In this note we describe conditions under which, in idempotent functional analysis (see [1–3]), linear operators have integral representations in terms of idempotent integral of V. P. Maslov. We define the notion of nuclear idempotent semimodule and describe idempotent analogs of the classical kernel theorems of L. Schwartz and A. Grothendieck (see, e.g., [4–6]). In [11], for the idempotent sem...

design methods for paraunitary matrices from complete orthogonal sets of idempotents and related matrix structures are presented. these include techniques for designing non-separable multidimensional paraunitary matrices. properties of the structures are obtained and proofs given. paraunitary matrices play a central role in signal processing, in particular in the areas of filterbanks and wavelets.

Assume Mn is the n-dimensional permutation module for the symmetric group Sn, and let M n be its k-fold tensor power. The partition algebra Pk(n) maps surjectively onto the centralizer algebra EndSn(M ⊗k n ) for all k, n ∈ Z≥1 and isomorphically when n ≥ 2k. We describe the image of the surjection Φk,n : Pk(n) → EndSn(M ⊗k n ) explicitly in terms of the orbit basis of Pk(n) and show that when 2...

We show that the class of locally finite varieties omitting type 1 has the following properties. This class (1) is definable by an idempotent, linear, strong Mal’cev condition in a language with one 4-ary function symbol. (2) is not definable by an idempotent, linear, strong Mal’cev condition in a language with only one function symbol of arity strictly less than 4. (3) is definable by an idemp...

An approach to schedule development in project management is developed within the framework of idempotent algebra. The approach offers a way to represent precedence relationships among activities in projects as linear vector equations in terms of an idempotent semiring. As a result, many issues in project scheduling reduce to solving computational problems in the idempotent algebra setting, inc...

Linear vector equations and inequalities are considered defined in terms of idempotent mathematics. To solve the equations, we apply an approach that is based on the analysis of distances between vectors in idempotent vector spaces. The approach reduces the solution of the equation to that of an optimization problem in the idempotent algebra setting. Based on the approach, existence and uniquen...

Both unconstrained and constrained minimax single facility location problems are considered in multidimensional space with Chebyshev distance. A new solution approach is proposed within the framework of idempotent algebra to reduce the problems to solving linear vector equations and minimizing functionals defined on some idempotent semimodule. The approach offers a solution in a closed form tha...

This is a survey paper on applications of mathematics of semirings to numerical analysis and computing. Concepts of universal algorithm and generic program are discussed. Relations between these concepts and mathematics of semirings are examined. A very brief introduction to mathematics of semirings (including idempotent and tropical mathematics) is presented. Concrete applications to optimizat...