Cyclotomic integers and finite geometry

Schmidt, Bernhard

doi:10.1090/s0894-0347-99-00298-2

Cited by 72 publications

(47 citation statements)

References 48 publications

(64 reference statements)

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“…They arise naturally in lattice statistical mechanics in relation with the Yang-Baxter equations for solving vertex models and star-triangle relation for solving spin model. This problem turns out to be related to many interesting field of mathematics: Bose-Mesner algebra, 8 association schemes, 9 Hadamard matrices ͑one of our result demonstrates a particular case of a conjecture about Hadamard matrices 10,11 ͒, Gauss identity, etc. 4 These transformations can, in fact, be considered per se as discrete dynamical system.…”

Section: Introduction and Presentationsupporting

confidence: 52%

“…b͔, 4 ͓a b c d e f b c d e g b c d e h b c d e i b c d e͔, 5 ͓a c d e e b f c f g b d g g d b g f c f b e e d c͔, 6 ͓a d e f g b h i j k c l m m l c k j i h b g f e d͔, 7 ͓a g h i j c k l n b d p m o q e r s t u f v w x y ͔.Finally the remaining patterns are computed from Z 5Ã = ͕1,−1 ͑K 1 ͒ = Z 25 Ã , P −1 ͑K 2 ͒ = ͕1,4,6,9,11,14,16,19,21,24͖, P −1 ͑K 3 ͒ = ͕1,6,11,16,21͖, yielding the six last patterns:8 ͓a b b b…”

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confidence: 99%

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Birational mappings and matrix subalgebra from the chiral Potts model

Preissmann

d'Auriac

Maillard

2009

Journal of Mathematical Physics

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We study birational transformations of the projective space originating from lattice statistical mechanics, specifically from various chiral Potts models. Associating these models with stable patterns and signed patterns, we give general results which allow us to find all chiral q-state spin-edge Potts models when the number of states q is a prime or the square of a prime, as well as several q-dependent family of models. We also prove the absence of monocolor stable signed pattern with more than four states. This demonstrates a conjecture about cyclic Hadamard matrices in a particular case. The birational transformations associated with these lattice spinedge models show complexity reduction. In particular, we recover a one-parameter family of integrable transformations, for which we give a matrix representation when the parameter has a suitable value.

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Section: Introduction and Presentationsupporting

confidence: 52%

mentioning

confidence: 99%

Birational mappings and matrix subalgebra from the chiral Potts model

Preissmann

d'Auriac

Maillard

2009

Journal of Mathematical Physics

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“…Such a sequence is called a (binary) Barker sequence; they exist for n ∈ {2, 3, 4, 5, 7, 11, 13} and for no other n 4 •10 12 . (See [15] or [17].) The autocorrelations of a binary sequence (a 0 , .…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Autocorrelations of Random Binary Sequences

Mercer

2006

Combinator. Probab. Comp.

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We define B n to be the set of n-tuples of the form (a 0 , . . . , a n−1 ) where a j = ±1. If A ∈ B n , then we call A a binary sequence and define the autocorrelations of A by c k := n−k−1 j=0 a j a j+k for 0 k n − 1. The problem of finding binary sequences with autocorrelations 'near zero' has arisen in communications engineering and is also relevant to conjectures of Littlewood and Erdős on 'flat' polynomials with ±1 coefficients. Following Turyn, we define b(n) := min A∈Bn max 1 k n−1 |c k |.The purpose of this article is to show that, using some known techniques from discrete probability, we can improve upon the best upper bound on b(n) appearing in the previous literature, and we can obtain both asymptotic and exact expressions for the expected value of c m k if the a j are independent ±1 random variables with mean 0. We also include some brief heuristic remarks in support of the unproved conjecture that b(n) = O( √ n).

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“…One may expect that an exponent bound on the group containing primitive formal dual pairs will rule out the group like

Z_{2}^{2} \times Z_{16}

. In this regard, some deep number‐theoretic approach may help.…”

Section: Discussionmentioning

confidence: 99%

Constructions of primitive formally dual pairs having subsets with unequal sizes

Pott

2019

J of Combinatorial Designs

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The concept of formal duality was proposed by Cohn, Kumar and Schürmann, which reflects a remarkable symmetry among energy‐minimizing periodic configurations. This formal duality was later translated into a purely combinatorial property by Cohn, Kumar, Reiher and Schürmann, where the corresponding combinatorial objects were called formally dual pairs. Motivated by this surprising application on the energy minimization problem, we focus on the algebraic constructions of primitive formally dual pairs. It is worth noting that almost all known examples of primitive formally dual pairs satisfy that the two subsets have the same size. Indeed, prior to this work, there was only one known example derived from computer search, which had subsets with unequal sizes in Z 2 × Z 4 2. Inspired by this example, we propose a lifting construction framework and a recursive construction framework, which generate new primitive formally dual pairs from known ones. As an application, for m ≥ 2, we obtain m + 1 pairwise inequivalent primitive formally dual pairs in Z 2 × Z 4 2 m, which have subsets with unequal sizes.

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Cyclotomic integers and finite geometry

Cited by 72 publications

References 48 publications

Birational mappings and matrix subalgebra from the chiral Potts model

Birational mappings and matrix subalgebra from the chiral Potts model

Autocorrelations of Random Binary Sequences

Constructions of primitive formally dual pairs having subsets with unequal sizes

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