This is a survey on trace constructions on various operator algebras with an emphasis on regularized traces on algebras of pseudodifferential operators. For motivation our point of departure is the classical Hilbert space trace which is the unique semifinite normal trace on the algebra of bounded operators on a separable Hilbert space. Dropping the normality assumption leads to the celebrated D...

The Hadamard conjecture is that Hadamard matrices exist for all orders 1,2, 4t where t 2 1 is an integer. We have obtained the following results which strongly support the conjecture: (i) Given any natural number q, there exists an Hadamard matrix of order 2 q for every s 2 [2log2 (q 3)]. (ii) Given any natural number q, there exists a regular symmetric Hadamard matrix with constant diagonal of...

Recently, Feng and Xiang [10] found a new construction of skew Hadamard difference sets in elementary abelian groups. In this paper, we introduce a new invariant for equivalence of skew Hadamard difference sets, namely triple intersection numbers modulo a prime, and discuss inequivalence between Feng-Xiang skew Hadamard difference sets and the Paley difference sets. As a consequence, we show th...

A Hadamard matrix of side n is an n×n matrix with every entry either 1 or −1, which satisfies HHT = nI. Two Hadamard matrices are called equivalent if one can be obtained from the other by some sequence of row and column permutations and negations. To identify the equivalence of two Hadamard matrices by a complete search is known to be an NP hard problem when n increases. In this paper, a new a...

In this paper we obtain several results on the rank properties of Hadamard matrices (including Sylvester Hadamard matrices) as well as (generalized) Hadamard matrices. These results are used to show that the classes of (generalized) Sylvester Hadamard matrices and of generalized pseudo-noise matrices are equivalent, i.e., they can be obtained from each other by means of row/column permutations....

This essay introduces a new featured section, to be published in Small Axe annually, that explores the critical vocabulary of field Caribbean studies.

The Z2s-additive and Z2Z4-additive codes are subgroups of Z n 2 and Z α 2 × Z β 4 , respectively. Both families can be seen as generalizations of linear codes over Z2 and Z4. A Z2s-linear (resp. Z2Z4-linear) Hadamard code is a binary Hadamard code which is the Gray map image of a Z2s-additive (resp. Z2Z4-additive) code. It is known that there are exactly ⌊ t−1 2 ⌋ and ⌊ t 2⌋ nonequivalent Z2Z4-...

A balanced incomplete block design (BIBD) [1] with parameters 2-(v, b, r, k, λ) (short 2-(v, k, λ)) is a pair (V,B) where V is a v-set (elements are called points) and B is a collection of b k-subsets (elements are called blocks) of V such that each point is contained in exactly r blocks and any pair of points is contained in exactly λ blocks. A Hadamard matrix of order n is an n × n (1,−1)-mat...