MODULAR GROUP IMAGES ARISING FROM DRINFELD DOUBLES OF DIHEDRAL GROUPS

Generic-case approach to algorithmic problems was suggested by A. Miasnikov, I. Kapovich, P. Schupp and V. Shpilrain in 2003. This approach studies behavior of an algo-rithm on typical (almost all) inputs and ignores the rest of inputs. In this paper, we prove that the subset sum problems for the monoid of integer positive unimodular matrices of the second order, the special linear group of the second order, and the modular group are generically solvable in polynomial time.

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Power graphs and exchange property for resolving sets

Open Mathematics ◽

10.1515/math-2019-0093 ◽

2019 ◽

Vol 17 (1) ◽

pp. 1303-1309 ◽

Cited By ~ 1

Author(s):

Ghulam Abbas ◽

Usman Ali ◽

Mobeen Munir ◽

Syed Ahtsham Ul Haq Bokhary ◽

Shin Min Kang

Keyword(s):

Present Article ◽

Linear Algebra ◽

Finite Groups ◽

Robot Navigation ◽

Simple Graph ◽

Exchange Property ◽

Metric Dimension ◽

Sufficient Condition ◽

Resolving Set ◽

Dihedral Groups

Abstract Classical applications of resolving sets and metric dimension can be observed in robot navigation, networking and pharmacy. In the present article, a formula for computing the metric dimension of a simple graph wihtout singleton twins is given. A sufficient condition for the graph to have the exchange property for resolving sets is found. Consequently, every minimal resolving set in the graph forms a basis for a matriod in the context of independence defined by Boutin [Determining sets, resolving set and the exchange property, Graphs Combin., 2009, 25, 789-806]. Also, a new way to define a matroid on finite ground is deduced. It is proved that the matroid is strongly base orderable and hence satisfies the conjecture of White [An unique exchange property for bases, Linear Algebra Appl., 1980, 31, 81-91]. As an application, it is shown that the power graphs of some finite groups can define a matroid. Moreover, we also compute the metric dimension of the power graphs of dihedral groups.

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Fermion masses and mixing from the double cover and metaplectic cover of the A5 modular group

Physical Review D ◽

10.1103/physrevd.103.095013 ◽

2021 ◽

Vol 103 (9) ◽

Author(s):

Chang-Yuan Yao ◽

Xiang-Gan Liu ◽

Gui-Jun Ding

Keyword(s):

Modular Group ◽

Double Cover ◽

Fermion Masses ◽

Metaplectic Cover

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Identities on poly-Dedekind sums

Advances in Difference Equations ◽

10.1186/s13662-020-03024-x ◽

2020 ◽

Vol 2020 (1) ◽

Cited By ~ 2

Author(s):

Taekyun Kim ◽

Dae San Kim ◽

Hyunseok Lee ◽

Lee-Chae Jang

Keyword(s):

Modular Group ◽

Reciprocity Relation ◽

Dedekind Sums ◽

Transformation Behavior ◽

Dedekind Eta Function ◽

Bernoulli Function ◽

Bernoulli Functions

Abstract Dedekind sums occur in the transformation behavior of the logarithm of the Dedekind eta-function under substitutions from the modular group. In 1892, Dedekind showed a reciprocity relation for the Dedekind sums. Apostol generalized Dedekind sums by replacing the first Bernoulli function appearing in them by any Bernoulli functions and derived a reciprocity relation for the generalized Dedekind sums. In this paper, we consider the poly-Dedekind sums obtained from the Dedekind sums by replacing the first Bernoulli function by any type 2 poly-Bernoulli functions of arbitrary indices and prove a reciprocity relation for the poly-Dedekind sums.

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Regular Cayley maps for dihedral groups

Journal of Combinatorial Theory Series B ◽

10.1016/j.jctb.2020.12.002 ◽

2021 ◽

Vol 148 ◽

pp. 84-124

Author(s):

István Kovács ◽

Young Soo Kwon

Keyword(s):

Dihedral Groups ◽

Cayley Maps

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On Fourier Coefficients of the Symmetric Square L-Function at Piatetski-Shapiro Prime Twins

Mathematics ◽

10.3390/math9111254 ◽

2021 ◽

Vol 9 (11) ◽

pp. 1254

Author(s):

Xue Han ◽

Xiaofei Yan ◽

Deyu Zhang

Keyword(s):

Asymptotic Formula ◽

Fourier Coefficient ◽

Cusp Form ◽

Riemann Hypothesis ◽

Modular Group ◽

Fourier Coefficients ◽

Mean Value ◽

Symmetric Square ◽

The Mean ◽

Almost All

Let Pc(x)={p≤x|p,[pc]areprimes},c∈R+∖N and λsym2f(n) be the n-th Fourier coefficient associated with the symmetric square L-function L(s,sym2f). For any A>0, we prove that the mean value of λsym2f(n) over Pc(x) is ≪xlog−A−2x for almost all c∈ε,(5+3)/8−ε in the sense of Lebesgue measure. Furthermore, it holds for all c∈(0,1) under the Riemann Hypothesis. Furthermore, we obtain that asymptotic formula for λf2(n) over Pc(x) is ∑p,qprimep≤x,q=[pc]λf2(p)=xclog2x(1+o(1)), for almost all c∈ε,(5+3)/8−ε, where λf(n) is the normalized n-th Fourier coefficient associated with a holomorphic cusp form f for the full modular group.

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The Picard group and the modular group

Proceedings of Groups – St Andrews 1985 ◽

10.1017/cbo9780511600647.014 ◽

2012 ◽

pp. 150-163

Author(s):

B. Fine

Keyword(s):

Modular Group ◽

Picard Group

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On *-clean non-commutative group rings

Journal of Algebra and Its Applications ◽

10.1142/s0219498816501504 ◽

2016 ◽

Vol 15 (08) ◽

pp. 1650150 ◽

Cited By ~ 4

Author(s):

Hongdi Huang ◽

Yuanlin Li ◽

Gaohua Tang

Keyword(s):

Finite Field ◽

Local Ring ◽

Group Algebra ◽

Rational Numbers ◽

Semisimple Group ◽

Group Algebras ◽

Group Rings ◽

Dihedral Groups ◽

Commutative Local Ring ◽

Generalized Quaternion

A ring with involution ∗ is called ∗-clean if each of its elements is the sum of a unit and a projection (∗-invariant idempotent). In this paper, we consider the group algebras of the dihedral groups [Formula: see text], and the generalized quaternion groups [Formula: see text] with standard involution ∗. For the non-semisimple group algebra case, we characterize the ∗-cleanness of [Formula: see text] with a prime [Formula: see text], and [Formula: see text] with [Formula: see text], where [Formula: see text] is a commutative local ring. For the semisimple group algebra case, we investigate when [Formula: see text] is ∗-clean, where [Formula: see text] is the field of rational numbers [Formula: see text] or a finite field [Formula: see text] and [Formula: see text] or [Formula: see text].

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