Chain models on hecke algebra for corner type representations
NASA Astrophysics Data System (ADS)
Isaev, A. P.; Ogievetsky, O. V.; Os'kin, A. F.
2008-04-01
We consider the integrable open chain models formulated in terms of generators of the Hecke algebra. The spectrum of Hamiltonians for the open Hecke chains of finite size with free boundary conditions is deduced for special (comer type) irreducible representations of the Hecke algebra.
On Ternary Quotients of Cubic Hecke Algebras
NASA Astrophysics Data System (ADS)
Cabanes, Marc; Marin, Ivan
2012-08-01
We prove that the quotient of the group algebra of the braid group introduced by Funar (Commun Math Phys 173:513-558, 1995) collapses in characteristic distinct from 2. In characteristic 2 we define several quotients of it, which are connected to the classical Hecke and Birman-Wenzl-Murakami quotients, but which admit in addition a symmetry of order 3. We also establish conditions on the possible Markov traces factorizing through it.
Baxter Operator and Archimedean Hecke Algebra
NASA Astrophysics Data System (ADS)
Gerasimov, A.; Lebedev, D.; Oblezin, S.
2008-12-01
In this paper we introduce Baxter integral {mathcal{Q}} -operators for finite-dimensional Lie algebras {mathfrak{gl}_{ell+1}} and {mathfrak{so}_{2ell+1}} . Whittaker functions corresponding to these algebras are eigenfunctions of the {mathcal{Q}}-operators with the eigenvalues expressed in terms of Gamma-functions. The appearance of the Gamma-functions is one of the manifestations of an interesting connection between Mellin-Barnes and Givental integral representations of Whittaker functions, which are in a sense dual to each other. We define a dual Baxter operator and derive a family of mixed Mellin-Barnes-Givental integral representations. Givental and Mellin-Barnes integral representations are used to provide a short proof of the Friedberg-Bump and Bump conjectures for G = GL( ℓ + 1) proved earlier by Stade. We also identify eigenvalues of the Baxter {mathcal{Q}}-operator acting on Whittaker functions with local Archimedean L-factors. The Baxter {mathcal{Q}}-operator introduced in this paper is then described as a particular realization of the explicitly defined universal Baxter operator in the spherical Hecke algebra {mathcal {H}(G(mathbb{R}), K)} , K being a maximal compact subgroup of G. Finally we stress an analogy between {mathcal{Q}}-operators and certain elements of the non-Archimedean Hecke algebra {mathcal {H}(G(mathbb{Q}_p),G(mathbb{Z}_p))}.
Eisenstein Hecke algebras and Iwasawa theory
NASA Astrophysics Data System (ADS)
Wake, Preston
We show that if an Eisenstein component of the p-adic Hecke algebra associated to modular forms is Gorenstein, then it is necessary that the plus-part of a certain ideal class group is trivial. We also show that this condition is sufficient whenever a conjecture of Sharifi holds. We also formulate a weaker Gorenstein property, and show that this weak Gorenstein property holds if and only if a weak form of Sharifi's conjecture and a weak form of Greenberg's conjecture hold.
Spherical Hecke algebra in the Nekrasov-Shatashvili limit
NASA Astrophysics Data System (ADS)
Bourgine, Jean-Emile
2015-01-01
The Spherical Hecke central (SHc) algebra has been shown to act on the Nekrasov instanton partition functions of gauge theories. Its presence accounts for both integrability and AGT correspondence. On the other hand, a specific limit of the Omega background, introduced by Nekrasov and Shatashvili (NS), leads to the appearance of TBA and Bethe like equations. To unify these two points of view, we study the NS limit of the SHc algebra. We provide an expression of the instanton partition function in terms of Bethe roots, and define a set of operators that generates infinitesimal variations of the roots. These operators obey the commutation relations defining the SHc algebra at first order in the equivariant parameter ɛ 2. Furthermore, their action on the bifundamental contributions reproduces the Kanno-Matsuo-Zhang transformation. We also discuss the connections with the Mayer cluster expansion approach that leads to TBA-like equations.
All exactly solvable U(1)-invariant quantum spin 1 chains from Hecke algebra
Alcarez, F.C. ); Koberle, R. ); Lima-Santos, A. )
1992-12-10
In this paper, the authors obtain all exactly integrable spin 1 quantum chains, which are U(1) invariant and satisfy the Hecke algebra. The authors present various generalizations for arbitrary spin S and discuss their solution via Bethe ansatz methods.
Topological Strings, Double Affine Hecke Algebras, and Exceptional Knot Homology
NASA Astrophysics Data System (ADS)
Elliot, Ross F.
In this thesis, we consider two main subjects: refined, composite invariants and exceptional knot homologies of torus knots. The main technical tools are double affine Hecke algebras ("DAHA") and various insights from topological string theory. In particular, we define and study the composite DAHA-superpolynomials of torus knots, which depend on pairs of Young diagrams and generalize the composite HOMFLY-PT polynomials from the full HOMFLY-PT skein of the annulus. We also describe a rich structure of differentials that act on homological knot invariants for exceptional groups. These follow from the physics of BPS states and the adjacencies/spectra of singularities associated with Landau-Ginzburg potentials. At the end, we construct two DAHA-hyperpolynomials which are closely related to the Deligne-Gross exceptional series of root systems. In addition to these main themes, we also provide new results connecting DAHA-Jones polynomials to quantum torus knot invariants for Cartan types A and D, as well as the first appearance of quantum E6 knot invariants in the literature.
NASA Astrophysics Data System (ADS)
Fu, Yuchen; Shelley-Abrahamson, Seth
2016-06-01
We give explicit constructions of some finite-dimensional representations of generalized double affine Hecke algebras (GDAHA) of higher rank using R-matrices for U_q(sl_N). Our construction is motivated by an analogous construction of Silvia Montarani in the rational case. Using the Drinfeld-Kohno theorem for Knizhnik-Zamolodchikov differential equations, we prove that the explicit representations we produce correspond to Montarani's representations under a monodromy functor introduced by Etingof, Gan, and Oblomkov.
On boundary fusion and functional relations in the Baxterized affine Hecke algebra
Babichenko, A.; Regelskis, V.
2014-04-15
We construct boundary type operators satisfying fused reflection equation for arbitrary representations of the Baxterized affine Hecke algebra. These operators are analogues of the fused reflection matrices in solvable half-line spin chain models. We show that these operators lead to a family of commuting transfer matrices of Sklyanin type. We derive fusion type functional relations for these operators for two families of representations.
On boundary fusion and functional relations in the Baxterized affine Hecke algebra
NASA Astrophysics Data System (ADS)
Babichenko, A.; Regelskis, V.
2014-04-01
We construct boundary type operators satisfying fused reflection equation for arbitrary representations of the Baxterized affine Hecke algebra. These operators are analogues of the fused reflection matrices in solvable half-line spin chain models. We show that these operators lead to a family of commuting transfer matrices of Sklyanin type. We derive fusion type functional relations for these operators for two families of representations.
On boundary fusion and functional relations in the Baxterized affine Hecke algebra
Babichenko, A.; Regelskis, V.
2014-04-15
We construct boundary type operators satisfying fused reflection equation for arbitrary representations of the Baxterized affine Hecke algebra. These operators are analogues of the fused reflection matrices in solvable half-line spin chain models. We show that these operators lead to a family of commuting transfer matrices of Sklyanin type. We derive fusion type functional relations for these operators for two families of representations.
Double Affine Hecke Algebras of Rank 1 and the Z_3-Symmetric Askey-Wilson Relations
NASA Astrophysics Data System (ADS)
Ito, Tatsuro; Terwilliger, Paul
2010-08-01
We consider the double affine Hecke algebra H=H(k0,k1,k0v,k1v;q) associated with the root system (C1v,C1). We display three elements x, y, z in H that satisfy essentially the Z3-symmetric Askey-Wilson relations. We obtain the relations as follows. We work with an algebra H^ that is more general than H, called the universal double affine Hecke algebra of type (C1v,C1). An advantage of H^ over H is that it is parameter free and has a larger automorphism group. We give a surjective algebra homomorphism H^ → H. We define some elements x, y, z in H^ that get mapped to their counterparts in H by this homomorphism. We give an action of Artin's braid group B3 on H^ that acts nicely on the elements x, y, z; one generator sends x → y → z → x and another generator interchanges x, y. Using the B3 action we show that the elements x, y, z in H^ satisfy three equations that resemble the Z3-symmetric Askey-Wilson relations. Applying the homomorphism H^ → H we find that the elements x, y, z in H satisfy similar relations.
Nekrasov and Argyres-Douglas theories in spherical Hecke algebra representation
NASA Astrophysics Data System (ADS)
Rim, Chaiho; Zhang, Hong
2017-06-01
AGT conjecture connects Nekrasov instanton partition function of 4D quiver gauge theory with 2D Liouville conformal blocks. We re-investigate this connection using the central extension of spherical Hecke algebra in q-coordinate representation, q being the instanton expansion parameter. Based on AFLT basis together with intertwiners we construct gauge conformal state and demonstrate its equivalence to the Liouville conformal state, with careful attention to the proper scaling behavior of the state. Using the colliding limit of regular states, we obtain the formal expression of irregular conformal states corresponding to Argyres-Douglas theory, which involves summation of functions over Young diagrams.
Evens, Sam; Bressler, Paul
1987-01-01
We examine rings that embed into the smash product of the group algebra of the Weyl group with the field of meromorphic functions on the Cartan subalgebra and are generated by elements that satisfy braid relations. We prove that every such ring is isomorphic to either the Hecke algebra, the nil Hecke ring, or the group algebra of the Weyl group. PMID:16593804
Twisted Trace Formula for Hecke Correspondences
NASA Astrophysics Data System (ADS)
Shokranian, Salahoddin
2004-12-01
By the twisted trace formula for the Hecke correspondences we understand the trace formulas for Hecke operators for a non-connected algebraic group acting on certain cohomology spaces. The traces of Hecke operators developed by Selberg in 1957 and the work of Arthur on the traces of Hecke operators of 1989 on L2 -cohomology are in some sense application of analysis and representation theory. On the other hand the work of Eichler in 1957 and of Goresky-MacPherson, Harder and Kottwitz from 1993 to present, reflects more geometric and topological applications. These may be seen as different solutions to the same problem, the calculation of the traces of Hecke operators. The present paper intends to be an introductory note about these operators and their applications.
Early Algebra with Graphics Software as a Type II Application of Technology
ERIC Educational Resources Information Center
Abramovich, Sergei
2006-01-01
This paper describes the use of Kid Pix-graphics software for creative activities of young children--in the context of early algebra as determined by the mathematics core curriculum of New York state. It shows how grade-two appropriate pedagogy makes it possible to bring about a qualitative change in the learning process of those commonly…
Heck's disease: diagnosis and susceptibility.
Bennett, Lindsey K; Hinshaw, Molly
2009-01-01
Focal epithelial hyperplasia, or Heck's disease, is an uncommon proliferation of oral mucosa that presents primarily in Native Central and South American populations. It presents as asymptomatic papules or nodules on the oral mucosa, gingiva, tongue, and lips. In the majority of cases, human papilloma virus 13 or 32 is detected. Factors that determine disease susceptibility are unclear, but genetics, and having the human lymphocytic antigen-DR4 (DRB1*0404) allele in particular, are thought to play a major role in disease vulnerability. We report another case of focal epithelial hyperplasia, hypothesize on disease susceptibility, and review the current understanding of this uncommon disorder.
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…
Cremmer-Gervais r-Matrices and the Cherednik Algebras of Type GL 2
NASA Astrophysics Data System (ADS)
Johnson, Garrett
2010-11-01
We give an interpretation of the Cremmer-Gervais r-matrices for {mathfrak{sl}_n} in terms of actions of elements in the rational and trigonometric Cherednik algebras of type GL 2 on certain subspaces of their polynomial representations. This is used to compute the nilpotency index of the Jordanian r-matrices, thus answering a question of Gerstenhaber and Giaquinto. We also give an interpretation of the Cremmer-Gervais quantization in terms of the corresponding double affine Hecke algebra.
NASA Astrophysics Data System (ADS)
Blumenhagen, Ralph; Fuchs, Michael; Traube, Matthias
2017-07-01
It is shown that the closure of the infinitesimal symmetry transformations underlying classical W algebras give rise to L∞ algebras with in general field dependent gauge parameters. Therefore, the class of well understood W algebras provides highly nontrivial examples of such strong homotopy Lie algebras. We develop the general formalism for this correspondence and apply it explicitly to the classical W_3 algebra.
Kinetic Study of the Heck Reaction: An Interdisciplinary Experiment
ERIC Educational Resources Information Center
Gozzi, Christel; Bouzidi, Naoual
2008-01-01
The aim of this experiment is to study and calculate the kinetic constant of a Heck reaction: the arylation of but-3-en-2-ol by iodobenzene catalyzed by palladium acetate in presence of triethylamine in DMF. The reaction leads to a mixture of two ketones. Students use GC analysis to quantify reagents and products of reaction. They control the…
Kinetic Study of the Heck Reaction: An Interdisciplinary Experiment
ERIC Educational Resources Information Center
Gozzi, Christel; Bouzidi, Naoual
2008-01-01
The aim of this experiment is to study and calculate the kinetic constant of a Heck reaction: the arylation of but-3-en-2-ol by iodobenzene catalyzed by palladium acetate in presence of triethylamine in DMF. The reaction leads to a mixture of two ketones. Students use GC analysis to quantify reagents and products of reaction. They control the…
[Focal epithelial hyperplasia (Heck's disease) in a Turkish family].
Weidner, F
1996-12-01
A 31-year-old Turkish patient and some family members suffered from multiple hyperplastic oral mucosal papules. Intralesional papilloma virus was not found but the patient had elevated levels of CD8 lymphocytes in his peripheral blood. We diagnosed focal epithelial hyperplasia of Heck.
Hecke Groups, Dessins d'Enfants and the Archimedean Solids
NASA Astrophysics Data System (ADS)
He, Yang-Hui; Read, James
2015-12-01
Grothendieck's dessins d'enfants arise with ever-increasing frequency in many areas of 21st century mathematical physics. In this paper, we review the connections between dessins and the theory of Hecke groups. Focussing on the restricted class of highly symmetric dessins corresponding to the so-called Archimedean solids, we apply this theory in order to provide a means of computing representatives of the associated conjugacy classes of Hecke subgroups in each case. The aim of this paper is to demonstrate that dessins arising in mathematical physics can point to new and hitherto unexpected directions for further research. In addition, given the particular ubiquity of many of the dessins corresponding to the Archimedean solids, the hope is that the computational results of this paper will prove useful in the further study of these objects in mathematical physics contexts.
Late-onset adenosine deaminase deficiency presenting with Heck's disease.
Artac, Hasibe; Göktürk, Bahar; Bozdemir, Sefika Elmas; Toy, Hatice; van der Burg, Mirjam; Santisteban, Ines; Hershfield, Michael; Reisli, Ismail
2010-08-01
Focal epithelial hyperplasia, also known as Heck's disease, is a rare but distinctive entity of viral etiology with characteristic clinical and histopathological features. It is a benign, asymptomatic disease of the oral mucosa caused by human papilloma viruses (HPV). Previous studies postulated an association between these lesions and immunodeficiency. Genetic deficiency of adenosine deaminase (ADA) results in varying degrees of immunodeficiency, including neonatal onset severe combined immunodeficiency (ADA-SCID), and milder, later onset immunodeficiency. We report a 12-year-old girl with the late onset-ADA deficiency presenting with Heck's disease. Our case report should draw attention to the possibility of immunodeficiency in patients with HPV-induced focal epithelial hyperplasia.
Non-Abelian Vortices, Hecke Modifications and Singular Monopoles
NASA Astrophysics Data System (ADS)
Baptista, J. M.
2010-06-01
In this note, we show that for the group G = U( N) the space of Hecke modifications of a rank N vector bundle over a Riemann surface C coincides with the moduli space of solutions of certain non-Abelian vortex equations over C. Through the recent work of Kapustin and Witten this then leads to an isomorphism between the moduli space of vortices and the moduli space of singular monopoles on the product of C with a closed interval I.
1998-06-01
on courses being taught at NPS. LIST OF REFERENCES [1] Anton , Howard , Elementary Linear Algebra , John Wiley and Sons, New York, New York, 1994...and computational techniques for solving systems of linear equations. The goal is to enhance current matrix algebra textbooks and help the beginning... algebra is the study of algebraic operations on matrices and of their applications, primarily for solving systems of linear equations. Systems of
Heck-type reactions of imine derivatives: a DFT study.
Li, Zhe; Fu, Yao; Zhang, Song-Lin; Guo, Qing-Xiang; Liu, Lei
2010-06-01
The mechanism of a recently discovered intramolecular Heck-type coupling of oximes with aryl halides (Angew. Chem. Int. Ed. 2007, 46, 6325) was systematically studied by using density functional methods enhanced with a polarized continuum solvation model. The overall catalytic cycle of the reaction was found to consist of four steps: oxidative addition, migratory insertion, beta-H elimination, and catalyst regeneration, whereas an alternative base-promoted C-H activation pathway was determined to be less favorable. Migratory insertion was found to be the rate determining step in the catalytic cycle. The apparent activation barrier of migratory insertion of the (E)-oxime was +20.5 kcal mol(-1), whereas the barrier of (Z)-oxime was as high as +32.7 kcal mol(-1). However, (Z)-oxime could isomerize to form the more active (E)-oxime with the assistance of K(2)CO(3), so that both the (E)- and (Z)-oxime substrates could be transformed to the desired product. Our calculations also indicated that the Z product was predominant in the equilibrium of the isomerization of the imine double bond, which constituted the reason for the good Z-selectivity observed for the reaction. Furthermore, we examined the difference between the intermolecular Heck-type reactions of imines and of olefins. It was found that in the intermolecular Heck-type coupling of imines, the apparent activation barrier of migratory insertion was as high as +35 kcal mol(-1), which should be the main obstacle of the reaction. The analysis also revealed the main problem for the intermolecular Heck-type reactions of imines, which was that the breaking of a C=N pi bond was much more difficult than the breaking of a C=C pi bond. After systematic examination of a series of substituted imines, (Z)-N-amino imine and N-acetyl imine were found to have relatively low barriers of migratory insertion, so that they might be possible substrates for intermolecular Heck-type coupling.
Synthesis of Phenylpropanoids via Matsuda-Heck Coupling of Arene Diazonium Salts.
Schmidt, Bernd; Wolf, Felix
2017-04-21
The Pd-catalyzed Heck-type coupling (Matsuda-Heck reaction) of electron rich arene diazonium salts with electron deficient olefins has been exploited for the synthesis of phenylpropanoid natural products. Examples described herein are the naturally occurring benzofurans methyl wutaifuranate, wutaifuranol, wutaifuranal, their 7-methoxy derivatives, and the O-prenylated natural products boropinols A and C.
On the Li Coefficients for the Hecke L-functions
NASA Astrophysics Data System (ADS)
Omar, Sami; Ouni, Raouf; Mazhouda, Kamel
2014-06-01
In this paper, we compute and verify the positivity of the Li coefficients for the Hecke L-functions using an arithmetic formula established in Omar and Mazhouda, J. Number Theory 125(1), 50-58 (2007) and J. Number Theory 130(4), 1098-1108 (2010) and the Serre trace formula. Additional results are presented, including new formulas for the Li coefficients and a formulation of a criterion for the partial Riemann hypothesis. Basing on the numerical computations made below, we conjecture that these coefficients are increasing in n.
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.…
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.…
ERIC Educational Resources Information Center
Amdahl, Kenn; Loats, Jim
This book discusses algebra in a non-threatening, fun way. It explains concepts, vocabulary, and strategies of algebra in understandable terms. Chapter titles include: "Numbers with Interesting Properties"; "Important Concepts"; "Fraction Refresher"; "Terms, Factors, and Polynomials"; "Rearranging Expressions"; "Handy Tricks and Magic Words";…
Magnetically Separable Fe3O4@DOPA-Pd: A Heterogeneous Catalyst for Aqueous Heck Reaction
Magnetically separable Fe3O4@DOPA-Pd catalyst has been synthesized via anchoring of palladium over dopamine-coated magnetite via non-covalent interaction and the catalyst is utilized for expeditious Heck coupling in aqueous media.
Magnetically Separable Fe3O4@DOPA-Pd: A Heterogeneous Catalyst for Aqueous Heck Reaction
Magnetically separable Fe3O4@DOPA-Pd catalyst has been synthesized via anchoring of palladium over dopamine-coated magnetite via non-covalent interaction and the catalyst is utilized for expeditious Heck coupling in aqueous media.
Ionic liquid effects on Mizoroki-Heck reactions: more than just carbene complex formation.
Gyton, Matthew R; Cole, Marcus L; Harper, Jason B
2011-08-28
Reaction profiles for a Mizoroki-Heck reaction in either an ionic liquid or a molecular solvent with different palladium sources demonstrate that the rate enhancements observed in ionic liquids cannot be solely attributed to Pd-carbene complex formation.
Sellinger, Alan; Tamaki, Ryo; Laine, Richard M; Ueno, Kazunori; Tanabe, Hiroshi; Williams, Evan; Jabbour, Ghassan E
2005-08-07
A new solution processable nanocomposite material has been prepared via the Heck coupling of octavinylsilsesquioxane with a selected bromoaromatic hole transport compound. Resultant electroluminescent devices show an 18% improvement in external quantum efficiencies over their small molecule analogues.
Focal epithelial hyperplasia (Heck's disease) in two Chinese females.
Liu, N; Li, Y; Zhou, Y; Zeng, X
2012-08-01
Focal epithelial hyperplasia, or Heck's disease, is a relatively rare virus-induced benign disease. To the best of the authors' knowledge it has not been reported in an ethnic Chinese population. The authors report two cases of focal epithelial hyperplasia (FEH) in Chinese patients, which were clinically and histologically in accord with FEH. In particular, the lesions in one case were located on the gingival mucosa, which is rarely affected by FEH. DNA extracted from paraffin-embedded specimens from the two patients was tested for the presence of human papilloma virus followed by speciﬁc polymerase chain reaction testing for 16, 18, 13, and 32 subtypes in order to conﬁrm the clinical diagnosis.
Miles, Kelsey C; Le, Chi Chip; Stambuli, James P
2014-09-01
The formation of exo-methylene indanones and indenones from simple ortho-allyl benzoic acid derivatives has been developed. Selective formation of the indanone or indenone products in these reactions is controlled by choice of ancillary ligand. This new process has a low environmental footprint as the products are formed in high yields using low catalyst loadings, while the only stoichiometric chemical waste generated from the reactants in the transformation is acetic acid. The conversion of the active cyclization catalyst into the Hermman-Beller palladacycle was exploited in a one-pot tandem acyl Heck-Heck (aHH) reaction, and utilized in the synthesis of donepezil.
NASA Technical Reports Server (NTRS)
Mulligan, Jeffrey B.
2017-01-01
A color algebra refers to a system for computing sums and products of colors, analogous to additive and subtractive color mixtures. We would like it to match the well-defined algebra of spectral functions describing lights and surface reflectances, but an exact correspondence is impossible after the spectra have been projected to a three-dimensional color space, because of metamerism physically different spectra can produce the same color sensation. Metameric spectra are interchangeable for the purposes of addition, but not multiplication, so any color algebra is necessarily an approximation to physical reality. Nevertheless, because the majority of naturally-occurring spectra are well-behaved (e.g., continuous and slowly-varying), color algebras can be formulated that are largely accurate and agree well with human intuition. Here we explore the family of algebras that result from associating each color with a member of a three-dimensional manifold of spectra. This association can be used to construct a color product, defined as the color of the spectrum of the wavelength-wise product of the spectra associated with the two input colors. The choice of the spectral manifold determines the behavior of the resulting system, and certain special subspaces allow computational efficiencies. The resulting systems can be used to improve computer graphic rendering techniques, and to model various perceptual phenomena such as color constancy.
Fan, Jian-Hong; Yang, Ji; Song, Ren-Jie; Li, Jin-Heng
2015-02-20
A new Pd(II)-catalyzed alkene oxidative difunctionalization initiated by Heck insertion has been developed for the selective synthesis of acyclic and cyclic all-carbon quaternary stereocenters, which achieves an oxidative Heck-type alkylation, aryl migration, and desulfonylation sequence and represents a different input from those previously used Heck coupling in synthesis is reported.
NASA Astrophysics Data System (ADS)
Mikhalev, A. V.; Pinchuk, I. A.
2005-06-01
The structure of Steinberg conformal algebras is studied; these are analogues of Steinberg groups (algebras, superalgebras).A Steinberg conformal algebra is defined as an abstract algebra by a system of generators and relations between the generators. It is proved that a Steinberg conformal algebra is the universal central extension of the corresponding conformal Lie algebra; the kernel of this extension is calculated.
ERIC Educational Resources Information Center
Capani, Antonio; De Dominicis, Gabriel
This paper proposes a model for a general interface between people and Computer Algebra Systems (CAS). The main features in the CAS interface are data navigation and the possibility of accessing powerful remote machines. This model is based on the idea of session management, in which the main engine of the tool enables interactions with the…
Phosphite-oxazole/imidazole ligands in asymmetric intermolecular Heck reaction.
Mazuela, Javier; Tolstoy, Paivi; Pàmies, Oscar; Andersson, Pher G; Diéguez, Montserrat
2011-02-07
We describe the application of a new class of ligands--the phosphite-oxazole/imidazole (L1-L5a-g)--in asymmetric intermolecular Pd-catalyzed Heck reactions under thermal and microwave conditions. These ligands combine the advantages of the oxazole/imidazole moiety with those of the phosphite moiety: they are more stable than their oxazoline counterparts, less sensitive to air and other oxidizing agents than phosphines and phosphinites, and easy to synthesize from readily available alcohols. The results indicate that activities, regio- and enantioselectivities, are highly influenced by the type of nitrogen donor group (oxazole or imidazole), the oxazole and biaryl-phosphite substituents and the axial chirality of the biaryl moiety of the ligand. By carefully selecting the ligand components, we achieved high activities, regio- (up to 99%) and enantioselectivities (up to 99%) using several triflate sources. Under microwave-irradiation conditions, reaction times were considerably shorter (from 24 h to 30 min) and regio- and enantioselectivities were still excellent.
Jiménez, Fermín; Fernández, Antonio; Boulifa, Ettahir; Mansour, Ahmed Ibn; Alvarez-Manzaneda, Ramón; Chahboun, Rachid; Alvarez-Manzaneda, Enrique
2017-09-15
The first synthesis of antifungal sesquiterpene quinol dasyscyphin E was achieved starting from trans-communic acid. The process described involves the first diastereoselective synthesis of this type of compound by cyclization of an aryl bicyclosesquiterpene. The acid was efficiently transformed into a sesquiterpene synthon, which was converted into the corresponding bromoaryl sesquiterpene. The key step of synthetic sequence was the cyclization of the latter under Heck reaction conditions, which yielded the tetracyclic skeleton of the target compound with complete diastereoselectivity. The participation of an acetate group is decisive, both for the course of the Heck reaction and for the stereoselectivity of the process.
A rapid microwave protocol for Heck vinylation of aryl chlorides under air.
Datta, Gopal K; Vallin, Karl S A; Larhed, Mats
2003-01-01
In modern high-throughput chemistry, the overall workflow is a crucial factor and much work is devoted to speeding up the process of chemistry development. Since automated microwave-based synthesizers are known to streamline the compound production and to accelerate slow organic transformations, this technology was implemented for Heck reactions with sluggish aryl chlorides. Furthermore, homogeneous palladium-catalyzed Heck vinylations of aryl chlorides can be performed under air under optimized conditions. Based on this finding, controlled microwave heating was utilized to accelerate model reactions down to 30 min employing a mixture of ionic liquid and 1,4-dioxane as solvent.
Magnetically separable Fe3O4@DOPA??Pd: a heterogeneous catalyst for aqueous Heck reaction
Magnetically separable Fe3O4@DOPA??Pd catalyst has been synthesized via anchoring of palladium over dopamine-coated magnetite via co-ordinate interactionand the catalyst is utilized for expeditious Heck coupling in aqueous media.This dataset is associated with the following publication:Baig, N., J. Leazer , and R. Varma. Magnetically Separable Fe3O4@DOPA-Pd: A Heterogeneous Catalyst for Aqueous Heck Reaction. CLEAN TECHNOLOGIES AND ENVIRONMENTAL POLICY. Springer-Verlag, New York, NY, USA, 17(7): 2073-2077, (2015).
NASA Astrophysics Data System (ADS)
Golenishcheva-Kutuzova, Maria I.; Kac, Victor G.
1998-04-01
Γ-conformal algebra is an axiomatic description of the operator product expansion of chiral fields with simple poles at finitely many points. We classify these algebras and their representations in terms of Lie algebras and their representations with an action of the group Γ. To every Γ-conformal algebra and a character of Γ we associate a Lie algebra generated by fields with the OPE with simple poles. Examples include twisted affine Kac-Moody algebras, the sin algebra (which is a "Γ-conformal" analogue of the general linear algebra) and its analogues, the algebra of pseudodifferential operators on the circle, etc.
The local Hurwitz constant and Diophantine approximation on Hecke groups
NASA Astrophysics Data System (ADS)
Lehner, J.
1990-10-01
Define the Hecke group by {G_q} = < {( {begin{array}{*{20}{c}} 1 & {{⪉mb... ...{array}{*{20}{c}} 0 & { - 1} 1 & 0 } )} rangle , {λ _q} = 2cos π /q , q = 3,4, ldots . We call {G_q}(∞ ) the {G_q} -rationals, and R - {G_q}(∞ ) the {G_q} -irrationals. The problem we treat here is the approximation of {G_q} -irrationals by {G_q} -rationals. Let M(α ) be the upper bound of numbers c for which \\vertα - k/m\\vert < 1/c{m^2} for all {G_q} -irrationals and infinitely many k/m in {G_q}(∞ ) . Set h_q'= {inf _α }M(α ) . We call h_q' the Hurwitz constant for {G_q} . It is known that h_q'= 2 , q even; h_q'= 2{(1 + {(1 - {λ _q}/2)^2})^{1/2}} , q odd. In this paper we prove this result by using {λ _q} -continued fractions, as developed previously by D. Rosen. Write α - frac{{{P_{n - 1}}}}{{{Q_{n - 1}}}} = frac{{{{( - 1)}^{... ...}{\\varepsilon _2} \\cdots {\\varepsilon _n}}}{{{m_{n - 1}}(α )Q_{n - 1}^2}}, where {\\varepsilon _i} = ± 1 and {P_i}/{Q_i} are the convergents of the {λ _q} -continued fraction for α . Then M(α ) = {overline {lim } _n}{m_n}(α ) . We call {m_n}(α ) the local Hurwitz constant. In the final section we prove some results on the local Hurwitz constant. For example (Theorem 4), it is shown that if q is odd and {\\varepsilon _{n + 1}} = {\\varepsilon _{n + 2}} = + 1 , then {m_i} ≥ {(λ _q^2 + 4)^{1/2}} > h_q' for at least one of i = n - 1,n,n + 1 .
Classification of Non-Affine Non-Hecke Dynamical R-Matrices
NASA Astrophysics Data System (ADS)
Avan, Jean; Billaud, Baptiste; Rollet, Geneviève
2012-09-01
A complete classification of non-affine dynamical quantum R-matrices obeying the Gl_n({C})-Gervais-Neveu-Felder equation is obtained without assuming either Hecke or weak Hecke conditions. More general dynamical dependences are observed. It is shown that any solution is built upon elementary blocks, which individually satisfy the weak Hecke condition. Each solution is in particular characterized by an arbitrary partition {{I}(i),; iin\\{1,dots,n}} of the set of indices {1,dots,n} into classes, {I}(i) being the class of the index i, and an arbitrary family of signs (ɛ_{I})_{{I}in{{I}(i), ; iin{1,dots,n}}} on this partition. The weak Hecke-type R-matrices exhibit the analytical behaviour R_{ij,ji}=f(ɛ_{{I}(i)}Λ_{{I}(i)}-ɛ_{{I}(j)}Λ_{{I}(j)}), where f is a particular trigonometric or rational function, Λ_{{I}(i)}=sumlimits_{jin{I}(i)}λ_j, and (λ_i)_{iin{1,dots,n}} denotes the family of dynamical coordinates.
Wang, Jian; Tang, Shi; Zhu, Qiang
2016-07-01
Efficient access to five- to seven-membered cyclic ketoimines, through palladium-catalyzed intramolecular imidoylative Heck reaction of alkene-containing isocyanides, has been developed. Consecutive isocyanide and alkene insertion into aryl or alkyl Pd(II) intermediates takes place in this process. No byproduct derived from monoinsertion or reversed sequence is detected.
Focal epithelial hyperplasia (Heck's disease) in three Kenyan girls: case reports.
Chindia, M L; Awange, D O; Guthua, S W; Mwaniki, D L
1993-09-01
We report the first three patients diagnosed with focal epithelial hyperplasia (Heck's disease) in Kenya. Clinically they presented as focal or diffuse papillomatous lesions in the oral mucosa. Histopathological features rule out other similar lesions inter alia multiple fibro-epithelial and viral warts.
Successful topical treatment of focal epithelial hyperplasia (Heck's disease) with interferon-beta.
Steinhoff, M; Metze, D; Stockfleth, E; Luger, T A
2001-05-01
We report the successful topical treatment of focal epithelial hyperplasia (Heck's disease) with interferon-beta (Fiblaferon gel). Topical treatment with interferon-beta appears to be an effective, simple, non-invasive, cheap and low-risk alternative to other invasive or surgical therapeutic modalities.
Ligand-free Heck reaction: Pd(OAc)2 as an active catalyst revisited.
Yao, Qingwei; Kinney, Elizabeth P; Yang, Zhi
2003-09-19
Palladium acetate was shown to be an extremely active catalyst for the Heck reaction of aryl bromides. Both the base and the solvent were found to have a fundamental influence on the efficiency of the reaction, with K(3)PO(4) and N,N-dimethylacetamide being the optimal base and solvent, respectively.
Preparation of Vinyl Silyl Ethers and Disiloxanes via the Silyl-Heck Reaction of Silyl Ditriflates
Martin, Sara E. S.; Watson, Donald A.
2013-01-01
Vinyl silyl ethers and disiloxanes can now be prepared from aryl-substituted alkenes and related substrates using a silyl-Heck reaction. The reaction employs a commercially available catalyst system and mild conditions. This work represents a highly practical means of accessing diverse classes of vinyl silyl ether substrates in an efficient and direct manner with complete regio- and geometric selectivity. PMID:23984876
NASA Astrophysics Data System (ADS)
Vaninsky, Alexander
2011-04-01
This article introduces a trigonometric field (TF) that extends the field of real numbers by adding two new elements: sin and cos - satisfying an axiom sin2 + cos2 = 1. It is shown that by assigning meaningful names to particular elements of the field, all known trigonometric identities may be introduced and proved. Two different interpretations of the TF are discussed with many others potentially possible. The main objective of this article is to introduce a broader view of trigonometry that can serve as motivation for mathematics students and teachers to study and teach abstract algebraic structures.
Ternary algebraic approach to extended superconformal algebras
NASA Astrophysics Data System (ADS)
Günaydin, Murat; Hyun, Seungjoon
1992-04-01
The construction of extended ( N = 2 and N = 4) superconformal algebras (SCA) over very general classes of ternary algebras (triple systems) is given. For N = 2 this construction leads to superconformal algebras corresponding to certain Kählerian coset spaces of Lie groups with non-vanishing torsion. In general, a given Lie group admits more than one coset space of this type. The construction and a complete classification of N = 2 SCAs over Kantor triple system is given. In particular, the division algebras and their tensor products lead to N = 2 superconformal algebras associated with the coset spaces of the groups of the Magic Square. For a very special class of ternary algebras, namely the Freudenthal triple (FT) systems, the N = 2 superconformal algebras can be extended to N = 4 superconformal algebras with the gauge group SU(2)×SU(2)×U(1). The realization and a complete classification of N = 2 and N = 4
Derive Workshop Matrix Algebra and Linear Algebra.
ERIC Educational Resources Information Center
Townsley Kulich, Lisa; Victor, Barbara
This document presents the course content for a workshop that integrates the use of the computer algebra system Derive with topics in matrix and linear algebra. The first section is a guide to using Derive that provides information on how to write algebraic expressions, make graphs, save files, edit, define functions, differentiate expressions,…
The Heck-type arylation of alkenes was achieved in aqueous polyethylene glycol using a magnetically recoverable heterogenized palladium catalyst employing diaryliodonium salts under ambient conditions. The benign reaction medium and the stability of the catalyst are the salient f...
The Heck-type arylation of alkenes was achieved in aqueous polyethylene glycol using a magnetically recoverable heterogenized palladium catalyst employing diaryliodonium salts under ambient conditions. The benign reaction medium and the stability of the catalyst are the salient f...
Yaşar, Sedat; Ozcan, Emine Ozge; Gürbüz, Nevin; Cetinkaya, Bekir; Ozdemir, Ismail
2010-01-28
An efficient and stereoselective catalytic system for the Heck cross coupling reaction using novel 1,3-dialkyl-3,4,5,6-tetrahydropyrimidinium salts (1, LHX) and Pd(OAc)2 loading has been reported. The palladium complexes derived from the salts 1a-f prepared in situ exhibit good catalytic activity in the Heck coupling reaction of aryl bromides under mild conditions.
ERIC Educational Resources Information Center
Edwards, Edgar L., Jr., Ed.
The fundamentals of algebra and algebraic thinking should be a part of the background of all citizens in society. The vast increase in the use of technology requires that school mathematics ensure the teaching of algebraic thinking as well as its use at both the elementary and secondary school levels. Algebra is a universal theme that runs through…
On Exceptional Superconformal Algebras
Poletaeva, Elena
2010-06-17
We obtain realizations of superconformal algebras K(2), K'(4) and the exceptional N = 6 superconformal algebra in matrices over a Weyl algebra of size 2x2, 4x4 and 8x8, respectively. A general construction of these realizations is based on the spin representations of the orthogonal Lie algebras. We give explicit descriptions of these matrix realizations.
NASA Astrophysics Data System (ADS)
Foulis, David J.; Jenčová, Anna; Pulmannová, Sylvia
2017-02-01
Different versions of the notion of a state have been formulated for various so-called quantum structures. In this paper, we investigate the interplay among states on synaptic algebras and on its sub-structures. A synaptic algebra is a generalization of the partially ordered Jordan algebra of all bounded self-adjoint operators on a Hilbert space. The paper culminates with a characterization of extremal states on a commutative generalized Hermitian algebra, a special kind of synaptic algebra.
Quantum cluster algebras and quantum nilpotent algebras.
Goodearl, Kenneth R; Yakimov, Milen T
2014-07-08
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.
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
Oxidative Heck vinylation for the synthesis of complex dienes and polyenes.
Delcamp, Jared H; Gormisky, Paul E; White, M Christina
2013-06-12
We introduce an oxidative Heck reaction for selective complex diene and polyene formation. The reaction proceeds via oxidative Pd(II)/sulfoxide catalysis that retards palladium-hydride isomerizations which previously limited the Heck manifold's capacity for furnishing stereodefined conjugated dienes. Limiting quantities of nonactivated terminal olefins (1 equiv) and slight excesses of vinyl boronic esters (1.5 equiv) that feature diverse functionality can be used to furnish complex dienes and polyenes in good yields and excellent selectivities (generally E:Z = >20:1; internal:terminal = >20:1). Because this reaction only requires prior activation of a single vinylic carbon, improvements in efficiency are observed for synthetic sequences relative to ones featuring reactions that require activation of both coupling partners.
Enantioselective Heck arylations of acyclic alkenyl alcohols using a redox-relay strategy.
Werner, Erik W; Mei, Tian-Sheng; Burckle, Alexander J; Sigman, Matthew S
2012-12-14
Progress in the development of asymmetric Heck couplings of arenes and acyclic olefins has been limited by a tenuous understanding of the factors that dictate selectivity in migratory insertion and β-hydride elimination. On the basis of key mechanistic insight recently garnered in the exploration of selective Heck reactions, we report here an enantioselective variant that delivers β-, γ-, or δ-aryl carbonyl products from acyclic alkenol substrates. The catalyst system imparts notable regioselectivity during migratory insertion and promotes the migration of the alkene's unsaturation toward the alcohol to ultimately form the ketone product. The reaction uses aryldiazonium salts as the arene source, yields enantiomeric products from opposite starting alkene configurations, and uses a readily accessible ligand. The racemic nature of the alkenol substrate does not bias the enantioselection.
A generalization of the Funk-Hecke theorem to the case of hyperbolic spaces
NASA Astrophysics Data System (ADS)
Shtepina, T. V.
2004-10-01
The well-known Funk-Hecke theorem states that for integral operators whose kernels depend only on the distance between points in spherical geometry and where the integral is taken over the surface of a hypersphere, every surface spherical harmonic is an eigenvector. In this paper we extend this theorem to the case of non-compact Lobachevsky spaces. We compute the corresponding eigenvalue in some physically important cases.
Palladium-catalyzed Heck-type cross-couplings of unactivated alkyl iodides.
McMahon, Caitlin M; Alexanian, Erik J
2014-06-02
A palladium-catalyzed, intermolecular Heck-type coupling of alkyl iodides and alkenes is described. This process is successful with a variety of primary and secondary unactivated alkyl iodides as reaction partners, including those with hydrogen atoms in the β position. The mild catalytic conditions enable intermolecular C-C bond formations with a diverse set of alkyl iodides and alkenes, including substrates containing base- or nucleophile-sensitive functionality.
Jayasooriya, P R; Abeyratne, S; Ranasinghe, A W; Tilakaratne, W M
2004-07-01
Focal epithelial hyperplasia (FEH) (Heck's disease) is essentially a benign oral infection produced by the human papillomavirus (HPV). Although this condition is known to exist in numerous populations and ethnic groups, it is relatively rare in South-East Asia. The following report is based on two cases of adult FEH with histopathological features in favour of the disease. In addition, polymerase chain reaction was performed to detect the presence of HPV DNA in the lesions in order to confirm the histopathological diagnosis.
Deformed Virasoro Algebras from Elliptic Quantum Algebras
NASA Astrophysics Data System (ADS)
Avan, J.; Frappat, L.; Ragoucy, E.
2017-09-01
We revisit the construction of deformed Virasoro algebras from elliptic quantum algebras of vertex type, generalizing the bilinear trace procedure proposed in the 1990s. It allows us to make contact with the vertex operator techniques that were introduced separately at the same period. As a by-product, the method pinpoints two critical values of the central charge for which the center of the algebra is extended, as well as (in the gl(2) case) a Liouville formula.
Zhang, Fengwei; Niu, Jianrui; Wang, Haibo; Yang, Honglei; Jin, Jun; Liu, Na; Zhang, Yubin; Li, Rong; Ma, Jiantai
2012-02-15
Highlights: Black-Right-Pointing-Pointer Palladium-based heterogeneous catalyst was prepared facilely via the co-precipitation method. Black-Right-Pointing-Pointer The particles are nearly spherical in shape with an average size of 20 {+-} 1.0 nm. Black-Right-Pointing-Pointer The developed magnetic catalyst showed high activity for Heck reaction. Black-Right-Pointing-Pointer The catalyst was easily recovered from the reaction mixture with external magnetic field. Black-Right-Pointing-Pointer The catalytic efficiency for Heck reaction remains unaltered even after 6 repeated cycles. -- Abstract: A novel and high-performance palladium-based catalyst for Heck reaction was prepared easily by the co-precipitation method. The catalyst was characterized by transmission electron microscopy (TEM), X-ray powder diffraction (XRD), vibrating sample magnetometry (VSM), X-ray photoelectron spectroscopy (XPS) and atomic absorption spectrophotometry (AAS). The catalyst afforded a fast conversion of the 4-bromonitrobenzene to 4-nitrostilbene at a catalyst loading of 5 mol%, and the efficiency of the catalyst remains unaltered even after 6 repeated cycles. The excellent catalytic performance of the Pd/Fe{sub 3}O{sub 4} catalyst might be attributed to the enhanced synergistic effect between Pd nanoparticles and magnetite.
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…
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…
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…
ERIC Educational Resources Information Center
Miller, L. Diane; England, David A.
1989-01-01
Describes a study in a large metropolitan high school to ascertain what influence the use of regular writing in algebra classes would have on students' attitudes towards algebra and their skills in algebra. Reports the simpler and more direct the writing topics the better. (MVL)
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…
Arithmetic: Prerequisite to Algebra?
ERIC Educational Resources Information Center
Rotman, Jack W.
Drawing from research and observations at Lansing Community College (Michigan) (LCC), this paper argues that typical arithmetic courses do little to prepare students to master algebra, and proposes an alternative set of arithmetic skills as actual prerequisites to algebra. The first section offers a description of the algebra sequence at LCC,…
ERIC Educational Resources Information Center
Mitchell, Sarah
2006-01-01
Many students find the leap from arithmetic to algebra a difficult one to make, and current literature suggests that a number of the common misconceptions held by students arise as a result of the methods which teachers use to present algebra. The abstract nature of algebra makes it difficult for students to grasp; students also struggle with the…
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,…
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.
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.
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.
Discrete Minimal Surface Algebras
NASA Astrophysics Data System (ADS)
Arnlind, Joakim; Hoppe, Jens
2010-05-01
We consider discrete minimal surface algebras (DMSA) as generalized noncommutative analogues of minimal surfaces in higher dimensional spheres. These algebras appear naturally in membrane theory, where sequences of their representations are used as a regularization. After showing that the defining relations of the algebra are consistent, and that one can compute a basis of the enveloping algebra, we give several explicit examples of DMSAs in terms of subsets of sln (any semi-simple Lie algebra providing a trivial example by itself). A special class of DMSAs are Yang-Mills algebras. The representation graph is introduced to study representations of DMSAs of dimension d ≤ 4, and properties of representations are related to properties of graphs. The representation graph of a tensor product is (generically) the Cartesian product of the corresponding graphs. We provide explicit examples of irreducible representations and, for coinciding eigenvalues, classify all the unitary representations of the corresponding algebras.
On Derivations Of Genetic Algebras
NASA Astrophysics Data System (ADS)
Mukhamedov, Farrukh; Qaralleh, Izzat
2014-11-01
A genetic algebra is a (possibly non-associative) algebra used to model inheritance in genetics. In application of genetics this algebra often has a basis corresponding to genetically different gametes, and the structure constant of the algebra encode the probabilities of producing offspring of various types. In this paper, we find the connection between the genetic algebras and evolution algebras. Moreover, we prove the existence of nontrivial derivations of genetic algebras in dimension two.
Zang, Qin; Javed, Salim; Porubsky, Patrick; Ullah, Farman; Neuenswander, Benjamin; Lushington, Gerald H.; Basha, Fatima Z.; Organ, Michael G.; Hanson, Paul R.
2012-01-01
The synthesis of a unique isoindoline- and tetrahydroisoquinoline (THIQ)-containing tricyclic sultam library, utilizing a Heck-aza-Michael (HaM) strategy is reported. Both isoindoline and THIQ rings are installed through a Heck reaction on a vinylsulfonamide, followed by one-pot deprotection and intramolecular aza-Michael reaction. Subsequent cyclization with either paraformaldehyde condensation or 1,1'-carbonyldiimidazole coupling generates a variety of tricyclic sultams. Overall, a 160-member library of these sultams, together with their isoindolines/THIQ and secondary sulfonamides precursors, were constructed using this strategy. PMID:22311745
Zang, Qin; Javed, Salim; Porubsky, Patrick; Ullah, Farman; Neuenswander, Benjamin; Lushington, Gerald H; Basha, Fatima Z; Organ, Michael G; Hanson, Paul R
2012-03-12
The synthesis of a unique isoindoline- and tetrahydroisoquinoline (THIQ)-containing tricyclic sultam library, utilizing a Heck-aza-Michael (HaM) strategy is reported. Both isoindoline and THIQ rings are installed through a Heck reaction on a vinylsulfonamide, followed by one-pot deprotection and intramolecular aza-Michael reaction. Subsequent cyclization with either paraformaldehyde condensation or 1,1'-carbonyldiimidazole coupling generates a variety of tricyclic sultams. Overall, a 160-member library of these sultams, together with their isoindolines/THIQ and secondary sulfonamides precursors, were constructed using this strategy.
Aurrecoechea, José M; Durana, Aritz; Pérez, Elena
2008-05-02
Palladium-catalyzed heterocyclization-coupling sequences have been developed starting from buta-1,2,3-trienyl carbinols and electron-deficient alkenes. Polysubstituted furans are formed where the heterocyclic ring originates from the elements of the butatrienyl carbinol while the electron-deficient olefin is incorporated as a C-3 substituent. In most cases, the reaction proceeds via a Heck-type pathway leading to the efficient formation of 3-vinylfurans. However, couplings with methyl vinyl ketone display a divergent behavior to afford selectively either Heck- or hydroarylation-type products depending on reaction conditions.
de Azambuja, Francisco; Carmona, Rafaela C; Chorro, Tomaz H D; Heerdt, Gabriel; Correia, Carlos Roque D
2016-08-01
S- and P-Stereogenic heterocycles were synthesized by a remarkably simple enantioselective Heck desymmetrization reaction based on the unprecedented noncovalent directing effect of S=O and P=O functionalities. Selected prochiral symmetric substrates were efficiently arylated using the recently disclosed chiral PyraBOx ligand under mild and open-flask reaction conditions. Several five-membered aryl- sulfones, sulfoxides, and phosphine oxides were synthesized in good to excellent yields, in good to high diastereoselectivity, and enantiomeric ratios up to 98:2. Theoretical calculations also support the noncovalent directing effect of the S=O and P=O functionalities during the arylation process.
Ourailidou, Maria E; Dockerty, Paul; Witte, Martin; Poelarends, Gerrit J; Dekker, Frank J
2015-03-28
The detection of protein lysine acylations remains a challenge due to lack of specific antibodies for acylations with various chain lengths. This problem can be addressed by metabolic labeling techniques using carboxylates with reactive functionalities. Subsequent chemoselective reactions with a complementary moiety connected to a detection tag enable the visualization and quantification of the protein lysine acylome. In this study, we present EDTA-Pd(II) as a novel catalyst for the oxidative Heck reaction on protein-bound alkenes, which allows employment of fully aqueous reaction conditions. We used this reaction to monitor histone lysine acylation in vitro after metabolic incorporation of olefinic carboxylates as chemical reporters.
Linear polystyrene-stabilized PdO nanoparticle-catalyzed Mizoroki-Heck reactions in water.
Ohtaka, Atsushi; Yamaguchi, Tomohiro; Teratani, Takuto; Shimomura, Osamu; Nomura, Ryôki
2011-10-27
Linear polystyrene-stabilized PdO nanoparticles (PS-PdONPs) were prepared by thermal decomposition of Pd(OAc)(2) in the presence of polystyrene. X-ray diffraction (XRD) and transmission electron microscopy (TEM) indicated the production of PdO nanoparticles. The loading of palladium was determined by inductively coupled plasma-atomic emission spectroscopy (ICP-AES). PS-PdONPs exhibited high catalytic activity for Mizoroki-Heck reactions under air in water and could be recycled without loss of activity.
Zheng, Changwu; Wang, Dian; Stahl, Shannon S.
2012-01-01
Pd-catalyzed aerobic oxidative coupling of vinylboronic acids and electronically unbiased alkyl olefins provides regioselective access to 1,3-disubstituted conjugated dienes. Catalyst-controlled regioselectivity is achieved by using 2,9-dimethylphenanthroline as a ligand. The observed regioselectivity is opposite to that observed from a traditional (non-oxidative) Heck reaction between a vinyl bromide and an alkene. DFT computational studies reveal that steric effects of the 2,9-dimethylphenanthroline ligand promote C–C bond-formation at the internal position of the alkene. PMID:22998540
Classification of Normal Subgroups of Hecke Group H6 in Terms of Parabolic Class Number
NASA Astrophysics Data System (ADS)
Yurttas, Aysun; Demirci, Musa; Cangul, I. Naci
2011-09-01
In [3], Greenberg showed that n≤6t3 so that μ = nt≤6t4 for a normal subgroup N of level n and index μ having t parabolic classes in the modular group Γ. Accola, [1], improved these to n≤6t2 always and n≤t2 if Γ/N is not abelian. Newman, [5], obtained another generalisation of these results. Hecke groups are generalisations of the modular group. We particularly deal with one of the most important cases, q = 6.
Rubina, Marina; Sherrill, William M; Barkov, Alexey Yu
2014-01-01
Summary A novel class of chiral phosphanyl-oxazoline (PHOX) ligands with a conformationally rigid cyclopropyl backbone was synthesized and tested in the intermolecular asymmetric Heck reaction. Mechanistic modelling and crystallographic studies were used to predict the optimal ligand structure and helped to design a very efficient and highly selective catalytic system. Employment of the optimized ligands in the asymmetric arylation of cyclic olefins allowed for achieving high enantioselectivities and significantly suppressing product isomerization. Factors affecting the selectivity and the rate of the isomerization were identified. It was shown that the nature of this isomerization is different from that demonstrated previously using chiral diphosphine ligands. PMID:25161709
Prediction of Algebraic Instabilities
NASA Astrophysics Data System (ADS)
Zaretzky, Paula; King, Kristina; Hill, Nicole; Keithley, Kimberlee; Barlow, Nathaniel; Weinstein, Steven; Cromer, Michael
2016-11-01
A widely unexplored type of hydrodynamic instability is examined - large-time algebraic growth. Such growth occurs on the threshold of (exponentially) neutral stability. A new methodology is provided for predicting the algebraic growth rate of an initial disturbance, when applied to the governing differential equation (or dispersion relation) describing wave propagation in dispersive media. Several types of algebraic instabilities are explored in the context of both linear and nonlinear waves.
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)
Infinite dimensional matrix algebras
NASA Astrophysics Data System (ADS)
Bordemann, M.; Hoppe, J.; Schaller, P.
1989-11-01
To each (finite dimensional) Lie algebra g we associate a class L λ(g) of infinite dimensional Lie algebras, induced by representations D λ(g). We show that in the case of sl(2, C) one obtains a series of pairwise non-isomorphic infinite dimensional Lie algebras depending continuously on a complex parameter λ. We connect this method with previous results on the relation between Diff AS 2 and su( N), and comment on a recent conjecture concerning higher spin algebras, and (2 + 1)-dimensional gravity.
Santos-Silva, Paulo Roberto; Fonseca, Alfredo José; Castro, Anita Weigand de; Greve, Júlia Maria D'Andréa; Hernandez, Arnaldo José
2007-08-01
To determine the degree of reproducibility of maximum oxygen consumption (VO2max) among soccer players, using a modified Heck protocol. 2 evaluations with an interval of 15 days between them were performed on 11 male soccer players. All the players were at a high performance level; they were training for an average of 10 hours per week, totaling 5 times a week. When they were evaluated, they were in the middle of the competitive season, playing 1 match per week. The soccer players were evaluated on an ergometric treadmill with velocity increments of 1.2 km.h-1 every 2 minutes and a fixed inclination of 3% during the test. VO2max was measured directly using a breath-by-breath metabolic gas analyzer. The maximum running speed and VO2max attained in the 2 tests were, respectively: (15.6 +/- 1.1 vs. 15.7 +/- 1.2 km.h-1; [P = .78]) and (54.5 +/- 3.9 vs. 55.2 +/- 4.4 ml.kg-1.min-1; [P = .88]). There was high and significant correlation of VO2max between the 2 tests with a 15-day interval between them [r = 0.97; P < .001]. The modified Heck protocol was reproducible, and the 15-day interval between the ergospirometric testing was insufficient to significantly modify the soccer players' VO2max values.
Xu, Liping; Hilton, Margaret J; Zhang, Xinhao; Norrby, Per-Ola; Wu, Yun-Dong; Sigman, Matthew S; Wiest, Olaf
2014-02-05
The enantioselective Pd-catalyzed redox-relay Heck arylation of acyclic alkenyl alcohols allows access to various useful chiral building blocks from simple olefinic substrates. Mechanistically, after the initial migratory insertion, a succession of β-hydride elimination and migratory insertion steps yields a saturated carbonyl product instead of the more general Heck product, an unsaturated alcohol. Here, we investigate the reaction mechanism, including the relay function, yielding the final carbonyl group transformation. M06 calculations predict a ΔΔG(‡) of 1 kcal/mol for the site selectivity and 2.5 kcal/mol for the enantioselectivity, in quantitative agreement with experimental results. The site selectivity is controlled by a remote electronic effect, where the developing polarization of the alkene in the migratory insertion transition state is stabilized by the C-O dipole of the alcohol moiety. The enantioselectivity is controlled by steric repulsion between the oxazoline substituent and the alcohol-bearing alkene substituent. The relay efficiency is due to an unusually smooth potential energy surface without high barriers, where the hydroxyalkyl-palladium species acts as a thermodynamic sink, driving the reaction toward the carbonyl product. Computational predictions of the relative reactivity and selectivity of the double bond isomers are validated experimentally.
The Virasoro vertex algebra and factorization algebras on Riemann surfaces
NASA Astrophysics Data System (ADS)
Williams, Brian
2017-08-01
This paper focuses on the connection of holomorphic two-dimensional factorization algebras and vertex algebras which has been made precise in the forthcoming book of Costello-Gwilliam. We provide a construction of the Virasoro vertex algebra starting from a local Lie algebra on the complex plane. Moreover, we discuss an extension of this factorization algebra to a factorization algebra on the category of Riemann surfaces. The factorization homology of this factorization algebra is computed as the correlation functions. We provide an example of how the Virasoro factorization algebra implements conformal symmetry of the beta-gamma system using the method of effective BV quantization.
Crawley, Matthew L; Phipps, Kristin M; Goljer, Igor; Mehlmann, John F; Lundquist, Joseph T; Ullrich, John W; Yang, Cuijian; Mahaney, Paige E
2009-03-05
An efficient and mild method to couple aryl bromides and activated non-allylic alcohols in a Heck reaction with tandem isomerization to selectively afford high yields of 1,5-diarylalkan-1-ones has been developed. Mechanistic insight was gained through NMR studies of products derived from deuterium-labeled intermediates.
Language Approaches to Beginning Algebra.
ERIC Educational Resources Information Center
Rotman, Jack W.
1990-01-01
Ideas which apply language concepts to the study of algebra are presented. Discussed are algebraic notation, vocabulary, vocalization, and written assignments. The careful use of algebraic language in mathematics classes is emphasized. (CW)
Bicovariant quantum algebras and quantum Lie algebras
NASA Astrophysics Data System (ADS)
Schupp, Peter; Watts, Paul; Zumino, Bruno
1993-10-01
A bicovariant calculus of differential operators on a quantum group is constructed in a natural way, using invariant maps from Fun(mathfrak{G}_q ) to U q g, given by elements of the pure braid group. These operators—the “reflection matrix” Y≡L + SL - being a special case—generate algebras that linearly close under adjoint actions, i.e. they form generalized Lie algebras. We establish the connection between the Hopf algebra formulation of the calculus and a formulation in compact matrix form which is quite powerful for actual computations and as applications we find the quantum determinant and an orthogonality relation for Y in SO q (N).
Seo, Young Joo; Kim, Young Hee
2016-01-01
In this paper we construct some real algebras by using elementary functions, and discuss some relations between several axioms and its related conditions for such functions. We obtain some conditions for real-valued functions to be a (edge) d-algebra.
ERIC Educational Resources Information Center
Merlin, Ethan M.
2013-01-01
This article describes how the author has developed tasks for students that address the missed "essence of the matter" of algebraic transformations. Specifically, he has found that having students practice "perceiving" algebraic structure--by naming the "glue" in the expressions, drawing expressions using…
ERIC Educational Resources Information Center
Levy, Alissa Beth
2012-01-01
The California Department of Education (CDE) has long asserted that success Algebra I by Grade 8 is the goal for all California public school students. In fact, the state's accountability system penalizes schools that do not require all of their students to take the Algebra I end-of-course examination by Grade 8 (CDE, 2009). In this dissertation,…
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
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…
Learning Activity Package, Algebra.
ERIC Educational Resources Information Center
Evans, Diane
A set of ten teacher-prepared Learning Activity Packages (LAPs) in beginning algebra and nine in intermediate algebra, these units cover sets, properties of operations, number systems, open expressions, solution sets of equations and inequalities in one and two variables, exponents, factoring and polynomials, relations and functions, radicals,…
NASA Technical Reports Server (NTRS)
Lawson, C. L.; Krogh, F. T.; Gold, S. S.; Kincaid, D. R.; Sullivan, J.; Williams, E.; Hanson, R. J.; Haskell, K.; Dongarra, J.; Moler, C. B.
1982-01-01
The Basic Linear Algebra Subprograms (BLAS) library is a collection of 38 FORTRAN-callable routines for performing basic operations of numerical linear algebra. BLAS library is portable and efficient source of basic operations for designers of programs involving linear algebriac computations. BLAS library is supplied in portable FORTRAN and Assembler code versions for IBM 370, UNIVAC 1100 and CDC 6000 series computers.
Algebraic Reasoning through Patterns
ERIC Educational Resources Information Center
Rivera, F. D.; Becker, Joanne Rossi
2009-01-01
This article presents the results of a three-year study that explores students' performance on patterning tasks involving prealgebra and algebra. The findings, insights, and issues drawn from the study are intended to help teach prealgebra and algebra. In the remainder of the article, the authors take a more global view of the three-year study on…
ERIC Educational Resources Information Center
Levy, Alissa Beth
2012-01-01
The California Department of Education (CDE) has long asserted that success Algebra I by Grade 8 is the goal for all California public school students. In fact, the state's accountability system penalizes schools that do not require all of their students to take the Algebra I end-of-course examination by Grade 8 (CDE, 2009). In this dissertation,…
Parastatistics Algebras and Combinatorics
NASA Astrophysics Data System (ADS)
Popov, T.
2005-03-01
We consider the algebras spanned by the creation parafermionic and parabosonic operators which give rise to generalized parastatistics Fock spaces. The basis of such a generalized Fock space can be labelled by Young tableaux which are combinatorial objects. By means of quantum deformations a nice combinatorial structure of the algebra of the plactic monoid that lies behind the parastatistics is revealed.
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…
NASA Astrophysics Data System (ADS)
Majewski, Władysław A.; Tylec, Tomasz I.
2010-12-01
Erik M. Alfsen and Frederic W. Shultz had recently developed the characterisation of state spaces of operator algebras. It established full equivalence (in the mathematical sense) between the Heisenberg and the Schrödinger picture, i.e. given a physical system we are able to construct its state space out of its observables as well as to construct algebra of observables from its state space. As an underlying mathematical structure they used the theory of duality of ordered linear spaces and obtained results are valid for various types of operator algebras (namely C *, von Neumann, JB and JBW algebras). Here, we show that the language they developed also admits a representation of an effect algebra.
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.
Overman, Larry E.; Rosen, Mark D.
2010-01-01
A catalytic intramolecular Heck reaction, followed by capture of the resulting η3-allylpalladium intermediate by a tethered diketopiperazine, is the central step in a concise synthetic route to (−)-spirotryprostatin B and three stereoisomers. This study demonstrates that an acyclic, chiral η3-allylpalladium fragment generated in a catalytic asymmetric Heck cyclization can be trapped by even a weakly nucleophilic diketopiperazine more rapidly than it undergoes diastereomeric equilibration. PMID:20725641
Algebraic Nonlinear Collective Motion
NASA Astrophysics Data System (ADS)
Troupe, J.; Rosensteel, G.
1998-11-01
Finite-dimensional Lie algebras of vector fields determine geometrical collective models in quantum and classical physics. Every set of vector fields on Euclidean space that generates the Lie algebra sl(3, R) and contains the angular momentum algebra so(3) is determined. The subset of divergence-free sl(3, R) vector fields is proven to be indexed by a real numberΛ. TheΛ=0 solution is the linear representation that corresponds to the Riemann ellipsoidal model. The nonlinear group action on Euclidean space transforms a certain family of deformed droplets among themselves. For positiveΛ, the droplets have a neck that becomes more pronounced asΛincreases; for negativeΛ, the droplets contain a spherical bubble of radius |Λ|1/3. The nonlinear vector field algebra is extended to the nonlinear general collective motion algebra gcm(3) which includes the inertia tensor. The quantum algebraic models of nonlinear nuclear collective motion are given by irreducible unitary representations of the nonlinear gcm(3) Lie algebra. These representations model fissioning isotopes (Λ>0) and bubble and two-fluid nuclei (Λ<0).
Algebraic invariants for homotopy types
NASA Astrophysics Data System (ADS)
Blanc, David
1999-11-01
We define a sequence of purely algebraic invariants - namely, classes in the Quillen cohomology of the [Pi]-algebra [pi][low asterisk]X - for distinguishing between different homotopy types of spaces. Another sequence of such cohomology classes allows one to decide whether a given abstract [Pi]-algebra can be realized as the homotopy [Pi]-algebra of a space.
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,…
Adaptive algebraic reconstruction technique
Lu Wenkai; Yin Fangfang
2004-12-01
Algebraic reconstruction techniques (ART) are iterative procedures for reconstructing objects from their projections. It is proven that ART can be computationally efficient by carefully arranging the order in which the collected data are accessed during the reconstruction procedure and adaptively adjusting the relaxation parameters. In this paper, an adaptive algebraic reconstruction technique (AART), which adopts the same projection access scheme in multilevel scheme algebraic reconstruction technique (MLS-ART), is proposed. By introducing adaptive adjustment of the relaxation parameters during the reconstruction procedure, one-iteration AART can produce reconstructions with better quality, in comparison with one-iteration MLS-ART. Furthermore, AART outperforms MLS-ART with improved computational efficiency.
Duan, Hui; Li, Mengyang; Zhang, Guanghui; Gallagher, James R.; Huang, Zhiliang; Sun, Yu; Luo, Zhong; Chen, Hongzhong; Miller, Jeffrey T.; Zou, Ruqiang; Lei, Aiwen; Zhao, Yanli
2015-01-01
ABSTRACT: The development of organometallic single-site catalysts (SSCs) has inspired the designs of new heterogeneous catalysts with high efficiency. Nevertheless, the application of SSCs in certain modern organic reactions, such as C-C bond formation reactions, has still been less investigated. In this study, a single-site Pd(II) catalyst was developed, where 2,2'-bipyridine-grafted periodic mesoporous organosilica (PMO) was employed as the support of a Pd(II) complex. The overall performance of the single-site Pd(II) catalyst in the oxidative Heck reaction was then investigated. The investigation results show that the catalyst displays over 99% selectivity for the product formation with high reaction yield. Kinetic profiles further confirm its high catalytic efficiency, showing that the rate constant is nearly 40 times higher than that for the free Pd(II) salt. X-ray absorption spectroscopy reveals that the catalyst has remarkable lifetime and recyclability.
Avila, Carolina M; Patel, Jigar S; Reddi, Yernaidu; Saito, Masato; Nelson, Hosea M; Shunatona, Hunter P; Sigman, Matthew S; Sunoj, Raghavan B; Toste, F Dean
2017-05-15
A mild, asymmetric Heck-Matsuda reaction of five-, six- and seven-membered ring alkenes and aryl diazonium salts is presented. High yields and enantioselectivities were achieved using Pd(0) and chiral anion co-catalysts, the latter functioning as a chiral anion phase-transfer (CAPT) reagent. For certain substrate classes, the chiral anion catalysts were modulated to minimize the formation of undesired by-products. More specifically, BINAM-derived phosphoric acid catalysts were shown to prevent alkene isomerization in cyclopentene and cycloheptene starting materials. DFT(B3LYP-D3) computations revealed that increased product selectivity resulted from a chiral anion dependent lowering of the activation barrier for the desired pathway. © 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim.
Andappan, Murugaiah M S; Nilsson, Peter; Larhed, Mats
2003-01-01
The useful and selective reactivity of arylboronic acids makes them favourite building blocks for many modern organic chemistry applications like the metal-mediated formation of C-C, C-O, C-N, and C-S bonds. This report describes oxidative Heck coupling reactions of arylboronic acids and olefins, which were conveniently and rapidly (5-30 min) carried out under air with temperature-controlled microwave heating. Different reaction conditions were investigated with regard to both microwave heating capability and chemical productivity. Copper(II) acetate was identified as a microwave compatible reoxidant of Pd(0). The scope and limitations of this high-speed chemistry protocol with diverse olefins and organoboronic acids are discussed.
Ozden, Bora; Gunduz, Kaan; Gunhan, Omer; Ozden, Feyza Otan
2011-12-01
Focal epithelial hyperplasia or Heck's disease, is a rare viral infection of the oral mucosa caused by human papillomavirus. The frequency of this disease varies widely from one geographic region to another. In Caucasians there have been only few cases reported. This paper reports a case of focal epithelial hyperplasia and demonstrates the association with HPV subtype 32 through polymerase chain reaction (PCR) and sequencing of PCR products. A 7-year-old Caucasian girl was admitted to our clinic for investigation of multiple oral mucosal lesions in the mouth. Lesion was excised under local anesthesia without any complication. The lesion was diagnosed as focal epithelial hyperplasia according to both clinical and histopathological features. Dental staff should be aware of these kind of lesions and histopathological examination together with a careful clinical observation should be carried out for a definitive diagnosis.
Upper Bounds for the Level of Normal Subgroups of Hecke Groups
NASA Astrophysics Data System (ADS)
Demirci, Musa; Yurttas, Aysun; Cangul, I. Naci
2011-09-01
In [4], Greenberg showed that n≤6t3 so that μ = nt≤6t4 for a normal subgroup N of level n and index μ having t parabolic classes in the modular group Γ. Accola, [1], improved these to n≤6t2 always and n≤t2 if Γ/N is not abelian. In this work we generalise these results to Hecke groups. We get results between three parameters of a normal subgroup, i.e. the index μ, the level n and the parabolic class number t. We deal with the case q = 4, and then obtain the generalisation to other q. Two main problems here are the calculation of the number of normal subgroups and the determination of the bounds on the level n for a given t.
Pawliczek, Martin; Milde, Bastian; Jones, Peter G; Werz, Daniel B
2015-08-24
An intramolecular domino process consisting of a formal anti-carbopalladation followed by Heck reaction is realized. Complex oligo(hetero)cyclic scaffolds are efficiently obtained in one synthetic step from easily obtainable enyne precursors. In contrast to common syn-carbopalladation reactions of alkyne units, the carbopalladation employed here is designed to afford an anti-arrangement of the two new substituents across the emerging double bond. A prerequisite is that the residues next to the alkyne should lack any β-hydrogen atoms. The method paves the way to tri- and tetrasubstituted double-bond systems that have not been accessible by conventional Pd catalysis. © 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.
Ekebergh, Andreas; Lingblom, Christine; Sandin, Peter; Wennerås, Christine; Mårtensson, Jerker
2015-03-21
Design of Experiments (DoE) has been used to optimize a diversity oriented palladium catalyzed cascade Heck-Suzuki reaction for the construction of 3-alkenyl substituted cyclopenta[b]indole compounds. The obtained DoE model revealed a reaction highly dependent on the ligand. Guided by the model, an optimal ligand was chosen that selectively delivered the desired products in high yields. The conditions were applicable with a variety of boronic acids and were used to synthesize a library of 3-alkenyl derivatized compounds. Focusing on inhibition of kinases relevant for combating melanoma, the library was used in an initial structure-activity survey. In line with the observed kinase inhibition, cellular studies revealed one of the more promising derivatives to inhibit cell proliferation via an apoptotic mechanism.
Pseudo-Riemannian Novikov algebras
NASA Astrophysics Data System (ADS)
Chen, Zhiqi; Zhu, Fuhai
2008-08-01
Novikov algebras were introduced in connection with the Poisson brackets of hydrodynamic-type and Hamiltonian operators in formal variational calculus. Pseudo-Riemannian Novikov algebras denote Novikov algebras with non-degenerate invariant symmetric bilinear forms. In this paper, we find that there is a remarkable geometry on pseudo-Riemannian Novikov algebras, and give a special class of pseudo-Riemannian Novikov algebras.
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.
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)
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)
Jordan Algebraic Quantum Categories
NASA Astrophysics Data System (ADS)
Graydon, Matthew; Barnum, Howard; Ududec, Cozmin; Wilce, Alexander
2015-03-01
State cones in orthodox quantum theory over finite dimensional complex Hilbert spaces enjoy two particularly essential features: homogeneity and self-duality. Orthodox quantum theory is not, however, unique in that regard. Indeed, all finite dimensional formally real Jordan algebras -- arenas for generalized quantum theories with close algebraic kinship to the orthodox theory -- admit homogeneous self-dual positive cones. We construct categories wherein these theories are unified. The structure of composite systems is cast from universal tensor products of the universal C*-algebras enveloping ambient spaces for the constituent state cones. We develop, in particular, a notion of composition that preserves the local distinction of constituent systems in quaternionic quantum theory. More generally, we explicitly derive the structure of hybrid quantum composites with subsystems of arbitrary Jordan algebraic type.
ERIC Educational Resources Information Center
Leron, Uri; Dubinsky, Ed
1995-01-01
Describes a constructivist, interactive approach for teaching undergraduate mathematics, abstract algebra in particular, using computer constructions programmed in ISETL to induce students' mental constructions and collaborative learning to help students reflect on these constructions. (18 references) (MKR)
Accounting Equals Applied Algebra.
ERIC Educational Resources Information Center
Roberts, Sondra
1997-01-01
Argues that students should be given mathematics credits for completing accounting classes. Demonstrates that, although the terminology is different, the mathematical concepts are the same as those used in an introductory algebra class. (JOW)
NASA Astrophysics Data System (ADS)
Markarian, Nikita
2017-03-01
We introduce Weyl n-algebras and show how their factorization complex may be used to define invariants of manifolds. In the appendix, we heuristically explain why these invariants must be perturbative Chern-Simons invariants.
Aprepro - Algebraic Preprocessor
2005-08-01
Aprepro is an algebraic preprocessor that reads a file containing both general text and algebraic, string, or conditional expressions. It interprets the expressions and outputs them to the output file along witht the general text. Aprepro contains several mathematical functions, string functions, and flow control constructs. In addition, functions are included that, with some additional files, implement a units conversion system and a material database lookup system.
Covariant deformed oscillator algebras
NASA Technical Reports Server (NTRS)
Quesne, Christiane
1995-01-01
The general form and associativity conditions of deformed oscillator algebras are reviewed. It is shown how the latter can be fulfilled in terms of a solution of the Yang-Baxter equation when this solution has three distinct eigenvalues and satisfies a Birman-Wenzl-Murakami condition. As an example, an SU(sub q)(n) x SU(sub q)(m)-covariant q-bosonic algebra is discussed in some detail.
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.
NASA Astrophysics Data System (ADS)
Fialowski, A.; Schlichenmaier, M.
2007-11-01
In two earlier articles we constructed algebraic-geometric families of genus one (i.e. elliptic) Lie algebras of Krichever Novikov type. The considered algebras are vector fields, current and affine Lie algebras. These families deform the Witt algebra, the Virasoro algebra, the classical current, and the affine Kac Moody Lie algebras respectively. The constructed families are not equivalent (not even locally) to the trivial families, despite the fact that the classical algebras are formally rigid. This effect is due to the fact that the algebras are infinite dimensional. In this article the results are reviewed and developed further. The constructions are induced by the geometric process of degenerating the elliptic curves to singular cubics. The algebras are of relevance in the global operator approach to the Wess Zumino Witten Novikov models appearing in the quantization of Conformal Field Theory.
Li, Hongfang; Lü, Jian; Lin, Jingxiang; Huang, Yuanbiao; Cao, Minna; Cao, Rong
2013-11-11
A series of MPdMe10 CB[5] (M=Li, Na, K, Rb, and Cs; Me10 CB[5]=decamethylcucurbit[5]uril) hybrid solid materials have been successfully synthesized for the first time through a simple diffusion method. These as-prepared hybrid solids have been applied as phosphine-free precatalysts for Heck cross-coupling reactions with excellent catalytic performance and good recyclability. In the processes of the catalytic reactions, the activated Pd(II) species were released from the crystalline hybrid precatalysts and transformed into catalytically active Pd nanoparticles, which have been demonstrated as key to carry on the catalytic reactions for the recoverable precatalysts MPdMe10 CB[5] (M=K, Rb, and Cs). It has also been rationalized that the introduction of different alkali metals afforded crystalline hybrid precatalysts with different crystal structures, which are responsible for their diversified stability and reusability presented in Heck reactions.
An in Situ Generated Palladium on Aluminum Oxide: Applications in Gram-Scale Matsuda-Heck Reactions.
Pape, Simon; Daukšaitė, Lauryna; Lucks, Sandra; Gu, Xiaoting; Brunner, Heiko
2016-12-16
In situ generated palladium on aluminum oxide provides an active catalytic system for Matsuda-Heck reactions in gram-scale. The novel catalyst proceeded through a significantly higher catalytic activity compared to the classical Pd/C system. Based on the high catalytic activity the first α,β,β-triarylation of methyl acrylate in good yields could be provided in one-step.
Zakrzewski, Jerzy; Huras, Bogumiła
2015-01-01
Cinnamic acid derivatives bearing a nitroxyl moiety (2,2,6,6-tetramethyl-1-oxyl-4-piperidyl 3-E-aryl acrylates) were synthesized in 30-100% yield using a Mizoroki-Heck cross-coupling reaction between 4-acryloyloxy-2,2,6,6-tetramethylpiperidine-1-oxyl and iodobenzene derivatives in the presence of palladium(II) acetate coordinated with a tri(o-tolyl)phosphine ligand immobilized in a polyurea matrix.
Huras, Bogumiła
2015-01-01
Summary Cinnamic acid derivatives bearing a nitroxyl moiety (2,2,6,6-tetramethyl-1-oxyl-4-piperidyl 3-E-aryl acrylates) were synthesized in 30–100% yield using a Mizoroki–Heck cross-coupling reaction between 4-acryloyloxy-2,2,6,6-tetramethylpiperidine-1-oxyl and iodobenzene derivatives in the presence of palladium(II) acetate coordinated with a tri(o-tolyl)phosphine ligand immobilized in a polyurea matrix. PMID:26199672
Feng, Zhang; Min, Qiao-Qiao; Zhao, Hai-Yang; Gu, Ji-Wei; Zhang, Xingang
2015-01-19
An efficient palladium-catalyzed Heck-type reaction of fluoroalkyl halides, including perfluoroalkyl bromides, trifluoromethyl iodides, and difluoroalkyl bromides, has been developed. The reaction proceeds under mild reaction conditions with high efficiency and broad substrate scope, and provides a general and straightforward access to fluoroalkylated alkenes which are of interest in life and material sciences. © 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.
McAtee, Jesse R.; Martin, Sara E. S.; Cinderella, Andrew P.; Reid, William B.; Johnson, Keywan A.
2014-01-01
For the first time, nickel-catalyzed silyl-Heck reactions are reported. Using simple phosphine-supported nickel catalysts, direct activation of silyl triflates has been achieved. These results contrast earlier palladium-catalyzed systems, which require iodide additives to activate silyl-triflates. These nickel-based catalysts exhibit good functional group tolerance in the preparation of vinyl silanes, and unlike earlier systems, allows for the incorporation of trialkylsilanes larger than Me3Si. PMID:24914247
Qian, Qun; Zang, Zhenhua; Chen, Yang; Tong, Weiqi; Gong, Hegui
2013-05-01
Cross-coupling of alkyl halides with alkenes leading to Heck-type and addition products is summarized. The development of Heck reaction with aliphatic halides although has made significant progress in the past decade and particularly recently, it was much less explored in comparison with the aryl halides. The use of Ni- and Co-catalyzed protocols allowed efficient Heck coupling of activated and unactivated alkenes with 1°, 2° and 3° alkyl halides. In addition, radical conjugate addition to activated alkenes has become a well-established method that has led to efficient construction of many natural products. The utilization of Ni- and Co-catalyzed strategies would avoid toxic tin reagents, and therefore worth exploring. The recent development of Ni- and Co-catalyzed addition of alkyl halides to alkenes displays much improved reactivity and functional group tolerance. In this mini-review, we also attempt to overview the mechanisms that are proposed in the reactions, aiming at providing insight into the nickel and cobalt-catalyzed coupling of alkyl halides with alkenes.
Bethe subalgebras in affine Birman-Murakami-Wenzl algebras and flat connections for q-KZ equations
NASA Astrophysics Data System (ADS)
Isaev, A. P.; Kirillov, A. N.; Tarasov, V. O.
2016-05-01
Commutative sets of Jucys-Murphy elements for affine braid groups of {A}(1),{B}(1),{C}(1),{D}(1) types were defined. Construction of R-matrix representations of the affine braid group of type {C}(1) and its distinguished commutative subgroup generated by the {C}(1)-type Jucys-Murphy elements are given. We describe a general method to produce flat connections for the two-boundary quantum Knizhnik-Zamolodchikov equations as necessary conditions for Sklyanin's type transfer matrix associated with the two-boundary multicomponent Zamolodchikov algebra to be invariant under the action of the {C}(1)-type Jucys-Murphy elements. We specify our general construction to the case of the Birman-Murakami-Wenzl algebras (BMW algebras for short). As an application we suggest a baxterization of the Dunkl-Cherednik elements {Y}\\prime {{s}} in the double affine Hecke algebra of type A. Dedicated to Professor Rodney Baxter on the occasion of his 75th Birthday.
Algebraic mesh quality metrics
KNUPP,PATRICK
2000-04-24
Quality metrics for structured and unstructured mesh generation are placed within an algebraic framework to form a mathematical theory of mesh quality metrics. The theory, based on the Jacobian and related matrices, provides a means of constructing, classifying, and evaluating mesh quality metrics. The Jacobian matrix is factored into geometrically meaningful parts. A nodally-invariant Jacobian matrix can be defined for simplicial elements using a weight matrix derived from the Jacobian matrix of an ideal reference element. Scale and orientation-invariant algebraic mesh quality metrics are defined. the singular value decomposition is used to study relationships between metrics. Equivalence of the element condition number and mean ratio metrics is proved. Condition number is shown to measure the distance of an element to the set of degenerate elements. Algebraic measures for skew, length ratio, shape, volume, and orientation are defined abstractly, with specific examples given. Combined metrics for shape and volume, shape-volume-orientation are algebraically defined and examples of such metrics are given. Algebraic mesh quality metrics are extended to non-simplical elements. A series of numerical tests verify the theoretical properties of the metrics defined.
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…
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…
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.
NASA Astrophysics Data System (ADS)
Durka, R.
2017-04-01
The S-expansion framework is analyzed in the context of a freedom in closing the multiplication tables for the abelian semigroups. Including the possibility of the zero element in the resonant decomposition, and associating the Lorentz generator with the semigroup identity element, leads to a wide class of the expanded Lie algebras introducing interesting modifications to the gauge gravity theories. Among the results, we find all the Maxwell algebras of type {{B}m} , {{C}m} , and the recently introduced {{D}m} . The additional new examples complete the resulting generalization of the bosonic enlargements for an arbitrary number of the Lorentz-like and translational-like generators. Some further prospects concerning enlarging the algebras are discussed, along with providing all the necessary constituents for constructing the gravity actions based on the obtained results.
Hatcher, N.; Restuccia, A.; Stephany, J.
2006-02-15
We present the complete set of N=1, D=4 quantum algebras associated to massive superparticles. We obtain the explicit solution of these algebras realized in terms of unconstrained operators acting on the Hilbert space of superfields. These solutions are expressed using the chiral, antichiral and tensorial projectors which define the three irreducible representations of the supersymmetry on the superfields. In each case the space-time variables are noncommuting and their commutators are proportional to the internal angular momentum of the representation. The quantum algebra associated to the chiral or the antichiral projector is the one obtained by the quantization of the Casalbuoni-Brink-Schwarz (superspin 0) massive superparticle. We present a new superparticle action for the (superspin 1/2) case and show that their wave functions are the ones associated to the irreducible tensor multiplet.
NASA Astrophysics Data System (ADS)
Roytenberg, Dmitry
2007-11-01
A Lie 2-algebra is a linear category equipped with a functorial bilinear operation satisfying skew-symmetry and Jacobi identity up to natural transformations which themselves obey coherence laws of their own. Functors and natural transformations between Lie 2-algebras can also be defined, yielding a 2-category. Passing to the normalized chain complex gives an equivalence of 2-categories between Lie 2-algebras and certain "up to homotopy" structures on the complex; for strictly skew-symmetric Lie 2-algebras these are L∞-algebras, by a result of Baez and Crans. Lie 2-algebras appear naturally as infinitesimal symmetries of solutions of the Maurer-Cartan equation in some differential graded Lie algebras and L∞-algebras. In particular, (quasi-) Poisson manifolds, (quasi-) Lie bialgebroids and Courant algebroids provide large classes of examples.
A Holistic Approach to Algebra.
ERIC Educational Resources Information Center
Barbeau, Edward J.
1991-01-01
Described are two examples involving recursive mathematical sequences designed to integrate a holistic approach to learning algebra. These examples promote pattern recognition with algebraic justification, full class participation, and mathematical values that can be transferred to other situations. (MDH)
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.
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.
Algebra for Gifted Third Graders.
ERIC Educational Resources Information Center
Borenson, Henry
1987-01-01
Elementary school children who are exposed to a concrete, hands-on experience in algebraic linear equations will more readily develop a positive mind-set and expectation for success in later formal, algebraic studies. (CB)
Yamazaki, Yasuomi; Ishitani, Osamu
2017-04-05
The addition of a tertiary phosphine and O2 to reaction solutions strongly affected the reactivity and selectivity of coupling reactions between transition metal complexes. The Mizoroki-Heck reaction between metal complexes with bromo and those with vinyl groups in the diimine ligand did not proceed using Pd(OAc)2 in the presence of 2-dicyclohexylphosphino-2',6'-dimethoxybiphenyl (Sphos) under Ar but proceeded selectively after injection of air into the reaction vessel. In the absence of the phosphine ligand, on the other hand, not only the Mizoroki-Heck reaction but also a homo-coupling reaction between the metal complexes with the bromo groups proceeded at the same time. Mechanistic investigation showed that nanoparticles of Pd species were produced in the absence of the phosphine ligand and worked as catalysts for both the Mizoroki-Heck and homo-coupling reactions. On the other hand, larger Pd particles, which were produced in the presence of Sphos but after addition of air for oxidising Sphos, selectively catalysed the Mizoroki-Heck reaction. 'Molecular' Pd species that were stabilised in the presence of non-oxidised Sphos could not catalyse both coupling reactions under the reaction conditions. Based on these results, reaction conditions were established for the selective progress of the Mizoroki-Heck and the homo-coupling reactions.
1985-03-13
AD-Ri55 296 ALGEBRA OF NEURON MATRICES(U) FOREIGN TECHNOLOGY DIV i/i WRIGHT-PATTERSON RF8 ON K~ G RGRBRBYRN 13 MAR 85 FTD-ID(RS)T-8@4i-85...ANCLASSIFIED F/G6/6 NL I. 1j.2 U .611111 ’’ K1*10 Vl( PIH OPY Pl (iLUTION TL T CHART I-" FTD-ID(RS )T-0041-85 FOREIGN TECHNOLOGY DIVISION In ALGEBRA OF NEURON ... NEURON MATRICES DTIC TAB Unannounced Q By: K.G. Agababyan JustLficatlon English pages: 10 By Distribution/ Source: Doklady Akademii Nauk SSSR, Vol. 199
An algebra of reversible computation.
Wang, Yong
2016-01-01
We design an axiomatization for reversible computation called reversible ACP (RACP). It has four extendible modules: basic reversible processes algebra, algebra of reversible communicating processes, recursion and abstraction. Just like process algebra ACP in classical computing, RACP can be treated as an axiomatization foundation for reversible computation.
Thomys, Janus; Zhang, Xiaohong
2013-01-01
We describe weak-BCC-algebras (also called BZ-algebras) in which the condition (x∗y)∗z = (x∗z)∗y is satisfied only in the case when elements x, y belong to the same branch. We also characterize ideals, nilradicals, and nilpotent elements of such algebras. PMID:24311983
Visual Salience of Algebraic Transformations
ERIC Educational Resources Information Center
Kirshner, David; Awtry, Thomas
2004-01-01
Information processing researchers have assumed that algebra symbol skills depend on mastery of the abstract rules presented in the curriculum (Matz, 1980; Sleeman, 1986). Thus, students' ubiquitous algebra errors have been taken as indicating the need to embed algebra in rich contextual settings (Kaput, 1995; National Council of Teachers of…
NASA Astrophysics Data System (ADS)
Estrada, Sandra E.; Ochoa-Puentes, Cristian; Sierra, Cesar A.
2017-04-01
In order to study the effect of the molecular structure on the optical properties of totally trans-trans phenylenevinylene oligomers (OPVs), sixteen 1,4-distyrylbenzene derivatives (1a-i and 2a-g) functionalized with different electron-donating (ED) and electron-withdrawing (EW) groups were synthesized by the Mizoroki-Heck cross coupling reaction in moderate to good yields (40-95%). The implemented methodology, with a small modification previously reported by our group, allows obtaining the desired vinyl configuration as well as one novel OPV compound (1h). After structural characterization by several techniques (e.g. FTIR, 1H, 13C and Solid-State NMR), particular emphasis was placed upon the investigation of their optical properties by UV-vis and fluorescence spectroscopies. The results showed that, with only one exception, the ED and EW groups at the ends of OPV systems lead to a bathochromic shift. This effect is intensified with the introduction of methoxy groups on the central ring. Consistent with these, the HOMO-LUMO gaps (ΔE) decreases as the strength of ED and EW substituents increases. The ED and EW substituents also lead to a decrease in the Φf values. This contribution in the area of organic electronics can be used as a reference to better select the most appropriate technological application for each OPV and this can be extrapolated to their respective structurally analogous segmented polymer.
[Imiquimod for the topical treatment of focal epithelial hyperplasia (Heck disease) in a child].
Maschke, Jan; Brauns, Tilmann C; Goos, Manfred
2004-10-01
Focal epithelial hyperplasia (FEH) or Heck disease is a rare skin disease caused by human papilloma viruses (HPV). The case of a 9-year old boy is presented to demonstrate the successful treatment of massive FEH with 5% imiquimod cream. Initially, the patient had noticed several separate papules, which spread and developed into multiple peri- and intraoral papillomatous nodules. The lesions were treated with carbon dioxide laser destruction. However, multiple, skin-coloured papillomatous nodules were found on the tongue, buccal mucosa and lips 1.5 years later. Treatment with imiquimod was initiated, because the patient suffered tremendously from the disease. 5% imiquimod cream was applied 3 times per week. Regression of lesions was obvious after 1 month of treatment. Complete clearance was achieved after 2 additional months of treatment and no recurrence was detected over a follow-up period of 5 months. Our case points out the clinical value of imiquimod for the non-traumatic and almost painless therapy of HPV-induced skin diseases in children.
Relative reactivity of alkenyl alcohols in the palladium-catalyzed redox-relay Heck reaction.
Hilton, Margaret J; Cheng, Bin; Buckley, Benjamin R; Xu, Liping; Wiest, Olaf; Sigman, Matthew S
2015-09-16
The relative rates of alkenyl alcohols in the Pd-catalyzed redox-relay Heck reaction were measured in order to examine the effect of their steric and electronic properties on the rate-determining step. Competition experiments between an allylic alkenyl alcohol and two substrates with differing chain lengths revealed that the allylic alcohol reacts 3-4 times faster in either case. Competition between di- and trisubstituted alkenyl alcohols provided an interesting scenario, in which the disubstituted alkene was consumed first followed by reaction of the trisubstituted alkene. Consistent with this observation, the transition structures for the migratory insertion of the aryl group into the di- and trisubstituted alkenes were calculated with a lower barrier for the former. An internal competition between a substrate containing two alcohols with differing chain lengths demonstrated the catalyst's preference for migrating towards the closest alcohol. Additionally, it was observed that increasing the electron density in the arene boronic acid promotes a faster reaction, which correlates with Hammett σp values to give a ρ of -0.87.
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.…
Algebraic Thinking through Origami.
ERIC Educational Resources Information Center
Higginson, William; Colgan, Lynda
2001-01-01
Describes the use of paper folding to create a rich environment for discussing algebraic concepts. Explores the effect that changing the dimensions of two-dimensional objects has on the volume of related three-dimensional objects. (Contains 13 references.) (YDS)
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…
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,…
Algebraic Thinking through Origami.
ERIC Educational Resources Information Center
Higginson, William; Colgan, Lynda
2001-01-01
Describes the use of paper folding to create a rich environment for discussing algebraic concepts. Explores the effect that changing the dimensions of two-dimensional objects has on the volume of related three-dimensional objects. (Contains 13 references.) (YDS)
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…
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…
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…
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…
Tasker, Sarah Z.; Gutierrez, Alicia C.
2014-01-01
Achieving high selectivity in the Heck reaction of electronically unbiased alkenes has been a longstanding challenge. Using a nickel-catalyzed cationic Heck reaction, we were able to achieve excellent selectivity for branched products (≥19:1 in all cases) over a wide range of aryl electrophiles and aliphatic olefins. A bidentate ligand with a suitable bite angle and steric profile was key to obtaining high branched/linear selectivity, while the appropriate base suppressed alkene isomerization of the product. Though aryl triflates are traditionally used to access the cationic Heck pathway, we have shown that by using triethylsilyl trifluoromethanesulfonate (TESOTf), we can effect a counterion exchange of the catalytic nickel complex such that cheaper and more stable aryl chlorides, mesylates, tosylates, and sulfamates can be used to yield the same branched products with high selectivity. PMID:24402966
Planar Para Algebras, Reflection Positivity
NASA Astrophysics Data System (ADS)
Jaffe, Arthur; Liu, Zhengwei
2017-05-01
We define a planar para algebra, which arises naturally from combining planar algebras with the idea of ZN para symmetry in physics. A subfactor planar para algebra is a Hilbert space representation of planar tangles with parafermionic defects that are invariant under para isotopy. For each ZN, we construct a family of subfactor planar para algebras that play the role of Temperley-Lieb-Jones planar algebras. The first example in this family is the parafermion planar para algebra (PAPPA). Based on this example, we introduce parafermion Pauli matrices, quaternion relations, and braided relations for parafermion algebras, which one can use in the study of quantum information. An important ingredient in planar para algebra theory is the string Fourier transform (SFT), which we use on the matrix algebra generated by the Pauli matrices. Two different reflections play an important role in the theory of planar para algebras. One is the adjoint operator; the other is the modular conjugation in Tomita-Takesaki theory. We use the latter one to define the double algebra and to introduce reflection positivity. We give a new and geometric proof of reflection positivity by relating the two reflections through the string Fourier transform.
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.
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.
On extensions of Leibniz algebras
NASA Astrophysics Data System (ADS)
Rakhimov, I. S.; Said Husain, Sh. K.; Mohammed, M. A.
2017-09-01
This paper is dedicated to the study extensions of Leibniz algebras using the annihilator approach. The extensions methods have been used earlier to classify certain classes of algebras. In the paper we first review and adjust theoretical background of the method for Leibniz algebras then apply it to classify four-dimensional Leibniz algebras over a field K. We obtain complete classification of four-dimensional nilpotent Leibniz algebras. The main idea of the method is to transfer the “base change” action to an action of automorphism group of the algebras of smaller dimension on cocycles constructed by the annihilator extensions. The method can be used to classify low-dimensional Leibniz algebras over other finite fields as well.
Alvarez, Rosana; Martínez, Claudio; Madich, Youssef; Denis, J Gabriel; Aurrecoechea, José M; de Lera, Angel R
2010-11-08
Structurally diverse C3-alkenylbenzofurans, C3-alkenylindoles, and C4-alkenylisoquinolones are efficiently prepared by using consecutive Sonogashira and cascade Pd-catalyzed heterocyclization/oxidative Heck couplings from readily available ortho-iodosubstituted phenol, aniline, and benzamide substrates, alkynes, and functionalized olefins. The cyclization of O- and N-heteronucleophiles follows regioselective 5-endo-dig- or 6-endo-dig-cyclization modes, whereas the subsequent Heck-type coupling with both mono- and disubstituted olefins takes place stereoselectively with exclusive formation of the E isomers in most cases.
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.
Fiebig, Lukas; Schlörer, Nils; Schmalz, Hans-Günther; Schäfer, Mathias
2014-04-22
The intramolecular aryl-phenyl scrambling reaction within palladium-DPPP-aryl complex (DPPP=1,3-bis(diphenylphosphino)propane) ions was analyzed by state-of-the-art tandem MS, including gas-phase ion/molecule reactions. The Mizoroki-Heck cross-coupling reaction was performed in the gas phase, and the intrinsic reactivity of important intermediates could be examined. Moreover, linear free-energy correlations were applied, and a mechanism for the scrambling reaction proceeding via phosphonium cations was assumed.
Kormos, Chad M; Leadbeater, Nicholas E
2008-05-16
We present here a strategy for the preparation of nonsymmetrically substituted stilbenes using a one-pot two-step double Heck strategy. First a protocol is developed for the selective preparation of a range of styrenes using ethene as the alkene coupling partner. Then conditions are found for the effective coupling of the styrenes with aryl halides using a 1:1 stoichiometric ratio of the two components. The use of the microwave apparatus to perform the reactions offers a convenient method for synthesis as well as for safely, easily, and accurately loading vessels with gaseous reagents.
Yao, Qingwei; Kinney, Elizabeth P; Zheng, Chong
2004-08-19
Three selenium-ligated Pd(II) complexes were readily synthesized and shown to be extremely active catalysts for the Heck reaction of various aryl bromides, including deactivated and heterocyclic ones. The catalytic activity of the selenide-based Pd(II) complexes not only rivals but vastly outperforms that of the corresponding phosphorus and sulfur analogues. Practical advantages of the selenium-based catalysts include their straightforward synthesis and high activity in the absence of any additives as well as the enhanced stability of the selenide ligands toward air oxidation.
NASA Astrophysics Data System (ADS)
Wu, Yan; Xie, Qiongtao; Zhong, Honghua; Wen, Linghua; Hai, Wenhua
2013-05-01
We investigate algebraic bright and vortex solitons in self-defocusing (SDF) media with a type of spatially inhomogeneous nonlinearity. For a specific choice of the nonlinearity parameters, certain exact analytical solutions for algebraic bright and vortex solitons have been constructed. By applying the linear stability analysis, the stability regions of these algebraic solitons are obtained numerically. In addition, we show analytically that a homogeneous SDF nonlinearity superposed by a localized self-focusing nonlinearity can support exact algebraic bright solitons under certain conditions.
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®,…
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®,…
Lee, Jaehoon; Wilczek, Frank
2013-11-27
Motivated by the problem of identifying Majorana mode operators at junctions, we analyze a basic algebraic structure leading to a doubled spectrum. For general (nonlinear) interactions the emergent mode creation operator is highly nonlinear in the original effective mode operators, and therefore also in the underlying electron creation and destruction operators. This phenomenon could open up new possibilities for controlled dynamical manipulation of the modes. We briefly compare and contrast related issues in the Pfaffian quantum Hall state.
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.
Palmkvist, Jakob
2014-01-15
We introduce an infinite-dimensional Lie superalgebra which is an extension of the U-duality Lie algebra of maximal supergravity in D dimensions, for 3 ⩽ D ⩽ 7. The level decomposition with respect to the U-duality Lie algebra gives exactly the tensor hierarchy of representations that arises in gauge deformations of the theory described by an embedding tensor, for all positive levels p. We prove that these representations are always contained in those coming from the associated Borcherds-Kac-Moody superalgebra, and we explain why some of the latter representations are not included in the tensor hierarchy. The most remarkable feature of our Lie superalgebra is that it does not admit a triangular decomposition like a (Borcherds-)Kac-Moody (super)algebra. Instead the Hodge duality relations between level p and D − 2 − p extend to negative p, relating the representations at the first two negative levels to the supersymmetry and closure constraints of the embedding tensor.
NASA 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.
Clifford Algebras and Their Decomposition into Conjugate Fermionic Heisenberg Algebras
NASA Astrophysics Data System (ADS)
Catto, Sultan; Gürcan, Yasemin; Khalfan, Amish; Kurt, Levent; Kato La, V.
2016-10-01
We discuss a construction scheme for Clifford numbers of arbitrary dimension. The scheme is based upon performing direct products of the Pauli spin and identity matrices. Conjugate fermionic algebras can then be formed by considering linear combinations of the Clifford numbers and the Hermitian conjugates of such combinations. Fermionic algebras are important in investigating systems that follow Fermi-Dirac statistics. We will further comment on the applications of Clifford algebras to Fueter analyticity, twistors, color algebras, M-theory and Leech lattice as well as unification of ancient and modern geometries through them.
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…
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.
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.
The applications of a higher-dimensional Lie algebra and its decomposed subalgebras.
Yu, Zhang; Zhang, Yufeng
2009-01-15
With the help of invertible linear transformations and the known Lie algebras, a higher-dimensional 6 x 6 matrix Lie algebra smu(6) is constructed. It follows a type of new loop algebra is presented. By using a (2 + 1)-dimensional partial-differential equation hierarchy we obtain the integrable coupling of the (2 + 1)-dimensional KN integrable hierarchy, then its corresponding Hamiltonian structure is worked out by employing the quadratic-form identity. Furthermore, a higher-dimensional Lie algebra denoted by E, is given by decomposing the Lie algebra smu(6), then a discrete lattice integrable coupling system is produced. A remarkable feature of the Lie algebras smu(6) and E is used to directly construct integrable couplings.
The applications of a higher-dimensional Lie algebra and its decomposed subalgebras
Yu, Zhang; Zhang, Yufeng
2009-01-01
With the help of invertible linear transformations and the known Lie algebras, a higher-dimensional 6 × 6 matrix Lie algebra sμ(6) is constructed. It follows a type of new loop algebra is presented. By using a (2 + 1)-dimensional partial-differential equation hierarchy we obtain the integrable coupling of the (2 + 1)-dimensional KN integrable hierarchy, then its corresponding Hamiltonian structure is worked out by employing the quadratic-form identity. Furthermore, a higher-dimensional Lie algebra denoted by E, is given by decomposing the Lie algebra sμ(6), then a discrete lattice integrable coupling system is produced. A remarkable feature of the Lie algebras sμ(6) and E is used to directly construct integrable couplings. PMID:20084092
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…
Applications of algebraic grid generation
NASA Technical Reports Server (NTRS)
Eiseman, Peter R.; Smith, Robert E.
1990-01-01
Techniques and applications of algebraic grid generation are described. The techniques are univariate interpolations and transfinite assemblies of univariate interpolations. Because algebraic grid generation is computationally efficient, the use of interactive graphics in conjunction with the techniques is advocated. A flexible approach, which works extremely well in an interactive environment, called the control point form of algebraic grid generation is described. The applications discussed are three-dimensional grids constructed about airplane and submarine configurations.
Higher level twisted Zhu algebras
NASA Astrophysics Data System (ADS)
van Ekeren, Jethro
2011-05-01
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 Γ /{Z} for some subgroup Γ 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 operatorname{Zhu}_{p, Γ }(V), and we prove the following theorems: For each p, there is a bijection between the irreducible operatorname{Zhu}_{p, Γ }(V)-modules and the irreducible Γ-twisted positive energy V-modules, and V is (Γ, H)-rational if and only if all its Zhu algebras operatorname{Zhu}_{p, Γ }(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 operatorname{Vir}^c and the universal affine Kac-Moody vertex algebra V^k({g}) at non-critical level. We also compute the inverse limits of these directed systems of algebras.
Epetra Linear Algebra Services Package
2011-09-09
Epetra provides the fundamental construction routines and service functions that are required for serial and paraliellinear algebra libraries using double precision scalar values and int ordinal values.
Lau, Phei Li; Allen, Ray W K
2013-01-01
Summary The palladium metal catalysed Heck reaction of 4-iodoanisole with styrene or methyl acrylate has been studied in a continuous plug flow reactor (PFR) using supercritical carbon dioxide (scCO2) as the solvent, with THF and methanol as modifiers. The catalyst was 2% palladium on silica and the base was diisopropylethylamine due to its solubility in the reaction solvent. No phosphine co-catalysts were used so the work-up procedure was simplified and the green credentials of the reaction were enhanced. The reactions were studied as a function of temperature, pressure and flow rate and in the case of the reaction with styrene compared against a standard, stirred autoclave reaction. Conversion was determined and, in the case of the reaction with styrene, the isomeric product distribution was monitored by GC. In the case of the reaction with methyl acrylate the reactor was scaled from a 1.0 mm to 3.9 mm internal diameter and the conversion and turnover frequency determined. The results show that the Heck reaction can be effectively performed in scCO2 under continuous flow conditions with a palladium metal, phosphine-free catalyst, but care must be taken when selecting the reaction temperature in order to ensure the appropriate isomer distribution is achieved. Higher reaction temperatures were found to enhance formation of the branched terminal alkene isomer as opposed to the linear trans-isomer. PMID:24367454
Pd loaded amphiphilic COF as catalyst for multi-fold Heck reactions, C-C couplings and CO oxidation
Mullangi, Dinesh; Nandi, Shyamapada; Shalini, Sorout; Sreedhala, Sheshadri; Vinod, Chathakudath P.; Vaidhyanathan, Ramanathan
2015-01-01
COFs represent a class of polymers with designable crystalline structures capable of interacting with active metal nanoparticles to form excellent heterogeneous catalysts. Many valuable ligands/monomers employed in making coordination/organic polymers are prepared via Heck and C-C couplings. Here, we report an amphiphilic triazine COF and the facile single-step loading of Pd0 nanoparticles into it. An 18–20% nano-Pd loading gives highly active composite working in open air at low concentrations (Conc. Pd(0) <0.05 mol%, average TON 1500) catalyzing simultaneous multiple site Heck couplings and C-C couplings using ‘non-boronic acid’ substrates, and exhibits good recyclability with no sign of catalyst leaching. As an oxidation catalyst, it shows 100% conversion of CO to CO2 at 150 °C with no loss of activity with time and between cycles. Both vapor sorptions and contact angle measurements confirm the amphiphilic character of the COF. DFT-TB studies showed the presence of Pd-triazine and Pd-Schiff bond interactions as being favorable. PMID:26057044
Pd loaded amphiphilic COF as catalyst for multi-fold Heck reactions, C-C couplings and CO oxidation.
Mullangi, Dinesh; Nandi, Shyamapada; Shalini, Sorout; Sreedhala, Sheshadri; Vinod, Chathakudath P; Vaidhyanathan, Ramanathan
2015-06-09
COFs represent a class of polymers with designable crystalline structures capable of interacting with active metal nanoparticles to form excellent heterogeneous catalysts. Many valuable ligands/monomers employed in making coordination/organic polymers are prepared via Heck and C-C couplings. Here, we report an amphiphilic triazine COF and the facile single-step loading of Pd(0) nanoparticles into it. An 18-20% nano-Pd loading gives highly active composite working in open air at low concentrations (Conc. Pd(0) <0.05 mol%, average TON 1500) catalyzing simultaneous multiple site Heck couplings and C-C couplings using 'non-boronic acid' substrates, and exhibits good recyclability with no sign of catalyst leaching. As an oxidation catalyst, it shows 100% conversion of CO to CO2 at 150 °C with no loss of activity with time and between cycles. Both vapor sorptions and contact angle measurements confirm the amphiphilic character of the COF. DFT-TB studies showed the presence of Pd-triazine and Pd-Schiff bond interactions as being favorable.
Asad, Naeem; Hanson, Paul R.; Long, Toby R.; Rayabarapu, Dinesh K.; Rolfe, Alan
2011-01-01
An atom-economical purification protocol, using solution phase processing via ring-opening metathesis polymerization (ROMP) has been developed for the synthesis of tricyclic sultams. This chromatography-free method allows for convenient isolation of reductive-Heck products and reclamation of excess starting material via sequestration involving metathesis catalysts and a catalyst-armed Si-surface. PMID:21727956
The first Pd-N-heterocyclic carbene (NHC) complex in the form of organic silica is prepared using sol-gel method and its application in Heck and Suzuki reactions are demonstrated. These C-C coupling reactions proceeded efficiently under the influence of microwave irradiation, wit...
The first Pd-N-heterocyclic carbene (NHC) complex in the form of organic silica is prepared using sol-gel method and its application in Heck and Suzuki reactions are demonstrated. These C-C coupling reactions proceeded efficiently under the influence of microwave irradiation, wit...
Schmidt, Bernd; Elizarov, Nelli; Berger, René; Hölter, Frank
2013-06-14
4-Phenol diazonium salts undergo Pd-catalyzed Heck reactions with various styrenes to 4'-hydroxy stilbenes. In almost all cases higher yields and fewer side products were observed, compared to the analogous 4-methoxy benzene diazonium salts. In contrast, the reaction fails completely with 2- and 3-phenol diazonium salts. For these substitution patterns the methoxy-substituted derivatives are superior.
Lee, Jun; Yu, Jihyun; Son, Seung Hwan; Heo, Jinyuk; Kim, Taelim; An, Ji-Young; Inn, Kyung-Soo; Kim, Nam-Jung
2016-01-14
A variety of flavones were expediently synthesized from readily accessible chromanones via a one-pot sequence involving Pd(II)-catalyzed dehydrogenation and oxidative boron-Heck coupling with arylboronic acid pinacol esters. In particular, the use of arylboronic acid pinacol esters was found to significantly improve the yield of the reaction.
Ruan, Jiwu; Xiao, Jianliang
2011-08-16
The Pd-catalyzed Mizoroki-Heck reaction of olefins with aryl halides, more often simply called the Heck reaction, was recently recognized with the 2010 Nobel Prize in chemistry. Although highly selective with electron-deficient olefins, which generally yield the linear β-arylated product exclusively, the Heck reaction is less satisfactory with electron-rich olefins. This substrate typically generates a mixture of both α- and β-arylated regioisomeric products, hampering wider application of the reaction in chemical synthesis. Pioneering studies by a number of researchers revealed that high α-regioselectivity could be obtained under Pd-diphosphine catalysis either through (i) the substitution of aryl triflates for halides or (ii) the addition of stoichiometric silver or thallium salts when aryl halides are used. Under these conditions, the arylation is believed to proceed via an ionic pathway. However, silver introduces added cost, thallium salts are toxic, and triflates are generally commercially unavailable, base sensitive, and thermally labile. Believing that the ionic pathway would be promoted in an ionic medium, in the early 2000s, we attempted the Pd-catalyzed arylation of the benchmark electron-rich olefin butyl vinyl ether with aryl bromides in an imidazolium ionic liquid. We were delighted to observe that highly regioselective α-arylation could readily be accomplished, with no need for silver additives, thallium additives, or aryl triflates. A range of other electron-rich olefins has since been shown to be viable as well. The high α-selectivity probably results from the high ionic strength of the medium, which facilitates the dissociation of halide anions from the [L(2)Pd(Ar)X] intermediate, channeling the arylation into the ionic pathway. Hydrogen bonding interactions may also play a role, however. We subsequently discovered that the α-arylation can indeed be significantly accelerated by a hydrogen bond donor salt, in both ionic liquids and common
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…
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
Patterns to Develop Algebraic Reasoning
ERIC Educational Resources Information Center
Stump, Sheryl L.
2011-01-01
What is the role of patterns in developing algebraic reasoning? This important question deserves thoughtful attention. In response, this article examines some differing views of algebraic reasoning, discusses a controversy regarding patterns, and describes how three types of patterns--in contextual problems, in growing geometric figures, and in…
Viterbi/algebraic hybrid decoder
NASA Technical Reports Server (NTRS)
Boyd, R. W.; Ingels, F. M.; Mo, C.
1980-01-01
Decoder computer program is hybrid between optimal Viterbi and optimal algebraic decoders. Tests have shown that hybrid decoder outperforms any strictly Viterbi or strictly algebraic decoder and effectively handles compound channels. Algorithm developed uses syndrome-detecting logic to direct two decoders to assume decoding load alternately, depending on real-time channel characteristics.
Teaching Algebra Conceptually: Student Achievement
ERIC Educational Resources Information Center
Linsell, Chris; Tozer, Lynn; Anakin, Megan; Cox, Anna; Jones, Rachel; McAuslan, Eric; Smith, Donna; Turner, Garry
2012-01-01
This paper reports findings from the second year of a two-year study designed to develop approaches to teaching algebra in years 9 and 10. The aim of the research was to explore and develop teaching approaches to assist students to acquire a conceptual understanding of algebra, and to document the impact of these approaches on student outcomes.…
SICs and Algebraic Number Theory
NASA Astrophysics Data System (ADS)
Appleby, Marcus; Flammia, Steven; McConnell, Gary; Yard, Jon
2017-08-01
We give an overview of some remarkable connections between symmetric informationally complete measurements (SIC-POVMs, or SICs) and algebraic number theory, in particular, a connection with Hilbert's 12th problem. The paper is meant to be intelligible to a physicist who has no prior knowledge of either Galois theory or algebraic number theory.
Patterns to Develop Algebraic Reasoning
ERIC Educational Resources Information Center
Stump, Sheryl L.
2011-01-01
What is the role of patterns in developing algebraic reasoning? This important question deserves thoughtful attention. In response, this article examines some differing views of algebraic reasoning, discusses a controversy regarding patterns, and describes how three types of patterns--in contextual problems, in growing geometric figures, and in…
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 the…
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.)
Linear algebra and image processing
NASA Astrophysics Data System (ADS)
Allali, Mohamed
2010-09-01
We use the computing technology digital image processing (DIP) to enhance the teaching of linear algebra so as to make the course more visual and interesting. Certainly, this visual approach by using technology to link linear algebra to DIP is interesting and unexpected to both students as well as many faculty.
Multiplier operator algebras and applications.
Blecher, David P; Zarikian, Vrej
2004-01-20
The one-sided multipliers of an operator space X are a key to "latent operator algebraic structure" in X. We begin with a survey of these multipliers, together with several of the applications that they have had to operator algebras. We then describe several new results on one-sided multipliers, and new applications, mostly to one-sided M-ideals.
ERIC Educational Resources Information Center
1997
Astro Algebra is one of six titles in the Mighty Math Series from Edmark, a comprehensive line of math software for students from kindergarten through ninth grade. Many of the activities in Astro Algebra contain a unique technology that uses the computer to help students make the connection between concrete and abstract mathematics. This software…
Elementary maps on nest algebras
NASA Astrophysics Data System (ADS)
Li, Pengtong
2006-08-01
Let , be algebras and let , be maps. An elementary map of is an ordered pair (M,M*) such that for all , . In this paper, the general form of surjective elementary maps on standard subalgebras of nest algebras is described. In particular, such maps are automatically additive.
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…
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.)
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.)
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…
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.
Symplectic Clifford Algebraic Field Theory.
NASA Astrophysics Data System (ADS)
Dixon, Geoffrey Moore
We develop a mathematical framework on which is built a theory of fermion, scalar, and gauge vector fields. This field theory is shown to be equivalent to the original Weinberg-Salam model of weak and electromagnetic interactions, but since the new framework is more rigid than that on which the original Weinberg-Salam model was built, a concomitant reduction in the number of assumptions lying outside of the framework has resulted. In particular, parity violation is actually hiding within our framework, and with little difficulty we are able to manifest it. The mathematical framework upon which we build our field theory is arrived at along two separate paths. The first is by the marriage of a Clifford algebra and a Lie superalgebra, the result being called a super Clifford algebra. The second is by providing a new characterization for a Clifford algebra employing its generators and a symmetric array of metric coefficients. Subsequently we generalize this characterization to the case of an antisymmetric array of metric coefficients, and we call the algebra which results a symplectic Clifford algebra. It is upon one of these that we build our field theory, and it is shown that this symplectic Clifford algebra is a particular subalgebra of a super Clifford algebra. The final ingredient is the operation of bracketing which involves treating the elements of our algebra as endomorphisms of a particular inner product space, and employing this space and its inner product to provide us with maps from our algebra to the reals. It is this operation which enables us to manifest the parity violation hiding in our algebra.
ERIC Educational Resources Information Center
Gonzalez-Vega, Laureano
1999-01-01
Using a Computer Algebra System (CAS) to help with the teaching of an elementary course in linear algebra can be one way to introduce computer algebra, numerical analysis, data structures, and algorithms. Highlights the advantages and disadvantages of this approach to the teaching of linear algebra. (Author/MM)
ERIC Educational Resources Information Center
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)
Macdonald index and chiral algebra
NASA Astrophysics Data System (ADS)
Song, Jaewon
2017-08-01
For any 4d N = 2 SCFT, there is a subsector described by a 2d chiral algebra. The vacuum character of the chiral algebra reproduces the Schur index of the corresponding 4d theory. The Macdonald index counts the same set of operators as the Schur index, but the former has one more fugacity than the latter. We conjecture a prescription to obtain the Macdonald index from the chiral algebra. The vacuum module admits a filtration, from which we construct an associated graded vector space. From this grading, we conjecture a notion of refined character for the vacuum module of a chiral algebra, which reproduces the Macdonald index. We test this prescription for the Argyres-Douglas theories of type ( A 1 , A 2 n ) and ( A 1 , D 2 n+1) where the chiral algebras are given by Virasoro and \\widehat{su}(2) affine Kac-Moody algebra. When the chiral algebra has more than one family of generators, our prescription requires a knowledge of the generators from the 4d.
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.
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.
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.
NASA Astrophysics Data System (ADS)
Grassi, P. A.; Hurth, T.; Steinhauser, M.
2001-09-01
Combining the effect of an intermediate renormalization prescription (zero momentum subtraction) and the background field method (BFM), we show that the algebraic renormalization procedure needed for the computation of radiative corrections within non-invariant regularization schemes is drastically simplified. The present technique is suitable for gauge models and, here, is applied to the Standard Model. The use of the BFM allows a powerful organization of the counterterms and avoids complicated Slavnov-Taylor identities. Furthermore, the Becchi-Rouet-Stora-Tyutin (BRST) variation of background fields plays a special role in disentangling Ward-Takahashi identities (WTI) and Slavnov-Taylor identities (STI). Finally, the strategy to be applied to physical processes is exemplified for the process b→ sγ.
A unique Pd-catalysed Heck arylation as a remote trigger for cyclopropane selective ring-opening.
Singh, Sukhdev; Bruffaerts, Jeffrey; Vasseur, Alexandre; Marek, Ilan
2017-02-07
Combining functionalization at a distant position from a reactive site with the creation of several consecutive stereogenic centres, including the formation of a quaternary carbon stereocentre, in acyclic system represents a pinnacle in organic synthesis. Here we report the regioselective Heck arylation of terminal olefins as a distant trigger for the ring-opening of cyclopropanes. This Pd-catalysed unfolding of the strained cycle, driving force of the chain-walking process, remarkably proved its efficiency and versatility, as the reaction proceeded regardless of the molecular distance between the initiation (double bond) and termination (alcohol) sites. Moreover, employing stereodefined polysubstituted cyclopropane vaults allowed to access sophisticated stereoenriched acyclic scaffolds in good yields. Conceptually, we demonstrated that merging catalytically a chain walking process with a selective C-C bond cleavage represents a powerful approach to construct linear skeleton possessing two stereogenic centres.
A unique Pd-catalysed Heck arylation as a remote trigger for cyclopropane selective ring-opening
NASA Astrophysics Data System (ADS)
Singh, Sukhdev; Bruffaerts, Jeffrey; Vasseur, Alexandre; Marek, Ilan
2017-02-01
Combining functionalization at a distant position from a reactive site with the creation of several consecutive stereogenic centres, including the formation of a quaternary carbon stereocentre, in acyclic system represents a pinnacle in organic synthesis. Here we report the regioselective Heck arylation of terminal olefins as a distant trigger for the ring-opening of cyclopropanes. This Pd-catalysed unfolding of the strained cycle, driving force of the chain-walking process, remarkably proved its efficiency and versatility, as the reaction proceeded regardless of the molecular distance between the initiation (double bond) and termination (alcohol) sites. Moreover, employing stereodefined polysubstituted cyclopropane vaults allowed to access sophisticated stereoenriched acyclic scaffolds in good yields. Conceptually, we demonstrated that merging catalytically a chain walking process with a selective C-C bond cleavage represents a powerful approach to construct linear skeleton possessing two stereogenic centres.
Tay, Daniel Weiliang; Jong, Howard; Lim, Yee Hwee; Wu, Wenqin; Chew, Xinying; Robins, Edward G; Johannes, Charles W
2015-04-17
The evolutionary meta-terarylphosphine ligand architecture of Cy*Phine was recently shown to be a key feature that imposed outstanding performance in palladium-catalyzed copper-free Sonogashira applications. Herein, the Pd-Cy*Phine combination has similarly proven to be a powerful catalyst system for the Mizoroki-Heck reaction. Using high-throughput screening (HTS) methodology, DMF and NaHCO3 were rapidly identified as the most effective solvent and base pair for the cross-coupling catalysis of challenging and industrially valuable substrates including highly electron-rich heteroaryl bromides and unactivated olefins. Unprotected functional groups were well tolerated using low catalyst loadings, and the simple protocol produced excellent yields (up to 99%) with unprecedented substrate diversity. The Pd-Cy*Phine system broadly outperformed many state-of-the-art commercial alternatives, which demonstrated its potential as a next-generation cross-coupling catalyst.
A unique Pd-catalysed Heck arylation as a remote trigger for cyclopropane selective ring-opening
Singh, Sukhdev; Bruffaerts, Jeffrey; Vasseur, Alexandre; Marek, Ilan
2017-01-01
Combining functionalization at a distant position from a reactive site with the creation of several consecutive stereogenic centres, including the formation of a quaternary carbon stereocentre, in acyclic system represents a pinnacle in organic synthesis. Here we report the regioselective Heck arylation of terminal olefins as a distant trigger for the ring-opening of cyclopropanes. This Pd-catalysed unfolding of the strained cycle, driving force of the chain-walking process, remarkably proved its efficiency and versatility, as the reaction proceeded regardless of the molecular distance between the initiation (double bond) and termination (alcohol) sites. Moreover, employing stereodefined polysubstituted cyclopropane vaults allowed to access sophisticated stereoenriched acyclic scaffolds in good yields. Conceptually, we demonstrated that merging catalytically a chain walking process with a selective C–C bond cleavage represents a powerful approach to construct linear skeleton possessing two stereogenic centres. PMID:28169276
Hilton, Margaret J; Xu, Li-Ping; Norrby, Per-Ola; Wu, Yun-Dong; Wiest, Olaf; Sigman, Matthew S
2014-12-19
The mechanism of the redox-relay Heck reaction was investigated using deuterium-labeled substrates. Results support a pathway through a low energy palladium-alkyl intermediate that immediately precedes product formation, ruling out a tautomerization mechanism. DFT calculations of the relevant transition structures at the M06/LAN2DZ+f/6-31+G* level of theory show that the former pathway is favored by 5.8 kcal/mol. Palladium chain-walking toward the alcohol, following successive β-hydride eliminations and migratory insertions, is also supported in this study. The stereochemistry of deuterium labels is determined, lending support that the catalyst remains bound to the substrate during the relay process and that both cis- and trans-alkenes form from β-hydride elimination.
Brehm, Mary A; Gordon, Katie; Firan, Miahil; Rady, Peter; Agim, Nnenna
2016-05-01
Focal epithelial hyperplasia (FEH), or Heck's disease, is an uncommon benign proliferation of oral mucosa caused by the human papillomavirus (HPV), particularly subtypes 13 and 32. The disease typically presents in young Native American patients and is characterized by multiple asymptomatic papules and nodules on the oral mucosa, lips, tongue, and gingiva. The factors that determine susceptibility to FEH are unknown, but the ethnic and geographic distribution of FEH suggests that genetic predisposition, particularly having the human lymphocytic antigen DR4 type, may be involved in pathogenesis. We report a case of FEH with polymerase chain reaction detection of HPV13 in a healthy 11-year-old Hispanic girl and discuss the current understanding of disease pathogenesis, susceptibility, and treatment.
Particle-like structure of Lie algebras
NASA Astrophysics Data System (ADS)
Vinogradov, A. M.
2017-07-01
If a Lie algebra structure 𝔤 on a vector space is the sum of a family of mutually compatible Lie algebra structures 𝔤i's, we say that 𝔤 is simply assembled from the 𝔤i's. Repeating this procedure with a number of Lie algebras, themselves simply assembled from the 𝔤i's, one obtains a Lie algebra assembled in two steps from 𝔤i's, and so on. We describe the process of modular disassembling of a Lie algebra into a unimodular and a non-unimodular part. We then study two inverse questions: which Lie algebras can be assembled from a given family of Lie algebras, and from which Lie algebras can a given Lie algebra be assembled. We develop some basic assembling and disassembling techniques that constitute the elements of a new approach to the general theory of Lie algebras. The main result of our theory is that any finite-dimensional Lie algebra over an algebraically closed field of characteristic zero or over R can be assembled in a finite number of steps from two elementary constituents, which we call dyons and triadons. Up to an abelian summand, a dyon is a Lie algebra structure isomorphic to the non-abelian 2-dimensional Lie algebra, while a triadon is isomorphic to the 3-dimensional Heisenberg Lie algebra. As an example, we describe constructions of classical Lie algebras from triadons.
Investigating Teacher Noticing of Student Algebraic Thinking
ERIC Educational Resources Information Center
Walkoe, Janet Dawn Kim
2013-01-01
Learning algebra is critical for students in the U.S. today. Algebra concepts provide the foundation for much advanced mathematical content. In addition, algebra serves as a gatekeeper to opportunities such as admission to college. Yet many students in the U.S. struggle in algebra classes. Researchers claim that one reason for these difficulties…
Investigating Teacher Noticing of Student Algebraic Thinking
ERIC Educational Resources Information Center
Walkoe, Janet Dawn Kim
2013-01-01
Learning algebra is critical for students in the U.S. today. Algebra concepts provide the foundation for much advanced mathematical content. In addition, algebra serves as a gatekeeper to opportunities such as admission to college. Yet many students in the U.S. struggle in algebra classes. Researchers claim that one reason for these difficulties…
Generalized Lie algebras of type An
NASA Astrophysics Data System (ADS)
Lyubashenko, Volodymyr; Sudbery, Anthony
1998-06-01
It is shown that the quantized enveloping algebra of sl(n) contains a generalized Lie algebra, defined by means of axioms similar to Woronowicz's. This gives rise to Lie algebra-like generators and relations for the locally finite part of the quantized enveloping algebra, and suggests a canonical Poincaré-Birkhoff-Witt basis.
Readiness and Preparation for Beginning Algebra.
ERIC Educational Resources Information Center
Rotman, Jack W.
Drawing from experience at Lansing Community College (LCC), this paper discusses how to best prepare students for success in a beginning algebra course. First, an overview is presented of LCC's developmental math sequence, which includes Basic Arithmetic (MTH 008), Pre-Algebra (MTH 009), Beginning Algebra (MTH 012), and Intermediate Algebra (MTH…
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.
Central extensions of Lax operator algebras
NASA Astrophysics Data System (ADS)
Schlichenmaier, M.; Sheinman, O. K.
2008-08-01
Lax operator algebras were introduced by Krichever and Sheinman as a further development of Krichever's theory of Lax operators on algebraic curves. These are almost-graded Lie algebras of current type. In this paper local cocycles and associated almost-graded central extensions of Lax operator algebras are classified. It is shown that in the case when the corresponding finite-dimensional Lie algebra is simple the two-cohomology space is one-dimensional. An important role is played by the action of the Lie algebra of meromorphic vector fields on the Lax operator algebra via suitable covariant derivatives.
Asymptotic aspect of derivations in Banach algebras.
Roh, Jaiok; Chang, Ick-Soon
2017-01-01
We prove that every approximate linear left derivation on a semisimple Banach algebra is continuous. Also, we consider linear derivations on Banach algebras and we first study the conditions for a linear derivation on a Banach algebra. Then we examine the functional inequalities related to a linear derivation and their stability. We finally take central linear derivations with radical ranges on semiprime Banach algebras and a continuous linear generalized left derivation on a semisimple Banach algebra.
Computing Matrix Representations of Filiform Lie Algebras
NASA Astrophysics Data System (ADS)
Ceballos, Manuel; Núñez, Juan; Tenorio, Ángel F.
In this paper, we compute minimal faithful unitriangular matrix representations of filiform Lie algebras. To do it, we use the nilpotent Lie algebra, g_n, formed of n ×n strictly upper-triangular matrices. More concretely, we search the lowest natural number n such that the Lie algebra g_n contains a given filiform Lie algebra, also computing a representative of this algebra. All the computations in this paper have been done using MAPLE 9.5.
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.
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.
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.
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.
An "Arithmetic" Thinker Tackles Algebra
ERIC Educational Resources Information Center
Armstrong, Alayne C.
2006-01-01
Working from Carolyn Kieran's categorization of "arithmetic" and "algebraic" thinkers, the article describes one eighth-grade "arithmetic" thinker's progress as she attempts to solve one- and two-step equations.
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).
Quantum Process Algebra with Priorities
NASA Astrophysics Data System (ADS)
Ren, Xingtian; Wang, Yong; Dai, Guiping
2017-08-01
One of the most fascinating characteristics is the modularity of ACP (Algebra of Communicating Processes), that is, ACP can be extended easily. qACP also inherents the modularity characteristics of ACP. By introducing new operators or new constants, qACP can have more properties. In this paper, we extend the quantum process algebra qACP with priorities support in an elegant way. And we obtain the soundness and completeness of the extension.
Coherent States for Hopf Algebras
NASA Astrophysics Data System (ADS)
Škoda, Zoran
2007-07-01
Families of Perelomov coherent states are defined axiomatically in the context of unitary representations of Hopf algebras. A global geometric picture involving locally trivial noncommutative fibre bundles is involved in the construction. If, in addition, the Hopf algebra has a left Haar integral, then a formula for noncommutative resolution of identity in terms of the family of coherent states holds. Examples come from quantum groups.
Multiplier operator algebras and applications
Blecher, David P.; Zarikian, Vrej
2004-01-01
The one-sided multipliers of an operator space X are a key to “latent operator algebraic structure” in X. We begin with a survey of these multipliers, together with several of the applications that they have had to operator algebras. We then describe several new results on one-sided multipliers, and new applications, mostly to one-sided M-ideals. PMID:14711990
Algebraic Systems and Pushdown Automata
NASA Astrophysics Data System (ADS)
Petre, Ion; Salomaa, Arto
We concentrate in this chapter on the core aspects of algebraic series, pushdown automata, and their relation to formal languages. We choose to follow here a presentation of their theory based on the concept of properness. We introduce in Sect. 2 some auxiliary notions and results needed throughout the chapter, in particular the notions of discrete convergence in semirings and C-cycle free infinite matrices. In Sect. 3 we introduce the algebraic power series in terms of algebraic systems of equations. We focus on interconnections with context-free grammars and on normal forms. We then conclude the section with a presentation of the theorems of Shamir and Chomsky-Schützenberger. We discuss in Sect. 4 the algebraic and the regulated rational transductions, as well as some representation results related to them. Section 5 is dedicated to pushdown automata and focuses on the interconnections with classical (non-weighted) pushdown automata and on the interconnections with algebraic systems. We then conclude the chapter with a brief discussion of some of the other topics related to algebraic systems and pushdown automata.
NASA Astrophysics Data System (ADS)
Sun, Yuanxu; Zhu, Xiaoqing; Guo, Dandan; Chen, Xiao; Dai, Jingtao
2017-07-01
Platinum and palladium bimetal nanoparticles on ferroferric oxide (PtPd/Fe3O4 NPs) nanocomposite catalysts were successfully synthesized with polyvinyl pyrrolidone as a stabilizing agent. The resultant samples were characterized by x-ray diffraction, x-ray photoelectron spectroscopy, transmission electron microscopy, high-resolution transmission electron microscopy, inductively coupled plasma, and magnetic studies. The catalytic performance of the PtPd/Fe3O4 NPs in the Heck and Suzuki coupling reactions was evaluated. Results showed that the cubic phase of Pt and Pd bimetal nanoparticles coexists with that of Fe3O4. The PtPd/Fe3O4 NP catalysts, which were approximately 22 nm in size, showed excellent catalytic activity in the Heck and Suzuki reactions. Moreover, the catalyst can be recovered with a magnet and reused several times without the significant loss of catalytic activity.
Trejos, Alejandro; Odell, Luke R; Larhed, Mats
2012-02-01
A stereoselective and 1,4-benzoquinone-mediated palladium(II)-catalyzed Heck/Suzuki domino reaction involving metal coordinating cyclic methylamino vinyl ethers and a number of electronically diverse arylboronic acids has been developed and studied. Diastereomeric ratios up to 39:1 and 78 % isolated yields were obtained. The stereoselectivity of the reaction was found to be highly dependent on the nature of the arylboronic acid and the amount of water present in the reaction mixture. Thus, a domino β,α-diarylation-reduction of chelating vinyl ethers can now be accomplished and stereochemically controlled, given that optimized conditions and an appropriate chiral auxiliary are used. To the best of our knowledge, this represents the first example of a stereoselective, oxidative Heck/Suzuki domino reaction in the literature.
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)
Oberholzer, Miriam; Frech, Christian M.
2014-01-01
Dichloro-bis(aminophosphine) complexes of palladium with the general formula of [(P{(NC5H10)3-n(C6H11)n})2Pd(Cl)2] (where n = 0-2), belong to a new family of easy accessible, very cheap, and air stable, but highly active and universally applicable C-C cross-coupling catalysts with an excellent functional group tolerance. Dichloro{bis[1,1',1''-(phosphinetriyl)tripiperidine]}palladium [(P(NC5H10)3)2Pd(Cl)2] (1), the least stable complex within this series towards protons; e.g. in the form of water, allows an eased nanoparticle formation and hence, proved to be the most active Heck catalyst within this series at 100 °C and is a very rare example of an effective and versatile catalyst system that efficiently operates under mild reaction conditions. Rapid and complete catalyst degradation under work-up conditions into phosphonates, piperidinium salts and other, palladium-containing decomposition products assure an easy separation of the coupling products from catalyst and ligands. The facile, cheap, and rapid synthesis of 1,1',1"-(phosphinetriyl)tripiperidine and 1 respectively, the simple and convenient use as well as its excellent catalytic performance in the Heck reaction at 100 °C make 1 to one of the most attractive and greenest Heck catalysts available. We provide here the visualized protocols for the ligand and catalyst syntheses as well as the reaction protocol for Heck reactions performed at 10 mmol scale at 100 °C and show that this catalyst is suitable for its use in organic syntheses. PMID:24686532
Trejos, Alejandro; Fardost, Ashkan; Yahiaoui, Samir; Larhed, Mats
2009-12-28
A mild and novel palladium(II)-catalyzed domino Heck/Suzuki alpha,beta-diarylation-reduction of a dimethylaminoethyl substituted chelating vinyl ether was developed by using electron-rich arylboronic acids in combination with p-benzoquinone. Based on the preparative results, a catalytic cycle is proposed. Further, highly regioselective palladium(II)-catalyzed alpha- or beta-monoarylation of the chelating vinyl ether was achieved using either a bidentate ligand or by employing ligand-less conditions.
Rolfe, Alan; Young, Kyle; Hanson, Paul R
2008-01-01
The development of a new method for the synthesis of 1,1-dioxido-1,2-benzisothiazoline-3-acetic acid by a domino process is reported whereby a classical Heck reaction is coupled to an intramolecular aza-Michael reaction. Ultimately, this method has been expanded to a one-pot, sequential three-component protocol to generate diverse benzofused γ-sultams from a range of commercially available α-bromobenzenesulfonyl chlorides, amines and Michael acceptors.
Demel, Jan; Lamac, Martin; Cejka, Jirí; Stepnicka, Petr
2009-01-01
A series of supported catalysts is prepared by treatment of SBA-15-type mesoporous molecular sieve bearing [triple chemical bond]SiCH(2)CH(2)CH(2)NHCH(2)CH(2)NEt(2) groups with palladium(II) acetate. These catalysts are studied in Suzuki biaryl couplings and in Heck reactions to establish the influence of metal loading and innocent surface modifications (trimethylsilylation). The Suzuki reaction proceeded efficiently with model and practically relevant substrates; the catalyst performance increasing with an increasing degree of metalation (decreasing N/Pd ratio). Catalyst poisoning tests revealed that the reaction takes place in the liquid phase with the catalyst serving as a reservoir of active metal species and also as a stabilizing support once the reaction is performed. In the Heck reactions, on the other hand, the catalyst performance strongly changed with the reaction temperature and with the N/Pd ratio. The material with the lowest metal loading (0.01 mmol palladium per gram of material, N/Pd ratio ca. 100:1) proved particularly attractive in the Heck coupling, being highly active at elevated temperatures, recyclable, and capable of acting as a bifunctional catalyst (i.e., functioning without any external base.
Zou, Gang; Guo, Jianping; Wang, Zhiyong; Huang, Wen; Tang, Jie
2007-07-28
The competition between Heck-type coupling and conjugate addition in phosphine-rhodium catalyzed reactions of aryl boronic acids with alpha,beta-unsaturated carbonyls has been systematically investigated in a toluene-H(2)O biphasic system. Aside from the intrinsic nature of rhodium and the enolization of carbonyls, the phosphine supporting ligand on rhodium, the ratio of aryl boronic acid to alpha,beta-unsaturated carbonyl and the pH value of the aqueous phase were found to affect the competition significantly. Highly selective rhodium-based catalyst systems have therefore been developed for both Heck-type coupling and conjugate addition by synergistically tuning the supporting ligand, the boronic acid to olefin ratio and other reaction conditions. Conjugate addition with selectivity >99% and Heck-type coupling with selectivity of up to 100%, 98% and 84% for acrylates, acrylamides and methyl vinyl ketone, respectively, could be achieved in the rhodium-catalyzed reactions of aryl boronic acids with alpha,beta-unsaturated carbonyls using the corresponding optimized rhodium-based catalyst systems.
Novikov algebras with associative bilinear forms
NASA Astrophysics Data System (ADS)
Zhu, Fuhai; Chen, Zhiqi
2007-11-01
Novikov algebras were introduced in connection with the Poisson brackets of hydrodynamic-type and Hamiltonian operators in formal variational calculus. The goal of this paper is to study Novikov algebras with non-degenerate associative symmetric bilinear forms, which we call quadratic Novikov algebras. Based on the classification of solvable quadratic Lie algebras of dimension not greater than 4 and Novikov algebras in dimension 3, we show that quadratic Novikov algebras up to dimension 4 are commutative. Furthermore, we obtain the classification of transitive quadratic Novikov algebras in dimension 4. But we find that not every quadratic Novikov algebra is commutative and give a non-commutative quadratic Novikov algebra in dimension 6.
Quantum Q systems: from cluster algebras to quantum current algebras
NASA Astrophysics Data System (ADS)
Di Francesco, Philippe; Kedem, Rinat
2017-02-01
This paper gives a new algebraic interpretation for the algebra generated by the quantum cluster variables of the A_r quantum Q-system (Di Francesco and Kedem in Int Math Res Not IMRN 10:2593-2642, 2014). We show that the algebra can be described as a quotient of the localization of the quantum algebra U_{√{q}}({n}[u,u^{-1}])subset U_{√{q}}(widehat{{sl}}_2), in the Drinfeld presentation. The generating current is made up of a subset of the cluster variables which satisfy the Q-system, which we call fundamental. The other cluster variables are given by a quantum determinant-type formula, and are polynomials in the fundamental generators. The conserved quantities of the discrete evolution (Di Francesco and Kedem in Adv Math 228(1):97-152, 2011) described by quantum Q-system generate the Cartan currents at level 0, in a non-standard polarization. The rest of the quantum affine algebra is also described in terms of cluster variables.
(Fuzzy) Ideals of BN-Algebras.
Dymek, Grzegorz; 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.
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.
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.
A Metric Conceptual Space Algebra
NASA Astrophysics Data System (ADS)
Adams, Benjamin; Raubal, Martin
The modeling of concepts from a cognitive perspective is important for designing spatial information systems that interoperate with human users. Concept representations that are built using geometric and topological conceptual space structures are well suited for semantic similarity and concept combination operations. In addition, concepts that are more closely grounded in the physical world, such as many spatial concepts, have a natural fit with the geometric structure of conceptual spaces. Despite these apparent advantages, conceptual spaces are underutilized because existing formalizations of conceptual space theory have focused on individual aspects of the theory rather than the creation of a comprehensive algebra. In this paper we present a metric conceptual space algebra that is designed to facilitate the creation of conceptual space knowledge bases and inferencing systems. Conceptual regions are represented as convex polytopes and context is built in as a fundamental element. We demonstrate the applicability of the algebra to spatial information systems with a proof-of-concept application.
Using Number Theory to Reinforce Elementary Algebra.
ERIC Educational Resources Information Center
Covillion, Jane D.
1995-01-01
Demonstrates that using the elementary number theory in algebra classes helps students to use acquired algebraic skills as well as helping them to more clearly understand concepts that are presented. Discusses factoring, divisibility rules, and number patterns. (AIM)
Colored Quantum Algebra and Its Bethe State
NASA Astrophysics Data System (ADS)
Wang, Jin-Zheng; Jia, Xiao-Yu; Wang, Shi-Kun
2014-12-01
We investigate the colored Yang—Baxter equation. Based on a trigonometric solution of colored Yang—Baxter equation, we construct a colored quantum algebra. Moreover we discuss its algebraic Bethe ansatz state and highest wight representation.
Comments on /N=4 superconformal algebras
NASA Astrophysics Data System (ADS)
Rasmussen, Jørgen
2001-01-01
We present a new and asymmetric N=4 superconformal algebra for arbitrary central charge, thus completing our recent work on its classical analogue with vanishing central charge. Besides the Virasoro generator and 4 supercurrents, the algebra consists of an internal SU(2)⊗U(1) Kac-Moody algebra in addition to two spin 1/2 fermions and a bosonic scalar. The algebra is shown to be invariant under a linear twist of the generators, except for a unique value of the continuous twist parameter. At this value, the invariance is broken and the algebra collapses to the small N=4 superconformal algebra. The asymmetric N=4 superconformal algebra may be seen as induced by an affine SL(2|2) current superalgebra. Replacing SL(2|2) with the coset SL(2|2)/U(1), results directly in the small N=4 superconformal algebra.
Tensor models and 3-ary algebras
NASA Astrophysics Data System (ADS)
Sasakura, Naoki
2011-10-01
Tensor models are the generalization of matrix models, and are studied as models of quantum gravity in general dimensions. In this paper, I discuss the algebraic structure in the fuzzy space interpretation of the tensor models which have a tensor with three indices as its only dynamical variable. The algebraic structure is studied mainly from the perspective of 3-ary algebras. It is shown that the tensor models have algebraic expressions, and that their symmetries are represented by 3-ary algebras. It is also shown that the 3-ary algebras of coordinates, which appear in the nonassociative fuzzy flat spacetimes corresponding to a certain class of configurations with Gaussian functions in the tensor models, form Lie triple systems, and the associated Lie algebras are shown to agree with those of the Snyder's noncommutative spacetimes. The Poincare transformations of the coordinates on the fuzzy flat spacetimes are shown to be generated by 3-ary algebras.
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.
Algebraic orbifold conformal field theories
Xu, Feng
2000-01-01
The unitary rational orbifold conformal field theories in the algebraic quantum field theory and subfactor theory framework are formulated. Under general conditions, it is shown that the orbifold of a given unitary rational conformal field theory generates a unitary modular category. Many new unitary modular categories are obtained. It is also shown that the irreducible representations of orbifolds of rank one lattice vertex operator algebras give rise to unitary modular categories and determine the corresponding modular matrices, which has been conjectured for some time. PMID:11106383
Entropic Forms and Related Algebras
NASA Astrophysics Data System (ADS)
Scarfone, Antonio
2013-02-01
Starting from a very general trace-form entropy, we introduce a pair of algebraic structures endowed by a generalized sum and a generalized product. These algebras form, respectively, two Abelian fields in the realm of the complex numbers isomorphic each other. We specify our results to several entropic forms related to distributions recurrently observed in social, economical, biological and physical systems including the stretched exponential, the power-law and the interpolating Bosons-Fermions distributions. Some potential applications in the study of complex systems are advanced.
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.
Symmetry algebras of linear differential equations
NASA Astrophysics Data System (ADS)
Shapovalov, A. V.; Shirokov, I. V.
1992-07-01
The local symmetries of linear differential equations are investigated by means of proven theorems on the structure of the algebra of local symmetries of translationally and dilatationally invariant differential equations. For a nonparabolic second-order equation, the absence of nontrivial nonlinear local symmetries is proved. This means that the local symmetries reduce to the Lie algebra of linear differential symmetry operators. For the Laplace—Beltrami equation, all local symmetries reduce to the enveloping algebra of the algebra of the conformal group.
Discrimination in a General Algebraic Setting
Fine, Benjamin; Lipschutz, Seymour; Spellman, Dennis
2015-01-01
Discriminating groups were introduced by G. Baumslag, A. Myasnikov, and V. Remeslennikov as an outgrowth of their theory of algebraic geometry over groups. Algebraic geometry over groups became the main method of attack on the solution of the celebrated Tarski conjectures. In this paper we explore the notion of discrimination in a general universal algebra context. As an application we provide a different proof of a theorem of Malcev on axiomatic classes of Ω-algebras. PMID:26171421
Post-Lie algebras and factorization theorems
NASA Astrophysics Data System (ADS)
Ebrahimi-Fard, Kurusch; Mencattini, Igor; Munthe-Kaas, Hans
2017-09-01
In this note we further explore the properties of universal enveloping algebras associated to a post-Lie algebra. Emphasizing the role of the Magnus expansion, we analyze the properties of group like-elements belonging to (suitable completions of) those Hopf algebras. Of particular interest is the case of post-Lie algebras defined in terms of solutions of modified classical Yang-Baxter equations. In this setting we will study factorization properties of the aforementioned group-like elements.
Twining characters and orbit Lie algebras
Fuchs, Jurgen; Ray, Urmie; Schellekens, Bert; Schweigert, Christoph
1996-12-05
We associate to outer automorphisms of generalized Kac-Moody algebras generalized character-valued indices, the twining characters. A character formula for twining characters is derived which shows that they coincide with the ordinary characters of some other generalized Kac-Moody algebra, the so-called orbit Lie algebra. Some applications to problems in conformal field theory, algebraic geometry and the theory of sporadic simple groups are sketched.
Discrimination in a General Algebraic Setting.
Fine, Benjamin; Gaglione, Anthony; Lipschutz, Seymour; Spellman, Dennis
2015-01-01
Discriminating groups were introduced by G. Baumslag, A. Myasnikov, and V. Remeslennikov as an outgrowth of their theory of algebraic geometry over groups. Algebraic geometry over groups became the main method of attack on the solution of the celebrated Tarski conjectures. In this paper we explore the notion of discrimination in a general universal algebra context. As an application we provide a different proof of a theorem of Malcev on axiomatic classes of Ω-algebras.
Applications of Algebraic Logic and Universal Algebra to Computer Science
1989-06-21
conference, with roughly equal representation from Mathematics and Computer Science . The conference consisted of eight invited lectures (60 minutes...each) and 26 contributed talks (20-40 minutes each). There was also a round-table discussion on the role of algebra and logic in computer science . Keywords
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"…
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…
Teacher Actions to Facilitate Early Algebraic Reasoning
ERIC Educational Resources Information Center
Hunter, Jodie
2015-01-01
In recent years there has been an increased emphasis on integrating the teaching of arithmetic and algebra in primary school classrooms. This requires teachers to develop links between arithmetic and algebra and use pedagogical actions that facilitate algebraic reasoning. Drawing on findings from a classroom-based study, this paper provides an…
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…
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…
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…
Cyclic homology for Hom-associative algebras
NASA Astrophysics Data System (ADS)
Hassanzadeh, Mohammad; Shapiro, Ilya; Sütlü, Serkan
2015-12-01
In the present paper we investigate the noncommutative geometry of a class of algebras, called the Hom-associative algebras, whose associativity is twisted by a homomorphism. We define the Hochschild, cyclic, and periodic cyclic homology and cohomology for this class of algebras generalizing these theories from the associative to the Hom-associative setting.
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…
Embedding Algebraic Thinking throughout the Mathematics Curriculum
ERIC Educational Resources Information Center
Vennebush, G. Patrick; Marquez, Elizabeth; Larsen, Joseph
2005-01-01
This article explores the algebra that can be uncovered in many middle-grades mathematics tasks that, on first inspection, do not appear to be algebraic. It shows connections to the other four Standards that occur in traditional algebra problems, and it offers strategies for modifying activities so that they can be used to foster algebraic…
Constraint-Referenced Analytics of Algebra Learning
ERIC Educational Resources Information Center
Sutherland, Scot M.; White, Tobin F.
2016-01-01
The development of the constraint-referenced analytics tool for monitoring algebra learning activities presented here came from the desire to firstly, take a more quantitative look at student responses in collaborative algebra activities, and secondly, to situate those activities in a more traditional introductory algebra setting focusing on…
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"…
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…
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…
Quadratic Dynamical Systems and Algebras
NASA Astrophysics Data System (ADS)
Kinyon, M. K.; Sagle, A. A.
1995-03-01
Quadratic dynamical systems come from differential or discrete systems of the form Ẋ = Q(X) or X(k+1)=Q(X(k)), where Q:Rn→Rn is homogeneous of degree 2; i.e., Q(αX) = α2Q(X) for all α∈R, X∈Rn. Defining a bilinear mapping β:Rn × Rn→Rn by β(X, Y) ≔ {1}/{2}[Q(X+Y)-Q(X)-Q(Y)], we view XY≡β(X, Y) as a multiplication, and thus consider A=(Rn, β) to be a commutative, nonassociative algebra. The quadratic systems are then studied with the general theme that the structure of the algebras helps determine the behavior of the solutions. For example, semisimple algebras give a decoupling of the original system into systems occurring in simple algebras, and solvable algebras give solutions to differential systems via linear differential equations; the general three-dimensional example of the latter phenomenon is described. There are many classical examples and the scope of quadratic systems is large; every polynomial system can be embedded into a higher dimensional quadratic system such that solutions of the original system are obtained from the quadratic system. For differential systems, nilpotents of index 2 (N2=0) are equilibria and idempotents (E2=E) give ray solutions. The origin is never asymptotically stable, and the existence of nonzero idempotents implies that the origin is actually unstable. Nonzero equilibria are not hyperbolic, but can be studied by standard algebra techniques using nondegenerate bilinear forms as Lyapunov functions. Periodic orbits lie on "cones." They cannot occur in dimension 2 or in power-associative algebras. No periodic orbit can be an attractor but "limit cycles" (invariant cones) can exist. Automorphisms of the algebra A leave equilibria, periodic orbits, and domains of attraction invariant. Also, explicit solutions can be given by the action of automorphisms on an initial point; the general three-dimensional example of this is described. Thus if there are sufficient automorphisms, Hilbert‧s sixteenth problem in R3 has
de Castro, Luciano Alberto; de Castro, Joao Gabriel Leite; da Cruz, Alexandre Duarte Lopes; Barbosa, Bruno Henrique de Sousa; de Spindula-Filho, Jose Vieira; Costa, Mauricio Barcelos
2016-04-01
Focal epithelial hyperplasia (FEH), or Heck's disease, is a rare disease of the oral mucosa associated with infection by some subtypes of human papilloma virus, especially subtypes 13 or 32. The disease is predominantly found in children and adolescents with indigenous heritage, but other ethnic groups can be affected worldwide. To the best of the authors' knowledge, it has not been reported in Brazil's elderly population. This article describes a case of FEH in a 57-year-old Brazilian patient presenting since childhood, with multiple lesions in the lips, buccal mucosa and tongue. The solitary tongue lesion underwent excisional biopsy and the histopathological analysis showed parakeratosis, acanthosis, rete pegs with a club-shaped appearance, koilocytosis and the presence of mitosoid cells. These microscopic findings in conjunction with clinical presentation were sufficient to establish the accurate diagnosis of FEH. Polymerase chain reaction (PCR) was performed, but no one human papillomavirus (HPV) subtype could be identified. Clinicians must be aware of this rare oral disease, which can even affect elderly patients, as we described here. Treatment may be indicated in selected cases due to esthetic and/or functional problems.
Batsanov, Andrei S; Knowles, Jonathan P; Whiting, Andrew
2007-03-30
Mechanistic studies of the Heck-Mizoroki reaction of a vinylboronate ester with electronically different (four-substituted) aryl iodides shows that electron donors accelerate the cross-coupling, demonstrating that the oxidative addition step is not rate determining and that there is development of some degree of positive charge in the rate determining step. These results were used as a basis to allow the development of reaction conditions for the Heck-Mizoroki coupling of a hindered vinylboronate ester with electron deficient methyl cis-2-iodoacrylate. The resulting dienylboronate ester was converted through a series of highly stereoselective iodo-deboronations and Heck-Mizoroki reactions into a trienyl iodide precursor for further application in the total synthesis of viridenomycin.
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…
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…
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…
Dimension independence in exterior algebra.
Hawrylycz, M
1995-03-14
The identities between homogeneous expressions in rank 1 vectors and rank n - 1 covectors in a Grassmann-Cayley algebra of rank n, in which one set occurs multilinearly, are shown to represent a set of dimension-independent identities. The theorem yields an infinite set of nontrivial geometric identities from a given identity.
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 contemporary…
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…
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.
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…
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…
Kleene Algebra and Bytecode Verification
2016-04-27
Languages, ACM SIGPLAN/SIGACT, 1998, pp. 149–160. [2] Coglio, A., Simple verification technique for complex Java bytecode subroutines, Concurrency and...of Programming Languages (POPL’73), ACM , 1973, pp. 194–206. [6] Kot, L. and D. Kozen, Second-order abstract interpretation via Kleene algebra
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…
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…
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 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.
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 contemporary…
Celestial mechanics with geometric algebra
NASA Technical Reports Server (NTRS)
Hestenes, D.
1983-01-01
Geometric algebra is introduced as a general tool for Celestial Mechanics. A general method for handling finite rotations and rotational kinematics is presented. The constants of Kepler motion are derived and manipulated in a new way. A new spinor formulation of perturbation theory is developed.
Algebra, 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…
Easing Students' Transition to Algebra
ERIC Educational Resources Information Center
Baroudi, Ziad
2006-01-01
Traditionally, students learn arithmetic throughout their primary schooling, and this is seen as the ideal preparation for the learning of algebra in the junior secondary school. The four operations are taught and rehearsed in the early years and from this, it is assumed, "children will induce the fundamental structure of arithmetic" (Warren &…
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…
Generalized electromagnetism and Dirac algebra
Fryberger, D.
1989-02-01
Using a framework of Dirac algebra, the Clifford algebra appropriate for Minkowski space-time, the formulation of classical electromagnetism including both electric and magnetic charge is explored. Employing the two-potential approach of Cabibbo and Ferrari, a Lagrangian is obtained that is dyality invariant and from which it is possible to derive by Hamilton's principle both the symmetrized Maxwell's equations and the equations of motion for both electrically and magnetically charged particles. This latter result is achieved by defining the variation of the action associated with the cross terms of the interaction Lagrangian in terms of a surface integral. The surface integral has an equivalent path-integral form, showing that the contribution of the cross terms is local in nature. The form of these cross terms derives in a natural way from a Dirac algebraic formulation, and, in fact, the use of the geometric product of Dirac algebra is an essential aspect of this derivation. No kinematic restrictions are associated with the derivation, and no relationship between magnetic and electric charge evolves from the (classical) formulations. However, it is indicated that in bound states quantum mechanical considerations will lead to a version of Dirac's quantization condition. A discussion of parity violation of the generalized electromagnetic theory is given, and a new approach to the incorporation of this violation into the formalism is suggested. Possibilities for extensions are mentioned.
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…
Carry Groups: Abstract Algebra Projects
ERIC Educational Resources Information Center
Miller, Cheryl Chute; Madore, Blair F.
2004-01-01
Carry Groups are a wonderful collection of groups to introduce in an undergraduate Abstract Algebra course. These groups are straightforward to define but have interesting structures for students to discover. We describe these groups and give examples of in-class group projects that were developed and used by Miller.
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…
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…
Dimension independence in exterior algebra.
Hawrylycz, M
1995-01-01
The identities between homogeneous expressions in rank 1 vectors and rank n - 1 covectors in a Grassmann-Cayley algebra of rank n, in which one set occurs multilinearly, are shown to represent a set of dimension-independent identities. The theorem yields an infinite set of nontrivial geometric identities from a given identity. PMID:11607520
Carry Groups: Abstract Algebra Projects
ERIC Educational Resources Information Center
Miller, Cheryl Chute; Madore, Blair F.
2004-01-01
Carry Groups are a wonderful collection of groups to introduce in an undergraduate Abstract Algebra course. These groups are straightforward to define but have interesting structures for students to discover. We describe these groups and give examples of in-class group projects that were developed and used by Miller.
Monitoring Student Learning in Algebra
ERIC Educational Resources Information Center
Accardo, Amy L.; Kuder, S. Jay
2017-01-01
Mr. Perez and Mrs. Peterson co-teach a ninth-grade algebra class. Perez and Peterson's class includes four students with individualized education programs (IEPs). In response to legislation, such as the No Child Left Behind (NCLB) Act (2001) and the Individuals with Disabilities Education Improvement Act (2006), an increasing number of students…
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…
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…
Math Sense: Algebra and Geometry.
ERIC Educational Resources Information Center
Howett, Jerry
This book is designed to help students gain the range of math skills they need to succeed in life, work, and on standardized tests; overcome math anxiety; discover math as interesting and purposeful; and develop good number sense. Topics covered in this book include algebra and geometry. Lessons are organized around four strands: (1) skill lessons…
Teachers' Understanding of Algebraic Generalization
NASA Astrophysics Data System (ADS)
Hawthorne, Casey Wayne
Generalization has been identified as a cornerstone of algebraic thinking (e.g., Lee, 1996; Sfard, 1995) and is at the center of a rich conceptualization of K-8 algebra (Kaput, 2008; Smith, 2003). Moreover, mathematics teachers are being encouraged to use figural-pattern generalizing tasks as a basis of student-centered instruction, whereby teachers respond to and build upon the ideas that arise from students' explorations of these activities. Although more and more teachers are engaging their students in such generalizing tasks, little is known about teachers' understanding of generalization and their understanding of students' mathematical thinking in this domain. In this work, I addressed this gap, exploring the understanding of algebraic generalization of 4 exemplary 8th-grade teachers from multiple perspectives. A significant feature of this investigation is an examination of teachers' understanding of the generalization process, including the use of algebraic symbols. The research consisted of two phases. Phase I was an examination of the teachers' understandings of the underlying quantities and quantitative relationships represented by algebraic notation. In Phase II, I observed the instruction of 2 of these teachers. Using the lens of professional noticing of students' mathematical thinking, I explored the teachers' enacted knowledge of algebraic generalization, characterizing how it supported them to effectively respond to the needs and queries of their students. Results indicated that teachers predominantly see these figural patterns as enrichment activities, disconnected from course content. Furthermore, in my analysis, I identified conceptual difficulties teachers experienced when solving generalization tasks, in particular, connecting multiple symbolic representations with the quantities in the figures. Moreover, while the teachers strived to overcome the challenges of connecting different representations, they invoked both productive and unproductive
Explicit field realizations of W algebras
NASA Astrophysics Data System (ADS)
Wei, Shao-Wen; Liu, Yu-Xiao; Zhang, Li-Jie; Ren, Ji-Rong
2009-06-01
The fact that certain nonlinear W2,s algebras can be linearized by the inclusion of a spin-1 current can provide a simple way to realize W2,s algebras from linear W1,2,s algebras. In this paper, we first construct the explicit field realizations of linear W1,2,s algebras with double scalar and double spinor, respectively. Then, after a change of basis, the realizations of W2,s algebras are presented. The results show that all these realizations are Romans-type realizations.
Roughness in lattice ordered effect algebras.
Xin, Xiao Long; Hua, Xiu Juan; Zhu, Xi
2014-01-01
Many authors have studied roughness on various algebraic systems. In this paper, we consider a lattice ordered effect algebra and discuss its roughness in this context. Moreover, we introduce the notions of the interior and the closure of a subset and give some of their properties in effect algebras. Finally, we use a Riesz ideal induced congruence and define a function e(a, b) in a lattice ordered effect algebra E and build a relationship between it and congruence classes. Then we study some properties about approximation of lattice ordered effect algebras.
Roughness in Lattice Ordered Effect Algebras
Xin, Xiao Long; Hua, Xiu Juan; Zhu, Xi
2014-01-01
Many authors have studied roughness on various algebraic systems. In this paper, we consider a lattice ordered effect algebra and discuss its roughness in this context. Moreover, we introduce the notions of the interior and the closure of a subset and give some of their properties in effect algebras. Finally, we use a Riesz ideal induced congruence and define a function e(a, b) in a lattice ordered effect algebra E and build a relationship between it and congruence classes. Then we study some properties about approximation of lattice ordered effect algebras. PMID:25170523
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.
On special classes of n-algebras
NASA Astrophysics Data System (ADS)
Vainerman, L.; Kerner, R.
1996-05-01
We define n-algebras as linear spaces on which the internal composition law involves n elements: m:V⊗n■V. It is known that such algebraic structures are interesting for their applications to problems of modern mathematical physics. Using the notion of a commutant of two subalgebras of an n-algebra, we distinguish certain classes of n-algebras with reasonable properties: semisimple, Abelian, nilpotent, solvable. We also consider a few examples of n-algebras of different types, and show their properties.
Infinite order decompositions of C*-algebras.
Nematjonovich, Arzikulov Farhodjon
2016-01-01
The present paper is devoted to infinite order decompositions of C*-algebras. It is proved that an infinite order decomposition (IOD) of a C*-algebra forms the complexification of an order unit space, and, if the C*-algebra is monotone complete (not necessarily weakly closed) then its IOD is also monotone complete ordered vector space. Also it is established that an IOD of a C*-algebra is a C*-algebra if and only if this C*-algebra is a von Neumann algebra. As a summary we obtain that the norm of an infinite dimensional matrix is equal to the supremum of norms of all finite dimensional main diagonal submatrices of this matrix and an infinite dimensional matrix is positive if and only if all finite dimensional main diagonal submatrices of this matrix are positive.
Deformed Kac Moody and Virasoro algebras
NASA Astrophysics Data System (ADS)
Balachandran, A. P.; Queiroz, A. R.; Marques, A. M.; Teotonio-Sobrinho, P.
2007-07-01
Whenever the group {\\bb R}^n acts on an algebra {\\cal A} , there is a method to twist \\cal A to a new algebra {\\cal A}_\\theta which depends on an antisymmetric matrix θ (θμν = -θνμ = constant). The Groenewold-Moyal plane {\\cal A}_\\theta({\\bb R}^{d+1}) is an example of such a twisted algebra. We give a general construction to realize this twist in terms of {\\cal A} itself and certain 'charge' operators Qμ. For {\\cal A}_\\theta({\\bb R}^{d+1}), Q_\\mu are translation generators. This construction is then applied to twist the oscillators realizing the Kac-Moody (KM) algebra as well as the KM currents. They give different deformations of the KM algebra. From one of the deformations of the KM algebra, we construct, via the Sugawara construction, the Virasoro algebra. These deformations have an implication for statistics as well.
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.
Recursion and feedback in image algebra
NASA Astrophysics Data System (ADS)
Ritter, Gerhard X.; Davidson, Jennifer L.
1991-04-01
Recursion and feedback are two important processes in image processing. Image algebra, a unified algebraic structure developed for use in image processing and image analysis, provides a common mathematical environment for expressing image processing transforms. It is only recently that image algebra has been extended to include recursive operations [1]. Recently image algebra was shown to incorporate neural nets [2], including a new type of neural net, the morphological neural net [3]. This paper presents the relationship of the recursive image algebra to the field of fractions of the ring of matrices, and gives the two dimensional moving average filter as an example. Also, the popular multilayer perceptron with back propagation and a morphology neural network with learning rule are presented in image algebra notation. These examples show that image algebra can express these important feedback concepts in a succinct way.
Algebraic complexities and algebraic curves over finite fields
Chudnovsky, D. V.; Chudnovsky, G. V.
1987-01-01
We consider the problem of minimal (multiplicative) complexity of polynomial multiplication and multiplication in finite extensions of fields. For infinite fields minimal complexities are known [Winograd, S. (1977) Math. Syst. Theory 10, 169-180]. We prove lower and upper bounds on minimal complexities over finite fields, both linear in the number of inputs, using the relationship with linear coding theory and algebraic curves over finite fields. PMID:16593816
Chen, Yen-Bo; Liu, Shi-Hao; Hsieh, Min-Tsang; Chang, Chih-Shiang; Lin, Chun-Hung; Chen, Chen-Yin; Chen, Po-Yen; Lin, Hui-Chang
2016-04-01
Spiro bis-C,C-α-arylglycosides were synthesized in three steps in 78-85% overall yields starting from exo-glycals. The initial Heck type C-aryl addition of exo-glycals with arylboronic acids afforded α-aryl-β-substituted C-glycosides with exclusive α-stereoselectivity. Among the products, β-ethanal α-aryl C-glycosides further reacted with alkylthiol in the presence of InCl3, followed by in situ Friedel-Crafts cyclization to yield the desirable final products. We proposed a mechanism to explain how the α-aryl group serves as a main determinant of the cyclization.
Wang, Jian-Xin; McCubbin, J Adam; Jin, Meizhong; Laufer, Radoslaw S; Mao, Yunyu; Crew, Andrew P; Mulvihill, Mark J; Snieckus, Victor
2008-07-17
A general and efficient synthesis of 5-aryl imidazo[1,5- a]pyrazines by palladium-catalyzed coupling of the corresponding 8-substituted derivatives with aryl halides is described. The scope of this new reaction for the imidazo[1,5- a]pyrazine ring system was explored using three readily available 8-substituted precursors, X = NH2, NMe2, and OMe, as well as 8-aryl derivatives, X = Ar'. On the basis of these results as well as studies using a deuterated derivative, a Heck-like mechanism is proposed for this transformation.
Oliveira, Caio C; Pfaltz, Andreas; Correia, Carlos Roque Duarte
2015-11-16
We describe herein a highly regio- and enantioselective Pd-catalyzed Heck arylation of unactivated trisubstituted acyclic olefins to provide all-carbon quaternary stereogenic centers. Chiral N,N ligands of the pyrimidine- and pyrazino-oxazoline class were developed for that purpose, providing the desired products in good to high yields with enantiomeric ratios up to >99:1. Both linear and branched substituents on the olefins were well-tolerated. The potential of this new method is demonstrated by the straightforward synthesis of several O-methyl lactols and lactones containing quaternary stereocenters, together with a concise enantioselective total synthesis of the calcium channel blocker verapamil.
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.
Surface defects and chiral algebras
Córdova, Clay; Gaiotto, Davide; Shao, Shu-Heng
2017-05-26
Here, we investigate superconformal surface defects in four-dimensional N = 2 superconformal theories. Each such defect gives rise to a module of the associated chiral algebra and the surface defect Schur index is the character of this module. Various natural chiral algebra operations such as Drinfield-Sokolov reduction and spectral flow can be interpreted as constructions involving four-dimensional surface defects. We compute the index of these defects in the free hypermultiplet theory and Argyres-Douglas theories, using both infrared techniques involving BPS states, as well as renormalization group flows onto Higgs branches. We find perfect agreement with the predicted characters, in eachmore » case.« less
Marulasiddeshwara, M B; Kumar, P Raghavendra
2016-02-01
Palladium(0) nanospheres with an average size of 1-5 nm were synthesized and stabilized by lignin in water without any reducing agent. The lignin supported palladium(0) nanoparticles (lignin@Pd-NPs) were characterized by UV-vis., FT-IR, SEM, TEM, HRICP-AES, EDX and PXRD. Absence of the peak at 430 nm in UV-vis., spectrum confirmed the reduction of Pd(II) to Pd(0). The five characteristic peaks at (111), (200), (220), (311) and (222) corresponding to the 2θ values 40°, 46.7°, 67.9°, 81.9° and 86.9°, respectively, appeared in PXRD spectrum indicated the crystallographic planes of Pd(0) with fcc structure. The Pd(0) loaded on lignin was 0.0467 mmol/g determined by HRICP-AES and 0.89% (w/w) by EDX. The performance of lignin@Pd-NPs catalyst have been investigated for the Mizoroki-Heck CC bond formation reactions between n-butyl propene-2-enoate and halobenzenes and substituted halobenzenes in polar to highly polar solvents as well as under solvent-free conditions in the presence of organic or inorganic bases. The lignin@Pd-NPs was found to be a highly efficient catalyst to yield the desired products of up to 94% under solvent-free conditions in short reaction times. The catalyst was heterogeneous and hence recovered by filtration and reused several times in the subsequent batches of the same reaction.
Algebraic Methods to Design Signals
2015-08-27
whose out-of-phase autocorrelation values are very small. We call the constructed sequences perfect sequences and they serve as perfect algebraic...matrices DISTRIBUTION A: Distribution approved for public release. 2. Gauss sum factorizations yield perfect sequences, (with John Dillon and...the p- ary case, the prefix " perfect " for the underlying sequence (i.e. 1-dimensional array) refers to the case when all the out-of-phase
Algebraic learning for language acquisition
NASA Astrophysics Data System (ADS)
Farrell, Kevin R.; Mammone, Richard J.; Gorin, Allen
1994-02-01
This paper explores the application of new algorithms to the adaptive language acquisition model formulated by Gorin. The new methods consists of incremental approaches for the algebraic learning of statistical associations proposed by Tishby. The incremental methods are evaluated on a text-based natural language experiment, namely the inward call manager task. Performance is evaluated with respect to the alternative methods, namely the smooth mutual information method and the pseudo-inverse solution.
Introduction to Image Algebra Ada
NASA Astrophysics Data System (ADS)
Wilson, Joseph N.
1991-07-01
Image Algebra Ada (IAA) is a superset of the Ada programming language designed to support use of the Air Force Armament Laboratory's image algebra in the development of computer vision application programs. The IAA language differs from other computer vision languages is several respects. It is machine independent, and an IAA translator has been implemented in the military standard Ada language. Its image operands and operations can be used to program a range of both low- and high-level vision algorithms. This paper provides an overview of the image algebra constructs supported in IAA and describes the embodiment of these constructs in the IAA extension of Ada. Examples showing the use of IAA for a range of computer vision tasks are given. The design of IAA as a superset of Ada and the implementation of the initial translator in Ada represent critical choices. The authors discuss the reasoning behind these choices as well as the benefits and drawbacks associated with them. Implementation strategies associated with the use of Ada as an implementation language for IAA are also discussed. While one can look on IAA as a program design language (PDL) for specifying Ada programs, it is useful to consider IAA as a separate language superset of Ada. This admits the possibility of directly translating IAA for implementation on special purpose architectures. This paper explores strategies for porting IAA to various architectures and notes the critical language and implementation features for porting to different architectures.
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.
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.
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.
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.…
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.…
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
Bilinear forms on fermionic Novikov algebras
NASA Astrophysics Data System (ADS)
Chen, Zhiqi; Zhu, Fuhai
2007-05-01
Novikov algebras were introduced in connection with the Poisson brackets of hydrodynamic type and Hamiltonian operators in formal variational calculus. Fermionic Novikov algebras correspond to a certain Hamiltonian super-operator in a super-variable. In this paper, we show that there is a remarkable geometry on fermionic Novikov algebras with non-degenerate invariant symmetric bilinear forms, which we call pseudo-Riemannian fermionic Novikov algebras. They are related to pseudo-Riemannian Lie algebras. Furthermore, we obtain a procedure to classify pseudo-Riemannian fermionic Novikov algebras. As an application, we give the classification in dimension <=4. Motivated by the one in dimension 4, we construct some examples in high dimensions.
Hopf algebras of rooted forests, cocyles, and free Rota-Baxter algebras
NASA Astrophysics Data System (ADS)
Zhang, Tianjie; Gao, Xing; Guo, Li
2016-10-01
The Hopf algebra and the Rota-Baxter algebra are the two algebraic structures underlying the algebraic approach of Connes and Kreimer to renormalization of perturbative quantum field theory. In particular, the Hopf algebra of rooted trees serves as the "baby model" of Feynman graphs in their approach and can be characterized by certain universal properties involving a Hochschild 1-cocycle. Decorated rooted trees have also been applied to study Feynman graphs. We will continue the study of universal properties of various spaces of decorated rooted trees with such a 1-cocycle, leading to the concept of a cocycle Hopf algebra. We further apply the universal properties to equip a free Rota-Baxter algebra with the structure of a cocycle Hopf algebra.
Lie algebras of classical and stochastic electrodynamics
NASA Astrophysics Data System (ADS)
Neto, J. J. Soares; Vianna, J. D. M.
1994-03-01
The Lie algebras associated with infinitesimal symmetry transformations of third-order differential equations of interest to classical electrodynamics and stochastic electrodynamics have been obtained. The structure constants for a general case are presented and the Lie algebra for each particular application is easily achieved. By the method used here it is not necessary to know the explicit expressions of the infinitesimal generators in order to determine the structure constants of the Lie algebra.
On classification of m-dimensional algebras
NASA Astrophysics Data System (ADS)
Bekbaev, U.
2017-03-01
A constructive approach to the classification and invariance problems, with respect to basis changes, of the finite dimensional algebras is offered. A construction of an invariant open, dense (in the Zariski topology) subset of the space of structure constants of algebras is given. A classification of all algebras with structure constants from this dense set is given by providing canonical representatives of their orbits. A finite system of generators for the corresponding field of invariant rational functions of structure constants is shown.
Symbolic Lie algebras manipulations using COMMON LISP
NASA Astrophysics Data System (ADS)
Cecchini, R.; Tarlini, M.
1989-01-01
We present a description and an implementation of a program in COMMON LISP to perform symbolic computations in a given Lie algebra. Using the general definitions of vector space Lie algebra and enveloping algebra, the program is able to compute commutators, to evaluate similarity transformations and the general Baker-Campbell-Hausdorff formula. All the computations are exact, including numerical coefficients. For the interactive user an optional menu facility and online help are available. LISP knowledge is unnecessary.
Numerical linear algebra algorithms and software
NASA Astrophysics Data System (ADS)
Dongarra, Jack J.; Eijkhout, Victor
2000-11-01
The increasing availability of advanced-architecture computers has a significant effect on all spheres of scientific computation, including algorithm research and software development in numerical linear algebra. Linear algebra - in particular, the solution of linear systems of equations - lies at the heart of most calculations in scientific computing. This paper discusses some of the recent developments in linear algebra designed to exploit these advanced-architecture computers. We discuss two broad classes of algorithms: those for dense, and those for sparse matrices.
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.
Sharma, Abhishek; Sharma, Naina; Kumar, Rakesh; Shard, Amit; Sinha, Arun K
2010-05-21
A new approach for one step olefination of benzaldehydes into hydroxy functionalized OPVs is achieved through the first domino Knoevenagel-decarboxylation-Heck sequence using a single catalyst system. The methodology also led to new oxygen based OPV scaffolds capable of selective and visible fluoride recognition in organic or aqueous medium.
Vertex representations of quantum affine algebras.
Frenkel, I B; Jing, N
1988-12-01
We construct vertex representations of quantum affine algebras of ADE type, which were first introduced in greater generality by Drinfeld and Jimbo. The limiting special case of our construction is the untwisted vertex representation of affine Lie algebras of Frenkel-Kac and Segal. Our representation is given by means of a new type of vertex operator corresponding to the simple roots and satisfying the defining relations. In the case of the quantum affine algebra of type A, we introduce vertex operators corresponding to all the roots and determine their commutation relations. This provides an analogue of a Chevalley basis of the affine Lie algebra [unk](n) in the basic representation.
Regular subalgebras of affine Kac Moody algebras
NASA Astrophysics Data System (ADS)
Felikson, Anna; Retakh, Alexander; Tumarkin, Pavel
2008-09-01
We classify regular subalgebras of Kac-Moody algebras in terms of their root systems. In the process, we establish that a root system of a subalgebra is always an intersection of the root system of the algebra with a sublattice of its root lattice. We also discuss applications to investigations of regular subalgebras of hyperbolic Kac-Moody algebras and conformally invariant subalgebras of affine Kac-Moody algebras. In particular, we provide explicit formulae for determining all Virasoro charges in coset constructions that involve regular subalgebras.
Dispersion Operators Algebra and Linear Canonical Transformations
NASA Astrophysics Data System (ADS)
Andriambololona, Raoelina; Ranaivoson, Ravo Tokiniaina; Hasimbola Damo Emile, Randriamisy; Rakotoson, Hanitriarivo
2017-02-01
This work intends to present a study on relations between a Lie algebra called dispersion operators algebra, linear canonical transformation and a phase space representation of quantum mechanics that we have introduced and studied in previous works. The paper begins with a brief recall of our previous works followed by the description of the dispersion operators algebra which is performed in the framework of the phase space representation. Then, linear canonical transformations are introduced and linked with this algebra. A multidimensional generalization of the obtained results is given.
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.
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.
Dispersion Operators Algebra and Linear Canonical Transformations
NASA Astrophysics Data System (ADS)
Andriambololona, Raoelina; Ranaivoson, Ravo Tokiniaina; Hasimbola Damo Emile, Randriamisy; Rakotoson, Hanitriarivo
2017-04-01
This work intends to present a study on relations between a Lie algebra called dispersion operators algebra, linear canonical transformation and a phase space representation of quantum mechanics that we have introduced and studied in previous works. The paper begins with a brief recall of our previous works followed by the description of the dispersion operators algebra which is performed in the framework of the phase space representation. Then, linear canonical transformations are introduced and linked with this algebra. A multidimensional generalization of the obtained results is given.
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.
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…
ERIC Educational Resources Information Center
Hitt, Fernando; Saboya, Mireille; Cortés Zavala, Carlos
2016-01-01
This paper presents an experiment that attempts to mobilise an arithmetic-algebraic way of thinking in order to articulate between arithmetic thinking and the early algebraic thinking, which is considered a prelude to algebraic thinking. In the process of building this latter way of thinking, researchers analysed pupils' spontaneous production…
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…
ERIC Educational Resources Information Center
Hitt, Fernando; Saboya, Mireille; Cortés Zavala, Carlos
2016-01-01
This paper presents an experiment that attempts to mobilise an arithmetic-algebraic way of thinking in order to articulate between arithmetic thinking and the early algebraic thinking, which is considered a prelude to algebraic thinking. In the process of building this latter way of thinking, researchers analysed pupils' spontaneous production…
A Hardy's Uncertainty Principle Lemma in Weak Commutation Relations of Heisenberg-Lie Algebra
NASA Astrophysics Data System (ADS)
Takaesu, Toshimitsu
2011-07-01
In this article we consider linear operators satisfying a generalized commutation relation of a type of the Heisenberg-Lie algebra. It is proven that a generalized inequality of the Hardy's uncertainty principle lemma follows. Its applications to time operators and abstract Dirac operators are also investigated.
The Growing Importance of Linear Algebra in Undergraduate Mathematics.
ERIC Educational Resources Information Center
Tucker, Alan
1993-01-01
Discusses the theoretical and practical importance of linear algebra. Presents a brief history of linear algebra and matrix theory and describes the place of linear algebra in the undergraduate curriculum. (MDH)
Classification of filiform Lie algebras of order 3
NASA Astrophysics Data System (ADS)
Navarro, Rosa María
2016-12-01
Lie algebras of order 3 constitute a generalization of Lie algebras and superalgebras. Throughout this paper the classification problem of filiform Lie algebras of order 3 is considered and therefore this work is a continuation papers seen in the literature. We approach this classification by extending Vergne's result for filiform Lie algebras and by considering algebras of order 3 of high nilindex. We find the expression of the law to which any elementary filiform Lie algebra of order 3 is isomorphic.
Centroids and derivations of low-dimensional Leibniz algebra
NASA Astrophysics Data System (ADS)
Husain, Sh. K. Said; Rakhimov, I. S.; Basri, W.
2017-08-01
In this paper we introduce the concept of centroid and derivation of Leibniz algebras. By using the classification results of Leibniz algebras obtained earlier, we describe the centroids and derivations of low-dimensional Leibniz algebras. We also study some properties of centroids of Leibniz algebras and use these properties to categorize the algebras to have so-called small centroids. The description of the derivations enables us to specify an important subclass of Leibniz algebras called characteristically nilpotent.
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.
Automated Angular Momentum Recoupling Algebra
NASA Astrophysics Data System (ADS)
Williams, H. T.; Silbar, Richard R.
1992-04-01
We present a set of heuristic rules for algebraic solution of angular momentum recoupling problems. The general problem reduces to that of finding an optimal path from one binary tree (representing the angular momentum coupling scheme for the reduced matrix element) to another (representing the sub-integrals and spin sums to be done). The method lends itself to implementation on a microcomputer, and we have developed such an implementation using a dialect of LISP. We describe both how our code, called RACAH, works and how it appears to the user. We illustrate the use of RACAH for several transition and scattering amplitude matrix elements occurring in atomic, nuclear, and particle physics.
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.
Non-relativistic Bondi–Metzner–Sachs algebra
NASA Astrophysics Data System (ADS)
Batlle, Carles; Delmastro, Diego; Gomis, Joaquim
2017-09-01
We construct two possible candidates for non-relativistic bms4 algebra in four space-time dimensions by contracting the original relativistic bms4 algebra. bms4 algebra is infinite-dimensional and it contains the generators of the Poincaré algebra, together with the so-called super-translations. Similarly, the proposed nrbms4 algebras can be regarded as two infinite-dimensional extensions of the Bargmann algebra. We also study a canonical realization of one of these algebras in terms of the Fourier modes of a free Schrödinger field, mimicking the canonical realization of relativistic bms4 algebra using a free Klein–Gordon field.
Global Geometric Deformations of Current Algebras as Krichever-Novikov Type Algebras
NASA Astrophysics Data System (ADS)
Fialowski, Alice; Schlichenmaier, Martin
2005-12-01
We construct algebraic-geometric families of genus one (i.e. elliptic) current and affine Lie algebras of Krichever-Novikov type. These families deform the classical current, respectively affine Kac-Moody Lie algebras. The construction is induced by the geometric process of degenerating the elliptic curve to singular cubics. If the finite-dimensional Lie algebra defining the infinite dimensional current algebra is simple then, even if restricted to local families, the constructed families are non-equivalent to the trivial family. In particular, we show that the current algebra is geometrically not rigid, despite its formal rigidity. This shows that in the infinite dimensional Lie algebra case the relations between geometric deformations, formal deformations and Lie algebra two-cohomology are not that close as in the finite-dimensional case. The constructed families are e.g. of relevance in the global operator approach to the Wess-Zumino-Witten-Novikov models appearing in the quantization of Conformal Field Theory. The algebras are explicitly given by generators and structure equations and yield new examples of infinite dimensional algebras of current and affine Lie algebra type.
Deriving the Regression Line with Algebra
ERIC Educational Resources Information Center
Quintanilla, John A.
2017-01-01
Exploration with spreadsheets and reliance on previous skills can lead students to determine the line of best fit. To perform linear regression on a set of data, students in Algebra 2 (or, in principle, Algebra 1) do not have to settle for using the mysterious "black box" of their graphing calculators (or other classroom technologies).…
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…
Learning from Student Approaches to Algebraic Proofs
ERIC Educational Resources Information Center
D'Ambrosio, Beatriz S.; Kastberg, Signe E.; Viola dos Santos, Joao Ricardo
2010-01-01
Many mathematics teachers struggle to support their students' developing understanding of proof as an essential element in investigations of mathematics. The area of mathematics where the development of an understanding of proof is most challenging is algebra. In the case of algebraic proof, analysis of student written work on tasks that demand…
Post-Lie Algebras and Isospectral Flows
NASA Astrophysics Data System (ADS)
Ebrahimi-Fard, Kurusch; Lundervold, Alexander; Mencattini, Igor; Munthe-Kaas, Hans Z.
2015-11-01
In this paper we explore the Lie enveloping algebra of a post-Lie algebra derived from a classical R-matrix. An explicit exponential solution of the corresponding Lie bracket flow is presented. It is based on the solution of a post-Lie Magnus-type differential equation.
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…
2-Supernilpotent Mal'cev algebras.
Mudrinski, Nebojša
In this note we prove that a Mal'cev algebra is 2-supernilpotent ([1, 1, 1] = 0) if and only if it is polynomially equivalent to a special expanded group. This generalizes Gumm's result that a Mal'cev algebra is abelian if and only if it is polynomially equivalent to a module over a ring.
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…
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)
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…
Algebraic Thinking through Koch Snowflake Constructions
ERIC Educational Resources Information Center
Ghosh, Jonaki B.
2016-01-01
Generalizing is a foundational mathematical practice for the algebra classroom. It entails an act of abstraction and forms the core of algebraic thinking. Kinach (2014) describes two kinds of generalization--by analogy and by extension. This article illustrates how exploration of fractals provides ample opportunity for generalizations of both…
Practicing Algebraic Skills: A Conceptual Approach
ERIC Educational Resources Information Center
Friedlander, Alex; Arcavi, Abraham
2012-01-01
Traditionally, a considerable part of teaching and learning algebra has focused on routine practice and the application of rules, procedures, and techniques. Although today's computerized environments may have decreased the need to master algebraic skills, procedural competence is still a central component in any mathematical activity. However,…
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…
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…
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).
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…
THE RADICAL OF A JORDAN ALGEBRA
McCrimmon, Kevin
1969-01-01
In this paper we define a Jacobson radical for Jordan algebras analogous to that for associative algebras and show that it enjoys many of the properties of the associative radical. We then relate the corresponding notion of “semisimplicity” to the previously defined notion of “nondegeneracy” (Jacobson, N., these Proceedings, 55, 243-251 (1966)). PMID:16591736
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.``
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.
SAYD Modules over Lie-Hopf Algebras
NASA Astrophysics Data System (ADS)
Rangipour, Bahram; Sütlü, Serkan
2012-11-01
In this paper a general van Est type isomorphism is proved. The isomorphism is between the Lie algebra cohomology of a bicrossed sum Lie algebra and the Hopf cyclic cohomology of its Hopf algebra. We first prove a one to one correspondence between stable-anti-Yetter-Drinfeld (SAYD) modules over the total Lie algebra and those modules over the associated Hopf algebra. In contrast to the non-general case done in our previous work, here the van Est isomorphism is proved at the first level of a natural spectral sequence, rather than at the level of complexes. It is proved that the Connes-Moscovici Hopf algebras do not admit any finite dimensional SAYD modules except the unique one-dimensional one found by Connes-Moscovici in 1998. This is done by extending our techniques to work with the infinite dimensional Lie algebra of formal vector fields. At the end, the one to one correspondence is applied to construct a highly nontrivial four dimensional SAYD module over the Schwarzian Hopf algebra. We then illustrate the whole theory on this example. Finally explicit representative cocycles of the cohomology classes for this example are calculated.
Parabolas: Connection between Algebraic and Geometrical Representations
ERIC Educational Resources Information Center
Shriki, Atara
2011-01-01
A parabola is an interesting curve. What makes it interesting at the secondary school level is the fact that this curve is presented in both its contexts: algebraic and geometric. Being one of Apollonius' conic sections, the parabola is basically a geometric entity. It is, however, typically known for its algebraic characteristics, in particular…
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)
From operator algebras to superconformal field theory
Kawahigashi, Yasuyuki
2010-01-15
We survey operator algebraic approach to (super)conformal field theory. We discuss representation theory, classification results, full and boundary conformal field theories, relations to supervertex operator algebras and Moonshine, connections to subfactor theory of Jones, and certain aspects of noncommutative geometry of Connes.
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…
Relational Algebra and SQL: Better Together
ERIC Educational Resources Information Center
McMaster, Kirby; Sambasivam, Samuel; Hadfield, Steven; Wolthuis, Stuart
2013-01-01
In this paper, we describe how database instructors can teach Relational Algebra and Structured Query Language together through programming. Students write query programs consisting of sequences of Relational Algebra operations vs. Structured Query Language SELECT statements. The query programs can then be run interactively, allowing students to…
Using Tables to Bridge Arithmetic and Algebra
ERIC Educational Resources Information Center
Brown, Susan A.; Mehilos, Megan
2010-01-01
Many students and adults feel that algebra is merely the shuffling of symbols. The three interrelated concepts of variable, expression, and equation are central to beginning algebra, and in recent years, helping students understand the idea of a variable has been emphasized. Although graphing calculators help students solve equations, it is also…
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)
Algebraic Thinking through Koch Snowflake Constructions
ERIC Educational Resources Information Center
Ghosh, Jonaki B.
2016-01-01
Generalizing is a foundational mathematical practice for the algebra classroom. It entails an act of abstraction and forms the core of algebraic thinking. Kinach (2014) describes two kinds of generalization--by analogy and by extension. This article illustrates how exploration of fractals provides ample opportunity for generalizations of both…
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…
Using Students' Interests as Algebraic Models
ERIC Educational Resources Information Center
Whaley, Kenneth A.
2012-01-01
Fostering algebraic thinking is an important goal for middle-grades mathematics teachers. Developing mathematical reasoning requires that teachers cultivate students' habits of mind. Teachers develop students' understanding of algebra by engaging them in tasks that involve modeling and representation. This study was designed to investigate how…
Symbolic Notations and Students' Achievements in Algebra
ERIC Educational Resources Information Center
Peter, Ebiendele E.; Olaoye, Adetunji A.
2013-01-01
This study focuses on symbolic notations and its impact on students' achievement in Algebra. The main reason for this study rests on the observation from personal and professional experiences on students' increasing hatred for Algebra. One hundred and fifty (150) Senior Secondary School Students (SSS) from Ojo Local Education District, Ojo, Lagos,…
Endomorphisms of Quantum Generalized Weyl Algebras
NASA Astrophysics Data System (ADS)
Kitchin, Andrew P.; Launois, Stéphane
2014-07-01
We prove that every endomorphism of a simple quantum generalized Weyl algebra A over a commutative Laurent polynomial ring in one variable is an automorphism. This is achieved by obtaining an explicit classification of all endomorphisms of A. Our main result applies to minimal primitive factors of the quantized enveloping algebra and certain minimal primitive quotients of the positive part of.
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 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)
Symbolic Notations and Students' Achievements in Algebra
ERIC Educational Resources Information Center
Peter, Ebiendele E.; Olaoye, Adetunji A.
2013-01-01
This study focuses on symbolic notations and its impact on students' achievement in Algebra. The main reason for this study rests on the observation from personal and professional experiences on students' increasing hatred for Algebra. One hundred and fifty (150) Senior Secondary School Students (SSS) from Ojo Local Education District, Ojo, Lagos,…
Upper Primary School Students' Algebraic Thinking
ERIC Educational Resources Information Center
Kamol, Natcha; Ban Har, Yeap
2010-01-01
This qualitative research study involving 128 students in grades 4-6 was conducted to develop a framework for characterizing upper primary school students' algebraic thinking. Four levels of algebraic thinking were identified from student responses to tasks involving patterns and open number sentences. Level 1 students failed to understand the…
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…
Gary M. Klingler Algebra Teacher Assistance Packages
ERIC Educational Resources Information Center
Klingler, Gary
2005-01-01
Several packages designed by Elizabeth Marquez for mathematics teachers of grades 6-12, officially entitled the Teacher Assistance Package in Support of Better Algebra Assessment, is a series of resources developed to accompany ET's End-of-Course Algebra Assessment. It is designed to enhance teachers classroom assessment by providing examples of…
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…
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…
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…
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…
The operator algebra approach to quantum groups
Kustermans, Johan; Vaes, Stefaan
2000-01-01
A relatively simple definition of a locally compact quantum group in the C*-algebra setting will be explained as it was recently obtained by the authors. At the same time, we put this definition in the historical and mathematical context of locally compact groups, compact quantum groups, Kac algebras, multiplicative unitaries, and duality theory. PMID:10639116
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…
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…
PREFACE: Infinite Dimensional Algebras and their Applications to Quantum Integrable Systems
NASA Astrophysics Data System (ADS)
Fring, Andreas; Kulish, Petr P.; Manojlović, Nenad; Nagy, Zoltán; Nunes da Costa, Joana; Samtleben, Henning
2008-05-01
This special issue is centred around the workshop Infinite Dimensional Algebras and Quantum Integrable Systems II—IDAQUIS 2007, held at the University of Algarve, Faro, Portugal in July 2007. It was the second workshop in the IDAQUIS series following a previous meeting at the same location in 2003. The latest workshop gathered around forty experts in the field reviewing recent developments in the theory and applications of integrable systems in the form of invited lectures and in a number of contributions from the participants. All contributions contain significant new results or provide a survey of the state of the art of the subject or a critical assessment of the present understanding of the topic and a discussion of open problems. Original contributions from non-participants are also included. The origins of the topic of this issue can be traced back a long way to the early investigations of completely integrable systems of classical mechanics in the fundamental papers by Euler, Lagrange, Jacobi, Liouville, Kowalevski and others. By the end of the nineteenth century all interesting examples seemed to have been exhausted. A revival in the study of integrable systems began with the development of the classical inverse scattering method, or the theory of solitons. Later developments led to the basic geometrical ideas of the theory, of which infinite dimensional algebras are a key ingredient. In a loose sense one may think that all integrable systems possess some hidden symmetry. In the quantum version of these systems the representation theory of these algebras may be exploited in the description of the structure of the Hilbert space of states. Modern examples of field theoretical systems such as conformal field theories, with the Liouville model being a prominent example, affine Toda field theories and the AdS/CFT correspondence are based on algebraic structures like quantum groups, modular doubles, global conformal invariance, Hecke algebras, Kac
Entanglement classification with algebraic geometry
NASA Astrophysics Data System (ADS)
Sanz, M.; Braak, D.; Solano, E.; Egusquiza, I. L.
2017-05-01
We approach multipartite entanglement classification in the symmetric subspace in terms of algebraic geometry, its natural language. We show that the class of symmetric separable states has the structure of a Veronese variety and that its k-secant varieties are SLOCC invariants. Thus SLOCC classes gather naturally into families. This classification presents useful properties such as a linear growth of the number of families with the number of particles, and nesting, i.e. upward consistency of the classification. We attach physical meaning to this classification through the required interaction length of parent Hamiltonians. We show that the states W N and GHZ N are in the same secant family and that, effectively, the former can be obtained in a limit from the latter. This limit is understood in terms of tangents, leading to a refinement of the previous families. We compute explicitly the classification of symmetric states with N≤slant4 qubits in terms of both secant families and its refinement using tangents. This paves the way to further use of projective varieties in algebraic geometry to solve open problems in entanglement theory.
MODEL IDENTIFICATION AND COMPUTER ALGEBRA.
Bollen, Kenneth A; Bauldry, Shawn
2010-10-07
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.
Irrational "Coefficients" in Renaissance Algebra.
Oaks, Jeffrey A
2017-06-01
Argument From the time of al-Khwārizmī in the ninth century to the beginning of the sixteenth century algebraists did not allow irrational numbers to serve as coefficients. To multiply by x, for instance, the result was expressed as the rhetorical equivalent of . The reason for this practice has to do with the premodern concept of a monomial. The coefficient, or "number," of a term was thought of as how many of that term are present, and not as the scalar multiple that we work with today. Then, in sixteenth-century Europe, a few algebraists began to allow for irrational coefficients in their notation. Christoff Rudolff (1525) was the first to admit them in special cases, and subsequently they appear more liberally in Cardano (1539), Scheubel (1550), Bombelli (1572), and others, though most algebraists continued to ban them. We survey this development by examining the texts that show irrational coefficients and those that argue against them. We show that the debate took place entirely in the conceptual context of premodern, "cossic" algebra, and persisted in the sixteenth century independent of the development of the new algebra of Viète, Decartes, and Fermat. This was a formal innovation violating prevailing concepts that we propose could only be introduced because of the growing autonomy of notation from rhetorical text.
LINPACK. Simultaneous Linear Algebraic Equations
Miller, M.A.
1990-05-01
LINPACK is a collection of FORTRAN subroutines which analyze and solve various classes of systems of simultaneous linear algebraic equations. The collection deals with general, banded, symmetric indefinite, symmetric positive definite, triangular, and tridiagonal square matrices, as well as with least squares problems and the QR and singular value decompositions of rectangular matrices. A subroutine-naming convention is employed in which each subroutine name consists of five letters which represent a coded specification (TXXYY) of the computation done by that subroutine. The first letter, T, indicates the matrix data type. Standard FORTRAN allows the use of three such types: S REAL, D DOUBLE PRECISION, and C COMPLEX. In addition, some FORTRAN systems allow a double-precision complex type: Z COMPLEX*16. The second and third letters of the subroutine name, XX, indicate the form of the matrix or its decomposition: GE General, GB General band, PO Positive definite, PP Positive definite packed, PB Positive definite band, SI Symmetric indefinite, SP Symmetric indefinite packed, HI Hermitian indefinite, HP Hermitian indefinite packed, TR Triangular, GT General tridiagonal, PT Positive definite tridiagonal, CH Cholesky decomposition, QR Orthogonal-triangular decomposition, SV Singular value decomposition. The final two letters, YY, indicate the computation done by the particular subroutine: FA Factor, CO Factor and estimate condition, SL Solve, DI Determinant and/or inverse and/or inertia, DC Decompose, UD Update, DD Downdate, EX Exchange. The LINPACK package also includes a set of routines to perform basic vector operations called the Basic Linear Algebra Subprograms (BLAS).
LINPACK. Simultaneous Linear Algebraic Equations
Dongarra, J.J.
1982-05-02
LINPACK is a collection of FORTRAN subroutines which analyze and solve various classes of systems of simultaneous linear algebraic equations. The collection deals with general, banded, symmetric indefinite, symmetric positive definite, triangular, and tridiagonal square matrices, as well as with least squares problems and the QR and singular value decompositions of rectangular matrices. A subroutine-naming convention is employed in which each subroutine name consists of five letters which represent a coded specification (TXXYY) of the computation done by that subroutine. The first letter, T, indicates the matrix data type. Standard FORTRAN allows the use of three such types: S REAL, D DOUBLE PRECISION, and C COMPLEX. In addition, some FORTRAN systems allow a double-precision complex type: Z COMPLEX*16. The second and third letters of the subroutine name, XX, indicate the form of the matrix or its decomposition: GE General, GB General band, PO Positive definite, PP Positive definite packed, PB Positive definite band, SI Symmetric indefinite, SP Symmetric indefinite packed, HI Hermitian indefinite, HP Hermitian indefinite packed, TR Triangular, GT General tridiagonal, PT Positive definite tridiagonal, CH Cholesky decomposition, QR Orthogonal-triangular decomposition, SV Singular value decomposition. The final two letters, YY, indicate the computation done by the particular subroutine: FA Factor, CO Factor and estimate condition, SL Solve, DI Determinant and/or inverse and/or inertia, DC Decompose, UD Update, DD Downdate, EX Exchange. The LINPACK package also includes a set of routines to perform basic vector operations called the Basic Linear Algebra Subprograms (BLAS).
Hexagonal tessellations in image algebra
NASA Astrophysics Data System (ADS)
Eberly, David H.; Wenzel, Dennis J.; Longbotham, Harold G.
1990-11-01
In image algebra '' the concept of a coordinate set X is general in that such a set is simply a subset of ndimensional Euclidean space . The standard applications in 2-dimensional image processing use coordinate sets which are rectangular arrays X 72 x ZZm. However some applications may require other geometries for the coordinate set. We look at three such related applications in the context of image algebra. The first application is the modeling of photoreceptors in primate retinas. These receptors are inhomogeneously distributed on the retina. The largest receptor density occurs in the center of the fovea and decreases radially outwards. One can construct a hexagonal tessellation of the retina such that each hexagon contains approximately the same number of receptors. The resulting tessellation called a sunflower heart2 consists of concentric rings of hexagons whose sizes increase as the radius of the ring increases. The second application is the modeling of the primary visual . The neurons are assumed to be uniformly distributed as a regular hexagonal lattice. Cortical neural image coding is modeled by a recursive convolution of the retinal neural image using a special set of filters. The third application involves analysis of a hexagonally-tessellated image where the pixel resolution is variable .
Beyond CFT: Deformed Virasoro and Elliptic Algebras
NASA Astrophysics Data System (ADS)
Odake, Satoru
Introduction Conformal Field Theory and Virasoro Algebra Conformal Field Theory Virasoro Algebra Free Field Realization Deformed Virasoro Algebra (A1(1) Type) Definition and Consistency Conformal Limit Representation Theory Free Field Realization Higher DVA Currents Solvable Lattice Models and Elliptic Algebras Solvable Lattice Models and Yang-Baxter Equation Corner Transfer Matrices and Vertex Operators Introduction to Quasi-Hopf Algebra Elliptic Quantum Groups Free Field Approach to ABF Model ABF Model Vertex Operators Local Height Probability Form Factor OPE and Trace Screening Operators and Vertex Operators DVA (A2(2) Type) and Dilute AL Models DVA (A2(2)) Free Field Realization Dilute AL Models Free Field Approach OPE and trace Conclusion References Some Formulas Some Functions Delta Function Some Summations Some Integrals Hausdorff Formula Trace Technique
Algebra in a man with severe aphasia.
Klessinger, Nicolai; Szczerbinski, Marcin; Varley, Rosemary
2007-04-09
We report a dissociation between higher order mathematical ability and language in the case of a man (SO) with severe aphasia. Despite severely impaired abilities in the language domain and difficulties with processing both phonological and orthographic number words, he was able to judge the equivalence of and to transform and simplify mathematical expressions in algebraic notation. SO was sensitive to structure-dependent properties of algebraic expressions and displayed considerable capacity to retrieve algebraic facts, rules and principles, and to apply them to novel problems. He demonstrated similar capacity in solving expressions containing either solely numeric or abstract algebraic symbols (e.g., 8-(3-5)+3 versus b-(a-c)+a). The results show the retention of elementary algebra despite severe aphasia and provide evidence for the preservation of symbolic capacity in one modality and hence against the notion of aphasia as asymbolia.
Generalization of n-ary Nambu algebras and beyond
Ataguema, H.; Makhlouf, A.; Silvestrov, S.
2009-08-15
The aim of this paper is to introduce n-ary Hom-algebra structures generalizing the n-ary algebras of Lie type including n-ary Nambu algebras, n-ary Nambu-Lie algebras and n-ary Lie algebras, and n-ary algebras of associative type including n-ary totally associative and n-ary partially associative algebras. We provide examples of the new structures and present some properties and construction theorems. We describe the general method allowing one to obtain an n-ary Hom-algebra structure starting from an n-ary algebra and an n-ary algebra endomorphism. Several examples are derived using this process. Also we initiate investigation of classification problems for algebraic structures introduced in the article and describe all ternary three-dimensional Hom-Nambu-Lie structures with diagonal homomorphism.
Generalization of n-ary Nambu algebras and beyond
NASA Astrophysics Data System (ADS)
Ataguema, H.; Makhlouf, A.; Silvestrov, S.
2009-08-01
The aim of this paper is to introduce n-ary Hom-algebra structures generalizing the n-ary algebras of Lie type including n-ary Nambu algebras, n-ary Nambu-Lie algebras and n-ary Lie algebras, and n-ary algebras of associative type including n-ary totally associative and n-ary partially associative algebras. We provide examples of the new structures and present some properties and construction theorems. We describe the general method allowing one to obtain an n-ary Hom-algebra structure starting from an n-ary algebra and an n-ary algebra endomorphism. Several examples are derived using this process. Also we initiate investigation of classification problems for algebraic structures introduced in the article and describe all ternary three-dimensional Hom-Nambu-Lie structures with diagonal homomorphism.
Song, Kunpeng; Liu, Peng; Wang, Jingyu; Pang, Lei; Chen, Jian; Hussain, Irshad; Tan, Bien; Li, Tao
2015-08-21
Novel dual-porous carbon-supported palladium nanoparticle (Pd NP) catalysts were prepared by sequential carbonization and reduction of microporous organic polymer-encaged PdCl2. The diverse pore structure of microporous organic polymers provides a reservoir for the palladium precursors and prevents Pd NPs from sintering during the carbonization and reaction. The microporous structure has a significant effect on the size and dispersion of palladium NPs. The average size of the Pd NPs (in the range of 4-6 nm) was tuned by changing the pore size distribution and the carbonization temperature. The resulting carbon-supported Pd NPs were characterized by TEM, BET, XRD, and XPS and the Pd loading was calculated by AAS. The encaged Pd NP catalysts prepared by this methodology exhibited outstanding stability and reusability in the Heck reaction and could be reused at least 10 times without appreciable loss of activity.
Menezes da Silva, Vitor H; de Lima Batista, Ana Paula; Navarro, Oscar; Braga, Ataualpa A C
2017-10-30
The regioselectivity of the NHC-Pd catalyzed Heck coupling reaction between phenyl bromide and styrene has been investigated using the density functional theory, wave-function (WF)-based methods and two different sizes of model ligands. In addition to the WF methods, the TPSS-D3, ω B97X-D, BP86-D3, and M06-L density functionals were reliable approaches to be applied, independently of the basis set. Moreover, the NCI analysis showed that weak interactions are important forces to be taken into account when exploring the regioselectivity of this reaction, mainly when a crowded NHC ligand is present. © 2017 Wiley Periodicals, Inc. © 2017 Wiley Periodicals, Inc.
Shakeel-u-Rehman; Rah, Bilal; Lone, Shabir H; Rasool, Reyaz Ur; Farooq, Saleem; Nayak, Debasis; Chikan, Naveed Anjum; Chakraborty, Souneek; Behl, Akanksha; Mondhe, Dilip Manikaro; Goswami, Anindya; Bhat, Khursheed Ahmad
2015-04-23
Sclareol, a promising anticancer labdane diterpene, was isolated from Salvia sclarea. Keeping the basic stereochemistry-rich framework of the molecule intact, a method for the synthesis of novel sclareol analogues was designed using palladium(II)-catalyzed oxidative Heck coupling reaction in order to study their structure-activity relationship. Both sclareol and its derivatives showed an interesting cytotoxicity profile, with 15-(4-fluorophenyl)sclareol (SS-12) as the most potent analogue, having IC50 = 0.082 μM against PC-3 cells. It was found that SS-12 commonly interacts with Bcl-2 and Beclin 1 BH3 domain proteins and enhances autophagic flux by modulating autophagy-related proteins. Moreover, inhibition of autophagy by autophagy inhibitors protected against SS-12-induced apoptosis. Finally, SS-12 effectively suppressed tumor growth in vivo in Ehrlich's ascitic and solid Sarcoma-180 mouse models.
NASA Astrophysics Data System (ADS)
Rafiee, Ezzat; Joshaghani, Mohammad; Abadi, Parvaneh Ghaderi-Shekhi
2016-06-01
The wicker-like Pd-PVP-Fe (palladium-poly(N-vinylpyrrolidone)-iron) was synthesized by the external magnetic field (EMF). The Pd-based catalyst with nano and the face-centered cubic (fcc) structure was obtained at room temperature without using any additive. The resulting composite was characterized. The results show that EMF has a great influence on morphology, particle size, and crystalline structure of the Pd-PVP-Fe composite. The resulting composite (Pd-PVP-Fe), was found to be an effective catalyst for the Mizoroki-Heck reaction while is exposed to EMF with the intensity at 486 μT. The reused catalyst for at least five repeating cycles, shows excellent activity.
Ortiz, Alba M; Sánchez-Méndez, Alberto; de Jesús, Ernesto; Flores, Juan C; Gómez-Sal, Pilar; Mendicuti, Francisco
2016-02-01
Bis(imidazolylidene)palladium complexes 9-12 containing a sterically hindered aryl group (mesityl or 2,6-diisopropylphenyl) and a poly(benzyl ether) dendron as N-substituents of the NHC ligand are accessible up to the third generation by transmetalation of the corresponding silver complexes. Complexes 9-12 are soluble, active, and very stable catalysts under Heck reaction conditions. The NHC ligand appears to be stably coordinated to the Pd during catalysis. The catalytic activity increases with generation number, although irregularly. The palladium site is not significantly congested in the reaction solvent by the increasing size of the dendritic substituents, as corroborated by X-ray diffraction, fluorescence and DOSY-NMR spectroscopy, and MD simulation studies. This is a consequence of the conformational semiflexibility of the poly(benzyl ether) dendrons and the benzylic link between these dendrons and the N-heterocyclic ligands.
Qin, Hua-Li; Zheng, Qinheng; Bare, Grant A L; Wu, Peng; Sharpless, K Barry
2016-11-02
A Heck-Matsuda process for the synthesis of the otherwise difficult to access compounds, β-arylethenesulfonyl fluorides, is described. Ethenesulfonyl fluoride (i.e., vinylsulfonyl fluoride, or ESF) undergoes β-arylation with stable and readily prepared arenediazonium tetrafluoroborates in the presence of the catalyst palladium(II) acetate to afford the E-isomer sulfonyl analogues of cinnamoyl fluoride in 43-97 % yield. The β-arylethenesulfonyl fluorides are found to be selectively addressable bis-electrophiles for sulfur(VI) fluoride exchange (SuFEx) click chemistry, in which either the alkenyl moiety or the sulfonyl fluoride group can be the exclusive site of nucleophilic attack under defined conditions, making these rather simple cores attractive for covalent drug discovery.
NASA Astrophysics Data System (ADS)
Wang, Ling-ling; Zhu, Lu-ping; Bing, Nai-ci; Wang, Li-jun
2017-08-01
Evenly dispersed Pd nanoparticles are facilely and successfully deposited on N-doped carbon nanotubes (Pd/N-CNTs) by employing sodium dodecyl sulfate (SDS) as a salt and polyvinylpyrrolidone (PVP) as both a surface modifier and a stabilizing agent. The presence of SDS and the amount of PVP have significant influences on the formation of evenly dispersed Pd/N-CNTs catalyst. No additional functionalization steps, reducing agents and stabilizers are required as usual to achieve the uniform deposition of Pd NPs over the N-CNTs surfaces. The as-synthesized Pd/N-CNTs catalyst is proved to be very active in the Heck reaction and can be reused at least for 5 times without significant loss of catalytic activity in the aerobic oxidation of benzyl alcohol.
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.
Conformal current algebra in two dimensions
NASA Astrophysics Data System (ADS)
Ashok, Sujay K.; Benichou, Raphael; Troost, Jan
2009-06-01
We construct a non-chiral current algebra in two dimensions consistent with conformal invariance. We show that the conformal current algebra is realized in non-linear sigma-models on supergroup manifolds with vanishing Killing form, with or without a Wess-Zumino term. The current algebra is computed using two distinct methods. First we exploit special algebraic properties of supergroups to compute the exact two- and three-point functions of the currents and from them we infer the current algebra. The algebra is also calculated by using conformal perturbation theory about the Wess-Zumino-Witten point and resumming the perturbation series. We also prove that these models realize a non-chiral Kac-Moody algebra and construct an infinite set of commuting operators that is closed under the action of the Kac-Moody generators. The supergroup models that we consider include models with applications to statistical mechanics, condensed matter and string theory. In particular, our results may help to systematically solve and clarify the quantum integrability of PSU(n|n) models and their cosets, which appear prominently in string worldsheet models on anti-deSitter spaces.
Light polarization: A geometric-algebra approach
NASA Astrophysics Data System (ADS)
Baylis, W. E.; Bonenfant, J.; Derbyshire, J.; Huschilt, J.
1993-06-01
The geometric algebra of three-dimensional space (the ``Pauli algebra'') is known to provide an efficient geometric description of electromagnetic phenomena. Here, it is applied to the three-dimensional Stokes subspace to describe the polarization of an approximately monochromatic collimated beam of electromagnetic radiation. The coherency density ρ is a real element of the algebra whose components are the four Stokes parameters: a scalar representing the total photon flux density plus a three-dimensional vector whose direction and length in the Poincaré sphere give the type and degree of polarization. The detection of the radiation and the incoherent and coherent modification of the polarization by various optical elements are calculated by algebraic multiplication which has faithful representations in 2×2 matrices. One matrix representation of ρ is the coherency matrix with which Jones and Mueller matrices are related whereas another representation is the spin density matrix. However, the calculations are simplest to perform and interpret in the algebraic form independent of any particular matrix representation. It is shown that any possible change in the Stokes parameters can be treated algebraically by a combination of attenuation, depolarization, polarization, and rotation transformations of ρ. The geometric algebra thus unifies Stokes parameters, the Poincaré sphere, Jones and Mueller matrices, and the coherency and density matrices in a single, simple formalism.
Working memory, worry, and algebraic ability.
Trezise, Kelly; Reeve, Robert A
2014-05-01
Math anxiety (MA)-working memory (WM) relationships have typically been examined in the context of arithmetic problem solving, and little research has examined the relationship in other math domains (e.g., algebra). Moreover, researchers have tended to examine MA/worry separate from math problem solving activities and have used general WM tasks rather than domain-relevant WM measures. Furthermore, it seems to have been assumed that MA affects all areas of math. It is possible, however, that MA is restricted to particular math domains. To examine these issues, the current research assessed claims about the impact on algebraic problem solving of differences in WM and algebraic worry. A sample of 80 14-year-old female students completed algebraic worry, algebraic WM, algebraic problem solving, nonverbal IQ, and general math ability tasks. Latent profile analysis of worry and WM measures identified four performance profiles (subgroups) that differed in worry level and WM capacity. Consistent with expectations, subgroup membership was associated with algebraic problem solving performance: high WM/low worry>moderate WM/low worry=moderate WM/high worry>low WM/high worry. Findings are discussed in terms of the conceptual relationship between emotion and cognition in mathematics and implications for the MA-WM-performance relationship.
Three-dimensional polarization algebra.
R Sheppard, Colin J; Castello, Marco; Diaspro, Alberto
2016-10-01
If light is focused or collected with a high numerical aperture lens, as may occur in imaging and optical encryption applications, polarization should be considered in three dimensions (3D). The matrix algebra of polarization behavior in 3D is discussed. It is useful to convert between the Mueller matrix and two different Hermitian matrices, representing an optical material or system, which are in the literature. Explicit transformation matrices for converting the column vector form of these different matrices are extended to the 3D case, where they are large (81×81) but can be generated using simple rules. It is found that there is some advantage in using a generalization of the Chandrasekhar phase matrix treatment, rather than that based on Gell-Mann matrices, as the resultant matrices are of simpler form and reduce to the two-dimensional case more easily. Explicit expressions are given for 3D complex field components in terms of Chandrasekhar-Stokes parameters.
Induction of Relational Algebra Expressions
NASA Astrophysics Data System (ADS)
Gillis, Joris J. M.; van den Bussche, Jan
We study the problem of inducing relational databases queries expressing quantification. Such queries express interesting multi-relational patterns in a database in a concise manner. Queries on relational databases can be expressed as Datalog programs. Inducing Datalog programs expressing quantification requires both negation and predicate invention. Predicate invention has been studied in the ILP literature. However, we propose a radical new approach to inducing quantification. We express queries in the relational algebra and we perform a heuristic search for the desired expression. A heuristic, which takes the complement operator into account, is proposed. We report some preliminary experimental results of a software prototype implementing our ideas. The results are compared with the results of FOIL and Tilde on the same examples.
Algebraic construction of twinlike models
NASA Astrophysics Data System (ADS)
Adam, C.; Queiruga, J. M.
2011-11-01
If the generalized dynamics of K field theories (i.e., field theories with a nonstandard kinetic term) is taken into account, then the possibility of so-called twinlike models opens up, that is, of different field theories which share the same topological defect solution with the same energy density. These twinlike models were first introduced in [M. Andrews, M. Lewandowski, M. Trodden, and D. Wesley, Phys. Rev. DPRVDAQ1550-7998 82, 105006 (2010)10.1103/PhysRevD.82.105006], where the authors also considered possible cosmological implications and gave a geometric characterization of twinlike models. A further analysis of the twinlike models was accomplished in [D. Bazeia, J. D. Dantas, A. R. Gomes, L. Losano, and R. Menezes, Phys. Rev. DPRVDAQ1550-7998 84, 045010 (2011)10.1103/PhysRevD.84.045010], with the help of the first order formalism, where also the case with gravitational self-interaction was considered. Here we show that by combining the geometric conditions of [M. Andrews, M. Lewandowski, M. Trodden, and D. Wesley, Phys. Rev. DPRVDAQ1550-7998 82, 105006 (2010)10.1103/PhysRevD.82.105006], with the first order formalism of [D. Bazeia, J. D. Dantas, A. R. Gomes, L. Losano, and R. Menezes, Phys. Rev. DPRVDAQ1550-7998 84, 045010 (2011)10.1103/PhysRevD.84.045010], one may easily derive a purely algebraic method to explicitly calculate an infinite number of twin field theories for a given theory. We determine this algebraic construction for the cases of scalar field theories, supersymmetric scalar field theories, and self-gravitating scalar fields. Further, we give several examples for each of these cases.
Noh, Ji-Hyang; Meijboom, Reinout
2014-02-01
Palladium nanoparticles (NPs) were prepared using a dendrimer-templated method using G4, G5 and G6 PAMAM-OH dendrimers as well as a reverse microemulsion method using the water/dioctyl sulfosuccinate sodium salt (aerosol-OT, AOT) surfactant/isooctane system with water to surfactant ratios (ω0) of 5, 10 and 13. These 6 catalysts were characterized by UV-Vis spectroscopy, TEM, EDX, and XRD. TEM micrographs showed that the average sizes of 2.74-3.32nm with narrower size distribution were achieved by using dendrimer-templated synthetic methods, whereas the reverse microemulsion method resulted in broad size distribution with an average size of 3.87-5.06nm. The influence of various reaction parameters such as base, catalyst dosing, alkene, aryl halide and temperature on the Heck C-C coupling reaction was evaluated. The activation parameters were derived from the reaction rate of each catalyst obtained at various temperatures. A correlation of catalytic activity, enthalpy of activation and particle size is discussed. Particle size changes of each catalyst were investigated after the catalytic reaction. Overall results indicated that dendrimer-templated Pd NP catalysts showed superior activity as compared to the Pd NPs synthesized by reverse microemulsions, with the dendrimer-templated G5-OH(Pd80) showing the best activity. These catalysts were also reusable for 3 cycles, retaining high yield and showing excellent yields under mild conditions. Therefore, the dendrimer-templated Pd NPs are efficient catalyst systems for the ligand-free Heck C-C coupling reaction.
Kumar, Anuj; Gangwar, Manoj Kumar; Prakasham, A P; Mhatre, Darshan; Kalita, Alok Ch; Ghosh, Prasenjit
2016-03-21
Well-defined palladium N-heterocyclic carbene (NHC) complexes were employed in the one-pot tandem Heck alkynylation/cyclization sequence for preparing biologically relevant benzofuran compounds under copper-free conditions in a time-efficient step-reduced fashion. In particular, a series of binuclear palladium complexes, 1b-1e and 2b-2e, of the alkyl-bridged NHC ligands, namely, {1,1'-di-R1-4,4'-R2-di-1,2,4-triazoline-5,5'-diylid-2-ene] (R1 = i-Pr; R2 = -(CH2)2-, -(CH2)3-), and their mononuclear analogues, trans-(NHC)PdBr2(pyridine) (3b) and cis-(NHC)PdBr2(PPh3) (3c), successfully catalyzed the one-pot tandem Heck alkynylation/cyclization reaction of 2-iodophenol with a variety of terminal alkyne substrates, yielding 2-substituted benzofuran derivatives. The mononuclear complexes 3b and 3c were nearly half as active as the representative dinuclear analogue 1c under analogous reaction conditions, thereby implying that, at the same mole percent of the palladium loading, the monometallic 3b and 3c and the bimetallic 1c complexes were equally effective as catalysts. The two sites of the bimetallic complex 1c performed as two separate independent catalytic sites, displaying no cooperativity effect in the catalysis. Finally, the practical utility of the aforementioned catalysts was demonstrated for a representative catalyst 1c through the convenient synthesis of a key intermediate, 3-[2-(benzo[d][1,3]dioxol-5-yl)-7-methoxybenzofuran-5-yl]propan-1-ol, in a total-synthesis protocol of the natural product Egonol.
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.
Jucys-Murphy elements for Birman-Murakami-Wenzl algebras
NASA Astrophysics Data System (ADS)
Isaev, A. P.; Ogievetsky, O. V.
2011-05-01
The Burman-Wenzl-Murakami algebra, considered as the quotient of the braid group algebra, possesses the commutative set of Jucys-Murphy elements. We show that the set of Jucys-Murphy elements is maximal commutative for the generic Birman-Wenzl-Murakami algebra and reconstruct the representation theory of the tower of Birman-Wenzl-Murakami algebras.
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.
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.
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)].
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.
Gauged Ads-Maxwell Algebra and Gravity
NASA Astrophysics Data System (ADS)
Durka, R.; Kowalski-Glikman, J.; Szczachor, M.
We deform the anti-de Sitter algebra by adding additional generators {Z}ab, forming in this way the negative cosmological constant counterpart of the Maxwell algebra. We gauge this algebra and construct a dynamical model with the help of a constrained BF theory. It turns out that the resulting theory is described by the Einstein-Cartan action with Holst term, and the gauge fields associated with the Maxwell generators {Z}ab appear only in topological terms that do not influence dynamical field equations. We briefly comment on the extension of this construction, which would lead to a nontrivial Maxwell fields dynamics.
Homomorphisms between C*-algebras and linear derivations on C*-algebras
NASA Astrophysics Data System (ADS)
Park, Choonkil; Boo, Deok-Hoon; An, Jong Su
2008-01-01
It is shown that every almost unital almost linear mapping of a unital C*-algebra to a unital C*-algebra is a homomorphism when h(3nuy)=h(3nu)h(y) holds for all unitaries , all , and all , and that every almost unital almost linear continuous mapping of a unital C*-algebra of real rank zero to a unital C*-algebra is a homomorphism when h(3nuy)=h(3nu)h(y) holds for all , and v is invertible}, all , and all . Furthermore, we prove the Hyers-Ulam-Rassias stability of *-homomorphisms between unital C*-algebras, and -linear *-derivations on unital C*-algebras. The concept of Hyers-Ulam-Rassias stability originated from the Th.M. Rassias' stability theorem that appeared in his paper: On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978) 297-300.
Lie algebra type noncommutative phase spaces are Hopf algebroids
NASA Astrophysics Data System (ADS)
Meljanac, Stjepan; Škoda, Zoran; Stojić, Martina
2016-11-01
For a noncommutative configuration space whose coordinate algebra is the universal enveloping algebra of a finite-dimensional Lie algebra, it is known how to introduce an extension playing the role of the corresponding noncommutative phase space, namely by adding the commuting deformed derivatives in a consistent and nontrivial way; therefore, obtaining certain deformed Heisenberg algebra. This algebra has been studied in physical contexts, mainly in the case of the kappa-Minkowski space-time. Here, we equip the entire phase space algebra with a coproduct, so that it becomes an instance of a completed variant of a Hopf algebroid over a noncommutative base, where the base is the enveloping algebra.
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
A double commutant theorem for Murray–von Neumann algebras
Liu, Zhe
2012-01-01
Murray–von Neumann algebras are algebras of operators affiliated with finite von Neumann algebras. In this article, we study commutativity and affiliation of self-adjoint operators (possibly unbounded). We show that a maximal abelian self-adjoint subalgebra of the Murray–von Neumann algebra associated with a finite von Neumann algebra is the Murray–von Neumann algebra , where is a maximal abelian self-adjoint subalgebra of and, in addition, is . We also prove that the Murray–von Neumann algebra with the center of is the center of the Murray–von Neumann algebra . Von Neumann’s celebrated double commutant theorem characterizes von Neumann algebras as those for which , where , the commutant of , is the set of bounded operators on the Hilbert space that commute with all operators in . At the end of this article, we present a double commutant theorem for Murray–von Neumann algebras. PMID:22543165
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.
Equivariant Gromov-Witten Invariants of Algebraic GKM Manifolds
NASA Astrophysics Data System (ADS)
Liu, Chiu-Chu Melissa; Sheshmani, Artan
2017-07-01
An algebraic GKM manifold is a non-singular algebraic variety equipped with an algebraic action of an algebraic torus, with only finitely many torus fixed points and finitely many 1-dimensional orbits. In this expository article, we use virtual localization to express equivariant Gromov-Witten invariants of any algebraic GKM manifold (which is not necessarily compact) in terms of Hodge integrals over moduli stacks of stable curves and the GKM graph of the GKM manifold.
The Vertex Algebra M(1)+ and Certain Affine Vertex Algebras of Level -1
NASA Astrophysics Data System (ADS)
Adamović, Dražen; Perše, Ozren
2012-07-01
We give a coset realization of the vertex operator algebra M(1)^+ with central charge ℓ. We realize M(1) ^+ as a commutant of certain affine vertex algebras of level -1 in the vertex algebra L_{C_{ℓ} ^{(1)}}(-1/2 Λ_0) ⊗ L_{C_{ℓ} ^{(1)}}(-1/2 Λ_0). We show that the simple vertex algebra L_{C_{ℓ} ^{(1)}}(-Λ_0) can be (conformally) embedded into L_{A_{2 ℓ -1} ^{(1)}} (-Λ_0) and find the corresponding decomposition. We also study certain coset subalgebras inside L_{C_{ℓ} ^{(1)}}(-Λ_0).
BiHom-Associative Algebras, BiHom-Lie Algebras and BiHom-Bialgebras
NASA Astrophysics Data System (ADS)
Graziani, Giacomo; Makhlouf, Abdenacer; Menini, Claudia; Panaite, Florin
2015-10-01
A BiHom-associative algebra is a (nonassociative) algebra A endowed with two commuting multiplicative linear maps α,β\\colon A→ A such that α (a)(bc)=(ab)β (c), for all a, b, cin A. This concept arose in the study of algebras in so-called group Hom-categories. In this paper, we introduce as well BiHom-Lie algebras (also by using the categorical approach) and BiHom-bialgebras. We discuss these new structures by presenting some basic properties and constructions (representations, twisted tensor products, smash products etc).
Cohomological invariants of central simple algebras
NASA Astrophysics Data System (ADS)
Merkurjev, A. S.
2016-10-01
We determine the indecomposable degree 3 cohomological invariants of tuples of central simple algebras with linear relations. Equivalently, we determine the degree 3 reductive cohomological invariants of all split semisimple groups of type A.
Using Schemas to Develop Algebraic Thinking
ERIC Educational Resources Information Center
Steele, Diana F.
2005-01-01
This article describes ways in which students develop schemas as they generalize and formalize patterns when solving related algebraic problems that involve size, shape, growth, and change. (Contains 7 figures and 3 tables.)
On fuzzy ideals of BL-algebras.
Meng, Biao Long; Xin, Xiao Long
2014-01-01
In this paper we investigate further properties of fuzzy ideals of a BL-algebra. The notions of fuzzy prime ideals, fuzzy irreducible ideals, and fuzzy Gödel ideals of a BL-algebra are introduced and their several properties are investigated. We give a procedure to generate a fuzzy ideal by a fuzzy set. We prove that every fuzzy irreducible ideal is a fuzzy prime ideal but a fuzzy prime ideal may not be a fuzzy irreducible ideal and prove that a fuzzy prime ideal ω is a fuzzy irreducible ideal if and only if ω(0) = 1 and |Im(ω)| = 2. We give the Krull-Stone representation theorem of fuzzy ideals in BL-algebras. Furthermore, we prove that the lattice of all fuzzy ideals of a BL-algebra is a complete distributive lattice. Finally, it is proved that every fuzzy Boolean ideal is a fuzzy Gödel ideal, but the converse implication is not true.
ALGEBRAIC DEPENDENCE THEOREMS ON COMPLEX PSEUDOCONCAVE SPACES
The notion of pseudoconcave space is introduced and classical theorems on algebraic dependence of meromorphic functions are extended for this new class of spaces and for sections in a coherent sheaf. (Author)
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)
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
Cyclic Cocycles on Twisted Convolution Algebras
NASA Astrophysics Data System (ADS)
Angel, Eitan
2013-01-01
We give a construction of cyclic cocycles on convolution algebras twisted by gerbes over discrete translation groupoids. For proper étale groupoids, Tu and Xu (Adv Math 207(2):455-483, 2006) provide a map between the periodic cyclic cohomology of a gerbe-twisted convolution algebra and twisted cohomology groups which is similar to the construction of Mathai and Stevenson (Adv Math 200(2):303-335, 2006). When the groupoid is not proper, we cannot construct an invariant connection on the gerbe; therefore to study this algebra, we instead develop simplicial techniques to construct a simplicial curvature 3-form representing the class of the gerbe. Then by using a JLO formula we define a morphism from a simplicial complex twisted by this simplicial curvature 3-form to the mixed bicomplex computing the periodic cyclic cohomology of the twisted convolution algebras.
An Oral-Intensive Abstract Algebra Course.
ERIC Educational Resources Information Center
Myers, Nadine C.
2000-01-01
Discusses the evolution of a method for engaging students in the study of abstract algebra and enhancing their depth of understanding. Emphasizes the impact of oral activities on student learning. (Contains 16 references.) (Author/ASK)
Representations of filtered solvable Lie algebras
Panov, Alexander N
2012-01-31
The representation theory of filtered solvable Lie algebras is constructed. In this framework a classification of irreducible representations is obtained and spectra of some reducible representations are found. Bibliography: 9 titles.
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)
Supergroups in Critical Dimensions and Division Algebras
NASA Astrophysics Data System (ADS)
Burdik, Čestmir; Catto, Sultan; Gürcan, Yasemin; Khalfan, Amish; Kurt, Levent; Kato La, V.
2017-01-01
We establish a link between classical heterotic strings and the groups of the magic square associated with Jordan algebras, allowing for a uniform treatment of the bosonic and superstring sectors of the heterotic string.