Narrow Results

Constructing 2m-variable Boolean functions with optimal algebraic immunity based on decomposition of additive group of the finite field seems to be a promising approach since Tu and Deng's work. In this paper, we consider the same problem in a new way. Based on polar decomposition of the multiplicative group of , we propose a new construction of Boolean functions with optimal algebraic immunity. By a slight modification of it, we obtain a class of balanced Boolean functions achieving optimal algebraic immunity, which also have optimal algebraic degree and high nonlinearity. Computer investigations imply that this class of functions also behaves well against fast algebraic attacks.

In this paper, we present a refined convergence analysis for a simple yet powerful method for computing a symmetric low-rank orthogonal approximation of a symmetric tensor proposed in the literature. The significance is that the assumption guaranteeing the global convergence is vastly relaxed to only on an input parameter of this algorithm.

The Lie algebra of the group SU₂ is constructed from two deformed oscillator algebras for which the deformation parameter is a root of unity. This leads to an unusual quantization scheme, the {J², U_r} scheme, an alternative to the familiar {J², J_z} quantization scheme corresponding to common eigenvectors of the Casimir operator J² and the Cartan operator J_z. A connection is established between the eigenvectors of the complete set of commuting operators {J², U_r} and mutually unbiased bases in spaces of constant angular momentum.

A simple recipe for generating a complete set of mutually unbiased bases in dimension 2j + 1, with 2j integer and 2j + 1 prime, is developed from a single matrix V_a acting on a space of constant angular momentum j and defined in terms of the irreducible characters of the cyclic group C_{2j + 1}. This recipe yields an (apparently new) compact formula for the vectors spanning the various mutually unbiased bases. In dimension (2j + 1)^e, with 2j integer, 2j + 1 prime and e positive integer, the use of direct products of matrices of type V_a makes it possible to generate mutually unbiased bases. As two pending results, the matrix V_a is used in the derivation of a polar decomposition of SU(2) and of a FFZ algebra.

In this paper, we establish a strong connection between groups and gyrogroups, which provides the machinery for studying gyrogroups via group theory. Specifically, we prove that there is a correspondence between the class of gyrogroups and a class of triples with components being groups and twisted subgroups. This in particular provides a construction of a gyrogroup from a group with an automorphism of order two that satisfies the uniquely 2-divisible property. We then present various examples of such groups, including the general linear groups over $ℝ$ and $ℂ$, the Clifford group of a Clifford algebra, the Heisenberg group on a module, and the group of units in a unital C${}^{\ast }$-algebra. As a consequence, we derive polar decompositions for the groups mentioned previously.

The main aim of this work is to build unitary phase operators and the corresponding temporally stable phase states for the $su(n+1)$ Lie algebra. We first introduce an irreducible finite-dimensional Hilbertian representation of the $su(n+1)$ Lie algebra which is suitable for our purpose. The phase operators obtained from the $su(n+1)$ generators are defined and the phase states are derived as eigenstates associated to these unitary phase operators. The special cases of $su(3)$ and $su(2)$ Lie algebras are also explicitly investigated.

Let $T\in B({\mathscr{H}})$ be a $(p,k)$-quasiposinormal operator on a complex Hilbert space ${\mathscr{H}}$. In this paper, we give basic properties for $T$ and we show that a $(p,k)$-quasiposinormal operator $T$ is polaroid. We also prove that all Weyl type theorems (generalized or not) hold and are equivalent for $f(T)$, where $f$ is an analytic function defined on a neighborhood of $\sigma (T)$.