Related papers: Extendable orthogonal sets of integral vectors
We introduce a total order and the absolute value function for dual numbers. The absolute value function of dual numbers are with dual number values, and have properties similar to the properties of the absolute value function of real…
I adapt a recently introduced method for integrating over the unitary group (S. Aubert and C.S. Lam, J.Math.Phys. 44, 6112-6131 (2003)) to the orthogonal group. I derive explicit formulas for a number of one, two and three-vector integrals,…
This paper studies two topics concerning on the orthogonal complement of one dimensional subspace with respect to a given quadratic form on a vector space over a number field. One is to determine the invariants for the isomorphism class of…
The tridiagonal representation approach is an algebraic method for solving second order differential wave equations. Using this approach in the solution of quantum mechanical problems, we encounter two new classes of orthogonal polynomials…
In the paper we investigate the theory of quantum optical systems. As an application we integrate and describe the quantum optical systems which are generically related to the classical orthogonal polynomials. The family of coherent states…
The random matrix ensembles are applied to the quantum statistical two-dimensional systems of electrons. The quantum systems are studied using the finite dimensional real, complex and quaternion Hilbert spaces of the eigenfunctions. The…
In this paper, we consider the new q-extension of Frobenius-Euler numbers and polynomials and we derive some interesting identities from the orthogonality type properties for the new q-extension of Frobenius-Euler polynomials. Finally we…
A new way of orthogonalizing ensembles of vectors by "lifting" them to higher dimensions is introduced. This method can potentially be utilized for solving quantum decision and computing problems.
A hypergeometric type equation satisfying certain conditions defines either a finite or an infinite system of orthogonal polynomials. The associated special functions are eigenfunctions of some shape invariant operators. These operators can…
The fundamental properties of biquaternions (complexified quaternions) are presented including several different representations, some of them new, and definitions of fundamental operations such as the scalar and vector parts, conjugates,…
This thesis is a study of large sets of unit vectors in $\cx^n$ such that the absolute value of their standard inner products takes on only a small number of values. We begin with bounds: what is the maximal size of a set of lines with only…
We introduce the triangulant of two matrices, and relate it to the existence of orthogonal eigenvectors. We also use it for a new characterization of mutually unbiased bases. Generalizing the notion, we introduce higher order triangulants…
The famous Hadwiger theorem classifies all rigid motion invariant continuous valuations on convex sets as linear conbinations of quermassintegrals. We prove much more general result. We classify continuous valuations which are invariant…
It is shown that the groups of Euclidian rotations, rigid motions, proper, orthochronous Lorentz transformations, and the complex rigid motions can be represented by the groups of unit-norm elements in the algebras of real, dual, complex,…
A hypergeometric type equation satisfying certain conditions defines either a finite or an infinite system of orthogonal polynomials. We present in a unified and explicit way all these systems of orthogonal polynomials, the associated…
The concept of balancedly splittable orthogonal designs is introduced along with a recursive construction. As an application, equiangular tight frames over the real, complex, and quaternions meeting the Delsarte-Goethals-Seidel upper bound…
We describe various properties of continued fraction expansions of complex numbers in terms of Gaussian integers. Numerous distinct such expansions are possible for a complex number. They can be arrived at through various algorithms, as…
In this text, based on elementary computations, we provide a perturbative expansion of the coordinates of the eigenvectors of a Hermitian matrix of large size perturbed by a random matrix with small operator norm whose entries in the…
We discuss the existence of an orthogonal basis consisting of decomposable vectors for some symmetry classes of tensors associated with Semi-Dihedral groups $SD_{8n}$. The dimensions of these symmetry classes of tensors are also computed.
The attitude space has been parameterized in various ways for practical purposes. Different representations gain preferences over others based on their intuitive understanding, ease of implementation, formulaic simplicity, and physical as…