On Commuting Exponentials in Low Dimensions

We investigate properties of differential and difference operators annihilating certain finite-dimensional subspaces of exponential functions in two variables that are connected to the representation of real-valued trigonometric and hyperbolic functions. Although exponential functions appear in a variety of contexts, the motivation behind this work comes from considering subdivision schemes with the capability of preserving those exponential functions required for an exact de...

We study the phase retrieval problem within the broader context of analytic functions. Given a parameter $\theta \in \mathbb{R}$, we derive the following characterization of uniqueness in terms of rigidity of a set $\Lambda \subseteq \mathbb{R}$: if $\mathcal{F}$ is a vector space of entire functions containing all exponentials $e^{\xi z}, \, \xi \in \mathbb{C} \setminus \{ 0 \}$, then every $F \in \mathcal{F}$ is uniquely determined up to a unimodular phase factor by $\{ |F(...

Let $\mathbb R_t[\theta]$ be the ring generated over $\mathbb R$ by $\cos\theta$ and $\sin\theta$, and $\mathbb R_t(\theta)$ be its quotient field. In this paper we study the ways in which an element p of $\mathbb R_t[\theta]$ can be decomposed into a composition of functions of the form $p=R(q),$ where $\mathbb R\in \mathbb R(x)$ and $q\in \mathbb R_t(\theta)$. In particular, we describe all possible solutions of the functional equation $R_1(q_1)=R_2(q_2)$, where $R_1, R_2 \...

In this work is proposed a method using orthogonal matrix transform properties to encrypt and decrypt a message. It will be showed how to use matrix functions to create complex encryptions. Because orthogonal matrix are always diagonalizable on R, and the exponential of a diagonal matrix is easy to compute, the exponential of orthogonal matrix will be used to encrypt text messages.

Polar commutative n-complex numbers of the form u=x_0+h_1x_1+h_2x_2+...+h_{n-1}x_{n-1} are introduced in n dimensions, the variables x_0,...,x_{n-1} being real numbers. The polar n-complex number can be represented, in an even number of dimensions, by the modulus d, by the amplitude \rho, by 2 polar angles \theta_+,\theta_-, by n/2-2 planar angles \psi_{k-1}, and by n/2-1 azimuthal angles \phi_k. In an odd number of dimensions, the polar n-complex number can be represented by...

These notes concern linear transformations on R^n and C^n, exponentials of linear transformations, and some related geometric questions.

We consider commutativity equations $F_i F_j =F_j F_i$ for a function $F(x^1, \dots, x^N),$ where $F_i$ is a matrix of the third order derivatives $F_{ikl}$. We show that under certain non-degeneracy conditions a solution $F$ satisfies the WDVV equations. Equivalently, the corresponding family of Frobenius algebras has the identity field $e$. We also study trigonometric solutions $F$ determined by a finite collection of vectors with multiplicities, and we give an explicit for...

We show an explicit formula, with a quite easy deduction, for the exponential matrix $e^{tA}$ of a real square matrix $A$ of order $n\times n$. The elementary method developed requires neither Jordan canonical form, nor eigenvectors, nor resolution of linear systems of differential equations, nor resolution of linear systems with constant coefficients, nor matrix inversion, nor complex integration, nor functional analysis. The basic tools are power series and the method of pa...

Gelfond and Khovanskii found a formula for the sum of the values of a Laurent polynomial over the zeros of a system of n Laurent polynomials in the algebraic n-torus. We expect that a similar formula holds in the case of exponential sums with real frequencies. Here we prove such a formula in dimension 1.

The aim of this note is to answer a question by Guoliang Yu of whether the group $EL_3(Z<x,y>)$, where $Z<x,y>$ is the free (non-commutative) ring, has any faithful linear representations over a field. We prove, in particular, that for every (unitary associative) ring $R$, the group $EL_3(R)$ has a faithful finite dimensional complex representation if and only if $R$ has a finite index ideal that has a faithful finite dimensional complex representation.