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.
NASA Astrophysics Data System (ADS)
Paullin, Katherine L.
Many of a 3-manifold's properties are determined by the surfaces they contain, and this knowledge leads to the foundation of decision algorithms for 3- manifolds. Popular work influencing the work of 3-manifold algorithms has it's roots in normal surface theory. In a triangulated 3-manifold, Haken and Kneser showed that we could put any incompressible surface into normal form. Expanding on those techniques, Rubinstein and Stocking later showed we could put any strongly irreducible surface into almost normal form. Walsh has more recently shown that in an ideal triangulation of a hyperbolic manifold many surfaces can be spun normalized. One unsolved problem in 3-manifold algorithms is studying the complexity of Lens Space Recognition. Spun almost normalization appears to be a part of solving this larger problem. In this dissertation, I will first discuss a nontraditional technique using graphs of equivalence classes of compressing disks that allows us to take a combinatorial approach to generalize the result of Walsh's to nonhyperbolic manifolds. Using that method, I'll also explore the conditions needed to show that a surface can be spun almost normalized.
Hom-Lie algebras with symmetric invariant nondegenerate bilinear forms
NASA Astrophysics Data System (ADS)
Benayadi, Saïd; Makhlouf, Abdenacer
2014-02-01
The aim of this paper is to introduce and study quadratic Hom-Lie algebras, which are Hom-Lie algebras equipped with symmetric invariant nondegenerate bilinear forms. We provide several constructions leading to examples and extend the Double Extension Theory to this class of nonassociative algebras. Elements of Representation Theory for Hom-Lie algebras, including adjoint and coadjoint representations, are supplied with application to quadratic Hom-Lie algebras. Centerless involutive quadratic Hom-Lie algebras are characterized. We reduce the case where the twist map is invertible to the study of involutive quadratic Lie algebras. Also, we establish a correspondence between the class of involutive quadratic Hom-Lie algebras and quadratic simple Lie algebras with symmetric involution.
Normal forms of Hopf-zero singularity
NASA Astrophysics Data System (ADS)
Gazor, Majid; Mokhtari, Fahimeh
2015-01-01
The Lie algebra generated by Hopf-zero classical normal forms is decomposed into two versal Lie subalgebras. Some dynamical properties for each subalgebra are described; one is the set of all volume-preserving conservative systems while the other is the maximal Lie algebra of nonconservative systems. This introduces a unique conservative-nonconservative decomposition for the normal form systems. There exists a Lie-subalgebra that is Lie-isomorphic to a large family of vector fields with Bogdanov-Takens singularity. This gives rise to a conclusion that the local dynamics of formal Hopf-zero singularities is well-understood by the study of Bogdanov-Takens singularities. Despite this, the normal form computations of Bogdanov-Takens and Hopf-zero singularities are independent. Thus, by assuming a quadratic nonzero condition, complete results on the simplest Hopf-zero normal forms are obtained in terms of the conservative-nonconservative decomposition. Some practical formulas are derived and the results implemented using Maple. The method has been applied on the Rössler and Kuramoto-Sivashinsky equations to demonstrate the applicability of our results.
The simplest normal form of Hopf bifurcation
NASA Astrophysics Data System (ADS)
Yu, P.; Leung, A. Y. T.
2003-01-01
Recently, further reduction on normal forms of differential equations leading to the simplest normal forms (SNFs) has received considerable attention. However, the computation of the SNF has been mainly restricted to systems which do not contain perturbation parameters (unfolding), since the computation of the SNF with unfolding is much more complicated than that of the SNF without unfolding. From the practical point of view, only the SNF with perturbation (bifurcation) parameters is useful in analysing physical or engineering problems. It is shown that the SNF with unfolding cannot be obtained using only near-identity transformation. Additional transformations such as time and parameter rescaling need to be introduced. An efficient computational method is presented for computing the algebraic equations that can be used to find the SNF. A physical example is given to show the applicability of the new method.
Diagonalization and Jordan Normal Form--Motivation through "Maple"[R
ERIC Educational Resources Information Center
Glaister, P.
2009-01-01
Following an introduction to the diagonalization of matrices, one of the more difficult topics for students to grasp in linear algebra is the concept of Jordan normal form. In this note, we show how the important notions of diagonalization and Jordan normal form can be introduced and developed through the use of the computer algebra package…
Forms and algebras in (half-)maximal supergravity theories
NASA Astrophysics Data System (ADS)
Howe, Paul; Palmkvist, Jakob
2015-05-01
The forms in D-dimensional (half-)maximal supergravity theories are discussed for 3 ≤ D ≤ 11. Superspace methods are used to derive consistent sets of Bianchi identities for all the forms for all degrees, and to show that they are soluble and fully compatible with supersymmetry. The Bianchi identities determine Lie superalgebras that can be extended to Borcherds superalgebras of a special type. It is shown that any Borcherds superalgebra of this type gives the same form spectrum, up to an arbitrary degree, as an associated Kac-Moody algebra. For maximal supergravity up to D-form potentials, this is the very extended Kac-Moody algebra E 11. It is also shown how gauging can be carried out in a simple fashion by deforming the Bianchi identities by means of a new algebraic element related to the embedding tensor. In this case the appropriate extension of the form algebra is a truncated version of the so-called tensor hierarchy algebra.
Box products in nilpotent normal form theory: The factoring method
NASA Astrophysics Data System (ADS)
Murdock, James
2016-01-01
Let N be a nilpotent matrix and consider vector fields x ˙ = Nx + v (x) in normal form. Then v is equivariant under the flow eN*t for the inner product normal form or eMt for the sl2 normal form. These vector equivariants can be found by finding the scalar invariants for the Jordan blocks in N* or M; taking the box product of these to obtain the invariants for N* or M itself; and then boosting the invariants to equivariants by another box product. These methods, developed by Murdock and Sanders in 2007, are here given a self-contained exposition with new foundations and new algorithms yielding improved (simpler) Stanley decompositions for the invariants and equivariants. Ideas used include transvectants (from classical invariant theory), Stanley decompositions (from commutative algebra), and integer cones (from integer programming). This approach can be extended to covariants of sl2k for k > 1, known as SLOCC in quantum computing.
Symmetric structure of field algebra of G-spin models determined by a normal subgroup
Xin, Qiaoling Jiang, Lining
2014-09-15
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.
Birkhoff Normal Form for Some Nonlinear PDEs
NASA Astrophysics Data System (ADS)
Bambusi, Dario
We consider the problem of extending to PDEs Birkhoff normal form theorem on Hamiltonian systems close to nonresonant elliptic equilibria. As a model problem we take the nonlinear wave equation
Normal Forms for Nonautonomous Differential Equations
NASA Astrophysics Data System (ADS)
Siegmund, Stefan
2002-01-01
We extend Henry Poincarés normal form theory for autonomous differential equations x=f(x) to nonautonomous differential equations x=f(t, x). Poincarés nonresonance condition λj-∑ni=1 ℓiλi≠0 for eigenvalues is generalized to the new nonresonance condition λj∩∑ni=1 ℓiλi=∅ for spectral intervals.
Bikchentaev, A M
2008-04-30
It is proved that every skew-Hermitian element of any properly infinite von Neumann algebra can be represented in the form of a finite sum of commutators of projections in this algebra. A new commutation condition for projections in terms of their upper (lower) bound in the lattice of all projections of the algebra is obtained. For the full matrix algebra the set of operators with canonical trace zero is described in terms of finite sums of commutators of projections and the domain in which the trace is positive is described in terms of finite sums of pairwise products of projections. Applications to AF-algebras are obtained. Bibliography: 33 titles.
A brief study of quasi-normal modes in relativistic stars using algebraic computation
Campos, M. de
2010-11-12
The damped oscillations in relativistic stars generate gravitational waves that in the literature appear under the general denomination of quasi-normal modes. In this brief note we want offer some information about the use of algebraic computation to obtain the field equations and the perturbed version of them, in the context of general relativity theory, that is the framework to study gravitational waves in this work.
Computer algebra and operators
NASA Technical Reports Server (NTRS)
Fateman, Richard; Grossman, Robert
1989-01-01
The symbolic computation of operator expansions is discussed. Some of the capabilities that prove useful when performing computer algebra computations involving operators are considered. These capabilities may be broadly divided into three areas: the algebraic manipulation of expressions from the algebra generated by operators; the algebraic manipulation of the actions of the operators upon other mathematical objects; and the development of appropriate normal forms and simplification algorithms for operators and their actions. Brief descriptions are given of the computer algebra computations that arise when working with various operators and their actions.
Resonant normal form and asymptotic normal form behaviour in magnetic bottle Hamiltonians
NASA Astrophysics Data System (ADS)
Efthymiopoulos, C.; Harsoula, M.; Contopoulos, G.
2015-04-01
We consider normal forms in ‘magnetic bottle’ type Hamiltonians of the form H=\\frac{1}{2}(ρ^2_ρ+ω^2_1ρ^2) +\\frac{1}{2}p^2_z+hot (second frequency ω2 equal to zero in the lowest order). Our main results are: (i) a novel method to construct the normal form in cases of resonance, and (ii) a study of the asymptotic behaviour of both the non-resonant and the resonant series. We find that, if we truncate the normal form series at order r, the series remainder in both constructions decreases with increasing r down to a minimum, and then it increases with r. The computed minimum remainder turns to be exponentially small in \\frac{1}{Δ E} , where ΔE is the mirror oscillation energy, while the optimal order scales as an inverse power of ΔE. We estimate numerically the exponents associated with the optimal order and the remainder's exponential asymptotic behaviour. In the resonant case, our novel method allows to compute a ‘quasi-integral’ (i.e. truncated formal integral) valid both for each particular resonance as well as away from all resonances. We applied these results to a specific magnetic bottle Hamiltonian. The non-resonant normal form yields theoretical invariant curves on a surface of section which fit well the empirical curves away from resonances. On the other hand the resonant normal form fits very well both the invariant curves inside the islands of a particular resonance as well as the non-resonant invariant curves. Finally, we discuss how normal forms allow to compute a critical threshold for the onset of global chaos in the magnetic bottle.
ADDENDUM: The classification of Novikov algebras in low dimensions: invariant bilinear forms
NASA Astrophysics Data System (ADS)
Bai, Chengming; Meng, Daoji
2001-10-01
In this note, we give a complete classification of the (non-degenerate) symmetric invariant bilinear forms on Novikov algebras in dimension 2 and 3, which can be regarded as an addendum of the classification of Novikov algebras in low dimensions given in our previous work (Bai C M and Meng D J 2001 J. Phys. A: Math. Gen. 34 1581-94).
Algebras Generated by Geometric Scalar Forms and their Applications in Physics and Social Sciences
Keller, Jaime
2008-09-17
The present paper analyzes the consequences of defining that the geometric scalar form is not necessarily quadratic, but in general K-atic, that is obtained from the K{sup th} power of the linear form, requiring {l_brace}e{sub i};i = 1,...,N;(e{sub i}){sup K} = 1{r_brace} and d-vector {sigma}{sub i}x{sub i}e{sub i}. We consider the algebras which are thus generated, for positive integer K, a generalization of the geometric algebras we know under the names of Clifford or Grassmann algebras. We then obtain a set of geometric K-algebras. We also consider the generalization of special functions of geometry which corresponds to the K-order scalar forms (as trigonometric functions and other related geometric functions which are based on the use of quadratic forms). We present an overview of the use of quadratic forms in physics as in our general theory, we have called START. And, in order to give an introduction to the use of the more general K-algebras and to the possible application to sciences other than physics, the application to social sciences is considered.For the applications to physics we show that quadratic spaces are a fundamental clue to understand the structure of theoretical physics (see, for example, Keller in ICNAAM 2005 and 2006)
ERIC Educational Resources Information Center
Lim, Kok Seng
2010-01-01
Introduction: This study aimed to investigate the errors made by 265 Form 2 male students in simplifying algebraic expressions. Method: A total of 265 Form 2 (Grade 7) male students were selected for this study. 10 high, medium and low ability students in each group were selected for the interviews. 40 items were administered to the respondents to…
Early universe cosmology, effective supergravity, and invariants of algebraic forms
NASA Astrophysics Data System (ADS)
Sinha, Kuver
2015-09-01
The presence of light scalars can have profound effects on early universe cosmology, influencing its thermal history as well as paradigms like inflation and baryogenesis. Effective supergravity provides a framework to make quantifiable, model-independent studies of these effects. The Riemannian curvature of the Kähler manifold spanned by scalars belonging to chiral superfields, evaluated along supersymmetry breaking directions, provides an order parameter (in the sense that it must necessarily take certain values) for phenomena as diverse as slow roll modular inflation, nonthermal cosmological histories, and the viability of Affleck-Dine baryogenesis. Within certain classes of UV completions, the order parameter for theories with n scalar moduli is conjectured to be related to invariants of n -ary cubic forms (for example, for models with three moduli, the order parameter is given by a function on the ring of invariants spanned by the Aronhold invariants). Within these completions, and under the caveats spelled out, this may provide an avenue to obtain necessary conditions for the above phenomena that are in principle calculable given nothing but the intersection numbers of a Calabi-Yau compactification geometry. As an additional result, abstract relations between holomorphic sectional and bisectional curvatures are utilized to constrain Affleck-Dine baryogenesis on a wide class of Kähler geometries.
Further Reductions of Normal Forms for Dynamical Systems
NASA Astrophysics Data System (ADS)
Chen, Guoting; Della Dora, Jean
2000-09-01
We propose in this paper a method for obtaining a significant refinement of normal forms for dynamical systems or vector fields, with concrete and interesting applications. We use lower order nonlinear terms in the normal form for the simplifications of higher order terms. Our approach is applicable for both the non nilpotent and the nilpotent cases. For dynamical systems of dimensions 2 and 3 we give an algorithm that leads to interesting finite order normal forms which are optimal (or unique) with respect to equivalence by formal near identity transformations. We can compute at the same time a formal diffeormorphism that realizes the normalization. Comparisons with other methods are given for several examples.
The use of normal forms for analysing nonlinear mechanical vibrations
Neild, Simon A.; Champneys, Alan R.; Wagg, David J.; Hill, Thomas L.; Cammarano, Andrea
2015-01-01
A historical introduction is given of the theory of normal forms for simplifying nonlinear dynamical systems close to resonances or bifurcation points. The specific focus is on mechanical vibration problems, described by finite degree-of-freedom second-order-in-time differential equations. A recent variant of the normal form method, that respects the specific structure of such models, is recalled. It is shown how this method can be placed within the context of the general theory of normal forms provided the damping and forcing terms are treated as unfolding parameters. The approach is contrasted to the alternative theory of nonlinear normal modes (NNMs) which is argued to be problematic in the presence of damping. The efficacy of the normal form method is illustrated on a model of the vibration of a taut cable, which is geometrically nonlinear. It is shown how the method is able to accurately predict NNM shapes and their bifurcations. PMID:26303917
The use of normal forms for analysing nonlinear mechanical vibrations.
Neild, Simon A; Champneys, Alan R; Wagg, David J; Hill, Thomas L; Cammarano, Andrea
2015-09-28
A historical introduction is given of the theory of normal forms for simplifying nonlinear dynamical systems close to resonances or bifurcation points. The specific focus is on mechanical vibration problems, described by finite degree-of-freedom second-order-in-time differential equations. A recent variant of the normal form method, that respects the specific structure of such models, is recalled. It is shown how this method can be placed within the context of the general theory of normal forms provided the damping and forcing terms are treated as unfolding parameters. The approach is contrasted to the alternative theory of nonlinear normal modes (NNMs) which is argued to be problematic in the presence of damping. The efficacy of the normal form method is illustrated on a model of the vibration of a taut cable, which is geometrically nonlinear. It is shown how the method is able to accurately predict NNM shapes and their bifurcations. PMID:26303917
Motility in normal and filamentous forms of Rhodospirillum rubrum.
Lee, A G; Fitzsimons, J T
1976-04-01
By suitable choice of medium, Rhodospirillum rubrum has been grown both in normal (length 2 mum) and filamentous (length up to 60 mum) forms. Both forms were highly motile, and negatively-stained preparations showed bipolar flagellated cells, with an average of seven flagella at each pole. Motion consisted of a series of runs and tumbles, the ditribution of run time-lengths being Poissonian. Both forms tumbled in response to dark shock and showed negative chemotaxis to oxygen. The observation that the motility pattern was very similar in normal and filamentous forms makes chemical control of tumbling unlikely and favours a system involving membrane potentials. PMID:819618
Parametric normal forms of vector fields and their further simplification
NASA Astrophysics Data System (ADS)
Gao, Bo; Zhang, Weinian
2010-10-01
Given a family of vector fields parametrized by ξ in its linear part, one usually obtains its versal unfolding by the so-called two-step approach, i.e. find a Poincaré normal form for the system with fixed ξ0 and then apply the obtained near-identity transformation to the system with general ξ. It is also a common practice to treat ξ as a component together with those state components and calculate normal forms of the extended system. In this paper we reformulate normal forms on modules of homogeneous polynomials over the ring of all continuous functions of ξ and give a direct computation of versal unfolding. Our procedure enables us to determine coefficients of all terms of a certain degree in the normal form before we give a near-identity transformation of this degree. We can give all available near-identity transformations and choose an appropriate one to eliminate more terms of higher degree for a simpler normal form. We prove that the normal form reduced in our procedure is the simplest and unique. We illustrate our method with systems of linear centre and nilpotent linear parts separately.
Implementation of control point form of algebraic grid-generation technique
NASA Technical Reports Server (NTRS)
Choo, Yung K.; Miller, David P.; Reno, Charles J.
1991-01-01
The control point form (CPF) provides explicit control of physical grid shape and grid spacing through the movement of the control points. The control point array, called a control net, is a space grid type arrangement of locations in physical space with an index for each direction. As an algebraic method CPF is efficient and works well with interactive computer graphics. A family of menu-driven, interactive grid-generation computer codes (TURBO) is being developed by using CPF. Key features of TurboI (a TURBO member) are discussed and typical results are presented. TurboI runs on any IRIS 4D series workstation.
Maximal supergravity in D = 10: forms, Borcherds algebras and superspace cohomology
NASA Astrophysics Data System (ADS)
Greitz, J.; Howe, P. S.
2011-08-01
We give a very simple derivation of the forms of N = 2, D = 10 supergravity theories from supersymmetry and {text{SL}}left( {2,mathbb{R}} right) (for IIB). Using superspace cohomology we show that, if the Bianchi identities for the physical fields are satisfied, the (consistent) Bianchi identities for all of the higher-rank forms must be identically satisfied, and that there are no possible gauge-trivial Bianchi identities (i.e. dF = 0) except for exact eleven-forms. We also show that the degrees of the forms can be extended beyond the spacetime limit, and that the representations they fall into agree with those predicted from Borcherds algebras. In IIA there are even-rank RR forms, including a non-zero twelve-form, while in IIB there are non-trivial Bianchi identities for thirteen-forms even though these forms are identically zero in supergravity. It is speculated that these higher-rank forms could be non-zero when higher-order string corrections are included.
Trojan dynamics well approximated by a new Hamiltonian normal form
NASA Astrophysics Data System (ADS)
Páez, Rocío Isabel; Locatelli, Ugo
2015-10-01
We revisit a classical perturbative approach to the Hamiltonian related to the motions of Trojan bodies, in the framework of the planar circular restricted three-body problem, by introducing a number of key new ideas in the formulation. In some sense, we adapt the approach of Garfinkel to the context of the normal form theory and its modern techniques. First, we make use of Delaunay variables for a physically accurate representation of the system. Therefore, we introduce a novel manipulation of the variables so as to respect the natural behaviour of the model. We develop a normalization procedure over the fast angle which exploits the fact that singularities in this model are essentially related to the slow angle. Thus, we produce a new normal form, i.e. an integrable approximation to the Hamiltonian. We emphasize some practical examples of the applicability of our normalizing scheme, e.g. the estimation of the stable libration region. Finally, we compare the level curves produced by our normal form with surfaces of section provided by the integration of the non-normalized Hamiltonian, with very good agreement. Further precision tests are also provided. In addition, we give a step-by-step description of the algorithm, allowing for extensions to more complicated models.
Cotangent bundle reduction and Poincaré-Birkhoff normal forms
NASA Astrophysics Data System (ADS)
Çiftçi, Ünver; Waalkens, Holger; Broer, Henk W.
2014-02-01
In this paper we study a systematic and natural construction of canonical coordinates for the reduced space of a cotangent bundle with a free Lie group action. The canonical coordinates enable us to compute Poincaré-Birkhoff normal forms of relative equilibria using standard algorithms. The case of simple mechanical systems with symmetries is studied in detail. As examples we compute Poincaré-Birkhoff normal forms for a Lagrangian equilateral triangle configuration of a three-body system with a Morse-type potential and the stretched-out configuration of a double spherical pendulum.
Normal forms near a symmetric planar saddle connection
NASA Astrophysics Data System (ADS)
Wynen, Jeroen
2016-05-01
The present paper studies vector fields of the form x ˙ = (q / 2 + O (1 -x2)) (1 -x2) + O (y), y ˙ = (px + O (1 -x2)) y + O (y2), which contain a separatrix connection between hyperbolic saddles with opposite eigenvalues where the connection is fixed. Smooth semi-local normal forms are provided in vicinity of the connection, both in the resonant and non-resonant case. First, a formal conjugacy is constructed near the separatrix. Then, a smooth change of coordinates is realized by generalizing known local results near the hyperbolic points.
Normal forms of fuzzy middle and fuzzy contradictions.
Turksen, I B; Kandel, A; Zhang, Y Q
1999-01-01
The expressions of "excluded middle" and "crisp contradiction" are reexamined starting with their original linguistic expressions which are first restated in propositional and then predicate forms. It is shown that, in order to generalize the truth tables and hence the normal forms, the membership assignments in predicate expressions must be separated from their truth qualification. In two-valued logic, there is no need to separate them from each other due to reductionist Aristotalean dichotomy. Whereas, in infinite (fuzzy) valued set and logic, the separation of membership assignments from their truth qualification forms the bases of a new reconstruction of the truth tables. The results obtained from these extended truth tables are reducible to their Boolean equivalents under the axioms of Boolean theory. Whereas, in fuzzy set and logic theory, we obtain a richer and more complex interpretations of the "fuzzy middle" and "fuzzy contradiction." PMID:18252295
Syntax and Meaning as Sensuous, Visual, Historical Forms of Algebraic Thinking
ERIC Educational Resources Information Center
Radford, Luis; Puig, Luis
2007-01-01
Before the advent of symbolism, i.e. before the end of the 16th Century, algebraic calculations were made using natural language. Through a kind of metaphorical process, a few terms from everyday life (e.g. thing, root) acquired a technical mathematical status and constituted the specialized language of algebra. The introduction of letters and…
ERIC Educational Resources Information Center
Ruthven, Kenneth; Deaney, Rosemary; Hennessy, Sara
2009-01-01
From preliminary analysis of teacher-nominated examples of successful technology-supported practice in secondary-school mathematics, the use of graphing software to teach about algebraic forms was identified as being an important archetype. Employing evidence from lesson observation and teacher interview, such practice was investigated in greater…
A new quantum scheme for normal-form games
NASA Astrophysics Data System (ADS)
Fraçkiewicz, Piotr
2015-06-01
We give a strict mathematical description for a refinement of the Marinatto-Weber quantum game scheme. The model allows the players to choose projector operators that determine the state on which they perform their local operators. The game induced by the scheme generalizes finite strategic-form game. In particular, it covers normal representations of extensive games, i.e., strategic games generated by extensive ones. We illustrate our idea with an example of extensive game and prove that rational choices in the classical game and its quantum counterpart may lead to significantly different outcomes.
NASA Technical Reports Server (NTRS)
Smialek, James L.
2002-01-01
A cyclic oxidation interfacial spalling model has been developed in Part 1. The governing equations have been simplified here by substituting a new algebraic expression for the series (Good-Smialek approximation). This produced a direct relationship between cyclic oxidation weight change and model input parameters. It also allowed for the mathematical derivation of various descriptive parameters as a function of the inputs. It is shown that the maximum in weight change varies directly with the parabolic rate constant and cycle duration and inversely with the spall fraction, all to the 1/2 power. The number of cycles to reach maximum and zero weight change vary inversely with the spall fraction, and the ratio of these cycles is exactly 1:3 for most oxides. By suitably normalizing the weight change and cycle number, it is shown that all cyclic oxidation weight change model curves can be represented by one universal expression for a given oxide scale.
Programmed First Course in Algebra, Revised Form H, Student's Text, Part I, Unit 60.
ERIC Educational Resources Information Center
Buck, R. Creighton; And Others
This is part one of a two-part SMSG Programed Algebra Text for high school students. The general plan of the course is to build upon the student's experience with arithmetic. The student is initially led to extract from his or her experience the fundamental properties of addition and multiplication. The text then introduces negative real numbers…
NASA Astrophysics Data System (ADS)
Gerzen, T.; Minkwitz, D.
2016-01-01
The accuracy and availability of satellite-based applications like GNSS positioning and remote sensing crucially depends on the knowledge of the ionospheric electron density distribution. The tomography of the ionosphere is one of the major tools to provide link specific ionospheric corrections as well as to study and monitor physical processes in the ionosphere. In this paper, we introduce a simultaneous multiplicative column-normalized method (SMART) for electron density reconstruction. Further, SMART+ is developed by combining SMART with a successive correction method. In this way, a balancing between the measurements of intersected and not intersected voxels is realised. The methods are compared with the well-known algebraic reconstruction techniques ART and SART. All the four methods are applied to reconstruct the 3-D electron density distribution by ingestion of ground-based GNSS TEC data into the NeQuick model. The comparative case study is implemented over Europe during two periods of the year 2011 covering quiet to disturbed ionospheric conditions. In particular, the performance of the methods is compared in terms of the convergence behaviour and the capability to reproduce sTEC and electron density profiles. For this purpose, independent sTEC data of four IGS stations and electron density profiles of four ionosonde stations are taken as reference. The results indicate that SMART significantly reduces the number of iterations necessary to achieve a predefined accuracy level. Further, SMART+ decreases the median of the absolute sTEC error up to 15, 22, 46 and 67 % compared to SMART, SART, ART and NeQuick respectively.
Normalization Of Thermal-Radiation Form-Factor Matrix
NASA Technical Reports Server (NTRS)
Tsuyuki, Glenn T.
1994-01-01
Report describes algorithm that adjusts form-factor matrix in TRASYS computer program, which calculates intraspacecraft radiative interchange among various surfaces and environmental heat loading from sources such as sun.
Carleman linearization and normal forms for differential systems with quasi-periodic coefficients.
Chermnykh, Sergey V
2016-01-01
We study the matrix representation of Poincaré normalization using the Carleman linearization technique for non-autonomous differential systems with quasi-periodic coefficients. We provide a rigorous proof of the validity of the matrix representation of the normalization and obtain a recursive algorithm for computing the normalizing transformation and the normal form of the differential systems. The algorithm provides explicit formulas for the coefficients of the normal form and the corresponding transformation. PMID:27588240
Realizations of Galilei algebras
NASA Astrophysics Data System (ADS)
Nesterenko, Maryna; Pošta, Severin; Vaneeva, Olena
2016-03-01
All inequivalent realizations of the Galilei algebras of dimensions not greater than five are constructed using the algebraic approach proposed by Shirokov. The varieties of the deformed Galilei algebras are discussed and families of one-parametric deformations are presented in explicit form. It is also shown that a number of well-known and physically interesting equations and systems are invariant with respect to the considered Galilei algebras or their deformations.
Teaching Algebra without Algebra
ERIC Educational Resources Information Center
Kalman, Richard S.
2008-01-01
Algebra is, among other things, a shorthand way to express quantitative reasoning. This article illustrates ways for the classroom teacher to convert algebraic solutions to verbal problems into conversational solutions that can be understood by students in the lower grades. Three reasonably typical verbal problems that either appeared as or…
Computation of Normal Forms of Bogdanov-Takens Singularities for High Dimensional Nonlinear Systems
Chen Shuping; Zhang Wei; Qian Youhua
2010-05-21
A new computation of the normal forms of Bogdanov-Takens singularities is developed in this paper. In the theoretical model for the nonplanar nonlinear oscillation of a cantilever beam, the computation method is applied to compute the coefficients of the normal forms for the case of one non-semisimple double zero and a pair of pure imaginary eigenvalues.
NASA Astrophysics Data System (ADS)
Jiang, Heping; Jiang, Jiao; Song, Yongli
In this paper, we firstly employ the normal form theory of delayed differential equations according to Faria and Magalhães to derive the normal form of saddle-node-Hopf bifurcation for the general retarded functional differential equations. Then, the dynamical behaviors of a Leslie-Gower predator-prey model with time delay and nonmonotonic functional response are considered. Specially, the dynamical classification near the saddle-node-Hopf bifurcation point is investigated by using the normal form and the center manifold approaches. Finally, the numerical simulations are employed to support the theoretical results.
The Normalized Reduced Form and Cell Mathematical Tools for Lattice Analysis—Symmetry and Similarity
Mighell, Alan D.
2003-01-01
To intelligently and effectively use crystallographic databases, mathematical and computer tools are required that can elucidate diverse types of intra- and interlattice relationships. Two such tools are the normalized reduced form and normalized reduced cell. Practical experience has revealed that the first tool—the normalized reduced form—is very helpful in establishing lattice metric symmetry as it enables one to readily deduce significant relationships between the elements of the reduced form. Likewise research with crystallographic databases has demonstrated that the second tool—the normalized reduced cell—plays a vital role in determining metrically similar lattices. Knowledge of similar lattices has practical value in solving structures, in assignment of structure types, in materials design, and in nano-technology. In addition to using the reduced cell, it is recommended that lattice-matching strategies based on the normalized reduced cell be routinely carried out in database searching, in data evaluation, and in experimental work.
NASA Astrophysics Data System (ADS)
Liu, Zhihua; Magal, Pierre; Ruan, Shigui
2014-08-01
Normal form theory is very important and useful in simplifying the forms of equations restricted on the center manifolds in studying nonlinear dynamical problems. In this paper, using the center manifold theorem associated with the integrated semigroup theory, we develop a normal form theory for semilinear Cauchy problems in which the linear operator is not densely defined and is not a Hille-Yosida operator and present procedures to compute the Taylor expansion and normal form of the reduced system restricted on the center manifold. We then apply the main results and computation procedures to determine the direction of the Hopf bifurcation and stability of the bifurcating periodic solutions in a structured evolutionary epidemiological model of influenza A drift and an age structured population model.
Normal form solutions of dynamical systems in the basin of attraction of their fixed points
NASA Astrophysics Data System (ADS)
Bountis, Tassos; Tsarouhas, George; Herman, Russell
1998-10-01
The normal form theory of Poincaré, Siegel and Arnol'd is applied to an analytically solvable Lotka-Volterra system in the plane, and a periodically forced, dissipative Duffing's equation with chaotic orbits in its 3-dimensional phase space. For the planar model, we determine exactly how the convergence region of normal forms about a nodal fixed point is limited by the presence of singularities of the solutions in the complex t-plane. Despite such limitations, however, we show, in the case of a periodically driven system, that normal forms can be used to obtain useful estimates of the basin of attraction of a stable fixed point of the Poincaré map, whose ``boundary'' is formed by the intersecting invariant manifolds of a second hyperbolic fixed point nearby.
Spinors in the hyperbolic algebra
NASA Astrophysics Data System (ADS)
Ulrych, S.
2006-01-01
The three-dimensional universal complex Clifford algebra Cbar3,0 is used to represent relativistic vectors in terms of paravectors. In analogy to the Hestenes spacetime approach spinors are introduced in an algebraic form. This removes the dependance on an explicit matrix representation of the algebra.
Fuzzy-algebra uncertainty assessment
Cooper, J.A.; Cooper, D.K.
1994-12-01
A significant number of analytical problems (for example, abnormal-environment safety analysis) depend on data that are partly or mostly subjective. Since fuzzy algebra depends on subjective operands, we have been investigating its applicability to these forms of assessment, particularly for portraying uncertainty in the results of PRA (probabilistic risk analysis) and in risk-analysis-aided decision-making. Since analysis results can be a major contributor to a safety-measure decision process, risk management depends on relating uncertainty to only known (not assumed) information. The uncertainties due to abnormal environments are even more challenging than those in normal-environment safety assessments; and therefore require an even more judicious approach. Fuzzy algebra matches these requirements well. One of the most useful aspects of this work is that we have shown the potential for significant differences (especially in perceived margin relative to a decision threshold) between fuzzy assessment and probabilistic assessment based on subtle factors inherent in the choice of probability distribution models. We have also shown the relation of fuzzy algebra assessment to ``bounds`` analysis, as well as a description of how analyses can migrate from bounds analysis to fuzzy-algebra analysis, and to probabilistic analysis as information about the process to be analyzed is obtained. Instructive examples are used to illustrate the points.
On the global structure of normal forms for slow-fast Hamiltonian systems
NASA Astrophysics Data System (ADS)
Avendaño Camacho, M.; Vorobiev, Yu.
2013-04-01
In the framework of Lie transforms and the global method of averaging, the normal forms of a multidimensional slow-fast Hamiltonian system are studied in the case when the flow of the unperturbed (fast) system is periodic and the induced {S}^1 1-action is not necessarily free and trivial. An intrinsic splitting of the second term in a {S}^1 1-invariant normal form of first order is derived in terms of the Hannay-Berry connection assigned to the periodic flow.
Design of a spatial data structure using the relational normal forms
van Roessel, Jan W.
1987-01-01
In previous work, a relational data structure aimed at the exchange of spatial data between systems was developed. As this data structure was relational it was of first normal form, but compliance with the higher normal forms was not investigated. Recently, a new procedural method for composing fully normalized data structures from the basic data fields has been developed by H. C. Smith, as an alternative to the process of non-loss decomposition which is difficult to understand. Smith's method has been applied to data fields required to store points, lines and polygons in a chain-node spatial data model. When geographic domain, coverage layer and map are also considered, the procedure naturally leads to a catalogue model, needed for the exchange of spatial data. Although the method produces a fully normalized data structure, it is not as easy to identify which normal forms are responsible for the ultimate arrangement of the data fields into relations, but the benefits of these criteria for data base development also apply to spatial data structures and related ancillary data.
NASA Astrophysics Data System (ADS)
Fortunati, Alessandro; Wiggins, Stephen
2014-05-01
The aim of this paper is to extend the result of Giorgilli and Zehnder for aperiodic time dependent systems to a case of nearly integrable convex analytic Hamiltonians. The existence of a normal form and then a stability result are shown in the case of a slow aperiodic time dependence that, under some smallness conditions, is independent of the size of the perturbation.
A semantic normal form for clinical drugs in the UMLS: early experiences with the VANDF.
Nelson, Stuart J.; Brown, Steven H.; Erlbaum, Mark S.; Olson, Nels; Powell, Tammy; Carlsen, Brian; Carter, John; Tuttle, Mark S.; Hole, William T.
2002-01-01
A semantic normal form (SNF) for a clinical drug, designed to represent the meaning of an expression typically seen in a practitioner's medication order, has been developed and is being created in the UMLS Metathesaurus. The long term goal is to establish a relationship for every concept in the Metathesaurus with semantic type "clinical drug" with one or more of these semantic normal forms. First steps have been taken using the Veterans Administration National Drug File (VANDF). 70% of the entries in the VANDF could be parsed algorithmically into the SNF. Next steps include parsing other drug vocabularies included in the UMLS Metathesaurus and performing human review of the parsed vocabularies. After machine parsed forms have been merged in the Metathesaurus Information Database (MID), editors will be able to edit matched SNFs for accuracy and establish relationships and relationship attributes with other clinical drug concepts PMID:12463886
Normal and quasinormal forms for systems of difference and differential-difference equations
NASA Astrophysics Data System (ADS)
Kashchenko, Ilya; Kaschenko, Sergey
2016-09-01
The local dynamics of systems of difference and singularly perturbed differential-difference equations is studied in the neighborhood of a zero equilibrium state. Critical cases in the problem of stability of its state of equilibrium have infinite dimension. Special nonlinear evolution equations, which act as normal forms, are set up. It is shown that their dynamics defines the behavior of solutions to the initial system.
NASA Astrophysics Data System (ADS)
Heikkila, S.
2007-08-01
In this paper we apply generalized iteration methods to prove comparison results which show how fixed points of a multifunction can be bounded by least and greatest fixed points of single-valued functions. As an application we prove existence and comparison results for fixed points of multifunctions. These results are applied to normal-form games, by proving existence and comparison results for pure and mixed Nash equilibria and their utilities.
Covariant deformed oscillator algebras
NASA Technical Reports Server (NTRS)
Quesne, Christiane
1995-01-01
The general form and associativity conditions of deformed oscillator algebras are reviewed. It is shown how the latter can be fulfilled in terms of a solution of the Yang-Baxter equation when this solution has three distinct eigenvalues and satisfies a Birman-Wenzl-Murakami condition. As an example, an SU(sub q)(n) x SU(sub q)(m)-covariant q-bosonic algebra is discussed in some detail.
Hopf normal form with SN symmetry and reduction to systems of nonlinearly coupled phase oscillators
NASA Astrophysics Data System (ADS)
Ashwin, Peter; Rodrigues, Ana
2016-06-01
Coupled oscillator models where N oscillators are identical and symmetrically coupled to all others with full permutation symmetry SN are found in a variety of applications. Much, but not all, work on phase descriptions of such systems consider the special case of pairwise coupling between oscillators. In this paper, we show this is restrictive-and we characterize generic multi-way interactions between oscillators that are typically present, except at the very lowest order near a Hopf bifurcation where the oscillations emerge. We examine a network of identical weakly coupled dynamical systems that are close to a supercritical Hopf bifurcation by considering two parameters, ɛ (the strength of coupling) and λ (an unfolding parameter for the Hopf bifurcation). For small enough λ > 0 there is an attractor that is the product of N stable limit cycles; this persists as a normally hyperbolic invariant torus for sufficiently small ɛ > 0. Using equivariant normal form theory, we derive a generic normal form for a system of coupled phase oscillators with SN symmetry. For fixed N and taking the limit 0 < ɛ ≪ λ ≪ 1, we show that the attracting dynamics of the system on the torus can be well approximated by a coupled phase oscillator system that, to lowest order, is the well-known Kuramoto-Sakaguchi system of coupled oscillators. The next order of approximation generically includes terms with up to four interacting phases, regardless of N. Using a normalization that maintains nontrivial interactions in the limit N → ∞, we show that the additional terms can lead to new phenomena in terms of coexistence of two-cluster states with the same phase difference but different cluster size.
Practical output tracking of switched nonlinear systems in p-normal form with unstable subsystems
NASA Astrophysics Data System (ADS)
Long, Lijun; Zhao, Jun
2016-08-01
This paper studies practical output tracking of switched nonlinear systems in p-normal form. No solvability of the practical output tracking problem for subsystems is required. A constructive scheme to solve the problem for a switched nonlinear system is set up by exploiting the single Lyapunov function method and the tool of adding a power integrator. Also, we design a proper switching law and construct state-feedback controllers of subsystems. A two inverted pendulums as a practical example, which cannot be handled by the existing approaches, illustrates our theoretical result.
Implementation of Boolean functions with a bounded number of zeros by disjunctive normal forms
NASA Astrophysics Data System (ADS)
Maximov, Yu. V.
2013-09-01
The problem of constructing simple disjunctive normal forms (DNFs) of Boolean functions with a small number of zeros is considered. The problem is of interest in the complexity analysis of Boolean functions and in its applications to data analysis. The method used is a further development of the reduction approach to the construction of DNFs of Boolean functions. A key idea of the reduction method is that a Boolean function is represented as a disjunction of Boolean functions with fewer zeros. In a number of practically important cases, this technique makes it possible to considerably reduce the complexity of DNF implementations of Boolean functions.
Algebraic integrability: a survey.
Vanhaecke, Pol
2008-03-28
We give a concise introduction to the notion of algebraic integrability. Our exposition is based on examples and phenomena, rather than on detailed proofs of abstract theorems. We mainly focus on algebraic integrability in the sense of Adler-van Moerbeke, where the fibres of the momentum map are affine parts of Abelian varieties; as it turns out, most examples from classical mechanics are of this form. Two criteria are given for such systems (Kowalevski-Painlevé and Lyapunov) and each is illustrated in one example. We show in the case of a relatively simple example how one proves algebraic integrability, starting from the differential equations for the integrable vector field. For Hamiltonian systems that are algebraically integrable in the generalized sense, two examples are given, which illustrate the non-compact analogues of Abelian varieties which typically appear in such systems. PMID:17588863
ERIC Educational Resources Information Center
Schaufele, Christopher; Zumoff, Nancy
Earth Algebra is an entry level college algebra course that incorporates the spirit of the National Council of Teachers of Mathematics (NCTM) Curriculum and Evaluation Standards for School Mathematics at the college level. The context of the course places mathematics at the center of one of the major current concerns of the world. Through…
ERIC Educational Resources Information Center
Cavanagh, Sean
2009-01-01
As educators and policymakers search for ways to prepare students for the rigors of algebra, teachers in the Helena, Montana, school system are starting early by attempting to nurture students' algebraic-reasoning ability, as well as their basic number skills, in early elementary school, rather than waiting until middle or early high school.…
Structural features of normal and complemented forms of the Neurospora isopropylmalate isomerase.
Reichenbecher, V E; Gross, S R
1978-01-01
The isopropylmalate isomerase (EC 4.2.1.33) of Neurospora crassa is a globular protein consisting of a single polypeptide chain with a molecular weight of about 90,000. The isomerase cannot easily be freed of a contaminating protease which cleaves the enzyme into two major fragments, one of approximately 56,000 and the other 37,000 daltons. This suggests that the folded polypeptide chain may contain some hinge point or loop exposed on the surface which makes it susceptible to proteolytic attack. Most of the isomerase activity extracted from the wild-type strain is in monomer form. However, a small fraction of the activity in crude extracts is found in multimeric aggregates, and the active isomerase extracted from complementing leu-2 heterokaryons consists entirely of dimers and higher multimers. These observations suggest that, though active as a monomer, a significant fraction of the normal enzyme might be organized in multimeric form within the cell. Images PMID:146703
Invertible linear transformations and the Lie algebras
NASA Astrophysics Data System (ADS)
Zhang, Yufeng; Tam, Honwah; Guo, Fukui
2008-07-01
With the help of invertible linear transformations and the known Lie algebras, a way to generate new Lie algebras is given. These Lie algebras obtained have a common feature, i.e. integrable couplings of solitary hierarchies could be obtained by using them, specially, the Hamiltonian structures of them could be worked out. Some ways to construct the loop algebras of the Lie algebras are presented. It follows that some various loop algebras are given. In addition, a few new Lie algebras are explicitly constructed in terms of the classification of Lie algebras proposed by Ma Wen-Xiu, which are bases for obtaining new Lie algebras by using invertible linear transformations. Finally, some solutions of a (2 + 1)-dimensional partial-differential equation hierarchy are obtained, whose Hamiltonian form-expressions are manifested by using the quadratic-form identity.
Unique form of rickets with low serum 25-hydroxyvitamin D in two normally nourished children.
Asami, T; Kawasaki, T; Uchiyama, M
1995-04-01
We present an unusual type of rickets involving two children: a 2 year old boy and a 15 month old boy, who presented with marked bowing of the lower extremities and bulging of costochondral junctions. Both children had normal growth, with their height and body weight greater than the 50th and 97th percentile for age. Roentgenograms of their extremities showed the typical changes of vitamin D refractory rickets. Serum alkaline phosphatase levels were elevated and serum levels of calcium and phosphate were both within the normal range. No primary cause for the rickets, including nutritional deficiencies, was found in the two patients. Characteristic findings were persistently low serum 25-hydroxyvitamin D (25-OH-D) and normal 1,25-dihydroxyvitamin D (1,25-(OH)2-D). Improvements in clinical and X-ray findings were observed after either oral administration of 1 alpha-(OH)-D3 (9-15 micrograms per day) or massive vitamin D2 therapy (600,000 IU single injection). The low serum levels of 25-OH-D did not increase unless massive vitamin D2 therapy was also given. These two cases represent a unique form of rickets that does not meet the criteria for any type of previously known rickets. PMID:7793252
Overexpression of wild-type or mutants forms of CEBPA alter normal human hematopoiesis.
Quintana-Bustamante, O; Lan-Lan Smith, S; Griessinger, E; Reyal, Y; Vargaftig, J; Lister, T A; Fitzgibbon, J; Bonnet, D
2012-07-01
CCAAT/enhancer-binding protein-α (C/EBPα/CEBPA) is mutated in approximately 8% of acute myeloid leukemia (AML) in both familial and sporadic AML and, with FLT3 and NPM1, has received most attention as a predictive marker of outcome in patients with normal karyotype disease. Mutations clustering to either the N- or C-terminal (N- and C-ter) portions of the protein have different consequences on the protein function. In familial cases, the N-ter form is inherited with patients exhibiting long latency period before the onset of overt disease, typically with the acquisition of a C-ter mutation. Despite the essential insights murine models provide the functional consequences of wild-type C/EBPα in human hematopoiesis and how different mutations are involved in AML development have received less attention. Our data underline the critical role of C/EBPα in human hematopoiesis and demonstrate that C/EBPα mutations (alone or in combination) are insufficient to convert normal human hematopoietic stem/progenitor cells into leukemic-initiating cells, although individually each altered normal hematopoiesis. It provides the first insight into the effects of N- and C-ter mutations acting alone and to the combined effects of N/C double mutants. Our results mimicked closely what happens in CEBPA mutated patients. PMID:22371011
Theory and praxis pf map analsys in CHEF part 1: Linear normal form
Michelotti, Leo; /Fermilab
2008-10-01
This memo begins a series which, put together, could comprise the 'CHEF Documentation Project' if there were such a thing. The first--and perhaps only--three will telegraphically describe theory, algorithms, implementation and usage of the normal form map analysis procedures encoded in CHEF's collection of libraries. [1] This one will begin the sequence by explaining the linear manipulations that connect the Jacobian matrix of a symplectic mapping to its normal form. It is a 'Reader's Digest' version of material I wrote in Intermediate Classical Dynamics (ICD) [2] and randomly scattered across technical memos, seminar viewgraphs, and lecture notes for the past quarter century. Much of its content is old, well known, and in some places borders on the trivial.1 Nevertheless, completeness requires their inclusion. The primary objective is the 'fundamental theorem' on normalization written on page 8. I plan to describe the nonlinear procedures in a subsequent memo and devote a third to laying out algorithms and lines of code, connecting them with equations written in the first two. Originally this was to be done in one short paper, but I jettisoned that approach after its first section exceeded a dozen pages. The organization of this document is as follows. A brief description of notation is followed by a section containing a general treatment of the linear problem. After the 'fundamental theorem' is proved, two further subsections discuss the generation of equilibrium distributions and issue of 'phase'. The final major section reviews parameterizations--that is, lattice functions--in two and four dimensions with a passing glance at the six-dimensional version. Appearances to the contrary, for the most part I have tried to restrict consideration to matters needed to understand the code in CHEF's libraries.
Vertex Algebras, Kac-Moody Algebras, and the Monster
NASA Astrophysics Data System (ADS)
Borcherds, Richard E.
1986-05-01
It is known that the adjoint representation of any Kac-Moody algebra A can be identified with a subquotient of a certain Fock space representation constructed from the root lattice of A. I define a product on the whole of the Fock space that restricts to the Lie algebra product on this subquotient. This product (together with a infinite number of other products) is constructed using a generalization of vertex operators. I also construct an integral form for the universal enveloping algebra of any Kac-Moody algebra that can be used to define Kac-Moody groups over finite fields, some new irreducible integrable representations, and a sort of affinization of any Kac-Moody algebra. The ``Moonshine'' representation of the Monster constructed by Frenkel and others also has products like the ones constructed for Kac-Moody algebras, one of which extends the Griess product on the 196884-dimensional piece to the whole representation.
Theory and praxis of map analsys in CHEF part 2: Nonlinear normal form
Michelotti, Leo; /FERMILAB
2009-04-01
This is the second of three memos describing how normal form map analysis is implemented in CHEF. The first [1] explained the manipulations required to assure that initial, linear transformations preserved Poincare invariants, thereby confirming correct normalization of action-angle coordinates. In this one, the transformation will be extended to nonlinear terms. The third, describing how the algorithms were implemented within the software of CHEF's libraries, most likely will never be written. The first section, Section 2, quickly lays out preliminary concepts and relationships. In Section 3, we shall review the perturbation theory - an iterative sequence of transformations that converts a nonlinear mapping into its normal form - and examine the equation which moves calculations from one step to the next. Following that is a section titled 'Interpretation', which identifies connections between the normalized mappings and idealized, integrable, fictitious Hamiltonian models. A final section contains closing comments, some of which may - but probably will not - preview work to be done later. My reasons for writing this memo and its predecessor have already been expressed. [1] To them can be added this: 'black box code' encourages users to proceed with little or no understanding of what it does or how it operates. So far, CHEF has avoided this trap admirably by failing to attract potential users. However, we reached a watershed last year: even I now have difficulty following the software through its maze of operations. Extensions to CHEF's physics functionalities, software upgrades, and even simple maintenance are becoming more difficult than they should. I hope these memos will mark parts of the maze for easier navigation in the future. Despite appearances to the contrary, I tried to include no (or very little) more than the minimum needed to understand what CHEF's nonlinear analysis modules do.1 As with the first memo, material has been lifted - and modified - from
Optimization of accelerator parameters using normal form methods on high-order transfer maps
Snopok, Pavel; /Michigan State U.
2007-05-01
Methods of analysis of the dynamics of ensembles of charged particles in collider rings are developed. The following problems are posed and solved using normal form transformations and other methods of perturbative nonlinear dynamics: (1) Optimization of the Tevatron dynamics: (a) Skew quadrupole correction of the dynamics of particles in the Tevatron in the presence of the systematic skew quadrupole errors in dipoles; (b) Calculation of the nonlinear tune shift with amplitude based on the results of measurements and the linear lattice information; (2) Optimization of the Muon Collider storage ring: (a) Computation and optimization of the dynamic aperture of the Muon Collider 50 x 50 GeV storage ring using higher order correctors; (b) 750 x 750 GeV Muon Collider storage ring lattice design matching the Tevatron footprint. The normal form coordinates have a very important advantage over the particle optical coordinates: if the transformation can be carried out successfully (general restrictions for that are not much stronger than the typical restrictions imposed on the behavior of the particles in the accelerator) then the motion in the new coordinates has a very clean representation allowing to extract more information about the dynamics of particles, and they are very convenient for the purposes of visualization. All the problem formulations include the derivation of the objective functions, which are later used in the optimization process using various optimization algorithms. Algorithms used to solve the problems are specific to collider rings, and applicable to similar problems arising on other machines of the same type. The details of the long-term behavior of the systems are studied to ensure the their stability for the desired number of turns. The algorithm of the normal form transformation is of great value for such problems as it gives much extra information about the disturbing factors. In addition to the fact that the dynamics of particles is represented
Molecular gas mass functions of normal star-forming galaxies since z ~ 3
NASA Astrophysics Data System (ADS)
Berta, S.; Lutz, D.; Nordon, R.; Genzel, R.; Magnelli, B.; Popesso, P.; Rosario, D.; Saintonge, A.; Wuyts, S.; Tacconi, L. J.
2013-07-01
We used deep far-infrared data from the PEP/GOODS-Herschel surveys and restframe ultraviolet photometry to study the evolution of the molecular gas mass function of normal star-forming galaxies. Computing the molecular gas mass, Mmol, by scaling star formation rates through depletion timescales, or combining infrared (IR) luminosity and obscuration properties as described in the literature, we obtained Mmol for roughly 700, z = 0.2-3.0 galaxies near the star-forming main sequence. The number density of galaxies follows a Schechter function of Mmol. The characteristic mass M ∗ is found to strongly evolve up to z ~ 1 and then flatten at earlier epochs, resembling the IR luminosity evolution of similar objects. At z ~ 1, our result is supported by an estimate based on the stellar mass function of star-forming galaxies and gas fraction scalings from the PHIBSS survey. We compared our measurements with results from current models, finding better agreement with those that are treating star formation laws directly rather than in post-processing. Integrating the mass function, we studied the evolution of the Mmol density and its density parameter Ωmol. Herschel is an ESA space observatory with science instruments provided by European-led Principal Investigator consortia and with important participation from NASA.
Locally finite dimensional Lie algebras
NASA Astrophysics Data System (ADS)
Hennig, Johanna
We prove that in a locally finite dimensional Lie algebra L, any maximal, locally solvable subalgebra is the stabilizer of a maximal, generalized flag in an integrable, faithful module over L. Then we prove two structure theorems for simple, locally finite dimensional Lie algebras over an algebraically closed field of characteristic p which give sufficient conditions for the algebras to be of the form [K(R, *), K( R, *)] / (Z(R) ∩ [ K(R, *), K(R, *)]) for a simple, locally finite dimensional associative algebra R with involution *. Lastly, we explore the noncommutative geometry of locally simple representations of the diagonal locally finite Lie algebras sl(ninfinity), o( ninfinity), and sp(n infinity).
Quantum computation using geometric algebra
NASA Astrophysics Data System (ADS)
Matzke, Douglas James
This dissertation reports that arbitrary Boolean logic equations and operators can be represented in geometric algebra as linear equations composed entirely of orthonormal vectors using only addition and multiplication Geometric algebra is a topologically based algebraic system that naturally incorporates the inner and anticommutative outer products into a real valued geometric product, yet does not rely on complex numbers or matrices. A series of custom tools was designed and built to simplify geometric algebra expressions into a standard sum of products form, and automate the anticommutative geometric product and operations. Using this infrastructure, quantum bits (qubits), quantum registers and EPR-bits (ebits) are expressed symmetrically as geometric algebra expressions. Many known quantum computing gates, measurement operators, and especially the Bell/magic operators are also expressed as geometric products. These results demonstrate that geometric algebra can naturally and faithfully represent the central concepts, objects, and operators necessary for quantum computing, and can facilitate the design and construction of quantum computing tools.
Normal form and limit cycle bifurcation of piecewise smooth differential systems with a center
NASA Astrophysics Data System (ADS)
Wei, Lijun; Zhang, Xiang
2016-07-01
In this paper we prove that any Σ-center (either nondegenerate or degenerate) of a planar piecewise Cr smooth vector field Z is topologically equivalent to that of Z0: (x ˙ , y ˙) = (- 1 , 2 x) for y ≥ 0, (x ˙ , y ˙) = (1 , 2 x) for y ≤ 0, and that the homeomorphism between Z and Z0 is Cr smoothness when restricted to each side of the switching line except at the center p. We illustrate by examples that there are degenerate Σ-centers whose flows are conjugate to that of Z0, and also there exist nondegenerate Σ-centers whose flows cannot be conjugate to that of Z0. Finally applying the normal form Z0 together with the piecewise smooth equivalence, we study the number of limit cycles which can be bifurcated from the Σ-center of Z.
Exact traveling wave solutions of the van der Waals normal form for fluidized granular matter
NASA Astrophysics Data System (ADS)
Abourabia, A. M.; Morad, A. M.
2015-11-01
Analytical solutions of the van der Waals normal form for fluidized granular media have been done to study the phase separation phenomenon by using two different exact methods. The Painlevé analysis is discussed to illustrate the integrability of the model equation. An auto-Bäcklund transformation is presented via the truncated expansion and symbolic computation. The results show that the exact solutions of the model introduce solitary waves of different types. The solutions of the hydrodynamic model and the van der Waals equation exhibit a behavior similar to the one observed in molecular dynamic simulations such that two pairs of shock and rarefaction waves appear and move away, giving rise to the bubbles. The dispersion properties and the relation between group and phase velocities of the model equation are studied using the plane wave assumption. The diagrams are drawn to illustrate the physical properties of the exact solutions, and indicate their stability and bifurcation.
Normal forms for linear mode conversion and Landau-Zener transitions in one dimension
Flynn, W.G.; Littlejohn, R.G.
1994-09-01
Standard eikonal methods for the asymptotic analysis of coupled linear wave equations may fail when two eigenvalues of a matrix (the dispersion matrix) associated with the wave operator are both small in the same region of wave phase space. In this region the two eikonal modes associated with the two small eigenvalues are coupled, leading to a process called linear mode conversion or Landau-Zener coupling. A theory of linear mode conversion is presented in which geometric structure is emphasized. This theory is then used to identify the most generic type of mode conversion which occurs in one dimension. Finally, a general solution for this generic mode conversion problem is derived by transforming an arbitrary equation exhibiting generic mode conversion into an easily solvable normal form. This solution is given as a connection rule, with which one may continue standard eikonal wave solutions through mode conversion regions. 51 refs., 13 figs.
Silber, M; Skeldon, A C
1999-05-01
Motivated by experimental observations of exotic free surface standing wave patterns in the two-frequency Faraday experiment, we investigate the role of normal form symmetries in the associated pattern-selection problem. With forcing frequency components in ratio m/n, where m and n are coprime integers that are not both odd, there is the possibility that both harmonic waves and subharmonic waves may lose stability simultaneously, each with a different wave number. We focus on this situation and compare the case where the harmonic waves have a longer wavelength than the subharmonic waves with the case where the harmonic waves have a shorter wavelength. We show that in the former case a normal form transformation can be used to remove all quadratic terms from the amplitude equations governing the relevant resonant triad interactions. Thus the role of resonant triads in the pattern-selection problem is greatly diminished in this situation. We verify our general bifurcation theoretic results within the example of one-dimensional surface wave solutions of the Zhang-Viñals model [J. Fluid Mech. 341, 225 (1997)] of the two-frequency Faraday problem. In one-dimension, a 1:2 spatial resonance takes the place of a resonant triad in our investigation. We find that when the bifurcating modes are in this spatial resonance, it dramatically effects the bifurcation to subharmonic waves in the case that the forcing frequencies are in ratio 1/2; this is consistent with the results of Zhang and Viñals. In sharp contrast, we find that when the forcing frequencies are in a ratio 2/3, the bifurcation to (sub)harmonic waves is insensitive to the presence of another spatially resonant bifurcating mode. This is consistent with the results of our general analysis. PMID:11969524
NASA Astrophysics Data System (ADS)
Roitman, Michael
2008-08-01
In this paper we prove that for any commutative (but in general non-associative) algebra A with an invariant symmetric non-degenerate bilinear form there is a graded vertex algebra V = V0 Å V2 Å V3 Å ¼, such that dim V0 = 1 and V2 contains A. We can choose V so that if A has a unit e, then 2e is the Virasoro element of V, and if G is a finite group of automorphisms of A, then G acts on V as well. In addition, the algebra V can be chosen with a non-degenerate invariant bilinear form, in which case it is simple.
Nemir, M.; DeVouge, M.W.; Mukherjee, B.B. )
1989-10-25
We have reported previously that the 69-kDa major phosphoprotein, secreted by normal rat kidney (NRK) cells, is osteopontin, a glycosylated bone matrix protein. Here we show that this 69-kDa osteopontin is secreted by NRK cells in both phosphorylated (pp69) and nonphosphorylated (np69) forms, with estimated isoelectric points of 3.8 and 4.5, respectively. Electrophoretic analysis of radioiodinated cell surface proteins immunoprecipitated with an anti-69-kDa osteopontin serum, demonstrates that the 69-kDa osteopontin is also present on the cell surface, but only its phosphorylated form (pp69) shows such cell surface association. Because osteopontin mediates cell adhesion and spreading, and contains an Arg-Gly-Asp-Ser cell-binding sequence, our observations strongly suggest that the cell surface localization of pp69 osteopontin is receptor-mediated, and the modification by phosphorylation may be crucial for its receptor binding activity. We also report that antisera directed against either fibronectin or 69-kDa osteopontin co-immunoprecipitate both np69 osteopontin and fibronectin as a heat-dissociable complex. In contrast, pp69 osteopontin does not co-precipitate with fibronectin. Furthermore, compared to NRK cells, vanadyl sulfate-treated NRK cells which acquire a reversible transformed phenotype, including anchorage-independent growth, show increased levels of pp69 on the cell surface, concomitant with significantly decreased levels of pp69 and elevated levels of np69 in the conditioned media. The data presented here establish transformation sensitivity of NRK cell-secreted osteopontin with respect to its secretion and cell surface localization, and demonstrate that phosphorylated and nonphosphorylated forms of osteopontin have different physiological properties, which may regulate the functional roles of this extracellular matrix protein.
Nonlinear control design for stressed power systems using normal forms of vector fields
NASA Astrophysics Data System (ADS)
Jang, Gilsoo
Large stressed interconnected power systems exhibit complicated dynamic behavior when subjected to disturbances. This nonlinear complex behavior is not well analyzed with present tools, and a complete theoretical analysis of this is not feasible in large systems. In stressed power systems, due to the presence of increased nonlinearity and the existence of nonlinear modal interactions, there exist some limitation to the use of conventional linear control design techniques. Therefore there is a need to understand the nature of nonlinear modal interactions and their influences on control performance for optimal controller setting. This work deals with control design in power systems using the method of normal forms. The objective of this work is to understand the effect of the nonlinear modal interaction on control performance and to develop a procedure to design controls incorporating the nonlinear information. For power systems equipped with fast exciters, the exciter gains have crucial influence on the system dynamic behavior. In order to be able to tune the exciter gains for optimal system performance, one has to understand, how the system response changes with different gain settings. In linear analysis, this consists of determining the eigenvalues for various gains, and computing the sensitivity of the eigenvalues under gain variations. If one takes into account the influence of the second order normal forms on the system response, then the corresponding interaction coefficients and their sensitivity with respect to gain variations has to be studied as well. This is the topic of the study presented here. The concept of nonlinear participation factors, and sensitivity of the normal forms coefficient, together with linear participation factors and eigenvalue sensitivity are used to vary control settings. The control settings are varied to obtain improved stability and to reduce the nonlinearity in the system. The proposed procedure was applied to the 50-generator
Vortices formed on the mitral valve tips aid normal left ventricular filling
NASA Astrophysics Data System (ADS)
Vlachos, Pavlos
2011-11-01
For the left ventricle to function as an effective pump it must be able to fill from a low left atrial pressure. However, this ability is lost in patients with heart failure. We investigated the fluid dynamics of the left ventricle filling by imaging the blood flow in patients with healthy and impaired diastolic function, using 2D phase contrast magnetic resonance imaging and we quantified the intraventricular pressure gradients and the strength and location of the formed vortices. We found that during early filling in normal subjects, prior to the opening of the mitral valve the flow moves towards the apex and subsequently at the time of the opening of the valve the rapid movement of the mitral annulus away from the left ventricle apex enhances the formation of a vortex ring at the mitral valve tips. Instead of being a passive byproduct of the process as was previously believed, this vortex ring facilitates filling by reducing convective losses and enhancing the function of the left ventricle as a suction pump. Impairment of this mechanism contributes to diastolic dysfunction, with the left ventricle filling becoming dependent on left atrial pressure, and eventually leading to heart failure. John R. Jones Professor
Curvature calculations with spacetime algebra
Hestenes, D.
1986-06-01
A new method for calculating the curvature tensor is developed and applied to the Scharzschild case. The method employs Clifford algebra and has definite advantages over conventional methods using differential forms or tensor analysis.
GCD, LCM, and Boolean Algebra?
ERIC Educational Resources Information Center
Cohen, Martin P.; Juraschek, William A.
1976-01-01
This article investigates the algebraic structure formed when the process of finding the greatest common divisor and the least common multiple are considered as binary operations on selected subsets of positive integers. (DT)
Invariants of triangular Lie algebras
NASA Astrophysics Data System (ADS)
Boyko, Vyacheslav; Patera, Jiri; Popovych, Roman
2007-07-01
Triangular Lie algebras are the Lie algebras which can be faithfully represented by triangular matrices of any finite size over the real/complex number field. In the paper invariants ('generalized Casimir operators') are found for three classes of Lie algebras, namely those which are either strictly or non-strictly triangular, and for so-called special upper triangular Lie algebras. Algebraic algorithm of Boyko et al (2006 J. Phys. A: Math. Gen.39 5749 (Preprint math-ph/0602046)), developed further in Boyko et al (2007 J. Phys. A: Math. Theor.40 113 (Preprint math-ph/0606045)), is used to determine the invariants. A conjecture of Tremblay and Winternitz (2001 J. Phys. A: Math. Gen.34 9085), concerning the number of independent invariants and their form, is corroborated.
Becchi-Rouet-Stora-Tyutin operators for W algebras
Isaev, A. P.; Krivonos, S. O.; Ogievetsky, O. V.
2008-07-15
The study of quantum Lie algebras motivates a use of noncanonical ghosts and antighosts for nonlinear algebras, such as W-algebras. This leads, for the W{sub 3} and W{sub 3}{sup (2)} algebras, to the Becchi-Rouet-Stora-Tyutin operator having the conventional cubic form.
Categorical Formulation of Finite-Dimensional Quantum Algebras
NASA Astrophysics Data System (ADS)
Vicary, Jamie
2011-06-01
We describe how †-Frobenius monoids give the correct categorical description of certain kinds of finite-dimensional `quantum algebras'. We develop the concept of an involution monoid, and use it to construct a correspondence between finite-dimensional C*-algebras and certain types of †-Frobenius monoids in the category of Hilbert spaces. Using this technology, we recast the spectral theorems for commutative C*-algebras and for normal operators into an explicitly categorical language, and we examine the case that the results of measurements do not form finite sets, but rather objects in a finite Boolean topos. We describe the relevance of these results for topological quantum field theory.
Walendziak, Andrzej
2015-01-01
The notions of an ideal and a fuzzy ideal in BN-algebras are introduced. The properties and characterizations of them are investigated. The concepts of normal ideals and normal congruences of a BN-algebra are also studied, the properties of them are displayed, and a one-to-one correspondence between them is presented. Conditions for a fuzzy set to be a fuzzy ideal are given. The relationships between ideals and fuzzy ideals of a BN-algebra are established. The homomorphic properties of fuzzy ideals of a BN-algebra are provided. Finally, characterizations of Noetherian BN-algebras and Artinian BN-algebras via fuzzy ideals are obtained. PMID:26125050
Long-term stability of dental arch form in normal occlusion from 13 to 31 years of age.
Henrikson, J; Persson, M; Thilander, B
2001-02-01
Based on observations of longitudinal changes in dental arch dimensions, it has been stated that an individuality of arch form and an integrity of this form exists. However, longitudinal studies evaluating arch form changes have rarely been reported in the literature. The purpose of this investigation was to use a computer-assisted method for the description and analysis of maxillary and mandibular arch form in a sample of normal occlusion subjects, and to evaluate the long-term stability in dental arch form from the age of 13-31 years. The study was carried out on 30 subjects of Scandinavian origin with normal occlusion, recorded at a mean age of 13.6 years and at follow-up at 31.1 years. Arch form analysis was based on a standardized photographic procedure, digitization of morphological landmarks, and a computerized form analysis in which arch form was described using eccentricity values of conics. No specific arch form could be found to represent the sample. Age changes occurred in arch form, although with large individual variations. For the mandible, a significant change to a more rounded arch form with age was found, which in males was accompanied by a significant increase in inter-molar distance and reduction in arch depth. There was also a significant correlation between change in mandibular arch form and increased irregularity of the lower incisors. These findings of lack of stability in arch form in normal occlusion subjects, when passing from adolescence into adulthood, further question the possibility of achieving stability post-orthodontically. PMID:11296510
2005-04-11
The ALGEBRA program allows the user to manipulate data from a finite element analysis before it is plotted. The finite element output data is in the form of variable values (e.g., stress, strain, and velocity components) in an EXODUS II database. The ALGEBRA program evaluates user-supplied functions of the data and writes the results to an output EXODUS II database that can be read by plot programs.
ERIC Educational Resources Information Center
Menn, Lise; And Others
This study examined the role of empathy in the choice of syntactic form and the degree of independence of pragmatic and syntactic abilities in a range of aphasic patients. Study 1 involved 9 English-speaking and 9 Japanese-speaking aphasic subjects with 10 English-speaking and 4 Japanese normal controls. Study 2 involved 14 English- and 6…
A local construction of the Smith normal form of a matrix polynomial
Wilkening, Jon; Yu, Jia
2008-09-01
We present an algorithm for computing a Smith form with multipliers of a regular matrix polynomial over a field. This algorithm differs from previous ones in that it computes a local Smith form for each irreducible factor in the determinant separately and then combines them into a global Smith form, whereas other algorithms apply a sequence of unimodular operations to the original matrix row by row (or column by column). The performance of the algorithm in exact arithmetic is reported for several test cases.
Direct Observation of the Interconversion of Normal and Toxic Forms of α-Synuclein
Cremades, Nunilo; Cohen, Samuel I.A.; Deas, Emma; Abramov, Andrey Y.; Chen, Allen Y.; Orte, Angel; Sandal, Massimo; Clarke, Richard W.; Dunne, Paul; Aprile, Francesco A.; Bertoncini, Carlos W.; Wood, Nicholas W.; Knowles, Tuomas P.J.; Dobson, Christopher M.; Klenerman, David
2012-01-01
Summary Here, we use single-molecule techniques to study the aggregation of α-synuclein, the protein whose misfolding and deposition is associated with Parkinson's disease. We identify a conformational change from the initially formed oligomers to stable, more compact proteinase-K-resistant oligomers as the key step that leads ultimately to fibril formation. The oligomers formed as a result of the structural conversion generate much higher levels of oxidative stress in rat primary neurons than do the oligomers formed initially, showing that they are more damaging to cells. The structural conversion is remarkably slow, indicating a high kinetic barrier for the conversion and suggesting that there is a significant period of time for the cellular protective machinery to operate and potentially for therapeutic intervention, prior to the onset of cellular damage. In the absence of added soluble protein, the assembly process is reversed and fibrils disaggregate to form stable oligomers, hence acting as a source of cytotoxic species. PMID:22632969
Twisted Quantum Toroidal Algebras
NASA Astrophysics Data System (ADS)
Jing, Naihuan; Liu, Rongjia
2014-09-01
We construct a principally graded quantum loop algebra for the Kac-Moody algebra. As a special case a twisted analog of the quantum toroidal algebra is obtained together with the quantum Serre relations.
Cartan calculus on quantum Lie algebras
Schupp, P.; Watts, P.; Zumino, B.
1993-12-09
A generalization of the differential geometry of forms and vector fields to the case of quantum Lie algebras is given. In an abstract formulation that incorporates many existing examples of differential geometry on quantum spaces we combine an exterior derivative, inner derivations, Lie derivatives, forms and functions au into one big algebra, the ``Cartan Calculus.``
Sequential products on effect algebras
NASA Astrophysics Data System (ADS)
Gudder, Stan; Greechie, Richard
2002-02-01
A sequential effect algebra (SEA) is an effect algebra on which a sequential product with natural properties is defined. The properties of sequential products on Hilbert space effect algebras are discussed. For a general SEA, relationships between sequential independence, coexistence and compatibility are given. It is shown that the sharp elements of a SEA form an orthomodular poset. The sequential center of a SEA is discussed and a characterization of when the sequential center is isomorphic to a fuzzy set system is presented. It is shown that the existence, of a sequential product is a strong restriction that eliminates many effect algebras from being SEA's. For example, there are no finite nonboolean SEA's, A measure of sharpness called the sharpness index is studied. The existence of horizontal sums of SEA's is characterized and examples of horizontal sums and tensor products are presented.
Algebraic vs physical N = 6 3-algebras
Cantarini, Nicoletta; Kac, Victor G.
2014-01-15
In our previous paper, we classified linearly compact algebraic simple N = 6 3-algebras. In the present paper, we classify their “physical” counterparts, which actually appear in the N = 6 supersymmetric 3-dimensional Chern-Simons theories.
Form of 15q proximal duplication appears to be a normal euchromatic variant
Jalal, S.M.; Persons, D.L.; DeWald, G.W.; Lindor, N.M.
1994-10-01
Deletions involving often leads to either Prader-Willi or Angelman syndrome, depending on the hereditary path of the deletion (paternal or maternal). A number of cases have been reported in which duplications involving 15q11.2-q13 have not been associated with any detectable phenotypic abnormalities. Ludowese et al. (1991) have summarized 25 such cases that include 10 of their own cases from 5 unrelated families. They conclude that duplication of 15q12-13 does not have an adverse phenotypic effect, though they do not completely rule out the possibility that, instead of 15q12-13 duplication, the extra material could be an insertion from another chromosome. Thus, the dilemma is when duplication of 15q11.2-q13 is clinically significant. We suggest that certain kinds of amplification or duplication involving distal 15q12 and 15q13 may represent a normal variant. 14 refs., 1 fig., 1 tab.
NASA Astrophysics Data System (ADS)
Dankova, T. S.; Rosensteel, G.
1998-10-01
Mean field theory has an unexpected group theoretic mathematical foundation. Instead of representation theory which applies to most group theoretic quantum models, Hartree-Fock and Hartree-Fock-Bogoliubov have been formulated in terms of coadjoint orbits for the groups U(n) and O(2n). The general theory of mean fields is formulated for an arbitrary Lie algebra L of fermion operators. The moment map provides the correspondence between the Hilbert space of microscopic wave functions and the dual space L^* of densities. The coadjoint orbits of the group in the dual space are phase spaces on which time-dependent mean field theory is equivalent to a classical Hamiltonian dynamical system. Indeed it forms a finite-dimensional Lax system. The mean field theories for the Elliott SU(3) and symplectic Sp(3,R) algebras are constructed explicitly in the coadjoint orbit framework.
Paranhos, Luiz Renato; Lima, Carolina Souto; da Silva, Ricardo Henrique Alves; Daruge Júnior, Eduardo; Torres, Fernando Cesar
2012-01-01
The aim of this study was to evaluate the correlation between the morphology of the mandibular dental arch and the maxillary central incisor crown. Cast models from 51 Caucasian individuals, older than 15 years, with optimal occlusion, no previous orthodontic treatment, featuring 4 of the 6 keys to normal occlusion by Andrews (the first being mandatory) were observed. The models were digitalized using a 3D scanner, and images of the maxillary central incisor and mandibular dental arch were obtained. These were printed and placed in an album below pre-set models of arches and dental crowns, and distributed to 12 dental surgeons, who were asked to choose which shape was most in accordance with the models and crown presented. The Kappa test was performed to evaluate the concordance among evaluators while the chi-square test was used to verify the association between the dental arch and central incisor morphology, at a 5% significance level. The Kappa test showed moderate agreement among evaluators for both variables of this study, and the chi-square test showed no significant association between tooth shape and mandibular dental arch morphology. It may be concluded that the use of arch morphology as a diagnostic method to determine the shape of the maxillary central incisor is not appropriate. Further research is necessary to assess tooth shape using a stricter scientific basis. PMID:22666773
NASA Astrophysics Data System (ADS)
Ellison, James A.; Heinemann, Klaus; Vogt, Mathias; Gooden, Matthew
2013-09-01
We present a mathematical analysis of planar motion of energetic electrons moving through a planar dipole undulator, excited by a fixed planar polarized plane wave Maxwell field in the x-ray free electron laser (FEL) regime. Our starting point is the 6D Lorentz system, which allows planar motions, and we examine this dynamical system as the wavelength λ of the traveling wave varies. By scalings and transformations the 6D system is reduced, without approximation, to a 2D system in a form for a rigorous asymptotic analysis using the method of averaging (MoA), a long-time perturbation theory. The two dependent variables are a scaled energy deviation and a generalization of the so-called ponderomotive phase. As λ varies the system passes through resonant and nonresonant (NonR) intervals and we develop NonR and near-to-resonant (NearR) MoA normal form approximations to the exact equations. The NearR normal forms contain a parameter which measures the distance from a resonance. For the planar motion, with the special initial condition that matches into the undulator design trajectory, and on resonance, the NearR normal form reduces to the well-known FEL pendulum system. We then state and prove NonR and NearR first-order averaging theorems which give explicit error bounds for the normal form approximations. We prove the theorems in great detail, giving the interested reader a tutorial on mathematically rigorous perturbation theory in a context where the proofs are easily understood. The proofs are novel in that they do not use a near-identity transformation and they use a system of differential inequalities. The NonR case is an example of quasiperiodic averaging where the small divisor problem enters in the simplest possible way. To our knowledge the planar problem has not been analyzed with the generality we aspire to here nor has the standard FEL pendulum system been derived with associated error bounds as we do here. We briefly discuss the low gain theory in light of
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.
ERIC Educational Resources Information Center
National Council of Teachers of Mathematics, Inc., Reston, VA.
This is a reprint of the historical capsules dealing with algebra from the 31st Yearbook of NCTM,"Historical Topics for the Mathematics Classroom." Included are such themes as the change from a geometric to an algebraic solution of problems, the development of algebraic symbolism, the algebraic contributions of different countries, the origin and…
ERIC Educational Resources Information Center
Padula, Janice
2014-01-01
If educators want to interest students in mathematics (and science), they must engage them in the lower forms of high school or even earlier (Fisher, 2012). So, teachers should always consider a topic's ability to interest students in the early years of instruction in high school and its topicality. Networks have come into prominence recently with…
Choh, Vivian; Lew, MinJung Y; Nadel, Michel W; Wildsoet, Christine F
2006-03-01
To test the hypothesis that the same mechanisms mediate form deprivation and lens-induced myopia, the ocular growth responses of chicks alternately exposed to lenses and diffusers at regular intervals (3h) were compared to those of chicks exposed to either negative lenses or diffusers alone. In total, there were four experiments: (1) -15 D lenses and/or diffusers on normal birds, (2) -15 D lenses and/or diffusers on optic nerve-sectioned (ONS) birds, (3) -5/-10/-15 D lenses (sequentially applied) and/or diffusers on normal birds and (4) -5/-10/-15 D lenses and/or diffusers on ONS birds. All treatments were monocular. In all experiments, optical axial lengths (cornea-to-retina distances) in treated eyes were greater than in fellow eyes, irrespective of the optical device (diffuser, lens or switch), lens power (fixed or incremented) and optic nerve condition (intact or severed). In normal chicks, optical axial length responses in the switch group were significantly reduced relative to those of the diffuser but not to those of the -15 D lens group. For both groups of ONS birds, diffusers exaggerated the optical axial length changes. For all groups, the responses to the switch and lens groups were most similar. These results together suggest that the mechanisms mediating form deprivation- and lens-induced myopia are different. PMID:16212999
Figueroa-O'Farrill, Jose Miguel
2009-11-15
We phrase deformations of n-Leibniz algebras in terms of the cohomology theory of the associated Leibniz algebra. We do the same for n-Lie algebras and for the metric versions of n-Leibniz and n-Lie algebras. We place particular emphasis on the case of n=3 and explore the deformations of 3-algebras of relevance to three-dimensional superconformal Chern-Simons theories with matter.
Role of division algebra in seven-dimensional gauge theory
NASA Astrophysics Data System (ADS)
Kalauni, Pushpa; Barata, J. C. A.
2015-03-01
The algebra of octonions 𝕆 forms the largest normed division algebra over the real numbers ℝ, complex numbers ℂ and quaternions ℍ. The usual three-dimensional vector product is given by quaternions, while octonions produce seven-dimensional vector product. Thus, octonionic algebra is closely related to the seven-dimensional algebra, therefore one can extend generalization of rotations in three dimensions to seven dimensions using octonions. An explicit algebraic description of octonions has been given to describe rotational transformation in seven-dimensional space. We have also constructed a gauge theory based on non-associative algebra to discuss Yang-Mills theory and field equation in seven-dimensional space.
Solving Absolute Value Equations Algebraically and Geometrically
ERIC Educational Resources Information Center
Shiyuan, Wei
2005-01-01
The way in which students can improve their comprehension by understanding the geometrical meaning of algebraic equations or solving algebraic equation geometrically is described. Students can experiment with the conditions of the absolute value equation presented, for an interesting way to form an overall understanding of the concept.
Quantum cluster algebras and quantum nilpotent algebras
Goodearl, Kenneth R.; Yakimov, Milen T.
2014-01-01
A major direction in the theory of cluster algebras is to construct (quantum) cluster algebra structures on the (quantized) coordinate rings of various families of varieties arising in Lie theory. We prove that all algebras in a very large axiomatically defined class of noncommutative algebras possess canonical quantum cluster algebra structures. Furthermore, they coincide with the corresponding upper quantum cluster algebras. We also establish analogs of these results for a large class of Poisson nilpotent algebras. Many important families of coordinate rings are subsumed in the class we are covering, which leads to a broad range of applications of the general results to the above-mentioned types of problems. As a consequence, we prove the Berenstein–Zelevinsky conjecture [Berenstein A, Zelevinsky A (2005) Adv Math 195:405–455] for the quantized coordinate rings of double Bruhat cells and construct quantum cluster algebra structures on all quantum unipotent groups, extending the theorem of Geiß et al. [Geiß C, et al. (2013) Selecta Math 19:337–397] for the case of symmetric Kac–Moody groups. Moreover, we prove that the upper cluster algebras of Berenstein et al. [Berenstein A, et al. (2005) Duke Math J 126:1–52] associated with double Bruhat cells coincide with the corresponding cluster algebras. PMID:24982197
Rees algebras, Monomial Subrings and Linear Optimization Problems
NASA Astrophysics Data System (ADS)
Dupont, Luis A.
2010-06-01
In this thesis we are interested in studying algebraic properties of monomial algebras, that can be linked to combinatorial structures, such as graphs and clutters, and to optimization problems. A goal here is to establish bridges between commutative algebra, combinatorics and optimization. We study the normality and the Gorenstein property-as well as the canonical module and the a-invariant-of Rees algebras and subrings arising from linear optimization problems. In particular, we study algebraic properties of edge ideals and algebras associated to uniform clutters with the max-flow min-cut property or the packing property. We also study algebraic properties of symbolic Rees algebras of edge ideals of graphs, edge ideals of clique clutters of comparability graphs, and Stanley-Reisner rings.
Asymptotic Expansion of the Heteroclinic Bifurcation for the Planar Normal Form of the 1:2 Resonance
NASA Astrophysics Data System (ADS)
Roberto, Luci A. F.; da Silva, Paulo R.; Torregrosa, Joan
We consider the family of planar differential systems depending on two real parameters ẋ = y,ẏ = δ1x + δ2y + x3 - x2y. This system corresponds to the normal form for the 1:2 resonance which exhibits a heteroclinic connection. The phase portrait of the system has a limit cycle which disappears in the heteroclinic connection for the parameter values on the curve δ2 = c(δ1) = -1 5δ1 + O(δ12), δ1 < 0. We significantly improve the knowledge of this curve in a neighborhood of the origin.
Bobodzhanov, A A; Safonov, V F
2013-07-31
The paper deals with extending the Lomov regularization method to classes of singularly perturbed Fredholm-type integro-differential systems, which have not so far been studied. In these the limiting operator is discretely noninvertible. Such systems are commonly known as problems with unstable spectrum. Separating out the essential singularities in the solutions to these problems presents great difficulties. The principal one is to give an adequate description of the singularities induced by 'instability points' of the spectrum. A methodology for separating singularities by using normal forms is developed. It is applied to the above type of systems and is substantiated in these systems. Bibliography: 10 titles.
Algebraic grid generation with corner singularities
NASA Technical Reports Server (NTRS)
Vinokur, M.; Lombard, C. K.
1983-01-01
A simple noniterative algebraic procedure is presented for generating smooth computational meshes on a quadrilateral topology. Coordinate distribution and normal derivative are provided on all boundaries, one of which may include a slope discontinuity. The boundary conditions are sufficient to guarantee continuity of global meshes formed of joined patches generated by the procedure. The method extends to 3-D. The procedure involves a synthesis of prior techniques stretching functions, cubic blending functions, and transfinite interpolation - to which is added the functional form of the corner solution. The procedure introduces the concept of generalized blending, which is implemented as an automatic scaling of the boundary derivatives for effective interpolation. Some implications of the treatment at boundaries for techniques solving elliptic PDE's are discussed in an Appendix.
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.
Ternary generalization of Heisenberg's algebra
NASA Astrophysics Data System (ADS)
Kerner, Richard
2015-06-01
A concise study of ternary and cubic algebras with Z3 grading is presented. We discuss some underlying ideas leading to the conclusion that the discrete symmetry group of permutations of three objects, S3, and its abelian subgroup Z3 may play an important role in quantum physics. We show then how most of important algebras with Z2 grading can be generalized with ternary composition laws combined with a Z3 grading. We investigate in particular a ternary, Z3-graded generalization of the Heisenberg algebra. It turns out that introducing a non-trivial cubic root of unity, , one can define two types of creation operators instead of one, accompanying the usual annihilation operator. The two creation operators are non-hermitian, but they are mutually conjugate. Together, the three operators form a ternary algebra, and some of their cubic combinations generate the usual Heisenberg algebra. An analogue of Hamiltonian operator is constructed by analogy with the usual harmonic oscillator, and some properties of its eigenfunctions are briefly discussed.
Algebraic Lattices in QFT Renormalization
NASA Astrophysics Data System (ADS)
Borinsky, Michael
2016-04-01
The structure of overlapping subdivergences, which appear in the perturbative expansions of quantum field theory, is analyzed using algebraic lattice theory. It is shown that for specific QFTs the sets of subdivergences of Feynman diagrams form algebraic lattices. This class of QFTs includes the standard model. In kinematic renormalization schemes, in which tadpole diagrams vanish, these lattices are semimodular. This implies that the Hopf algebra of Feynman diagrams is graded by the coradical degree or equivalently that every maximal forest has the same length in the scope of BPHZ renormalization. As an application of this framework, a formula for the counter terms in zero-dimensional QFT is given together with some examples of the enumeration of primitive or skeleton diagrams.
Algebraic Lattices in QFT Renormalization
NASA Astrophysics Data System (ADS)
Borinsky, Michael
2016-07-01
The structure of overlapping subdivergences, which appear in the perturbative expansions of quantum field theory, is analyzed using algebraic lattice theory. It is shown that for specific QFTs the sets of subdivergences of Feynman diagrams form algebraic lattices. This class of QFTs includes the standard model. In kinematic renormalization schemes, in which tadpole diagrams vanish, these lattices are semimodular. This implies that the Hopf algebra of Feynman diagrams is graded by the coradical degree or equivalently that every maximal forest has the same length in the scope of BPHZ renormalization. As an application of this framework, a formula for the counter terms in zero-dimensional QFT is given together with some examples of the enumeration of primitive or skeleton diagrams.
Gatti, G; Barzaghi, N; Attardo Parrinello, G; Vitiello, B; Perucca, E
1989-01-01
The pharmacokinetic profile of an innovative formulation of soluble aspirin (l-ornithine acetylsalicylate, ldB 1003) was compared with that of conventional tablets and two other soluble dosage forms (d, l-lysine acetylsalicylate and a buffered effervescent formulation of acetylsalicylic acid) after administration of single oral doses in six normal volunteers. All soluble forms showed a rapid absorption profile, peak plasma salicylic acid levels being attained after about 30 min on average and without statistically significant differences among the solutions tested. As compared to the soluble formulations, acetylsalicylic acid given as tablets resulted in slower absorption, with peak plasma salicylic acid levels being reached more than 1 h after dosing. Despite these differences in time course of plasma level profiles, the extent of absorption was similar for all formulations. Apart from the potential advantages in terms of improved gastric tolerability, the increased rate of absorption of aspirin solutions is therapeutically useful whenever a rapid onset of action is required. In this respect, the kinetic pattern of the innovative formulation compares favourably with that of other available soluble dosage forms. PMID:2517497
Learning Algebra in a Computer Algebra Environment
ERIC Educational Resources Information Center
Drijvers, Paul
2004-01-01
This article summarises a doctoral thesis entitled "Learning algebra in a computer algebra environment, design research on the understanding of the concept of parameter" (Drijvers, 2003). It describes the research questions, the theoretical framework, the methodology and the results of the study. The focus of the study is on the understanding of…
NASA Technical Reports Server (NTRS)
Iachello, Franco
1995-01-01
An algebraic formulation of quantum mechanics is presented. In this formulation, operators of interest are expanded onto elements of an algebra, G. For bound state problems in nu dimensions the algebra G is taken to be U(nu + 1). Applications to the structure of molecules are presented.
Orientation in operator algebras
Alfsen, Erik M.; Shultz, Frederic W.
1998-01-01
A concept of orientation is relevant for the passage from Jordan structure to associative structure in operator algebras. The research reported in this paper bridges the approach of Connes for von Neumann algebras and ourselves for C*-algebras in a general theory of orientation that is of geometric nature and is related to dynamics. PMID:9618457
Developing Thinking in Algebra
ERIC Educational Resources Information Center
Mason, John; Graham, Alan; Johnson-Wilder, Sue
2005-01-01
This book is for people with an interest in algebra whether as a learner, or as a teacher, or perhaps as both. It is concerned with the "big ideas" of algebra and what it is to understand the process of thinking algebraically. The book has been structured according to a number of pedagogic principles that are exposed and discussed along the way,…
Connecting Arithmetic to Algebra
ERIC Educational Resources Information Center
Darley, Joy W.; Leapard, Barbara B.
2010-01-01
Algebraic thinking is a top priority in mathematics classrooms today. Because elementary school teachers lay the groundwork to develop students' capacity to think algebraically, it is crucial for teachers to have a conceptual understanding of the connections between arithmetic and algebra and be confident in communicating these connections. Many…
Applied Algebra Curriculum Modules.
ERIC Educational Resources Information Center
Texas State Technical Coll., Marshall.
This collection of 11 applied algebra curriculum modules can be used independently as supplemental modules for an existing algebra curriculum. They represent diverse curriculum styles that should stimulate the teacher's creativity to adapt them to other algebra concepts. The selected topics have been determined to be those most needed by students…
Profiles of Algebraic Competence
ERIC Educational Resources Information Center
Humberstone, J.; Reeve, R.A.
2008-01-01
The algebraic competence of 72 12-year-old female students was examined to identify profiles of understanding reflecting different algebraic knowledge states. Beginning algebraic competence (mapping abilities: word-to-symbol and vice versa, classifying, and solving equations) was assessed. One week later, the nature of assistance required to map…
Ternary Virasoro - Witt algebra.
Zachos, C.; Curtright, T.; Fairlie, D.; High Energy Physics; Univ. of Miami; Univ. of Durham
2008-01-01
A 3-bracket variant of the Virasoro-Witt algebra is constructed through the use of su(1,1) enveloping algebra techniques. The Leibniz rules for 3-brackets acting on other 3-brackets in the algebra are discussed and verified in various situations.
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.
Kuznetsov, Sergei A.; Mankani, Mahesh H.; Bianco, Paolo; Robey, Pamela G.
2009-01-01
Bone marrow stromal cell populations, containing a subset of multipotential skeletal stem cells, are increasingly contemplated for use in tissue engineering and stem cell therapy, whereas their involvement in the pathogenetic mechanisms of skeletal disorders is far less recognized. We compared the concentrations of stromal clonogenic cells, colony forming units–fibroblast (CFU-Fs), in norm and pathology. Initially, culture conditions were optimized by demonstrating that fetal bovine serum heat inactivation could significantly repress colony formation. Using non-heat-inactivated fetal bovine serum, the concentration of CFU-Fs (colony-forming efficiency, CFE) ranged from 3.5 ± 1.0 to 11.5 ± 4.0 per 1 × 105 nucleated cells in five inbred mouse strains. In four transgenic lines with profound bone involvement, CFE was either significantly reduced or increased compared to wild-type littermates. In normal human donors, CFE decreased slightly with age and averaged 52.2 ± 4.1 for children and 32.3 ± 3.0 for adults. CFE was significantly altered in patients with several skeletal, metabolic, and hematological disorders: reduced in congenital generalized lipodystrophy, achondroplasia (SADDAN), pseudoachondroplasia, and Paget disease of bone and elevated in alcaptonuria and sickle cell anemia. Our findings indicate that under appropriate culture conditions, CFE values may provide useful insights into bone/bone marrow pathophysiology. PMID:19383412
Krukowska, Anna; Tarkowski, Andrzej K
2005-11-01
A mouse spermatozoon was injected into mouse secondary oocytes (ICSI) in the vicinity of the metaphase spindle. In 22% of oocytes injected successfully, the maternal chromatin (the haploid chromatids formed after the second meiotic division) and paternal chromatin (from the sperm nucleus) were surrounded by a common nuclear envelope to form one diploid bi-parental pronucleus. However, the use of spermatozoa in which BrdU had been incorporated into DNA during spermatogenesis revealed, that maternal and paternal chromatin occupied two separate compartments within the one pronucleus. In the living state, the diploid pronucleus could be distinguished from a haploid one by its distinctly larger size and by a greater number of "nucleolus-like bodies"-criteria confirmed karylogically at the 1st cleavage division. Such zygotes with one diploid pronucleus were able to develop in vitro into blastocysts as often as those with two haploid pronuclei [11/29 (38%) vs. 14/35 (40%)]. Seventy nine 2-cell embryos developing in vitro from zygotes with one diploid pronucleus were transplanted to the oviducts of pseudopregnant recipients: two females had six foetuses when killed on the 17th day, and two females gave birth to nine young, eight of which survived and developed into normal fertile animals. PMID:16047392
Structure of classical affine and classical affine fractional W-algebras
Suh, Uhi Rinn
2015-01-15
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 of 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.
Exponential growth of codimensions of identities of algebras with unity
NASA Astrophysics Data System (ADS)
Zaicev, M. V.; Repovš, D.
2015-10-01
The asymptotic behaviour is studied of exponentially bounded sequences of codimensions of identities of algebras with unity. A series of algebras is constructed for which the base of the exponential increases by exactly 1 when an outer unity is adjoined to the original algebra. It is shown that the PI-exponents of unital algebras can take any value greater than 2, and the exponents of finite-dimensional unital algebras form a dense subset in the domain \\lbrack 2,∞). Bibliography: 34 titles.
Constraint algebra in bigravity
Soloviev, V. O.
2015-07-15
The number of degrees of freedom in bigravity theory is found for a potential of general form and also for the potential proposed by de Rham, Gabadadze, and Tolley (dRGT). This aim is pursued via constructing a Hamiltonian formalismand studying the Poisson algebra of constraints. A general potential leads to a theory featuring four first-class constraints generated by general covariance. The vanishing of the respective Hessian is a crucial property of the dRGT potential, and this leads to the appearance of two additional second-class constraints and, hence, to the exclusion of a superfluous degree of freedom—that is, the Boulware—Deser ghost. The use of a method that permits avoiding an explicit expression for the dRGT potential is a distinctive feature of the present study.
NASA Astrophysics Data System (ADS)
Matus-Vargas, Antonio; González-Hernandez, Hugo G.; Chan, Bernard S.; Palacios, Antonio; Buono, Pietro-Luciano; in, Visarath; Naik, Suketu; Phipps, Alex; Longhini, Patrick
Modeling and bifurcation analysis of an energy harvesting system composed of coupled resonators using the Galfenol-based magnetostrictive material are presented. The analysis in this work should be broad enough to be applicable to a large class of vibratory-based energy harvesting systems since various types of vibratory harvesters share the same normal forms, e.g. magnetostrictive and piezoelectric materials. A combined model of the mechanical and electrical domains of a single energy harvester is discussed first. Building on this model, the governing equations of the coupled system are derived, leading to a system of differential equations with an all-to-all coupling between the resonators. A bifurcation analysis of the system equations reveals different patterns of collective oscillations. Among the many different patterns, a synchronous state exists and it is stable over a broad region of parameter space. This pattern has the potential to yield significant increases in power output and it will be used as a starting point to guide future experimental work. A Hamiltonian approach is employed to study analytically the nature of the bifurcations and to calculate an expression for the onset of synchronization valid for any number of harvesters.
A Richer Understanding of Algebra
ERIC Educational Resources Information Center
Foy, Michelle
2008-01-01
Algebra is one of those hard-to-teach topics where pupils seem to struggle to see it as more than a set of rules to learn, but this author recently used the software "Grid Algebra" from ATM, which engaged her Year 7 pupils in exploring algebraic concepts for themselves. "Grid Algebra" allows pupils to experience number, pre-algebra, and algebra…
2003-06-03
The ALGEBRA II program allows the user to manipulate data from a finite element analysis before it is plotted by evaluating algebraic expressions. The equation variables are dependent on the input database variable names. The finite element output data is in the form of variable values (e.g., stress, strain, and velocity components) in an EXODUS II database which can be read by plot programs. Code is written in a portable form as possible. Fortran codemore » is written in ANSI Standard FORTRAN-77. Machine-specific routines are limited in number and are grouped together to minimize the time required to adapt them to a new system. SEACAS codes has been ported to several Unix systems.« less
Algebraic Sub-Structuring for Electromagnetic Applications
Yang, C.; Gao, W.G.; Bai, Z.J.; Li, X.Y.S.; Lee, L.Q.; Husbands, P.; Ng, E.G.; /LBL, Berkeley /UC, Davis /SLAC
2006-06-30
Algebraic sub-structuring refers to the process of applying matrix reordering and partitioning algorithms to divide a large sparse matrix into smaller submatrices from which a subset of spectral components are extracted and combined to form approximate solutions to the original problem. In this paper, they show that algebraic sub-structuring can be effectively used to solve generalized eigenvalue problems arising from the finite element analysis of an accelerator structure.
Algebraic sub-structuring for electromagnetic applications
Yang, Chao; Gao, Weiguo; Bai, Zhaojun; Li, Xiaoye; Lee, Lie-Quan; Husbands, Parry; Ng, Esmond G.
2004-09-14
Algebraic sub-structuring refers to the process of applying matrix reordering and partitioning algorithms to divide a large sparse matrix into smaller submatrices from which a subset of spectral components are extracted and combined to form approximate solutions to the original problem. In this paper, we show that algebraic sub-structuring can be effectively used to solve generalized eigenvalue problems arising from the finite element analysis of an accelerator structure.
Algebraic operator approach to gas kinetic models
NASA Astrophysics Data System (ADS)
Il'ichov, L. V.
1997-02-01
Some general properties of the linear Boltzmann kinetic equation are used to present it in the form ∂ tϕ = - Â†Âϕ with the operators ÂandÂ† possessing some nontrivial algebraic properties. When applied to the Keilson-Storer kinetic model, this method gives an example of quantum ( q-deformed) Lie algebra. This approach provides also a natural generalization of the “kangaroo model”.
Connecting Algebra and Chemistry.
ERIC Educational Resources Information Center
O'Connor, Sean
2003-01-01
Correlates high school chemistry curriculum with high school algebra curriculum and makes the case for an integrated approach to mathematics and science instruction. Focuses on process integration. (DDR)
Comparing the Effectiveness of Collaborative Instructional Practices in Algebra
ERIC Educational Resources Information Center
Triaga, Russell D.
2014-01-01
The use of multiple forms of collaborative instruction to teach integrated algebra makes it difficult for teachers to determine which collaborative form is best suited for the curriculum. An inconsistent approach to integrated algebra instruction at the study school needed to be addressed for the benefit of teacher effectiveness and student…
Twisted vertex algebras, bicharacter construction and boson-fermion correspondences
Anguelova, Iana I.
2013-12-15
The boson-fermion correspondences are an important phenomena on the intersection of several areas in mathematical physics: representation theory, vertex algebras and conformal field theory, integrable systems, number theory, cohomology. Two such correspondences are well known: the types A and B (and their super extensions). As a main result of this paper we present a new boson-fermion correspondence of type D-A. Further, we define a new concept of twisted vertex algebra of order N, which generalizes super vertex algebra. We develop the bicharacter construction which we use for constructing classes of examples of twisted vertex algebras, as well as for deriving formulas for the operator product expansions, analytic continuations, and normal ordered products. By using the underlying Hopf algebra structure we prove general bicharacter formulas for the vacuum expectation values for two important groups of examples. We show that the correspondences of types B, C, and D-A are isomorphisms of twisted vertex algebras.
ERIC Educational Resources Information Center
Merlin, Ethan M.
2013-01-01
This article describes how the author has developed tasks for students that address the missed "essence of the matter" of algebraic transformations. Specifically, he has found that having students practice "perceiving" algebraic structure--by naming the "glue" in the expressions, drawing expressions using…
ERIC Educational Resources Information Center
Levy, Alissa Beth
2012-01-01
The California Department of Education (CDE) has long asserted that success Algebra I by Grade 8 is the goal for all California public school students. In fact, the state's accountability system penalizes schools that do not require all of their students to take the Algebra I end-of-course examination by Grade 8 (CDE, 2009). In this…
NASA Technical Reports Server (NTRS)
Lawson, C. L.; Krogh, F. T.; Gold, S. S.; Kincaid, D. R.; Sullivan, J.; Williams, E.; Hanson, R. J.; Haskell, K.; Dongarra, J.; Moler, C. B.
1982-01-01
The Basic Linear Algebra Subprograms (BLAS) library is a collection of 38 FORTRAN-callable routines for performing basic operations of numerical linear algebra. BLAS library is portable and efficient source of basic operations for designers of programs involving linear algebriac computations. BLAS library is supplied in portable FORTRAN and Assembler code versions for IBM 370, UNIVAC 1100 and CDC 6000 series computers.
ERIC Educational Resources Information Center
Cavanagh, Sean
2008-01-01
A popular humorist and avowed mathphobe once declared that in real life, there's no such thing as algebra. Kathie Wilson knows better. Most of the students in her 8th grade class will be thrust into algebra, the definitive course that heralds the beginning of high school mathematics, next school year. The problem: Many of them are about three…
Polynomial Extensions of the Weyl C*-Algebra
NASA Astrophysics Data System (ADS)
Accardi, Luigi; Dhahri, Ameur
2015-09-01
We introduce higher order (polynomial) extensions of the unique (up to isomorphisms) nontrivial central extension of the Heisenberg algebra, which can be concretely realized as sub-Lie algebras of the polynomial algebra generated by the creation and annihilation operators in the Schrödinger representation. The simplest nontrivial of these extensions (the quadratic one) is isomorphic to the Galilei algebra, widely studied in quantum physics. By exponentiation of this representation we construct the corresponding polynomial analogue of the Weyl C*-algebra and compute the polynomial Weyl relations. From this we deduce the explicit form of the composition law of the associated nonlinear extensions of the 1-dimensional Heisenberg group. The above results are used to calculate a simple explicit form of the vacuum characteristic functions of the nonlinear field operators of the Galilei algebra, as well as of their moments. The corresponding measures turn out to be an interpolation family between Gaussian and Meixner, in particular Gamma.
On the algebra of gauge invariants for one-flavour chromodynamics
NASA Astrophysics Data System (ADS)
Kijowski, J.; Rudolph, G.; Rudolph, M.
1997-08-01
The structure of the algebra of gauge invariant differential forms built from SU(3)-gauge potentials as well as (Grassmann algebra-valued) quark and antiquark fields is discussed. The relevance to one-flavour chromodynamics is outlined.
Highest weight representation for Sklyanin algebra sl(3)(u) with application to the Gaudin model
Burdik, C.; Navratil, O.
2011-06-15
We study the infinite-dimensional Sklyanin algebra sl(3)(u). Specifically we construct the highest weight representation for this algebra in an explicit form. Its application to the Gaudin model is mentioned.
Titration Calculations with Computer Algebra Software
ERIC Educational Resources Information Center
Lachance, Russ; Biaglow, Andrew
2012-01-01
This article examines the symbolic algebraic solution of the titration equations for a diprotic acid, as obtained using "Mathematica," "Maple," and "Mathcad." The equilibrium and conservation equations are solved symbolically by the programs to eliminate the approximations that normally would be performed by the student. Of the three programs,…
Semigroups and computer algebra in algebraic structures
NASA Astrophysics Data System (ADS)
Bijev, G.
2012-11-01
Some concepts in semigroup theory can be interpreted in several algebraic structures. A generalization fA,B,fA,B(X) = A(X')B of the complement operator (') on Boolean matrices is made, where A and B denote any rectangular Boolean matrices. While (') is an isomorphism between Boolean semilattices, the generalized complement operator is homomorphism in the general case. The map fA,B and its general inverse (fA,B)+ have quite similar properties to those in the linear algebra and are useful for solving linear equations in Boolean matrix algebras. For binary relations on a finite set, necessary and sufficient conditions for the equation αξβ = γ to have a solution ξ are proved. A generalization of Green's equivalence relations in semigroups for rectangular matrices is proposed. Relationships between them and the Moore-Penrose inverses are investigated. It is shown how any generalized Green's H-class could be constructed by given its corresponding linear subspaces and converted into a group isomorphic to a linear group. Some information about using computer algebra methods concerning this paper is given.
Lie algebra extensions of current algebras on S3
NASA Astrophysics Data System (ADS)
Kori, Tosiaki; Imai, Yuto
2015-06-01
An affine Kac-Moody algebra is a central extension of the Lie algebra of smooth mappings from S1 to the complexification of a Lie algebra. In this paper, we shall introduce a central extension of the Lie algebra of smooth mappings from S3 to the quaternization of a Lie algebra and investigate its root space decomposition. We think this extension of current algebra might give a mathematical tool for four-dimensional conformal field theory as Kac-Moody algebras give it for two-dimensional conformal field theory.
Leibniz algebras associated with representations of filiform Lie algebras
NASA Astrophysics Data System (ADS)
Ayupov, Sh. A.; Camacho, L. M.; Khudoyberdiyev, A. Kh.; Omirov, B. A.
2015-12-01
In this paper we investigate Leibniz algebras whose quotient Lie algebra is a naturally graded filiform Lie algebra nn,1. We introduce a Fock module for the algebra nn,1 and provide classification of Leibniz algebras L whose corresponding Lie algebra L / I is the algebra nn,1 with condition that the ideal I is a Fock nn,1-module, where I is the ideal generated by squares of elements from L. We also consider Leibniz algebras with corresponding Lie algebra nn,1 and such that the action I ×nn,1 → I gives rise to a minimal faithful representation of nn,1. The classification up to isomorphism of such Leibniz algebras is given for the case of n = 4.
Coreflections in Algebraic Quantum Logic
NASA Astrophysics Data System (ADS)
Jacobs, Bart; Mandemaker, Jorik
2012-07-01
Various generalizations of Boolean algebras are being studied in algebraic quantum logic, including orthomodular lattices, orthomodular po-sets, orthoalgebras and effect algebras. This paper contains a systematic study of the structure in and between categories of such algebras. It does so via a combination of totalization (of partially defined operations) and transfer of structure via coreflections.
NASA Astrophysics Data System (ADS)
Fortunati, Alessandro; Wiggins, Stephen
2016-06-01
The paper deals with the problem of the existence of a normal form for a nearly-integrable real-analytic Hamiltonian with aperiodically time-dependent perturbation decaying (slowly) in time. In particular, in the case of an isochronous integrable part, the system can be cast in an exact normal form, regardless of the properties of the frequency vector. The general case is treated by a suitable adaptation of the finite order normalization techniques usually used for Nekhoroshev arguments. The key point is that the so called "geometric part" is not necessary in this case. As a consequence, no hypotheses on the integrable part are required, apart from analyticity. The work, based on two different perturbative approaches developed by Giorgilli et al., is a generalisation of the techniques used by the same authors to treat more specific aperiodically time-dependent problems.
A process algebra model of QED
NASA Astrophysics Data System (ADS)
Sulis, William
2016-03-01
The process algebra approach to quantum mechanics posits a finite, discrete, determinate ontology of primitive events which are generated by processes (in the sense of Whitehead). In this ontology, primitive events serve as elements of an emergent space-time and of emergent fundamental particles and fields. Each process generates a set of primitive elements, using only local information, causally propagated as a discrete wave, forming a causal space termed a causal tapestry. Each causal tapestry forms a discrete and finite sampling of an emergent causal manifold (space-time) M and emergent wave function. Interactions between processes are described by a process algebra which possesses 8 commutative operations (sums and products) together with a non-commutative concatenation operator (transitions). The process algebra possesses a representation via nondeterministic combinatorial games. The process algebra connects to quantum mechanics through the set valued process and configuration space covering maps, which associate each causal tapestry with sets of wave functions over M. Probabilities emerge from interactions between processes. The process algebra model has been shown to reproduce many features of the theory of non-relativistic scalar particles to a high degree of accuracy, without paradox or divergences. This paper extends the approach to a semi-classical form of quantum electrodynamics.
FAST TRACK COMMUNICATION: Kac Moody algebras and controlled chaos
NASA Astrophysics Data System (ADS)
Wesley, Daniel H.
2007-02-01
Compactification can control chaotic Mixmaster behaviour in gravitational systems with p-form matter: we consider this in light of the connection between supergravity models and Kac Moody algebras. We show that different compactifications define 'mutations' of the algebras associated with the noncompact theories. We list the algebras obtained in this way, and find novel examples of wall systems determined by Lorentzian (but not hyperbolic) algebras. Cosmological models with a smooth pre-big bang phase require that chaos is absent: we show that compactification alone cannot eliminate chaos in the simplest compactifications of the heterotic string on a Calabi Yau, or M theory on a manifold of G2 holonomy.
Traditional vectors as an introduction to geometric algebra
NASA Astrophysics Data System (ADS)
Carroll, J. E.
2003-07-01
The 2002 Oersted Medal Lecture by David Hestenes concerns the many advantages for education in physics if geometric algebra were to replace standard vector algebra. However, such a change has difficulties for those who have been taught traditionally. A new way of introducing geometric algebra is presented here using a four-element array composed of traditional vector and scalar products. This leads to an explicit 4 × 4 matrix representation which contains key requirements for three-dimensional geometric algebra. The work can be extended to include Maxwell's equations where it is found that curl and divergence appear naturally together. However, to obtain an explicit representation of space-time algebra with the correct behaviour under Lorentz transformations, an 8 × 8 matrix representation has to be formed. This leads to a Dirac representation of Maxwell's equations showing that space-time algebra has hidden within its formalism the symmetry of 'parity, charge conjugation and time reversal'.
Developing Algebraic Thinking.
ERIC Educational Resources Information Center
Alejandre, Suzanne
2002-01-01
Presents a teaching experience that resulted in students getting to a point of full understanding of the kinesthetic activity and the algebra behind it. Includes a lesson plan for a traffic jam activity. (KHR)
Algebraic Semantics for Narrative
ERIC Educational Resources Information Center
Kahn, E.
1974-01-01
This paper uses discussion of Edmund Spenser's "The Faerie Queene" to present a theoretical framework for explaining the semantics of narrative discourse. The algebraic theory of finite automata is used. (CK)
NASA Astrophysics Data System (ADS)
Lannes, A.; Teunissen, P. J. G.
2011-05-01
The first objective of this paper is to show that some basic concepts used in global navigation satellite systems (GNSS) are similar to those introduced in Fourier synthesis for handling some phase calibration problems. In experimental astronomy, the latter are at the heart of what is called `phase closure imaging.' In both cases, the analysis of the related structures appeals to the algebraic graph theory and the algebraic number theory. For example, the estimable functions of carrier-phase ambiguities, which were introduced in GNSS to correct some rank defects of the undifferenced equations, prove to be `closure-phase ambiguities:' the so-called `closure-delay' (CD) ambiguities. The notion of closure delay thus generalizes that of double difference (DD). The other estimable functional variables involved in the phase and code undifferenced equations are the receiver and satellite pseudo-clock biases. A related application, which corresponds to the second objective of this paper, concerns the definition of the clock information to be broadcasted to the network users for their precise point positioning (PPP). It is shown that this positioning can be achieved by simply having access to the satellite pseudo-clock biases. For simplicity, the study is restricted to relatively small networks. Concerning the phase for example, these biases then include five components: a frequency-dependent satellite-clock error, a tropospheric satellite delay, an ionospheric satellite delay, an initial satellite phase, and an integer satellite ambiguity. The form of the PPP equations to be solved by the network user is then similar to that of the traditional PPP equations. As soon as the CD ambiguities are fixed and validated, an operation which can be performed in real time via appropriate decorrelation techniques, estimates of these float biases can be immediately obtained. No other ambiguity is to be fixed. The satellite pseudo-clock biases can thus be obtained in real time. This is
Aprepro - Algebraic Preprocessor
2005-08-01
Aprepro is an algebraic preprocessor that reads a file containing both general text and algebraic, string, or conditional expressions. It interprets the expressions and outputs them to the output file along witht the general text. Aprepro contains several mathematical functions, string functions, and flow control constructs. In addition, functions are included that, with some additional files, implement a units conversion system and a material database lookup system.
Geometric Algebra for Physicists
NASA Astrophysics Data System (ADS)
Doran, Chris; Lasenby, Anthony
2007-11-01
Preface; Notation; 1. Introduction; 2. Geometric algebra in two and three dimensions; 3. Classical mechanics; 4. Foundations of geometric algebra; 5. Relativity and spacetime; 6. Geometric calculus; 7. Classical electrodynamics; 8. Quantum theory and spinors; 9. Multiparticle states and quantum entanglement; 10. Geometry; 11. Further topics in calculus and group theory; 12. Lagrangian and Hamiltonian techniques; 13. Symmetry and gauge theory; 14. Gravitation; Bibliography; Index.
Towards a cladistics of double Yangians and elliptic algebras*
NASA Astrophysics Data System (ADS)
Arnaudon, D.; Avan, J.; Frappat, L.; Ragoucy, E.; Rossi, M.
2000-09-01
A self-contained description of algebraic structures, obtained by combinations of various limit procedures applied to vertex and face sl(2) elliptic quantum affine algebras, is given. New double Yangian structures of dynamical type are defined. Connections between these structures are established. A number of them take the form of twist-like actions. These are conjectured to be evaluations of universal twists.
NASA Astrophysics Data System (ADS)
Hiley, B. J.
In this chapter, we examine in detail the non-commutative symplectic algebra underlying quantum dynamics. By using this algebra, we show that it contains both the Weyl-von Neumann and the Moyal quantum algebras. The latter contains the Wigner distribution as the kernel of the density matrix. The underlying non-commutative geometry can be projected into either of two Abelian spaces, so-called `shadow phase spaces'. One of these is the phase space of Bohmian mechanics, showing that it is a fragment of the basic underlying algebra. The algebraic approach is much richer, giving rise to two fundamental dynamical time development equations which reduce to the Liouville equation and the Hamilton-Jacobi equation in the classical limit. They also include the Schrödinger equation and its wave-function, showing that these features are a partial aspect of the more general non-commutative structure. We discuss briefly the properties of this more general mathematical background from which the non-commutative symplectic algebra emerges.
DG Poisson algebra and its universal enveloping algebra
NASA Astrophysics Data System (ADS)
Lü, JiaFeng; Wang, XingTing; Zhuang, GuangBin
2016-05-01
In this paper, we introduce the notions of differential graded (DG) Poisson algebra and DG Poisson module. Let $A$ be any DG Poisson algebra. We construct the universal enveloping algebra of $A$ explicitly, which is denoted by $A^{ue}$. We show that $A^{ue}$ has a natural DG algebra structure and it satisfies certain universal property. As a consequence of the universal property, it is proved that the category of DG Poisson modules over $A$ is isomorphic to the category of DG modules over $A^{ue}$. Furthermore, we prove that the notion of universal enveloping algebra $A^{ue}$ is well-behaved under opposite algebra and tensor product of DG Poisson algebras. Practical examples of DG Poisson algebras are given throughout the paper including those arising from differential geometry and homological algebra.
On a Equation in Finite Algebraically Structures
ERIC Educational Resources Information Center
Valcan, Dumitru
2013-01-01
Solving equations in finite algebraically structures (semigroups with identity, groups, rings or fields) many times is not easy. Even the professionals can have trouble in such cases. Therefore, in this paper we proposed to solve in the various finite groups or fields, a binomial equation of the form (1). We specify that this equation has been…
Generalizing: The Core of Algebraic Thinking
ERIC Educational Resources Information Center
Kinach, Barbara M.
2014-01-01
Generalizing--along with conjecturing, representing, justifying, and refuting--are forms of mathematical reasoning important in all branches of mathematics (Lannin, Ellis, and Elliott 2011). Increasingly, however, generalizing is recognized as the essence of thinking in algebra (Mason, Graham, and Johnston-Wilder 2010; Kaput, Carraher, and Blanton…
A Linear Algebraic Approach to Teaching Interpolation
ERIC Educational Resources Information Center
Tassa, Tamir
2007-01-01
A novel approach for teaching interpolation in the introductory course in numerical analysis is presented. The interpolation problem is viewed as a problem in linear algebra, whence the various forms of interpolating polynomial are seen as different choices of a basis to the subspace of polynomials of the corresponding degree. This approach…
Modules as Learning Tools in Linear Algebra
ERIC Educational Resources Information Center
Cooley, Laurel; Vidakovic, Draga; Martin, William O.; Dexter, Scott; Suzuki, Jeff; Loch, Sergio
2014-01-01
This paper reports on the experience of STEM and mathematics faculty at four different institutions working collaboratively to integrate learning theory with curriculum development in a core undergraduate linear algebra context. The faculty formed a Professional Learning Community (PLC) with a focus on learning theories in mathematics and…
Mora, Maximilian; Bellack, Annett; Ugele, Matthias; Hopf, Johann
2014-01-01
To date, the behavior of hyperthermophilic microorganisms in their biotope has been studied only to a limited degree; this is especially true for motility. One reason for this lack of knowledge is the requirement for high-temperature microscopy—combined, in most cases, with the need for observations under strictly anaerobic conditions—for such studies. We have developed a custom-made, low-budget device that, for the first time, allows analyses in temperature gradients up to 40°C over a distance of just 2 cm (a biotope-relevant distance) with heating rates up to ∼5°C/s. Our temperature gradient-forming device can convert any upright light microscope into one that works at temperatures as high as 110°C. Data obtained by use of this apparatus show how very well hyperthermophiles are adapted to their biotope: they can react within seconds to elevated temperatures by starting motility—even after 9 months of storage in the cold. Using the temperature gradient-forming device, we determined the temperature ranges for swimming, and the swimming speeds, of 15 selected species of the genus Thermococcus within a few months, related these findings to the presence of cell surface appendages, and obtained the first evidence for thermotaxis in Archaea. PMID:24858087
SLAPP: A systolic linear algebra parallel processor
Drake, B.L.; Luk, F.T.; Speiser, J.M.; Symanski, J.J.
1987-07-01
Systolic array computer architectures provide a means for fast computation of the linear algebra algorithms that form the building blocks of many signal-processing algorithms, facilitating their real-time computation. For applications to signal processing, the systolic array operates on matrices, an inherently parallel view of the data, using numerical linear algebra algorithms that have been suitably parallelized to efficiently utilize the available hardware. This article describes work currently underway at the Naval Ocean Systems Center, San Diego, California, to build a two-dimensional systolic array, SLAPP, demonstrating efficient and modular parallelization of key matric computations for real-time signal- and image-processing problems.
Weak Lie symmetry and extended Lie algebra
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).
Boolean Operations with Prism Algebraic Patches.
Bajaj, Chandrajit; Paoluzzi, Alberto; Portuesi, Simone; Lei, Na; Zhao, Wenqi
2008-01-01
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 C(1)-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
Adaptive Algebraic Multigrid Methods
Brezina, M; Falgout, R; MacLachlan, S; Manteuffel, T; McCormick, S; Ruge, J
2004-04-09
Our ability to simulate physical processes numerically is constrained by our ability to solve the resulting linear systems, prompting substantial research into the development of multiscale iterative methods capable of solving these linear systems with an optimal amount of effort. Overcoming the limitations of geometric multigrid methods to simple geometries and differential equations, algebraic multigrid methods construct the multigrid hierarchy based only on the given matrix. While this allows for efficient black-box solution of the linear systems associated with discretizations of many elliptic differential equations, it also results in a lack of robustness due to assumptions made on the near-null spaces of these matrices. This paper introduces an extension to algebraic multigrid methods that removes the need to make such assumptions by utilizing an adaptive process. The principles which guide the adaptivity are highlighted, as well as their application to algebraic multigrid solution of certain symmetric positive-definite linear systems.
Abstract Algebra for Algebra Teaching: Influencing School Mathematics Instruction
ERIC Educational Resources Information Center
Wasserman, Nicholas H.
2016-01-01
This article explores the potential for aspects of abstract algebra to be influential for the teaching of school algebra (and early algebra). Using national standards for analysis, four primary areas common in school mathematics--and their progression across elementary, middle, and secondary mathematics--where teaching may be transformed by…
Computer Program For Linear Algebra
NASA Technical Reports Server (NTRS)
Krogh, F. T.; Hanson, R. J.
1987-01-01
Collection of routines provided for basic vector operations. Basic Linear Algebra Subprogram (BLAS) library is collection from FORTRAN-callable routines for employing standard techniques to perform basic operations of numerical linear algebra.
NASA Technical Reports Server (NTRS)
Shahshahani, M.
1991-01-01
The performance characteristics are discussed of certain algebraic geometric codes. Algebraic geometric codes have good minimum distance properties. On many channels they outperform other comparable block codes; therefore, one would expect them eventually to replace some of the block codes used in communications systems. It is suggested that it is unlikely that they will become useful substitutes for the Reed-Solomon codes used by the Deep Space Network in the near future. However, they may be applicable to systems where the signal to noise ratio is sufficiently high so that block codes would be more suitable than convolutional or concatenated codes.
NASA Astrophysics Data System (ADS)
Bouwknegt, Peter
1988-06-01
We investigate extensions of the Virasoro algebra by a single primary field of integer or halfinteger conformal dimension Δ. We argue that for vanishing structure constant CΔΔΔ, the extended conformal algebra can only be associative for a generic c-value if Δ=1/2, 1, 3/2, 2 or 3. For the other Δ<=5 we compute the finite set of allowed c-values and identify the rational solutions. The case CΔΔΔ≠0 is also briefly discussed. I would like to thank Kareljan Schoutens for discussions and Sander Bais for a careful reading of the manuscript.
Closed form evaluation of symmetric two-sided complex integrals
NASA Technical Reports Server (NTRS)
Winkelstein, R.
1981-01-01
Evaluation of two-sided complex integrals is often required when analyzing linear systems to determine signal variances resulting from stochastic inputs and system noise bandwidths. Algebraic solutions of integrals in a closed matrix equation form, using coefficients of the numerator and denominator polynomials, are presented. The closed forms provide the possibility of obtaining some insight into parameter sensitivity in addition to greatly reducing the computational complexity required by the normal method of evaluation by residues.
González-Salgado, Amaia; Steinmann, Michael; Major, Louise L; Sigel, Erwin; Reymond, Jean-Louis; Smith, Terry K; Bütikofer, Peter
2015-06-01
myo-Inositol is a building block for all inositol-containing phospholipids in eukaryotes. It can be synthesized de novo from glucose-6-phosphate in the cytosol and endoplasmic reticulum. Alternatively, it can be taken up from the environment via Na(+)- or H(+)-linked myo-inositol transporters. While Na(+)-coupled myo-inositol transporters are found exclusively in the plasma membrane, H(+)-linked myo-inositol transporters are detected in intracellular organelles. In Trypanosoma brucei, the causative agent of human African sleeping sickness, myo-inositol metabolism is compartmentalized. De novo-synthesized myo-inositol is used for glycosylphosphatidylinositol production in the endoplasmic reticulum, whereas the myo-inositol taken up from the environment is used for bulk phosphatidylinositol synthesis in the Golgi complex. We now provide evidence that the Golgi complex-localized T. brucei H(+)-linked myo-inositol transporter (TbHMIT) is essential in bloodstream-form T. brucei. Downregulation of TbHMIT expression by RNA interference blocked phosphatidylinositol production and inhibited growth of parasites in culture. Characterization of the transporter in a heterologous expression system demonstrated a remarkable selectivity of TbHMIT for myo-inositol. It tolerates only a single modification on the inositol ring, such as the removal of a hydroxyl group or the inversion of stereochemistry at a single hydroxyl group relative to myo-inositol. PMID:25888554
González-Salgado, Amaia; Steinmann, Michael; Major, Louise L.; Sigel, Erwin; Reymond, Jean-Louis
2015-01-01
myo-Inositol is a building block for all inositol-containing phospholipids in eukaryotes. It can be synthesized de novo from glucose-6-phosphate in the cytosol and endoplasmic reticulum. Alternatively, it can be taken up from the environment via Na+- or H+-linked myo-inositol transporters. While Na+-coupled myo-inositol transporters are found exclusively in the plasma membrane, H+-linked myo-inositol transporters are detected in intracellular organelles. In Trypanosoma brucei, the causative agent of human African sleeping sickness, myo-inositol metabolism is compartmentalized. De novo-synthesized myo-inositol is used for glycosylphosphatidylinositol production in the endoplasmic reticulum, whereas the myo-inositol taken up from the environment is used for bulk phosphatidylinositol synthesis in the Golgi complex. We now provide evidence that the Golgi complex-localized T. brucei H+-linked myo-inositol transporter (TbHMIT) is essential in bloodstream-form T. brucei. Downregulation of TbHMIT expression by RNA interference blocked phosphatidylinositol production and inhibited growth of parasites in culture. Characterization of the transporter in a heterologous expression system demonstrated a remarkable selectivity of TbHMIT for myo-inositol. It tolerates only a single modification on the inositol ring, such as the removal of a hydroxyl group or the inversion of stereochemistry at a single hydroxyl group relative to myo-inositol. PMID:25888554
NASA Astrophysics Data System (ADS)
Dobrev, V. K.
2013-02-01
In the present paper we continue the project of systematic construction of invariant differential operators for non-compact semisimple Lie groups. Our starting points is the class of algebras, which we call 'conformal Lie algebras' (CLA), which have very similar properties to the conformal algebras of Minkowski space-time, though our aim is to go beyond this class in a natural way. For this we introduce the new notion of parabolic relation between two non-compact semisimple Lie algebras G and G ' that have the same complexification and possess maximal parabolic subalgebras with the same complexification. Thus, we consider the exceptional algebra E 7(7) which is parabolically related to the CLA E 7(-25) , the parabolic subalgebras including E 6(6) and E 6(-26). Other interesting examples are the orthogonal algebras so(p, q) all of which are parabolically related to the conformal algebra so( n, 2) with p + q = n + 2, the parabolic subalgebras including the Lorentz subalgebra so( n - 1, 1) and its analogs so( p - 1, q - 1). We consider also E6(6) and E6(2) which are parabolically related to the hermitian symmetric case E6(-14) , the parabolic subalgebras including real forms of sl(6). We also give a formula for the number of representations in the main multiplets valid for CLAs and all algebras that are parabolically related to them. In all considered cases we give the main multiplets of indecomposable elementary representations including the necessary data for all relevant invariant differential operators. In the case of so( p, q) we give also the reduced multiplets. We should stress that the multiplets are given in the most economic way in pairs of shadow fields. Furthermore we should stress that the classification of all invariant differential operators includes as special cases all possible conservation laws and conserved currents, unitary or not.
Paving the Way To Algebraic Thought Using Residue Designs.
ERIC Educational Resources Information Center
Johnson, Iris DeLoach
1998-01-01
Presents a brief definition and examples of residue designs while sharing some of the algebraic thought that a student used to form generalizations about the patterns discovered during the investigations of residue designs. (ASK)
A Hard X-Ray Study of the Normal Star-forming Galaxy M83 with NuSTAR
NASA Astrophysics Data System (ADS)
Yukita, M.; Hornschemeier, A. E.; Lehmer, B. D.; Ptak, A.; Wik, D. R.; Zezas, A.; Antoniou, V.; Maccarone, T. J.; Replicon, V.; Tyler, J. B.; Venters, T.; Argo, M. K.; Bechtol, K.; Boggs, S.; Christensen, F. E.; Craig, W. W.; Hailey, C.; Harrison, F.; Krivonos, R.; Kuntz, K.; Stern, D.; Zhang, W. W.
2016-06-01
We present the results from sensitive, multi-epoch NuSTAR observations of the late-type star-forming galaxy M83 (d = 4.6 Mpc). This is the first investigation to spatially resolve the hard (E\\gt 10 keV) X-ray emission of this galaxy. The nuclear region and ˜20 off-nuclear point sources, including a previously discovered ultraluminous X-ray source, are detected in our NuSTAR observations. The X-ray hardnesses and luminosities of the majority of the point sources are consistent with hard X-ray sources resolved in the starburst galaxy NGC 253. We infer that the hard X-ray emission is most likely dominated by intermediate accretion state black hole binaries and neutron star low-mass X-ray binaries (Z-sources). We construct the X-ray binary luminosity function (XLF) in the NuSTAR band for an extragalactic environment for the first time. The M83 XLF has a steeper XLF than the X-ray binary XLF in NGC 253, which is consistent with previous measurements by Chandra at softer X-ray energies. The NuSTAR integrated galaxy spectrum of M83 drops quickly above 10 keV, which is also seen in the starburst galaxies NGC 253, NGC 3310, and NGC 3256. The NuSTAR observations constrain any active galactic nucleus (AGN) to be either highly obscured or to have an extremely low luminosity of ≲1038 erg s‑1 (10–30 keV), implying that it is emitting at a very low Eddington ratio. An X-ray point source that is consistent with the location of the nuclear star cluster with an X-ray luminosity of a few times 1038 erg s‑1 may be a low-luminosity AGN but is more consistent with being an X-ray binary.
A Hard X-Ray Study of the Normal Star-forming Galaxy M83 with NuSTAR
NASA Astrophysics Data System (ADS)
Yukita, M.; Hornschemeier, A. E.; Lehmer, B. D.; Ptak, A.; Wik, D. R.; Zezas, A.; Antoniou, V.; Maccarone, T. J.; Replicon, V.; Tyler, J. B.; Venters, T.; Argo, M. K.; Bechtol, K.; Boggs, S.; Christensen, F. E.; Craig, W. W.; Hailey, C.; Harrison, F.; Krivonos, R.; Kuntz, K.; Stern, D.; Zhang, W. W.
2016-06-01
We present the results from sensitive, multi-epoch NuSTAR observations of the late-type star-forming galaxy M83 (d = 4.6 Mpc). This is the first investigation to spatially resolve the hard (E\\gt 10 keV) X-ray emission of this galaxy. The nuclear region and ∼20 off-nuclear point sources, including a previously discovered ultraluminous X-ray source, are detected in our NuSTAR observations. The X-ray hardnesses and luminosities of the majority of the point sources are consistent with hard X-ray sources resolved in the starburst galaxy NGC 253. We infer that the hard X-ray emission is most likely dominated by intermediate accretion state black hole binaries and neutron star low-mass X-ray binaries (Z-sources). We construct the X-ray binary luminosity function (XLF) in the NuSTAR band for an extragalactic environment for the first time. The M83 XLF has a steeper XLF than the X-ray binary XLF in NGC 253, which is consistent with previous measurements by Chandra at softer X-ray energies. The NuSTAR integrated galaxy spectrum of M83 drops quickly above 10 keV, which is also seen in the starburst galaxies NGC 253, NGC 3310, and NGC 3256. The NuSTAR observations constrain any active galactic nucleus (AGN) to be either highly obscured or to have an extremely low luminosity of ≲1038 erg s‑1 (10–30 keV), implying that it is emitting at a very low Eddington ratio. An X-ray point source that is consistent with the location of the nuclear star cluster with an X-ray luminosity of a few times 1038 erg s‑1 may be a low-luminosity AGN but is more consistent with being an X-ray binary.
Teaching Arithmetic and Algebraic Expressions
ERIC Educational Resources Information Center
Subramaniam, K.; Banerjee, Rakhi
2004-01-01
A teaching intervention study was conducted with sixth grade students to explore the interconnections between students' growing understanding of arithmetic expressions and beginning algebra. Three groups of students were chosen, with two groups receiving instruction in arithmetic and algebra, and one group in algebra without arithmetic. Students…
Assessing Elementary Algebra with STACK
ERIC Educational Resources Information Center
Sangwin, Christopher J.
2007-01-01
This paper concerns computer aided assessment (CAA) of mathematics in which a computer algebra system (CAS) is used to help assess students' responses to elementary algebra questions. Using a methodology of documentary analysis, we examine what is taught in elementary algebra. The STACK CAA system, http://www.stack.bham.ac.uk/, which uses the CAS…
ERIC Educational Resources Information Center
Glick, David
1995-01-01
Presents a technique that helps students concentrate more on the science and less on the mechanics of algebra while dealing with introductory physics formulas. Allows the teacher to do complex problems at a lower level and not be too concerned about the mathematical abilities of the students. (JRH)
ERIC Educational Resources Information Center
Ketterlin-Geller, Leanne R.; Jungjohann, Kathleen; Chard, David J.; Baker, Scott
2007-01-01
Much of the difficulty that students encounter in the transition from arithmetic to algebra stems from their early learning and understanding of arithmetic. Too often, students learn about the whole number system and the operations that govern that system as a set of procedures to solve addition, subtraction, multiplication, and division problems.…
Computer Algebra versus Manipulation
ERIC Educational Resources Information Center
Zand, Hossein; Crowe, David
2004-01-01
In the UK there is increasing concern about the lack of skill in algebraic manipulation that is evident in students entering mathematics courses at university level. In this note we discuss how the computer can be used to ameliorate some of the problems. We take as an example the calculations needed in three dimensional vector analysis in polar…
ERIC Educational Resources Information Center
Boiteau, Denise; Stansfield, David
This document describes mathematical programs on the basic concepts of algebra produced by Louisiana Public Broadcasting. Programs included are: (1) "Inverse Operations"; (2) "The Order of Operations"; (3) "Basic Properties" (addition and multiplication of numbers and variables); (4) "The Positive and Negative Numbers"; and (5) "Using Positive…
Thinking Visually about Algebra
ERIC Educational Resources Information Center
Baroudi, Ziad
2015-01-01
Many introductions to algebra in high school begin with teaching students to generalise linear numerical patterns. This article argues that this approach needs to be changed so that students encounter variables in the context of modelling visual patterns so that the variables have a meaning. The article presents sample classroom activities,…
ERIC Educational Resources Information Center
Kennedy, John
This text provides information and exercises on arithmetic topics which should be mastered before a student enrolls in an Elementary Algebra course. Section I describes the fundamental properties and relationships of whole numbers, focusing on basic operations, divisibility tests, exponents, order of operations, prime numbers, greatest common…
ERIC Educational Resources Information Center
Nwabueze, Kenneth K.
2004-01-01
The current emphasis on flexible modes of mathematics delivery involving new information and communication technology (ICT) at the university level is perhaps a reaction to the recent change in the objectives of education. Abstract algebra seems to be one area of mathematics virtually crying out for computer instructional support because of the…
NASA Astrophysics Data System (ADS)
Zhang, Ming; Yao, JingTao
2004-04-01
The XML is a new standard for data representation and exchange on the Internet. There are studies on XML query languages as well as XML algebras in literature. However, attention has not been paid to research on XML algebras for data mining due to partially the fact that there is no widely accepted definition of XML mining tasks. This paper tries to examine the XML mining tasks and provide guidelines to design XML algebras for data mining. Some summarization and comparison have been done to existing XML algebras. We argue that by adding additional operators for mining tasks, XML algebras may work well for data mining with XML documents.
On Dunkl angular momenta algebra
NASA Astrophysics Data System (ADS)
Feigin, Misha; Hakobyan, Tigran
2015-11-01
We consider the quantum angular momentum generators, deformed by means of the Dunkl operators. Together with the reflection operators they generate a subalgebra in the rational Cherednik algebra associated with a finite real reflection group. We find all the defining relations of the algebra, which appear to be quadratic, and we show that the algebra is of Poincaré-Birkhoff-Witt (PBW) type. We show that this algebra contains the angular part of the Calogero-Moser Hamiltonian and that together with constants it generates the centre of the algebra. We also consider the gl( N ) version of the subalge-bra of the rational Cherednik algebra and show that it is a non-homogeneous quadratic algebra of PBW type as well. In this case the central generator can be identified with the usual Calogero-Moser Hamiltonian associated with the Coxeter group in the harmonic confinement.
Algebraic connectivity and graph robustness.
Feddema, John Todd; Byrne, Raymond Harry; Abdallah, Chaouki T.
2009-07-01
Recent papers have used Fiedler's definition of algebraic connectivity to show that network robustness, as measured by node-connectivity and edge-connectivity, can be increased by increasing the algebraic connectivity of the network. By the definition of algebraic connectivity, the second smallest eigenvalue of the graph Laplacian is a lower bound on the node-connectivity. In this paper we show that for circular random lattice graphs and mesh graphs algebraic connectivity is a conservative lower bound, and that increases in algebraic connectivity actually correspond to a decrease in node-connectivity. This means that the networks are actually less robust with respect to node-connectivity as the algebraic connectivity increases. However, an increase in algebraic connectivity seems to correlate well with a decrease in the characteristic path length of these networks - which would result in quicker communication through the network. Applications of these results are then discussed for perimeter security.
A q-Virasoro algebra at roots of unity, free fermions, and Temperley-Lieb hamiltonians
NASA Astrophysics Data System (ADS)
Nigro, Alessandro
2016-04-01
In this work, we consider the q-deformation of the Virasoro algebra [M. Chaichian and P. Presnajder, Phys. Lett. B 277, 109 (1992)] expressed in terms of free fermions, and we then realize this algebra, when the deformation parameter is a root of unity, on the lattice in a truncated form in terms of the Clifford algebra of Γ matrices. For this finite size truncation, the commutation relations of the Deformed algebra hold exactly albeit without central extension term. We then study the relations existing between this lattice truncation of the deformed Virasoro algebra at roots of unity and the tower of commuting Temperley-Lieb hamiltonians introduced in a previous work.
Koudinov, A; Matsubara, E; Frangione, B; Ghiso, J
1994-12-15
The amyloid fibrils of Alzheimer's neuritic plaques and cerebral blood vessels are mainly composed of aggregated forms of a 39 to 44 amino acids peptide, named amyloid beta (A beta). A similar although soluble form of A beta (sA beta) has been identified in plasma, cerebrospinal fluid and cell culture supernatants, indicating that it is produced under physiologic conditions. We report here that sA beta in normal human plasma is associated with lipoprotein particles, in particular to the HDL3 and VHDL fractions where it is complexed to ApoJ and, to a lesser extent, to ApoAI. This was assessed by immunoprecipitation experiments of purified plasma lipoproteins and lipoprotein-depleted plasma and confirmed by means of amino acid sequence analysis. Moreover, biotinylated synthetic peptide A beta 1-40 was traced in normal human plasma in in vitro experiments. As in the case of sA beta, biotinylated A beta 1-40 was specifically recovered in the HDL3 and VHDL fractions. This data together with the previous demonstration that A beta 1-40 is taken up into the brain via a specific mechanism and possibly as an A beta 1-40-ApoJ complex indicate a role for HDL3- and VHDL-containing ApoJ in the transport of the peptide in circulation and suggest their involvement in the delivery of sA beta across the blood-brain barrier. PMID:7802646
A Geometric Construction of Cyclic Cocycles on Twisted Convolution Algebras
NASA Astrophysics Data System (ADS)
Angel, Eitan
2010-09-01
In this thesis we give a construction of cyclic cocycles on convolution algebras twisted by gerbes over discrete translation groupoids. In his seminal book, Connes constructs a map from the equivariant cohomology of a manifold carrying the action of a discrete group into the periodic cyclic cohomology of the associated convolution algebra. Furthermore, for proper étale groupoids, J.-L. Tu and P. Xu provide a map between the periodic cyclic cohomology of a gerbe twisted convolution algebra and twisted cohomology groups. Our focus will be the convolution algebra with a product defined by a gerbe over a discrete translation groupoid. When the action is not proper, we cannot construct an invariant connection on the gerbe; therefore to study this algebra, we instead develop simplicial notions related to ideas of J. Dupont to construct a simplicial form representing the Dixmier-Douady class of the gerbe. Then by using a JLO formula we define a morphism from a simplicial complex twisted by this simplicial Dixmier-Douady form to the mixed bicomplex of certain matrix algebras. Finally, we define a morphism from this complex to the mixed bicomplex computing the periodic cyclic cohomology of the twisted convolution algebras.
Marquette, Ian
2013-07-15
We introduce the most general quartic Poisson algebra generated by a second and a fourth order integral of motion of a 2D superintegrable classical system. We obtain the corresponding quartic (associative) algebra for the quantum analog, extend Daskaloyannis construction obtained in context of quadratic algebras, and also obtain the realizations as deformed oscillator algebras for this quartic algebra. We obtain the Casimir operator and discuss how these realizations allow to obtain the finite-dimensional unitary irreducible representations of quartic algebras and obtain algebraically the degenerate energy spectrum of superintegrable systems. We apply the construction and the formula obtained for the structure function on a superintegrable system related to type I Laguerre exceptional orthogonal polynomials introduced recently.
ERIC Educational Resources Information Center
Beigie, Darin
2014-01-01
Most people who are attracted to STEM-related fields are drawn not by a desire to take mathematics tests but to create things. The opportunity to create an algebra drawing gives students a sense of ownership and adventure that taps into the same sort of energy that leads a young person to get lost in reading a good book, building with Legos®,…
2013-05-06
AMG2013 is a parallel algebraic multigrid solver for linear systems arising from problems on unstructured grids. It has been derived directly from the Boomer AMG solver in the hypre library, a large linear solvers library that is being developed in the Center for Applied Scientific Computing (CASC) at LLNL. The driver provided in the benchmark can build various test problems. The default problem is a Laplace type problem on an unstructured domain with various jumps and an anisotropy in one part.
The Propositional Logic Induced by Means of Basic Algebras
NASA Astrophysics Data System (ADS)
Chajda, I.
2015-12-01
A propositional logic induced by means of commutative basic algebras was already described by M. Botur and R. Halaš. It turns out that this is a kind of non-associative fuzzy logic which can be used e.g. in expert systems. Unfortunately, there are other important classes of basic algebras which are not commutative, e.g. orthomodular lattices which are used as an axiomatization of the logic of quantum mechanics. This motivated us to develop another axioms and derivation rules which form a propositional logic induced by basic algebras in general. We show that this logic is algebraizable in the sense of W. J. Blok and D. Pigozzi.
NASA Astrophysics Data System (ADS)
Palmkvist, Jakob
2014-01-01
We introduce an infinite-dimensional Lie superalgebra which is an extension of the U-duality Lie algebra of maximal supergravity in D dimensions, for 3 ⩽ D ⩽ 7. The level decomposition with respect to the U-duality Lie algebra gives exactly the tensor hierarchy of representations that arises in gauge deformations of the theory described by an embedding tensor, for all positive levels p. We prove that these representations are always contained in those coming from the associated Borcherds-Kac-Moody superalgebra, and we explain why some of the latter representations are not included in the tensor hierarchy. The most remarkable feature of our Lie superalgebra is that it does not admit a triangular decomposition like a (Borcherds-)Kac-Moody (super)algebra. Instead the Hodge duality relations between level p and D - 2 - p extend to negative p, relating the representations at the first two negative levels to the supersymmetry and closure constraints of the embedding tensor.
NASA Technical Reports Server (NTRS)
Cleaveland, Rance; Luettgen, Gerald; Natarajan, V.
1999-01-01
This paper surveys the semantic ramifications of extending traditional process algebras with notions of priority that allow for some transitions to be given precedence over others. These enriched formalisms allow one to model system features such as interrupts, prioritized choice, or real-time behavior. Approaches to priority in process algebras can be classified according to whether the induced notion of preemption on transitions is global or local and whether priorities are static or dynamic. Early work in the area concentrated on global pre-emption and static priorities and led to formalisms for modeling interrupts and aspects of real-time, such as maximal progress, in centralized computing environments. More recent research has investigated localized notions of pre-emption in which the distribution of systems is taken into account, as well as dynamic priority approaches, i.e., those where priority values may change as systems evolve. The latter allows one to model behavioral phenomena such as scheduling algorithms and also enables the efficient encoding of real-time semantics. Technically, this paper studies the different models of priorities by presenting extensions of Milner's Calculus of Communicating Systems (CCS) with static and dynamic priority as well as with notions of global and local pre- emption. In each case the operational semantics of CCS is modified appropriately, behavioral theories based on strong and weak bisimulation are given, and related approaches for different process-algebraic settings are discussed.
Palmkvist, Jakob
2014-01-15
We introduce an infinite-dimensional Lie superalgebra which is an extension of the U-duality Lie algebra of maximal supergravity in D dimensions, for 3 ⩽ D ⩽ 7. The level decomposition with respect to the U-duality Lie algebra gives exactly the tensor hierarchy of representations that arises in gauge deformations of the theory described by an embedding tensor, for all positive levels p. We prove that these representations are always contained in those coming from the associated Borcherds-Kac-Moody superalgebra, and we explain why some of the latter representations are not included in the tensor hierarchy. The most remarkable feature of our Lie superalgebra is that it does not admit a triangular decomposition like a (Borcherds-)Kac-Moody (super)algebra. Instead the Hodge duality relations between level p and D − 2 − p extend to negative p, relating the representations at the first two negative levels to the supersymmetry and closure constraints of the embedding tensor.
D-algebra structure of topological insulators
NASA Astrophysics Data System (ADS)
Estienne, B.; Regnault, N.; Bernevig, B. A.
2012-12-01
In the quantum Hall effect, the density operators at different wave vectors generally do not commute and give rise to the Girvin-MacDonald-Plazmann (GMP) algebra, with important consequences such as ground-state center-of-mass degeneracy at fractional filling fraction, and W1+∞ symmetry of the filled Landau levels. We show that the natural generalization of the GMP algebra to higher-dimensional topological insulators involves the concept of a D commutator. For insulators in even-dimensional space, the D commutator is isotropic and closes, and its structure factors are proportional to the D/2 Chern number. In odd dimensions, the algebra is not isotropic, contains the weak topological insulator index (layers of the topological insulator in one fewer dimension), and does not contain the Chern-Simons θ form. This algebraic structure paves the way towards the identification of fractional topological insulators through the counting of their excitations. The possible relation to D-dimensional volume-preserving diffeomorphisms and parallel transport of extended objects is also discussed.
Compactly Generated de Morgan Lattices, Basic Algebras and Effect Algebras
NASA Astrophysics Data System (ADS)
Paseka, Jan; Riečanová, Zdenka
2010-12-01
We prove that a de Morgan lattice is compactly generated if and only if its order topology is compatible with a uniformity on L generated by some separating function family on L. Moreover, if L is complete then L is (o)-topological. Further, if a basic algebra L (hence lattice with sectional antitone involutions) is compactly generated then L is atomic. Thus all non-atomic Boolean algebras as well as non-atomic lattice effect algebras (including non-atomic MV-algebras and orthomodular lattices) are not compactly generated.
Verburgt, Lukas M
2016-01-01
This paper provides a detailed account of the period of the complex history of British algebra and geometry between the publication of George Peacock's Treatise on Algebra in 1830 and William Rowan Hamilton's paper on quaternions of 1843. During these years, Duncan Farquharson Gregory and William Walton published several contributions on 'algebraical geometry' and 'geometrical algebra' in the Cambridge Mathematical Journal. These contributions enabled them not only to generalize Peacock's symbolical algebra on the basis of geometrical considerations, but also to initiate the attempts to question the status of Euclidean space as the arbiter of valid geometrical interpretations. At the same time, Gregory and Walton were bound by the limits of symbolical algebra that they themselves made explicit; their work was not and could not be the 'abstract algebra' and 'abstract geometry' of figures such as Hamilton and Cayley. The central argument of the paper is that an understanding of the contributions to 'algebraical geometry' and 'geometrical algebra' of the second generation of 'scientific' symbolical algebraists is essential for a satisfactory explanation of the radical transition from symbolical to abstract algebra that took place in British mathematics in the 1830s-1840s. PMID:26806075
On the cohomology of Leibniz conformal algebras
NASA Astrophysics Data System (ADS)
Zhang, Jiao
2015-04-01
We construct a new cohomology complex of Leibniz conformal algebras with coefficients in a representation instead of a module. The low-dimensional cohomology groups of this complex are computed. Meanwhile, we construct a Leibniz algebra from a Leibniz conformal algebra and prove that the category of Leibniz conformal algebras is equivalent to the category of equivalence classes of formal distribution Leibniz algebras.
Assessing Algebraic Solving Ability: A Theoretical Framework
ERIC Educational Resources Information Center
Lian, Lim Hooi; Yew, Wun Thiam
2012-01-01
Algebraic solving ability had been discussed by many educators and researchers. There exists no definite definition for algebraic solving ability as it can be viewed from different perspectives. In this paper, the nature of algebraic solving ability in terms of algebraic processes that demonstrate the ability in solving algebraic problem is…
Pawlak Algebra and Approximate Structure on Fuzzy Lattice
Zhuang, Ying; Liu, Wenqi; Wu, Chin-Chia; Li, Jinhai
2014-01-01
The aim of this paper is to investigate the general approximation structure, weak approximation operators, and Pawlak algebra in the framework of fuzzy lattice, lattice topology, and auxiliary ordering. First, we prove that the weak approximation operator space forms a complete distributive lattice. Then we study the properties of transitive closure of approximation operators and apply them to rough set theory. We also investigate molecule Pawlak algebra and obtain some related properties. PMID:25152922
Asymptotics of bivariate generating functions with algebraic singularities
NASA Astrophysics Data System (ADS)
Greenwood, Torin
Flajolet and Odlyzko (1990) derived asymptotic formulae the coefficients of a class of uni- variate generating functions with algebraic singularities. Gao and Richmond (1992) and Hwang (1996, 1998) extended these results to classes of multivariate generating functions, in both cases by reducing to the univariate case. Pemantle and Wilson (2013) outlined new multivariate ana- lytic techniques and used them to analyze the coefficients of rational generating functions. After overviewing these methods, we use them to find asymptotic formulae for the coefficients of a broad class of bivariate generating functions with algebraic singularities. Beginning with the Cauchy integral formula, we explicity deform the contour of integration so that it hugs a set of critical points. The asymptotic contribution to the integral comes from analyzing the integrand near these points, leading to explicit asymptotic formulae. Next, we use this formula to analyze an example from current research. In the following chapter, we apply multivariate analytic techniques to quan- tum walks. Bressler and Pemantle (2007) found a (d + 1)-dimensional rational generating function whose coefficients described the amplitude of a particle at a position in the integer lattice after n steps. Here, the minimal critical points form a curve on the (d + 1)-dimensional unit torus. We find asymptotic formulae for the amplitude of a particle in a given position, normalized by the number of steps n, as n approaches infinity. Each critical point contributes to the asymptotics for a specific normalized position. Using Groebner bases in Maple again, we compute the explicit locations of peak amplitudes. In a scaling window of size the square root of n near the peaks, each amplitude is asymptotic to an Airy function.
ERIC Educational Resources Information Center
Novotna, Jarmila; Hoch, Maureen
2008-01-01
Many students have difficulties with basic algebraic concepts at high school and at university. In this paper two levels of algebraic structure sense are defined: for high school algebra and for university algebra. We suggest that high school algebra structure sense components are sub-components of some university algebra structure sense…
Higher level twisted Zhu algebras
Ekeren, Jethro van
2011-05-15
The study of twisted representations of graded vertex algebras is important for understanding orbifold models in conformal field theory. In this paper, we consider the general setup of a vertex algebra V, graded by {Gamma}/Z for some subgroup {Gamma} of R containing Z, and with a Hamiltonian operator H having real (but not necessarily integer) eigenvalues. We construct the directed system of twisted level p Zhu algebras Zhu{sub p,{Gamma}}(V), and we prove the following theorems: For each p, there is a bijection between the irreducible Zhu{sub p,{Gamma}}(V)-modules and the irreducible {Gamma}-twisted positive energy V-modules, and V is ({Gamma}, H)-rational if and only if all its Zhu algebras Zhu{sub p,{Gamma}}(V) are finite dimensional and semisimple. The main novelty is the removal of the assumption of integer eigenvalues for H. We provide an explicit description of the level p Zhu algebras of a universal enveloping vertex algebra, in particular of the Virasoro vertex algebra Vir{sup c} and the universal affine Kac-Moody vertex algebra V{sup k}(g) at non-critical level. We also compute the inverse limits of these directed systems of algebras.
Handheld Computer Algebra Systems in the Pre-Algebra Classroom
ERIC Educational Resources Information Center
Gantz, Linda Ann Galofaro
2010-01-01
This mixed method analysis sought to investigate several aspects of student learning in pre-algebra through the use of computer algebra systems (CAS) as opposed to non-CAS learning. This research was broken into two main parts, one which compared results from both the experimental group (instruction using CAS, N = 18) and the control group…
Abstract Algebra to Secondary School Algebra: Building Bridges
ERIC Educational Resources Information Center
Christy, Donna; Sparks, Rebecca
2015-01-01
The authors have experience with secondary mathematics teacher candidates struggling to make connections between the theoretical abstract algebra course they take as college students and the algebra they will be teaching in secondary schools. As a mathematician and a mathematics educator, the authors collaborated to create and implement a…
Algebra and Algebraic Thinking in School Math: 70th YB
ERIC Educational Resources Information Center
National Council of Teachers of Mathematics, 2008
2008-01-01
Algebra is no longer just for college-bound students. After a widespread push by the National Council of Teachers of Mathematics (NCTM) and teachers across the country, algebra is now a required part of most curricula. However, students' standardized test scores are not at the level they should be. NCTM's seventieth yearbook takes a look at the…
Philip, Bobby; Chartier, Dr Timothy
2012-01-01
methods based on Local Sensitivity Analysis (LSA). The method can be used in the context of geometric and algebraic multigrid methods for constructing smoothers, and in the context of Krylov methods for constructing block preconditioners. It is suitable for both constant and variable coecient problems. Furthermore, the method can be applied to systems arising from both scalar and coupled system partial differential equations (PDEs), as well as linear systems that do not arise from PDEs. The simplicity of the method will allow it to be easily incorporated into existing multigrid and Krylov solvers while providing a powerful tool for adaptively constructing methods tuned to a problem.
Statecharts Via Process Algebra
NASA Technical Reports Server (NTRS)
Luttgen, Gerald; vonderBeeck, Michael; Cleaveland, Rance
1999-01-01
Statecharts is a visual language for specifying the behavior of reactive systems. The Language extends finite-state machines with concepts of hierarchy, concurrency, and priority. Despite its popularity as a design notation for embedded system, precisely defining its semantics has proved extremely challenging. In this paper, a simple process algebra, called Statecharts Process Language (SPL), is presented, which is expressive enough for encoding Statecharts in a structure-preserving and semantic preserving manner. It is establish that the behavioral relation bisimulation, when applied to SPL, preserves Statecharts semantics
2013-05-06
AMG2013 is a parallel algebraic multigrid solver for linear systems arising from problems on unstructured grids. It has been derived directly from the Boomer AMG solver in the hypre library, a large linear solvers library that is being developed in the Center for Applied Scientific Computing (CASC) at LLNL. The driver provided in the benchmark can build various test problems. The default problem is a Laplace type problem on an unstructured domain with various jumpsmore » and an anisotropy in one part.« less
Richgels, M A; Biffle, J H
1980-09-01
ALGEBRA is a program that allows the user to process output data from finite-element analysis codes before they are sent to plotting routines. These data take the form of variable values (stress, strain, and velocity components, etc.) on a tape that is both the output tape from the analyses code and the input tape to ALGEBRA. The ALGEBRA code evaluates functions of these data and writes the function values on an output tape that can be used as input to plotting routines. Convenient input format and error detection capabilities aid the user in providing ALGEBRA with the functions to be evaluated. 1 figure.
Constitutive relations in optics in terms of geometric algebra
NASA Astrophysics Data System (ADS)
Dargys, A.
2015-11-01
To analyze the electromagnetic wave propagation in a medium the Maxwell equations should be supplemented by constitutive relations. At present the classification of linear constitutive relations is well established in tensorial-matrix and exterior p-form calculus. Here the constitutive relations are found in the context of Clifford geometric algebra. For this purpose Cl1,3 algebra that conforms with relativistic 4D Minkowskian spacetime is used. It is shown that the classification of linear optical phenomena with the help of constitutive relations in this case comes from the structure of Cl1,3 algebra itself. Concrete expressions for constitutive relations which follow from this algebra are presented. They can be applied in calculating the propagation properties of electromagnetic waves in any anisotropic, linear and nondissipative medium.
Generalized Kaluza-Klein monopole, quadratic algebras and ladder operators
NASA Astrophysics Data System (ADS)
Marquette, Ian
2011-06-01
We present a generalized Kaluza-Klein monopole system. We solve this quantum superintegrable system on a Euclidean Taub Nut manifold using the separation of variables of the corresponding Schrödinger equation in spherical and parabolic coordinates. We present the integrals of motion of this system, the quadratic algebra generated by these integrals, the realization in terms of a deformed oscillator algebra using the Daskaloyannis construction and the energy spectrum. The structure constants and the Casimir operator are functions not only of the Hamiltonian but also of other two integrals commuting with all generators of the quadratic algebra and forming an Abelian subalgebra. We present another algebraic derivation of the energy spectrum of this system using the factorization method and ladder operators.
Open-closed homotopy algebra in mathematical physics
Kajiura, Hiroshige; Stasheff, Jim
2006-02-15
In this paper we discuss various aspects of open-closed homotopy algebras (OCHAs) presented in our previous paper, inspired by Zwiebach's open-closed string field theory, but that first paper concentrated on the mathematical aspects. Here we show how an OCHA is obtained by extracting the tree part of Zwiebach's quantum open-closed string field theory. We clarify the explicit relation of an OCHA with Kontsevich's deformation quantization and with the B-models of homological mirror symmetry. An explicit form of the minimal model for an OCHA is given as well as its relation to the perturbative expansion of open-closed string field theory. We show that our open-closed homotopy algebra gives us a general scheme for deformation of open string structures (A{sub {infinity}} algebras) by closed strings (L{sub {infinity}} algebras)
The Algebra of Complex Numbers.
ERIC Educational Resources Information Center
LePage, Wilbur R.
This programed text is an introduction to the algebra of complex numbers for engineering students, particularly because of its relevance to important problems of applications in electrical engineering. It is designed for a person who is well experienced with the algebra of real numbers and calculus, but who has no experience with complex number…
Algebraic Squares: Complete and Incomplete.
ERIC Educational Resources Information Center
Gardella, Francis J.
2000-01-01
Illustrates ways of using algebra tiles to give students a visual model of competing squares that appear in algebra as well as in higher mathematics. Such visual representations give substance to the symbolic manipulation and give students who do not learn symbolically a way of understanding the underlying concepts of completing the square. (KHR)
ERIC Educational Resources Information Center
Buerman, Margaret
2007-01-01
Finding real-world examples for middle school algebra classes can be difficult but not impossible. As we strive to accomplish teaching our students how to solve and graph equations, we neglect to teach the big ideas of algebra. One of those big ideas is functions. This article gives three examples of functions that are found in Arches National…
Online Algebraic Tools for Teaching
ERIC Educational Resources Information Center
Kurz, Terri L.
2011-01-01
Many free online tools exist to complement algebraic instruction at the middle school level. This article presents findings that analyzed the features of algebraic tools to support learning. The findings can help teachers select appropriate tools to facilitate specific topics. (Contains 1 table and 4 figures.)
Condensing Algebra for Technical Mathematics.
ERIC Educational Resources Information Center
Greenfield, Donald R.
Twenty Algebra-Packets (A-PAKS) were developed by the investigator for technical education students at the community college level. Each packet contained a statement of rationale, learning objectives, performance activities, performance test, and performance test answer key. The A-PAKS condensed the usual sixteen weeks of algebra into a six-week…
Algebraic Thinking in Adult Education
ERIC Educational Resources Information Center
Manly, Myrna; Ginsburg, Lynda
2010-01-01
In adult education, algebraic thinking can be a sense-making tool that introduces coherence among mathematical concepts for those who previously have had trouble learning math. Further, a modeling approach to algebra connects mathematics and the real world, demonstrating the usefulness of math to those who have seen it as just an academic…
Linear Algebra and Image Processing
ERIC Educational Resources Information Center
Allali, Mohamed
2010-01-01
We use the computing technology digital image processing (DIP) to enhance the teaching of linear algebra so as to make the course more visual and interesting. Certainly, this visual approach by using technology to link linear algebra to DIP is interesting and unexpected to both students as well as many faculty. (Contains 2 tables and 11 figures.)
ERIC Educational Resources Information Center
Instructional Objectives Exchange, Los Angeles, CA.
A complete set of behavioral objectives for first-year algebra taught in any of grades 8 through 12 is presented. Three to six sample test items and answers are provided for each objective. Objectives were determined by surveying the most used secondary school algebra textbooks. Fourteen major categories are included: (1) whole numbers--operations…
Exploring Algebraic Patterns through Literature.
ERIC Educational Resources Information Center
Austin, Richard A.; Thompson, Denisse R.
1997-01-01
Presents methods for using literature to develop algebraic thinking in an environment that connects algebra to various situations. Activities are based on the book "Anno's Magic Seeds" with additional resources listed. Students express a constant function, exponential function, and a recursive function in their own words as well as writing about…
Learning Algebra from Worked Examples
ERIC Educational Resources Information Center
Lange, Karin E.; Booth, Julie L.; Newton, Kristie J.
2014-01-01
For students to be successful in algebra, they must have a truly conceptual understanding of key algebraic features as well as the procedural skills to complete a problem. One strategy to correct students' misconceptions combines the use of worked example problems in the classroom with student self-explanation. "Self-explanation" is…
Solutions in bosonic string field theory and higher spin algebras in AdS
NASA Astrophysics Data System (ADS)
Polyakov, Dimitri
2015-11-01
We find a class of analytic solutions in open bosonic string field theory, parametrized by the chiral copy of higher spin algebra in AdS3. The solutions are expressed in terms of the generating function for the products of Bell polynomials in derivatives of bosonic space-time coordinates Xm(z ) of the open string, the form of which is determined in this work. The products of these polynomials form a natural operator algebra realizations of w∞ (area-preserving diffeomorphisms), enveloping algebra of SU(2) and higher spin algebra in AdS3. The class of string field theory solutions found can, in turn, be interpreted as the "enveloping of enveloping," or the enveloping of AdS3 higher spin algebra. We also discuss the extensions of this class of solutions to superstring theory and their relations to higher spin algebras in higher space-time dimensions.
A natural history of mathematics: George Peacock and the making of English algebra.
Lambert, Kevin
2013-06-01
In a series of papers read to the Cambridge Philosophical Society through the 1820s, the Cambridge mathematician George Peacock laid the foundation for a natural history of arithmetic that would tell a story of human progress from counting to modern arithmetic. The trajectory of that history, Peacock argued, established algebraic analysis as a form of universal reasoning that used empirically warranted operations of mind to think with symbols on paper. The science of counting would suggest arithmetic, arithmetic would suggest arithmetical algebra, and, finally, arithmetical algebra would suggest symbolic algebra. This philosophy of suggestion provided the foundation for Peacock's "principle of equivalent forms," which justified the practice of nineteenth-century English symbolic algebra. Peacock's philosophy of suggestion owed a considerable debt to the early Cambridge Philosophical Society culture of natural history. The aim of this essay is to show how that culture of natural history was constitutively significant to the practice of nineteenth-century English algebra. PMID:23961689
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…
Vertex operator algebras and conformal field theory
Huang, Y.Z. )
1992-04-20
This paper discusses conformal field theory, an important physical theory, describing both two-dimensional critical phenomena in condensed matter physics and classical motions of strings in string theory. The study of conformal field theory will deepen the understanding of these theories and will help to understand string theory conceptually. Besides its importance in physics, the beautiful and rich mathematical structure of conformal field theory has interested many mathematicians. New relations between different branches of mathematics, such as representations of infinite-dimensional Lie algebras and Lie groups, Riemann surfaces and algebraic curves, the Monster sporadic group, modular functions and modular forms, elliptic genera and elliptic cohomology, Calabi-Yau manifolds, tensor categories, and knot theory, are revealed in the study of conformal field theory. It is therefore believed that the study of the mathematics involved in conformal field theory will ultimately lead to new mathematical structures which would be important to both mathematics and physics.
Lie Triple Derivations of CSL Algebras
NASA Astrophysics Data System (ADS)
Yu, Weiyan; Zhang, Jianhua
2013-06-01
Let [InlineEquation not available: see fulltext.] be a commutative subspace lattice generated by finite many commuting independent nests on a complex separable Hilbert space [InlineEquation not available: see fulltext.] with [InlineEquation not available: see fulltext.], and [InlineEquation not available: see fulltext.] the associated CSL algebra. It is proved that every Lie triple derivation from [InlineEquation not available: see fulltext.] into any σ-weakly closed algebra [InlineEquation not available: see fulltext.] containing [InlineEquation not available: see fulltext.] is of the form X→ XT- TX+ h( X) I, where [InlineEquation not available: see fulltext.] and h is a linear mapping from [InlineEquation not available: see fulltext.] into ℂ such that h([[ A, B], C])=0 for all [InlineEquation not available: see fulltext.].
Twisting algebraically special solutions in five dimensions
NASA Astrophysics Data System (ADS)
Bernardi de Freitas, Gabriel; Godazgar, Mahdi; Reall, Harvey S.
2016-05-01
We determine the general form of the solutions of the five-dimensional vacuum Einstein equations with cosmological constant for which (i) the Weyl tensor is everywhere type II or more special in the null alignment classification of Coley et al, and (ii) the 3 × 3 matrix encoding the expansion, shear and twist of the aligned null direction has rank 2. The dependence of the solution on two coordinates is determined explicitly, so the Einstein equation reduces to PDEs in the three remaining coordinates, just as for four-dimensional (4d) algebraically special solutions. The solutions fall into several families. One of these consists of warped products of 4d algebraically special solutions. The others are new.
Studies in Mathematics, Volume VIII. Concepts of Algebra. Preliminary Edition.
ERIC Educational Resources Information Center
Clarkson, Donald R., Ed.; And Others
This volume is designed to provide information for teachers and prospective teachers who will teach the basic concepts of algebra normally taught in grade 9. Each section of the book contains background information, suggestions for instruction, and problems. Sections in the book include: (1) Numerals and Variables; (2) Open Sentences and English…
Algebraic grid generation with control points
NASA Technical Reports Server (NTRS)
Eiseman, Peter R.; Choo, Yung K.; Smith, Robert E.
1992-01-01
The control-point form (CPF) formulation is an algebraically defined class of coordinate transformations by means of which the interior form of the coordinates can be manipulated in the local fashion, and any boundary can be either specified or manipulated in a similar manner. Currently, the most intense activity involving CPF is with such graphic interactive codes as TurboI and TurboT, for which detailed illustrative examples are given; these have furnished experience on whose basis future interactive strategies can be developed.
Vector fields and nilpotent Lie algebras
NASA Technical Reports Server (NTRS)
Grayson, Matthew; Grossman, Robert
1987-01-01
An infinite-dimensional family of flows E is described with the property that the associated dynamical system: x(t) = E(x(t)), where x(0) is a member of the set R to the Nth power, is explicitly integrable in closed form. These flows E are of the form E = E1 + E2, where E1 and E2 are the generators of a nilpotent Lie algebra, which is either free, or satisfies some relations at a point. These flows can then be used to approximate the flows of more general types of dynamical systems.
An algebraic approach to BCJ numerators
NASA Astrophysics Data System (ADS)
Fu, Chih-Hao; Du, Yi-Jian; Feng, Bo
2013-03-01
One important discovery in recent years is that the total amplitude of gauge theory can be written as BCJ form where kinematic numerators satisfy Jacobi identity. Although the existence of such kinematic numerators is no doubt, the simple and explicit construction is still an important problem. As a small step, in this note we provide an algebraic approach to construct these kinematic numerators. Under our Feynman-diagram-like construction, the Jacobi identity is manifestly satisfied. The corresponding color ordered amplitudes satisfy off-shell KK-relation and off-shell BCJ relation similar to the color ordered scalar theory. Using our construction, the dual DDM form is also established.
Tamatani, T; Tsunoda, H; Iwasaki, H; Kaneko, M; Hashimoto, T; Onozaki, K
1988-01-01
We investigated the possible existence of IL-1 in human amniotic fluid (AF). Since AF from most full-term deliveries appeared to contain an inhibitor(s) for thymocyte proliferation, AFs were fractionated by gel filtration prior to IL-1 assay. IL-1 activities eluted in two peaks at positions of 90,000-60,000 MW and 20,000-15,000 MW. Growth inhibitory activity eluted at the position of 70,000-50,000 MW, and its effect appeared to be non-specific because these fractions inhibited the growth of various cell lines. Using isoelectric focusing (IEF) techniques, pI values of 6.8-7.3 for the higher MW IL-1 as well as 4.9-5.5 and 6.7-7.0 for the lower MW IL-1 were obtained. Antibody against human IL-1 alpha partially neutralized the activity of the lower MW IL-1, though it exhibited little effect on the higher MW IL-1. In contrast, antibody against human IL-1 beta almost completely neutralized the activity of the higher MW IL-1 and partially neutralized the activity of the lower MW IL-1. These results suggest that most of the higher MW IL-1 is beta-type, and the lower MW IL-1 is a mixture of alpha and beta-types. IL-1 beta appeared to exist as a complex (combined with AF components) or as an aggregate of the lower MW IL-1 forms. These findings indicate that both IL-1 alpha and IL-1 beta are present in normal human AF from full-term deliveries, though IL-1 beta exists as a higher MW form aggregated with an unknown molecule. PMID:3264804
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)
Chen, J.; Safro, I.
2011-01-01
Measuring the connection strength between a pair of vertices in a graph is one of the most important concerns in many graph applications. Simple measures such as edge weights may not be sufficient for capturing the effects associated with short paths of lengths greater than one. In this paper, we consider an iterative process that smooths an associated value for nearby vertices, and we present a measure of the local connection strength (called the algebraic distance; see [D. Ron, I. Safro, and A. Brandt, Multiscale Model. Simul., 9 (2011), pp. 407-423]) based on this process. The proposed measure is attractive in that the process is simple, linear, and easily parallelized. An analysis of the convergence property of the process reveals that the local neighborhoods play an important role in determining the connectivity between vertices. We demonstrate the practical effectiveness of the proposed measure through several combinatorial optimization problems on graphs and hypergraphs.
Quantum algebra of N superspace
Hatcher, Nicolas; Restuccia, A.; Stephany, J.
2007-08-15
We identify the quantum algebra of position and momentum operators for a quantum system bearing an irreducible representation of the super Poincare algebra in the N>1 and D=4 superspace, both in the case where there are no central charges in the algebra, and when they are present. This algebra is noncommutative for the position operators. We use the properties of superprojectors acting on the superfields to construct explicit position and momentum operators satisfying the algebra. They act on the projected wave functions associated to the various supermultiplets with defined superspin present in the representation. We show that the quantum algebra associated to the massive superparticle appears in our construction and is described by a supermultiplet of superspin 0. This result generalizes the construction for D=4, N=1 reported recently. For the case N=2 with central charges, we present the equivalent results when the central charge and the mass are different. For the {kappa}-symmetric case when these quantities are equal, we discuss the reduction to the physical degrees of freedom of the corresponding superparticle and the construction of the associated quantum algebra.
Using Homemade Algebra Tiles To Develop Algebra and Prealgebra Concepts.
ERIC Educational Resources Information Center
Leitze, Annette Ricks; Kitt, Nancy A.
2000-01-01
Describes how to use homemade tiles, sketches, and the box method to reach a broader group of students for successful algebra learning. Provides a list of concepts appropriate for such an approach. (KHR)
Distance geometry and geometric algebra
NASA Astrophysics Data System (ADS)
Dress, Andreas W. M.; Havel, Timothy F.
1993-10-01
As part of his program to unify linear algebra and geometry using the language of Clifford algebra, David Hestenes has constructed a (well-known) isomorphism between the conformal group and the orthogonal group of a space two dimensions higher, thus obtaining homogeneous coordinates for conformal geometry.(1) In this paper we show that this construction is the Clifford algebra analogue of a hyperbolic model of Euclidean geometry that has actually been known since Bolyai, Lobachevsky, and Gauss, and we explore its wider invariant theoretic implications. In particular, we show that the Euclidean distance function has a very simple representation in this model, as demonstrated by J. J. Seidel.(18)
Loop Virasoro Lie conformal algebra
Wu, Henan Chen, Qiufan; Yue, Xiaoqing
2014-01-15
The Lie conformal algebra of loop Virasoro algebra, denoted by CW, is introduced in this paper. Explicitly, CW is a Lie conformal algebra with C[∂]-basis (L{sub i} | i∈Z) and λ-brackets [L{sub i} {sub λ} L{sub j}] = (−∂−2λ)L{sub i+j}. Then conformal derivations of CW are determined. Finally, rank one conformal modules and Z-graded free intermediate series modules over CW are classified.
SD-CAS: Spin Dynamics by Computer Algebra System.
Filip, Xenia; Filip, Claudiu
2010-11-01
A computer algebra tool for describing the Liouville-space quantum evolution of nuclear 1/2-spins is introduced and implemented within a computational framework named Spin Dynamics by Computer Algebra System (SD-CAS). A distinctive feature compared with numerical and previous computer algebra approaches to solving spin dynamics problems results from the fact that no matrix representation for spin operators is used in SD-CAS, which determines a full symbolic character to the performed computations. Spin correlations are stored in SD-CAS as four-entry nested lists of which size increases linearly with the number of spins into the system and are easily mapped into analytical expressions in terms of spin operator products. For the so defined SD-CAS spin correlations a set of specialized functions and procedures is introduced that are essential for implementing basic spin algebra operations, such as the spin operator products, commutators, and scalar products. They provide results in an abstract algebraic form: specific procedures to quantitatively evaluate such symbolic expressions with respect to the involved spin interaction parameters and experimental conditions are also discussed. Although the main focus in the present work is on laying the foundation for spin dynamics symbolic computation in NMR based on a non-matrix formalism, practical aspects are also considered throughout the theoretical development process. In particular, specific SD-CAS routines have been implemented using the YACAS computer algebra package (http://yacas.sourceforge.net), and their functionality was demonstrated on a few illustrative examples. PMID:20843716
Hopf algebras and Dyson-Schwinger equations
NASA Astrophysics Data System (ADS)
Weinzierl, Stefan
2016-06-01
In this paper I discuss Hopf algebras and Dyson-Schwinger equations. This paper starts with an introduction to Hopf algebras, followed by a review of the contribution and application of Hopf algebras to particle physics. The final part of the paper is devoted to the relation between Hopf algebras and Dyson-Schwinger equations.
Invariants of triangular Lie algebras with one nil-independent diagonal element
NASA Astrophysics Data System (ADS)
Boyko, Vyacheslav; Patera, Jiri; Popovych, Roman
2007-08-01
The invariants of solvable triangular Lie algebras with one nil-independent diagonal element are studied exhaustively. Bases of the invariant sets of all such algebras are constructed using an original algebraic algorithm based on Cartan's method of moving frames and the special technique developed for triangular and closed algebras in Boyko et al (J. Phys. A: Math. Theor. 2007 40 7557). The conjecture of Tremblay and Winternitz (J. Phys. A: Math. Gen. 2001 34 9085) on the number and form of elements in the bases is completed and proved.
The Weyl realizations of Lie algebras, and left-right duality
NASA Astrophysics Data System (ADS)
Meljanac, Stjepan; Krešić-Jurić, Saša; Martinić, Tea
2016-05-01
We investigate dual realizations of non-commutative spaces of Lie algebra type in terms of formal power series in the Weyl algebra. To each realization of a Lie algebra 𝔤 we associate a star-product on the symmetric algebra S(𝔤) and an ordering on the enveloping algebra U(𝔤). Dual realizations of 𝔤 are defined in terms of left-right duality of the star-products on S(𝔤). It is shown that the dual realizations are related to an extension problem for 𝔤 by shift operators whose action on U(𝔤) describes left and right shift of the generators of U(𝔤) in a given monomial. Using properties of the extended algebra, in the Weyl symmetric ordering we derive closed form expressions for the dual realizations of 𝔤 in terms of two generating functions for the Bernoulli numbers. The theory is illustrated by considering the κ-deformed space.
Description of DASSL: a differential/algebraic system solver
Petzold, L.R.
1982-09-01
This paper describes a new code DASSL, for the numerical solution of implicit systems of differential/algebraic equations. These equations are written in the form F(t,y,y') = 0, and they can include systems which are substantially more complex than standard form ODE systems y' = f(t,y). Differential/algebraic equations occur in several diverse applications in the physical world. We outline the algorithms and strategies used in DASSL, and explain some of the features of the code. In addition, we outline briefly what needs to be done to solve a problem using DASSL.
Normal faults, normal friction?
NASA Astrophysics Data System (ADS)
Collettini, Cristiano; Sibson, Richard H.
2001-10-01
Debate continues as to whether normal faults may be seismically active at very low dips (δ < 30°) in the upper continental crust. An updated compilation of dip estimates (n = 25) has been prepared from focal mechanisms of shallow, intracontinental, normal-slip earthquakes (M > 5.5; slip vector raking 90° ± 30° in the fault plane) where the rupture plane is unambiguously discriminated. The dip distribution for these moderate-to-large normal fault ruptures extends from 65° > δ > 30°, corresponding to a range, 25° < θr < 60°, for the reactivation angle between the fault and inferred vertical σ1. In a comparable data set previously obtained for reverse fault ruptures (n = 33), the active dip distribution is 10° < δ = θr < 60°. For vertical and horizontal σ1 trajectories within extensional and compressional tectonic regimes, respectively, dip-slip reactivation is thus restricted to faults oriented at θr ≤ 60° to inferred σ1. Apparent lockup at θr ≈ 60° in each dip distribution and a dominant 30° ± 5° peak in the reverse fault dip distribution, are both consistent with a friction coefficient μs ≈ 0.6, toward the bottom of Byerlee's experimental range, though localized fluid overpressuring may be needed for reactivation of less favorably oriented faults.
Ota, Kazuaki; Walter, Fabian; Da Cunha, Elisabete; González-López, Jorge; Decarli, Roberto; Hodge, Jacqueline A.; Ohta, Kouji; Hatsukade, Bunyo; Nagai, Hiroshi; Iye, Masanori; Kashikawa, Nobunari; Carilli, Chris L.; Egami, Eiichi; Jiang, Linhua; Riechers, Dominik A.; Bertoldi, Frank; Cox, Pierre; Neri, Roberto; Weiss, Axel
2014-09-01
We present ALMA observations of the [C II] line and far-infrared (FIR) continuum of a normally star-forming galaxy in the reionization epoch, the z = 6.96 Lyα emitter (LAE) IOK-1. Probing to sensitivities of σ{sub line} = 240 μJy beam{sup –1} (40 km s{sup –1} channel) and σ{sub cont} = 21 μJy beam{sup –1}, we found the galaxy undetected in both [C II] and continuum. Comparison of ultraviolet (UV)-FIR spectral energy distribution (SED) of IOK-1, including our ALMA limit, with those of several types of local galaxies (including the effects of the cosmic microwave background, CMB, on the FIR continuum) suggests that IOK-1 is similar to local dwarf/irregular galaxies in SED shape rather than highly dusty/obscured galaxies. Moreover, our 3σ FIR continuum limit, corrected for CMB effects, implies intrinsic dust mass M {sub dust} < 6.4 × 10{sup 7} M {sub ☉}, FIR luminosity L {sub FIR} < 3.7 × 10{sup 10} L {sub ☉} (42.5-122.5 μm), total IR luminosity L {sub IR} < 5.7 × 10{sup 10} L {sub ☉} (8-1000 μm), and dust-obscured star formation rate (SFR) < 10 M {sub ☉} yr{sup –1}, if we assume that IOK-1 has a dust temperature and emissivity index typical of local dwarf galaxies. This SFR is 2.4 times lower than one estimated from the UV continuum, suggesting that <29% of the star formation is obscured by dust. Meanwhile, our 3σ [C II] flux limit translates into [C II] luminosity, L {sub [C} {sub II]} < 3.4 × 10{sup 7} L {sub ☉}. Locations of IOK-1 and previously observed LAEs on the L {sub [C} {sub II]} versus SFR and L {sub [C} {sub II]}/L {sub FIR} versus L {sub FIR} diagrams imply that LAEs in the reionization epoch have significantly lower gas and dust enrichment than AGN-powered systems and starbursts at similar/lower redshifts, as well as local star-forming galaxies.
Cartooning in Algebra and Calculus
ERIC Educational Resources Information Center
Moseley, L. Jeneva
2014-01-01
This article discusses how teachers can create cartoons for undergraduate math classes, such as college algebra and basic calculus. The practice of cartooning for teaching can be helpful for communication with students and for students' conceptual understanding.
NASA Technical Reports Server (NTRS)
Klumpp, A. R.; Lawson, C. L.
1988-01-01
Routines provided for common scalar, vector, matrix, and quaternion operations. Computer program extends Ada programming language to include linear-algebra capabilities similar to HAS/S programming language. Designed for such avionics applications as software for Space Station.
Semiclassical states on Lie algebras
Tsobanjan, Artur
2015-03-15
The effective technique for analyzing representation-independent features of quantum systems based on the semiclassical approximation (developed elsewhere) has been successfully used in the context of the canonical (Weyl) algebra of the basic quantum observables. Here, we perform the important step of extending this effective technique to the quantization of a more general class of finite-dimensional Lie algebras. The case of a Lie algebra with a single central element (the Casimir element) is treated in detail by considering semiclassical states on the corresponding universal enveloping algebra. Restriction to an irreducible representation is performed by “effectively” fixing the Casimir condition, following the methods previously used for constrained quantum systems. We explicitly determine the conditions under which this restriction can be consistently performed alongside the semiclassical truncation.
Hopf algebras and topological recursion
NASA Astrophysics Data System (ADS)
Esteves, João N.
2015-11-01
We consider a model for topological recursion based on the Hopf algebra of planar binary trees defined by Loday and Ronco (1998 Adv. Math. 139 293-309 We show that extending this Hopf algebra by identifying pairs of nearest neighbor leaves, and thus producing graphs with loops, we obtain the full recursion formula discovered by Eynard and Orantin (2007 Commun. Number Theory Phys. 1 347-452).
Gamma Stability in Free Product von Neumann Algebras
NASA Astrophysics Data System (ADS)
Houdayer, Cyril
2015-06-01
Let be a free product of arbitrary von Neumann algebras endowed with faithful normal states. Assume that the centralizer is diffuse. We first show that any intermediate subalgebra which has nontrivial central sequences in M is necessarily equal to M 1. Then we obtain a general structural result for all the intermediate subalgebras with expectation. We deduce that any diffuse amenable von Neumann algebra can be concretely realized as a maximal amenable subalgebra with expectation inside a full nonamenable type III1 factor. This provides the first class of concrete maximal amenable subalgebras in the framework of type III factors. We finally strengthen all these results in the case of tracial free product von Neumann algebras.
Spontaneous Meta-Arithmetic as the First Step toward School Algebra
ERIC Educational Resources Information Center
Caspi, Shai; Sfard, Anna
2012-01-01
Taking as a point of departure the vision of school algebra as a formalized meta-discourse of arithmetic, we have been following six pairs of 7th-grade students (12-13 years old) as they gradually modify their spontaneous meta-arithmetic toward the "official" algebraic form of talk. In this paper we take a look at the very beginning of…
NASA Astrophysics Data System (ADS)
Kuzmin, Dmitri; Möller, Matthias; Gurris, Marcel
Flux limiting for hyperbolic systems requires a careful generalization of the design principles and algorithms introduced in the context of scalar conservation laws. In this chapter, we develop FCT-like algebraic flux correction schemes for the Euler equations of gas dynamics. In particular, we discuss the construction of artificial viscosity operators, the choice of variables to be limited, and the transformation of antidiffusive fluxes. An a posteriori control mechanism is implemented to make the limiter failsafe. The numerical treatment of initial and boundary conditions is discussed in some detail. The initialization is performed using an FCT-constrained L 2 projection. The characteristic boundary conditions are imposed in a weak sense, and an approximate Riemann solver is used to evaluate the fluxes on the boundary. We also present an unconditionally stable semi-implicit time-stepping scheme and an iterative solver for the fully discrete problem. The results of a numerical study indicate that the nonlinearity and non-differentiability of the flux limiter do not inhibit steady state convergence even in the case of strongly varying Mach numbers. Moreover, the convergence rates improve as the pseudo-time step is increased.
The algebraic structure of quantum partial isometries
NASA Astrophysics Data System (ADS)
Banica, Teodor
2016-03-01
The partial isometries of ℝN, ℂN form compact semigroups O˜N,U˜N. We discuss here the liberation question for these semigroups, and for their discrete versions H˜N,K˜N. Our main results concern the construction of half-liberations H˜N×,K˜ N×,O˜ N×,U˜ N× and of liberations H˜N+,K˜ N+,O˜ N+,U˜ N+. We include a detailed algebraic and probabilistic study of all these objects, justifying our “half-liberation” and “liberation” claims.
Nonnumeric Computer Applications to Algebra, Trigonometry and Calculus.
ERIC Educational Resources Information Center
Stoutemyer, David R.
1983-01-01
Described are computer program packages requiring little or no knowledge of computer programing for college algebra, calculus, and abstract algebra. Widely available computer algebra systems are listed. (MNS)
NASA Astrophysics Data System (ADS)
Daddi, E.; Dannerbauer, H.; Liu, D.; Aravena, M.; Bournaud, F.; Walter, F.; Riechers, D.; Magdis, G.; Sargent, M.; Béthermin, M.; Carilli, C.; Cibinel, A.; Dickinson, M.; Elbaz, D.; Gao, Y.; Gobat, R.; Hodge, J.; Krips, M.
2015-05-01
We investigate the CO excitation of normal (near-IR selected BzK) star-forming (SF) disk galaxies at z = 1.5 using IRAM Plateau de Bure observations of the CO[2-1], CO[3-2], and CO[5-4] transitions for four galaxies, including VLA observations of CO[1-0] for three of them, with the aim of constraining the average state of H2 gas. By exploiting previous knowledge of the velocity range, spatial extent, and size of the CO emission, we measure reliable line fluxes with a signal-to-noise ratio >4-7 for individual transitions. While the average CO spectral line energy distribution (SLED) has a subthermal excitation similar to the Milky Way (MW) up to CO[3-2], we show that the average CO[5-4] emission is four times stronger than assuming MW excitation. This demonstrates that there is an additional component of more excited, denser, and possibly warmer molecular gas. The ratio of CO[5-4] to lower-J CO emission is lower than in local (ultra-)luminous infrared galaxies (ULIRGs) and high-redshift starbursting submillimeter galaxies, however, and appears to be closely correlated with the average intensity of the radiation field ⟨ U ⟩ and with the star formation surface density, but not with the star formation efficiency. The luminosity of the CO[5-4] transition is found to be linearly correlated with the bolometric infrared luminosity over four orders of magnitudes. For this transition, z = 1.5 BzK galaxies follow the same linear trend as local spirals and (U)LIRGs and high-redshift star-bursting submillimeter galaxies. The CO[5-4] luminosity is thus empirically related to the dense gas and might be a more convenient way to probe it than standard high-density tracers that are much fainter than CO. We see excitation variations among our sample galaxies that can be linked to their evolutionary state and clumpiness in optical rest-frame images. In one galaxy we see spatially resolved excitation variations, where the more highly excited part of the galaxy corresponds to the
Virasoro algebra in the KN algebra; Bosonic string with fermionic ghosts on Riemann surfaces
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.
Beyond Dirac - a Unified Algebra
NASA Astrophysics Data System (ADS)
Lundberg, Wayne R.
2001-10-01
A introductory insight will be shared regarding a 'separation of variables' approach to understanding the relationship between QCD and the origins of cosmological and particle mass. The discussion will then build upon work presented at DFP 2000, focussing on the formal basis for using 3x3x3 matrix algebra as it underlies and extends Dirac notation. A set of restrictions are established which break the multiple symmetries of the 3x3x3 matrix algebra, yielding Standard Model QCD objects and interactions. It will be shown that the 3x3x3 matrix representation unifies the algebra of strong and weak (and by extension, electromagnetic) interactions. A direct correspondence to string theoretic objects is established by considering the string to be partitioned in thirds. Rubik's cube is used as a graphical means of handling algebraic manipulation of 3x3x3 algebra. Further, its potential utility for advancing pedagogical methods through active engagement is discussed. A simulated classroom exercize will be conducted.
The Progressive Development of Early Embodied Algebraic Thinking
ERIC Educational Resources Information Center
Radford, Luis
2014-01-01
In this article I present some results from a 5-year longitudinal investigation with young students about the genesis of embodied, non-symbolic algebraic thinking and its progressive transition to culturally evolved forms of symbolic thinking. The investigation draws on a cultural-historical theory of teaching and learning--the theory of…
Excel Spreadsheets for Algebra: Improving Mental Modeling for Problem Solving
ERIC Educational Resources Information Center
Engerman, Jason; Rusek, Matthew; Clariana, Roy
2014-01-01
This experiment investigates the effectiveness of Excel spreadsheets in a high school algebra class. Students in the experiment group convincingly outperformed the control group on a post lesson assessment. The student responses, teacher observations involving Excel spreadsheet revealed that it operated as a mindtool, which formed the users'…
The Differential Graded Odd NilHecke Algebra
NASA Astrophysics Data System (ADS)
Ellis, Alexander P.; Qi, You
2016-05-01
We equip the odd nilHecke algebra and its associated thick calculus category with diagrammatically local differentials. The resulting differential graded Grothendieck groups are isomorphic to two different forms of the positive part of quantum {{{sl}_2}} at a fourth root of unity.
Programmed Math Continuum, Level One, Algebra, Volume 2.
ERIC Educational Resources Information Center
New York Inst. of Tech., Old Westbury.
This programed instruction study guide is one of a series that form a first-year algebra course. Structured in a multiple-choice question-answer format with scrambled pages, it is intended to be used in conjunction with a computer-managed instructional system. The following topics are covered in Volume 2: punctuation marks; order of operations;…
Programmed Math Continuum, Level One, Algebra, Volume 3.
ERIC Educational Resources Information Center
New York Inst. of Tech., Old Westbury.
This programed instruction study guide is one of a series that form a first-year algebra course. Structured in a multiple-choice question-answer format with scrambled pages, it is intended to be used in conjunction with a computer-managed instructional system. The following topics are covered in Volume 3: solving problems with open sentences;…
The coquaternion algebra and complex partial differential equations
NASA Astrophysics Data System (ADS)
Dimiev, Stancho; Konstantinov, Mihail; Todorov, Vladimir
2009-11-01
In this paper we consider the problem of differentiation of coquaternionic functions. Let us recall that coquaternions are elements of an associative non-commutative real algebra with zero divisor, introduced by James Cockle (1849) under the name of split-quaternions or coquaternions. Developing two type complex representations for Cockle algebra (complex and paracomplex ones) we present the problem in a non-commutative form of the δ¯-type holomorphy. We prove that corresponding differentiable coquaternionic functions, smooth and analytic, satisfy PDE of complex, and respectively of real variables. Applications for coquaternionic polynomials are sketched.
A Flexible Variable Truncated Power Series Algebra in Zlib
Yan, Y.T.; /SLAC
2011-08-25
Zlib is a numerical library for Truncated Power Series Algebra (TPSA) and Lie Algebra for application to nonlinear analysis of single particle dynamics. The first version was developed in 1990 with the use of the One-Step Index Pointers (OSIP's). The OSIP's form the Zlib nerve that offers optimal computation and alloworder grading as well as flexible initialization of the global number of variables for the TPSA. While the OSIP's are still kept for minimum index passing to achieve efficient computation, Zlib has been being upgraded to allow flexible and gradable local number of variables in each C++ object of the Truncated Power Series (Tps) class. Possible applications using Zlib are discussed.
Ternary Z3 -graded generalization of Heisenberg's algebra
NASA Astrophysics Data System (ADS)
Kerner, Richard
2015-04-01
We investigate a ternary, Z3-graded generalization of the Heisenberg algebra. It turns out that introducing a non-trivial cubic root of unity, j = e 2πi/3, one can define two types of creation operators instead of one, accompanying the usual annihilation operator. The two creation operators are non-hermitian, but they are mutually conjugate. Together, the three operators form a ternary algebra, and some of their cubic combinations generate the usual Heisenberg algebra. A cubic analogue of Hamiltonian operator is constructed by analogy with the usual harmonic oscillator. A set of eigenstates in coordinate representation is constructed in terms of functions satisfying linear differential equation of third order.
Quadratic algebras for three-dimensional superintegrable systems
Daskaloyannis, C. Tanoudis, Y.
2010-02-15
The three-dimensional superintegrable systems with quadratic integrals of motion have five functionally independent integrals, one among them is the Hamiltonian. Kalnins, Kress, and Miller have proved that in the case of nondegenerate potentials with quadratic integrals of motion there is a sixth quadratic integral, which is linearly independent of the other integrals. The existence of this sixth integral implies that the integrals of motion form a ternary parafermionic-like quadratic Poisson algebra with five generators. In this contribution we investigate the structure of this algebra. We show that in all the nondegenerate cases there is at least one subalgebra of three integrals having a Poisson quadratic algebra structure, which is similar to the two-dimensional case.
Infinite-dimensional Lie algebras, classical r-matrices, and Lax operators: Two approaches
NASA Astrophysics Data System (ADS)
Skrypnyk, T.
2013-10-01
For each finite-dimensional simple Lie algebra {g}, starting from a general {g}⊗ {g}-valued solutions r(u, v) of the generalized classical Yang-Baxter equation, we construct infinite-dimensional Lie algebras widetilde{{g}}-_r of {g}-valued meromorphic functions. We outline two ways of embedding of the Lie algebra widetilde{{g}}-_r into a larger Lie algebra with Kostant-Adler-Symmes decomposition. The first of them is an embedding of widetilde{{g}}-_r into Lie algebra widetilde{{g}}(u^{-1},u)) of formal Laurent power series. The second is an embedding of widetilde{{g}}-_r as a quasigraded Lie subalgebra into a quasigraded Lie algebra widetilde{{g}}_r: widetilde{{g}}_r=widetilde{{g}}-_r+widetilde{{g}}+_r, such that the Kostant-Adler-Symmes decomposition is consistent with a chosen quasigrading. We construct dual spaces widetilde{{g}}^*_r, (widetilde{{g}}^{± }_r)^* and explicit form of the Lax operators L(u), L±(u) as elements of these spaces. We develop a theory of integrable finite-dimensional hamiltonian systems and soliton hierarchies based on Lie algebras widetilde{{g}}_r, widetilde{{g}}^{± }_r. We consider examples of such systems and soliton equations and obtain the most general form of integrable tops, Kirchhoff-type integrable systems, and integrable Landau-Lifshitz-type equations corresponding to the Lie algebra {g}.
Generalized Galilean algebras and Newtonian gravity
NASA Astrophysics Data System (ADS)
González, N.; Rubio, G.; Salgado, P.; Salgado, S.
2016-04-01
The non-relativistic versions of the generalized Poincaré algebras and generalized AdS-Lorentz algebras are obtained. These non-relativistic algebras are called, generalized Galilean algebras of type I and type II and denoted by GBn and GLn respectively. Using a generalized Inönü-Wigner contraction procedure we find that the generalized Galilean algebras of type I can be obtained from the generalized Galilean algebras type II. The S-expansion procedure allows us to find the GB5 algebra from the Newton Hooke algebra with central extension. The procedure developed in Ref. [1] allows us to show that the nonrelativistic limit of the five dimensional Einstein-Chern-Simons gravity is given by a modified version of the Poisson equation. The modification could be compatible with the effects of Dark Matter, which leads us to think that Dark Matter can be interpreted as a non-relativistic limit of Dark Energy.
Computer Algebra Systems in Undergraduate Instruction.
ERIC Educational Resources Information Center
Small, Don; And Others
1986-01-01
Computer algebra systems (such as MACSYMA and muMath) can carry out many of the operations of calculus, linear algebra, and differential equations. Use of them with sketching graphs of rational functions and with other topics is discussed. (MNS)
Motivating Activities that Lead to Algebra
ERIC Educational Resources Information Center
Menon, Ramakrishnan
2004-01-01
Four activities consisting of puzzles are introduced, which help students to recognize the strength of algebraic generalizations. They also assist them to comprehend algebraic concepts, and enable them to develop their individual puzzles and games.
Scalable Parallel Algebraic Multigrid Solvers
Bank, R; Lu, S; Tong, C; Vassilevski, P
2005-03-23
The authors propose a parallel algebraic multilevel algorithm (AMG), which has the novel feature that the subproblem residing in each processor is defined over the entire partition domain, although the vast majority of unknowns for each subproblem are associated with the partition owned by the corresponding processor. This feature ensures that a global coarse description of the problem is contained within each of the subproblems. The advantages of this approach are that interprocessor communication is minimized in the solution process while an optimal order of convergence rate is preserved; and the speed of local subproblem solvers can be maximized using the best existing sequential algebraic solvers.
Computational triadic algebras of signs
Zadrozny, W.
1996-12-31
We present a finite model of Peirce`s ten classes of signs. We briefly describe Peirce`s taxonomy of signs; we prove that any finite collection of signs can be extended to a finite algebra of signs in which all interpretants are themselves being interpreted; and we argue that Peirce`s ten classes of signs can be defined using constraints on algebras of signs. The paper opens the possibility of defining multimodal cognitive agents using Peirce`s classes of signs, and is a first step towards building a computational logic of signs based on Peirce`s taxonomies.
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…
Spatial-Operator Algebra For Robotic Manipulators
NASA Technical Reports Server (NTRS)
Rodriguez, Guillermo; Kreutz, Kenneth K.; Milman, Mark H.
1991-01-01
Report discusses spatial-operator algebra developed in recent studies of mathematical modeling, control, and design of trajectories of robotic manipulators. Provides succinct representation of mathematically complicated interactions among multiple joints and links of manipulator, thereby relieving analyst of most of tedium of detailed algebraic manipulations. Presents analytical formulation of spatial-operator algebra, describes some specific applications, summarizes current research, and discusses implementation of spatial-operator algebra in the Ada programming language.
Bilinear covariants and spinor fields duality in quantum Clifford algebras
Abłamowicz, Rafał; Gonçalves, Icaro; Rocha, Roldão da
2014-10-15
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's 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.
Bilinear covariants and spinor fields duality in quantum Clifford algebras
NASA Astrophysics Data System (ADS)
Abłamowicz, Rafał; Gonçalves, Icaro; da Rocha, Roldão
2014-10-01
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's 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.
The weak Hopf algebras related to generalized Kac-Moody algebra
Wu Zhixiang
2006-06-15
We define a kind of quantized enveloping algebra of a generalized Kac-Moody algebra G by adding a generator J satisfying J{sup m}=J{sup m-1} for some integer m. We denote this algebra by wU{sub q}{sup {tau}}(G). This algebra is a weak Hopf algebra if and only if m=2. In general, it is a bialgebra, and contains a Hopf subalgebra. This Hopf subalgebra is isomorphic to the usually quantum envelope algebra U{sub q}(G) of a generalized Kac-Moody algebra G.
The Role of Proof in Comprehending and Teaching Elementary Linear Algebra.
ERIC Educational Resources Information Center
Uhlig, Frank
2002-01-01
Describes how elementary linear algebra can be taught successfully while introducing students to the concept and practice of mathematical proof. Suggests exploring the concept of solvability of linear systems first via the row echelon form (REF). (Author/KHR)
A New Reynolds Stress Algebraic Equation Model
NASA Technical Reports Server (NTRS)
Shih, Tsan-Hsing; Zhu, Jiang; Lumley, John L.
1994-01-01
A general turbulent constitutive relation is directly applied to propose a new Reynolds stress algebraic equation model. In the development of this model, the constraints based on rapid distortion theory and realizability (i.e. the positivity of the normal Reynolds stresses and the Schwarz' inequality between turbulent velocity correlations) are imposed. Model coefficients are calibrated using well-studied basic flows such as homogeneous shear flow and the surface flow in the inertial sublayer. The performance of this model is then tested in complex turbulent flows including the separated flow over a backward-facing step and the flow in a confined jet. The calculation results are encouraging and point to the success of the present model in modeling turbulent flows with complex geometries.
Algebra? A Gate! A Barrier! A Mystery!
ERIC Educational Resources Information Center
Mathematics Educatio Dialogues, 2000
2000-01-01
This issue of Mathematics Education Dialogues focuses on the nature and the role of algebra in the K-14 curriculum. Articles on this theme include: (1) "Algebra For All? Why?" (Nel Noddings); (2) "Algebra For All: It's a Matter of Equity, Expectations, and Effectiveness" (Dorothy S. Strong and Nell B. Cobb); (3) "Don't Delay: Build and Talk about…
UCSMP Algebra. What Works Clearinghouse Intervention Report
ERIC Educational Resources Information Center
What Works Clearinghouse, 2007
2007-01-01
"University of Chicago School Mathematics Project (UCSMP) Algebra," designed to increase students' skills in algebra, is appropriate for students in grades 7-10, depending on the students' incoming knowledge. This one-year course highlights applications, uses statistics and geometry to develop the algebra of linear equations and inequalities, and…
Graphing Calculator Use in Algebra Teaching
ERIC Educational Resources Information Center
Dewey, Brenda L.; Singletary, Ted J.; Kinzel, Margaret T.
2009-01-01
This study examines graphing calculator technology availability, characteristics of teachers who use it, teacher attitudes, and how use reflects changes to algebra curriculum and instructional practices. Algebra I and Algebra II teachers in 75 high school and junior high/middle schools in a diverse region of a northwestern state were surveyed.…
New family of Maxwell like algebras
NASA Astrophysics Data System (ADS)
Concha, P. K.; Durka, R.; Merino, N.; Rodríguez, E. K.
2016-08-01
We introduce an alternative way of closing Maxwell like algebras. We show, through a suitable change of basis, that resulting algebras are given by the direct sums of the AdS and the Maxwell algebras already known in the literature. Casting the result into the S-expansion method framework ensures the straightaway construction of the gravity theories based on a found enlargement.
Build an Early Foundation for Algebra Success
ERIC Educational Resources Information Center
Knuth, Eric; Stephens, Ana; Blanton, Maria; Gardiner, Angela
2016-01-01
Research tells us that success in algebra is a factor in many other important student outcomes. Emerging research also suggests that students who are started on an algebra curriculum in the earlier grades may have greater success in the subject in secondary school. What's needed is a consistent, algebra-infused mathematics curriculum all…
A Balancing Act: Making Sense of Algebra
ERIC Educational Resources Information Center
Gavin, M. Katherine; Sheffield, Linda Jensen
2015-01-01
For most students, algebra seems like a totally different subject than the number topics they studied in elementary school. In reality, the procedures followed in arithmetic are actually based on the properties and laws of algebra. Algebra should be a logical next step for students in extending the proficiencies they developed with number topics…
Difficulties in Initial Algebra Learning in Indonesia
ERIC Educational Resources Information Center
Jupri, Al; Drijvers, Paul; van den Heuvel-Panhuizen, Marja
2014-01-01
Within mathematics curricula, algebra has been widely recognized as one of the most difficult topics, which leads to learning difficulties worldwide. In Indonesia, algebra performance is an important issue. In the Trends in International Mathematics and Science Study (TIMSS) 2007, Indonesian students' achievement in the algebra domain was…
Teaching Strategies to Improve Algebra Learning
ERIC Educational Resources Information Center
Zbiek, Rose Mary; Larson, Matthew R.
2015-01-01
Improving student learning is the primary goal of every teacher of algebra. Teachers seek strategies to help all students learn important algebra content and develop mathematical practices. The new Institute of Education Sciences[IES] practice guide, "Teaching Strategies for Improving Algebra Knowledge in Middle and High School Students"…
Lessons for Algebraic Thinking. Grades K-2.
ERIC Educational Resources Information Center
von Rotz, Leyani; Burns, Marilyn
Algebra is one of the top priorities of mathematics instruction for the elementary and middle grades. This book is designed to help K-2 teachers meet the challenge of making algebra an integral part of their mathematics instruction and realize both what to teach and how to teach central algebraic concepts. Classroom-tested lessons help teachers…
Unifying the Algebra for All Movement
ERIC Educational Resources Information Center
Eddy, Colleen M.; Quebec Fuentes, Sarah; Ward, Elizabeth K.; Parker, Yolanda A.; Cooper, Sandi; Jasper, William A.; Mallam, Winifred A.; Sorto, M. Alejandra; Wilkerson, Trena L.
2015-01-01
There exists an increased focus on school mathematics, especially first-year algebra, due to recent efforts for all students to be college and career ready. In addition, there are calls, policies, and legislation advocating for all students to study algebra epitomized by four rationales of the "Algebra for All" movement. In light of this…
Weaving Geometry and Algebra Together
ERIC Educational Resources Information Center
Cetner, Michelle
2015-01-01
When thinking about student reasoning and sense making, teachers must consider the nature of tasks given to students along with how to plan to use the tasks in the classroom. Students should be presented with tasks in a way that encourages them to draw connections between algebraic and geometric concepts. This article focuses on the idea that it…
Inequalities, Assessment and Computer Algebra
ERIC Educational Resources Information Center
Sangwin, Christopher J.
2015-01-01
The goal of this paper is to examine single variable real inequalities that arise as tutorial problems and to examine the extent to which current computer algebra systems (CAS) can (1) automatically solve such problems and (2) determine whether students' own answers to such problems are correct. We review how inequalities arise in…
ERIC Educational Resources Information Center
Bosse, Michael J.; Ries, Heather; Chandler, Kayla
2012-01-01
Secondary school mathematics teachers often need to answer the "Why do we do that?" question in such a way that avoids confusion and evokes student interest. Understanding the properties of number systems can provide an avenue to better grasp algebraic structures, which in turn builds students' conceptual knowledge of secondary mathematics. This…
Implementing Change in College Algebra
ERIC Educational Resources Information Center
Haver, William E.
2007-01-01
In this paper, departments are urged to consider implementing the type of changes proposed in Beyond Crossroads in College Algebra. The author of this paper is chair of the Curriculum Renewal Across the First Two Years (CRAFTY) Committee of the Mathematical Association of America. The committee has members from two-year colleges, four-year…
Algebraic Activities Aid Discovery Lessons
ERIC Educational Resources Information Center
Wallace-Gomez, Patricia
2013-01-01
After a unit on the rules for positive and negative numbers and the order of operations for evaluating algebraic expressions, many students believe that they understand these principles well enough, but they really do not. They clearly need more practice, but not more of the same kind of drill. Wallace-Gomez provides three graphing activities that…
Entropy algebras and Birkhoff factorization
NASA Astrophysics Data System (ADS)
Marcolli, Matilde; Tedeschi, Nicolas
2015-11-01
We develop notions of Rota-Baxter structures and associated Birkhoff factorizations, in the context of min-plus semirings and their thermodynamic deformations, including deformations arising from quantum information measures such as the von Neumann entropy. We consider examples related to Manin's renormalization and computation program, to Markov random fields and to counting functions and zeta functions of algebraic varieties.
Algebra for All. Research Brief
ERIC Educational Resources Information Center
Bleyaert, Barbara
2009-01-01
The call for "algebra for all" is not a recent phenomenon. Concerns about the inadequacy of math (and science) preparation in America's high schools have been a steady drumbeat since the 1957 launch of Sputnik; a call for raising standards and the number of math (and science) courses required for graduation has been a part of countless national…
ERIC Educational Resources Information Center
Oishi, Lindsay
2011-01-01
"Solve for x." While many people first encountered this enigmatic instruction in high school, the last 20 years have seen a strong push to get students to take algebra in eighth grade or even before. Today, concerns about the economy highlight a familiar worry: American eighth-graders trailed their peers in five Asian countries on the 2007 TIMSS…
Exploring Algebraic Misconceptions with Technology
ERIC Educational Resources Information Center
Sakow, Matthew; Karaman, Ruveyda
2015-01-01
Many students struggle with algebra, from simplifying expressions to solving systems of equations. Students also have misconceptions about the meaning of variables. In response to the question "Can x + y + z ever equal x + p + z?" during a student interview, the student claimed, "Never . . . because p has to have a different value…
An Introduction to Algebraic Multigrid
Falgout, R D
2006-04-25
Algebraic multigrid (AMG) solves linear systems based on multigrid principles, but in a way that only depends on the coefficients in the underlying matrix. The author begins with a basic introduction to AMG methods, and then describes some more recent advances and theoretical developments
Elementary Algebra Connections to Precalculus
ERIC Educational Resources Information Center
Lopez-Boada, Roberto; Daire, Sandra Arguelles
2013-01-01
This article examines the attitudes of some precalculus students to solve trigonometric and logarithmic equations and systems using the concepts of elementary algebra. With the goal of enticing the students to search for and use connections among mathematical topics, they are asked to solve equations or systems specifically designed to allow…
Adventures in Flipping College Algebra
ERIC Educational Resources Information Center
Van Sickle, Jenna
2015-01-01
This paper outlines the experience of a university professor who implemented flipped learning in two sections of college algebra courses for two semesters. It details how the courses were flipped, what technology was used, advantages, challenges, and results. It explains what students do outside of class, what they do inside class, and discusses…
Kinds of Knowledge in Algebra.
ERIC Educational Resources Information Center
Lewis, Clayton
Solving equations in elementary algebra requires knowledge of the permitted operations, and knowledge of what operation to use at a given point in the solution process. While just these kinds of knowledge would be adequate for an ideal solver, human solvers appear to need and use other kinds of knowledge. First, many errors seem to indicate that…
Algebra, Home Mortgages, and Recessions
ERIC Educational Resources Information Center
Mariner, Jean A. Miller; Miller, Richard A.
2009-01-01
The current financial crisis and recession in the United States present an opportunity to discuss relevant applications of some topics in typical first-and second-year algebra and precalculus courses. Real-world applications of percent change, exponential functions, and sums of finite geometric sequences can help students understand the problems…
Algebra from Chips and Chopsticks
ERIC Educational Resources Information Center
Yun, Jeong Oak; Flores, Alfinio
2012-01-01
Students can use geometric representations of numbers as a way to explore algebraic ideas. With the help of these representations, students can think about the relations among the numbers, express them using their own words, and represent them with letters. The activities discussed here can stimulate students to try to find various ways of solving…
Celestial mechanics with geometric algebra
NASA Technical Reports Server (NTRS)
Hestenes, D.
1983-01-01
Geometric algebra is introduced as a general tool for Celestial Mechanics. A general method for handling finite rotations and rotational kinematics is presented. The constants of Kepler motion are derived and manipulated in a new way. A new spinor formulation of perturbation theory is developed.
Algebraic methods in system theory
NASA Technical Reports Server (NTRS)
Brockett, R. W.; Willems, J. C.; Willsky, A. S.
1975-01-01
Investigations on problems of the type which arise in the control of switched electrical networks are reported. The main results concern the algebraic structure and stochastic aspects of these systems. Future reports will contain more detailed applications of these results to engineering studies.
Principals + Algebra (- Fear) = Instructional Leadership
ERIC Educational Resources Information Center
Carver, Cynthia L.
2010-01-01
Recent state legislation in Michigan mandates that all graduating seniors successfully pass algebra I and II. Numerous initiatives have been enacted to help mathematics teachers meet this challenge, yet school principals have had little preparation for the necessary curricular and instructional changes. To address this unmet need, university-based…
Experts Question California's Algebra Edict
ERIC Educational Resources Information Center
Cavanagh, Sean
2008-01-01
Business leaders from important sectors of the American economy have been urging schools to set higher standards in math and science--and California officials, in mandating that 8th graders be tested in introductory algebra, have responded with one of the highest such standards in the land. Still, many California educators and school…
Algebraic Davis Decomposition and Asymmetric Doob Inequalities
NASA Astrophysics Data System (ADS)
Hong, Guixiang; Junge, Marius; Parcet, Javier
2016-04-01
In this paper we investigate asymmetric forms of Doob maximal inequality. The asymmetry is imposed by noncommutativity. Let {({M}, τ)} be a noncommutative probability space equipped with a filtration of von Neumann subalgebras {({M}_n)_{n ≥ 1}} , whose union {bigcup_{n≥1}{M}_n} is weak-* dense in {{M}} . Let {{E}_n} denote the corresponding family of conditional expectations. As an illustration for an asymmetric result, we prove that for {1 < p < 2} and {x in L_p({M},τ)} one can find {a, b in L_p({M},τ)} and contractions {u_n, v_n in {M}} such that {E}_n(x) = a u_n + v_n b quad and quad max big{ |a|_p,|b|_p big} ≤ c_p |x|_p. Moreover, it turns out that {a u_n} and {v_n b} converge in the row/column Hardy spaces {{H}_p^r({M})} and {{H}_p^c({M})} respectively. In particular, this solves a problem posed by the Defant and Junge in 2004. In the case p = 1, our results establish a noncommutative form of the Davis celebrated theorem on the relation betwe en martingale maximal and square functions in L 1, whose noncommutative form has remained open for quite some time. Given {1 ≤ p ≤ 2} , we also provide new weak type maximal estimates, which imply in turn left/right almost uniform convergence of {{E}_n(x)} in row/column Hardy spaces. This improves the bilateral convergence known so far. Our approach is based on new forms of Davis martingale decomposition which are of independent interest, and an algebraic atomic description for the involved Hardy spaces. The latter results are new even for commutative von Neumann algebras.
A uniform algebraically-based approach to computational physics and efficient programming
NASA Astrophysics Data System (ADS)
Raynolds, James; Mullin, Lenore
2007-03-01
We present an approach to computational physics in which a common formalism is used both to express the physical problem as well as to describe the underlying details of how computation is realized on arbitrary multiprocessor/memory computer architectures. This formalism is the embodiment of a generalized algebra of multi-dimensional arrays (A Mathematics of Arrays) and an efficient computational implementation is obtained through the composition of of array indices (the psi-calculus) of algorithms defined using matrices, tensors, and arrays in general. The power of this approach arises from the fact that multiple computational steps (e.g. Fourier Transform followed by convolution, etc.) can be algebraically composed and reduced to an simplified expression (i.e. Operational Normal Form), that when directly translated into computer code, can be mathematically proven to be the most efficient implementation with the least number of temporary variables, etc. This approach will be illustrated in the context of a cache-optimized FFT that outperforms or is competitive with established library routines: ESSL, FFTW, IMSL, NAG.
The Exocenter of a Generalized Effect Algebra
NASA Astrophysics Data System (ADS)
Foulis, David J.; Pulmannová, Sylvia
2011-12-01
Elements of the exocenter of a generalized effect algebra (GEA) correspond to decompositions of the GEA as a direct sum and thus the exocenter is a generalization to GEAs of the center of an effect algebra. The exocenter of a GEA is shown to be a boolean algebra, and the notion of a hull mapping for an effect algebra is generalized to a hull system for a GEA. We study Dedekind orthocompleteness of GEAs and extend to GEAs the notion of a centrally orthocomplete effect algebra.
Array algebra estimation in signal processing
NASA Astrophysics Data System (ADS)
Rauhala, U. A.
A general theory of linear estimators called array algebra estimation is interpreted in some terms of multidimensional digital signal processing, mathematical statistics, and numerical analysis. The theory has emerged during the past decade from the new field of a unified vector, matrix and tensor algebra called array algebra. The broad concepts of array algebra and its estimation theory cover several modern computerized sciences and technologies converting their established notations and terminology into one common language. Some concepts of digital signal processing are adopted into this language after a review of the principles of array algebra estimation and its predecessors in mathematical surveying sciences.
NASA Astrophysics Data System (ADS)
Sánchez-Castellanos, M.; Amezcua-Eccius, C. A.; Álvarez-Bajo, O.; Lemus, R.
2008-02-01
A general description of vibrational excitations of pyramidal molecules in both local and normal representations is presented. This study is restricted to the case when no tunneling motion is allowed. The Hamiltonian is first written in terms of curvilinear internal coordinates. The Wilson's G matrix as well as the potential are expanded in terms of Morse variables, which allows the identification of a set of six Morse oscillators as zeroth-order Hamiltonian. An algebraic realization of the Hamiltonian is obtained by introducing a linear expansion of the coordinates and momenta in terms of creation and annihilation operators of Morse functions. This algebraic realization provides in natural form the representation of the Hamiltonian in terms of local interactions. The normal interactions are constructed by successive couplings of tensors defined as linear combinations of the ladder operators. The matrix transformation between the local and normal interactions is obtained for the complete Hamiltonian. This analysis provides the spectroscopic parameters in both local and normal schemes in explicit form as functions of the force constants and structure parameters. To exemplify, the analysis of the vibrational excitations of stibine and arsine is presented. Force constants as well as the corresponding x,K relations are given. A comparison with the results obtained using the U(ν+1) unitary group approach is included.
Exploring Novel Cyclic Extensions of Hamilton's Dual-Quaternion Algebra
NASA Astrophysics Data System (ADS)
Amoroso, Richard L.; Rowlands, Peter; Kauffman, Louis H.
2013-09-01
We make a preliminary exploratory study of higher dimensional (HD) orthogonal forms of the quaternion algebra in order to explore putative novel Nilpotent/Idempotent/Dirac symmetry properties. Stage-1 transforms the dual quaternion algebra in a manner that extends the standard anticommutative 3-form, i, j, k into a 5D/6D triplet. Each is a copy of the others and each is self-commutative and believed to represent spin or different orientations of a 3-cube. The triplet represents a copy of the original that contains no new information other than rotational perspective and maps back to the original quaternion vertex or to a second point in a line element. In Stage-2 we attempt to break the inherent quaternionic property of algebraic closure by stereographic projection of the Argand plane onto rotating Riemann 4-spheres. Finally, we explore the properties of various topological symmetries in order to study anticommutative - commutative cycles in the periodic rotational motions of the quaternion algebra in additional HD dualities.
Filiform Lie algebras of order 3
Navarro, R. M.
2014-04-15
The aim of this work is to generalize a very important type of Lie algebras and superalgebras, i.e., filiform Lie (super)algebras, into the theory of Lie algebras of order F. Thus, the concept of filiform Lie algebras of order F is obtained. In particular, for F = 3 it has been proved that by using infinitesimal deformations of the associated model elementary Lie algebra it can be obtained families of filiform elementary lie algebras of order 3, analogously as that occurs into the theory of Lie algebras [M. Vergne, “Cohomologie des algèbres de Lie nilpotentes. Application à l’étude de la variété des algèbres de Lie nilpotentes,” Bull. Soc. Math. France 98, 81–116 (1970)]. Also we give the dimension, using an adaptation of the sl(2,C)-module Method, and a basis of such infinitesimal deformations in some generic cases.
Atomic effect algebras with compression bases
Caragheorgheopol, Dan; Tkadlec, Josef
2011-01-15
Compression base effect algebras were recently introduced by Gudder [Demonstr. Math. 39, 43 (2006)]. They generalize sequential effect algebras [Rep. Math. Phys. 49, 87 (2002)] and compressible effect algebras [Rep. Math. Phys. 54, 93 (2004)]. The present paper focuses on atomic compression base effect algebras and the consequences of atoms being foci (so-called projections) of the compressions in the compression base. Part of our work generalizes results obtained in atomic sequential effect algebras by Tkadlec [Int. J. Theor. Phys. 47, 185 (2008)]. The notion of projection-atomicity is introduced and studied, and several conditions that force a compression base effect algebra or the set of its projections to be Boolean are found. Finally, we apply some of these results to sequential effect algebras and strengthen a previously established result concerning a sufficient condition for them to be Boolean.
Atomic effect algebras with compression bases
NASA Astrophysics Data System (ADS)
Caragheorgheopol, Dan; Tkadlec, Josef
2011-01-01
Compression base effect algebras were recently introduced by Gudder [Demonstr. Math. 39, 43 (2006)]. They generalize sequential effect algebras [Rep. Math. Phys. 49, 87 (2002)] and compressible effect algebras [Rep. Math. Phys. 54, 93 (2004)]. The present paper focuses on atomic compression base effect algebras and the consequences of atoms being foci (so-called projections) of the compressions in the compression base. Part of our work generalizes results obtained in atomic sequential effect algebras by Tkadlec [Int. J. Theor. Phys. 47, 185 (2008)]. The notion of projection-atomicity is introduced and studied, and several conditions that force a compression base effect algebra or the set of its projections to be Boolean are found. Finally, we apply some of these results to sequential effect algebras and strengthen a previously established result concerning a sufficient condition for them to be Boolean.
2013-01-01
Background Limited intrinsic healing potential of the meniscus and a strong correlation between meniscal injury and osteoarthritis have prompted investigation of surgical repair options, including the implantation of functional bioengineered constructs. Cell-based constructs appear promising, however the generation of meniscal constructs is complicated by the presence of diverse cell populations within this heterogeneous tissue and gaps in the information concerning their response to manipulation of oxygen tension during cell culture. Methods Four human lateral menisci were harvested from patients undergoing total knee replacement. Inner and outer meniscal fibrochondrocytes (MFCs) were expanded to passage 3 in growth medium supplemented with basic fibroblast growth factor (FGF-2), then embedded in porous collagen type I scaffolds and chondrogenically stimulated with transforming growth factor β3 (TGF-β3) under 21% (normal or normoxic) or 3% (hypoxic) oxygen tension for 21 days. Following scaffold culture, constructs were analyzed biochemically for glycosaminoglycan production, histologically for deposition of extracellular matrix (ECM), as well as at the molecular level for expression of characteristic mRNA transcripts. Results Constructs cultured under normal oxygen tension expressed higher levels of collagen type II (p = 0.05), aggrecan (p < 0.05) and cartilage oligomeric matrix protein, (COMP) (p < 0.05) compared to hypoxic expanded and cultured constructs. Accumulation of ECM rich in collagen type II and sulfated proteoglycan was evident in normoxic cultured scaffolds compared to those under low oxygen tension. There was no significant difference in expression of these genes between scaffolds seeded with MFCs isolated from inner or outer regions of the tissue following 21 days chondrogenic stimulation (p > 0.05). Conclusions Cells isolated from inner and outer regions of the human meniscus demonstrated equivalent differentiation potential
Partially-massless higher-spin algebras and their finite-dimensional truncations
NASA Astrophysics Data System (ADS)
Joung, Euihun; Mkrtchyan, Karapet
2016-01-01
The global symmetry algebras of partially-massless (PM) higher-spin (HS) fields in (A)dS d+1 are studied. The algebras involving PM generators up to depth 2 ( ℓ - 1) are defined as the maximal symmetries of free conformal scalar field with 2 ℓ order wave equation in d dimensions. We review the construction of these algebras by quotienting certain ideals in the universal enveloping algebra of ( A) dS d+1 isometries. We discuss another description in terms of Howe duality and derive the formula for computing trace in these algebras. This enables us to explicitly calculate the bilinear form for this one-parameter family of algebras. In particular, the bilinear form shows the appearance of additional ideal for any non-negative integer values of ℓ - d/2 , which coincides with the annihilator of the one-row ℓ-box Young diagram representation of s{o}_{d+2} . Hence, the corresponding finite-dimensional coset algebra spanned by massless and PM generators is equivalent to the symmetries of this representation.
Hecke-Clifford Algebras and Spin Hecke Algebras IV: Odd Double Affine Type
NASA Astrophysics Data System (ADS)
Khongsap, Ta; Wang, Weiqiang
2009-01-01
We introduce an odd double affine Hecke algebra (DaHa) generated by a classical Weyl group W and two skew-polynomial subalgebras of anticommuting generators. This algebra is shown to be Morita equivalent to another new DaHa which are generated by W and two polynomial-Clifford subalgebras. There is yet a third algebra containing a spin Weyl group algebra which is Morita (super)equivalent to the above two algebras. We establish the PBW properties and construct Verma-type representations via Dunkl operators for these algebras.
Single axioms for Boolean algebra.
McCune, W.
2000-06-30
Explicit single axioms are presented for Boolean algebra in terms of (1) the Sheffer stroke; (2) disjunction and negation; (3) disjunction, conjunction, and negation; and (4) disjunction, conjunction, negation, 0, and 1. It was previously known that single axioms exist for these systems, but the procedures to generate them are exponential, producing huge equations. Automated deduction techniques were applied to find axioms of lengths 105, 131, 111, and 127, respectively, each with six variables.
The algebras of large N matrix mechanics
Halpern, M.B.; Schwartz, C.
1999-09-16
Extending early work, we formulate the large N matrix mechanics of general bosonic, fermionic and supersymmetric matrix models, including Matrix theory: The Hamiltonian framework of large N matrix mechanics provides a natural setting in which to study the algebras of the large N limit, including (reduced) Lie algebras, (reduced) supersymmetry algebras and free algebras. We find in particular a broad array of new free algebras which we call symmetric Cuntz algebras, interacting symmetric Cuntz algebras, symmetric Bose/Fermi/Cuntz algebras and symmetric Cuntz superalgebras, and we discuss the role of these algebras in solving the large N theory. Most important, the interacting Cuntz algebras are associated to a set of new (hidden!) local quantities which are generically conserved only at large N. A number of other new large N phenomena are also observed, including the intrinsic nonlocality of the (reduced) trace class operators of the theory and a closely related large N field identification phenomenon which is associated to another set (this time nonlocal) of new conserved quantities at large N.
Alternative algebraic approaches in quantum chemistry
Mezey, Paul G.
2015-01-22
Various algebraic approaches of quantum chemistry all follow a common principle: the fundamental properties and interrelations providing the most essential features of a quantum chemical representation of a molecule or a chemical process, such as a reaction, can always be described by algebraic methods. Whereas such algebraic methods often provide precise, even numerical answers, nevertheless their main role is to give a framework that can be elaborated and converted into computational methods by involving alternative mathematical techniques, subject to the constraints and directions provided by algebra. In general, algebra describes sets of interrelations, often phrased in terms of algebraic operations, without much concern with the actual entities exhibiting these interrelations. However, in many instances, the very realizations of two, seemingly unrelated algebraic structures by actual quantum chemical entities or properties play additional roles, and unexpected connections between different algebraic structures are often giving new insight. Here we shall be concerned with two alternative algebraic structures: the fundamental group of reaction mechanisms, based on the energy-dependent topology of potential energy surfaces, and the interrelations among point symmetry groups for various distorted nuclear arrangements of molecules. These two, distinct algebraic structures provide interesting interrelations, which can be exploited in actual studies of molecular conformational and reaction processes. Two relevant theorems will be discussed.
NASA Astrophysics Data System (ADS)
Riechers, Dominik A.; Carilli, Chris Luke; Capak, Peter L.; COSMOS, HerMES
2016-01-01
Cold molecular and atomic gas plays a central role in our understanding of early galaxy formation and evolution. It represents the material that stars form out of, and its mass, distribution, excitation, and dynamics provide crucial insight into the physical processes that support the ongoing star formation and stellar mass buildup. We present some of the most recent progress in studies of gas-rich galaxies out to the highest redshifts through detailed investigations of the cold gas and dust with the most powerful facilities, i.e., the Karl G. Jansky Very Large Array (VLA), the NOrthern Extended Millimeter Array (NOEMA) and the Atacama Large (sub-) Millimeter Array (ALMA). Facilitating the impressive sensitivity of ALMA, this investigation encompasses a systematic study of the star-forming interstellar medium, gas dynamics, and dust obscuration in massive dusty starbursts and (much less luminous and massive) "typical" galaxies at such early epochs. These new results show that "typical" z>5 galaxies are significantly metal-enriched, but not heavily dust-obscured, consistent with a decreasing contribution of dust-obscured star formation to the star formation history of the universe towards the earliest cosmic epochs.
Shneider, Neil A.; Mentis, George Z.; Schustak, Joshua; O’Donovan, Michael J.
2009-01-01
Summary The mechanisms controlling the formation of synaptic connections between muscle spindle afferents and spinal motor neurons are believed to be regulated by factors originating from muscle spindles. Here, we find that the connections form with appropriate specificity in mice with abnormal spindle development caused by the conditional elimination of the neuregulin1 receptor ErbB2 from muscle precursors. However, despite a modest (~30%) decrease in the number of afferent terminals on motor neuron somata, the amplitude of afferent-evoked synaptic potentials recorded in motor neurons was reduced by ~80%, suggesting that many of the connections that form are functionally silent. The selective elimination of neurotrophin 3 (NT3) from muscle spindles had no effect on the amplitude of afferent-evoked ventral root potentials until the second postnatal week, revealing a late role for spindle-derived NT3 in the functional maintenance of the connections. These findings indicate that spindle-derived factors regulate the strength of the connections, but not their initial formation or their specificity. PMID:19369542
Block algebra in two-component BKP and D type Drinfeld-Sokolov hierarchies
Li, Chuanzhong He, Jingsong
2013-11-15
We construct generalized additional symmetries of a two-component BKP hierarchy defined by two pseudo-differential Lax operators. These additional symmetry flows form a Block type algebra with some modified (or additional) terms because of a B type reduction condition of this integrable hierarchy. Further we show that the D type Drinfeld-Sokolov hierarchy, which is a reduction of the two-component BKP hierarchy, possess a complete Block type additional symmetry algebra. That D type Drinfeld-Sokolov hierarchy has a similar algebraic structure as the bigraded Toda hierarchy which is a differential-discrete integrable system.
Computer algebra and transport theory.
Warsa, J. S.
2004-01-01
Modern symbolic algebra computer software augments and complements more traditional approaches to transport theory applications in several ways. The first area is in the development and enhancement of numerical solution methods for solving the Boltzmann transport equation. Typically, special purpose computer codes are designed and written to solve specific transport problems in particular ways. Different aspects of the code are often written from scratch and the pitfalls of developing complex computer codes are numerous and well known. Software such as MAPLE and MATLAB can be used to prototype, analyze, verify and determine the suitability of numerical solution methods before a full-scale transport application is written. Once it is written, the relevant pieces of the full-scale code can be verified using the same tools I that were developed for prototyping. Another area is in the analysis of numerical solution methods or the calculation of theoretical results that might otherwise be difficult or intractable. Algebraic manipulations are done easily and without error and the software also provides a framework for any additional numerical calculations that might be needed to complete the analysis. We will discuss several applications in which we have extensively used MAPLE and MATLAB in our work. All of them involve numerical solutions of the S{sub N} transport equation. These applications encompass both of the two main areas in which we have found computer algebra software essential.
BLAS- BASIC LINEAR ALGEBRA SUBPROGRAMS
NASA Technical Reports Server (NTRS)
Krogh, F. T.
1994-01-01
The Basic Linear Algebra Subprogram (BLAS) library is a collection of FORTRAN callable routines for employing standard techniques in performing the basic operations of numerical linear algebra. The BLAS library was developed to provide a portable and efficient source of basic operations for designers of programs involving linear algebraic computations. The subprograms available in the library cover the operations of dot product, multiplication of a scalar and a vector, vector plus a scalar times a vector, Givens transformation, modified Givens transformation, copy, swap, Euclidean norm, sum of magnitudes, and location of the largest magnitude element. Since these subprograms are to be used in an ANSI FORTRAN context, the cases of single precision, double precision, and complex data are provided for. All of the subprograms have been thoroughly tested and produce consistent results even when transported from machine to machine. BLAS contains Assembler versions and FORTRAN test code for any of the following compilers: Lahey F77L, Microsoft FORTRAN, or IBM Professional FORTRAN. It requires the Microsoft Macro Assembler and a math co-processor. The PC implementation allows individual arrays of over 64K. The BLAS library was developed in 1979. The PC version was made available in 1986 and updated in 1988.
Commutation of Projections and Characterization of Traces on von Neumann Algebras. III
NASA Astrophysics Data System (ADS)
Bikchentaev, A. M.
2015-12-01
We obtain new necessary and sufficient commutation conditions for nonnegative operators and projections in terms of operator inequalities. It is shown that in the general case in this inequalities the projections cannot be replaced by arbitrary nonnegative operators with preservation of operators commutativity. We also present new necessary and sufficient commutation conditions for projections in terms of operator inequalities. These inequalities are applied for trace characterization on von Neumann algebras in the class of all positive normal functionals. We also consider the following problems: I. Characterization of traces among arbitrary weights on von Neumann algebras. II. Characterization of tracial functionals among all positive linear functionals on C ∗-algebras. III. Characterization of commutativity for C ∗-algebras.
Kogan, A Kh; Losev, N I; Biriukov, Iu V; Sandrikov, V A; Tsypin, A B; Al'-Khadidi, M; Manuĭlov, B M; Syrkin, A L; Krotovskiĭ, G S; Pogromov, A P
1991-01-01
Studies conducted in the clinic (in patients with cardiac diseases) and experiments (performed on intact dogs) by means of the hemiluminescent method and the nitroblue tetrazolium test showed that the lungs, in distinction to other organs (heart and others), have a stimulating effect on the generation of active oxygen forms (AOF) by the leukocytes. In this way the lungs may probably play a double role in the organism: potentiate its defence (by intensifying the microbicidal activity of the phagocytes) and facilitate damage (by secretion of AOF by the phagocytes beyond them--into the tissues); the resultant effect depends on the balance of these two types of action. In carcinoma of the lung the stimulating effect of its involved lobe (part) on the leukocytes diminishes. PMID:2057236
Boundary algebras and Kac modules for logarithmic minimal models
NASA Astrophysics Data System (ADS)
Morin-Duchesne, Alexi; Rasmussen, Jørgen; Ridout, David
2015-10-01
Virasoro Kac modules were originally introduced indirectly as representations whose characters arise in the continuum scaling limits of certain transfer matrices in logarithmic minimal models, described using Temperley-Lieb algebras. The lattice transfer operators include seams on the boundary that use Wenzl-Jones projectors. If the projectors are singular, the original prescription is to select a subspace of the Temperley-Lieb modules on which the action of the transfer operators is non-singular. However, this prescription does not, in general, yield representations of the Temperley-Lieb algebras and the Virasoro Kac modules have remained largely unidentified. Here, we introduce the appropriate algebraic framework for the lattice analysis as a quotient of the one-boundary Temperley-Lieb algebra. The corresponding standard modules are introduced and examined using invariant bilinear forms and their Gram determinants. The structures of the Virasoro Kac modules are inferred from these results and are found to be given by finitely generated submodules of Feigin-Fuchs modules. Additional evidence for this identification is obtained by comparing the formalism of lattice fusion with the fusion rules of the Virasoro Kac modules. These are obtained, at the character level, in complete generality by applying a Verlinde-like formula and, at the module level, in many explicit examples by applying the Nahm-Gaberdiel-Kausch fusion algorithm.
Lax operator algebras and integrable systems
NASA Astrophysics Data System (ADS)
Sheinman, O. K.
2016-02-01
A new class of infinite-dimensional Lie algebras, called Lax operator algebras, is presented, along with a related unifying approach to finite-dimensional integrable systems with a spectral parameter on a Riemann surface such as the Calogero-Moser and Hitchin systems. In particular, the approach includes (non-twisted) Kac-Moody algebras and integrable systems with a rational spectral parameter. The presentation is based on quite simple ideas about the use of gradings of semisimple Lie algebras and their interaction with the Riemann-Roch theorem. The basic properties of Lax operator algebras and the basic facts about the theory of the integrable systems in question are treated (and proved) from this general point of view. In particular, the existence of commutative hierarchies and their Hamiltonian properties are considered. The paper concludes with an application of Lax operator algebras to prequantization of finite-dimensional integrable systems. Bibliography: 51 titles.
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.…
Algebraic structure of general electromagnetic fields and energy flow
Hacyan, Shahen
2011-08-15
Highlights: > Algebraic structure of general electromagnetic fields in stationary spacetime. > Eigenvalues and eigenvectors of the electomagnetic field tensor. > Energy-momentum in terms of eigenvectors and Killing vector. > Explicit form of reference frame with vanishing Poynting vector. > Application of formalism to Bessel beams. - Abstract: The algebraic structures of a general electromagnetic field and its energy-momentum tensor in a stationary space-time are analyzed. The explicit form of the reference frame in which the energy of the field appears at rest is obtained in terms of the eigenvectors of the electromagnetic tensor and the existing Killing vector. The case of a stationary electromagnetic field is also studied and a comparison is made with the standard short-wave approximation. The results can be applied to the general case of a structured light beams, in flat or curved spaces. Bessel beams are worked out as example.
Non-Abelian gerbes and enhanced Leibniz algebras
NASA Astrophysics Data System (ADS)
Strobl, Thomas
2016-07-01
We present the most general gauge-invariant action functional for coupled 1- and 2-form gauge fields with kinetic terms in generic dimensions, i.e., dropping eventual contributions that can be added in particular space-time dimensions only such as higher Chern-Simons terms. After appropriate field redefinitions it coincides with a truncation of the Samtleben-Szegin-Wimmer action. In the process one sees explicitly how the existence of a gauge-invariant functional enforces that the most general semistrict Lie 2-algebra describing the bundle of a non-Abelian gerbe gets reduced to a very particular structure, which, after the field redefinition, can be identified with the one of an enhanced Leibniz algebra. This is the first step towards a systematic construction of such functionals for higher gauge theories, with kinetic terms for a tower of gauge fields up to some highest form degree p , solved here for p =2 .
Very extended Kac Moody algebras and their interpretation at low levels
NASA Astrophysics Data System (ADS)
Kleinschmidt, Axel; Schnakenburg, Igor; West, Peter
2004-05-01
We analyse the very extended Kac Moody algebras as representations in terms of certain Ad-1 subalgebras and find the generators at low levels. Our results for low levels agree precisely with the bosonic field content of the theories containing gravity, forms and scalars which upon reduction to three dimensions can be described by a nonlinear realization. We explain how the Dynkin diagrams of the very extended algebras encode information about the field content and generalized T-duality transformations.
Quantum Algebra Symmetry of the ASEP with Second-Class Particles
NASA Astrophysics Data System (ADS)
Belitsky, V.; Schütz, G. M.
2015-11-01
We consider a two-component asymmetric simple exclusion process (ASEP) on a finite lattice with reflecting boundary conditions. For this process, which is equivalent to the ASEP with second-class particles, we construct the representation matrices of the quantum algebra U_q[{gl}(3)] that commute with the generator. As a byproduct we prove reversibility and obtain in explicit form the reversible measure. A review of the algebraic techniques used in the proofs is given.
Pauli theorem in the description of n-dimensional spinors in the Clifford algebra formalism
NASA Astrophysics Data System (ADS)
Shirokov, D. S.
2013-04-01
We discuss a generalized Pauli theorem and its possible applications for describing n-dimensional (Dirac, Weyl, Majorana, and Majorana-Weyl) spinors in the Clifford algebra formalism. We give the explicit form of elements that realize generalizations of Dirac, charge, and Majorana conjugations in the case of arbitrary space dimensions and signatures, using the notion of the Clifford algebra additional signature to describe conjugations. We show that the additional signature can take only certain values despite its dependence on the matrix representation
Coverings of topological semi-abelian algebras
NASA Astrophysics Data System (ADS)
Mucuk, Osman; Demir, Serap
2016-08-01
In this work, we study on a category of topological semi-abelian algebras which are topological models of given an algebraic theory T whose category of models is semi-abelian; and investigate some results on the coverings of topological models of such theories yielding semi-abelian categories. We also consider the internal groupoid structure in the semi-abelian category of T-algebras, and give a criteria for the lifting of internal groupoid structure to the covering groupoids.
Stability of algebraically unstable dispersive flows
NASA Astrophysics Data System (ADS)
King, Kristina; Zaretzky, Paula; Weinstein, Steven; Cromer, Michael; Barlow, Nathaniel
2015-11-01
A widely unexplored type of hydrodynamic instability is examined - large-time algebraic growth. Such growth occurs on the threshold of (exponentially) neutral stability. A methodology is provided for predicting the algebraic growth rate of an initial disturbance, when applied to a class of partial differential equations describing wave propagation in dispersive media. There are several morphological differences between algebraically growing disturbances and the exponentially growing wave packets inherent to classical linear stability analysis, and these are elucidated in this study.
Explicit travelling waves and invariant algebraic curves
NASA Astrophysics Data System (ADS)
Gasull, Armengol; Giacomini, Hector
2015-06-01
We introduce a precise definition of algebraic travelling wave solution of n-th order partial differential equations and prove that the only algebraic travelling waves solutions for the celebrated Fisher-Kolmogorov equation are the ones found in 1979 by Ablowitz and Zeppetella. This question is equivalent to study when an associated one-parameter family of planar ordinary differential systems has invariant algebraic curves.
Finite-dimensional simple graded algebras
Bahturin, Yu A; Zaicev, M V; Sehgal, S K
2008-08-31
Let R be a finite-dimensional algebra over an algebraically closed field F graded by an arbitrary group G. In the paper it is proved that if the characteristic of F is zero or does not divide the order of any finite subgroup of G, then R is graded simple if and only if it is isomorphic to a matrix algebra over a finite-dimensional graded skew field. Bibliography: 24 titles.
NASA Astrophysics Data System (ADS)
Manerowska, Anna; Nieznański, Edward; Mulawka, Jan
2013-10-01
Our aim is to present the algebra of concepts in two formal languages. First, after introducing a primary relation between concepts, which is subsumption, we shall specify in a language that uses quantifiers, the Boolean algebra of general concepts. Next, we shall note down the same algebra in simplified non-quantifying language, in order to use it as basis for two specific implementations, i.e. to create the Boolean algebras of deontic concepts and axiological concepts.
Representations of Super Yang-Mills Algebras
NASA Astrophysics Data System (ADS)
Herscovich, Estanislao
2013-06-01
We study in this article the representation theory of a family of super algebras, called the super Yang-Mills algebras, by exploiting the Kirillov orbit method à la Dixmier for nilpotent super Lie algebras. These super algebras are an extension of the so-called Yang-Mills algebras, introduced by A. Connes and M. Dubois-Violette in (Lett Math Phys 61(2):149-158, 2002), and in fact they appear as a "background independent" formulation of supersymmetric gauge theory considered in physics, in a similar way as Yang-Mills algebras do the same for the usual gauge theory. Our main result states that, under certain hypotheses, all Clifford-Weyl super algebras {{Cliff}q(k) ⊗ Ap(k)}, for p ≥ 3, or p = 2 and q ≥ 2, appear as a quotient of all super Yang-Mills algebras, for n ≥ 3 and s ≥ 1. This provides thus a family of representations of the super Yang-Mills algebras.
Difficulties in initial algebra learning in Indonesia
NASA Astrophysics Data System (ADS)
Jupri, Al; Drijvers, Paul; van den Heuvel-Panhuizen, Marja
2014-12-01
Within mathematics curricula, algebra has been widely recognized as one of the most difficult topics, which leads to learning difficulties worldwide. In Indonesia, algebra performance is an important issue. In the Trends in International Mathematics and Science Study (TIMSS) 2007, Indonesian students' achievement in the algebra domain was significantly below the average student performance in other Southeast Asian countries such as Thailand, Malaysia, and Singapore. This fact gave rise to this study which aims to investigate Indonesian students' difficulties in algebra. In order to do so, a literature study was carried out on students' difficulties in initial algebra. Next, an individual written test on algebra tasks was administered, followed by interviews. A sample of 51 grade VII Indonesian students worked the written test, and 37 of them were interviewed afterwards. Data analysis revealed that mathematization, i.e., the ability to translate back and forth between the world of the problem situation and the world of mathematics and to reorganize the mathematical system itself, constituted the most frequently observed difficulty in both the written test and the interview data. Other observed difficulties concerned understanding algebraic expressions, applying arithmetic operations in numerical and algebraic expressions, understanding the different meanings of the equal sign, and understanding variables. The consequences of these findings on both task design and further research in algebra education are discussed.
Multicloning and Multibroadcasting in Operator Algebras
NASA Astrophysics Data System (ADS)
Kaniowski, Krzysztof; Lubnauer, Katarzyna; Łuczak, Andrzej
2015-12-01
We investigate multicloning and multibroadcasting in the general operator algebra framework in arbitrary dimension, generalizing thus results obtained in this framework for simple cloning and broadcasting.
On Realization of Generalized Effect Algebras
NASA Astrophysics Data System (ADS)
Paseka, Jan
2012-12-01
A well-known fact is that there is a finite orthomodular lattice with an order determining set of states which is not representable in the standard quantum logic, the lattice L(H) of all closed subspaces of a separable complex Hilbert space. We show that a generalized effect algebra is representable in the operator generalized effect algebra G(H) of effects of a complex Hilbert space H iff it has an order determining set of generalized states. This extends the corresponding results for effect algebras of Riečanová and Zajac. Further, any operator generalized effect algebra G(H) possesses an order determining set of generalized states.
Literal algebra for satellite dynamics. [perturbation analysis
NASA Technical Reports Server (NTRS)
Gaposchkin, E. M.
1975-01-01
A description of the rather general class of operations available is given and the operations are related to problems in satellite dynamics. The implementation of an algebra processor is discussed. The four main categories of symbol processors are related to list processing, string manipulation, symbol manipulation, and formula manipulation. Fundamental required operations for an algebra processor are considered. It is pointed out that algebra programs have been used for a number of problems in celestial mechanics with great success. The advantage of computer algebra is its accuracy and speed.
Banach Algebras Associated to Lax Pairs
NASA Astrophysics Data System (ADS)
Glazebrook, James F.
2015-04-01
Lax pairs featuring in the theory of integrable systems are known to be constructed from a commutative algebra of formal pseudodifferential operators known as the Burchnall- Chaundy algebra. Such pairs induce the well known KP flows on a restricted infinite-dimensional Grassmannian. The latter can be exhibited as a Banach homogeneous space constructed from a Banach *-algebra. It is shown that this commutative algebra of operators generating Lax pairs can be associated with a commutative C*-subalgebra in the C*-norm completion of the *-algebra. In relationship to the Bose-Fermi correspondence and the theory of vertex operators, this C*-algebra has an association with the CAR algebra of operators as represented on Fermionic Fock space by the Gelfand-Naimark-Segal construction. Instrumental is the Plücker embedding of the restricted Grassmannian into the projective space of the associated Hilbert space. The related Baker and tau-functions provide a connection between these two C*-algebras, following which their respective state spaces and Jordan-Lie-Banach algebras structures can be compared.
Type-Decomposition of an Effect Algebra
NASA Astrophysics Data System (ADS)
Foulis, David J.; Pulmannová, Sylvia
2010-10-01
Effect algebras (EAs), play a significant role in quantum logic, are featured in the theory of partially ordered Abelian groups, and generalize orthoalgebras, MV-algebras, orthomodular posets, orthomodular lattices, modular ortholattices, and boolean algebras. We study centrally orthocomplete effect algebras (COEAs), i.e., EAs satisfying the condition that every family of elements that is dominated by an orthogonal family of central elements has a supremum. For COEAs, we introduce a general notion of decomposition into types; prove that a COEA factors uniquely as a direct sum of types I, II, and III; and obtain a generalization for COEAs of Ramsay’s fourfold decomposition of a complete orthomodular lattice.
NASA Astrophysics Data System (ADS)
Chajda, Ivan
2014-10-01
Commutative BCI-algebras can be considered as semilattices whose sections are equipped with certain involutions. A similar view can be applied to commutative BCK-algebras. However, for general BCK-algebras a certain construction was settled by the author and J. Kühr (Miskolc Math. Notes 8:11-21, 2007) showing that they can be considered as structures essentially weaker than semilattices but still with certain involutions in sections. The aim of this paper is to involve a similar approach for BCI-algebras.
NASA Astrophysics Data System (ADS)
Korf, Lisa A.; Schroeck, Franklin E.
2015-12-01
We consider an effect algebra of phase space localization operators for a quantum mechanical Hilbert space that contains no non-trivial projections, and the C*-algebra generated by it. This C∗-algebra forms an informationally complete set in the original Hilbert space. Its elements are shown to have singular-value-based decompositions that permit their characterization in terms of limits of linear combinations of products of pairs of the phase space fuzzy localization operators. Through these results, it is shown that the informational completeness of the C*-algebra can be greatly reduced to the informational completeness of the set of products of pairs formed from the elements of the effect algebra.
ERIC Educational Resources Information Center
Ozgun-Koca, S. Ash
2010-01-01
Although growing numbers of secondary school mathematics teachers and students use calculators to study graphs, they mainly rely on paper-and-pencil when manipulating algebraic symbols. However, the Computer Algebra Systems (CAS) on computers or handheld calculators create new possibilities for teaching and learning algebraic manipulation. This…
Results of Using Algebra Tiles as Meaningful Representations of Algebra Concepts.
ERIC Educational Resources Information Center
Sharp, Janet M.
Mathematical meanings can be developed when individuals construct translations between algebra symbol systems and physical systems that represent one another. Previous research studies indicated (1) few high school students connect whole number manipulations to algebraic manipulations and (2) students who encounter algebraic ideas through…
Some C∗-algebras which are coronas of non-C∗-Banach algebras
NASA Astrophysics Data System (ADS)
Voiculescu, Dan-Virgil
2016-07-01
We present results and motivating problems in the study of commutants of hermitian n-tuples of Hilbert space operators modulo normed ideals. In particular, the C∗-algebras which arise in this context as coronas of non-C∗-Banach algebras, the connections with normed ideal perturbations of operators, the hyponormal operators and the bidual Banach algebras one encounters are discussed.
Leibniz algebras associated with some finite-dimensional representation of Diamond Lie algebra
NASA Astrophysics Data System (ADS)
Camacho, Luisa M.; Ladra, Manuel; Karimjanov, Iqboljon A.; Omirov, Bakhrom A.
2016-03-01
In this paper we classify Leibniz algebras whose associated Lie algebra is four-dimensional Diamond Lie algebra 𝕯 and the ideal generated by squares of elements is represented by one of the finite-dimensional indecomposable D-modules Un 1, Un 2 or Wn 1 or Wn 2.
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.
Commutative n-ary superalgebras with an invariant skew-symmetric form
NASA Astrophysics Data System (ADS)
Vishnyakova, E. G.
2015-12-01
We study n-ary commutative superalgebras and L∞-algebras that possess a skew-symmetric invariant form, using the derived bracket formalism. This class of superalgebras includes for instance Lie algebras and their n-ary generalizations, commutative associative and Jordan algebras with an invariant form. We give a classification of anti-commutative m-dimensional (m - 3) -ary algebras with an invariant form, and a classification of real simple m-dimensional Lie (m - 3) -algebras with a positive definite invariant form up to isometry. Furthermore, we develop the Hodge Theory for L∞-algebras with a symmetric invariant form, and we describe quasi-Frobenius structures on skew-symmetric n-ary algebras.
The Algebra of Lexical Semantics
NASA Astrophysics Data System (ADS)
Kornai, András
The current generative theory of the lexicon relies primarily on tools from formal language theory and mathematical logic. Here we describe how a different formal apparatus, taken from algebra and automata theory, resolves many of the known problems with the generative lexicon. We develop a finite state theory of word meaning based on machines in the sense of Eilenberg [11], a formalism capable of describing discrepancies between syntactic type (lexical category) and semantic type (number of arguments). This mechanism is compared both to the standard linguistic approaches and to the formalisms developed in AI/KR.
Strengthening Effect Algebras in a Logical Perspective: Heyting-Wajsberg Algebras
NASA Astrophysics Data System (ADS)
Konig, Martinvaldo
2014-10-01
Heyting effect algebras are lattice-ordered pseudoboolean effect algebras endowed with a pseudocomplementation that maps on the center (i.e. Boolean elements). They are the algebraic counterpart of an extension of both Łukasiewicz many-valued logic and intuitionistic logic. We show that Heyting effect algebras are termwise equivalent to Heyting-Wajsberg algebras where the two different logical implications are defined as primitive operators. We prove this logic to be decidable, to be strongly complete and to have the deduction-detachment theorem.
Automorphisms and Derivations of the Insertion-Elimination Algebra and Related Graded Lie Algebras
NASA Astrophysics Data System (ADS)
Ondrus, Matthew; Wiesner, Emilie
2016-07-01
This paper addresses several structural aspects of the insertion-elimination algebra {mathfrak{g}}, a Lie algebra that can be realized in terms of tree-inserting and tree-eliminating operations on the set of rooted trees. In particular, we determine the finite-dimensional subalgebras of {mathfrak{g}}, the automorphism group of {mathfrak{g}}, the derivation Lie algebra of {mathfrak{g}}, and a generating set. Several results are stated in terms of Lie algebras admitting a triangular decomposition and can be used to reproduce results for the generalized Virasoro algebras.
Realizations of conformal current-type Lie algebras
Pei Yufeng; Bai Chengming
2010-05-15
In this paper we obtain the realizations of some infinite-dimensional Lie algebras, named 'conformal current-type Lie algebras', in terms of a two-dimensional Novikov algebra and its deformations. Furthermore, Ovsienko and Roger's loop cotangent Virasoro algebra, which can be regarded as a nice generalization of the Virasoro algebra with two space variables, is naturally realized as an affinization of the tensor product of a deformation of the two-dimensional Novikov algebra and the Laurent polynomial algebra. These realizations shed new light on various aspects of the structure and representation theory of the corresponding infinite-dimensional Lie algebras.
Is Algebra Really Difficult for All Students?
ERIC Educational Resources Information Center
Egodawatte, Gunawardena
2009-01-01
Research studies have shown that students encounter difficulties in transitioning from arithmetic to algebra. Errors made by high school students were analyzed for patterns and their causes. The origins of errors were: intuitive assumptions, failure to understand the syntax of algebra, analogies with other familiar symbol systems such as the…
Some Applications of Algebraic System Solving
ERIC Educational Resources Information Center
Roanes-Lozano, Eugenio
2011-01-01
Technology and, in particular, computer algebra systems, allows us to change both the way we teach mathematics and the mathematical curriculum. Curiously enough, unlike what happens with linear system solving, algebraic system solving is not widely known. The aim of this paper is to show that, although the theory lying behind the "exact solve"…
A Technology-Intensive Approach to Algebra.
ERIC Educational Resources Information Center
Heid, M. Kathleen; Zbiek, Rose Mary
1995-01-01
Computer-Intensive Algebra (CIA) focuses on the use of technology to help develop a rich understanding of fundamental algebraic concepts in real-world settings using computing tools for easy access to numerical, graphical, and symbolic representations of mathematical ideas. (MKR)
An Inquiry-Based Linear Algebra Class
ERIC Educational Resources Information Center
Wang, Haohao; Posey, Lisa
2011-01-01
Linear algebra is a standard undergraduate mathematics course. This paper presents an overview of the design and implementation of an inquiry-based teaching material for the linear algebra course which emphasizes discovery learning, analytical thinking and individual creativity. The inquiry-based teaching material is designed to fit the needs of a…
Algebra in the Early Years? Yes!
ERIC Educational Resources Information Center
Taylor-Cox, Jennifer
2003-01-01
Suggests ways early years educators can begin teaching young children to think algebraically and prepare them for success in algebra. Focuses on ways to promote mathematical patterns, mathematical situations and structures, models of quantitative relationship, and change. Describes how first-graders used real object representations to better…
Algebraic Thinking: A Problem Solving Approach
ERIC Educational Resources Information Center
Windsor, Will
2010-01-01
Algebraic thinking is a crucial and fundamental element of mathematical thinking and reasoning. It initially involves recognising patterns and general mathematical relationships among numbers, objects and geometric shapes. This paper will highlight how the ability to think algebraically might support a deeper and more useful knowledge, not only of…
New directions in algebraic dynamical systems
NASA Astrophysics Data System (ADS)
Schmidt, Klaus; Verbitskiy, Evgeny
2011-02-01
The logarithmic Mahler measure of certain multivariate polynomials occurs frequently as the entropy or the free energy of solvable lattice models (especially dimer models). It is also known that the entropy of an algebraic dynamical system is the logarithmic Mahler measure of the defining polynomial. The connection between the lattice models and the algebraic dynamical systems is still rather mysterious.
Low Performers Found Unready to Take Algebra
ERIC Educational Resources Information Center
Cavanagh, Sean
2008-01-01
As state and school leaders across the country push to have more students take algebra in 8th grade, a new study argues that middle schoolers struggling the most in math are being enrolled in that course despite being woefully unprepared. "The Misplaced Math Student: Lost in Eighth Grade Algebra," scheduled for release by the Brookings Institution…
An algebraic approach to the scattering equations
NASA Astrophysics Data System (ADS)
Huang, Rijun; Rao, Junjie; Feng, Bo; He, Yang-Hui
2015-12-01
We employ the so-called companion matrix method from computational algebraic geometry, tailored for zero-dimensional ideals, to study the scattering equations. The method renders the CHY-integrand of scattering amplitudes computable using simple linear algebra and is amenable to an algorithmic approach. Certain identities in the amplitudes as well as rationality of the final integrand become immediate in this formalism.
Calif. Laws Shift Gears on Algebra, Textbooks
ERIC Educational Resources Information Center
Robelen, Erik W.
2012-01-01
New laws in California have set the state on a course for some potentially significant changes to the curriculum, including a measure that revisits the matter of teaching Algebra 1 in 8th grade and another that revamps the state's textbook-adoption process and hands districts greater leeway in choosing instructional materials. The algebra-related…
Success in Algebra among Community College Students
ERIC Educational Resources Information Center
Reyes, Czarina
2010-01-01
College algebra is a required course for most majors, but is viewed by many as a gatekeeper course for degree completion by students. With almost half a million students taking college algebra each year, faculty are experimenting with new course lengths of time that might result in higher success, completion, and retention rates for college…
Using the Internet To Investigate Algebra.
ERIC Educational Resources Information Center
Sherwood, Walter
The lesson plans in this book engage students by using a tool they enjoy--the Internet--to explore key concepts in algebra. Working either individually or in groups, students learn to approach algebra from a problem solving perspective. Each lesson shows learners how to use the Internet as a resource for gathering facts, data, and other…
Algebraic Geodesics on Three-Dimensional Quadrics
NASA Astrophysics Data System (ADS)
Kai, Yue
2015-12-01
By Hamilton-Jacobi method, we study the problem of algebraic geodesics on the third-order surface. By the implicit function theorem, we proved the existences of the real geodesics which are the intersections of two algebraic surfaces, and we also give some numerical examples.
Algebraic Formulas for Areas between Curves.
ERIC Educational Resources Information Center
Gabai, Hyman
1982-01-01
Korean secondary school students preparing for college learn about a simple algebraic formula for area bounded by a parabola and line. The approach does not seem well-known among American students. It is noted that, while the formula derivations rely on integration, algebra students could use the formulas without proofs. (MP)
Classical and quantum Kummer shape algebras
NASA Astrophysics Data System (ADS)
Odzijewicz, A.; Wawreniuk, E.
2016-07-01
We study a family of integrable systems of nonlinearly coupled harmonic oscillators on the classical and quantum levels. We show that the integrability of these systems follows from their symmetry characterized by algebras, here called Kummer shape algebras. The resolution of identity for a wide class of reproducing kernels is found. A number of examples, illustrating this theory, are also presented.
Young Mathematicians at Work: Constructing Algebra
ERIC Educational Resources Information Center
Fosnot, Catherine Twomey; Jacob, Bill
2010-01-01
This book provides a landscape of learning that helps teachers recognize, support, and celebrate their students' capacity to structure their worlds algebraically. It identifies the models, contexts, and landmarks that facilitate algebraic thinking in young students and provides insightful and practical methods for teachers, math supervisors, and…
Focus on Fractions to Scaffold Algebra
ERIC Educational Resources Information Center
Ooten, Cheryl Thomas
2013-01-01
Beginning algebra is a gatekeeper course into the pipeline to higher mathematics courses required for respected professions in engineering, science, statistics, mathematics, education, and technology. Beginning algebra can also be a perfect storm if the necessary foundational skills are not within a student's grasp. What skills ensure beginning…
Fourier theory and C∗-algebras
NASA Astrophysics Data System (ADS)
Bédos, Erik; Conti, Roberto
2016-07-01
We discuss a number of results concerning the Fourier series of elements in reduced twisted group C∗-algebras of discrete groups, and, more generally, in reduced crossed products associated to twisted actions of discrete groups on unital C∗-algebras. A major part of the article gives a review of our previous work on this topic, but some new results are also included.
Situated Learning in an Abstract Algebra Classroom
ERIC Educational Resources Information Center
Ticknor, Cindy S.
2012-01-01
Advisory committees of mathematics consider abstract algebra as an essential component of the mathematical preparation of secondary teachers, yet preservice teachers find it challenging to connect the topics addressed in this advanced course with the high school algebra they must someday teach. This study analyzed the mathematical content…
Teaching Algebra to Students with Learning Disabilities
ERIC Educational Resources Information Center
Impecoven-Lind, Linda S.; Foegen, Anne
2010-01-01
Algebra is a gateway to expanded opportunities, but it often poses difficulty for students with learning disabilities. Consequently, it is essential to identify evidence-based instructional strategies for these students. The authors begin by identifying three areas of algebra difficulty experienced by students with disabilities: cognitive…
Arithmetic and Cognitive Contributions to Algebra
ERIC Educational Resources Information Center
Cirino, Paul T.; Tolar, Tammy D.; Fuchs, Lynn S.
2013-01-01
Algebra is a prerequisite for access to STEM careers and occupational success (NMAP, 2008a), yet algebra is difficult for students through high school (US DOE, 2008). Growth in children's conceptual and procedural arithmetical knowledge is reciprocal, although conceptual knowledge has more impact on procedural knowledge than the reverse…
Just Say Yes to Early Algebra!
ERIC Educational Resources Information Center
Stephens, Ana; Blanton, Maria; Knuth, Eric; Isler, Isil; Gardiner, Angela Murphy
2015-01-01
Mathematics educators have argued for some time that elementary school students are capable of engaging in algebraic thinking and should be provided with rich opportunities to do so. Recent initiatives like the Common Core State Standards for Mathematics (CCSSM) (CCSSI 2010) have taken up this call by reiterating the place of early algebra in…
Fusion rule algebras from graph theory
NASA Astrophysics Data System (ADS)
Caselle, M.; Ponzano, G.
1989-06-01
We describe a new class of fusion algebras related to graph theory which bear intriguing connections with group algebras. The structure constants and the matrix S, which diagonalizes the fusion rules, are explicitly computed in terms of SU(2) coupling coefficients.
NINTH YEAR MATHEMATICS. COURSE I, ALGEBRA.
ERIC Educational Resources Information Center
New York State Education Dept., Albany.
THIS GUIDE OUTLINES THE MINIMUM MATERIAL FOR WHICH STUDENTS OF NINTH YEAR MATHEMATICS - COURSE 1 - ALGEBRA WERE HELD RESPONSIBLE ON THE REGENTS EXAMINATIONS BEGINNING IN JUNE, 1966. THE REPORT ALSO PRESENTS THE SCOPE AND CONTENT OF THE ALGEBRA COURSE AND POSSIBLE SUGGESTIONS FOR TEACHING THE MATERIAL TO STUDENTS. (RP)
Modern Algebra, Mathematics: 5293.36.
ERIC Educational Resources Information Center
Edwards, Raymond J.
This guidebook covers Boolean algebra, matrices, linear transformations of the plane, characteristic values, vectors, and algebraic structures. Overall course goals and performance objectives for each unit are specified; sequencing of units and various time schedules are suggested. A sample pretest and posttest are given, and an annotated list of…
The Structural Algebra Option: A Discussion Paper.
ERIC Educational Resources Information Center
Kirshner, David
The goal of this paper is to renew interest in the structural option to algebra instruction. Concern for the usual secondary school algebra curriculum related to simplifying expressions, solving equations, and rationalizing numerators and denominators is viewed from three pedagogical approaches: (1) structural approach, (2) empirical approach, and…
ERIC Educational Resources Information Center
Rickard, Caroline
2008-01-01
Shortly after starting work for the University of Chichester in the School of Teacher Education, the author was planning a session relating to algebra and found herself inspired by an article in MT182: "Algebraic Infants" by Andrews and Sayers (2003). Based on the making of families of "Multilink" animals, Andrews and Sayers (2003) claim that…
Teaching Modeling and Axiomatization with Boolean Algebra.
ERIC Educational Resources Information Center
De Villiers, Michael D.
1987-01-01
Presented is an alternative approach to the traditional teaching of Boolean algebra for secondary school mathematics. The main aim of the approach is to use Boolean algebra to teach pupils such mathematical processes as modeling and axiomatization. A course using the approach is described. (RH)
Loop realizations of quantum affine algebras
Cautis, Sabin; Licata, Anthony
2012-12-15
We give a simplified description of quantum affine algebras in their loop presentation. This description is related to Drinfeld's new realization via halves of vertex operators. We also define an idempotent version of the quantum affine algebra which is suitable for categorification.
Deforming the Maxwell-Sim algebra
Gibbons, G. W.; Gomis, Joaquim; Pope, C. N.
2010-09-15
The Maxwell algebra is a noncentral extension of the Poincare algebra, in which the momentum generators no longer commute, but satisfy [P{sub {mu}},P{sub {nu}}]=Z{sub {mu}{nu}}. The charges Z{sub {mu}{nu}} commute with the momenta, and transform tensorially under the action of the angular momentum generators. If one constructs an action for a massive particle, invariant under these symmetries, one finds that it satisfies the equations of motion of a charged particle interacting with a constant electromagnetic field via the Lorentz force. In this paper, we explore the analogous constructions where one starts instead with the ISim subalgebra of Poincare, this being the symmetry algebra of very special relativity. It admits an analogous noncentral extension, and we find that a particle action invariant under this Maxwell-Sim algebra again describes a particle subject to the ordinary Lorentz force. One can also deform the ISim algebra to DISim{sub b}, where b is a nontrivial dimensionless parameter. We find that the motion described by an action invariant under the corresponding Maxwell-DISim algebra is that of a particle interacting via a Finslerian modification of the Lorentz force. In an appendix is it shown that the DISim{sub b} algebra is isomorphic to the extended Schroedinger algebra with its standard deformation parameter z, when b=(1/1-z).
MODEL IDENTIFICATION AND COMPUTER ALGEBRA.
Bollen, Kenneth A; Bauldry, Shawn
2010-10-01
Multiequation models that contain observed or latent variables are common in the social sciences. To determine whether unique parameter values exist for such models, one needs to assess model identification. In practice analysts rely on empirical checks that evaluate the singularity of the information matrix evaluated at sample estimates of parameters. The discrepancy between estimates and population values, the limitations of numerical assessments of ranks, and the difference between local and global identification make this practice less than perfect. In this paper we outline how to use computer algebra systems (CAS) to determine the local and global identification of multiequation models with or without latent variables. We demonstrate a symbolic CAS approach to local identification and develop a CAS approach to obtain explicit algebraic solutions for each of the model parameters. We illustrate the procedures with several examples, including a new proof of the identification of a model for handling missing data using auxiliary variables. We present an identification procedure for Structural Equation Models that makes use of CAS and that is a useful complement to current methods. PMID:21769158
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.
Algebraic Apect of Helicities in Hadron Physics
NASA Astrophysics Data System (ADS)
An, Murat; Ji, Chueng
2015-04-01
We examined the relation of polarization vectors and spinors of (1 , 0) ⊕(0 , 1) representation of Lorentz group in Clifford algebra Cl1 , 3 , their relation with standard algebra, and properties of these spinors. Cl1 , 3 consists of different grades:e.g. the first and the second grades represent (1 / 2 , 1 / 2) and (1 , 0) ⊕(0 , 1) representation of spin groups respectively with 4 and 6 components. However, these Clifford numbers are not the helicity eigenstates and thus we transform them into combinations of helicity eigenstates by expressing them as spherical harmonics. We relate the spin-one polarization vectors and (1 , 0) ⊕(0 , 1) spinors under one simple transformation with the spin operators. We also link our work with Winnberg's work of a superfield of a spinors of Clifford algebra by giving a physical meaning to Grassmann variables and discuss how Grassman algebra is linked with Clifford algebra.
Spinon-phonon interaction in algebraic spin liquids
NASA Astrophysics Data System (ADS)
Serbyn, Maksym; Lee, Patrick A.
2013-05-01
Motivated by a search for experimental probes to access the physics of fractionalized excitations called spinons in spin liquids, we study the interaction of spinons with lattice vibrations. We consider the case of algebraic spin liquid, when spinons have fermionic statistics and a Dirac-like dispersion. We establish the general procedure for deriving spinon-phonon interactions, which is based on symmetry considerations. The procedure is illustrated for four different algebraic spin liquids: π-flux and staggered-flux phases on a square lattice, π-flux phase on a kagome lattice, and zero-flux phase on a honeycomb lattice. Although the low-energy description is similar for all these phases, different underlying symmetry groups lead to a distinct form of spinon-phonon interaction Hamiltonian. The explicit form of the spinon-phonon interaction is used to estimate the attenuation of ultrasound in an algebraic spin liquid. The prospects of the sound attenuation as a probe of spinons are discussed.
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.
Approximating smooth functions using algebraic-trigonometric polynomials
NASA Astrophysics Data System (ADS)
Sharapudinov, Idris I.
2011-01-01
The problem under consideration is that of approximating classes of smooth functions by algebraic-trigonometric polynomials of the form p_n(t)+\\tau_m(t), where p_n(t) is an algebraic polynomial of degree n and \\tau_m(t)=a_0+\\sum_{k=1}^ma_k\\cos k\\pi t+b_k\\sin k\\pi t is a trigonometric polynomial of order m. The precise order of approximation by such polynomials in the classes W^r_\\infty(M) and an upper bound for similar approximations in the class W^r_p(M) with \\frac43 are found. The proof of these estimates uses mixed series in Legendre polynomials which the author has introduced and investigated previously. Bibliography: 13 titles.
Bicervical Normal Uterus with Normal Vagina
Okeke, CE; Anele, TI; Onyejelam, CC
2014-01-01
This is a report of the form of uterine anomaly involving a dual cervical canal in a side-by-side disposition with normal uterine cavity and normal vagina. It portrays a form of congenital uterine anomaly not explicable by the existing classical theory of mullerian anomalies. However, there has been a proposed reclassification of mullerian anomalies, which includes this type of anomaly under Type IIIc. The patient was a 31-year-old woman being managed for “secondary infertility.” To report a case of uterine anomaly that is not explicable by the existing classical theory of mullerian anomalies. To the best of our knowledge, only few cases of bicervical normal uterus with normal vagina exist in the literature; one of the cases had an anterior-posterior disposition. This form of uterine abnormality is not explicable by the existing classical theory of mullerian anomalies and suggests that a complex interplay of events beyond the classical postulate gives rise to the female genital tract. PMID:25364608
Teaching Linear Algebra: Proceeding More Efficiently by Staying Comfortably within Z
ERIC Educational Resources Information Center
Beaver, Scott
2015-01-01
For efficiency in a linear algebra course the instructor may wish to avoid the undue arithmetical distractions of rational arithmetic. In this paper we explore how to write fraction-free problems of various types including elimination, matrix inverses, orthogonality, and the (non-normalizing) Gram-Schmidt process.
NASA Astrophysics Data System (ADS)
Pramanik, Souvik; Ghosh, Subir
2013-10-01
We have developed a unified scheme for studying noncommutative algebras based on generalized uncertainty principle (GUP) and Snyder form in a relativistically covariant point particle Lagrangian (or symplectic) framework. Even though the GUP-based algebra and Snyder algebra are very distinct, the more involved latter algebra emerges from an approximation of the Lagrangian model of the former algebra. Deformed Poincaré generators for the systems that keep space-time symmetries of the relativistic particle models have been studied thoroughly. From a purely constrained dynamical analysis perspective the models studied here are very rich and provide insights on how to consistently construct approximate models from the exact ones when nonlinear constraints are present in the system. We also study dynamics of the GUP particle in presence of external electromagnetic field.
NASA Astrophysics Data System (ADS)
Pramanik, Souvik; Ghosh, Subir
2013-08-01
We have developed a unified scheme for studying noncommutative algebras based on generalized uncertainty principle (GUP) and Snyder form in a relativistically covariant point particle Lagrangian (or symplectic) framework. Even though the GUP-based algebra and Snyder algebra are very distinct, the more involved latter algebra emerges from an approximation of the Lagrangian model of the former algebra. Deformed Poincaré generators for the systems that keep space-time symmetries of the relativistic particle models have been studied thoroughly. From a purely constrained dynamical analysis perspective the models studied here are very rich and provide insights on how to consistently construct approximate models from the exact ones when nonlinear constraints are present in the system. We also study dynamics of the GUP particle in presence of external electromagnetic field.
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.
NASA Technical Reports Server (NTRS)
Crutcher, H. L.; Falls, L. W.
1976-01-01
Sets of experimentally determined or routinely observed data provide information about the past, present and, hopefully, future sets of similarly produced data. An infinite set of statistical models exists which may be used to describe the data sets. The normal distribution is one model. If it serves at all, it serves well. If a data set, or a transformation of the set, representative of a larger population can be described by the normal distribution, then valid statistical inferences can be drawn. There are several tests which may be applied to a data set to determine whether the univariate normal model adequately describes the set. The chi-square test based on Pearson's work in the late nineteenth and early twentieth centuries is often used. Like all tests, it has some weaknesses which are discussed in elementary texts. Extension of the chi-square test to the multivariate normal model is provided. Tables and graphs permit easier application of the test in the higher dimensions. Several examples, using recorded data, illustrate the procedures. Tests of maximum absolute differences, mean sum of squares of residuals, runs and changes of sign are included in these tests. Dimensions one through five with selected sample sizes 11 to 101 are used to illustrate the statistical tests developed.
PC Basic Linear Algebra Subroutines
1992-03-09
PC-BLAS is a highly optimized version of the Basic Linear Algebra Subprograms (BLAS), a standardized set of thirty-eight routines that perform low-level operations on vectors of numbers in single and double-precision real and complex arithmetic. Routines are included to find the index of the largest component of a vector, apply a Givens or modified Givens rotation, multiply a vector by a constant, determine the Euclidean length, perform a dot product, swap and copy vectors, andmore » find the norm of a vector. The BLAS have been carefully written to minimize numerical problems such as loss of precision and underflow and are designed so that the computation is independent of the interface with the calling program. This independence is achieved through judicious use of Assembly language macros. Interfaces are provided for Lahey Fortran 77, Microsoft Fortran 77, and Ryan-McFarland IBM Professional Fortran.« less
NASA Astrophysics Data System (ADS)
Gainutdinov, A. M.; Read, N.; Saleur, H.
2016-01-01
We develop in this paper the principles of an associative algebraic approach to bulk logarithmic conformal field theories (LCFTs). We concentrate on the closed {gl(1|1)} spin-chain and its continuum limit—the {c=-2} symplectic fermions theory—and rely on two technical companion papers, Gainutdinov et al. (Nucl Phys B 871:245-288, 2013) and Gainutdinov et al. (Nucl Phys B 871:289-329, 2013). Our main result is that the algebra of local Hamiltonians, the Jones-Temperley-Lieb algebra JTL N , goes over in the continuum limit to a bigger algebra than {V}, the product of the left and right Virasoro algebras. This algebra, {S}—which we call interchiral, mixes the left and right moving sectors, and is generated, in the symplectic fermions case, by the additional field {S(z,bar{z})≡ S_{αβ} ψ^α(z)bar{ψ}^β(bar{z})}, with a symmetric form {S_{αβ}} and conformal weights (1,1). We discuss in detail how the space of states of the LCFT (technically, a Krein space) decomposes onto representations of this algebra, and how this decomposition is related with properties of the finite spin-chain. We show that there is a complete correspondence between algebraic properties of finite periodic spin chains and the continuum limit. An important technical aspect of our analysis involves the fundamental new observation that the action of JTL N in the {gl(1|1)} spin chain is in fact isomorphic to an enveloping algebra of a certain Lie algebra, itself a non semi-simple version of {sp_{N-2}}. The semi-simple part of JTL N is represented by {U sp_{N-2}}, providing a beautiful example of a classical Howe duality, for which we have a non semi-simple version in the full JTL N image represented in the spin-chain. On the continuum side, simple modules over {S} are identified with "fundamental" representations of {sp_∞}.
Weak homological dimensions and biflat Koethe algebras
Pirkovskii, A Yu
2008-06-30
The homological properties of metrizable Koethe algebras {lambda}(P) are studied. A criterion for an algebra A={lambda}(P) to be biflat in terms of the Koethe set P is obtained, which implies, in particular, that for such algebras the properties of being biprojective, biflat, and flat on the left are equivalent to the surjectivity of the multiplication operator A otimes-hat A{yields}A. The weak homological dimensions (the weak global dimension w.dg and the weak bidimension w.db) of biflat Koethe algebras are calculated. Namely, it is shown that the conditions w.db {lambda}(P)<=1 and w.dg {lambda}(P)<=1 are equivalent to the nuclearity of {lambda}(P); and if {lambda}(P) is non-nuclear, then w.dg {lambda}(P)=w.db {lambda}(P)=2. It is established that the nuclearity of a biflat Koethe algebra {lambda}(P), under certain additional conditions on the Koethe set P, implies the stronger estimate db {lambda}(P), where db is the (projective) bidimension. On the other hand, an example is constructed of a nuclear biflat Koethe algebra {lambda}(P) such that db {lambda}(P)=2 (while w.db {lambda}(P)=1). Finally, it is shown that many biflat Koethe algebras, while not being amenable, have trivial Hochschild homology groups in positive degrees (with arbitrary coefficients). Bibliography: 37 titles.
Phase Boundaries in Algebraic Conformal QFT
NASA Astrophysics Data System (ADS)
Bischoff, Marcel; Kawahigashi, Yasuyuki; Longo, Roberto; Rehren, Karl-Henning
2016-02-01
We study the structure of local algebras in relativistic conformal quantum field theory with phase boundaries. Phase boundaries are instances of a more general notion of boundaries that give rise to a variety of algebraic structures. These can be formulated in a common framework originating in Algebraic QFT, with the principle of Einstein Causality playing a prominent role. We classify the phase boundary conditions by the centre of a certain universal construction, which produces a reducible representation in which all possible boundary conditions are realized. For a large class of models, the classification reproduces results obtained in a different approach by Fuchs et al. before.
Toward robust scalable algebraic multigrid solvers.
Waisman, Haim; Schroder, Jacob; Olson, Luke; Hiriyur, Badri; Gaidamour, Jeremie; Siefert, Christopher; Hu, Jonathan Joseph; Tuminaro, Raymond Stephen
2010-10-01
This talk highlights some multigrid challenges that arise from several application areas including structural dynamics, fluid flow, and electromagnetics. A general framework is presented to help introduce and understand algebraic multigrid methods based on energy minimization concepts. Connections between algebraic multigrid prolongators and finite element basis functions are made to explored. It is shown how the general algebraic multigrid framework allows one to adapt multigrid ideas to a number of different situations. Examples are given corresponding to linear elasticity and specifically in the solution of linear systems associated with extended finite elements for fracture problems.
Algebraic method for finding equivalence groups
NASA Astrophysics Data System (ADS)
Bihlo, Alexander; Dos Santos Cardoso-Bihlo, Elsa; Popovych, Roman O.
2015-06-01
The algebraic method for computing the complete point symmetry group of a system of differential equations is extended to finding the complete equivalence group of a class of such systems. The extended method uses the knowledge of the corresponding equivalence algebra. Two versions of the method are presented, where the first involves the automorphism group of this algebra and the second is based on a list of its megaideals. We illustrate the megaideal-based version of the method with the computation of the complete equivalence group of a class of nonlinear wave equations with applications in nonlinear elasticity.
The nth root of sequential effect algebras
NASA Astrophysics Data System (ADS)
Shen, Jun; Wu, Junde
2010-06-01
In 2005, Gudder [Int. J. Theor. Phys. 44, 2219 (2005)] presented 25 problems of sequential effect algebras, the 20th problem asked: In a sequential effect algebra, if the square root of some element exists, is it unique? In this paper, we show that for each given positive integer n >1, there is a sequential effect algebra such that the nth root of its some element c is not unique, and the nth root of c is not the kth root of c (k
On computational complexity of Clifford algebra
NASA Astrophysics Data System (ADS)
Budinich, Marco
2009-05-01
After a brief discussion of the computational complexity of Clifford algebras, we present a new basis for even Clifford algebra Cl(2m) that simplifies greatly the actual calculations and, without resorting to the conventional matrix isomorphism formulation, obtains the same complexity. In the last part we apply these results to the Clifford algebra formulation of the NP-complete problem of the maximum clique of a graph introduced by Budinich and Budinich ["A spinorial formulation of the maximum clique problem of a graph," J. Math. Phys. 47, 043502 (2006)].
Imperfect Cloning Operations in Algebraic Quantum Theory
NASA Astrophysics Data System (ADS)
Kitajima, Yuichiro
2015-01-01
No-cloning theorem says that there is no unitary operation that makes perfect clones of non-orthogonal quantum states. The objective of the present paper is to examine whether an imperfect cloning operation exists or not in a C*-algebraic framework. We define a universal -imperfect cloning operation which tolerates a finite loss of fidelity in the cloned state, and show that an individual system's algebra of observables is abelian if and only if there is a universal -imperfect cloning operation in the case where the loss of fidelity is less than . Therefore in this case no universal -imperfect cloning operation is possible in algebraic quantum theory.
Contractions of affine Kac-Moody algebras
NASA Astrophysics Data System (ADS)
Daboul, J.; Daboul, C.; de Montigny, M.
2008-08-01
I review our recent work on contractions of affine Kac-Moody algebras (KMA) and present new results. We study generalized contractions of KMA with respect to their twisted and untwisted KM subalgebras. As a concrete example, we discuss contraction of D(1)4 and D(3)4, based on Z3-grading. We also describe examples of 'level-dependent' contractions, which are based on Z-gradings of KMA. Our work generalizes the Inönü-Wigner contraction of P. Majumdar in several directions. We also give an algorithm for constructing Kac-Moody-like algebras hat g for any Lie algebra g.
Kinematical superalgebras and Lie algebras of order 3
Campoamor-Stursberg, R.; Rausch de Traubenberg, M.
2008-06-15
We study and classify kinematical algebras which appear in the framework of Lie superalgebras or Lie algebras of order 3. All these algebras are related through generalized Inonue-Wigner contractions from either the orthosymplectic superalgebra or the de Sitter Lie algebra of order 3.
Alternative algebras admitting derivations with invertible values and invertible derivations
NASA Astrophysics Data System (ADS)
Kaygorodov, I. B.; Popov, Yu S.
2014-10-01
We prove an analogue of the Bergen-Herstein-Lanski theorem for alternative algebras: describe all alternative algebras that admit derivations with invertible values. We also prove an analogue of Moens' theorem for alternative algebras (a finite-dimensional alternative algebra over a field of characteristic zero is nilpotent if and only if it admits an invertible Leibniz derivation).
Spinor-vector supersymmetry algebra in three dimensions
NASA Astrophysics Data System (ADS)
Shima, Kazunari; Tsuda, Motomu
2006-06-01
We focus on a spin-3/2 supersymmetry (SUSY) algebra of Baaklini in D = 3 and explicitly show a nonlinear realization of the SUSY algebra. The unitary representation of the spin-3/2 SUSY algebra is discussed and compared with the ordinary (spin-1/2) SUSY algebra.
Lie bialgebra structures on the Schroedinger-Virasoro Lie algebra
Han Jianzhi; Su Yucai; Li Junbo
2009-08-15
In this paper we shall investigate Lie bialgebra structures on the Schroedinger-Virasoro algebra L. We found out that not all Lie bialgebra structures on the Schroedinger-Virasoro algebra are triangular coboundary, which is different from the related known results of some other Lie algebras related to the Virasoro algebra.
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.
The algebra of two dimensional generalized Chebyshev-Koornwinder oscillator
Borzov, V. V.; Damaskinsky, E. V.
2014-10-15
In the previous works of Borzov and Damaskinsky [“Chebyshev-Koornwinder oscillator,” Theor. Math. Phys. 175(3), 765–772 (2013)] and [“Ladder operators for Chebyshev-Koornwinder oscillator,” in Proceedings of the Days on Diffraction, 2013], the authors have defined the oscillator-like system that is associated with the two variable Chebyshev-Koornwinder polynomials. We call this system the generalized Chebyshev-Koornwinder oscillator. In this paper, we study the properties of infinite-dimensional Lie algebra that is analogous to the Heisenberg algebra for the Chebyshev-Koornwinder oscillator. We construct the exact irreducible representation of this algebra in a Hilbert space H of functions that are defined on a region which is bounded by the Steiner hypocycloid. The functions are square-integrable with respect to the orthogonality measure for the Chebyshev-Koornwinder polynomials and these polynomials form an orthonormalized basis in the space H. The generalized oscillator which is studied in the work can be considered as the simplest nontrivial example of multiboson quantum system that is composed of three interacting oscillators.
Top Element Problem and Macneille Completions of Generalized Effect Algebras
NASA Astrophysics Data System (ADS)
RieČanová, Z.; Kalina, M.
2014-10-01
Effect algebras (EAs), introduced by D. J. Foulis and M. K. Bennett, as common generalizations of Boolean algebras, orthomodular lattices and MV-algebras, are nondistributive algebraic structures including unsharp elements. Their unbounded versions, called generalized effect algebras, are posets which may have or may have not an EA-MacNeille completion, or cannot be embedded into any complete effect algebra. We give a necessary and sufficient condition for a generalized effect algebra to have an EA-MacNeille completion. Some examples are provided.
The Progressive Development of Early Embodied Algebraic Thinking
NASA Astrophysics Data System (ADS)
Radford, Luis
2014-06-01
In this article I present some results from a 5-year longitudinal investigation with young students about the genesis of embodied, non-symbolic algebraic thinking and its progressive transition to culturally evolved forms of symbolic thinking. The investigation draws on a cultural-historical theory of teaching and learning—the theory of objectification. Within this theory, thinking is conceived of as a form of reflection and action that is simultaneously material and ideal: It includes inner and outer speech, sensuous forms of imagination and visualisation, gestures, rhythm, and their intertwinement with material culture (symbols, artifacts, etc.). The theory articulates a cultural view of development as an unfolding dialectic process between culturally and historically constituted forms of mathematical knowing and semiotically mediated classroom activity. Looking at the experimental data through these theoretical lenses reveals a developmental path where embodied forms of thinking are sublated or subsumed into more sophisticated ones through the mediation of properly designed classroom activity.
I CAN Learn[R] Pre-Algebra and Algebra. What Works Clearinghouse Intervention Report
ERIC Educational Resources Information Center
What Works Clearinghouse, 2009
2009-01-01
The I CAN Learn[R] Education System is an interactive, self-paced, mastery-based software system that includes the I CAN Learn[R] Fundamentals of Math (5th-6th grade math) curriculum, the I CAN Learn[R] Pre-Algebra curriculum, and the I CAN Learn[R] Algebra curriculum. College algebra credit is also available to students in participating schools…
Clifford Algebras in Symplectic Geometry and Quantum Mechanics
NASA Astrophysics Data System (ADS)
Binz, Ernst; de Gosson, Maurice A.; Hiley, Basil J.
2013-04-01
The necessary appearance of Clifford algebras in the quantum description of fermions has prompted us to re-examine the fundamental role played by the quaternion Clifford algebra, C 0,2. This algebra is essentially the geometric algebra describing the rotational properties of space. Hidden within this algebra are symplectic structures with Heisenberg algebras at their core. This algebra also enables us to define a Poisson algebra of all homogeneous quadratic polynomials on a two-dimensional sub-space, {F}a of the Euclidean three-space. This enables us to construct a Poisson Clifford algebra, ℍ F , of a finite dimensional phase space which will carry the dynamics. The quantum dynamics appears as a realisation of ℍ F in terms of a Clifford algebra consisting of Hermitian operators.
A note on derivations of Murray–von Neumann algebras
Kadison, Richard V.; Liu, Zhe
2014-01-01
A Murray–von Neumann algebra is the algebra of operators affiliated with a finite von Neumann algebra. In this article, we first present a brief introduction to the theory of derivations of operator algebras from both the physical and mathematical points of view. We then describe our recent work on derivations of Murray–von Neumann algebras. We show that the “extended derivations” of a Murray–von Neumann algebra, those that map the associated finite von Neumann algebra into itself, are inner. In particular, we prove that the only derivation that maps a Murray–von Neumann algebra associated with a factor of type II1 into that factor is 0. Those results are extensions of Singer’s seminal result answering a question of Kaplansky, as applied to von Neumann algebras: The algebra may be noncommutative and may even contain unbounded elements. PMID:24469831
Griffith, M J; Breitkreutz, L; Trapp, H; Briet, E; Noyes, C M; Lundblad, R L; Roberts, H R
1985-01-01
Two structurally different forms of activated human Factor IX (Factor IXa alpha and IXa beta) have been previously reported to have essentially identical clotting activity in vitro. Although it has been shown that activated Factor IX Chapel Hill, an abnormal Factor IX isolated from the plasma of a patient with mild hemophilia B, and normal Factor IXa alpha are structurally very similar, the clotting activity of activated Factor IX Chapel Hill is much lower (approximately fivefold) than that of normal Factor IXa beta. In the present study we have prepared activated Factor IX by incubating human Factor IX with calcium and Russell's viper venom covalently bound to agarose. Fractionation of the activated Factor IX by high-performance liquid chromatography demonstrated the presence of both Factors IXa alpha and IXa beta. On the basis of active site concentration, determined by titration with antithrombin III, the clotting activities of activated Factor IX Chapel Hill and IXa alpha were similar, but both activities were less than 20% of the clotting activity of Factor IXa beta. Activated Factor IX activity was also measured in the absence of calcium, phospholipid, and Factor VIII, by determination of the rate of Factor X activation in the presence of polylysine. In the presence of polylysine, the rates of Factor X activation by activated Factor IX Chapel Hill, Factor IXa alpha, and Factor IXa beta were essentially identical. We conclude that the clotting activity of activated Factor IX Chapel Hill is reduced when compared with that of Factor IXa beta but essentially normal when compared with that of Factor IXa alpha. PMID:3871202
Finding the Axis of Revolution of an Algebraic Surface of Revolution.
Alcazar, Juan G; Goldman, Ron
2016-09-01
We present an algorithm for extracting the axis of revolution from the implicit equation of an algebraic surface of revolution based on three distinct computational methods: factoring the highest order form into quadrics, contracting the tensor of the highest order form, and using univariate resultants and gcds. We compare and contrast the advantages and disadvantages of each of these three techniques and we derive conditions under which each technique is most appropriate. In addition, we provide several necessary conditions for an implicit algebraic equation to represent a surface of revolution. PMID:26561460
Algebraic Multiscale Solver for Elastic Geomechanical Deformation
NASA Astrophysics Data System (ADS)
Castelletto, N.; Hajibeygi, H.; Tchelepi, H.
2015-12-01
Predicting the geomechanical response of geological formations to thermal, pressure, and mechanical loading is important in many engineering applications. The mathematical formulation that describes deformation of a reservoir coupled with flow and transport entails heterogeneous coefficients with a wide range of length scales. Such detailed heterogeneous descriptions of reservoir properties impose severe computational challenges for the study of realistic-scale (km) reservoirs. To deal with these challenges, we developed an Algebraic Multiscale Solver for ELastic geomechanical deformation (EL-AMS). Constructed on finite element fine-scale system, EL-AMS imposes a coarse-scale grid, which is a non-overlapping decomposition of the domain. Then, local (coarse) basis functions for the displacement vector are introduced. These basis functions honor the elastic properties of the local domains subject to the imposed local boundary conditions. The basis form the Restriction and Prolongation operators. These operators allow for the construction of accurate coarse-scale systems for the displacement. While the multiscale system is efficient for resolving low-frequency errors, coupling it with a fine-scale smoother, e.g., ILU(0), leads to an efficient iterative solver. Numerical results for several test cases illustrate that EL-AMS is quite efficient and applicable to simulate elastic deformation of large-scale heterogeneous reservoirs.
Using computer algebra and SMT solvers in algebraic biology
NASA Astrophysics Data System (ADS)
Pineda Osorio, Mateo
2014-05-01
Biologic processes are represented as Boolean networks, in a discrete time. The dynamics within these networks are approached with the help of SMT Solvers and the use of computer algebra. Software such as Maple and Z3 was used in this case. The number of stationary states for each network was calculated. The network studied here corresponds to the immune system under the effects of drastic mood changes. Mood is considered as a Boolean variable that affects the entire dynamics of the immune system, changing the Boolean satisfiability and the number of stationary states of the immune network. Results obtained show Z3's great potential as a SMT Solver. Some of these results were verified in Maple, even though it showed not to be as suitable for the problem approach. The solving code was constructed using Z3-Python and Z3-SMT-LiB. Results obtained are important in biology systems and are expected to help in the design of immune therapies. As a future line of research, more complex Boolean network representations of the immune system as well as the whole psychological apparatus are suggested.
Highest-weight representations of Brocherd`s algebras
Slansky, R.
1997-01-01
General features of highest-weight representations of Borcherd`s algebras are described. to show their typical features, several representations of Borcherd`s extensions of finite-dimensional algebras are analyzed. Then the example of the extension of affine- su(2) to a Borcherd`s algebra is examined. These algebras provide a natural way to extend a Kac-Moody algebra to include the hamiltonian and number-changing operators in a generalized symmetry structure.
On \\delta-derivations of n-ary algebras
NASA Astrophysics Data System (ADS)
Kaygorodov, Ivan B.
2012-12-01
We give a description of \\delta-derivations of (n+1)-dimensional n-ary Filippov algebras and, as a consequence, of simple finite-dimensional Filippov algebras over an algebraically closed field of characteristic zero. We also give new examples of non-trivial \\delta-derivations of Filippov algebras and show that there are no non-trivial \\delta-derivations of the simple ternary Mal'tsev algebra M_8.
Excision in algebraic K-theory and Karoubi's conjecture.
Suslin, A A; Wodzicki, M
1990-12-15
We prove that the property of excision in algebraic K-theory is for a Q-algebra A equivalent to the H-unitality of the latter. Our excision theorem, in particular, implies Karoubi's conjecture on the equality of algebraic and topological K-theory groups of stable C*-algebras. It also allows us to identify the algebraic K-theory of the symbol map in the theory of pseudodifferential operators. PMID:11607130
Excision in algebraic K-theory and Karoubi's conjecture.
Suslin, A A; Wodzicki, M
1990-01-01
We prove that the property of excision in algebraic K-theory is for a Q-algebra A equivalent to the H-unitality of the latter. Our excision theorem, in particular, implies Karoubi's conjecture on the equality of algebraic and topological K-theory groups of stable C*-algebras. It also allows us to identify the algebraic K-theory of the symbol map in the theory of pseudodifferential operators. PMID:11607130
On algebraic endomorphisms of the Einstein gyrogroup
NASA Astrophysics Data System (ADS)
Molnár, Lajos; Virosztek, Dániel
2015-08-01
We describe the structure of all continuous algebraic endomorphisms of the open unit ball B of ℝ3 equipped with the Einstein velocity addition. We show that any nonzero such transformation originates from an orthogonal linear transformation on ℝ3.
Clifford algebras and physical and engineering sciences
NASA Astrophysics Data System (ADS)
Furui, Sadataka
2013-10-01
Clifford algebra in physical and engineering science are studied. Roles of triality symmetry of Cartan's spinor in axial anomaly of particle physics and quaternion and octonion in the memristic circuits are discussed.
Positive basis for surface skein algebras
Thurston, Dylan Paul
2014-01-01
We show that the twisted SL2 skein algebra of a surface has a natural basis (the bracelets basis) that is positive, in the sense that the structure constants for multiplication are positive integers. PMID:24982193
Lisa's Lemonade Stand: Exploring Algebraic Ideas.
ERIC Educational Resources Information Center
Billings, Esther M. H.; Lakatos, Tracy
2003-01-01
Presents an activity, "Lisa's Lemonade Stand," that actively engages students in algebraic thinking as they analyze change by investigating relationships between variables and gain experience describing and representing these relationships graphically. (YDS)
Lima Beans, Paper Cups, and Algebra.
ERIC Educational Resources Information Center
Loewen, A. C.
1991-01-01
An activity in which students use manipulative materials to help solve simple algebraic equations using the operations of adding inverses, removing opposites, and sharing equally is presented. Directions, examples, the rationale, and cautions are included. (KR)
Supersymmetric extension of Galilean conformal algebras
Bagchi, Arjun; Mandal, Ipsita
2009-10-15
The Galilean conformal algebra has recently been realized in the study of the nonrelativistic limit of the AdS/CFT conjecture. This was obtained by a systematic parametric group contraction of the parent relativistic conformal field theory. In this paper, we extend the analysis to include supersymmetry. We work at the level of the coordinates in superspace to construct the N=1 super-Galilean conformal algebra. One of the interesting outcomes of the analysis is that one is able to naturally extend the finite algebra to an infinite one. This looks structurally similar to the N=1 superconformal algebra in two dimensions, but is different. We also comment on the extension of our construction to cases of higher N.
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)
Semigroups And Computer Algebra In Discrete Structures
NASA Astrophysics Data System (ADS)
Bijev, G.
2010-10-01
Some concepts in semigroup theory are interpreted in discrete structures such as finite lattices, binary relations, and finite semilattices. An algebraic approach to the pseudoinverse generalization problem in Boolean vector spaces is used. By analogy with the linear spaces in the linear algebra semilattice homomorphisms, isomorphisms, projections on Boolean vector spaces are defined and some properties of them are investigated in detail. Maps, corresponding to them in the linear algebra, are connected with matrices and their pseudouinverse. Important properties of these maps, which are essential for solving linear systems, remain the same in the Boolean vector spaces. Stochastic experiments using the maps defined and computer algebra methods have been made for solving linear equations Ax = b. The Hamming distance between b and the projection p(b) = Ax of b is equal or close to the least possible one, if the system has no solutions.
Twisted Logarithmic Modules of Vertex Algebras
NASA Astrophysics Data System (ADS)
Bakalov, Bojko
2016-07-01
Motivated by logarithmic conformal field theory and Gromov-Witten theory, we introduce a notion of a twisted module of a vertex algebra under an arbitrary (not necessarily semisimple) automorphism. Its main feature is that the twisted fields involve the logarithm of the formal variable. We develop the theory of such twisted modules and, in particular, derive a Borcherds identity and commutator formula for them. We investigate in detail the examples of affine and Heisenberg vertex algebras.
Edge covers and independence: Algebraic approach
NASA Astrophysics Data System (ADS)
Kalinina, E. A.; Khitrov, G. M.; Pogozhev, S. V.
2016-06-01
In this paper, linear algebra methods are applied to solve some problems of graph theory. For ordinary connected graphs, edge coverings and independent sets are considered. Some results concerning minimum edge covers and maximum matchings are proved with the help of linear algebraic approach. The problem of finding a maximum matching of a graph is fundamental both practically and theoretically, and has numerous applications, e.g., in computational chemistry and mathematical chemistry.
Filtering Algebraic Multigrid and Adaptive Strategies
Nagel, A; Falgout, R D; Wittum, G
2006-01-31
Solving linear systems arising from systems of partial differential equations, multigrid and multilevel methods have proven optimal complexity and efficiency properties. Due to shortcomings of geometric approaches, algebraic multigrid methods have been developed. One example is the filtering algebraic multigrid method introduced by C. Wagner. This paper proposes a variant of Wagner's method with substantially improved robustness properties. The method is used in an adaptive, self-correcting framework and tested numerically.
Sharply Dominating MV-Effect Algebras
NASA Astrophysics Data System (ADS)
Kalina, Martin; Olejček, Vladimír; Paseka, Jan; Riečanová, Zdenka
2011-04-01
Some open questions on Archimedean atomic MV-effect algebras are answered. Namely we prove that there are Archimedean atomic MV-effect algebras which are not sharply dominating. Equivalently, they don't have a basic decomposition of elements. Moreover, if their set of sharp elements (their center) is a complete lattice then they need not be complete lattices. The existence of infinite orthogonal sums of their elements is discussed.
Stability of Lie groupoid C∗-algebras
NASA Astrophysics Data System (ADS)
Debord, Claire; Skandalis, Georges
2016-07-01
In this paper we generalize a theorem of M. Hilsum and G. Skandalis stating that the C∗-algebra of any foliation of nonzero dimension is stable. Precisely, we show that the C∗-algebra of a Lie groupoid is stable whenever the groupoid has no orbit of dimension zero. We also prove an analogous theorem for singular foliations for which the holonomy groupoid as defined by I. Androulidakis and G. Skandalis is not Lie in general.
Vague Congruences and Quotient Lattice Implication Algebras
Qin, Xiaoyan; Xu, Yang
2014-01-01
The aim of this paper is to further develop the congruence theory on lattice implication algebras. Firstly, we introduce the notions of vague similarity relations based on vague relations and vague congruence relations. Secondly, the equivalent characterizations of vague congruence relations are investigated. Thirdly, the relation between the set of vague filters and the set of vague congruences is studied. Finally, we construct a new lattice implication algebra induced by a vague congruence, and the homomorphism theorem is given. PMID:25133207
One-Equation Algebraic Model Of Turbulence
NASA Technical Reports Server (NTRS)
Baldwin, B. S.; Barth, T. J.
1993-01-01
One-equation model of turbulence based on standard equations of k-epsilon model of turbulence, where k is turbulent energy and e is rate of dissipation of k. Derivation of one-equation model motivated partly by inaccuracies of flows computed by some Navier-Stokes-equations-solving algorithms incorporating algebraic models of turbulence. Satisfies need to avoid having to determine algebraic length scales.
The arithmetic theory of algebraic groups
NASA Astrophysics Data System (ADS)
Platonov, V. P.
1982-06-01
CONTENTS Introduction § 1. Arithmetic groups § 2. Adèle groups § 3. Tamagawa numbers § 4. Approximations in algebraic groups § 5. Class numbers and class groups of algebraic groups § 6. The genus problem in arithmetic groups § 7. Classification of maximal arithmetic subgroups § 8. The congruence problem § 9. Groups of rational points over global fields § 10. Galois cohomology and the Hasse principle § 11. Cohomology of arithmetic groups References
Algorithmic Questions for Linear Algebraic Groups. Ii
NASA Astrophysics Data System (ADS)
Sarkisjan, R. A.
1982-04-01
It is proved that, given a linear algebraic group defined over an algebraic number field and satisfying certain conditions, there exists an algorithm which determines whether or not two double cosets of a special type coincide in its adele group, and which enumerates all such double cosets. This result is applied to the isomorphism problem for finitely generated nilpotent groups, and also to other problems.Bibliography: 18 titles.
Conn, Vicki S; Zerwic, Julie; Jefferson, Urmeka; Anderson, Cindy M; Killion, Cheryl M; Smith, Carol E; Cohen, Marlene Z; Fahrenwald, Nancy L; Herrick, Linda; Topp, Robert; Benefield, Lazelle E; Loya, Julio
2016-02-01
Getting turned down for grant funding or having a manuscript rejected is an uncomfortable but not unusual occurrence during the course of a nurse researcher's professional life. Rejection can evoke an emotional response akin to the grieving process that can slow or even undermine productivity. Only by "normalizing" rejection, that is, by accepting it as an integral part of the scientific process, can researchers more quickly overcome negative emotions and instead use rejection to refine and advance their scientific programs. This article provides practical advice for coming to emotional terms with rejection and delineates methods for working constructively to address reviewer comments. PMID:26041785
Clifford algebras and Hestenes spinors
NASA Astrophysics Data System (ADS)
Lounesto, Pertti
1993-09-01
This article reviews Hestenes' work on the Dirac theory, where his main achievement is a real formulation of the theory within the real Clifford algebra Cl 1,3 ≃ M2 (H). Hestenes invented first in 1966 his ideal spinorsφ in Cl_{1,3 _2}^1 (1 - γ _{03} ) and later 1967/75 he recognized the importance of his operator spinors ψ ∈ Cl{/1,3 + } ≃ M2 (C). This article starts from the conventional Dirac equation as presented with matrices by Bjorken-Drell. Explicit mappings are given for a passage between Hestenes' operator spinors and Dirac's column spinors. Hestenes' operator spinors are seen to be multiples of even parts of real parts of Dirac spinors (real part in the decomposition C ⊗ Cl 1,3 and not in C ⊗ M4 (R)=M4 (C)). It will become apparent that the standard matrix formulation contains superfluous parts, which ought to be cut out by Occam's razor. Fierz identities of bilinear covariants are known to be sufficient to study the non-null case but are seen to be insufficient for the null case ψ†γ0ψ=0, ψ†γ0γ0123ψ=0. The null case is thoroughly scrutinized for the first time with a new concept called boomerang. This permits a new intrinsically geometric classification of spinors. This in turn reveals a new class of spinors which has not been discussed before. This class supplements the spinors of Dirac, Weyl, and Majorana; it describes neither the electron nor the neutron; it is awaiting a physical interpretation and a possible observation. Projection operators P±, Σ± are resettled among their new relatives in End(Cl 1,3 ). Finally, a new mapping, called tilt, is introduced to enable a transition from Cl 1,3 to the (graded) opposite algebra Cl 3,1 without resorting to complex numbers, that is, not using a replacement γμ → iγμ.
Cluster Algebras from Dualities of 2d = (2, 2) Quiver Gauge Theories
NASA Astrophysics Data System (ADS)
Benini, Francesco; Park, Daniel S.; Zhao, Peng
2015-11-01
We interpret certain Seiberg-like dualities of two-dimensional = (2,2) quiver gauge theories with unitary groups as cluster mutations in cluster algebras, originally formulated by Fomin and Zelevinsky. In particular, we show how the complexified Fayet-Iliopoulos parameters of the gauge group factors transform under those dualities and observe that they are in fact related to the dual cluster variables of cluster algebras. This implies that there is an underlying cluster algebra structure in the quantum Kähler moduli space of manifolds constructed from the corresponding Kähler quotients. We study the S 2 partition function of the gauge theories, showing that it is invariant under dualities/mutations, up to an overall normalization factor, whose physical origin and consequences we spell out in detail. We also present similar dualities in = (2,2)* quiver gauge theories, which are related to dualities of quantum integrable spin chains.
From Atiyah Classes to Homotopy Leibniz Algebras
NASA Astrophysics Data System (ADS)
Chen, Zhuo; Stiénon, Mathieu; Xu, Ping
2016-01-01
A celebrated theorem of Kapranov states that the Atiyah class of the tangent bundle of a complex manifold X makes T X [-1] into a Lie algebra object in D + ( X), the bounded below derived category of coherent sheaves on X. Furthermore, Kapranov proved that, for a Kähler manifold X, the Dolbeault resolution {Ω^{bullet-1}(T_X^{1, 0})} of T X [-1] is an L ∞ algebra. In this paper, we prove that Kapranov's theorem holds in much wider generality for vector bundles over Lie pairs. Given a Lie pair ( L, A), i.e. a Lie algebroid L together with a Lie subalgebroid A, we define the Atiyah class α E of an A-module E as the obstruction to the existence of an A- compatible L-connection on E. We prove that the Atiyah classes α L/ A and α E respectively make L/ A[-1] and E[-1] into a Lie algebra and a Lie algebra module in the bounded below derived category {D^+(A)} , where {A} is the abelian category of left {U(A)} -modules and {U(A)} is the universal enveloping algebra of A. Moreover, we produce a homotopy Leibniz algebra and a homotopy Leibniz module stemming from the Atiyah classes of L/ A and E, and inducing the aforesaid Lie structures in {D^+(A)}.
Black brane solutions related to non-singular Kac-Moody algebras
NASA Astrophysics Data System (ADS)
Ivashchuk, V. D.; Melnikov, V. N.
2011-01-01
A multidimensional gravitational model containing scalar fields and antisymmetric forms is considered. The manifold is chosen in the form M = M_0 x M_1 x ... x M_n, where M_i are Einstein spaces (i > 0). The sigma-model approach and exact solutions with intersecting composite branes (e.g., solutions with harmonic functions and black brane ones) with intersection rules related to non-singular Kac-Moody (KM) algebras (e.g. hyperbolic ones) are considered. Some examples of black brane solutions are presented, e.g., those corresponding to hyperbolic KM algebras: H_2(q,q) (q > 2), HA_2^(1) = A_2^{++} and to the Lorentzian KM algebra P_{10}.
ERIC Educational Resources Information Center
Srinivasan, V. K.
2013-01-01
Given a parabola in the standard form y[superscript 2] = 4ax, corresponding to three points on the parabola, such that the normals at these three points P, Q, R concur at a point M = (h, k), the equation of the circumscribing circle through the three points P, Q, and R provides a tremendous opportunity to illustrate "The Art of Algebraic…
Some algebraic properties of crystallographic sublattices.
Rutherford, John S
2006-03-01
In this article, a number of the results relevant to the concept of sublattices of a basic crystallographic lattice are reviewed, emphasizing particularly previously unpublished work on the algebraic aspects. A three-dimensional geometric lattice L can be considered as an infinite Abelian group under addition. A sublattice S of L, which is also three-dimensional, is a subgroup of L such that the finite quotient group, G approximately equals L/S, is an Abelian group of order the index of S in L. The sublattice itself in its standard form is represented by an upper triangular matrix. The index of the sublattice is given by the determinant of this matrix. It is first noted that a sublattice described by an arbitrary basis set in L may be converted to this standard form. Next the sublattice is expressed as the intersection of a set of sublattices of individual index a power of a distinct prime, i.e. S(n = p(a)(1)p(b)(2)...) = S(1)(p(a)(1)[cap]S(2)(p(b)(2)...[cap]... = [bigcap](i)S(i)(p(alpha(i)), where p(1), p(2) etc. are prime numbers and n = Pi(i)p(alpha)(i) is the Euclidean factorization of n. This decomposition is important because it corresponds to the Sylow decomposition of the corresponding quotient group G approximately equals (i)[sign: see text] A(p)(i). It is also useful to be able to carry out two commutative binary operations on sublattices of L; these are to find their common sublattice of lowest index in L, which is their intersection S(cap) = S(a)(m)[cap]S(b)(n) and their common superlattice of highest index in L, given by S(< >) = , where < > indicates the span of the sublattices. PMID:16489245
A Z{sub 3} generalization of Pauli's principle, quark algebra and the Lorentz invariance
Kerner, Richard
2012-09-24
The fundamental difference between bosons and fermions is that they obey two alternative representations of the Z{sub 2} group, resulting in symmetric or anti-symmetric binary commutation relations. Our aim is to explore possibilities offered by ternary Z{sub 3} generalization commutation relations. This leads to cubic and ternary algebras which are a direct generalization of usual commutation relations, with Z{sub 3}-grading replacing the usual Z{sub 2}-grading. Properties and structure of such algebras are discussed, with special interest in a low-dimensional one, with two generators. Invariant cubic forms on such algebras are introduced, and it is shown how the SL(2,C) group arises naturally as the symmetry group preserving these forms. In the case of lowest dimension, with only two generators, it is shown how the cubic combinations of elements of the same Z{sub 3} grade behave like Lorentz spinors, while binary products of elements of this algebra with an element of the conjugate algebra behave like Lorentz vectors. The wave equation generalizing the Dirac operator to the Z{sub 3}-graded case is introduced, whose diagonalization leads to a third-order equation. The solutions of this equation cannot propagate because their exponents always contain non-oscillating real damping factor. We show how certain cubic products can propagate nevertheless. The model suggests the origin of the color SU(3) symmetry obeyed by quark states.
Girard, Nadine; Koob, Meriam; Brunel, Herv
2016-01-01
Numerous events are involved in brain development, some of which are detected by neuroimaging. Major changes in brain morphology are depicted by brain imaging during the fetal period while changes in brain composition can be demonstrated in both pre- and postnatal periods. Although ultrasonography and computed tomography can show changes in brain morphology, these techniques are insensitive to myelination that is one of the most important events occurring during brain maturation. Magnetic resonance imaging (MRI) is therefore the method of choice to evaluate brain maturation. MRI also gives insight into the microstructure of brain tissue through diffusion-weighted imaging and diffusion tensor imaging. Metabolic changes are also part of brain maturation and are assessed by proton magnetic resonance spectroscopy. Understanding and knowledge of the different steps in brain development are required to be able to detect morphologic and structural changes on neuroimaging. Consequently alterations in normal development can be depicted. PMID:27430460
Automorphism groups of composition algebras and quark models
Bjerregard, P.A.; Gonzalez, C.M.
1996-12-01
In this the authors study the automorphisms and derivations of real composition algebras with a view to its physical interpretations. They obtain canonical forms with a special stress in the four and eight dimensional cases. Also, using this description, they work with two mathematical models which describe some particles with certain observables in a surprising way. A first model, split g{sub 2}, describes two observables for three quarks, their antiquarks, and eight mesons combining the quarks involved. A second one, so(4,4) {circle_plus} so(2,2), describes all the observables for all quarks (u, d, s, c, b and t).
Negative base encoding in optical linear algebra processors
NASA Technical Reports Server (NTRS)
Perlee, C.; Casasent, D.
1986-01-01
In the digital multiplication by analog convolution algorithm, the bits of two encoded numbers are convolved to form the product of the two numbers in mixed binary representation; this output can be easily converted to binary. Attention is presently given to negative base encoding, treating base -2 initially, and then showing that the negative base system can be readily extended to any radix. In general, negative base encoding in optical linear algebra processors represents a more efficient technique than either sign magnitude or 2's complement encoding, when the additions of digitally encoded products are performed in parallel.
ERIC Educational Resources Information Center
Okpube, Nnaemeka Michael; Anugwo, M. N.
2016-01-01
This study investigated the Card Games and Algebra tic-Tacmatics on Junior Secondary II Students' Achievement in Algebraic Expressions. Three research questions and three null hypotheses guided the study. The study adopted the pre-test, post-test control group design. A total of two hundred and forty (240) Junior Secondary School II students were…
ERIC Educational Resources Information Center
Davies Gomez, Lisa
2012-01-01
Algebra is the gatekeeper of access to higher-level math and science courses, higher education and future earning opportunities. Unequal numbers of African-American males drop out of Algebra and mathematics courses and underperform on tests of mathematical competency and are thus denied both essential skills and a particularly important pathway to…
Slower Algebra Students Meet Faster Tools: Solving Algebra Word Problems with Graphing Software
ERIC Educational Resources Information Center
Yerushalmy, Michal
2006-01-01
The article discusses the ways that less successful mathematics students used graphing software with capabilities similar to a basic graphing calculator to solve algebra problems in context. The study is based on interviewing students who learned algebra for 3 years in an environment where software tools were always present. We found differences…
ERIC Educational Resources Information Center
Ormond, Christine
2012-01-01
Primary teachers play a key role in their students' future mathematical success in the early secondary years. While the word "algebra" may make some primary teachers feel uncomfortable or worried, the basic arithmetic ideas underlying algebra are vitally important for older primary students as they are increasingly required to use "algebraic…
C∗-completions and the DFR-algebra
NASA Astrophysics Data System (ADS)
Forger, Michael; Paulino, Daniel V.
2016-02-01
The aim of this paper is to present the construction of a general family of C∗-algebras which includes, as a special case, the "quantum spacetime algebra" introduced by Doplicher, Fredenhagen, and Roberts. It is based on an extension of the notion of C∗-completion from algebras to bundles of algebras, compatible with the usual C∗-completion of the appropriate algebras of sections, combined with a novel definition for the algebra of the canonical commutation relations using Rieffel's theory of strict deformation quantization. Taking the C∗-algebra of continuous sections vanishing at infinity, we arrive at a functor associating a C∗-algebra to any Poisson vector bundle and recover the original DFR-algebra as a particular example.
Three-algebra for supermembrane and two-algebra for superstring
NASA Astrophysics Data System (ADS)
Lee, Kanghoon; Park, Jeong-Hyuck
2009-04-01
While string or Yang-Mills theories are based on Lie algebra or two-algebra structure, recent studies indicate that Script M-theory may require a one higher, three-algebra structure. Here we construct a covariant action for a supermembrane in eleven dimensions, which is invariant under global supersymmetry, local fermionic symmetry and worldvolume diffeomorphism. Our action is classically on-shell equivalent to the celebrated Bergshoeff-Sezgin-Townsend action. However, the novelty is that we spell the action genuinely in terms of Nambu three-brackets: All the derivatives appear through Nambu brackets and hence it manifests the three-algebra structure. Further the double dimensional reduction of our action gives straightforwardly to a type IIA string action featuring two-algebra. Applying the same method, we also construct a covariant action for type IIB superstring, leading directly to the IKKT matrix model.
NASA Astrophysics Data System (ADS)
Kimura, Yusuke
2015-07-01
It has been understood that correlation functions of multi-trace operators in SYM can be neatly computed using the group algebra of symmetric groups or walled Brauer algebras. On the other hand, such algebras have been known to construct 2D topological field theories (TFTs). After reviewing the construction of 2D TFTs based on symmetric groups, we construct 2D TFTs based on walled Brauer algebras. In the construction, the introduction of a dual basis manifests a similarity between the two theories. We next construct a class of 2D field theories whose physical operators have the same symmetry as multi-trace operators constructed from some matrices. Such field theories correspond to non-commutative Frobenius algebras. A matrix structure arises as a consequence of the noncommutativity. Correlation functions of the Gaussian complex multi-matrix models can be translated into correlation functions of the two-dimensional field theories.
Algebraic theory of recombination spaces.
Stadler, P F; Wagner, G P
1997-01-01
A new mathematical representation is proposed for the configuration space structure induced by recombination, which we call "P-structure." It consists of a mapping of pairs of objects to the power set of all objects in the search space. The mapping assigns to each pair of parental "genotypes" the set of all recombinant genotypes obtainable from the parental ones. It is shown that this construction allows a Fourier decomposition of fitness landscapes into a superposition of "elementary landscapes." This decomposition is analogous to the Fourier decomposition of fitness landscapes on mutation spaces. The elementary landscapes are obtained as eigenfunctions of a Laplacian operator defined for P-structures. For binary string recombination, the elementary landscapes are exactly the p-spin functions (Walsh functions), that is, the same as the elementary landscapes of the string point mutation spaces (i.e., the hypercube). This supports the notion of a strong homomorphism between string mutation and recombination spaces. However, the effective nearest neighbor correlations on these elementary landscapes differ between mutation and recombination and among different recombination operators. On average, the nearest neighbor correlation is higher for one-point recombination than for uniform recombination. For one-point recombination, the correlations are higher for elementary landscapes with fewer interacting sites as well as for sites that have closer linkage, confirming the qualitative predictions of the Schema Theorem. We conclude that the algebraic approach to fitness landscape analysis can be extended to recombination spaces and provides an effective way to analyze the relative hardness of a landscape for a given recombination operator. PMID:10021760
TBGG- INTERACTIVE ALGEBRAIC GRID GENERATION
NASA Technical Reports Server (NTRS)
Smith, R. E.
1994-01-01
TBGG, Two-Boundary Grid Generation, applies an interactive algebraic grid generation technique in two dimensions. The program incorporates mathematical equations that relate the computational domain to the physical domain. TBGG has application to a variety of problems using finite difference techniques, such as computational fluid dynamics. Examples include the creation of a C-type grid about an airfoil and a nozzle configuration in which no left or right boundaries are specified. The underlying two-boundary technique of grid generation is based on Hermite cubic interpolation between two fixed, nonintersecting boundaries. The boundaries are defined by two ordered sets of points, referred to as the top and bottom. Left and right side boundaries may also be specified, and call upon linear blending functions to conform interior interpolation to the side boundaries. Spacing between physical grid coordinates is determined as a function of boundary data and uniformly spaced computational coordinates. Control functions relating computational coordinates to parametric intermediate variables that affect the distance between grid points are embedded in the interpolation formulas. A versatile control function technique with smooth cubic spline functions is also presented. The TBGG program is written in FORTRAN 77. It works best in an interactive graphics environment where computational displays and user responses are quickly exchanged. The program has been implemented on a CDC Cyber 170 series computer using NOS 2.4 operating system, with a central memory requirement of 151,700 (octal) 60 bit words. TBGG requires a Tektronix 4015 terminal and the DI-3000 Graphics Library of Precision Visuals, Inc. TBGG was developed in 1986.
Hidden Q-structure and Lie 3-algebra for non-abelian superconformal models in six dimensions
NASA Astrophysics Data System (ADS)
Lavau, Sylvain; Samtleben, Henning; Strobl, Thomas
2014-12-01
We disclose the mathematical structure underlying the gauge field sector of the recently constructed non-abelian superconformal models in six space-time dimensions. This is a coupled system of 1-form, 2-form, and 3-form gauge fields. We show that the algebraic consistency constraints governing this system permit to define a Lie 3-algebra, generalizing the structural Lie algebra of a standard Yang-Mills theory to the setting of a higher bundle. Reformulating the Lie 3-algebra in terms of a nilpotent degree 1 BRST-type operator Q, this higher bundle can be compactly described by means of a Q-bundle; its fiber is the shifted tangent of the Q-manifold corresponding to the Lie 3-algebra and its base the odd tangent bundle of space-time equipped with the de Rham differential. The generalized Bianchi identities can then be retrieved concisely from Q2 = 0, which encode all the essence of the structural identities. Gauge transformations are identified as vertical inner automorphisms of such a bundle, their algebra being determined from a Q-derived bracket.
Algebraic multigrid domain and range decomposition (AMG-DD / AMG-RD)*
Bank, R.; Falgout, R. D.; Jones, T.; Manteuffel, T. A.; McCormick, S. F.; Ruge, J. W.
2015-10-29
In modern large-scale supercomputing applications, algebraic multigrid (AMG) is a leading choice for solving matrix equations. However, the high cost of communication relative to that of computation is a concern for the scalability of traditional implementations of AMG on emerging architectures. This paper introduces two new algebraic multilevel algorithms, algebraic multigrid domain decomposition (AMG-DD) and algebraic multigrid range decomposition (AMG-RD), that replace traditional AMG V-cycles with a fully overlapping domain decomposition approach. While the methods introduced here are similar in spirit to the geometric methods developed by Brandt and Diskin [Multigrid solvers on decomposed domains, in Domain Decomposition Methods inmore » Science and Engineering, Contemp. Math. 157, AMS, Providence, RI, 1994, pp. 135--155], Mitchell [Electron. Trans. Numer. Anal., 6 (1997), pp. 224--233], and Bank and Holst [SIAM J. Sci. Comput., 22 (2000), pp. 1411--1443], they differ primarily in that they are purely algebraic: AMG-RD and AMG-DD trade communication for computation by forming global composite “grids” based only on the matrix, not the geometry. (As is the usual AMG convention, “grids” here should be taken only in the algebraic sense, regardless of whether or not it corresponds to any geometry.) Another important distinguishing feature of AMG-RD and AMG-DD is their novel residual communication process that enables effective parallel computation on composite grids, avoiding the all-to-all communication costs of the geometric methods. The main purpose of this paper is to study the potential of these two algebraic methods as possible alternatives to existing AMG approaches for future parallel machines. As a result, this paper develops some theoretical properties of these methods and reports on serial numerical tests of their convergence properties over a spectrum of problem parameters.« less
Algebraic multigrid domain and range decomposition (AMG-DD / AMG-RD)*
Bank, R.; Falgout, R. D.; Jones, T.; Manteuffel, T. A.; McCormick, S. F.; Ruge, J. W.
2015-10-29
In modern large-scale supercomputing applications, algebraic multigrid (AMG) is a leading choice for solving matrix equations. However, the high cost of communication relative to that of computation is a concern for the scalability of traditional implementations of AMG on emerging architectures. This paper introduces two new algebraic multilevel algorithms, algebraic multigrid domain decomposition (AMG-DD) and algebraic multigrid range decomposition (AMG-RD), that replace traditional AMG V-cycles with a fully overlapping domain decomposition approach. While the methods introduced here are similar in spirit to the geometric methods developed by Brandt and Diskin [Multigrid solvers on decomposed domains, in Domain Decomposition Methods in Science and Engineering, Contemp. Math. 157, AMS, Providence, RI, 1994, pp. 135--155], Mitchell [Electron. Trans. Numer. Anal., 6 (1997), pp. 224--233], and Bank and Holst [SIAM J. Sci. Comput., 22 (2000), pp. 1411--1443], they differ primarily in that they are purely algebraic: AMG-RD and AMG-DD trade communication for computation by forming global composite “grids” based only on the matrix, not the geometry. (As is the usual AMG convention, “grids” here should be taken only in the algebraic sense, regardless of whether or not it corresponds to any geometry.) Another important distinguishing feature of AMG-RD and AMG-DD is their novel residual communication process that enables effective parallel computation on composite grids, avoiding the all-to-all communication costs of the geometric methods. The main purpose of this paper is to study the potential of these two algebraic methods as possible alternatives to existing AMG approaches for future parallel machines. As a result, this paper develops some theoretical properties of these methods and reports on serial numerical tests of their convergence properties over a spectrum of problem parameters.
Boundary Lax pairs from non-ultra-local Poisson algebras
Avan, Jean; Doikou, Anastasia
2009-11-15
We consider non-ultra-local linear Poisson algebras on a continuous line. Suitable combinations of representations of these algebras yield representations of novel generalized linear Poisson algebras or 'boundary' extensions. They are parametrized by a boundary scalar matrix and depend, in addition, on the choice of an antiautomorphism. The new algebras are the classical-linear counterparts of the known quadratic quantum boundary algebras. For any choice of parameters, the non-ultra-local contribution of the original Poisson algebra disappears. We also systematically construct the associated classical Lax pair. The classical boundary principal chiral model is examined as a physical example.
Classification of central extensions of Lax operator algebras
Schlichenmaier, Martin
2008-11-18
Lax operator algebras were introduced by Krichever and Sheinman as further developments of Krichever's theory of Lax operators on algebraic curves. They are infinite dimensional Lie algebras of current type with meromorphic objects on compact Riemann surfaces (resp. algebraic curves) as elements. Here we report on joint work with Oleg Sheinman on the classification of their almost-graded central extensions. It turns out that in case that the finite-dimensional Lie algebra on which the Lax operator algebra is based on is simple there is a unique almost-graded central extension up to equivalence and rescaling of the central element.
Permutation centralizer algebras and multimatrix invariants
NASA Astrophysics Data System (ADS)
Mattioli, Paolo; Ramgoolam, Sanjaye
2016-03-01
We introduce a class of permutation centralizer algebras which underly the combinatorics of multimatrix gauge-invariant observables. One family of such noncommutative algebras is parametrized by two integers. Its Wedderburn-Artin decomposition explains the counting of restricted Schur operators, which were introduced in the physics literature to describe open strings attached to giant gravitons and were subsequently used to diagonalize the Gaussian inner product for gauge invariants of two-matrix models. The structure of the algebra, notably its dimension, its center and its maximally commuting subalgebra, is related to Littlewood-Richardson numbers for composing Young diagrams. It gives a precise characterization of the minimal set of charges needed to distinguish arbitrary matrix gauge invariants, which are related to enhanced symmetries in gauge theory. The algebra also gives a star product for matrix invariants. The center of the algebra allows efficient computation of a sector of multimatrix correlators. These generate the counting of a certain class of bicoloured ribbon graphs with arbitrary genus.
Spinor representations of affine Lie algebras
Frenkel, I. B.
1980-01-01
Let [unk] be an infinite-dimensional Kac-Moody Lie algebra of one of the types Dl+1(2), Bl(1), or Dl(1). These algebras are characterized by the property that an elimination of any endpoint of their Dynkin diagrams gives diagrams of types Bl or Dl of classical orthogonal Lie algebras. We construct two representations of a Lie algebra [unk], which we call spinor representations, following the analogy with the classical case. We obtain that every spinor representation is either irreducible or has two irreducible components. This provides us with an explicit construction of fundamental representations of [unk], two for the type Dl+1(2), three for Bl(1), and four for Dl(1). We note the profound connection of our construction with quantum field theory—in particular, with fermion fields. Comparing the character formulas of our representations with another construction of the fundamental representations of Kac-Moody Lie algebras of types Al(1), Dl(1), El(1), we obtain classical Jacobi identities and addition formulas for elliptic θ-functions. PMID:16592912
The kinematic algebras from the scattering equations
NASA Astrophysics Data System (ADS)
Monteiro, Ricardo; O'Connell, Donal
2014-03-01
We study kinematic algebras associated to the recently proposed scattering equations, which arise in the description of the scattering of massless particles. In particular, we describe the role that these algebras play in the BCJ duality between colour and kinematics in gauge theory, and its relation to gravity. We find that the scattering equations are a consistency condition for a self-dual-type vertex which is associated to each solution of those equations. We also identify an extension of the anti-self-dual vertex, such that the two vertices are not conjugate in general. Both vertices correspond to the structure constants of Lie algebras. We give a prescription for the use of the generators of these Lie algebras in trivalent graphs that leads to a natural set of BCJ numerators. In particular, we write BCJ numerators for each contribution to the amplitude associated to a solution of the scattering equations. This leads to a decomposition of the determinant of a certain kinematic matrix, which appears naturally in the amplitudes, in terms of trivalent graphs. We also present the kinematic analogues of colour traces, according to these algebras, and the associated decomposition of that determinant.
The algebraic Bethe ansatz for scalar products in SU(3)-invariant integrable models
NASA Astrophysics Data System (ADS)
Belliard, S.; Pakuliak, S.; Ragoucy, E.; Slavnov, N. A.
2012-10-01
We study SU(3)-invariant integrable models solvable by a nested algebraic Bethe ansatz. We obtain a determinant representation for the particular case of scalar products of Bethe vectors. This representation can be used for the calculation of form factors and correlation functions of the XXX SU(3)-invariant Heisenberg chain.
Algebraic Bethe Ansatz for Open XXX Model with Triangular Boundary Matrices
NASA Astrophysics Data System (ADS)
Belliard, Samuel; Crampé, Nicolas; Ragoucy, Eric
2013-05-01
We consider an open XXX spin chain with two general boundary matrices whose entries obey a relation, which is equivalent to the possibility to put simultaneously the two matrices in a upper-triangular form. We construct Bethe vectors by means of a generalized algebraic Bethe ansatz. As usual, the method uses Bethe equations and provides transfer matrix eigenvalues.
On the algebraic K-theory of the complex K-theory spectrum
NASA Astrophysics Data System (ADS)
Ausoni, Christian
2010-03-01
Let p>3 be a prime, let ku be the connective complex K-theory spectrum, and let K(ku) be the algebraic K-theory spectrum of ku. We study the p-primary homotopy type of the spectrum K(ku) by computing its mod (p,v_1) homotopy groups. We show that up to a finite summand, these groups form a finitely generated free module over a polynomial algebra F_p[b], where b is a class of degree 2p+2 defined as a higher Bott element.
Schwinger's Measurement Algebra, Preons and the Lepton Masses
NASA Astrophysics Data System (ADS)
Brannen, Carl
2006-04-01
In the 1950s and 1960s, Julian Schwinger developed an elegant general scheme for quantum kinematics and dynamics appropriate to systems with a finite number of dynamical variables, now knowns as ``Schwinger's Measurement Algebra'' (SMA). The SMA has seen little use, largely because it is non relativistic in that it does not allow for particle creation. In this paper, we apply the SMA to the problem of modeling tightly bound subparticles (preons) of the leptons and quarks. We discuss the structure of the ideals of Clifford algebras and, applying this to the elementary fermions, derive a preon substructure for the quarks and leptons. We show that matrices of SMA type elements can be used to model the quarks and leptons under the assumption that the preons are of such high energy that they cannot be created in normal interactions. This gives a definition of the SMA for the composite particle in terms of the SMA of its constituents. We solve the resulting matrix equation for the quarks and leptons. We show that the mass operator for the charged leptons is related to the democratic mass matrix used in the Koide mass formula.
NASA Astrophysics Data System (ADS)
Orantin, N.
2007-09-01
The 2-matrix model has been introduced to study Ising model on random surfaces. Since then, the link between matrix models and combinatorics of discrete surfaces has strongly tightened. This manuscript aims to investigate these deep links and extend them beyond the matrix models, following my work's evolution. First, I take care to define properly the hermitian 2 matrix model which gives rise to generating functions of discrete surfaces equipped with a spin structure. Then, I show how to compute all the terms in the topological expansion of any observable by using algebraic geometry tools. They are obtained as differential forms on an algebraic curve associated to the model: the spectral curve. In a second part, I show how to define such differentials on any algebraic curve even if it does not come from a matrix model. I then study their numerous symmetry properties under deformations of the algebraic curve. In particular, I show that these objects coincide with the topological expansion of the observable of a matrix model if the algebraic curve is the spectral curve of this model. Finally, I show that fine tuning the parameters ensure that these objects can be promoted to modular invariants and satisfy the holomorphic anomaly equation of the Kodaira-Spencer theory. This gives a new hint that the Dijkgraaf-Vafa conjecture is correct.
Computational algebraic geometry of epidemic models
NASA Astrophysics Data System (ADS)
Rodríguez Vega, Martín.
2014-06-01
Computational Algebraic Geometry is applied to the analysis of various epidemic models for Schistosomiasis and Dengue, both, for the case without control measures and for the case where control measures are applied. The models were analyzed using the mathematical software Maple. Explicitly the analysis is performed using Groebner basis, Hilbert dimension and Hilbert polynomials. These computational tools are included automatically in Maple. Each of these models is represented by a system of ordinary differential equations, and for each model the basic reproductive number (R0) is calculated. The effects of the control measures are observed by the changes in the algebraic structure of R0, the changes in Groebner basis, the changes in Hilbert dimension, and the changes in Hilbert polynomials. It is hoped that the results obtained in this paper become of importance for designing control measures against the epidemic diseases described. For future researches it is proposed the use of algebraic epidemiology to analyze models for airborne and waterborne diseases.
An algebra of discrete event processes
NASA Technical Reports Server (NTRS)
Heymann, Michael; Meyer, George
1991-01-01
This report deals with an algebraic framework for modeling and control of discrete event processes. The report consists of two parts. The first part is introductory, and consists of a tutorial survey of the theory of concurrency in the spirit of Hoare's CSP, and an examination of the suitability of such an algebraic framework for dealing with various aspects of discrete event control. To this end a new concurrency operator is introduced and it is shown how the resulting framework can be applied. It is further shown that a suitable theory that deals with the new concurrency operator must be developed. In the second part of the report the formal algebra of discrete event control is developed. At the present time the second part of the report is still an incomplete and occasionally tentative working paper.
Optical systolic solutions of linear algebraic equations
NASA Technical Reports Server (NTRS)
Neuman, C. P.; Casasent, D.
1984-01-01
The philosophy and data encoding possible in systolic array optical processor (SAOP) were reviewed. The multitude of linear algebraic operations achievable on this architecture is examined. These operations include such linear algebraic algorithms as: matrix-decomposition, direct and indirect solutions, implicit and explicit methods for partial differential equations, eigenvalue and eigenvector calculations, and singular value decomposition. This architecture can be utilized to realize general techniques for solving matrix linear and nonlinear algebraic equations, least mean square error solutions, FIR filters, and nested-loop algorithms for control engineering applications. The data flow and pipelining of operations, design of parallel algorithms and flexible architectures, application of these architectures to computationally intensive physical problems, error source modeling of optical processors, and matching of the computational needs of practical engineering problems to the capabilities of optical processors are emphasized.
Automatic code generation in SPARK: Applications of computer algebra and compiler-compilers
Nataf, J.M.; Winkelmann, F.
1992-09-01
We show how computer algebra and compiler-compilers are used for automatic code generation in the Simulation Problem Analysis and Research Kernel (SPARK), an object oriented environment for modeling complex physical systems that can be described by differential-algebraic equations. After a brief overview of SPARK, we describe the use of computer algebra in SPARK's symbolic interface, which generates solution code for equations that are entered in symbolic form. We also describe how the Lex/Yacc compiler-compiler is used to achieve important extensions to the SPARK simulation language, including parametrized macro objects and steady-state resetting of a dynamic simulation. The application of these methods to solving the partial differential equations for two-dimensional heat flow is illustrated.
Automatic code generation in SPARK: Applications of computer algebra and compiler-compilers
Nataf, J.M.; Winkelmann, F.
1992-09-01
We show how computer algebra and compiler-compilers are used for automatic code generation in the Simulation Problem Analysis and Research Kernel (SPARK), an object oriented environment for modeling complex physical systems that can be described by differential-algebraic equations. After a brief overview of SPARK, we describe the use of computer algebra in SPARK`s symbolic interface, which generates solution code for equations that are entered in symbolic form. We also describe how the Lex/Yacc compiler-compiler is used to achieve important extensions to the SPARK simulation language, including parametrized macro objects and steady-state resetting of a dynamic simulation. The application of these methods to solving the partial differential equations for two-dimensional heat flow is illustrated.