Results 1  10
of
26
Enumerative tropical algebraic geometry in R 2
 J. Amer. Math. Soc
, 2005
"... Abstract. The paper establishes a formula for enumeration of curves of arbitrary genus in toric surfaces. It turns out that such curves can be counted by means of certain lattice paths in the Newton polygon. The formula was announced earlier in [18]. The result is established with the help of the so ..."
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Cited by 72 (3 self)
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Abstract. The paper establishes a formula for enumeration of curves of arbitrary genus in toric surfaces. It turns out that such curves can be counted by means of certain lattice paths in the Newton polygon. The formula was announced earlier in [18]. The result is established with the help of the socalled tropical algebraic geometry. This geometry allows to replace complex toric varieties with the real space R n and holomorphic curves with certain piecewiselinear graphs there. 1.
The Maslov Dequantization, Idempotent and Tropical Mathematics: a Very Brief Introduction
, 2005
"... ..."
Enumerative tropical algebraic geometry
 in R2 , preprint 2003, math.AG/0312530
"... Abstract. The paper establishes a formula for enumeration of curves of arbitrary genus in toric surfaces. It turns out that such curves can be counted by means of certain lattice paths in the Newton polygon. The formula was announced earlier in [17]. The result is established with the help of the so ..."
Abstract

Cited by 25 (0 self)
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Abstract. The paper establishes a formula for enumeration of curves of arbitrary genus in toric surfaces. It turns out that such curves can be counted by means of certain lattice paths in the Newton polygon. The formula was announced earlier in [17]. The result is established with the help of the socalled tropical algebraic geometry. This geometry allows to replace complex toric varieties with the real space R n and holomorphic curves with certain piecewiselinear graphs there. 1.
Idempotent Interval Analysis and Optimization Problems
 RELIABLE COMPUTING
, 2001
"... 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. ..."
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Cited by 12 (1 self)
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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 NPhard in the traditional interval linear algebra; in the idempotent case it is polynomial. A generalization to the case of positive semirings is outlined.
Tensor products of idempotent semimodules. An algebraic approach
 Mathematical Notes
, 1999
"... Abstract. 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. Key words: idempotent functional analysis, idempotent semiri ..."
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Cited by 7 (5 self)
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Abstract. 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. Key words: idempotent functional analysis, idempotent semiring, idempotent semimodule, tensor product, polylinear mapping, nuclear operator. Dedicated to S.G. Krein on the occasion of his 80th birthday
Tropical algebraic geometry
, 2005
"... The purpose of the lectures is to make a brief introduction to tropical algebraic geometry and to present several important applications of tropical geometry in enumerative geometry. ..."
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Cited by 7 (0 self)
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The purpose of the lectures is to make a brief introduction to tropical algebraic geometry and to present several important applications of tropical geometry in enumerative geometry.
Universal numerical algorithms and their software implementation
 Programming and Computer Software
"... 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 univer ..."
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Cited by 4 (2 self)
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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.
DNAbased computation times
 In: Proc. Tenth International Meeting on DNA Computing
, 2004
"... Speed of computation and power consumption are the two main parameters of conventional computing devices implemented in microelectronic circuits. As performance of such devices approaches physical limits, new computing paradigms are emerging. Two paradigms receiving great attention are quantum and D ..."
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Cited by 4 (2 self)
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Speed of computation and power consumption are the two main parameters of conventional computing devices implemented in microelectronic circuits. As performance of such devices approaches physical limits, new computing paradigms are emerging. Two paradigms receiving great attention are quantum and DNAbased molecular computing. This paper focuses on DNAbased computing. This paradigm can be abstracted to growth models where computational elements called tiles are selfassembled one by one, subject to some simple hierarchical rules, to fill a given template encoding a Boolean formula. While DNAbased computational devices are known to be extremely energy efficient, little is known concerning the fundamental question of computation times. In particular, given a function, we study the time required to determine its value for a given input. In the simplest instance, the analysis has interesting connections with interacting particle systems and variational problems. 1
Maxplus definite matrix closures and their eigenspaces
, 2006
"... In this paper we introduce the definite closure operation for matrices with finite permanent, reveal inner structures of definite eigenspaces, and establish some facts about Hilbert distances between these inner structures and the boundary of the definite eigenspace. ..."
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Cited by 3 (2 self)
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In this paper we introduce the definite closure operation for matrices with finite permanent, reveal inner structures of definite eigenspaces, and establish some facts about Hilbert distances between these inner structures and the boundary of the definite eigenspace.
Idempotent Mathematics and Interval Analysis
, 1998
"... A brief introduction into Idempotent Mathematics and an idempotent version of Interval Analysis are presented. Some applications are discussed. ..."
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Cited by 2 (0 self)
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A brief introduction into Idempotent Mathematics and an idempotent version of Interval Analysis are presented. Some applications are discussed.