Grigory L. Litvinov

Bio: Grigory L. Litvinov is an academic researcher from National Research University – Higher School of Economics. The author has contributed to research in topics: Idempotence & Idempotent matrix. The author has an hindex of 6, co-authored 9 publications receiving 273 citations. Previous affiliations of Grigory L. Litvinov include Independent University of Moscow.

Idempotent functional analysis: An algebraic approach

Abstract: 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.

Linear functionals on idempotent spaces : an algebraic approach

Abstract: 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.

Idempotent functional analysis: an algebraic approach

Abstract: In this paper we consider Idempotent Functional Analysis, an `abstract' version of Idempotent Analysis developed by V. P. Maslov and his collaborators. We give a review of the basic ideas of Idempotent Analysis. The correspondence between concepts and theorems of the traditional Functional Analysis and its idempotent version is discussed; this correspondence is similar to N. Bohr's correspondence principle in quantum theory. We present an algebraical approach to Idempotent Functional Analysis. Basic notions and results are formulated in algebraical 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 main theorems of linear functional analysis and results concerning the general form of a linear functional and scalar products in idempotent spaces.

Idempotent/Tropical Analysis, the Hamilton–Jacobi and Bellman Equations

Abstract: Tropical and idempotent analysis with their relations to the Hamilton–Jacobi and matrix Bellman equations are discussed. Some dequantization procedures are important in tropical and idempotent mathematics. In particular, the Hamilton–Jacobi–Bellman equation is treated as a result of the Maslov dequantization applied to the Schrodinger equation. This leads to a linearity of the Hamilton–Jacobi–Bellman equation over tropical algebras. The correspondence principle and the superposition principle of idempotent mathematics are formulated and examined. The matrix Bellman equation and its applications to optimization problems on graphs are discussed. Universal algorithms for numerical algorithms in idempotent mathematics are investigated. In particular, an idempotent version of interval analysis is briefly discussed.

Kernel theorems and nuclearity in idempotent mathematics. An algebraic approach

Abstract: 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.

Introduction to Tropical Geometry

Abstract: Tropical islands Building blocks Tropical varieties Tropical rain forest Tropical garden Toric connections Bibliography Index

Lie Algebras and Lie Groups

Abstract: In this crucial lecture we introduce the definition of the Lie algebra associated to a Lie group and its relation to that group. All three sections are logically necessary for what follows; §8.1 is essential. We use here a little more manifold theory: specifically, the differential of a map of manifolds is used in a fundamental way in §8.1, the notion of the tangent vector to an arc in a manifold is used in §8.2 and §8.3, and the notion of a vector field is introduced in an auxiliary capacity in §8.3. The Campbell-Hausdorff formula is introduced only to establish the First and Second Principles of §8.1 below; if you are willing to take those on faith the formula (and exercises dealing with it) can be skimmed. Exercises 8.27–8.29 give alternative descriptions of the Lie algebra associated to a Lie group, but can be skipped for now.

Maslov dequantization, idempotent and tropical mathematics: A brief introduction

Abstract: This paper is a brief introduction to idempotent and tropical mathematics. Tropical mathematics can be treated as the result of the so-called Maslov dequantization of the traditional mathematics over numerical fields as the Planck constant ℏ tends to zero taking imaginary values. Bibliography: 187 titles.

Linear independence over tropical semirings and beyond

Abstract: We investigate different notions of linear independence and of ma- trix rank that are relevant for max-plus or tropical semirings. The factor rank and tropical rank have already received attention, we compare them with the ranks defined in terms of signed tropical determinants or arising from a no- tion of linear independence introduced by Gondran and Minoux. To do this, we revisit the symmetrization of the max-plus algebra, establishing properties of linear spaces, linear systems, and matrices over the symmetrized max-plus algebra. In parallel we develop some general technique to prove combinatorial and polynomial identities for matrices over semirings that we illustrate by a number of examples.