Nondigital Implementation of the Arithmetic of Real Numbers

In the framework of a model for quantum computer media, a nondigital implementation of the arithmetic of the real numbers is described. For this model, an elementary storage cell is an ensemble of qubits (quantum bits). It is found that to store an arbitrary real number it is sufficient to use four of these ensembles and the arithmetical operations can be implemented by fixed quantum circuits.     

An eigenvector existence theorem in idempotent analysis

In this paper, we prove an eigenvector existence theorem for linear operators on abstract idempotent spaces in the framework of the algebraic approach. Earlier, an algebraic version of a similar statement was known only for operators in free finite-dimensional semimodules. The corresponding result for compact operators in semimodules of real continuous functions is known in the case of topological semimodules.     

Tensor products of idempotent semimodules. An algebraic approach

We study idempotent analogs of topological tensor products in the sense of A. Grothendieck. The basic concepts and results are expressed in purely algebraic terms. This is one of a series of papers on idempotent functional analysis. Dedicate to S. G. Krein on the occasion of his 80th birthday Translated fromMatermaticheskie Zametki, Vol. 65, No. 4, pp. 572–585, April, 1999.     

Idempotent Functional Analysis: An Algebraic Approach

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 traditional Functional Analysis and its idempotent version is discussed in the spirit of N. Bohr's correspondence principle in quantum theory. We present an algebraic approach to Idempotent Functional Analysis. Basic notions and results are formulated in algebraic terms; the essential point is that the operation of idempotent addition can be defined for arbitrary infinite sets of summands. We study idempotent analogs of the basic principles of linear functional analysis and results on the general form of a linear functional and scalar products in idempotent spaces.     

Double-valued representations of the four-dimensional cubic lattice rotation group

The double-valued representations of the rotation symmetry group of the four-dimensional cubic lattice are described. Their connections with double-valued representations of the three-dimensional cubic lattice rotation group and of the continuous O(3) and O(4) groups are given in detail.     

Kernel theorems and nuclearity in idempotent mathematics. An algebraic approach

In the framework of idempotent mathematics, analogs of the classical kernel theorems of L. Schwartz and A. Grothendieck are studied. Idempotent versions of nuclear spaces (in the sense of A. Grothendieck) are discussed. We use the so-called algebraic approach, which means that the basic concepts and results (including those of “topological” nature) are simulated in purely algebraic terms. Bibliography: 33 titles. In dear memory of F. A. Berezin __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 331, 2006, pp. 60–83.