The Maslov dequantization, idempotent and tropical mathematics: a very brief introduction
This paper is a very brief introduction to idempotent mathematics and related topics.
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arXiv subjects
Grigori Litvinov
Publications and source records attributed to Grigori Litvinov.
Exact interval solutions to the discrete Bellman equation and polynomial complexity of problems in interval idempotent linear algebra
In this note we construct a solution of a matrix interval linear equation of the form X=AX+B (the discrete stationary Bellman equation) over partially ordered semirings, including the semiring of nonnegative real numbers and all idempotent semirings. We discuss also the computational complexity of problems in interval idempotent linear algebra. In the traditional Interval Analysis problems of this kind are generally NP-hard. In the note we consider matrix equations over positive semirings; in this case the computational complexity of the problem is polynomial. Idempotent and other positive semirings arise naturally in optimization problems. Many of these problems turn out to be linear over appropriate idempotent semirings. In this case, the system of equations X=AX+B appears to be a natural analog of a usual linear system in the traditional linear algebra over fields. B. A. Carre showed that many of the well-known algorithms of discrete optimization are analogous to standard algorithms of the traditional computational linear algebra.
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Idempotent (Asymptotic) Mathematics and the Representation Theory
A brief survey of some basic ideas of the so-called Idempotent Mathematics is presented; an "idempotent" version of the representation theory is discussed. The Idempotent Mathematics can be treated as a result of a dequantization of the traditional mathematics over numerical fields in the limit of the vanishing "imaginary Planck constant"; there is a correspondence, in the spirit of N. Bohr's correspondence principle, between constructions and results in traditional mathematics over the fields of real and complex numbers and similar constructions and results over idempotent semirings. In particular, there is an "idempotent" version of the theory of linear representations of groups. Some basic concepts and results of the "idempotent" representation theory are presented. In the framework of this theory the well-known Legendre transform can be treated as an idempotent version of the traditional Fourier transform. Some unexpected versions of the Engel theorem are given.
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Nuclear semimodules and kernel theorems in idempotent analysis. An algebraic approach
In this note we describe conditions under which, in idempotent functional analysis, 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. Our results provide a general description of a class of subsemimodules of the semimodule of all bounded functions with values in the Max-Plus algebra where some kind of kernel theorem holds, thus addressing an open problem posed by J. Gunawardena. Previously, some theorems on integral representations were obtained for a number of specific semimodules consisting of continuous or bounded functions taking values mostly in the Max-Plus algebra. In this work, a rather general case of semimodules over boundedly complete idempotent semirings is considered.
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Nondigital implementation of the arithmetic of real numbers by means of quantum computer media
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 arithmetic operations can be implemented by fixed quantum circuits.
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Universal numerical algorithms and their software implementation
The concept of a universal algorithm is discussed. Examples of this kind of algorithms are presented. Software implementations of such algorithms in C++ type languages are discussed together with means that provide for computations with an arbitrary accuracy. Particular emphasis is placed on universal algorithms of linear algebra over semirings.
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Idempotent mathematics and interval analysis
A brief introduction into Idempotent Mathematics and an idempotent version of Interval Analysis are presented. Some applications are discussed.
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Approximate rational arithmetics and arbitrary precision computations
We describe an approximate rational arithmetic with round-off errors (both absolute and relative) controlled by the user. The rounding procedure is based on the continued fraction expansion of real numbers. Results of computer experiments are given in order to compare efficiency and accuracy of different types of approximate arithmetics and rounding procedures.
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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 simulated on the algebraic level. This is one of a series of papers on idempotent functional analysis.
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Idempotent interval analysis and optimization problems
Many problems in optimization theory are strongly nonlinear in the traditional sense but possess a hidden linear structure over suitable idempotent semirings. After an overview of `Idempotent Mathematics' with an emphasis on matrix theory, interval analysis over idempotent semirings is developed. The theory is applied to construction of exact interval solutions to the interval discrete stationary Bellman equation. Solution of an interval system is typically NP-hard in the traditional interval linear algebra; in the idempotent case it is polynomial. A generalization to the case of positive semirings is outlined.
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A unifying approach to software and hardware design for scientific calculations and idempotent mathematics
A unifying approach to software and hardware design generated by ideas of Idempotent Mathematics is discussed. The so-called idempotent correspondence principle for algorithms, programs and hardware units is described. A software project based on this approach is presented.
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Approximate construction of rational approximations and the effect of error autocorrection. Applications
Several construction methods for rational approximations to functions of one real variable are described in the present paper; the computational results that characterize the comparative accuracy of these methods are presented; an effect of error autocorrection is considered. This effect occurs in efficient methods of rational approximation (e.g., Pade approximations, linear and nonlinear Pade-Chebyshev approximations) where very significant errors in the coefficients do not affect the accuracy of the approximation. The matter of import is that the errors in the numerator and the denominator of a fractional rational approximant compensate each other. This effect is related to the fact that the errors in the coefficients of a rational approximant are not distributed in an arbitrary way but form the coefficients of a new approximant to the approximated function. Understanding of the error autocorrection mechanism allows to decrease this error by varying the approximation procedure depending on the form of the approximant. Some applications are described in the paper. In particular, a method of implementation of basic calculations on decimal computers that uses the technique of rational approximations is described in the Appendix. To a considerable extent the paper is a survey and the exposition is as elementary as possible.
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Correspondence principle for idempotent calculus and some computer applications
This paper is devoted to heuristic aspects of the so-called idempotent calculus. There is a correspondence between important, useful and interesting constructions and results over the field of real (or complex) numbers and similar constructions and results over idempotent semirings in the spirit of N. Bohr's correspondence principle in Quantum Mechanics. Some problems nonlinear in the traditional sense (for example, the Bellman equation and its generalizations) turn out to be linear over a suitable semiring; this linearity considerably simplifies the explicit construction of solutions. The theory is well advanced and includes, in particular, new integration theory, new linear algebra, spectral theory and functional analysis. It has a wide range of applications. Besides a survey of the subject, in this paper the correspondence principle is used to develop an approach to object-oriented software and hardware design for algorithms of idempotent calculus.
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Linear functionals on idempotent spaces: an algebraic approach
In this paper, we present an algebraic approach to idempotent functional analysis, which is an abstract version of idempotent analysis. The basic concepts and results are expressed in purely algebraic terms. We consider idempotent versions of certain basic results of linear functional analysis, including the theorem on the general form of a linear functional and the Hahn-Banach and Riesz-Fischer theorems.
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