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Journal articles on the topic 'Real conjugacy classes'

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Author: Grafiati

Published: 10 March 2023

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1

Chan, Kai-Cheong, and Dragomir Ž. Đoković. "Conjugacy Classes of Subalgebras of the Real Sedenions." Canadian Mathematical Bulletin 49, no. 4 (December 1, 2006): 492–507. http://dx.doi.org/10.4153/cmb-2006-048-6.

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AbstractBy applying the Cayley–Dickson process to the division algebra of real octonions, one obtains a 16-dimensional real algebra known as (real) sedenions. We denote this algebra by A4. It is a flexible quadratic algebra (with unit element 1) but not a division algebra.We classify the subalgebras of A4 up to conjugacy (i.e., up to the action of the automorphismgroup G of A4) with one exception: we leave aside the more complicated case of classifying the quaternion subalgebras. Any nonzero subalgebra contains 1 and we show that there are no proper subalgebras of dimension 5, 7 or > 8. The proper non-division subalgebras have dimensions 3, 6 and 8. We show that in each of these dimensions there is exactly one conjugacy class of such subalgebras. There are infinitely many conjugacy classes of subalgebras in dimensions 2 and 4, but only 4 conjugacy classes in dimension 8.

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2

Beltrán, A., M. J. Felipe, and C. Melchor. "Squares of real conjugacy classes in finite groups." Annali di Matematica Pura ed Applicata (1923 -) 197, no. 2 (July 18, 2017): 317–28. http://dx.doi.org/10.1007/s10231-017-0681-0.

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3

Doković, Dragomir Ž., and Nguyêñ Quôć Thăńg. "Conjugacy Classes of Maximal Tori in Simple Real Algebraic Groups and Applications." Canadian Journal of Mathematics 46, no. 4 (August 1, 1994): 699–717. http://dx.doi.org/10.4153/cjm-1994-039-5.

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AbstractLet G be an almost simple complex algebraic group defined over R, and let G(R) be the group of real points of G. We enumerate the G(R)-conjugacy classes of maximal R-tori of G. Each of these conjugacy classes is also a single G(R)˚-conjugacy class, where G(R)˚ is the identity component of G(R), viewed as a real Lie group. As a consequence we also obtain a new and short proof of the Kostant-Sugiura's theorem on conjugacy classes of Cartan subalgebras in simple real Lie algebras.A connected real Lie group P is said to be weakly exponential (w.e.) if the image of its exponential map is dense in P. This concept was introduced in [HM] where also the question of identifying all w.e. almost simple real Lie groups was raised. By using a theorem of A. Borel and our classification of maximal R-tori we answer the above question when P is of the form G(R)˚.

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4

AMPARO, ELENA, and C. RYAN VINROOT. "ON THE NUMBER OF REAL CLASSES IN THE FINITE PROJECTIVE LINEAR AND UNITARY GROUPS." Glasgow Mathematical Journal 62, no. 1 (January 31, 2019): 93–107. http://dx.doi.org/10.1017/s0017089518000551.

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AbstractWe show that for any n and q, the number of real conjugacy classes in $ \rm{PGL}(\it{n},\mathbb{F}_q) $ is equal to the number of real conjugacy classes of $ \rm{GL}(\it{n},\mathbb{F}_q) $ which are contained in $ \rm{SL}(\it{n},\mathbb{F}_q) $, refining a result of Lehrer [J. Algebra36(2) (1975), 278–286] and extending the result of Gill and Singh [J. Group Theory14(3) (2011), 461–489] that this holds when n is odd or q is even. Further, we show that this quantity is equal to the number of real conjugacy classes in $ \rm{PGU}(\it{n},\mathbb{F}_q) $, and equal to the number of real conjugacy classes of $ \rm{U}(\it{n},\mathbb{F}_q) $ which are contained in $ \rm{SU}(\it{n},\mathbb{F}_q) $, refining results of Gow [Linear Algebra Appl.41 (1981), 175–181] and Macdonald [Bull. Austral. Math. Soc.23(1) (1981), 23–48]. We also give a generating function for this common quantity.

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5

Dolfi, Silvio, Emanuele Pacifici, and Lucia Sanus. "Finite groups with real conjugacy classes of prime size." Israel Journal of Mathematics 175, no. 1 (January 2010): 179–89. http://dx.doi.org/10.1007/s11856-010-0008-y.

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6

Robati, S. M. "Real-Imaginary Conjugacy Classes and Real-Imaginary Irreducible Characters in Finite Groups." Mathematical Notes 103, no. 1-2 (January 1, 2017): 251–58. http://dx.doi.org/10.1134/s0001434618010261.

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7

Sangroniz, Josu, and Joan Tent. "2-Groups with a fixed number of real conjugacy classes." Journal of Algebra 392 (October 2013): 42–51. http://dx.doi.org/10.1016/j.jalgebra.2013.05.031.

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8

Gongopadhyay, Krishnendu. "Algebraic Characterization of Isometries of the Hyperbolic 4-Space." ISRN Geometry 2012 (February 29, 2012): 1–12. http://dx.doi.org/10.5402/2012/757489.

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We classify isometries of the real hyperbolic 4-space by their conjugacy classes of centralizers. We use the representation of the isometries by quaternionic matrices to obtain this characterization. Another characterization in terms of conjugacy invariants is also given.

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9

Beltrán, A., and S. M. Robati. "Erratum to: Real-Imaginary Conjugacy Classes and Real-Imaginary Irreducible Characters in Finite Groups." Mathematical Notes 110, no. 3-4 (September 2021): 638–39. http://dx.doi.org/10.1134/s0001434621090352.

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10

Beltran, A., and Sajjad Mahmood Robati. "Erratum to: Real-Imaginary Conjugacy Classes and Real-Imaginary Irreducible Characters in Finite Groups." Matematicheskie Zametki 110, no. 4 (2021): 638–39. http://dx.doi.org/10.4213/mzm12922.

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11

Filom, Khashayar. "Real entropy rigidity under quasi-conformal deformations." Conformal Geometry and Dynamics of the American Mathematical Society 25, no. 1 (March 9, 2021): 1–33. http://dx.doi.org/10.1090/ecgd/356.

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We set up a real entropy function h R h_\Bbb {R} on the space M d ′ \mathcal {M}’_d of Möbius conjugacy classes of real rational maps of degree d d by assigning to each class the real entropy of a representative f ∈ R ( z ) f\in \Bbb {R}(z) ; namely, the topological entropy of its restriction f ↾ R ^ f\restriction _{\hat {\Bbb {R}}} to the real circle. We prove a rigidity result stating that h R h_\Bbb {R} is locally constant on the subspace determined by real maps quasi-conformally conjugate to f f . As examples of this result, we analyze real analytic stable families of hyperbolic and flexible Lattès maps with real coefficients along with numerous families of degree d d real maps of real entropy log ⁡ ( d ) \log (d) . The latter discussion moreover entails a complete classification of maps of maximal real entropy.

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12

Rakhimov, A. A., and Sh M. Usmanov. "Outer Conjugacy Classes of Automorphisms and Antiautomorphisms of Real and Complex Injective Factors." Journal of Functional Analysis 144, no. 2 (March 1997): 475–85. http://dx.doi.org/10.1006/jfan.1996.2993.

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13

Finashin, S., and V. Kharlamov. "On the deformation chirality of real cubic fourfolds." Compositio Mathematica 145, no. 5 (September 2009): 1277–304. http://dx.doi.org/10.1112/s0010437x09004126.

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AbstractAccording to our previous results, the conjugacy class of the involution induced by the complex conjugation in the homology of a real non-singular cubic fourfold determines the fourfold up to projective equivalence and deformation. Here, we show how to eliminate the projective equivalence and obtain a pure deformation classification, that is, how to respond to the chirality problem: which cubics are not deformation equivalent to their image under a mirror reflection. We provide an arithmetical criterion of chirality, in terms of the eigen-sublattices of the complex conjugation involution in homology, and show how this criterion can be effectively applied taking as examples M-cubics (that is, those for which the real locus has the richest topology) and (M−1)-cubics (the next case with respect to complexity of the real locus). It happens that there is one chiral class of M-cubics and three chiral classes of (M−1)-cubics, in contrast to two achiral classes of M-cubics and three achiral classes of (M−1)-cubics.

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14

Doković, D. Ž., and Nguyên Q. Thăńg. "Correction To: Conjugacy Classes of Maximal Tori in Simple Real Algebraic Groups and Applications." Canadian Journal of Mathematics 46, no. 06 (December 1994): 1208–10. http://dx.doi.org/10.4153/cjm-1994-068-7.

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15

LÓPEZ PEÑA, J., S. MAJID, and K. RIETSCH. "Lie theory of finite simple groups and the Roth property." Mathematical Proceedings of the Cambridge Philosophical Society 163, no. 2 (January 9, 2017): 301–40. http://dx.doi.org/10.1017/s030500411600102x.

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AbstractIn noncommutative geometry a ‘Lie algebra’ or bidirectional bicovariant differential calculus on a finite group is provided by a choice of an ad-stable generating subset $\mathcal{C}$ stable under inversion. We study the associated Killing form K. For the universal calculus associated to $\mathcal{C}$ = G \ {e} we show that the magnitude $\mu=\sum_{a,b\in\mathcal{C}}(K^{-1})_{a,b}$ of the Killing form is defined for all finite groups (even when K is not invertible) and that a finite group is Roth, meaning its conjugation representation contains every irreducible, iff μ ≠ 1/(N − 1) where N is the number of conjugacy classes. We show further that the Killing form is invertible in the Roth case, and that the Killing form restricted to the (N − 1)-dimensional subspace of invariant vectors is invertible iff the finite group is an almost-Roth group (meaning its conjugation representation has at most one missing irreducible). It is known [9, 10] that most nonabelian finite simple groups are Roth and that all are almost Roth. At the other extreme from the universal calculus we prove that the 2-cycles conjugacy class in any Sn has invertible Killing form, and the same for the generating conjugacy classes in the case of the dihedral groups D2n with n odd. We verify invertibility of the Killing forms of all real conjugacy classes in all nonabelian finite simple groups to order 75,000, by computer, and we conjecture this to extend to all nonabelian finite simple groups.

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16

Clarke, S. P., and J. F. Cornwell. "Real forms and conjugacy classes of involutive automorphisms of theDell(1)affine Kac-Moody algebras." Journal of Physics A: Mathematical and General 32, no. 25 (January 1, 1999): 4655–70. http://dx.doi.org/10.1088/0305-4470/32/25/308.

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17

Ballas, Samuel, Daryl Cooper, and Arielle Leitner. "The moduli space of marked generalized cusps in real projective manifolds." Conformal Geometry and Dynamics of the American Mathematical Society 26, no. 7 (August 17, 2022): 111–64. http://dx.doi.org/10.1090/ecgd/367.

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ln this paper, a generalized cusp is a properly convex manifold with strictly convex boundary that is diffeomorphic to M × [ 0 , ∞ ) M\times [0,\infty ) where M M is a closed Euclidean manifold. These are classified by Ballas, Cooper, and Leitner [J. Topol. 13 (2020), pp. 1455-1496]. The marked moduli space is homeomorphic to a subspace of the space of conjugacy classes of representations of π 1 M \pi _1M . It has one description as a generalization of a trace-variety, and another description involving weight data that is similar to that used to describe semi-simple Lie groups. It is also a bundle over the space of Euclidean similarity (conformally flat) structures on M M , and the fiber is a closed cone in the space of cubic differentials. For 3 3 -dimensional orientable generalized cusps, the fiber is homeomorphic to a cone on a solid torus.

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18

Bujalance, E., F. J. Cirre, and J. M. Gamboa. "Automorphism groups of the real projective plane with holes and their conjugacy classes within its mapping class group." Mathematische Annalen 332, no. 2 (February 18, 2005): 253–75. http://dx.doi.org/10.1007/s00208-004-0621-3.

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19

Pawałowski, Krzysztof, and Toshio Sumi. "The Laitinen Conjecture for finite non-solvable groups." Proceedings of the Edinburgh Mathematical Society 56, no. 1 (December 5, 2012): 303–36. http://dx.doi.org/10.1017/s0013091512000223.

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AbstractFor any finite group G, we impose an algebraic condition, the Gnil-coset condition, and prove that any finite Oliver group G satisfying the Gnil-coset condition has a smooth action on some sphere with isolated fixed points at which the tangent G-modules are not isomorphic to each other. Moreover, we prove that, for any finite non-solvable group G not isomorphic to Aut(A6) or PΣL(2, 27), the Gnil-coset condition holds if and only if rG ≥ 2, where rG is the number of real conjugacy classes of elements of G not of prime power order. As a conclusion, the Laitinen Conjecture holds for any finite non-solvable group not isomorphic to Aut(A6).

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20

de Melo, W., and S. J. van Strien. "Diffeomorphisms on surfaces with a finite number of moduli." Ergodic Theory and Dynamical Systems 7, no. 3 (September 1987): 415–62. http://dx.doi.org/10.1017/s0143385700004120.

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AbstractThis paper generalises the well known structural stability theorem which says that a diffeomorphism is structurally stable if it is Axiom A and if all invariant manifolds are transversal to each other.If these transversality conditions are not satisfied then the diffeomorphism not only fails to be stable, but also this gives rise to the appearance of moduli. That is, one needs several real parameters to parameterise all conjugacy classes of nearby diffeomorphisms. (The minimum number of parameters needed is called the number of moduli).Here we deal with diffeomorphisms on two dimensional manifolds, whose asymptotic dynamics are well understood (the class of Axiom A diffeomorphisms). The main result characterises those Axiom A diffeomorphisms which have a finite number of moduli. This result can be regarded as a generalisation of the structural stability theorem. From the proofs it follows that the dynamics of these diffeomorphisms can also be well understood.In the proof of our main theorem we need certain invariant foliations to be quite smooth. In an appendix we prove a differentiable version of the Lambda Lemma.

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21

Gongopadhyay, Krishnendu, Sudip Mazumder, and Sujit Kumar Sardar. "Conjugate real classes in general linear groups." Journal of Algebra and Its Applications 18, no. 03 (March 2019): 1950054. http://dx.doi.org/10.1142/s0219498819500543.

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Let [Formula: see text] be a field with a nontrivial involution [Formula: see text]. An element [Formula: see text] is called [Formula: see text]-real if it is conjugate to [Formula: see text]. We prove that for [Formula: see text], [Formula: see text] is [Formula: see text]-real if and only if it has a representation in some unitary group of degree [Formula: see text] over [Formula: see text].

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22

COUTINHO, Sabrine Mantuan dos Santos, Zeidi Araújo TRINDADE, Maria Cristina Smith MENANDRO, and Paulo Rogério Meira MENANDRO. "Sonoridades da vida conjugal registradas em versos de canções brasileiras produzidas entre 1940 e 1960." Estudos de Psicologia (Campinas) 32, no. 3 (September 2015): 461–73. http://dx.doi.org/10.1590/0103-166x2015000300011.

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O objetivo do presente estudo foi identificar elementos de representações sociais de casamento, marido e esposa, presentes em letras de canções populares gravadas nos anos de 1940 e 1950. Foram selecionadas 306 letras de canções, a partir de consulta a acervos particulares. Na análise dos dados foi utilizado o softwareAlceste, que apontou a existência de quatro classes, que diferentes formas de discurso acerca das relações conjugais e dos papéis de gênero no casamento, indicando elementos de representações sociais. Um primeiro grupo, constituído pelas classes 1, 3 e 4, pode ser considerado um eixo na organização das representações sociais de casamento, marido e esposa. Esse eixo, aqui denominado Papel feminino e masculino no casamento: real versus ideal, caracteriza-se por apresentar um discurso predominantemente masculino quanto aos aspectos do cotidiano conjugal, com indicações da esposa ideal e da esposa inadequada, retratando o marido como trabalhador/provedor financeiro. Já o outro eixo temático, constituído pela classe 2, tem como tema a desilusão amorosa e a separação. Os resultados exemplificam o papel dos fatores sociais, históricos e culturais na construção das configurações reais com que as relações conjugais são vividas. Apontam, também, como tais relações são orientadas e justificadas por representações sociais flexíveis, associadas aos contextos nos quais foram produzidas, consolidadas e/ou transformadas.

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23

HOLT, DEREK F., SARAH REES, and CLAAS E. RÖVER. "GROUPS WITH CONTEXT-FREE CONJUGACY PROBLEMS." International Journal of Algebra and Computation 21, no. 01n02 (February 2011): 193–216. http://dx.doi.org/10.1142/s0218196711006133.

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The conjugacy problem and the inverse conjugacy problem of a finitely generated group are defined, from a language theoretic point of view, as sets of pairs of words. An automaton might be obliged to read the two input words synchronously, or could have the option to read asynchronously. Hence each class of languages gives rise to four classes of groups; groups whose (inverse) conjugacy problem is an (a)synchronous language in the given class. For regular languages all these classes are identical with the class of finite groups. We show that the finitely generated groups with asynchronously context-free inverse conjugacy problem are precisely the virtually free groups. Moreover, the other three classes arising from context-free languages are shown all to coincide with the class of virtually cyclic groups, which is precisely the class of groups with synchronously one-counter (inverse) conjugacy problem. It is also proved that, for a δ-hyperbolic group and any λ ≥ 1, ϵ ≥ 0, the intersection of the inverse conjugacy problem with the set of pairs of (λ, ϵ)-quasigeodesics is context-free. Finally we show that the conjugacy problem of a virtually free group is an asynchronously indexed language.

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24

Ballico, E. "Discreteness For the Set of Complex Structures On a Real Variety." Canadian Mathematical Bulletin 46, no. 3 (September 1, 2003): 321–22. http://dx.doi.org/10.4153/cmb-2003-033-7.

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AbstractLet X, Y be reduced and irreducible compact complex spaces and S the set of all isomorphism classes of reduced and irreducible compact complex spaces W such that X × Y ≅ X × W. Here we prove that S is at most countable. We apply this result to show that for every reduced and irreducible compact complex space X the set S(X) of all complex reduced compact complex spaces W with X × Xσ ≅ W × Wσ (where Aσ denotes the complex conjugate of any variety A) is at most countable.

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25

Borsos, Benjámin. "AN INEXACT NEWTON METHOD WITH INNER PRECONDITIONED CG FOR NON-UNIFORMLY MONOTONE ELLIPTIC PROBLEMS." Mathematical Modelling and Analysis 26, no. 3 (July 13, 2021): 383–94. http://dx.doi.org/10.3846/mma.2021.12899.

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The present paper introduces an inexact Newton method, coupled with a preconditioned conjugate gradient method in inner iterations, for elliptic operators with non-uniformly monotone upper and lower bounds. Convergence is proved in Banach space level. The results cover real-life classes of elliptic problems. Numerical experiments reinforce the convergence results.

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26

Bakhyt, A., and N. T. Tleukhanova. "The problem of trigonometric Fourier series multipliers of classes in λp,q spaces." BULLETIN OF THE KARAGANDA UNIVERSITY-MATHEMATICS 100, no. 4 (December 30, 2020): 17–25. http://dx.doi.org/10.31489/2020m4/17-25.

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In this article, we consider weighted spaces of numerical sequences λp,q, which are defined as sets of sequences a = {ak}^∞_k=1, for which the norm ||a||λp,q :=\sum^∞_k=1|ak|^q k^(q/p −1)^1/q<∞ is finite. In the case of non-increasing sequences, the norm of the space λp,q coincides with the norm of the classical Lorentz space lp,q. Necessary and sufficient conditions are obtained for embeddings of the space λp,q into the space λp1,q1. The interpolation properties of these spaces with respect to the real interpolation method are studied. It is shown that the scale of spaces λp,q is closed in the relative real interpolation method, as well as in relative to the complex interpolation method. A description of the dual space to the weighted space λp,q is obtained. Specifically, it is shown that the space is reflective, where p', q' are conjugate to the parameters p and q. The paper also studies the properties of the convolution operator in these spaces. The main result of this work is an O’Neil type inequality. The resulting inequality generalizes the classical Young-O’Neil inequality. The research methods are based on the interpolation theorems proved in this paper for the spaces λp,q.

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27

Liu, Zhongyun, Rui Ralha, Yulin Zhang, and Carla Ferreira. "Minimization problems for certain structured matrices." Electronic Journal of Linear Algebra 30 (February 8, 2015): 613–31. http://dx.doi.org/10.13001/1081-3810.3144.

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For given Z, B â C^{n\times k}, the problem of finding A â C^{n\times n}, in some prescribed class W, that minimizes ||AZ â B|| (Frobenius norm) has been considered by different authors for distinct classes W. Here, this minimization problem is studied for two other classes, which include the symmetric Hamiltonian, symmetric skew-Hamiltonian, real orthogonal symplectic and unitary conjugate symplectic matrices. The problem of minimizing ||A â AË||, where AË is given and A is a solution of the previous problem, is also considered (as has been done by others, for different classes W). The key idea of this contribution is the reduction of each one of the above minimization problems to two independent subproblems in orthogonal subspaces of C^{n\times n}. This is possible due to the special structures under consideration. Matlab codes are developed, and numerical results of some tests are presented.

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28

ZARIFZODA, Sarvar K., and Raim N. ODINAEV. "INVESTIGATION OF SOME CLASSES OF SECOND ORDER PARTIAL INTEGRO-DIFFERENTIAL EQUATIONS WITH A POWERLOGARITHMIC SINGULARITY IN THE KERNEL." Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika, no. 67 (2020): 40–54. http://dx.doi.org/10.17223/19988621/67/4.

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For a class of second-order partial integro-differential equations with a power singularity and logarithmic singularity in the kernel, integral representations of the solution manifold in terms of arbitrary constants are obtained in the class of functions vanishing with a certain asymptotic behavior. Although the kernel of the given equation is not a Fredholm type kernel, the solution of the studied equation in a class of vanishing functions is found in an explicit form. We represent a second-order integro-differential equation as a product of two first-order integro-differential operators. For these one-dimensional integro-differential operators, in the cases when the roots of the corresponding characteristic equations are real and different, real and equal and complex and conjugate, the inverse operators are found. It is found that the presence of power singularity and logarithmic singularity in the kernel affects the number of arbitrary constants in the general solution. This number, depending on the roots of the corresponding characteristic equations, can reach nine. Also, the cases when the given integro-differential equation has a unique solution are found. The correctness of the obtained results with the help of the detailed solutions of concrete examples are shown. The method of solving the given problem can be used for solving model and nonmodel integro-differential equations with a higher order power singularity and logarithmic singularity in the kernel.

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29

PECKHAM, BRUCE B., and JAMES MONTALDI. "REAL CONTINUATION FROM THE COMPLEX QUADRATIC FAMILY: FIXED-POINT BIFURCATION SETS." International Journal of Bifurcation and Chaos 10, no. 02 (February 2000): 391–414. http://dx.doi.org/10.1142/s0218127400000256.

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This paper is primarily a numerical study of the fixed-point bifurcation loci — saddle-node, period-doubling and Hopf bifurcations — present in the family: [Formula: see text] where z is a complex dynamic (phase) variable, [Formula: see text] its complex conjugate, and C and A are complex parameters. We treat the parameter C as a primary parameter and A as a secondary parameter, asking how the bifurcation loci projected to the C plane change as the auxiliary parameter A is varied. For A=0, the resulting two-real-parameter family is a familiar one-complex-parameter quadratic family, and the local fixed-point bifurcation locus is the main cardioid of the Mandlebrot set. For A ≠ 0, the resulting two-real-parameter families are not complex analytic, but are still analytic (quadratic) when viewed as a map of ℛ2. Saddle-node and period-doubling loci evolve from points on the main cardioid for A=0 into closed curves for A ≠ 0. As A is varied further from 0 in the complex plane, the three sets interact in a variety of interesting ways. More generally, we discuss bifurcations of families of maps with some parameters designated as primary and the rest as auxiliary. The auxiliary parameter space is then divided into equivalence classes with respect to a specified set of bifurcation loci. This equivalence is defined by the existence of a diffeomorphism of corresponding primary parameter spaces which preserves the specified set of specified bifurcation loci. In our study there is a huge amount of complexity added by specifying the three fixed-point bifurcation loci together, rather than one at a time. We also provide a preliminary classification of the types of codimension-one bifurcations one should expect in general studies of families of two-parameter families of maps of the plane. Comments on numerical continuation techniques are provided as well.

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30

Huang, Wentao, Chengcheng Cao, and Dongping He. "Quasi-Periodic Motion and Hopf Bifurcation of a Two-Dimensional Aeroelastic Airfoil System in Supersonic Flow." International Journal of Bifurcation and Chaos 31, no. 02 (February 2021): 2150018. http://dx.doi.org/10.1142/s0218127421500188.

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In this article, the complex dynamic behavior of a nonlinear aeroelastic airfoil model with cubic nonlinear pitching stiffness is investigated by applying a theoretical method and numerical simulation method. First, through calculating the Jacobian of the nonlinear system at equilibrium, we obtain necessary and sufficient conditions when this system has two classes of degenerated equilibria. They are described as: (1) one pair of purely imaginary roots and one pair of conjugate complex roots with negative real parts; (2) two pairs of purely imaginary roots under nonresonant conditions. Then, with the aid of center manifold and normal form theories, we not only derive the stability conditions of the initial and nonzero equilibria, but also get the explicit expressions of the critical bifurcation lines resulting in static bifurcation and Hopf bifurcation. Specifically, quasi-periodic motions on 2D and 3D tori are found in the neighborhoods of the initial and nonzero equilibria under certain parameter conditions. Finally, the numerical simulations performed by the fourth-order Runge–Kutta method provide a good agreement with the results of theoretical analysis.

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31

Gorodetskiy, V., R. Kolisnyk, and O. Martynyuk. "ON A NONLOCAL PROBLEM FOR PARTIAL DIFFERENTIAL EQUATIONS OF PARABOLIC TYPE." Bukovinian Mathematical Journal 8, no. 2 (2020): 24–39. http://dx.doi.org/10.31861/bmj2020.02.03.

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Spaces of $S$ type, introduced by I.Gelfand and G.Shilov, as well as spaces of type $S'$, topologically conjugate with them, are natural sets of the initial data of the Cauchy problem for broad classes of equations with partial derivatives of finite and infinite orders, in which the solutions are integer functions over spatial variables. Functions from spaces of $S$ type on the real axis together with all their derivatives at $|x|\to \infty$ decrease faster than $\exp\{-a|x|^{1/\alpha}\}$, $\alpha > 0$, $a > 0$, $x\in \mathbb{R}$. The paper investigates a nonlocal multipoint by time problem for equations with partial derivatives of parabolic type in the case when the initial condition is given in a certain space of generalized functions of the ultradistribution type ($S'$ type). Moreover, results close to the Cauchy problem known in theory for such equations with an initial condition in the corresponding spaces of generalized functions of $S'$ type were obtained. The properties of the fundamental solution of a nonlocal multipoint by time problem are investigated, the correct solvability of the problem is proved, the image of the solution in the form of a convolution of the fundamental solution with the initial generalized function, which is an element of the space of generalized functions of $S'$ type.

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Ferreira, Johannes T., Tanita Botha, and Andriette Bekker. "Tsallis and Other Generalised Entropy Forms Subject to Dirichlet Mixture Priors." Symmetry 14, no. 6 (May 28, 2022): 1110. http://dx.doi.org/10.3390/sym14061110.

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Entropy indicates a measure of information contained in a complex system, and its estimation continues to receive ongoing focus in the case of multivariate data, particularly that on the unit simplex. Oftentimes the Dirichlet distribution is employed as choice of prior in a Bayesian framework conjugate to the popular multinomial likelihood with K distinct classes, where consideration of Shannon- and Tsallis entropy is of interest for insight detection within the data on the simplex. However, this prior choice only accounts for negatively correlated data, therefore this paper incorporates previously unconsidered mixtures of Dirichlet distributions as potential priors for the multinomial likelihood which addresses the drawback of negative correlation. The power sum functional, as the product moment of the mixture of Dirichlet distributions, is of direct interest in the multivariate case to conveniently access the Tsallis- and other generalized entropies that is incorporated within an estimation perspective of the posterior distribution using real economic data. A prior selection method is implemented to suggest a suitable prior for the consideration of the practitioner; empowering the user in future for consideration of suitable priors incorporating entropy within the estimation environment as well as having the option of certain mixture of Dirichlet distributions that may require positive correlation.

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Ali Hakami, Nada Ali, and Hanan Ahmed Hosni Hosni Mahmoud. "Deep Learning Classification of Crystal Structures Utilizing Wyckoff Positions." Crystals 12, no. 10 (October 16, 2022): 1460. http://dx.doi.org/10.3390/cryst12101460.

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In materials science, crystal lattice structures are the primary metrics used to measure the structure–property paradigm of a crystal structure. Crystal compounds are understood by the number of various atomic chemical settings, which are associated with Wyckoff sites. In crystallography, a Wyckoff site is a point of conjugate symmetry. Therefore, features associated with the various atomic settings in a crystal can be fed into the input layers of deep learning models. Methods to analyze crystals using Wyckoff sites can help to predict crystal structures. Hence, the main contribution of our article is the classification of crystal classes using Wyckoff sites. The presented model classifies crystals using diffraction images and a deep learning method. The model extracts feature groups including crystal Wyckoff features and crystal geometry. In this article, we present a deep learning model to predict the stage of the crystal structure–property. The lattice parameters and the structure–property commotion values are used as inputs into the deep learning model for training. The structure–property value of a crystal with a lattice width value of one-half millimeter on average is used for learning. The model attains a considerable increase in speed and precision for the real structure–property prediction. The experimental results prove that our proposed model has a fast learning curve, and can have a key role in predicting the structure–property of compound structures.

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Boytsov, Natalie, Karen Stockl, Chi-Chang Chen, Feng Wang, Xin Wang, Yao Cao, Allison Doherty, and Prani Paka. "Real-World Assessment of Prior Treatment Patterns in Patients Receiving Belantamab Mafodotin Using a Longitudinal Pharmacy and Medical Open Source Claims Database." Blood 138, Supplement 1 (November 5, 2021): 4973. http://dx.doi.org/10.1182/blood-2021-152242.

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Abstract Introduction: Belantamab mafodotin (belamaf) is a first-in-class B-cell maturation antigen-targeting antibody-drug conjugate, with a manageable adverse event profile, that has shown durable efficacy in a broad range of patients with relapsed or refractory multiple myeloma (RRMM) (Lonial et al., Lancet Oncol. 2020). As belamaf is an approved therapy, it is important to understand how it is being integrated into the management of patients with RRMM. The aim of this analysis was to examine baseline demographic and treatment patterns of these patients prior to belamaf in a real-world clinical practice. Methods: This descriptive, retrospective cohort study used IQVIA's longitudinal pharmacy and medical open source claims database, a nationally representative real-world database. The study covered the period from January 2001 to April 2021. Patient inclusion into the study required belamaf treatment initiation and ≥18 years of age at initiation. The first claim date for belamaf during the index period (August 2020 to April 2021) was the index date (ID). Key measures included patient demographics and prior MM treatments. Patient demographics were identified at the ID. Prior MM treatments were assessed during the prior treatment period, which extended from the database start date (January 2001) until the ID. The definition of a line of therapy (LOT) used in this analysis was consistent with published guidance (Rajkumar SV, et al. Blood, 2015;126:921-922). Treatments of interest were identified within the first 28 days of the first observed claim for MM treatment. A new LOT was identified if a new treatment was added after the initial 28 days after the start of the LOT, ≥1 treatment was discontinued (defined by a coverage gap >60 days), or the same LOT was restarted at a later date if ≥1 regimen was administered in between. Different therapeutic classes (e.g. PIs, immunomodulatory drugs, anti-CD38 antibodies) could be used in each LOT either singly or in combinations. Addition/deletion of corticosteriods did not constitute a new LOT. Results: 304 patients with a claim for belamaf were identified in the database and included in the analysis; median age was 70 years and 50.3% were female. Median (range) time from the earliest observed claim for MM treatment during the prior treatment period to the ID was 1,812 (104─6,676) days. Median (range) time from the earliest observed diagnosis for MM during the prior treatment period to the ID was 2,123 (22-7,186) days (~5.8 years). The median number of MM therapeutic classes, overall, received during the prior treatment period was 5; 82.9% of patients with MM received ≥4 classes of MM therapy. Prior therapy for MM was observed for 303 patients; the therapeutic classes of those MM treatments are shown in the Table. The median number of observed LOTs received by patients in the prior treatment period was 5 and 73.0% had ≥4 LOTs. LOT for belamaf may be over/under estimated as reporting on some of the MM medications was embargoed by manufacturers and/or specialty pharmacies for certain periods of time, leading to some incompleteness of data. Embargoed products included lenalidomide, pomalidomide, selinexor, thalidomide, ixazomib, and bortezomib. Although data capture for these MM treatments may not be complete in open source claims, some of the claims data for these medications were included in this analysis, i.e., medical claims for infused medications and pharmacy claims from switch suppliers for oral medications. Conclusions: The real-world evidence analyzed in this study is in line with the baseline characteristics reported for the DREAMM-1 and DREAMM-2 trials (Lonial et al., Lancet Oncol. 2020, Trudel et al., Blood Cancer J., 2019). Patients receiving belamaf have triple-class refractory myeloma with multiple LOTs generally aligned with the current indication. The most frequent prior treatments included proteasome inhibitors, anti-CD38 monoclonal antibodies, immunomodulatory drugs, and corticosteroids. Corticosteroid use is high as it is required to be prescribed in combination with other drugs according to their indications. This study identified patients in the US receiving belamaf and provides early insights into the patient demographic and clinical characteristics as well as number and range of treatments prior to treatment with belamaf. Funding: GSK (Study 214283). Figure 1 Figure 1. Disclosures Boytsov: GlaxoSmithKline: Current Employment, Current equity holder in publicly-traded company. Stockl: IQVIA: Current Employment; GlaxoSmithKline: Research Funding. Chen: GlaxoSmithKline: Research Funding; IQVIA: Current Employment. Wang: GlaxoSmithKline: Current Employment, Current equity holder in publicly-traded company. Wang: IQVIA: Current Employment; GlaxoSmithKline: Research Funding. Cao: IQVIA: Current Employment. Doherty: GlaxoSmithKline: Current Employment, Current equity holder in publicly-traded company. Paka: GlaxoSmithKline: Current Employment, Current equity holder in publicly-traded company.

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Fialho Sampaio, João Márcio, and Paulo Rogério Barbosa do Nascimento. "Possibilidades didáticas nas aulas de educação física: o conteúdo “exercício físico e saúde” no ensino médio." Caderno de Educação Física e Esporte 16, no. 2 (September 25, 2018): 113–18. http://dx.doi.org/10.36453/2318-5104.2018.v16.n2.p113.

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O estudo é resultado de uma experiência de tematização do conteúdo “exercício físico e saúde”, desenvolvido por integrantes do PIBID em uma escola de educação profissional de ensino médio, localizada no município do Crato (CE). O objetivo do estudo foi descrever e analisar uma unidade didática de ensino e a relação ensino-aprendizagem levada a efeito com o conteúdo “exercício físico e saúde”. O relato de experiência, de natureza qualitativa e exploratória, foi pautado a partir de uma ação colaborativa entre professor da disciplina de educação física e acadêmicos bolsistas do PIBID. Como resultados, pudemos observar que há possibilidade de conjugar teoria e prática e envolver o aluno de maneira ativa no processo de ensino-aprendizagem. A demonstração das dificuldades dos alunos revelou a necessidade de adaptações relativas às exigências motoras que lhes são feitas, o que faz com que a intervenção pedagógica seja atrelada à leitura da realidade. Como conclusão, identificou-se que é possível conduzir um processo de intervenção pedagógica em educação física pautada na colaboração, compartilhamento de experiências, criticidade, indissociabilidade entre teoria e prática e noção de movimento humano para além dos aspectos biomecânicos.ABSTRACT. Didactic possibilities in physical education classes: the topic of “physical exercise and health” in high school. The study results from an experience on thematizing the topic “physical exercise and health”, as developed by members of the PIBID in a school which offers professional education in high school in the city of Crato, in Ceará, Brazil. The purpose of the study was to describe and analyze a Didactic Unit used for teaching as well as the teaching-learning relationship which was taken to class under the topic of “Physical Exercise and Health”. This is a report from a qualitative, exploratory in nature, experience grounded on a collaborative-like action involving the physical education teacher and the trainees participating in the PIBID project. There are possibilities in combining theory and practice and involving the high school students in an active way in the teaching-learning process. The demonstration of the students’ difficulties revealed the necessity of making adaptations regarding the motor demands requested from them, which makes it necessary to read the reality and the pedagogical intervention as complementary. It was identified that it is possible to conduct a pedagogical intervention process in physical education which is grounded on collaborating, on sharing experiences, on being critical, and on associating theory and practice and the notion of human movement going further than the bio mechanical aspects.

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Gongopadhyay, Krishnendu. "The z-classes of quaternionic hyperbolic isometries." Journal of Group Theory 16, no. 6 (January 1, 2013). http://dx.doi.org/10.1515/jgt-2013-0013.

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Abstract.LetIn Appendix A, we show that our methods can be used to obtain the centralizers up to conjugacy in real and complex hyperbolic geometries as well. This provides a unified approach to the determination of the conjugacy classes of centralizers in hyperbolic geometries.

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Gates, Zachary, Anupam Singh, and C. Ryan Vinroot. "Strongly real classes in finite unitary groups of odd characteristic." Journal of Group Theory 17, no. 4 (July 1, 2014). http://dx.doi.org/10.1515/jgt-2014-0010.

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Beltrán, Antonio. "Conjugacy classes and union of cosets of normal subgroups." Monatshefte für Mathematik, May 27, 2022. http://dx.doi.org/10.1007/s00605-022-01726-w.

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AbstractLet G be a finite group, N a normal subgroup of G and K a conjugacy class of G. We prove that if $$K\cup K^{-1}$$ K ∪ K - 1 is union of cosets of N, then N is soluble, K is a real-imaginary class, that is, every irreducible character of G takes real or purely imaginary values on K, and if, in addition, the elements of K are p-elements for some prime p, then N has normal p-complement. We also prove that if $$K\cup K^{-1}$$ K ∪ K - 1 is a single coset of N, then $$\langle K\rangle $$ ⟨ K ⟩ has a normal 2-complement.

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Tiep, Pham Huu, and A. E. Zalesski. "Real conjugacy classes in algebraic groups and finite groups of Lie type." Journal of Group Theory 8, no. 3 (January 25, 2005). http://dx.doi.org/10.1515/jgth.2005.8.3.291.

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40

Warfield, Simon. "FFT Complex to Complex filters and helper classes." Insight Journal, October 22, 2006. http://dx.doi.org/10.54294/h4j7t7.

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ITK currently supports real to complex conjugate forward and inverse Fourier transformation’s through dedicated filters. In turn, these utilize one of a number of Fourier transform implementations, such as that available through VNL, through FFTW or through Intel’s implementation of the FFTW API. When dealing directly with complex data, such as may be acquired by an MRI scanner, it is convenient to have dedicated filters supporting Complex to Complex transformations. It is also convenient to have filter classes that can convert between complex data and real and imaginary component and magnitude and phase component representations.

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Kalliongis, John, and Ryo Ohashi. "Cyclic p-group actions on ℝℙ3." Journal of Algebra and Its Applications, August 18, 2020, 2150192. http://dx.doi.org/10.1142/s0219498821501929.

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In this paper, we classify the smooth orientation preserving cyclic [Formula: see text]-group actions on the real projective space [Formula: see text] up equivalence, where two actions are equivalent if their images are conjugate in the group of self-diffeomorphisms. We view [Formula: see text] as the lens space [Formula: see text]. We show that any such action on [Formula: see text] is conjugate to a standard action explicitly defined, and we identify the quotient spaces of these actions. In addition, we enumerate the equivalence classes.

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Liu, Jiangfan, Leonardo Modesto, and Gianluca Calcagni. "Quantum field theory with ghost pairs." Journal of High Energy Physics 2023, no. 2 (February 14, 2023). http://dx.doi.org/10.1007/jhep02(2023)140.

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Abstract We explicitly show that general local higher-derivative theories with only complex conjugate ghosts and normal real particles are unitary at any perturbative order in the loop expansion. The proof presented here relies on integrating the loop energies on complex paths resulting from the deformation of the purely imaginary paths, when the external energies are continued from imaginary to real values. Contrary to the case of nonlocal theories, where the same integration path was first proposed, for the classes of theories studied here the same procedure is not analytic, but the resulting theory is unitary and unique when the complex ghosts are present in pairs. As an explicit application, a special class of higher-derivative super-renormalizable or finite gravitational and gauge theories turns out to be unitary at any perturbative order if we exclude the complex ghosts from the spectrum of the theory, as it is normally accepted for Becchi-Rouet-Stora-Tyutin (BRST) ghosts. Finally, we propose an analogy between confined gluons in quantum Yang-Mills theory and classical complex pairs in local higher-derivative theories. According to such interpretation, complex ghosts will not appear on shell as asymptotic states because confined in what is natural to name “ghostballs.”

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