implementation of diffeomorphism invariance in purely Hamiltonian formulations of general relativity. We observe that, if a constrained Hamiltonian formulation derives from a manifestly covariant Lagrangian, the diffeomorphism invariance of the Lagrangian results in the following properties of the constrained Hamiltonian theory: the diffeomorphisms are generated by constraints on the phase space so that: (a) the algebra of the generators reflects the algebra of the diffeomorphism group; (b) the Poisson brackets of the basic fields with the generators reflects the spacetime transformation properties of these basic fields. This suggests that in a purely Hamiltonian approach the requirement of diffeomorphism invariance should be interpreted to include (b) and not just (a) as one might naively suppose. Giving up (b) amounts to giving up objective histories, even at the classical level. This observation has implications for loop quantum gravity which are spelled out in a companion paper. We also describe an analogy between canonical gravity and relativistic particle dynamics to illustrate our main point.

Generalizing the bms3 and 2D-conformal algebras by expanding the Virasoro algebra

NASA Astrophysics Data System (ADS)

Caroca, Ricardo; Concha, Patrick; Rodríguez, Evelyn; Salgado-Rebolledo, Patricio

2018-03-01

By means of the Lie algebra expansion method, the centrally extended conformal algebra in two dimensions and the bms3 algebra are obtained from the Virasoro algebra. We extend this result to construct new families of expanded Virasoro algebras that turn out to be infinite-dimensional lifts of the so-called Bk, Ck and Dk algebras recently introduced in the literature in the context of (super)gravity. We also show how some of these new infinite-dimensional symmetries can be obtained from expanded Kač-Moody algebras using modified Sugawara constructions. Applications in the context of three-dimensional gravity are briefly discussed.

DOE Office of Scientific and Technical Information (OSTI.GOV)

Suh, Uhi Rinn, E-mail: uhrisu1@math.snu.ac.kr

We introduce a classical BRST complex (See Definition 3.2.) and show that one can construct a classical affine W-algebra via the complex. This definition clarifies that classical affine W-algebras can be considered as quasi-classical limits of quantum affine W-algebras. We also give a definition of a classical affine fractional W-algebra as a Poisson vertex algebra. As in the classical affine case, a classical affine fractional W-algebra has two compatible λ-brackets and is isomorphic to an algebra of differential polynomials as a differential algebra. When a classical affine fractional W-algebra is associated to a minimal nilpotent, we describe explicit forms ofmore » free generators and compute λ-brackets between them. Provided some assumptions on a classical affine fractional W-algebra, we find an infinite sequence of integrable systems related to the algebra, using the generalized Drinfel’d and Sokolov reduction.« less

A note on derivations of Murray–von Neumann algebras

PubMed Central

Kadison, Richard V.; Liu, Zhe

2014-01-01

A Murray–von Neumann algebra is the algebra of operators affiliated with a finite von Neumann algebra. In this article, we first present a brief introduction to the theory of derivations of operator algebras from both the physical and mathematical points of view. We then describe our recent work on derivations of Murray–von Neumann algebras. We show that the “extended derivations” of a Murray–von Neumann algebra, those that map the associated finite von Neumann algebra into itself, are inner. In particular, we prove that the only derivation that maps a Murray–von Neumann algebra associated with a factor of type II1 into that factor is 0. Those results are extensions of Singer’s seminal result answering a question of Kaplansky, as applied to von Neumann algebras: The algebra may be noncommutative and may even contain unbounded elements. PMID:24469831

A double commutant theorem for Murray–von Neumann algebras

PubMed Central

Liu, Zhe

2012-01-01

Murray–von Neumann algebras are algebras of operators affiliated with finite von Neumann algebras. In this article, we study commutativity and affiliation of self-adjoint operators (possibly unbounded). We show that a maximal abelian self-adjoint subalgebra of the Murray–von Neumann algebra associated with a finite von Neumann algebra is the Murray–von Neumann algebra , where is a maximal abelian self-adjoint subalgebra of and, in addition, is . We also prove that the Murray–von Neumann algebra with the center of is the center of the Murray–von Neumann algebra . Von Neumann’s celebrated double commutant theorem characterizes von Neumann algebras as those for which , where , the commutant of , is the set of bounded operators on the Hilbert space that commute with all operators in . At the end of this article, we present a double commutant theorem for Murray–von Neumann algebras. PMID:22543165

Determination of key parameters of vector multifractal vector fields

NASA Astrophysics Data System (ADS)

Schertzer, D. J. M.; Tchiguirinskaia, I.

2017-12-01

For too long time, multifractal analyses and simulations have been restricted to scalar-valued fields (Schertzer and Tchiguirinskaia, 2017a,b). For instance, the wind velocity multifractality has been mostly analysed in terms of scalar structure functions and with the scalar energy flux. This restriction has had the unfortunate consequences that multifractals were applicable to their full extent in geophysics, whereas it has inspired them. Indeed a key question in geophysics is the complexity of the interactions between various fields or they components. Nevertheless, sophisticated methods have been developed to determine the key parameters of scalar valued fields. In this communication, we first present the vector extensions of the universal multifractal analysis techniques to multifractals whose generator belong to a Levy-Clifford algebra (Schertzer and Tchiguirinskaia, 2015). We point out further extensions noting the increased complexity. For instance, the (scalar) index of multifractality becomes a matrice. Schertzer, D. and Tchiguirinskaia, I. (2015) `Multifractal vector fields and stochastic Clifford algebra', Chaos: An Interdisciplinary Journal of Nonlinear Science, 25(12), p. 123127. doi: 10.1063/1.4937364. Schertzer, D. and Tchiguirinskaia, I. (2017) `An Introduction to Multifractals and Scale Symmetry Groups', in Ghanbarian, B. and Hunt, A. (eds) Fractals: Concepts and Applications in Geosciences. CRC Press, p. (in press). Schertzer, D. and Tchiguirinskaia, I. (2017b) `Pandora Box of Multifractals: Barely Open ?', in Tsonis, A. A. (ed.) 30 Years of Nonlinear Dynamics in Geophysics. Berlin: Springer, p. (in press).

Model predictive control of P-time event graphs

NASA Astrophysics Data System (ADS)

Hamri, H.; Kara, R.; Amari, S.

2016-12-01

This paper deals with model predictive control of discrete event systems modelled by P-time event graphs. First, the model is obtained by using the dater evolution model written in the standard algebra. Then, for the control law, we used the finite-horizon model predictive control. For the closed-loop control, we used the infinite-horizon model predictive control (IH-MPC). The latter is an approach that calculates static feedback gains which allows the stability of the closed-loop system while respecting the constraints on the control vector. The problem of IH-MPC is formulated as a linear convex programming subject to a linear matrix inequality problem. Finally, the proposed methodology is applied to a transportation system.

Which Q-analogue of the squeezed oscillator?

NASA Technical Reports Server (NTRS)

Solomon, Allan I.

1993-01-01

The noise (variance squared) of a component of the electromagnetic field - considered as a quantum oscillator - in the vacuum is equal to one half, in appropriate units (taking Planck's constant and the mass and frequency of the oscillator all equal to 1). A practical definition of a squeezed state is one for which the noise is less than the vacuum value - and the amount of squeezing is determined by the appropriate ratio. Thus the usual coherent (Glauber) states are not squeezed, as they produce the same variance as the vacuum. However, it is not difficult to define states analogous to coherent states which do have this noise-reducing effect. In fact, they are coherent states in the more general group sense but with respect to groups other than the Heisenberg-Weyl Group which defines the Glauber states. The original, conventional squeezed state in quantum optics is that associated with the group SU(1,1). Just as the annihilation operator a of a single photon mode (and its hermitian conjugate a, the creation operator) generates the Heisenberg Weyl algebra, so the pair-photon operator a(sup 2) and its conjugate generates the algebra of the group SU(1,1). Another viewpoint, more productive from the calculational stance, is to note that the automorphism group of the Heisenberg-Weyl algebra is SU(1,1). Needless to say, each of these viewpoints generalizes differently to the quantum group context. Both are discussed. The following topics are addressed: conventional coherent and squeezed states; eigenstate definitions; exponential definitions; algebra (group) definitions; automorphism group definition; example: signal-to-noise ratio; q-coherent and q-squeezed states; M and P q-bosons; eigenstate definitions; exponential definitions; algebra (q-group) definitions; and automorphism q-group definition.

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 intersection of irreducible components of the space of finite-dimensional Lie algebras

DOE Office of Scientific and Technical Information (OSTI.GOV)

Gorbatsevich, Vladimir V

2012-07-31

The irreducible components of the space of n-dimensional Lie algebras are investigated. The properties of Lie algebras belonging to the intersection of all the irreducible components of this kind are studied (these Lie algebras are said to be basic or founding Lie algebras). It is proved that all Lie algebras of this kind are nilpotent and each of these Lie algebras has an Abelian ideal of codimension one. Specific examples of founding Lie algebras of arbitrary dimension are described and, to describe the Lie algebras in general, we state a conjecture. The concept of spectrum of a Lie algebra ismore » considered and some of the most elementary properties of the spectrum are studied. Bibliography: 6 titles.« less

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

PubMed

Verburgt, Lukas M

2016-01-01

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

From simplicial Lie algebras and hypercrossed complexes to differential graded Lie algebras via 1-jets

NASA Astrophysics Data System (ADS)

Jurčo, Branislav

2012-12-01

Let g be a simplicial Lie algebra with Moore complex Ng of length k. Let G be the simplicial Lie group integrating g, such that each Gn is simply connected. We use the 1-jet of the classifying space W¯ G to construct, starting from g, a Lie k-algebra L. The so constructed Lie k-algebra L is actually a differential graded Lie algebra. The differential and the brackets are explicitly described in terms (of a part) of the corresponding k-hypercrossed complex structure of Ng. The result can be seen as a geometric interpretation of Quillen's (purely algebraic) construction of the adjunction between simplicial Lie algebras and dg-Lie algebras.

Spectral geometry of {kappa}-Minkowski space

DOE Office of Scientific and Technical Information (OSTI.GOV)

D'Andrea, Francesco

After recalling Snyder's idea [Phys. Rev. 71, 38 (1947)] of using vector fields over a smooth manifold as 'coordinates on a noncommutative space', we discuss a two-dimensional toy-model whose 'dual' noncommutative coordinates form a Lie algebra: this is the well-known {kappa}-Minkowski space [Phys. Lett. B 334, 348 (1994)]. We show how to improve Snyder's idea using the tools of quantum groups and noncommutative geometry. We find a natural representation of the coordinate algebra of {kappa}-Minkowski as linear operators on an Hilbert space (a major problem in the construction of a physical theory), study its 'spectral properties', and discuss how tomore » obtain a Dirac operator for this space. We describe two Dirac operators. The first is associated with a spectral triple. We prove that the cyclic integral of Dimitrijevic et al. [Eur. Phys. J. C 31, 129 (2003)] can be obtained as Dixmier trace associated to this triple. The second Dirac operator is equivariant for the action of the quantum Euclidean group, but it has unbounded commutators with the algebra.« less

On the modelling of non-reactive and reactive turbulent combustor flows

NASA Technical Reports Server (NTRS)

Nikjooy, Mohammad; So, Ronald M. C.

1987-01-01

A study of non-reactive and reactive axisymmetric combustor flows with and without swirl is presented. Closure of the Reynolds equations is achieved by three models: kappa-epsilon, algebraic stress and Reynolds stress closure. Performance of two locally nonequilibrium and one equilibrium algebraic stress models is analyzed assuming four pressure strain models. A comparison is also made of the performance of a high and a low Reynolds number model for combustor flow calculations using Reynolds stress closures. Effects of diffusion and pressure-strain models on these closures are also investigated. Two models for the scalar transport are presented. One employs the second-moment closure which solves the transport equations for the scalar fluxes, while the other solves the algebraic equations for the scalar fluxes. In addition, two cases of non-premixed and one case of premixed combustion are considered. Fast- and finite-rate chemistry models are applied to non-premixed combustion. Both show promise for application in gas turbine combustors. However, finite rate chemistry models need to be examined to establish a suitable coupling of the heat release effects on turbulence field and rate constants.

Baby de Sitter black holes and dS3/CFT2

NASA Astrophysics Data System (ADS)

de Buyl, Sophie; Detournay, Stéphane; Giribet, Gaston; Ng, Gim Seng

2014-02-01

Unlike three-dimensional Einstein gravity, three-dimensional massive gravity admits asymptotically de Sitter space (dS) black hole solutions. These black holes present interesting features and provide us with toy models to study the dS/CFT correspondence. A remarkable property of these black holes is that they are always in thermal equilibrium with the cosmological horizon of the space that hosts them. This invites us to study the thermodynamics of these solutions within the context of dS/CFT. We study the asymptotic symmetry group of the theory and find that it indeed coincides with the local two-dimensional conformal algebra. The charge algebra associated to the asymptotic Killing vectors consists of two copies of the Virasoro algebra with non-vanishing central extension. We compute the mass and angular momentum of the dS black holes and verify that a naive application of Cardy's formula exactly reproduces the entropy of both the black hole and the cosmological horizon. By adapting the holographic renormalization techniques to the case of dS space, we define the boundary stress tensor of the dual Euclidean conformal field theory.

Spline-based high-accuracy piecewise-polynomial phase-to-sinusoid amplitude converters.

PubMed

Petrinović, Davor; Brezović, Marko

2011-04-01

We propose a method for direct digital frequency synthesis (DDS) using a cubic spline piecewise-polynomial model for a phase-to-sinusoid amplitude converter (PSAC). This method offers maximum smoothness of the output signal. Closed-form expressions for the cubic polynomial coefficients are derived in the spectral domain and the performance analysis of the model is given in the time and frequency domains. We derive the closed-form performance bounds of such DDS using conventional metrics: rms and maximum absolute errors (MAE) and maximum spurious free dynamic range (SFDR) measured in the discrete time domain. The main advantages of the proposed PSAC are its simplicity, analytical tractability, and inherent numerical stability for high table resolutions. Detailed guidelines for a fixed-point implementation are given, based on the algebraic analysis of all quantization effects. The results are verified on 81 PSAC configurations with the output resolutions from 5 to 41 bits by using a bit-exact simulation. The VHDL implementation of a high-accuracy DDS based on the proposed PSAC with 28-bit input phase word and 32-bit output value achieves SFDR of its digital output signal between 180 and 207 dB, with a signal-to-noise ratio of 192 dB. Its implementation requires only one 18 kB block RAM and three 18-bit embedded multipliers in a typical field-programmable gate array (FPGA) device. © 2011 IEEE

Excitonic couplings between molecular crystal pairs by a multistate approximation

NASA Astrophysics Data System (ADS)

Aragó, Juan; Troisi, Alessandro

2015-04-01

In this paper, we present a diabatization scheme to compute the excitonic couplings between an arbitrary number of states in molecular pairs. The method is based on an algebraic procedure to find the diabatic states with a desired property as close as possible to that of some reference states. In common with other diabatization schemes, this method captures the physics of the important short-range contributions (exchange, overlap, and charge-transfer mediated terms) but it becomes particularly suitable in presence of more than two states of interest. The method is formulated to be usable with any level of electronic structure calculations and to diabatize different types of states by selecting different molecular properties. These features make the diabatization scheme presented here especially appropriate in the context of organic crystals, where several excitons localized on the same molecular pair may be found close in energy. In this paper, the method is validated on the tetracene crystal dimer, a well characterized case where the charge transfer (CT) states are closer in energy to the Frenkel excitons (FE). The test system was studied as a function of an external electric field (to explore the effect of changing the relative energy of the CT excited state) and as a function of different intermolecular distances (to probe the strength of the coupling between FE and CT states). Additionally, we illustrate how the approximation can be used to include the environment polarization effect.

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.…

Making Algebra Work: Instructional Strategies that Deepen Student Understanding, within and between Algebraic Representations

ERIC Educational Resources Information Center

Star, Jon R.; Rittle-Johnson, Bethany

2009-01-01

Competence in algebra is increasingly recognized as a critical milestone in students' middle and high school years. The transition from arithmetic to algebra is a notoriously difficult one, and improvements in algebra instruction are greatly needed (National Research Council, 2001). Algebra historically has represented students' first sustained…

- XSUMMER- Transcendental functions and symbolic summation in FORM

NASA Astrophysics Data System (ADS)

Moch, S.; Uwer, P.

2006-05-01

Harmonic sums and their generalizations are extremely useful in the evaluation of higher-order perturbative corrections in quantum field theory. Of particular interest have been the so-called nested sums, where the harmonic sums and their generalizations appear as building blocks, originating for example, from the expansion of generalized hypergeometric functions around integer values of the parameters. In this paper we discuss the implementation of several algorithms to solve these sums by algebraic means, using the computer algebra system FORM. Program summaryTitle of program:XSUMMER Catalogue identifier:ADXQ_v1_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADXQ_v1_0 Program obtainable from:CPC Program Library, Queen's University of Belfast, N. Ireland License:GNU Public License and FORM License Computers:all Operating system:all Program language:FORM Memory required to execute:Depending on the complexity of the problem, recommended at least 64 MB RAM No. of lines in distributed program, including test data, etc.:9854 No. of bytes in distributed program, including test data, etc.:126 551 Distribution format:tar.gz Other programs called:none External files needed:none Nature of the physical problem:Systematic expansion of higher transcendental functions in a small parameter. The expansions arise in the calculation of loop integrals in perturbative quantum field theory. Method of solution:Algebraic manipulations of nested sums. Restrictions on complexity of the problem:Usually limited only by the available disk space. Typical running time:Dependent on the complexity of the problem.

Using Spherical-Harmonics Expansions for Optics Surface Reconstruction from Gradients.

PubMed

Solano-Altamirano, Juan Manuel; Vázquez-Otero, Alejandro; Khikhlukha, Danila; Dormido, Raquel; Duro, Natividad

2017-11-30

In this paper, we propose a new algorithm to reconstruct optics surfaces (aka wavefronts) from gradients, defined on a circular domain, by means of the Spherical Harmonics. The experimental results indicate that this algorithm renders the same accuracy, compared to the reconstruction based on classical Zernike polynomials, using a smaller number of polynomial terms, which potentially speeds up the wavefront reconstruction. Additionally, we provide an open-source C++ library, released under the terms of the GNU General Public License version 2 (GPLv2), wherein several polynomial sets are coded. Therefore, this library constitutes a robust software alternative for wavefront reconstruction in a high energy laser field, optical surface reconstruction, and, more generally, in surface reconstruction from gradients. The library is a candidate for being integrated in control systems for optical devices, or similarly to be used in ad hoc simulations. Moreover, it has been developed with flexibility in mind, and, as such, the implementation includes the following features: (i) a mock-up generator of various incident wavefronts, intended to simulate the wavefronts commonly encountered in the field of high-energy lasers production; (ii) runtime selection of the library in charge of performing the algebraic computations; (iii) a profiling mechanism to measure and compare the performance of different steps of the algorithms and/or third-party linear algebra libraries. Finally, the library can be easily extended to include additional dependencies, such as porting the algebraic operations to specific architectures, in order to exploit hardware acceleration features.

Using Spherical-Harmonics Expansions for Optics Surface Reconstruction from Gradients

PubMed Central

Solano-Altamirano, Juan Manuel; Khikhlukha, Danila

2017-01-01

In this paper, we propose a new algorithm to reconstruct optics surfaces (aka wavefronts) from gradients, defined on a circular domain, by means of the Spherical Harmonics. The experimental results indicate that this algorithm renders the same accuracy, compared to the reconstruction based on classical Zernike polynomials, using a smaller number of polynomial terms, which potentially speeds up the wavefront reconstruction. Additionally, we provide an open-source C++ library, released under the terms of the GNU General Public License version 2 (GPLv2), wherein several polynomial sets are coded. Therefore, this library constitutes a robust software alternative for wavefront reconstruction in a high energy laser field, optical surface reconstruction, and, more generally, in surface reconstruction from gradients. The library is a candidate for being integrated in control systems for optical devices, or similarly to be used in ad hoc simulations. Moreover, it has been developed with flexibility in mind, and, as such, the implementation includes the following features: (i) a mock-up generator of various incident wavefronts, intended to simulate the wavefronts commonly encountered in the field of high-energy lasers production; (ii) runtime selection of the library in charge of performing the algebraic computations; (iii) a profiling mechanism to measure and compare the performance of different steps of the algorithms and/or third-party linear algebra libraries. Finally, the library can be easily extended to include additional dependencies, such as porting the algebraic operations to specific architectures, in order to exploit hardware acceleration features. PMID:29189722

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.

Integrand Reduction Reloaded: Algebraic Geometry and Finite Fields

NASA Astrophysics Data System (ADS)

Sameshima, Ray D.; Ferroglia, Andrea; Ossola, Giovanni

2017-01-01

The evaluation of scattering amplitudes in quantum field theory allows us to compare the phenomenological prediction of particle theory with the measurement at collider experiments. The study of scattering amplitudes, in terms of their symmetries and analytic properties, provides a theoretical framework to develop techniques and efficient algorithms for the evaluation of physical cross sections and differential distributions. Tree-level calculations have been known for a long time. Loop amplitudes, which are needed to reduce the theoretical uncertainty, are more challenging since they involve a large number of Feynman diagrams, expressed as integrals of rational functions. At one-loop, the problem has been solved thanks to the combined effect of integrand reduction, such as the OPP method, and unitarity. However, plenty of work is still needed at higher orders, starting with the two-loop case. Recently, integrand reduction has been revisited using algebraic geometry. In this presentation, we review the salient features of integrand reduction for dimensionally regulated Feynman integrals, and describe an interesting technique for their reduction based on multivariate polynomial division. We also show a novel approach to improve its efficiency by introducing finite fields. Supported in part by the National Science Foundation under Grant PHY-1417354.

Higher spin gauge theory on fuzzy \\boldsymbol {S^4_N}

NASA Astrophysics Data System (ADS)

Sperling, Marcus; Steinacker, Harold C.

2018-02-01

We examine in detail the higher spin fields which arise on the basic fuzzy sphere S^4N in the semi-classical limit. The space of functions can be identified with functions on classical S 4 taking values in a higher spin algebra associated to \

Computers as Cognitive Tools.

ERIC Educational Resources Information Center

Lajoie, Susanne P., Ed.; Derry, Sharon J., Ed.

This book provides exemplars of the types of computer-based learning environments represented by the theoretical camps within the field and the practical applications of the theories. The contributors discuss a variety of computer applications to learning, ranging from school-related topics such as geometry, algebra, biology, history, physics, and…

Coherent state quantization of quaternions

DOE Office of Scientific and Technical Information (OSTI.GOV)

Muraleetharan, B., E-mail: bbmuraleetharan@jfn.ac.lk, E-mail: santhar@gmail.com; Thirulogasanthar, K., E-mail: bbmuraleetharan@jfn.ac.lk, E-mail: santhar@gmail.com

Parallel to the quantization of the complex plane, using the canonical coherent states of a right quaternionic Hilbert space, quaternion field of quaternionic quantum mechanics is quantized. Associated upper symbols, lower symbols, and related quantities are analyzed. Quaternionic version of the harmonic oscillator and Weyl-Heisenberg algebra are also obtained.

G-theory: The generator of M-theory and supersymmetry

NASA Astrophysics Data System (ADS)

Sepehri, Alireza; Pincak, Richard

2018-04-01

In string theory with ten dimensions, all Dp-branes are constructed from D0-branes whose action has two-dimensional brackets of Lie 2-algebra. Also, in M-theory, with 11 dimensions, all Mp-branes are built from M0-branes whose action contains three-dimensional brackets of Lie 3-algebra. In these theories, the reason for difference between bosons and fermions is unclear and especially in M-theory there is not any stable object like stable M3-branes on which our universe would be formed on it and for this reason it cannot help us to explain cosmological events. For this reason, we construct G-theory with M dimensions whose branes are formed from G0-branes with N-dimensional brackets. In this theory, we assume that at the beginning there is nothing. Then, two energies, which differ in their signs only, emerge and produce 2M degrees of freedom. Each two degrees of freedom create a new dimension and then M dimensions emerge. M-N of these degrees of freedom are removed by symmetrically compacting half of M-N dimensions to produce Lie-N-algebra. In fact, each dimension produces a degree of freedom. Consequently, by compacting M-N dimensions from M dimensions, N dimensions and N degrees of freedom is emerged. These N degrees of freedoms produce Lie-N-algebra. During this compactification, some dimensions take extra i and are different from other dimensions, which are known as time coordinates. By this compactification, two types of branes, Gp and anti-Gp-branes, are produced and rank of tensor fields which live on them changes from zero to dimension of brane. The number of time coordinates, which are produced by negative energy in anti-Gp-branes, is more sensible to number of times in Gp-branes. These branes are compactified anti-symmetrically and then fermionic superpartners of bosonic fields emerge and supersymmetry is born. Some of gauge fields play the role of graviton and gravitino and produce the supergravity. The question may arise that what is the physical reason which shows that this theory is true. We shown that G-theory can be reduced to other theories like nonlinear gravity theories in four dimensions. Also, this theory, can explain the physical properties of fermions and bosons. On the other hand, this theory explains the origin of supersymmetry. For this reason, we can prove that this theory is true. By reducing the dimension of algebra to three and dimension of world to 11 and dimension of brane to four, G-theory is reduced to F(R)-gravity.

Collision Dynamics of Rydberg Atoms and Molecules at Ultralow Energies

DTIC Science & Technology

2005-12-31

body recombination between electrons, ions and neural gas atoms. We wish to study the interaction and collisions between two Rydberg atoms in the...transitions, Exact solutions of Stark mixing in atomic hydro- where Ekjn is the Levi - Civita antisymmetric symbol gen induced by the time-dependent...L and U do not close under commutation to form a Lie algebra because [Ui, Uj] = (-2g)iCijkLk, where cijk is the Levi - Civita antisymmetric symbol for

An Evaluation of a Modified Simulated Annealing Algorithm for Various Formulations

DTIC Science & Technology

1990-08-01

trials of the K"h Markov chain, is sufficiently close to q(c, ), the stationary distribution at ck la (Lk,c,,) - q(c.) < epsilon Requiring the final...Wiley and Sons . Aarts, E. H. L., & Van Laarhoven, P. J. M. (1985). Statistical cooling: A general approach to combinatorial optimization problems...Birkhoff, G. (1946). Tres observaciones sobre el algebra lineal, Rev. Univ. Nac. TucumanSer. A, 5, 147-151. Bohr, Niels (1913). Old quantum theory

An analogue of the Berry phase for simple harmonic oscillators

NASA Astrophysics Data System (ADS)

Suslov, S. K.

2013-03-01

We evaluate a variant of Berry's phase for a ‘missing’ family of the square integrable wavefunctions for the linear harmonic oscillator, which cannot be derived by the separation of variables (in a natural way). Instead, it is obtained by the action of the maximal kinematical invariance group on the standard solutions. A simple closed formula for the phase (in terms of elementary functions) is found here by integration with the help of a computer algebra system.

A cluster bootstrap for two-loop MHV amplitudes

DOE PAGES

Golden, John; Spradlin, Marcus

2015-02-02

We apply a bootstrap procedure to two-loop MHV amplitudes in planar N=4 super-Yang-Mills theory. We argue that the mathematically most complicated part (the Λ 2 B 2 coproduct component) of the n-particle amplitude is uniquely determined by a simple cluster algebra property together with a few physical constraints (dihedral symmetry, analytic structure, supersymmetry, and well-defined collinear limits). Finally, we present a concise, closed-form expression which manifests these properties for all n.

Colliding holes in Riemann surfaces and quantum cluster algebras

NASA Astrophysics Data System (ADS)

Chekhov, Leonid; Mazzocco, Marta

2018-01-01

In this paper, we describe a new type of surgery for non-compact Riemann surfaces that naturally appears when colliding two holes or two sides of the same hole in an orientable Riemann surface with boundary (and possibly orbifold points). As a result of this surgery, bordered cusps appear on the boundary components of the Riemann surface. In Poincaré uniformization, these bordered cusps correspond to ideal triangles in the fundamental domain. We introduce the notion of bordered cusped Teichmüller space and endow it with a Poisson structure, quantization of which is achieved with a canonical quantum ordering. We give a complete combinatorial description of the bordered cusped Teichmüller space by introducing the notion of maximal cusped lamination, a lamination consisting of geodesic arcs between bordered cusps and closed geodesics homotopic to the boundaries such that it triangulates the Riemann surface. We show that each bordered cusp carries a natural decoration, i.e. a choice of a horocycle, so that the lengths of the arcs in the maximal cusped lamination are defined as λ-lengths in Thurston-Penner terminology. We compute the Goldman bracket explicitly in terms of these λ-lengths and show that the groupoid of flip morphisms acts as a generalized cluster algebra mutation. From the physical point of view, our construction provides an explicit coordinatization of moduli spaces of open/closed string worldsheets and their quantization.

Generalized Clifford Algebras as Algebras in Suitable Symmetric Linear Gr-Categories

NASA Astrophysics Data System (ADS)

Cheng, Tao; Huang, Hua-Lin; Yang, Yuping

2016-01-01

By viewing Clifford algebras as algebras in some suitable symmetric Gr-categories, Albuquerque and Majid were able to give a new derivation of some well known results about Clifford algebras and to generalize them. Along the same line, Bulacu observed that Clifford algebras are weak Hopf algebras in the aforementioned categories and obtained other interesting properties. The aim of this paper is to study generalized Clifford algebras in a similar manner and extend the results of Albuquerque, Majid and Bulacu to the generalized setting. In particular, by taking full advantage of the gauge transformations in symmetric linear Gr-categories, we derive the decomposition theorem and provide categorical weak Hopf structures for generalized Clifford algebras in a conceptual and simpler manner.

Positive-entropy Hamiltonian systems on Nilmanifolds via scattering

NASA Astrophysics Data System (ADS)

Butler, Leo T.

2014-10-01

Let Σ be a compact quotient of T4, the Lie group of 4 × 4 upper triangular matrices with unity along the diagonal. The Lie algebra {\\mathfrak t}4 of T4 has the standard basis {Xij} of matrices with 0 everywhere but in the (i, j) entry, which is unity. Let g be the Carnot metric, a sub-Riemannian metric, on T4 for which Xi, i+1, (i = 1, 2, 3), is an orthonormal basis. Montgomery, Shapiro and Stolin showed that the geodesic flow of g is algebraically non-integrable. This paper proves that the geodesic flow of that Carnot metric on TΣ has positive topological entropy and its Euler field is real-analytically non-integrable. It extends earlier work by Butler and Gelfreich.

A real signal and its states

NASA Astrophysics Data System (ADS)

Basiladze, S. G.

2017-05-01

The paper describes the general physical theory of signals, carriers of information, which supplements Shannon's abstract classical theory and is applicable in much broader fields, including nuclear physics. It is shown that in the absence of classical noise its place should be taken by the physical threshold of signal perception for objects of both macrocosm and microcosm. The signal perception threshold allows the presence of subthreshold (virtual) signal states. For these states, Boolean algebra of logic ( A = 0/1) is transformed into the "algebraic logic" of probabilities (0 ≤ a ≤ 1). The similarity and difference of virtual states of macroand microsignals are elucidated. "Real" and "quantum" information for computers is considered briefly. The maximum information transmission rate is estimated based on physical constants.

An arena for model building in the Cohen-Glashow very special relativity

DOE Office of Scientific and Technical Information (OSTI.GOV)

Sheikh-Jabbari, M. M., E-mail: jabbari@theory.ipm.ac.i; Tureanu, A., E-mail: anca.tureanu@helsinki.f

2010-02-15

The Cohen-Glashow Very Special Relativity (VSR) algebra is defined as the part of the Lorentz algebra which upon addition of CP or T invariance enhances to the full Lorentz group, plus the space-time translations. We show that noncommutative space-time, in particular noncommutative Moyal plane, with light- like noncommutativity provides a robust mathematical setting for quantum field theories which are VSR invariant and hence set the stage for building VSR invariant particle physics models. In our setting the VSR invariant theories are specified with a single deformation parameter, the noncommutativity scale {Lambda}{sub NC}. Preliminary analysis with the available data leads tomore » {Lambda}{sub NC} {>=} 1-10 TeV.« less

Quantum field theory in generalised Snyder spaces

NASA Astrophysics Data System (ADS)

Meljanac, S.; Meljanac, D.; Mignemi, S.; Štrajn, R.

2017-05-01

We discuss the generalisation of the Snyder model that includes all possible deformations of the Heisenberg algebra compatible with Lorentz invariance and investigate its properties. We calculate perturbatively the law of addition of momenta and the star product in the general case. We also undertake the construction of a scalar field theory on these noncommutative spaces showing that the free theory is equivalent to the commutative one, like in other models of noncommutative QFT.

Groenewold-Moyal product, α*-cohomology, and classification of translation-invariant non-commutative structures

DOE Office of Scientific and Technical Information (OSTI.GOV)

Varshovi, Amir Abbass

2013-07-15

The theory of α*-cohomology is studied thoroughly and it is shown that in each cohomology class there exists a unique 2-cocycle, the harmonic form, which generates a particular Groenewold-Moyal star product. This leads to an algebraic classification of translation-invariant non-commutative structures and shows that any general translation-invariant non-commutative quantum field theory is physically equivalent to a Groenewold-Moyal non-commutative quantum field theory.

Coupling of c  =  ‑2 and c =\\frac{1}{2} and c  =  0 conformal field theories: the geometrical point of view

NASA Astrophysics Data System (ADS)

Najafi, M. N.

2018-04-01

The coupling of the c  =  ‑2, c=\\frac{1}{2} and c  =  0 conformal field theories are numerically considered in this paper. As the prototypes of the couplings, (c_1=-2)\\oplus (c_2=0) and (c_1=-2)\\oplus (c_2=\\frac{1}{2}) , we consider the Bak–Tang–Weisenfeld (BTW) model on the 2D square critical site-percolation and the BTW model on Ising-correlated percolation lattices respectively. Some geometrical techniques are used to characterize the presumable conformal symmetry of the resultant systems. Based on the numerical analysis of the diffusivity parameter (κ) in the Schramm–Loewner evolution (SLE) theory we propose that the algebra of the central charges of the coupled models is closed. This result is based on the analysis of the conformal loop ensemble (CLE) analysis. The diffusivity parameter in each case is obtained by calculating the fractal dimension of loops (and the corresponding exponent of mean-square root distance), the direct SLE mapping method, the left passage probability and the winding angle analysis. More precisely we numerically show that the coupling (c_1=-2)\\oplus (c_2=\\frac{1}{2}) results to 2D self-avoiding walk (SAW) fixed point corresponding to c  =  0 conformal field theory, whereas the coupling (c_1=-2)\\oplus (c_2=0) results to the 2D critical Ising fixed point corresponding to the c=\\frac{1}{2} conformal field theory.

A near-wall turbulence model and its application to fully developed turbulent channel and pipe flows

NASA Technical Reports Server (NTRS)

Kim, S.-W.

1988-01-01

A near wall turbulence model and its incorporation into a multiple-time-scale turbulence model are presented. In the method, the conservation of mass, momentum, and the turbulent kinetic energy equations are integrated up to the wall; and the energy transfer rate and the dissipation rate inside the near wall layer are obtained from algebraic equations. The algebraic equations for the energy transfer rate and the dissipation rate inside the near wall layer were obtained from a k-equation turbulence model and the near wall analysis. A fully developed turbulent channel flow and fully developed turbulent pipe flows were solved using a finite element method to test the predictive capability of the turbulence model. The computational results compared favorably with experimental data. It is also shown that the present turbulence model could resolve the over shoot phenomena of the turbulent kinetic energy and the dissipation rate in the region very close to the wall.

Closure of the operator product expansion in the non-unitary bootstrap

DOE Office of Scientific and Technical Information (OSTI.GOV)

Esterlis, Ilya; Fitzpatrick, A. Liam; Ramirez, David M.

We use the numerical conformal bootstrap in two dimensions to search for finite, closed sub-algebras of the operator product expansion (OPE), without assuming unitarity. We find the minimal models as special cases, as well as additional lines of solutions that can be understood in the Coulomb gas formalism. All the solutions we find that contain the vacuum in the operator algebra are cases where the external operators of the bootstrap equation are degenerate operators, and we argue that this follows analytically from the expressions in arXiv:1202.4698 for the crossing matrices of Virasoro conformal blocks. Our numerical analysis is a specialmore » case of the “Gliozzi” bootstrap method, and provides a simpler setting in which to study technical challenges with the method. In the supplementary material, we provide a Mathematica notebook that automates the calculation of the crossing matrices and OPE coefficients for degenerate operators using the formulae of Dotsenko and Fateev.« less

Applications of computer algebra to distributed parameter systems

NASA Technical Reports Server (NTRS)

Storch, Joel A.

1993-01-01

In the analysis of vibrations of continuous elastic systems, one often encounters complicated transcendental equations with roots directly related to the system's natural frequencies. Typically, these equations contain system parameters whose values must be specified before a numerical solution can be obtained. The present paper presents a method whereby the fundamental frequency can be obtained in analytical form to any desired degree of accuracy. The method is based upon truncation of rapidly converging series involving inverse powers of the system natural frequencies. A straightforward method to developing these series and summing them in closed form is presented. It is demonstrated how Computer Algebra can be exploited to perform the intricate analytical procedures which otherwise would render the technique difficult to apply in practice. We illustrate the method by developing two analytical approximations to the fundamental frequency of a vibrating cantilever carrying a rigid tip body. The results are compared to the numerical solution of the exact (transcendental) frequency equation over a range of system parameters.

Closure of the operator product expansion in the non-unitary bootstrap

DOE PAGES

Esterlis, Ilya; Fitzpatrick, A. Liam; Ramirez, David M.

2016-11-07

We use the numerical conformal bootstrap in two dimensions to search for finite, closed sub-algebras of the operator product expansion (OPE), without assuming unitarity. We find the minimal models as special cases, as well as additional lines of solutions that can be understood in the Coulomb gas formalism. All the solutions we find that contain the vacuum in the operator algebra are cases where the external operators of the bootstrap equation are degenerate operators, and we argue that this follows analytically from the expressions in arXiv:1202.4698 for the crossing matrices of Virasoro conformal blocks. Our numerical analysis is a specialmore » case of the “Gliozzi” bootstrap method, and provides a simpler setting in which to study technical challenges with the method. In the supplementary material, we provide a Mathematica notebook that automates the calculation of the crossing matrices and OPE coefficients for degenerate operators using the formulae of Dotsenko and Fateev.« less

Octupolar tensors for liquid crystals

NASA Astrophysics Data System (ADS)

Chen, Yannan; Qi, Liqun; Virga, Epifanio G.

2018-01-01

A third-rank three-dimensional symmetric traceless tensor, called the octupolar tensor, has been introduced to study tetrahedratic nematic phases in liquid crystals. The octupolar potential, a scalar-valued function generated on the unit sphere by that tensor, should ideally have four maxima (on the vertices of a tetrahedron), but it was recently found to possess an equally generic variant with three maxima instead of four. It was also shown that the irreducible admissible region for the octupolar tensor in a three-dimensional parameter space is bounded by a dome-shaped surface, beneath which is a separatrix surface connecting the two generic octupolar states. The latter surface, which was obtained through numerical continuation, may be physically interpreted as marking a possible intra-octupolar transition. In this paper, by using the resultant theory of algebraic geometry and the E-characteristic polynomial of spectral theory of tensors, we give a closed-form, algebraic expression for both the dome-shaped surface and the separatrix surface. This turns the envisaged intra-octupolar transition into a quantitative, possibly observable prediction.

Dynamical systems defined on infinite dimensional lie algebras of the ''current algebra'' or ''Kac-Moody'' type

NASA Astrophysics Data System (ADS)

Hermann, Robert

1982-07-01

Recent work by Morrison, Marsden, and Weinstein has drawn attention to the possibility of utilizing the cosymplectic structure of the dual of the Lie algebra of certain infinite dimensional Lie groups to study hydrodynamical and plasma systems. This paper treats certain models arising in elementary particle physics, considered by Lee, Weinberg, and Zumino; Sugawara; Bardacki, Halpern, and Frishman; Hermann; and Dolan. The lie algebras involved are associated with the ''current algebras'' of Gell-Mann. This class of Lie algebras contains certain of the algebras that are called ''Kac-Moody algebras'' in the recent mathematics and mathematical physics literature.

Entanglement control in a superconducting qubit system by an electromagnetic field

NASA Astrophysics Data System (ADS)

Zhang, Y. Q.; Xu, J. B.

2011-08-01

By making use of the dynamical algebraic method we investigate a quantum system consisting of superconducting qubits interacting with data buses, where the qubits are driven by time-dependent electromagnetic field and obtain an explicit expression of time evolution operator. Furthermore, we explore the entanglement dynamics and the influence of the time-dependent electromagnetic field and the initial state on the entanglement sudden death and birth for the system. It is shown that the entanglement between the qubit and bus as well as the entanglement sudden death and birth can be controlled by the time-dependent electromagnetic field.

THE SMALLEST FIELD OF DEFINITION OF A SUBGROUP OF THE GROUP \\mathrm{PSL}_2

NASA Astrophysics Data System (ADS)

Vinberg, È. B.

1995-02-01

As previously proved by the author, for each semisimple algebraic group of adjoint type that is dense in the Zariski topology there exists a smallest field of definition which is an invariant of the class of commensurable subgroups. In the present paper an algorithm is given for finding the smallest field of definition of a dense finitely generated subgroup of the group \\mathrm{PSL}_2(\\mathbb{C}). A criterion for arithmeticity of a lattice in \\mathrm{PSL}_2(\\mathbb{R}) or \\mathrm{PSL}_2(\\mathbb{C}) in terms of this field is presented.Bibliography: 7 titles.

Conformal field theories from deformations of theories with Wn symmetry

NASA Astrophysics Data System (ADS)

Babaro, Juan Pablo; Giribet, Gaston; Ranjbar, Arash

2016-10-01

We construct a set of nonrational conformal field theories that consist of deformations of Toda field theory for s l (n ). In addition to preserving conformal invariance, the theories may still exhibit a remnant infinite-dimensional affine symmetry. The case n =3 is used to illustrate this phenomenon, together with further deformations that yield enhanced Kac-Moody symmetry algebras. For generic n we compute N -point correlation functions on the Riemann sphere and show that these can be expressed in terms of s l (n ) Toda field theory ((N -2 )n +2 ) -point correlation functions.

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…

A Model for Math Modeling

ERIC Educational Resources Information Center

Lin, Tony; Erfan, Sasan

2016-01-01

Mathematical modeling is an open-ended research subject where no definite answers exist for any problem. Math modeling enables thinking outside the box to connect different fields of studies together including statistics, algebra, calculus, matrices, programming and scientific writing. As an integral part of society, it is the foundation for many…

Causal Attributions for Success or Failure of Students in College Algebra

ERIC Educational Resources Information Center

Cortes-Suarez, Gina; Sandiford, Janice R.

2008-01-01

Research in the field of attribution theory and academic achievement suggests a relationship between a student's attributional style and achievement. Theorists and researchers contend that attributions influence individual reactions to success and failure. They also report that individuals use attributions to explain and justify their performance.…

Math for Electronics; Industrial Electronics 1: 9323.04.

ERIC Educational Resources Information Center

Dade County Public Schools, Miami, FL.

This curriculum guide is designed for the student interested in preparing for vocational electronics and related fields of electricity, emphasizing the mathematics necessary for an indepth study of electronics. Included in the course content are goals, specific block objectives, basic algebra, powers of 10, the slide rule, basic trigonometry…

76 FR 65187 - Notice of Proposed Information Collection Requests

Federal Register 2010, 2011, 2012, 2013, 2014

2011-10-20

... careers. This study includes a new student assessment in algebraic skills, reasoning, and problem solving... School Longitudinal Study of 2009 (HSLS:09) High School Transcript Collection and College Update Field... Longitudinal Study of 2009 (HSLS:09) is a nationally representative, longitudinal study of more than 20,000...