In the Wigner-Moyal approach to quantum mechanics, we show that Moyal’s starting point, the characteristic function M(τ, θ) = ∫ ψ∗(x)ei(τp̂+θx̂)ψ(x)dx, is essentially the primitive idempotent used by von Neumann in his classic paper “Die Eindeutigkeit der Schrödingerschen Operatoren”. This paper provides the original proof of the Stone-von Neumann equation. Thus the mathematical structure Moyal d...

Abstract Basic matrices are defined which provide unique building blocks for the class of normal include classes unitary and Hermitian matrices. Unique builders quantum logic gates hence derived as a is represented by, or said to be, matrix. An efficient algorithm expressing an idempotent sum rank 1 idempotents with increasing initial zeros derived. This used derive form mixed A number (further...

This paper is devoted to Idempotent Functional Analysis, which is an “abstract” version of Idempotent Analysis developed by V. P. Maslov and his collaborators. We give a brief survey of the basic ideas of Idempotent Analysis. The correspondence between concepts and theorems of the traditional Functional Analysis and its idempotent version is discussed in the spirit of N. Bohr’s correspondence p...

In graph-based clustering, a relevant affinity matrix is crucial for good results. Double stochasticity of the has been shown to be an important condition, both in theory and practice. this paper, we emphasize idempotency as another key condition. fact, theorem from Sinkhorn, R. (1968) allows us exhibit bijective relationship between set doubly stochastic idempotent matrices order n (modulo per...

We consider a multidimensional optimization problem in the framework of tropical mathematics. The problem is formulated to minimize a nonlinear objective function that is defined on vectors in a finitedimensional semimodule over an idempotent semifield and calculated by means of multiplicative conjugate transposition. We offer two complete direct solutions to the problem. The first solution con...

We consider a multidimensional extremal problem formulated in terms of tropical mathematics. The problem is to minimize a nonlinear objective function, which is defined on a finite-dimensional semimodule over an idempotent semifield, subject to linear inequality constraints. An efficient solution approach is developed which reduces the problem to that of solving a linear inequality with an exte...

1. Introduction. Idempotent Mathematics is based on replacing the usual arithmetic operations by a new set of basic operations (e.g., such as maximum or minimum), that is on replacing numerical fields by idempotent semirings and semifields. Typical (and the most common) examples are given by the so-called (max, +) algebra R max and (min, +) algebra R min. Let R be the field of real numbers. The...

Algebraic theories are called Morita equivalent provided that the corresponding varieties of algebras are equivalent. Generalizing Dukarm’s result from one-sorted theories to many-sorted ones, we prove that all theories Morita equivalent to an S-sorted theory T are obtained as idempotent modifications of T . This is analogous to the classical result of Morita that all rings Morita equivalent to...

begin{abstract} The relations between the spectrum of the matrix $Q+R$ and the spectra of the matrices $(gamma + delta)Q+(alpha + beta)R-QR-RQ$, $QR-RQ$, $alpha beta R-QRQ$, $alpha RQR-(QR)^{2}$, and $beta R-QR$ have been given on condition that the matrix $Q+R$ is diagonalizable, where $Q$, $R$ are ${alpha, beta}$-quadratic matrix and ${gamma, delta}$-quadratic matrix, respectively, of ord...