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Analytic aspects of the circulant Hadamard conjecture

Teodor Banica, Ion Nechita, Jean-Marc Schlenker (2014)

Annales mathématiques Blaise Pascal

We investigate the problem of counting the real or complex Hadamard matrices which are circulant, by using analytic methods. Our main observation is the fact that for | q 0 | = ... = | q N - 1 | = 1 the quantity Φ = ∑ i + k = j + l q i q k q j q l satisfies Φ ≥ N 2 , with equality if and only if q = ( q i ) is the eigenvalue vector of a rescaled circulant complex Hadamard matrix. This suggests three analytic problems, namely: (1) the brute-force minimization of Φ , (2) the study of the critical points of Φ , and (3) the computation of the moments of Φ . We explore here these questions,...

Coalescing Fiedler and core vertices

Didar A. Ali, John Baptist Gauci, Irene Sciriha, Khidir R. Sharaf (2016)

Czechoslovak Mathematical Journal

The nullity of a graph G is the multiplicity of zero as an eigenvalue in the spectrum of its adjacency matrix. From the interlacing theorem, derived from Cauchy’s inequalities for matrices, a vertex of a graph can be a core vertex if, on deleting the vertex, the nullity decreases, or a Fiedler vertex, otherwise. We adopt a graph theoretical approach to determine conditions required for the identification of a pair of prescribed types of root vertices of two graphs to form a cut-vertex of unique...

Complex Hadamard Matrices contained in a Bose–Mesner algebra

Takuya Ikuta, Akihiro Munemasa (2015)

Special Matrices

Acomplex Hadamard matrix is a square matrix H with complex entries of absolute value 1 satisfying HH* = nI, where * stands for the Hermitian transpose and I is the identity matrix of order n. In this paper, we first determine the image of a certain rational map from the d-dimensional complex projective space to Cd(d+1)/2. Applying this result with d = 3, we give constructions of complex Hadamard matrices, and more generally, type-II matrices, in the Bose–Mesner algebra of a certain 3-class symmetric...

Constructing a Canonical form of a Matrix in Several Problems about Combinatorial Designs

Mateva, Zlatka (2008)

Serdica Journal of Computing

Partially supported by the Bulgarian Science Fund contract with TU Varna, No 487.The author developed computer programs needed for the classification of designs with certain automorphisms by the local approach method. All these programs use canonicity test or/and construction of canonical form of an integer matrix. Their efficiency substantially influences the speed of the whole computation. The present paper deals with the implemented canonicity algorithm. It is based on ideas used by McKay, Meringer,...

Control treatments in designs with split units generated by Latin squares

Shinji Kuriki, Iwona Mejza, Kazuhiro Ozawa, Stanisław Mejza (2014)

Biometrical Letters

This paper deals with two-factor experiments with split units. The whole plot treatments occur in a repeated Latin square, modified Latin square or Youden square, while subplot treatments occur in a block design within the whole plots. The statistical properties of the considered designs are examined. Special attention is paid to the case where one of the treatments is an individual control or an individual standard treatment. In addition, we give a brief overview of work on the design of experiments...

Covering energy of posets and its bounds

Vandana P. Bhamre, Madhukar M. Pawar (2023)

Mathematica Bohemica

The concept of covering energy of a poset is known and its McClelland type bounds are available in the literature. In this paper, we establish formulas for the covering energy of a crown with 2 n elements and a fence with n elements. A lower bound for the largest eigenvalue of a poset is established. Using this lower bound, we improve the McClelland type bounds for the covering energy for some special classes of posets.

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