Submathematics and Tropical Mathematics

2021 ◽  
Vol 109 (1-2) ◽  
pp. 241-246
Author(s):  
V. P. Maslov
Keyword(s):  
Tropical Mathematics

scholarly journals Layered tropical mathematics

2014 ◽  
Vol 416 ◽  
pp. 200-273 ◽  
Author(s):  
Zur Izhakian ◽  
Manfred Knebusch ◽  
Louis Rowen
Keyword(s):  
Tropical Mathematics

scholarly journals Idempotent Analog of the Legendre Transformation and lts Application in Digital Processing of Signals

2020 ◽  
pp. 96-101
Author(s):  
M.V. Kurkina ◽  
S.P. Semenov ◽  
V.V. Slavsky ◽  
O.V. Samarina ◽  
O.A. Petuhova ◽  
...  
Keyword(s):  
Convex Functions ◽  
Riemannian Geometry ◽  
Duality Theory ◽  
Digital Processing ◽  
Riemannian Metric ◽  
Theoretical Physics ◽  
Vector Spaces ◽  
Legendre Transformation ◽  
Conformally Flat ◽  
Tropical Mathematics

In recent years, a new area of mathematics — idempotent or “tropical” mathematics — has been intensively developed within the framework of the Sofus Lee international center, which is reflected in the works of V.P. Maslov, G.L. Litvinov, and A.N. Sobolevsky. The Legendre transformation plays an important role in theoretical physics, classical and statistical mechanics, and thermodynamics. In mathematics and its applications, the Legendre transformation is based on the concept of duality of vector spaces and duality theory for convex functions and subsets of a vector space. The purpose of this paper is to go beyond linear vector spaces using similar notions of duality in conformally flat Riemannian geometry and in idempotent algebra.An abstract idempotent analog of the Legendre transformation is constructed in a way similar to the polar transformation of the conformally flat Riemannian metric introduced in the works of E.D. Rodionov and V.V. Slavsky. Its capabilities for digital processing of signals and images are being investigated

Tropical Mathematics

2009 ◽  
Vol 82 (3) ◽  
pp. 163-173 ◽  
Author(s):  
David Speyer ◽  
Bernd Sturmfels
Keyword(s):  
Tropical Mathematics

scholarly journals Approximate factorization of positive matrices by using methods of tropical optimization

2020 ◽  
Vol 16 (4) ◽  
pp. 357-374
Author(s):  
Nikolai K. Krivulin ◽  
◽  
Elizaveta Yu. Romanova ◽  
Keyword(s):  
Approximation Problem ◽  
Logarithmic Scale ◽  
Box Constraints ◽  
Approximate Factorization ◽  
Tropical Mathematics ◽  
Tropical Optimization ◽  
The Matrix ◽  
Rank One ◽  
Positive Matrices ◽  
Factor Decomposition

The problem of rank-one factorization of positive matrices with missing (unspecified) entries is considered where a matrix is approximated by a product of column and row vectors that are subject to box constraints. The problem is reduced to the constrained approximation of the matrix, using the Chebyshev metric in logarithmic scale, by a matrix of unit rank. Furthermore, the approximation problem is formulated in terms of tropical mathematics that deals with the theory and applications of algebraic systems with idempotent addition. By using methods of tropical optimization, direct analytical solutions to the problem are derived for the case of an arbitrary positive matrix and for the case when the matrix does not contain columns (rows) with all entries missing. The results obtained allow one to find the vectors of the factor decomposition by using expressions in a parametric form which is ready for further analysis and immediate calculation. In conclusion, an example of approximate rank-one factorization of a matrix with missing entries is provided.

scholarly journals IDEALS OF POLYNOMIAL SEMIRINGS IN TROPICAL MATHEMATICS

2012 ◽  
Vol 12 (02) ◽  
pp. 1250143
Author(s):  
ZUR IZHAKIAN ◽  
LOUIS ROWEN
Keyword(s):  
Prime Ideal ◽  
Principal Ideal ◽  
Prime Ideals ◽  
Hilbert Basis ◽  
Basis Theorem ◽  
Tropical Mathematics

We describe the ideals, especially the prime ideals, of semirings of polynomials over layered domains, and in particular over supertropical domains. Since there are so many of them, special attention is paid to the ideals arising from layered varieties, for which we prove that every prime ideal is a consequence of finitely many binomials. We also obtain layered tropical versions of the classical Principal Ideal Theorem and Hilbert Basis Theorem.

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