Related papers: Analytic Functions of a General Matrix Variable
We present theory for general partial derivatives of matrix functions on the form $f(A(x))$ where $A(x)$ is a matrix path of several variables ($x=(x_1,\dots,x_j)$). Building on results by Mathias [SIAM J. Matrix Anal. Appl., 17 (1996), pp.…
Fractal sets, by definition, are non-differentiable, however their dimension can be continuous, differentiable, and arithmetically manipulable as function of their construction parameters. A new arithmetic for fractal dimension of polyadic…
Motivated by applications to stochastic programming, we introduce and study the expected-integral functionals, which are mappings given in an integral form depending on two variables, the first a finite dimensional decision vector and the…
Let G be a piecewise constant $n\times n$ matrix function which is defined on a smooth closed curve $\Gamma$ in the complex sphere and which has m jumps. We consider the problem of determining the partial indices of the factorization of the…
A generalized matrix function is a generalization of determinant and permanent function. In this paper, we introduced the formula for the value of a generalized matrix function of a linear sum of permutation matrices. We show that a linear…
We show that certain determinantal functions of multiple matrices, when summed over the symmetries of the cube, decompose into functions of the original matrices. These are shown to be true in complete generality; that is, no properties of…
Functions of several quaternion variables are investigated and integral representation theorems for them are proved. With the help of them solutions of the $\tilde \partial $-equations are studied. Moreover, quaternion Stein manifolds are…
The paper addresses the study and applications of a broad class of extended-real-valued functions, known as optimal value or marginal functions, which are frequently appeared in variational analysis, parametric optimization, and a variety…
The generalised Gegenbauer functions of fractional degree (GGF-Fs), denoted by ${}^{r\!}G^{(\lambda)}_\nu(x)$ (right GGF-Fs) and ${}^{l}G^{(\lambda)}_\nu(x)$ (left GGF-Fs) with $x\in (-1,1),$ $\lambda>-1/2$ and real $\nu\ge 0,$ are special…
The correspondence between a high-order non symmetric difference operator with complex coefficients and the evolution of an operator defined by a Lax pair is established. The solution of the discrete dynamical system is studied, giving…
We consider the notion of the matrix (tensor) distribution of a measurable function of several variables. On the one hand, it is an invariant of this function with respect to a certain group of transformations of variables; on the other…
This paper proposes the density and characteristic functions of a general matrix quadratic form $\mathbf{X}^{*}\mathbf{AX}$, when $\mathbf{A} = \mathbf{A}^{*}$, $\mathbf{X}$ has a matrix multivariate elliptical distribution and…
We study a class of matrix function algebras, here denoted $\mathcal{T}^{+}(\mathcal{C}_n)$. We introduce a notion of point derivations, and classify the point derivations for certain finite dimensional representations of…
Many different types of fractional calculus have been proposed, which can be organised into some general classes of operators. For a unified mathematical theory, results should be proved in the most general possible setting. Two important…
The present research deals with generalizations of the Salem function with arguments defined in terms of certain alternating expansions of real numbers. The special attention is given to modelling such functions by systems of functional…
Global dynamics of a non-linear Cellular Automata is, in general irregular, asymmetric and unpredictable as opposed to that of a linear CA, which is highly systematic and tractable. In the past efforts have been made to systematize…
An explicit formula is obtained for the generalized Macdonald functions on the $N$-fold Fock tensor spaces, calculating a certain matrix element of a composition of several screened vertex operators. As an application, we prove the…
Under certain general conditions, an explicit formula to compute the greatest delta-epsilon function of a continuous function is given. From this formula, a new way to analyze the uniform continuity of a continuous function is given.…
Recently, there emerges different versions of beta function and hypergeometric functions containing extra parameters. Gaining enlightenment from these ideas, we will first introduce a new extension of generalized hypergeometric function and…
This short note provides an explicit description of the Fr\'echet derivatives of the principal square root matrix functional at any order. We present an original formulation that allows to compute sequentially the Fr\'echet derivatives of…