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Journal articles on the topic 'Symmetry groups – Asymptotic theory'

Author: Grafiati

Published: 4 June 2021

Last updated: 14 February 2022

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1

GIULINI, DOMENICO. "ASYMPTOTIC SYMMETRY GROUPS OF LONG-RANGED GAUGE CONFIGURATIONS." Modern Physics Letters A 10, no. 28 (1995): 2059–70. http://dx.doi.org/10.1142/s0217732395002210.

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We make some general remarks on long-ranged configurations in gauge or diffeomorphism invariant theories where the fields are allowed to assume some nonvanishing values at spatial infinity. In this case the Gauss constraint only eliminates those gauge degrees of freedom which lie in the connected component of asymptotically trivial gauge transformations. This implies that proper physical symmetries arise either from gauge transformations that reach to infinity or those that are asymptotically trivial but do not lie in the connected component of transformations within that class. The latter transformations form a discrete subgroup of all symmetries whose position in the ambient group has proven to have interesting implications. We explain this for the dyon configuration in the SO(3) Yang-Mills-Higgs theory, where we prove that the asymptotic symmetry group is Z|m|×ℝ where m is the monopole number. We also discuss the application of the general setting to general relativity and show that here the only implication of discrete symmetries for the continuous part is a possible extension of the rotation group SO(3) to SU(2).

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2

Boyer, Robert. "Character theory of infinite wreath products." International Journal of Mathematics and Mathematical Sciences 2005, no. 9 (2005): 1365–79. http://dx.doi.org/10.1155/ijmms.2005.1365.

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The representation theory of infinite wreath product groups is developed by means of the relationship between their group algebras and conjugacy classes with those of the infinite symmetric group. Further, since these groups are inductive limits of finite groups, their finite characters can be classified as limits of normalized irreducible characters of prelimit finite groups. This identification is called the “asymptotic character formula.” TheK0-invariant of the groupC∗-algebra is also determined.

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3

Antoniadis, Ignatios, and Spiros Cotsakis. "Geodesic Incompleteness and Partially Covariant Gravity." Universe 7, no. 5 (2021): 126. http://dx.doi.org/10.3390/universe7050126.

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We study the issue of length renormalization in the context of fully covariant gravity theories as well as non-relativistic ones such as Hořava–Lifshitz gravity. The difference in their symmetry groups implies a relation among the lengths of paths in spacetime in the two types of theory. Provided that certain asymptotic conditions hold, this relation allows us to transfer analytic criteria for the standard spacetime length to be finite and the Perelman length to be likewise finite, and therefore formulate conditions for geodesic incompleteness in partially covariant theories. We also discuss implications of this result for the issue of singularities in the context of such theories.

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4

Baranov, A. A., A. S. Kleshchev, and A. E. Zalesskii. "Asymptotic Results on Modular Representations of Symmetric Groups and Almost Simple Modular Group Algebras." Journal of Algebra 219, no. 2 (1999): 506–30. http://dx.doi.org/10.1006/jabr.1999.7923.

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5

Koren, Zvi, and Igor Ravve. "Fourth-order normal moveout velocity in elastic layered orthorhombic media — Part 2: Offset-azimuth domain." GEOPHYSICS 82, no. 3 (2017): C113—C132. http://dx.doi.org/10.1190/geo2016-0222.1.

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Based on the theory derived in part 1, in which we obtained the azimuthally dependent fourth-order normal-moveout (NMO) velocity functions for layered orthorhombic media in the slowness-azimuth/slowness and the slowness-azimuth/offset domains, in part 2, we extend the theory to the offset-azimuth/slowness and offset-azimuth/offset domains. We reemphasize that this paper does not suggest a new nonhyperbolic traveltime approximation; rather, it provides exact expressions of the NMO series coefficients, computed for normal-incidence rays, which can then be further used within known azimuthally dependent traveltime approximations for short to moderate offsets. The same type of models as in part 1 are considered, in which the layers share a common horizontal plane of symmetry, but the azimuths of their vertical symmetry planes are different. The same eight local (single-layer) and global (overburden multilayer) effective parameters are used. In addition, we have developed an alternative set of global effective parameters in which the “anisotropic” effective parameters are normalized, classified into two groups: two “azimuthally isotropic” parameters and six “azimuthally anisotropic” parameters. These parameters have a clearer physical interpretation and they are suitable for inversion purposes because they can be controlled and constrained. Next, we propose a special case, referred to as “weak azimuthal anisotropy,” in which only the azimuthally anisotropic effective parameters are assumed to be weak. The resulting NMO velocity functions are considerably simplified, reduced to the form of the slowness-azimuth/slowness formula. We verify the correctness of our method by applying it to a multilayer orthorhombic medium with strong anisotropy. We introduce our derived, fourth-order slowness-azimuth/offset domain NMO velocity function into the well-known nonhyperbolic asymptotic traveltime approximation, and we compare the approximate traveltimes with exact traveltimes obtained by two-point ray tracing. The comparison shows an accurate match up to moderate offsets. Although the accuracy with the weak azimuthal anisotropic formula is inferior, it can still be considered reasonable for practical use.

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6

SEKIGUCHI, HIDEKO. "BRANCHING RULES OF SINGULAR UNITARY REPRESENTATIONS WITH RESPECT TO SYMMETRIC PAIRS (A2n-1, Dn)." International Journal of Mathematics 24, no. 04 (2013): 1350011. http://dx.doi.org/10.1142/s0129167x13500110.

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The irreducible decomposition of scalar holomorphic discrete series representations when restricted to semisimple symmetric pairs (G, H) is explicitly known by Schmid [Die Randwerte holomorphe funktionen auf hermetisch symmetrischen Raumen, Invent. Math.9 (1969–1970) 61–80] for H compact and by Kobayashi [Multiplicity-Free Theorems of the Restrictions of Unitary Highest Weight Modules with Respect to Reductive Symmetric Pairs, Progress in Mathematics, Vol. 255 (Birhäuser, 2007), pp. 45–109] for H non-compact. In this paper, we deal with the symmetric pair (U(n, n), SO* (2n)), and extend the Kobayashi–Schmid formula to certain non-tempered unitary representations which are realized in Dolbeault cohomology groups over open Grassmannian manifolds with indefinite metric. The resulting branching rule is multiplicity-free and discretely decomposable, which fits in the framework of the general theory of discrete decomposable restrictions by Kobayashi [Discrete decomposability of the restriction of A𝔮(λ) with respect to reductive subgroups II — micro-local analysis and asymptotic K-support, Ann. Math.147 (1998), 709–729].

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7

Bauer, Thomas, Sandra Di Rocco, Brian Harbourne, Jack Huizenga, Alexandra Seceleanu, and Tomasz Szemberg. "Negative Curves on Symmetric Blowups of the Projective Plane, Resurgences, and Waldschmidt Constants." International Mathematics Research Notices 2019, no. 24 (2018): 7459–514. http://dx.doi.org/10.1093/imrn/rnx329.

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Abstract The Klein and Wiman configurations are highly symmetric configurations of lines in the projective plane arising from complex reflection groups. One noteworthy property of these configurations is that all the singularities of the configuration have multiplicity at least 3. In this paper we study the surface X obtained by blowing up $\mathbb{P}^{2}$ in the singular points of one of these line configurations. We study invariant curves on X in detail, with a particular emphasis on curves of negative self-intersection. We use the representation theory of the stabilizers of the singular points to discover several invariant curves of negative self-intersection on X, and use these curves to study Nagata-type questions for linear series on X. The homogeneous ideal I of the collection of points in the configuration is an example of an ideal where the symbolic cube of the ideal is not contained in the square of the ideal; ideals with this property are seemingly quite rare. The resurgence and asymptotic resurgence are invariants which were introduced to measure such failures of containment. We use our knowledge of negative curves on X to compute the resurgence of I exactly. We also compute the asymptotic resurgence and Waldschmidt constant exactly in the case of the Wiman configuration of lines, and provide estimates on both for the Klein configuration.

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8

Bary-Soroker, Lior, and Tomer M. Schlank. "SIEVES AND THE MINIMAL RAMIFICATION PROBLEM." Journal of the Institute of Mathematics of Jussieu 19, no. 3 (2018): 919–45. http://dx.doi.org/10.1017/s1474748018000257.

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The minimal ramification problem may be considered as a quantitative version of the inverse Galois problem. For a nontrivial finite group $G$, let $m(G)$ be the minimal integer $m$ for which there exists a $G$-Galois extension $N/\mathbb{Q}$ that is ramified at exactly $m$ primes (including the infinite one). So, the problem is to compute or to bound $m(G)$.In this paper, we bound the ramification of extensions $N/\mathbb{Q}$ obtained as a specialization of a branched covering $\unicode[STIX]{x1D719}:C\rightarrow \mathbb{P}_{\mathbb{Q}}^{1}$. This leads to novel upper bounds on $m(G)$, for finite groups $G$ that are realizable as the Galois group of a branched covering. Some instances of our general results are: $$\begin{eqnarray}1\leqslant m(S_{k})\leqslant 4\quad \text{and}\quad n\leqslant m(S_{k}^{n})\leqslant n+4,\end{eqnarray}$$ for all $n,k>0$. Here $S_{k}$ denotes the symmetric group on $k$ letters, and $S_{k}^{n}$ is the direct product of $n$ copies of $S_{k}$. We also get the correct asymptotic of $m(G^{n})$, as $n\rightarrow \infty$ for a certain class of groups $G$.Our methods are based on sieve theory results, in particular on the Green–Tao–Ziegler theorem on prime values of linear forms in two variables, on the theory of specialization in arithmetic geometry, and on finite group theory.

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9

McCarthy, Patrick J. "Geometry of generalised asymptotic symmetry groups or asymptotic symmetries, product bundles and wreath products." Physics Letters A 174, no. 1-2 (1993): 25–28. http://dx.doi.org/10.1016/0375-9601(93)90536-9.

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10

Iwasa, Masatomo. "Derivation of Asymptotic Dynamical Systems with Partial Lie Symmetry Groups." Journal of Applied Mathematics 2015 (2015): 1–8. http://dx.doi.org/10.1155/2015/601657.

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Lie group analysis has been applied to singular perturbation problems in both ordinary differential and difference equations and has allowed us to find the reduced dynamics describing the asymptotic behavior of the dynamical system. The present study provides an extended method that is also applicable to partial differential equations. The main characteristic of the extended method is the restriction of the manifold by some constraint equations on which we search for a Lie symmetry group. This extension makes it possible to find a partial Lie symmetry group, which leads to a reduced dynamics describing the asymptotic behavior.

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11

Steigmann, David J. "Asymptotic theory for thin two-ply shells." Vietnam Journal of Mechanics 42, no. 3 (2020): 269–82. http://dx.doi.org/10.15625/0866-7136/15337.

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We develop an asymptotic model for the finite-deformation, small-strain response of thin laminated shells composed of two perfectly bonded laminae that exhibit reflection symmetry of the material properties with respect to an interfacial surface. No a priori hypotheses are made concerning the kinematics of deformation. The asymptotic procedure culminates in a generalization of Koiter's well-known shell theory to accommodate the laminated structure, and incorporates a rigorous limit model for pure bending.

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12

Chen, Robin Ming, Yujin Guo, and Daniel Spirn. "Asymptotic behavior and symmetry of condensate solutions in electroweak theory." Journal d'Analyse Mathématique 117, no. 1 (2012): 47–85. http://dx.doi.org/10.1007/s11854-012-0014-6.

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13

LÜST, DIETER, and STEFAN THEISEN. "EXCEPTIONAL GROUPS IN STRING THEORY." International Journal of Modern Physics A 04, no. 17 (1989): 4513–33. http://dx.doi.org/10.1142/s0217751x89001916.

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We review the occurrence of exceptional groups in string theory: their dual role as gauge symmetry and as a symmetry unifying space-time, superconformal ghost and internal degrees of freedom. In both cases the relation to the extended world-sheet supersymmetries is discussed in detail. This is used to construct the supermultiplet structure of the massless sectors of all supergravity theories possible in string theory, in even space-time dimensions between four and ten.

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14

Dranishnikov, A., and J. Smith. "Asymptotic dimension of discrete groups." Fundamenta Mathematicae 189, no. 1 (2006): 27–34. http://dx.doi.org/10.4064/fm189-1-2.

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15

Black, Sarah. "Asymptotic Growth of Finite Groups." Journal of Algebra 209, no. 2 (1998): 402–26. http://dx.doi.org/10.1006/jabr.1998.7502.

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16

FINKELSTEIN, R. J. "QUANTUM GROUPS AND FIELD THEORY." Modern Physics Letters A 15, no. 28 (2000): 1709–15. http://dx.doi.org/10.1142/s0217732300002218.

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When the symmetry of a physical theory describing a finite system is deformed by replacing its Lie group by the corresponding quantum group, the operators and state function will lie in a new algebra describing new degrees of freedom. If the symmetry of a field theory is deformed in this way, the enlarged state space will again describe additional degrees of freedom, and the energy levels will acquire fine structure. The massive particles will have a stringlike spectrum lifting the degeneracy of the point-particle theory, and the resulting theory will have a nonlocal description. Theories of this kind naturally contain two sectors with one sector lying close to the standard theory while the second sector describes particles that should be more difficult to observe.

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17

Dehaye, Paul-Olivier. "On an Identity due to Bump and Diaconis, and Tracy and Widom." Canadian Mathematical Bulletin 54, no. 2 (2011): 255–69. http://dx.doi.org/10.4153/cmb-2011-011-5.

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AbstractA classical question for a Toeplitz matrix with given symbol is how to compute asymptotics for the determinants of its reductions to finite rank. One can also consider how those asymptotics are affected when shifting an initial set of rows and columns (or, equivalently, asymptotics of their minors). Bump and Diaconis obtained a formula for such shifts involving Laguerre polynomials and sums over symmetric groups. They also showed how the Heine identity extends for such minors, which makes this question relevant to Random Matrix Theory. Independently, Tracy and Widom used the Wiener–Hopf factorization to express those shifts in terms of products of infinite matrices. We show directly why those two expressions are equal and uncover some structure in both formulas that was unknown to their authors. We introduce a mysterious differential operator on symmetric functions that is very similar to vertex operators. We show that the Bump–Diaconis–Tracy–Widom identity is a differentiated version of the classical Jacobi–Trudi identity.

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18

Pyber, L., and A. Shalev. "Asymptotic Results for Primitive Permutation Groups." Journal of Algebra 188, no. 1 (1997): 103–24. http://dx.doi.org/10.1006/jabr.1996.6818.

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19

Blagojevic, M., and M. Vasilic. "Asymptotic symmetry and conserved quantities in the Poincare gauge theory of gravity." Classical and Quantum Gravity 5, no. 9 (1988): 1241–57. http://dx.doi.org/10.1088/0264-9381/5/9/009.

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20

Hurth, Tobias. "Non-Abelian Gauge Symmetry in the Causal Epstein–Glaser Approach." International Journal of Modern Physics A 12, no. 24 (1997): 4461–76. http://dx.doi.org/10.1142/s0217751x97002437.

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Non-Abelian gauge symmetry in (3 + 1)-dimensional space–time is analyzed in the causal Epstein–Glaser framework. In this formalism, the technical details concerning the well-known UV and IR problem in quantum field theory are separated and reduced to well-defined problems, namely the causal splitting and the adiabatic switching of operator-valued distributions. Non-Abelian gauge invariance in perturbation theory is completely discussed in the well-defined Fock space of free asymptotic fields. The LSZ formalism is not used in this construction. The linear operator condition of asymptotic gauge invariance is sufficient for the unitarity of the S matrix in the physical subspace and the usual Slavnov–Taylor identities. We explicitly derive the most general specific coupling compatible with this condition. By analyzing only tree graphs in the second order of perturbation theory we show that the well-known Yang–Mills couplings with anticommuting ghosts are the only ones which are compatible with asymptotic gauge invariance. The required generalizations for linear gauges are given.

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21

Krupa, Martin, and Ian Melbourne. "Asymptotic stability of heteroclinic cycles in systems with symmetry. II." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 134, no. 6 (2004): 1177–97. http://dx.doi.org/10.1017/s0308210500003693.

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Systems possessing symmetries often admit robust heteroclinic cycles that persist under perturbations that respect the symmetry. In previous work, we began a systematic investigation into the asymptotic stability of such cycles. In particular, we found a sufficient condition for asymptotic stability, and we gave algebraic criteria for deciding when this condition is also necessary. These criteria are satisfied for cycles in R3.Field and Swift, and Hofbauer, considered examples in R4 for which our sufficient condition for stability is not optimal. They obtained necessary and sufficient conditions for asymptotic stability using a transition-matrix technique.In this paper, we combine our previous methods with the transition-matrix technique and obtain necessary and sufficient conditions for asymptotic stability for a larger class of heteroclinic cycles. In particular, we obtain a complete theory for ‘simple’ heteroclinic cycles in R4 (thereby proving and extending results for homoclinic cycles that were stated without proof by Chossat, Krupa, Melbourne and Scheel). A partial classification of simple heteroclinic cycles in R4 is also given. Finally, our stability results generalize naturally to higher dimensions and many of the higher-dimensional examples in the literature are covered by this theory.

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22

Thomsen, Klaus. "Discrete asymptotic homomorphisms in E-theory and KK-theory." MATHEMATICA SCANDINAVICA 92, no. 1 (2003): 103. http://dx.doi.org/10.7146/math.scand.a-14396.

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23

Lucchini, A., F. Menegazzo, and M. Morigi. "Asymptotic Results for Primitive Permutation Groups and Irreducible Linear Groups." Journal of Algebra 223, no. 1 (2000): 154–70. http://dx.doi.org/10.1006/jabr.1999.8081.

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24

MIELKE, ECKEHARD W. "SYMMETRY BREAKING IN TOPOLOGICAL QUANTUM GRAVITY." International Journal of Modern Physics D 22, no. 05 (2013): 1330009. http://dx.doi.org/10.1142/s0218271813300097.

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A SL (5, ℝ) gauge-invariant topological field theory of gravity and possible gauge unifications are considered in four-dimensions (4D). The problem of quantization is evaluated in the asymptotic safety scenario. "Minimal" BF type models for the high energy limit are physically not quite realistic, a tiny symmetry breaking is needed to recover standard Einsteinian gravity for the macroscopic metrical background with induced cosmological constant.

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25

Bonnafé, Cédric, and Raphaël Rouquier. "An asymptotic cell category for cyclic groups." Journal of Algebra 558 (September 2020): 102–28. http://dx.doi.org/10.1016/j.jalgebra.2019.12.015.

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26

Masuoka, Akira. "Formal groups and unipotent affine groups in non-categorical symmetry." Journal of Algebra 317, no. 1 (2007): 226–49. http://dx.doi.org/10.1016/j.jalgebra.2007.03.006.

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27

Kisielewicz, Andrzej. "Symmetry Groups of Boolean Functions and Constructions of Permutation Groups." Journal of Algebra 199, no. 2 (1998): 379–403. http://dx.doi.org/10.1006/jabr.1997.7198.

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28

Crumeyrolle, Albert. "Physical symmetry groups and associated bundles in field theory." Reports on Mathematical Physics 23, no. 1 (1986): 115–28. http://dx.doi.org/10.1016/0034-4877(86)90072-8.

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29

Oberlack, M. "Asymptotic Expansion, Symmetry Groups, and Invariant Solutions of Laminar and Turbulent Wall-Bounded Flows." ZAMM 80, no. 11-12 (2000): 791–800. http://dx.doi.org/10.1002/1521-4001(200011)80:11/12<791::aid-zamm791>3.0.co;2-5.

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30

HENRY, SHAWN R., and JEFFREY R. WEEKS. "SYMMETRY GROUPS OF HYPERBOLIC KNOTS AND LINKS." Journal of Knot Theory and Its Ramifications 01, no. 02 (1992): 185–201. http://dx.doi.org/10.1142/s0218216592000100.

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Canonical triangulations provide an algorithm to compute the symmetry groups of hyperbolic links and their complements, along with information on how each symmetry acts on the link components. The latter information determines the chirality and invertibility of the link. Symmetry groups of knots to ten crossings and multicomponent hyperbolic links to nine crossing are listed, with some exceptions. A Macintosh computer program, available free of charge, computes symmetry groups interactively.

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31

ARPONEN, HEIKKI. "EASY HOLOGRAPHY." International Journal of Modern Physics D 22, no. 12 (2013): 1342013. http://dx.doi.org/10.1142/s0218271813420133.

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It is argued that the role of infinite-dimensional asymptotic symmetry groups in gravity theories are essential for a holographic description of gravity and possibly to a resolution of the black hole information paradox. I present a simple toy model in two-dimensional hyperbolic/anti-de Sitter (AdS) space and describe, by very elementary considerations, how the asymptotic symmetry group is responsible for the entropy area law. Similar results apply also in three-dimensional AdS space. The failure of the approach in higher-dimensional AdS spaces is explained and resolved by considering other asymptotically noncompact homogeneous spaces.

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32

Sapir, Mark. "On groups with locally compact asymptotic cones." International Journal of Algebra and Computation 25, no. 01n02 (2015): 37–40. http://dx.doi.org/10.1142/s0218196715400020.

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We show how a recent result of Hrushovsky [Stable group theory and approximate subgroups, J. Amer. Math. Soc.25(1) (2012) 189–243] implies that if an asymptotic cone of a finitely generated group is locally compact, then the group is virtually nilpotent.

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33

Keller, Thomas Michael. "On the Asymptotic Taketa Bound for A-Groups." Journal of Algebra 191, no. 1 (1997): 127–40. http://dx.doi.org/10.1006/jabr.1996.6893.

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34

BRINK, LARS. "MAXIMAL SUPERSYMMETRY AND EXCEPTIONAL GROUPS." Modern Physics Letters A 25, no. 32 (2010): 2715–25. http://dx.doi.org/10.1142/s0217732310034262.

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The article is a tribute to my old mentor, collaborator and friend Murray Gell-Mann. In it I describe work by P. Ramond, S.-S. Kim and myself where we describe the [Formula: see text] supergravity in the light-cone formalism. We show how the Cremmer–Julia E7(7) nonlinear symmetry is implemented and how the full supermultiplet is a representation of the E7(7) symmetry. I also show how the E7(7) symmetry is a key to understand the higher order couplings in the theory and is very useful when we discuss possible counterterms for this theory.

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35

Pengitore, Mark. "Residual dimension of nilpotent groups." Journal of Group Theory 23, no. 5 (2020): 801–29. http://dx.doi.org/10.1515/jgth-2019-0117.

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AbstractThe function {\mathrm{F}_{G}(n)} gives the maximum order of a finite group needed to distinguish a nontrivial element of G from the identity with a surjective group morphism as one varies over nontrivial elements of word length at most n. In previous work [M. Pengitore, Effective separability of finitely generated nilpotent groups, New York J. Math. 24 2018, 83–145], the author claimed a characterization for {\mathrm{F}_{N}(n)} when N is a finitely generated nilpotent group. However, a counterexample to the above claim was communicated to the author, and consequently, the statement of the asymptotic characterization of {\mathrm{F}_{N}(n)} is incorrect. In this article, we introduce new tools to provide lower asymptotic bounds for {\mathrm{F}_{N}(n)} when N is a finitely generated nilpotent group. Moreover, we introduce a class of finitely generated nilpotent groups for which the upper bound of the above article can be improved. Finally, we construct a class of finitely generated nilpotent groups N for which the asymptotic behavior of {\mathrm{F}_{N}(n)} can be fully characterized.

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36

Blagojevic, M., M. Vasilic, and T. Vukasinac. "Asymptotic symmetry and conservation laws in the two-dimensional Poincaré gauge theory of gravity." Classical and Quantum Gravity 13, no. 11 (1996): 3003–19. http://dx.doi.org/10.1088/0264-9381/13/11/016.

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37

Holmes, Randall R., and Tin-Yau Tam. "Symmetry classes of tensors associated with certain groups." Linear and Multilinear Algebra 32, no. 1 (1992): 21–31. http://dx.doi.org/10.1080/03081089208818144.

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38

Gatermann, Karin, and Pablo A. Parrilo. "Symmetry groups, semidefinite programs, and sums of squares." Journal of Pure and Applied Algebra 192, no. 1-3 (2004): 95–128. http://dx.doi.org/10.1016/j.jpaa.2003.12.011.

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39

Heng Fong, Wan, Aqilahfarhana Abdul Rahman, and Nor Haniza Sarmin. "Isomorphism and matrix representation of point groups." Malaysian Journal of Fundamental and Applied Sciences 15, no. 1 (2019): 88–92. http://dx.doi.org/10.11113/mjfas.v15n2019.1087.

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In chemistry, point group is a type of group used to describe the symmetry of molecules. It is a collection of symmetry elements controlled by a form or shape which all go through one point in space, which consists of all symmetry operations that are possible for every molecule. Next, a set of number or matrices which assigns to the elements of a group and represents the multiplication of the elements is said to constitute representation of a group. Here, each individual matrix is called a representative that corresponds to the symmetry operations of point groups, and the complete set of matrices is called a matrix representation of the group. This research was aimed to relate the symmetry in point groups with group theory in mathematics using the concept of isomorphism, where elements of point groups and groups were mapped such that the isomorphism properties were fulfilled. Then, matrix representations of point groups were found based on the multiplication table where symmetry operations were represented by matrices. From this research, point groups of order less than eight were shown to be isomorphic with groups in group theory. In addition, the matrix representation corresponding to the symmetry operations of these point groups wasis presented. This research would hence connect the field of mathematics and chemistry, where the relation between groups in group theory and point groups in chemistry were shown.

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40

BELLIARD, JEAN-ROBERT. "ASYMPTOTIC COHOMOLOGY OF CIRCULAR UNITS." International Journal of Number Theory 05, no. 07 (2009): 1205–19. http://dx.doi.org/10.1142/s179304210900264x.

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Let F be a number field, abelian over ℚ, and fix a prime p ≠ 2. Consider the cyclotomic ℤp-extension F∞/F and denote Fn the nth finite subfield and Cn its group of circular units as defined by Sinnott. Then the Galois groups Gm,n = Gal (Fm/Fn) act naturally on the Cm 's (for any m ≥ n ≫ 0). We compute the Tate cohomology groups [Formula: see text] for i = -1,0 without assuming anything else neither on F nor on p.

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41

Roman’kov, V. A. "Algorithmic theory of solvable groups." Prikladnaya Diskretnaya Matematika, no. 52 (2021): 16–64. http://dx.doi.org/10.17223/20710410/52/2.

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The purpose of this survey is to give some picture of what is known about algorithmic and decision problems in the theory of solvable groups. We will provide a number of references to various results, which are presented without proof. Naturally, the choice of the material reported on reflects the author’s interests and many worthy contributions to the field will unfortunately go without mentioning. In addition to achievements in solving classical algorithmic problems, the survey presents results on other issues. Attention is paid to various aspects of modern theory related to the complexity of algorithms, their practical implementation, random choice, asymptotic properties. Results are given on various issues related to mathematical logic and model theory. In particular, a special section of the survey is devoted to elementary and universal theories of solvable groups. Special attention is paid to algorithmic questions regarding rational subsets of groups. Results on algorithmic problems related to homomorphisms, automorphisms, and endomorphisms of groups are presented in sufficient detail.

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42

CADONI, MARIANO. "STATISTICAL ENTROPY OF THE SCHWARZSCHILD BLACK HOLE." Modern Physics Letters A 21, no. 24 (2006): 1879–87. http://dx.doi.org/10.1142/s0217732306021165.

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We derive the statistical entropy of the Schwarzschild black hole by considering the asymptotic symmetry algebra near the [Formula: see text] boundary of the spacetime at past null infinity. Using a two-dimensional description and the Weyl invariance of black hole thermodynamics this symmetry algebra can be mapped into the Virasoro algebra generating asymptotic symmetries of anti-de Sitter spacetime. Using Lagrangian methods we identify the stress–energy tensor of the boundary conformal field theory and calculate the central charge of the Virasoro algebra. The Bekenstein–Hawking result for the black hole entropy is regained using Cardy's formula. Our result strongly supports a nonlocal realization of the holographic principle.

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43

GAO, YA-JUN. "NEW INFINITE-DIMENSIONAL DOUBLE SYMMETRY GROUPS FOR THE EINSTEIN–KALB–RAMOND THEORY." International Journal of Modern Physics A 23, no. 10 (2008): 1593–612. http://dx.doi.org/10.1142/s0217751x08039694.

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The symmetry structures of the dimensionally reduced Einstein–Kalb–Ramond (EKR) theory is further studied. By using a so-called extended double (ED)-complex method, the usual Riemann–Hilbert (RH) problem is extended to an ED-complex formulation. A pair of ED RH transformations are constructed and they are verified to give infinite-dimensional double symmetry groups of the EKR theory, each of these symmetry groups has the structure of semidirect product of Kac–Moody group [Formula: see text] and Virasoro group. Moreover, the infinitesimal forms of these RH transformations are calculated out and they are found to give exactly the same results as previous, these demonstrate that the pair of ED RH transformations in this paper provide exponentiations of all the infinitesimal symmetries in our previous paper. The finite forms of symmetry transformations given in the present paper are more important and useful for theoretic studies and new solution generation, etc.

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44

Berget, Andrew. "Critical groups of graphs with reflective symmetry." Journal of Algebraic Combinatorics 39, no. 1 (2013): 209–24. http://dx.doi.org/10.1007/s10801-013-0445-x.

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Balasubramanian, R., and Gyan Prakash. "Asymptotic formula for sum-free sets in abelian groups." Acta Arithmetica 127, no. 2 (2007): 115–24. http://dx.doi.org/10.4064/aa127-2-2.

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Benbenisty, Carmit, and Aner Shalev. "Asymptotic behavior of finite permutation groups acting on subsets." Journal of Algebra 287, no. 2 (2005): 310–18. http://dx.doi.org/10.1016/j.jalgebra.2005.02.008.

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47

Carbone, A. "Asymptotic Cyclic Expansion and Bridge Groups of Formal Proofs." Journal of Algebra 242, no. 1 (2001): 109–45. http://dx.doi.org/10.1006/jabr.2000.8700.

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48

KANG, KYUNGSIK, and ALAN R. WHITE. "A UNIQUE SU(5) AND SO(10) UNIFICATION WITH COMPLETE DYNAMICAL SYMMETRY BREAKING." International Journal of Modern Physics A 02, no. 02 (1987): 409–42. http://dx.doi.org/10.1142/s0217751x87000168.

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A unique asymptotically free, anomaly-free, SU(5) gauge theory is proposed as a possible complete unification of the standard model in which all symmetry-breaking is dynamical. The asymptotic freedom constraint is saturated, removing renormalon divergences and leaving well-defined instanton interactions as the only nonperturbative ingredient of the theory. Consequently, it is argued, topological vacuum polarization of a very heavy, unconventional quantum number, quark sector dominates the dynamics, producing SU(5) symmetry breaking and a three generation low energy spectrum. Electroweak symmetry breaking is due to a chiral condensate of color sextet quarks. The embedding of the theory in a single SO(10) representation is used for the dynamical analysis and may also have physical significance.

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Mitsotakis, Dimitrios, Denys Dutykh, and Qian Li. "Asymptotic nonlinear and dispersive pulsatile flow in elastic vessels with cylindrical symmetry." Computers & Mathematics with Applications 75, no. 11 (2018): 4022–47. http://dx.doi.org/10.1016/j.camwa.2018.03.011.

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Degenhardt, Sheldon L., and Stephen C. Milne. "Weighted-Inversion Statistics and Their Symmetry Groups." Journal of Combinatorial Theory, Series A 90, no. 1 (2000): 49–103. http://dx.doi.org/10.1006/jcta.1999.3019.

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