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Journal articles on the topic 'Group theory and generalizations'

Author: Grafiati

Published: 4 June 2021

Last updated: 29 July 2024

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1

Isaacs, I. M., and M. R. Pournaki. "Generalizations of Fermat's Little Theorem via Group Theory." American Mathematical Monthly 112, no. 8 (October 1, 2005): 734. http://dx.doi.org/10.2307/30037576.

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2

Isaacs, I. M., and M. R. Pournaki. "Generalizations of Fermat's Little Theorem via Group Theory." American Mathematical Monthly 112, no. 8 (October 2005): 734–40. http://dx.doi.org/10.1080/00029890.2005.11920246.

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3

Wang, Zichuan. "Proofs, Generalizations and Applications of Fermat’s Little Theorem." Highlights in Science, Engineering and Technology 47 (May 11, 2023): 32–36. http://dx.doi.org/10.54097/hset.v47i.8161.

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This paper introduces Fermat’s little theorem (FLT), which says that any integer raised to power is congruent to modulo . This paper will give several proofs of FLT, using methods including number theory and group theory, together with generalizations of FLT in different directions. FLT is an important result in number theory and group theory. It has multiple generalizations and corollaries, and one of its corollaries is the foundation of RSA cryptography. The effort made trying to prove FLT stimulated researches in many fields in mathematics, and FLT is crucial and fundamental in research of modern cryptography.

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4

MARCOLLI, MATILDE, and VARGHESE MATHAI. "TWISTED INDEX THEORY ON GOOD ORBIFOLDS, I: NONCOMMUTATIVE BLOCH THEORY." Communications in Contemporary Mathematics 01, no. 04 (November 1999): 553–87. http://dx.doi.org/10.1142/s0219199799000213.

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We study the twisted index theory of elliptic operators on orbifold covering spaces of compact good orbifolds, which are invariant under a projective action of the orbifold fundamental group. We apply these results to obtain qualitative results on real and complex hyperbolic spaces in two and four dimensions, related to generalizations of the Bethe–Sommerfeld conjecture and the Ten Martini Problem, on the spectrum of self adjoint elliptic operators which are invariant under a projective action of a discrete cocompact group.

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5

GRISHKOV, ALEXANDER, and EUGENE LOGINOV. "ON SOME GENERALIZATIONS OF GROUPS WITH TRIALITY." International Journal of Algebra and Computation 22, no. 02 (March 2012): 1250015. http://dx.doi.org/10.1142/s0218196711006820.

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In the present paper we generalize the concept of groups with triality and apply it to the theory of the Moufang, Bol and Bruck loops. Such generalizations allow us to reduce certain problems from the loop theory to problems in the theory of groups.

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6

Morato, Elisson. "GENERALIZAÇÃO DISCURSIVA E ENUNCIAÇÃO NAS CANTIGAS DE CONGADO EM MINAS GERAIS: UMA ABORDAGEM SEMIOLINGUÍSTICA." Entremeios, Revista de Estudos do Discurso 22, no. 22 (December 29, 2020): 236–54. http://dx.doi.org/10.20337/issn2179-3514revistaentremeiosvol22pagina236a254.

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This paper studies the discursive generalizations in the congado songs of the Brazilian state of Minas Gerais through the Discourse Analysis of French line. From the concept of discursive generalization, developed by Ali-Bouacha (1993, [2007] 2020) and by Moirand (1990), and throught the concepts of language competences, discursive strategies and enunciation, from the Semiolinguistic theory developed by Charaudeau (1983, [2000] 2020a, 2004a, 2004b, 2005a, 2005b, [2006] 2020b, 2008), we understand that discursive generalizations would be effects of meaning based on the assignation of this discourse to a meta-enunciator, building effects of universal truth. This procedure contributes for that a knowledge set forms the identity of a social group deeply identified with the Afro-Brazilian cultural traditions.

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7

Uuye, Otgonbayar. "The Baum-Connes conjecture for KK-theory." Journal of K-Theory 8, no. 1 (April 7, 2010): 3–29. http://dx.doi.org/10.1017/is010003012jkt114.

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AbstractWe define and compare two bivariant generalizations of the topological K - group Ktop(G). We consider the Baum-Connes conjecture in this context and study its relation to the usual Baum-Connes conjecture.

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8

Baytaş, Bekir, Martin Bojowald, and Sean Crowe. "Equivalence of Models in Loop Quantum Cosmology and Group Field Theory." Universe 5, no. 2 (January 23, 2019): 41. http://dx.doi.org/10.3390/universe5020041.

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The paradigmatic models often used to highlight cosmological features of loop quantum gravity and group field theory are shown to be equivalent, in the sense that they are different realizations of the same model given by harmonic cosmology. The loop version of harmonic cosmology is a canonical realization, while the group-field version is a bosonic realization. The existence of a large number of bosonic realizations suggests generalizations of models in group field cosmology.

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Dobrynina, Irina V., and Evgeny V. Manokhin. "ON THE SOLVABILITY OF GENERALIZATION OF THE CONJUGACY PROBLEM OF SUBGROUPS IN A CERTAIN CLASS OF ARTIN GROUPS." SOFT MEASUREMENTS AND COMPUTING 4, no. 65 (2023): 25–34. http://dx.doi.org/10.36871/2618-9976.2023.04.003.

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At the beginning of the twentieth century, the fundamental problems in group theory from the point of view of the existence of algorithms took shape. These include the problems of words, as well as the conjugacy of arbitrary words (formulated by M. Den). In addition, the problem of isomorphism for groups is posed (formulated by G. Titze). Russian algebraists P.S. Novikov and S.I. Adyan were able to obtain a proof of the unsolvability of these problems for all groups belonging to finitely defined groups. After the publication of these results, algorithmic problems, together with their various generalizations, are investigated already in fixed classes of groups. As one of the generalizations of the problem for conjugacy of arbitrary words, the question of conjugacy of arbitrary subgroups is posed, namely, the question of finding an algorithm that, by arbitrary subgroups given by a finite set of generating and defining words of some group, could determine whether they are conjugate in this group or not conjugate. As another generalization of the conjugacy of words, we can consider generalized conjugacy for words, which allows us to determine an algorithm capable of determining from arbitrary finite sets of words from a certain group whether they will be conjugated in it or not. If both of these generalizations are combined into one, then we get a generalization of the conjugacy problem of subgroups. Artin groups appeared a long time ago, they include wellknown braid groups that have been studied from an algebraic point of view since the twenties of the last century. The solution of the problems under consideration in Artin's groups caused great difficulties, which led to the allocation of various subclasses. The article proves the solvability of generalization of the conjugacy problem of subgroups in Artin groups on two generating ones.

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10

Guo, Xiao-Kan. "Thermofield double states in group field theory." International Journal of Modern Physics A 36, no. 02 (January 20, 2021): 2150008. http://dx.doi.org/10.1142/s0217751x21500081.

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Group field theories are higher-rank generalizations of matrix/tensor models, and encode the simplicial geometries of quantum gravity. In this paper, we study the thermofield double states in group field theories. The starting point is the equilibrium Gibbs states in group field theory recently found by Kotecha and Oriti, based on which we construct the thermofield double state as a “thermal” vacuum respecting the Kubo–Martin–Schwinger condition. We work with the Weyl [Formula: see text]-algebra of group fields, and a particular type of thermofield double states with single type of symmetry is obtained from the squeezed states on this Weyl algebra. The thermofield double states, when viewed as states on the group field theory Fock vacuum, are condensate states at finite flow parameter [Formula: see text]. We suggest that the equilibrium flow parameters [Formula: see text] of this type of thermofield double states in the group field theory condensate pictures of black hole horizon and quantum cosmology are related to the inverse temperatures in gravitational thermodynamics.

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11

DELGADO, MANUEL, STUART MARGOLIS, and BENJAMIN STEINBERG. "COMBINATORIAL GROUP THEORY, INVERSE MONOIDS, AUTOMATA, AND GLOBAL SEMIGROUP THEORY." International Journal of Algebra and Computation 12, no. 01n02 (February 2002): 179–211. http://dx.doi.org/10.1142/s0218196702000924.

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This paper explores various connections between combinatorial group theory, semigroup theory, and formal language theory. Let G = <A|R> be a group presentation and ℬA, R its standard 2-complex. Suppose X is a 2-complex with a morphism to ℬA, R which restricts to an immersion on the 1-skeleton. Then we associate an inverse monoid to X which algebraically encodes topological properties of the morphism. Applications are given to separability properties of groups. We also associate an inverse monoid M(A, R) to the presentation <A|R> with the property that pointed subgraphs of covers of ℬA, R are classified by closed inverse submonoids of M(A, R). In particular, we obtain an inverse monoid theoretic condition for a subgroup to be quasiconvex allowing semigroup theoretic variants on the usual proofs that the intersection of such subgroups is quasiconvex and that such subgroups are finitely generated. Generalizations are given to non-geodesic combings. We also obtain a formal language theoretic equivalence to quasiconvexity which holds even for groups which are not hyperbolic. Finally, we illustrate some applications of separability properties of relatively free groups to finite semigroup theory. In particular, we can deduce the decidability of various semidirect and Mal/cev products of pseudovarieties of monoids with equational pseudovarieties of nilpotent groups and with the pseudovariety of metabelian groups.

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12

Lenart, Cristian. "Formal Group-Theoretic Generalizations of the Necklace Algebra, Including aq-Deformation." Journal of Algebra 199, no. 2 (January 1998): 703–32. http://dx.doi.org/10.1006/jabr.1997.7203.

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13

TWAREQUE ALI, S., J. P. ANTOINE, J. P. GAZEAU, and U. A. MUELLER. "COHERENT STATES AND THEIR GENERALIZATIONS: A MATHEMATICAL OVERVIEW." Reviews in Mathematical Physics 07, no. 07 (October 1995): 1013–104. http://dx.doi.org/10.1142/s0129055x95000396.

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We present a survey of the theory of coherent states (CS) and some of their generalizations, with emphasis on the mathematical structure, rather than on physical applications. Starting from the standard theory of CS over Lie groups, we develop a general formalism, in which CS are associated to group representations which are square integrable over a homogeneous space. A further step allows us to dispense with the group context altogether, and thus obtain the so-called reproducing triples and continuous frames introduced in some earlier work. We discuss in detail a number of concrete examples, namely semisimple Lie groups, the relativity groups and various types of wavelets. Finally we turn to some physical applications, centering on quantum measurement and the quantization/dequantization problem, that is, the transition from the classical to the quantum level and vice versa.

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14

K., Appi Reddy, Kurmayya T., and K. Sivakumar. "Group inverse extensions of certain $M$-matrix properties." Electronic Journal of Linear Algebra 31 (February 5, 2016): 686–705. http://dx.doi.org/10.13001/1081-3810.3233.

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In this article, generalizations of certain $M$-matrix properties are proved for the group generalized inverse. The proofs use the notion of proper splittings of one type or the other. In deriving certain results, we make use of a recently introduced notion of a $B_{\#}$-splitting. Applications in obtaining comparison results for the spectral radii of matrices are presented.

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15

Montagard, P. L., B. Pasquier, and N. Ressayre. "Two generalizations of the PRV conjecture." Compositio Mathematica 147, no. 4 (March 18, 2011): 1321–36. http://dx.doi.org/10.1112/s0010437x10005233.

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AbstractLet G be a complex connected reductive group. The Parthasarathy–Ranga Rao–Varadarajan (PRV) conjecture, which was proved independently by S. Kumar and O. Mathieu in 1989, gives explicit irreducible submodules of the tensor product of two irreducible G-modules. This paper has three aims. First, we simplify the proof of the PRV conjecture, then we generalize it to other branching problems. Finally, we find other irreducible components of the tensor product of two irreducible G-modules that appear for ‘the same reason’ as the PRV ones.

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16

BALACHANDRAN, A. P., G. BIMONTE, and P. TEOTONIO-SOBRINHO. "EDGE STATES IN 4D AND THEIR 3D GROUPS AND FIELDS." Modern Physics Letters A 08, no. 14 (May 10, 1993): 1305–13. http://dx.doi.org/10.1142/s0217732393001033.

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It is known that the Lagrangian for the edge states of a Chern-Simons theory describes a coadjoint orbit of a Kac-Moody (KM) group with its associated Kirillov symplectic form and group representation. It can also be obtained from a chiral sector of a non-chiral field theory. We study the edge states of the Abelian BF system in four dimensions (4D) and show the following results in almost exact analogy: 1) The Lagrangian for these states is associated with a certain 2D generalization of the KM group. It describes a coadjoint orbit of this group as a Kirillov symplectic manifold and also the corresponding group representation. 2) It can be obtained from with a "self-dual" or "anti-self-dual" sector of a Lagrangian describing a massless scalar and a Maxwell field (the phrase "self-dual" here being used essentially in its sense in monopole theory). There are similar results for the non-Abelian BF system as well. These shared features of edge states in 3D and 4D suggest that the edge Lagrangians for BF systems are certain natural generalizations of field theory Lagrangians related to KM groups.

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17

Gaglione, A. M., and D. Spellman. "Generalizations of free groups: some questions." Communications in Algebra 22, no. 8 (January 1994): 3159–69. http://dx.doi.org/10.1080/00927879408825019.

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18

Johnson, Clifford V. "Physical generalizations of the Rényi entropy." International Journal of Modern Physics D 28, no. 07 (May 2019): 1950091. http://dx.doi.org/10.1142/s0218271819500913.

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We present a new type of generalization of the Rényi entropy that follows naturally from its representation as a thermodynamic quantity. We apply it to the case of [Formula: see text]-dimensional conformal field theories (CFTs) reduced on a region bounded by a sphere. It is known how to compute their Rényi entropy as an integral of the thermal entropy of hyperbolic black holes in [Formula: see text]-dimensional anti-de Sitter spacetime. We show how this integral fits into the framework of extended gravitational thermodynamics, and then point out the natural generalization of the Rényi entropy that suggests itself in that light. In the field theory terms, the new generalization employs aspects of the physics of Renormalization Group (RG) flow to define a refined version of the reduced vacuum density matrix. For [Formula: see text], it can be derived directly in terms of twist operators in field theory. The framework presented here may have applications beyond this context, perhaps in studies of both quantum and classical information theoretic properties of a variety of systems.

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19

Krupiński, Krzysztof. "Generalizations of small profinite structures." Journal of Symbolic Logic 75, no. 4 (December 2010): 1147–75. http://dx.doi.org/10.2178/jsl/1286198141.

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AbstractWe generalize the model theory of small profinite structures developed by Newelski to the case of compact metric spaces considered together with compact groups of homeomorphisms and satisfying the existence of m-independent extensions (we call them compact e-structures). We analyze the relationships between smallness and different versions of the assumption of the existence of m-independent extensions and we obtain some topological consequences of these assumptions. Using them, we adopt Newelski's proofs of various results about small profinite structures to compact e-structures. In particular, we notice that a variant of the group configuration theorem holds in this context.A general construction of compact structures is described. Using it, a class of examples of compact e-structures which are not small is constructed.It is also noticed that in an m-stable compact e-structure every orbit is equidominant with a product of m-regular orbits.

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20

de Vega, H. J., and N. Sánchez. "Quantum group generalization of string theory." Physics Letters B 216, no. 1-2 (January 1989): 97–102. http://dx.doi.org/10.1016/0370-2693(89)91375-0.

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21

Birget, Jean-Camille. "Monoid generalizations of the Richard Thompson groups." Journal of Pure and Applied Algebra 213, no. 2 (February 2009): 264–78. http://dx.doi.org/10.1016/j.jpaa.2008.06.012.

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22

AHN, CHANGHYUN. "MORE ON META-STABLE BRANE CONFIGURATIONS BY DUALIZING THE MULTIPLE GAUGE GROUPS." International Journal of Modern Physics A 25, no. 04 (February 10, 2010): 861–902. http://dx.doi.org/10.1142/s0217751x10047907.

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We reexamine the [Formula: see text] supersymmetric gauge theories with product gauge groups by adding the mass terms and the quartic terms for the flavors: two-gauge group theory with fundamentals, bifundamentals and adjoints, three-gauge group theory with fundamentals and bifundamentals, and their orientifold 4-plane generalizations. By moving the branes appropriately, we obtain the corresponding dual gauge theories. By analyzing the dual superpotentials, we present the type IIA nonsupersymmetric meta-stable brane configurations.

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23

JACKSON, DAVID M., IAIN MOFFATT, and ALEJANDRO MORALES. "ON THE GROUP-LIKE BEHAVIOR OF THE LE–MURAKAMI–OHTSUKI INVARIANT." Journal of Knot Theory and Its Ramifications 16, no. 06 (August 2007): 699–718. http://dx.doi.org/10.1142/s0218216507005452.

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We study the effect of Feynman integration and diagrammatic differential operators on the structure of group-like elements in the algebra generated by colored vertex-oriented uni-trivalent graphs. We provide applications of our results to the study of the LMO invariant, a quantum invariant of manifolds. We also indicate further situations in which our results apply and may prove useful. The enumerative approach that we adopt has a clarity that has enabled us to perceive a number of generalizations.

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24

Ragland, Matthew F. "Generalizations of Groups in which Normality Is Transitive." Communications in Algebra 35, no. 10 (September 21, 2007): 3242–52. http://dx.doi.org/10.1080/00914030701410302.

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25

Keef, P. "On Generalizations of Purity in Primary Abelian Groups." Journal of Algebra 167, no. 2 (July 1994): 309–29. http://dx.doi.org/10.1006/jabr.1994.1187.

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26

Tornier, Stephan. "Prime localizations of Burger–Mozes-type groups." Journal of Group Theory 21, no. 2 (March 1, 2018): 229–40. http://dx.doi.org/10.1515/jgth-2017-0036.

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AbstractThis article concerns Burger–Mozes universal groups acting on regular trees locally like a given permutation group of finite degree. We also consider locally isomorphic generalizations of the former due to Le Boudec and Lederle. For a large class of such permutation groups and primespwe determine their localp-Sylow subgroups as well as subgroups of theirp-localization, which is identified as a group of the same type in certain cases.

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27

Lukashova, T., and M. Drushlyak. "Generalized norms of groups: retrospective review and current status." Algebra and Discrete Mathematics 34, no. 1 (2022): 105–31. http://dx.doi.org/10.12958/adm1968.

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In this survey paper the authors specify all the known findings related to the norms of a~group and their generalizations (since 2016 in more details). Special attention is paid to the analysis of their own study of different generalized norms, particularly the norm of non-cyclic subgroups, the norm of Abelian non-cyclic subgroups, the norm of decomposable subgroups and relations between them.

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28

Carfora, Mauro. "The Wasserstein geometry of nonlinear σ models and the Hamilton–Perelman Ricci flow." Reviews in Mathematical Physics 29, no. 01 (January 10, 2017): 1750001. http://dx.doi.org/10.1142/s0129055x17500015.

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Nonlinear sigma models are quantum field theories describing, in the large deviation sense, random fluctuations of harmonic maps between a Riemann surface and a Riemannian manifold. Via their formal renormalization group analysis, they provide a framework for possible generalizations of the Hamilton–Perelman Ricci flow. By exploiting the heat kernel embedding introduced by Gigli and Mantegazza, we show that the Wasserstein geometry of the space of probability measures over Riemannian metric measure spaces provides a natural setting for discussing the relation between nonlinear sigma models and Ricci flow theory. In particular, we analyze the embedding of Ricci flow into a heat kernel renormalization group flow for dilatonic nonlinear sigma models, and characterize a non-trivial generalization of the Hamilton–Perelman version of the Ricci flow. We discuss in detail the monotonicity and gradient flow properties of this extended flow.

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29

Dougherty, Aria L., Hwajin Park, and David N. Yetter. "On 2-dimensional Dijkgraaf-Witten theory with defects." Journal of Knot Theory and Its Ramifications 25, no. 05 (April 2016): 1650021. http://dx.doi.org/10.1142/s0218216516500218.

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In this paper, we provide a construction of a state-sum model for finite gauge-group Dijkgraaf-Witten theory on surfaces with codimension 1 defects. The construction requires not only that the triangulation be subordinate to the filtration, but flag-like: each simplex of the triangulation is either disjoint from the defect curve, or intersects it in a closed face. The construction allows internal degrees of freedom in the defect curves by introducing a second gauge-group from which edges of the curve are labeled in the state-sum construction. Edges incident with the defect, but not lying in it, have states lying in a set with commuting actions of the two gauge-groups. We determine the appropriate generalizations of the 2-cocycles specifying twistings of defect-free 2D Dijkgraaf-Witten theory. Examples arising by restriction of group 2-cocycles, and constructed from characters of the 2-dimensional gauge group are presented.

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30

Hart, Sarah B., and Peter J. Rowley. "Poincaré Series of Cosets in Coxeter Groups." Algebra Colloquium 16, no. 04 (December 2009): 567–74. http://dx.doi.org/10.1142/s1005386709000534.

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The Poincaré series of a Coxeter group W measures its growth relative to the generating set R. A well-known formula gives an inductive method of calculating the Poincaré series of W when W is finite. The aim of this paper is to obtain generalizations of this formula.

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31

Chekhlov, Andrey R., Peter V. Danchev, and Patrick W. Keef. "Generalizations of fully transitive and valuated Abelian p-groups." Journal of Algebra 566 (January 2021): 187–204. http://dx.doi.org/10.1016/j.jalgebra.2020.09.014.

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32

Ahsanullah, T. M. G., and Gunther Jäger. "Quantale-Valued Generalizations of Approach Groups." New Mathematics and Natural Computation 15, no. 01 (December 25, 2018): 1–30. http://dx.doi.org/10.1142/s1793005719500017.

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The motive behind this paper is to generalize the concept of approach groups. In so doing, we introduce various categories, specifically, the categories of quantale-valued convergence groups, quantale-valued approach groups, quantale-valued gauge groups and quantale-valued approach system groups, and study their functorial relationships including the fact that the category of quantale-valued gauge groups as well as the category of quantale-valued approach system groups is topological over the category of groups. Besides obtaining a variety of categorical connections, we note that if the quantale is linearly ordered satisfying certain conditions, then the categories of quantale-valued approach system groups and quantale-valued gauge groups are isomorphic. Finally, we look into the embeddings of the category of quantale-valued metric groups into the categories of quantale-valued generalizations of approach groups including commutativity of some diagrams.

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REUTER, M. "QUANTUM MECHANICS AS A GAUGE THEORY OF METAPLECTIC SPINOR FIELDS." International Journal of Modern Physics A 13, no. 22 (September 10, 1998): 3835–83. http://dx.doi.org/10.1142/s0217751x98001803.

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A hidden gauge theory structure of quantum mechanics which is invisible in its conventional formulation is uncovered. Quantum mechanics is shown to be equivalent to a certain Yang–Mills theory with an infinite-dimensional gauge group and a nondynamical connection. It is defined over an arbitrary symplectic manifold which constitutes the phase space of the system under consideration. The "matter fields" are local generalizations of states and observables; they assume values in a family of local Hilbert spaces (and their tensor products) which are attached to the points of phase space. Under local frame rotations they transform in the spinor representation of the metaplectic group Mp(2N), the double covering of Sp(2N). The rules of canonical quantization are replaced by two independent postulates with a simple group-theoretical and differential-geometrical interpretation. A novel background-quantum split symmetry plays a central role.

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34

BELLON, M., J.-M. MAILLARD, and C. VIALLET. "ON THE SYMMETRIES OF INTEGRABILITY." International Journal of Modern Physics B 06, no. 11n12 (June 1992): 1881–903. http://dx.doi.org/10.1142/s021797929200092x.

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We show that the Yang-Baxter equations for two-dimensional models admit as a group of symmetry the infinite discrete group [Formula: see text]. The existence of this symmetry explains the presence of a spectral parameter in the solutions of the equations. We show that similarly, for three-dimensional vertex models and the associated tetrahedron equations, there also exists an infinite discrete group of symmetry. Although generalizing naturally the previous one, it is a much bigger hyperbolic Coxeter group. We indicate how this symmetry can help to resolve the Yang-Baxter equations and their higher-dimensional generalizations and initiate the study of three-dimensional vertex models. These symmetries are naturally represented as birational projective transformations. They may preserve non-trivial algebraic varieties, and lead to proper parametrizations of the models, be they integrable or not. We mention the relation existing between spin models and the Bose-Messner algebras of algebraic combinatorics. Our results also yield the generalization of the condition qn=1 so often mentioned in the theory of quantum groups, when no q parameter is available.

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35

Kato, Motoko. "On groups whose actions on finite-dimensional CAT(0) spaces have global fixed points." Journal of Group Theory 22, no. 6 (November 1, 2019): 1089–99. http://dx.doi.org/10.1515/jgth-2018-0116.

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Abstract We give a criterion for group elements to have fixed points with respect to a semi-simple action on a complete CAT(0) space of finite topological dimension. As an application, we show that Thompson’s group T and various generalizations of Thompson’s group V have global fixed points when they act semi-simply on finite-dimensional complete CAT(0) spaces, while it is known that T and V act properly on infinite-dimensional CAT(0) cube complexes.

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36

Qureshi, Mohd Idris, Junesang Choi, and Tafaz Rahman Shah. "Certain Generalizations of Quadratic Transformations of Hypergeometric and Generalized Hypergeometric Functions." Symmetry 14, no. 5 (May 23, 2022): 1073. http://dx.doi.org/10.3390/sym14051073.

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There have been numerous investigations on the hypergeometric series 2F1 and the generalized hypergeometric series pFq such as differential equations, integral representations, analytic continuations, asymptotic expansions, reduction cases, extensions of one and several variables, continued fractions, Riemann’s equation, group of the hypergeometric equation, summation, and transformation formulae. Among the various approaches to these functions, the transformation formulae for the hypergeometric series 2F1 and the generalized hypergeometric series pFq are significant, both in terms of applications and theory. The purpose of this paper is to establish a number of transformation formulae for pFq, whose particular cases would include Gauss’s and Kummer’s quadratic transformation formulae for 2F1, as well as their two extensions for 3F2, by making advantageous use of a recently introduced sequence and some techniques commonly used in dealing with pFq theory. The pFq function, which is the most significant function investigated in this study, exhibits natural symmetry.

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37

TSEYTLIN, A. A. "COSMOLOGICAL SOLUTIONS WITH DILATON AND MAXIMALLY SYMMETRIC SPACE IN STRING THEORY." International Journal of Modern Physics D 01, no. 01 (January 1992): 223–45. http://dx.doi.org/10.1142/s0218271892000112.

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Time-dependent solutions of the leading-order string effective equations for a nonzero central charge deficit and curved maximally-symmetric space are studied. Some regular solutions are found for the case of nontrivial antisymmetric tensor and vector backgrounds (in various dimensions) and negative spatial curvature. It remains an open question which conformal theories are exact generalizations of these solutions. An analogy between the string cosmological solutions and the solutions of the standard first order renormalisation group equations interpolating between “static” conformal theories is discussed.

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Bermejo-Vega, Juan, and Maarten Van den Nest. "Classical simulations of Abelian-group normalizer circuits with intermediate measurements." Quantum Information and Computation 14, no. 3&4 (March 2014): 181–216. http://dx.doi.org/10.26421/qic14.3-4-1.

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Quantum normalizer circuits were recently introduced as generalizations of Clifford circuits: a normalizer circuit over a finite Abelian group G is composed of the quantum Fourier transform (QFT) over G, together with gates which compute quadratic functions and automorphisms. In \cite{VDNest_12_QFTs} it was shown that every normalizer circuit can be simulated efficiently classically. This result provides a nontrivial example of a family of quantum circuits that cannot yield exponential speed-ups in spite of usage of the QFT, the latter being a central quantum algorithmic primitive. Here we extend the aforementioned result in several ways. Most importantly, we show that normalizer circuits supplemented with intermediate measurements can also be simulated efficiently classically, even when the computation proceeds adaptively. This yields a generalization of the Gottesman-Knill theorem (valid for n-qubit Clifford operations) to quantum circuits described by arbitrary finite Abelian groups. Moreover, our simulations are twofold: we present efficient classical algorithms to sample the measurement probability distribution of any adaptive-normalizer computation, as well as to compute the amplitudes of the state vector in every step of it. Finally we develop a generalization of the stabilizer formalism relative to arbitrary finite Abelian groups: for example we characterize how to update stabilizers under generalized Pauli measurements and provide a normal form of the amplitudes of generalized stabilizer states using quadratic functions and subgroup cosets.

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Bergelson, Vitaly, and Joseph Rosenblatt. "Joint ergodicity for group actions." Ergodic Theory and Dynamical Systems 8, no. 3 (September 1988): 351–64. http://dx.doi.org/10.1017/s0143385700004508.

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AbstractLet T1,…,Tn be continuous representations of a σ-compact separable locally compact amenable group G as measure-preserving transformations of a non-atomic separable probability space (X, β, m). Let (Kn) be a right Følner sequence of compact sets in G. If T1,…,Tn are pairwise commuting in the sense that Ti(g)Tj(h) = Tj(h)Ti(g) for i ≠ j and g, h ∈ G, then necessary and sufficient conditions can be given, in terms of the ergodicity of certain tensor products, for the following to hold: for all F1,…,Fn∈L∞, the sequence AN(x) whereconverges in L2(X) to . The necessary and sufficient conditions are that each of the following representations are ergodic: Tn, Tn−1⊗Tn−1Tn,…,T2⊗T2T3⊗…⊗T2…Tn, T1⊗T1T2⊗…⊗T1…Tn.In order to prove this theorem, specific properties of the decomposition of L2(X) into its weakly mixing and compact subspaces with respect to a representation Ti are needed. These properties are also used to prove some generalizations of wellknown facts from ergodic theory in the case where G is the integer group Z.

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Ahsanullah, T. M. G., and Gunther Jäger. "Quantale-Valued Uniformizations of Quantale-Valued Generalizations of Approach Groups." New Mathematics and Natural Computation 15, no. 03 (October 7, 2019): 517–38. http://dx.doi.org/10.1142/s1793005719500303.

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We introduce the categories of quantale-valued approach uniform spaces and quantale-valued uniform gauge spaces, and prove that they are topological categories. We first show that the category of quantale-valued uniform gauge spaces is a full bireflective subcategory of the category of quantale-valued approach uniform spaces and; second, we prove that only under strong restrictions on the quantale these two categories are isomorphic. Besides presenting embeddings of the category of quantale-valued metric spaces into the categories of quantale-valued approach uniform spaces as well as quantale-valued uniform gauge spaces, we show that every quantale-valued approach system group and quantale-valued gauge group has a natural underlying quantale-valued approach uniform space, respectively, a quantale-valued uniform gauge space.

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FLOREANINI, R., and R. PERCACCI. "ANOMALOUS GAUSS LAW ALGEBRAS." International Journal of Modern Physics A 04, no. 17 (October 20, 1989): 4581–91. http://dx.doi.org/10.1142/s0217751x89001953.

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Supplementing the Gauss law operator of an anomalous gauge theory with a certain set of functionals of the gauge potentials, one obtains a closed algebra. The algebras obtained in this way are Abelian extensions of the Lie algebra of the group of gauge transformations, and are natural generalizations of Kac-Moody algebras, both in two and four dimensions.

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Shum, Michael, Max Kleiman-Weiner, Michael L. Littman, and Joshua B. Tenenbaum. "Theory of Minds: Understanding Behavior in Groups through Inverse Planning." Proceedings of the AAAI Conference on Artificial Intelligence 33 (July 17, 2019): 6163–70. http://dx.doi.org/10.1609/aaai.v33i01.33016163.

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Human social behavior is structured by relationships. We form teams, groups, tribes, and alliances at all scales of human life. These structures guide multi-agent cooperation and competition, but when we observe others these underlying relationships are typically unobservable and hence must be inferred. Humans make these inferences intuitively and flexibly, often making rapid generalizations about the latent relationships that underlie behavior from just sparse and noisy observations. Rapid and accurate inferences are important for determining who to cooperate with, who to compete with, and how to cooperate in order to compete. Towards the goal of building machine-learning algorithms with human-like social intelligence, we develop a generative model of multiagent action understanding based on a novel representation for these latent relationships called Composable Team Hierarchies (CTH). This representation is grounded in the formalism of stochastic games and multi-agent reinforcement learning. We use CTH as a target for Bayesian inference yielding a new algorithm for understanding behavior in groups that can both infer hidden relationships as well as predict future actions for multiple agents interacting together. Our algorithm rapidly recovers an underlying causal model of how agents relate in spatial stochastic games from just a few observations. The patterns of inference made by this algorithm closely correspond with human judgments and the algorithm makes the same rapid generalizations that people do.

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GAIFULLIN, ALEXANDER A., and VASSILY O. MANTUROV. "ON THE RECOGNITION OF BRAIDS." Journal of Knot Theory and Its Ramifications 11, no. 08 (December 2002): 1193–209. http://dx.doi.org/10.1142/s0218216502002207.

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A simple complete combinatorial invariant for elements of the braid group is found. It admits some generalizations, e.g. a complete invariant of spherical braids and a complete invariant of cylindrical braids. Values of the invariants are well recognizable, i.e., they provide the complete algorithmic classification of elements in the named braid groups.

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Spiess, Michael, and Takao Yamazaki. "A counterexample to generalizations of the Milnor-Bloch-Kato conjecture." Journal of K-Theory 4, no. 1 (August 2009): 77–90. http://dx.doi.org/10.1017/is008008014jkt066.

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AbstractWe construct an example of a torus T over a field K for which the Galois symbol K(K;T,T)/nK(K;T,T) → H2(K,T[n] ⊗ T[n]) is not injective for some n. Here K(K;T,T) is the Milnor K-group attached to T introduced by Somekawa. We show also that the motive M(T × T) gives a counterexample to another generalization of the Milnor-Bloch-Kato conjecture (proposed by Beilinson).

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BELLINGERI, P., and A. CATTABRIGA. "HILDEN BRAID GROUPS." Journal of Knot Theory and Its Ramifications 21, no. 03 (March 2012): 1250029. http://dx.doi.org/10.1142/s0218216511009534.

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Let H g be a genus g handlebody and MCG 2n( T g) be the group of the isotopy classes of orientation preserving homeomorphisms of T g = ∂ H g, fixing a given set of 2n points. In this paper we study two particular subgroups of MCG 2n( T g) which generalize Hilden groups defined by Hilden in [Generators for two groups related to the braid groups, Pacific J. Math.59 (1975) 475–486]. As well as Hilden groups are related to plat closures of braids, these generalizations are related to Heegaard splittings of manifolds and to bridge decompositions of links. Connections between these subgroups and motion groups of links in closed 3-manifolds are also provided.

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Manturov, Vassily Olegovich. "Reidemeister moves and groups." Journal of Knot Theory and Its Ramifications 24, no. 10 (September 2015): 1540006. http://dx.doi.org/10.1142/s0218216515400064.

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Recently, the author discovered an interesting class of knot-like objects called free knots. These purely combinatorial objects are equivalence classes of Gauss diagrams modulo Reidemeister moves (the same notion in the language of words was introduced by Turaev [Topology of words, Proc. Lond. Math. Soc.95(3) (2007) 360–412], who thought all free knots to be trivial). As it turned out, these new objects are highly nontrivial, see [V. O. Manturov, Parity in knot theory, Mat. Sb.201(5) (2010) 65–110], and even admit nontrivial cobordism classes [V. O. Manturov, Parity and cobordisms of free knots, Mat. Sb.203(2) (2012) 45–76]. An important issue is the existence of invariants where a diagram evaluates to itself which makes such objects "similar" to free groups: An element has its minimal representative which "lives inside" any representative equivalent to it. In this paper, we consider generalizations of free knots by means of (finitely presented) groups. These new objects have lots of nontrivial properties coming from both knot theory and group theory. This connection allows one not only to apply group theory to various problems in knot theory but also to apply Reidemeister moves to the study of (finitely presented) groups. Groups appear naturally in this setting when graphs are embedded in surfaces.

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Shi, Jiangtao, Klavdija Kutnar, and Cui Zhang. "A Note on Special Local 2-Nilpotent Groups and the Solvability of Finite Groups." Algebra Colloquium 25, no. 04 (December 2018): 541–46. http://dx.doi.org/10.1142/s1005386718000378.

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A finite group G is called a special local 2-nilpotent group if G is not 2-nilpotent, the Sylow 2-subgroup P of G has a section isomorphic to the quaternion group of order 8, [Formula: see text] and NG(P) is 2-nilpotent. In this paper, it is shown that SL2(q), [Formula: see text], is a special local 2-nilpotent group if and only if [Formula: see text], and that GL2(q), [Formula: see text], is a special local 2-nilpotent group if and only if q is odd. Moreover, the solvability of finite groups is also investigated by giving two generalizations of a result from [A note on p-nilpotence and solvability of finite groups, J. Algebra 321 (2009) 1555–1560].

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Chekhlov, Andrey R., and Peter V. Danchev. "On commutator Krylov transitive and commutator weakly transitive Abelian p-groups." Forum Mathematicum 31, no. 6 (November 1, 2019): 1607–23. http://dx.doi.org/10.1515/forum-2019-0066.

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AbstractWe define the concepts of commutator (Krylov) transitive and strongly commutator (Krylov) transitive Abelian p-groups. These two innovations are respectively non-trivial generalizations of the notions of commutator fully transitive and strongly commutator fully transitive p-groups from a paper of Chekhlov and Danchev (J. Group Theory, 2015). They are also commutator socle-regular in the sense of Danchev and Goldsmith (J. Group Theory, 2014). Various results from there and from a paper of Goldsmith and Strüngmann (Houston J. Math., 2007) are considerably extended to this new point of view. We also define and explore the concept of a commutator weakly transitive Abelian p-group, comparing its properties with those of the aforementioned two group classes. Some affirmations, sounding quite curiously, are detected in order to illustrate the pathology of the commutators in the endomorphism rings of p-primary Abelian groups.

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HAYASHI, Takahiro. "Face algebras I---A generalization of quantum group theory." Journal of the Mathematical Society of Japan 50, no. 2 (April 1998): 293–315. http://dx.doi.org/10.2969/jmsj/05020293.

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BALANDRAUD, ÉRIC. "THE ISOPERIMETRIC METHOD IN NON-ABELIAN GROUPS WITH AN APPLICATION TO OPTIMALLY SMALL SUMSETS." International Journal of Number Theory 04, no. 06 (December 2008): 927–58. http://dx.doi.org/10.1142/s1793042108001821.

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Set addition theory is born a few decades ago from additive number theory. Several difficult issues, more combinatorial in nature than algebraic, have been revealed. In particular, computing the values taken by the function: [Formula: see text] where G is a given group does not seem easy in general. Some successive results, using Kneser's Theorem, allowed the determination of the values of this function, provided that the group G is abelian. Recently, a method called isoperimetric, has been developed by Hamidoune and allowed new proofs and generalizations of the classical theorems in additive number theory. For instance, a new interpretation of the isoperimetric method has been able to give a new proof of Kneser's Theorem. The purpose of this article is to adapt this last proof in a non-abelian group, in order to give new values of the function μG, for some solvable groups and alternating groups. These values allow us in particular to answer negatively a question asked in the literature on the μG functions.

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