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Eigenvalues of Finite Projective Planes with an Abelian Cartesian Group

2004
*
Designs, Codes and Cryptography
*

Let M be

doi:10.1023/b:desi.0000035469.24046.f3
fatcat:62t5pxnugna7hc23gqycfkhpb4
*an*incidence matrix for a*projective**plane**of*order n. ... Under these conditions it is shown that G is a 2-*group*and n ¼ 2 r when the*eigenvalues**of*A are real. ... Preliminary Discussion Let ðp; lÞ be a*finite**projective**plane**of*order n and set N ¼ n 2 þ n þ 1ð¼ jpj ¼ jljÞ. ...##
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Distance-regular Cayley graphs with least eigenvalue $$-2$$ - 2

2016
*
Designs, Codes and Cryptography
*

Besides sporadic examples, these comprise

doi:10.1007/s10623-016-0209-4
fatcat:v2y4byyn5ngmlk5a3g22nwluyi
*of*the lattice graphs, certain triangular graphs, and line graphs*of*incidence graphs*of*certain*projective**planes*. ... We classify the distance-regular Cayley graphs*with*least*eigenvalue*-2 and diameter at most three. ... Mojtaba Jazaeri thanks the Graduate Studies*of*University*of*Isfahan and Tilburg University since this paper was partly written during his visit at Tilburg University as part*of*his PhD program in Isfahan ...##
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Page 3773 of Mathematical Reviews Vol. , Issue 81I
[page]

1981
*
Mathematical Reviews
*

Authors’ summary: “After a brief review

*of*matrix representations*of**finite**abelian**groups*,*projection*operators are defined and used to compute symmetry coordinates for systems*of*coupled harmonic oscillators ... N. 811:81034 Representation theory*of**finite**abelian**groups*applied to a linear diatomic crystal. Internat. J. Math. Math. Sci. 3 (1980), no. 3, 559-574. ...##
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Cayley sum graphs and eigenvalues of (3,6)-fullerenes
[article]

2007
*
arXiv
*
pre-print

In particular we prove a conjecture

arXiv:0712.1631v2
fatcat:2eft7mzuhbauhmqwn5uuwi6l6e
*of*Fowler, which asserts that all their*eigenvalues*come in pairs*of*the form {λ,-λ} except for the four*eigenvalues*{3,-1,-1,-1}. ... We determine the spectra*of*cubic*plane*graphs whose faces have sizes 3 and 6. Such graphs, "(3,6)-fullerenes", have been studied by chemists who are interested in their energy spectra. ... We have constructed infinite families*of*Cayley sum graphs whose spectra have the form M ∪ L ∪ −L, where M is a fixed*finite*multiset. ...##
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The Bott cofiber sequence in deformation K-theory and simultaneous similarity in U( n)

2008
*
Mathematical proceedings of the Cambridge Philosophical Society (Print)
*

We show that there is a homotopy cofiber sequence

doi:10.1017/s0305004108001928
fatcat:j6lbkdvzdvcshjm6zf6ae4wmqy
*of*spectra relating Carlsson's deformation K-theory*of*a*group*G to its "deformation representation ring," analogous to the Bott periodicity sequence relating ... We then apply this to study simultaneous similarity*of*unitary matrices. ... Given*an*n 1 -dimensional*plane*V and*an*n 2 -dimensional*plane*W in C n , let p be the orthogonal*projection*from W to V and q = p T the*projection*from V to W . ...##
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Eigenvalues of Cayley graphs
[article]

2019
*
arXiv
*
pre-print

We survey some

arXiv:1809.09829v2
fatcat:57hq3pcaurhybgfuzzh6qi74nu
*of*the known results on*eigenvalues**of*Cayley graphs and their applications, together*with*related results on*eigenvalues**of*Cayley digraphs and generalizations*of*Cayley graphs. ... It is well known that for any*finite**projective**plane*π there exists*an*integer n ≥ 1, called the order*of*π, such that each line contains n + 1 points, each point is on n + 1 lines, and the number*of*... Ramanujan*finite*upper half*plane*graphs Let q = p r be*an*odd prime power and δ a non-square*of*the*finite*field F q . ...##
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The combinatorial-motivic nature of F_1-schemes
[article]

2014
*
arXiv
*
pre-print

certain

arXiv:1406.5447v1
fatcat:rmzkge2lrveq3pukotqzu7zatm
*group*actions on*projective*spaces, and brand new results in the latter context are described. ... We review Deitmar's theory*of*monoidal schemes to start*with*, and have a detailed look at the standard examples. ... The reader recalls a basic classical result for*finite**projective**planes*which states that if a*finite**projective**plane*Γ*of*order n admits the*projective*general linear*group*PGL 3 (n) as*an*automorphism ...##
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Page 3903 of Mathematical Reviews Vol. , Issue 83i
[page]

1983
*
Mathematical Reviews
*

The authors consider a system S

*of*first-order differential equations describing*an*interaction*of**an*arbitrary*finite*number*of*scalar and spinor fields. ... Gauging the model introduces a third*Cartesian*coordinate z. ...##
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Four-band non-Abelian topological insulator and its experimental realization
[article]

2021
*
arXiv
*
pre-print

Furthermore, a system

arXiv:2106.16080v1
fatcat:djnwmo5k3jhutm5dwx6r4hpb3m
*with*even or odd number*of*bands will exhibit significant difference in non-*Abelian*topological classifications. ... We show the evolution*of*bulk topology by extending the 1D Hamiltonian onto a 2D*plane*and provide the accompanying edge state distributions following*an*analytical method. ...*Eigenvalue**of*such*an*edge state is determined from Eq. (S25). ...##
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Fake projective planes, automorphic forms, exceptional collections
[article]

2019
*
arXiv
*
pre-print

The novelty stems from the idea that if some line bundle is non-linearizable

arXiv:1602.06107v2
fatcat:qbqafqycurdljpiq2rsnv353mm
*with*respect to a*finite**abelian**group*, then it should be linearized by a*finite*, non-*abelian*, Heisenberg*group*. ... In the present note we give two short proofs*of*this statement and show acyclicity*of*some other line bundles on the fake*projective**planes**with*at least 9 automorphisms. ...*with**an*action*of*a*finite**group*G : X (not necessarily faithful). ...##
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Distance-regular Cayley graphs with small valency

2019
*
Ars Mathematica Contemporanea
*

We obtain that the incidence graphs

doi:10.26493/1855-3974.1964.297
fatcat:pu2ispk3drfhpfj6mr3ar4yhby
*of*Desarguesian affine*planes*minus a parallel class*of*lines are Cayley graphs. ... We consider the problem*of*which distance-regular graphs*with*small valency are Cayley graphs. ... Let G be a*finite**group*and S be*an*inverse-closed subset*of*G not containing the identity element e*of*G. ...##
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Distance-regular Cayley graphs with small valency
[article]

2019
*
arXiv
*
pre-print

We obtain that the incidence graphs

arXiv:1808.01428v2
fatcat:acxku3se6ndihgakxsrkdpazaq
*of*Desarguesian affine*planes*minus a parallel class*of*lines are Cayley graphs. ... We consider the problem*of*which distance-regular graphs*with*small valency are Cayley graphs. ... Acknowledgements The research*of*Mojtaba Jazaeri was in part supported by a grant from School*of*Mathematics, Institute for Research in Fundamental Sciences (IPM) (No. 95050039). ...##
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Applications of the Weyl-Wigner formalism to noncommutative geometry
[article]

2005
*
arXiv
*
pre-print

It is presented

arXiv:hep-th/0505271v1
fatcat:hmmk5ptygvhmdewukzfgwtlqzy
*an*extension*of*this formalism to the case*of*a more general classical phase space, namely one whose configuration space is a compact simple Lie*group*. ... In this dissertation the Weyl-Wigner approach is presented as a map between functions on a real*cartesian*symplectic vector space and a set*of*operators on a Hilbert space, to analyse some aspects*of*the ... L is identified*with*the manifold representing the*abelian*Lie*group**of*translations, then a Weyl system can be seen as a unitary,*projective*representation*of*the translation*group*. ...##
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Integral form of Yang-Mills equations and its gauge invariant conserved charges

2012
*
Physical Review D
*

Our results are important in the understanding

doi:10.1103/physrevd.86.085039
fatcat:zokygymp6jelfajjcn5qvpo7ve
*of*global properties*of*non-*abelian*gauge theories. ... Despite the fact that the integral form*of*the equations*of*classical electrodynamics is well known, the same is not true for non-*abelian*gauge theories. The aim*of*the present paper is threefold. ... It might connect to the so-called*Abelian**projection*and arguments for confinement. B. ...##
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Remarks on endomorphisms and rational points

2011
*
Compositio Mathematica
*

AbstractLetXbe

doi:10.1112/s0010437x11005537
fatcat:awv72cmgqvgjvm4ute5tnkxmp4
*an*algebraic variety and letf:X−−→Xbe a rational self-map*with*a fixed pointq, where everything is defined over a number fieldK. ... As*an*application, we give a simplified proof*of*the potential density*of*rational points on the variety*of*lines*of*a cubic fourfold, originally proved by Claire Voisin and the first author in 2007. ... Acknowledgements: The starting ideas for this article have emerged during the special trimester "*Groups*and Geometry" at Centro Di Giorgi, Pisa, where the three authors were present in October 2008. ...
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