Related papers: Exponential of a Matrix, a Nonlinear Problem and Q…
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…
This paper investigates the generalized convexity properties of the Lambert $W$ function, defined as the solution to $W(z)e^{W(z)}=z$. Focusing on $H_{p,q}$-convexity and concavity with respect to H\"older means, we derive necessary and…
In this paper we study generalized J-inner matrix valued functions which appear as resolvent matrices in various indefinite interpolation problems. Reproducing kernel indefinite inner product spaces associated with a generalized J-inner…
A theorem on the solutions of the problem $U'(w)=\gamma F(U(w),w),\ U(w_1)=u_2,\ U(w_2)=u_2$ is applied for finding the functional solutions of the system of partial differential equations \begin{equation} \nabla\cdot(a(u,w)\nabla u)=0,\…
The properties of the normal distribution under linear transformation, as well the easy way to compute the covariance matrix of marginals and conditionals, offer a unique opportunity to get an insight about several aspects of uncertainties…
We analyze the joint numerical range $W$ of three hermitian matrices of order four. In the generic case, this three-dimensional convex set has a smooth boundary. We analyze non-generic structures. Fifteen possible classes regarding the…
The Lambert W function gives the solutions of a simple exponential polynomial. The generalized Lambert W function was defined by Mez\"{o} and Baricz, and has found applications in delay differential equations and physics. In this article we…
In this article, we introduce a notion of an exponential matrix, which is a polynomial matrix with exponential properties, and a notion of an equivalence relation of two exponential matrices, and then we initiate to study classifying…
An efficient algorithm is presented for computation of the limit of exp(t A) for t towards infinity where A denotes an intensity matrix of finite dimension.
We study the computability of the operator norm of a matrix with respect to norms induced by linear operators. Our findings reveal that this problem can be solved exactly in polynomial time in certain situations, and we discuss how it can…
We present substantially generalized and improved quantum algorithms over prior work for inhomogeneous linear and nonlinear ordinary differential equations (ODE). Specifically, we show how the norm of the matrix exponential characterizes…
Herein, we present a canonical form for a natural and necessary generalization of the Lambert W function, natural in that it requires minimal mathematical definitions for this generalization, and necessary in that it provides a means of…
We consider quantum group theory on the Hilbert space level. We find all solutions for scalar and general exponential equations for the quantum ``az+b'' group. It turns out that there is a simple formula for all of them involving the…
We use the theory of symmetric functions to enumerate various classes of alternating permutations w of {1,2,...,n}. These classes include the following: (1) both w and w^{-1} are alternating, (2) w has certain special shapes, such as…
The tomographic invertable map of the Wigner function onto the positive probability distribution function is studied. Alternatives to the Schr\"odinger evolution equation and to the energy level equation written for the positive probability…
In this paper, by use of matrix inversions, we establish a general $q$-expansion formula of arbitrary formal power series $F(z)$ with respect to the base $$\left\{z^n\frac{(az:q)_n}{(bz:q)_n}\bigg|n=0,1,2\cdots\right\}.$$ Some concrete…
This is an exposition of some basic mathematical aspects of quantum logic gates. At first we established some general formulas for the case of arbitrary quantum gate A with unique restriction A^2=I. The explicit form of the generators and…
Let $r$ be a nonconstant noncommutative rational function in $m$ variables over an algebraically closed field $K$ of characteristic 0. We show that for $n$ large enough, there exists an $X\in M_n(K)^m$ such that $r(X)$ has $n$ distinct and…
Large-scale eigenvalue problems pose a significant challenge to classical computers. While there are efficient quantum algorithms for unitary or Hermitian matrices, eigenvalue problems for non-normal matrices remain open in quantum…
An equivalence relation in the symmetric group, where is a positive integer has been considered. An algorithm for calculation of the number of the equivalence classes by this relation for arbitrary integer has been described.