Related papers: Two-parameter complex Hadamard matrices for N=6
We prove that if $A$ and $B$ are Hadamard matrices which are both of size $4 \times 4$ or $5 \times 5$ and in dephased form, then $tr(A) = tr(B)$ implies that $A$ and $B$ have the same eigenvalues, including multiplicity. We calculate…
A real symmetric matrix $M$ is completely positive semidefinite if it admits a Gram representation by (Hermitian) positive semidefinite matrices of any size $d$. The smallest such $d$ is called the (complex) completely positive semidefinite…
Two doubly indexed families of polynomials in several indeterminates are considered. They are related to the falling and rising factorials in a similar way as the potential polynomials (introduced by L. Comtet) are related to the ordinary…
Three algorithms looking for pretty large partial Hadamard matrices are described. Here "large" means that hopefully about a third of a Hadamard matrix (which is the best asymptotic result known so far, [dLa00]) is achieved. The first one…
We consider the alternating sign matrices of the odd order that have some kind of central symmetry. Namely, we deal with matrices invariant under the half-turn, quarter-turn and flips in both diagonals. In all these cases, there are two…
If $N=2^k > 8$ then there exist exactly $[(k-1)/2]$ pairwise nonequivalent $Z_4$-linear Hadamard $(N,2N,N/2)$-codes and $[(k+1)/2]$ pairwise nonequivalent $Z_4$-linear extended perfect $(N,2^N/2N,4)$-codes. A recurrent construction of…
We give a description in terms of square matrices of the family of group-like algebras with $S*id=id*S=u\epsilon$. In the case that $S=id$ and $char\Bbbk$ is not 2 and does not divide the dimension of the algebra, this translation take us…
Equiangular tight frames provide optimal packings of lines through the origin. We combine Steiner triple systems with Hadamard matrices to produce a new infinite family of equiangular tight frames. This in turn leads to new constructions of…
We define several operations that switch substructures of Hadamard matrices thereby producing new, generally inequivalent, Hadamard matrices. These operations have application to the enumeration and classification of Hadamard matrices. To…
Finite dimensional modular Lie superalgebras over algebraically closed fields with indecomposable Cartan matrices are classified under some technical, most probably inessential, hypotheses. If the Cartan matrix is invertible, the…
We derive families of Newton-like inequalities involving the elementary symmetric functions of sets of self-conjugate complex numbers in the right half-plane. These are the first known inequalities of this type which are independent of the…
We survey some recent developments in the theory of orthogonal polynomials defined by differential equations. The key finding is that there exist orthogonal polynomials defined by 2nd order differential equations that fall outside the…
We give a new characterization of skew Hadamard matrices of size $n$ in terms of the data of the spectra of tournaments of size $n-2$.
Our main result is the construction of symmetric Hadamard matrices of order q(1 + q) where q is a prime power congruent to 3 mod 8.
The object of this paper is to describe an explicit two--parameter family of logarithmic flat connections over the complex projective plane. These connections have dihedral monodromy and their polar locus is a prescribed quintic composed of…
We find and study a six (resp. seven, eight)-parameter family of polynomial Hamiltonian systems of second order, respectively. This system admits the affine Weyl group symmetry of type $E_6^{(1)}$ (resp. $E_7^{(1)}, E_8^{(1)}$) as the group…
Recently there has been a renewed interest in an extension of the notion of orthogonal polynomials known as multiple orthogonal polynomials. This notion comes from simultaneous rational approximation (Hermite-Pade approximation) of a system…
We introduce a new family of finite posets which we call 2-chains. These first arose in the study of 0-Hecke algebras, but they admit a variety of different characterisations. We give these characterisations, prove that they are equivalent…
This paper proposes a new class of copulas which characterize the set of all twice continuously differentiable copulas. We show that our proposed new class of copulas is a new generalized copula family that include not only asymmetric…
In this paper, we generalize classical constructions of skew Hadamard difference families with two or four blocks in the additive groups of finite fields given by Szekeres (1969, 1971), Whiteman (1971) and Wallis-Whiteman (1972). In…