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## On the max-min and tropical CP-rank conjectures

Shitov Y.

In press

We give a tropical proof of a recent result of Mohindru on the CP-rank conjecture over min-max semiring.

Shitov Y., Linear and Multilinear Algebra 2016 Vol. 64 No. 2 P. 219-220

We give a tropical proof of a recent result of Mohindru on the CP-rank conjecture over min-max semiring. ...

Added: December 16, 2015

Shitov Y., International Journal of Algebra and Computation 2014 Vol. 24 No. 8 P. 1183-1189

We present a number of new results on the multiplicative structure of univariate polynomials over the Boolean and tropical semirings.We answer the question asked by Kim and Roush in 2005 by proving that almost all Boolean polynomials with nonzero constant term are irreducible.We also give a lower bound for the number of reducible polynomials, and ...

Added: January 8, 2015

Providence : American Mathematical Society, 2014

This volume contains the proceedings of the International Workshop on Tropical and Idempotent Mathematics, held at the Independent University of Moscow, Russia, from August 26-31, 2012. The main purpose of the conference was to bring together and unite researchers and specialists in various areas of tropical and idempotent mathematics and applications. This volume contains articles ...

Added: February 1, 2015

Shitov Y., Journal of Mathematical Sciences 2013 Vol. 193 No. 5 P. 802-808

We consider the rank functions of matrices over semirings, functions that generalize the classical notion of the rank of a matrix over a field. We study semirings over which the factor and Gondran–Minoux ranks of any matrix coincide. It is shown that every semiring satisfying that condition is a subsemiring of a field. We provide ...

Added: January 20, 2014

Shitov Y., Journal of Combinatorial Theory, Series A 2014 Vol. 126 P. 166-176

The smallest integer k for which the elements of a real matrix A can be expressed as A_ij = min B_it + C_tj with B an m-by-k matrix and C an k-by-n matrix is called the combinatorial rank of A. This notion was introduced by Barvinok in 1993, and he posed a problem on the ...

Added: May 24, 2014

Shitov Y., Advances in Mathematics 2014 Vol. 254 P. 138-156

The tropical arithmetic operations on R are defined by a⊕b=min{a, b} and a⊗b=a+b. Let A be a tropical matrix and k a positive integer, the problem of Tropical Matrix Factorization (TMF) asks whether there exist tropical matrices B∈R^{m×k} and C∈R^{k×n} satisfying B⊗C=A. We show that no algorithm for TMF is likely to work in polynomial ...

Added: January 10, 2014

Shitov Y., / Cornell University. Series math "arxiv.org". 2014. No. 1406.2601.

We construct a nontrivial identity which holds in the semigroup of tropical 3-by-3 matrices. ...

Added: March 14, 2015

Shitov Y., St Petersburg Mathematical Journal 2014 Vol. 26 No. 2 P. 216-228

The tropical arithmetic operations on R are defined as (a,b) -> min{a,b} and (a,b) -> a+b. We are interested in the concept of a semimodule, which is rather ill-behaved in tropical mathematics. In our paper we study the semimodules S in R^n having topological dimension two, and we show that any such S has always ...

Added: May 24, 2014

СПб. : ВВМ, 2014

Тропическая (идемпотентная) математика представляет собой быстро развивающуюся область прикладной математики, которая связана с изучением полумодулей над полукольцами с идемпотентным сложением и имеет много приложений, включая задачи экономики и управления. Использование языка тропической математики позволяет некоторые нелинейные в обычном смысле задачи превращать в линейные, что в ряде случаев упрощает процедуру решения этих задач, а ткже облегчает ...

Added: December 29, 2014

Shitov Y., SIAM Review 2017 Vol. 59 No. 4 P. 794-800

Using elementary linear algebra, we develop a technique that leads to solutions of two widely known problems on nonnegative matrices. First, we give a short proof of the result by Vavasis stating that the nonnegative rank of a matrix is NP-hard to compute. This proof is essentially contained in the paper by Jiang and Ravikumar, ...

Added: November 9, 2017

Shitov Y., Доклады Академии наук 2015 Т. 465 № 3 С. 287-290

Обсуждаются результаты, связанные с расширенными представлениями выпуклых многоугольников: оказывается, в частности, что для задания выпуклого семиугольника с точностью до линейной проекции достаточно шести линейных неравенств. ...

Added: December 29, 2015

Matveenko V. D., В кн. : Модели и методы тропической математики в прикладных задачах экономики и управления. Вып. 2.: СПб. : ВВМ, 2014. Гл. 1. С. 4-23.

Рассматриваются различные варианты CES-функций, получившие распространение в экономике. С точки зрения тропической математики CES-функция без весов интересна тем, что при изменении параметра, связывает идемпотентные операции max и min со стандартной динейно-алгебраической операцией +. Мы рассматриваем CES-функции с позиций тропической математики на основе введения операции обобщенного сложения. Исследуются свойства монотонности, взаимосвязь различных видов CES-функции, а также ...

Added: December 29, 2014

Polishchuk A., Arnold Mathematical Journal 2019 Vol. 5 No. 1 P. 23-35

We present an improved construction of the fundamental matrix factorization in the FJRW-theory given in Polishchuk and Vaintrob (J Reine Angew Math 714:1—22, 2016). The revised construction is coordinate-free and works for a possibly nonabelian finite group of symmetries. One of the new ingrediants is the category of dg-matrix factorizations over a dg-scheme. ...

Added: September 4, 2019

Shitov Y., Proceedings of the American Mathematical Society 2014 Vol. 142 P. 15-19

We show that neither the Barvinok rank nor the Kapranov rank of a tropical matrix M can be defined in terms of the regular mixed subdivision produced by M. This answers a question asked by Develin, Santos and Sturmfels. ...

Added: October 5, 2013

Polishchuk A., Positselski L., Transactions of the American Mathematical Society 2012 Vol. 364 No. 10 P. 5311-5368

We define and study the Hochschild (co)homology of the second kind (known also as the Borel-Moore Hochschild homology and the compactly supported Hochschild cohomology) for curved DG categories. An isomorphism between the Hochschild (co)homology of the second kind of a CDG-category B and the same of the DG category C of right CDG-modules over B, ...

Added: June 27, 2012

Guterman A., Shitov Y., Linear Algebra and its Applications 2012 Vol. 437 No. 7 P. 1793-1811

We introduce the notion of the tropical matrix pattern, which provides a powerful tool to investigate tropical matrices. The above
approach is then illustrated by the application to the study of the properties of the Gondran–Minoux rank function. Our main result states that up to a multiplication of matrix rows by non-zero constants the Gondran–Minoux independence ...

Added: November 9, 2012

Pavlov A., Mathematische Zeitschrift 2021 No. 297 P. 223-254

We show that for maximal Cohen–Macaulay modules over the homogeneous coordinate ring of a smooth Calabi–Yau varieties X, the computation of Betti numbers can be reduced to computations of dimensions of certain HomHom spaces in the bounded derived category Db(X). In the simplest case of a smooth elliptic curve E embedded in P2 as a smooth cubic, we get explicit values for Betti ...

Added: October 31, 2020

Chebotarev A., Teretenkov A. E., Applied Mathematics and Computation 2014 Vol. 234 P. 380-384

We describe a simple implementation of the Takagi factorization of symmetric matrices $A = U\Lambda U^T$ with unitary $U$ and diagonal $\Lambda$ in terms of the square root of an auxiliary unitary matrix and the singular value decomposition of $A$. The method is based on an algebraically exact expression. For parameterized family $A_\epsilon = A ...

Added: June 4, 2014

Matveenko V. D., , in : Contemporary Mathematics. Vol. 616: Tropical and idempotent mathematics and applications.: Providence : American Mathematical Society, 2014. Ch. 11. P. 211-220.

Properties of increasing positively homogeneous functions are studied; in particular, their representations by use of tropical inner products with coefficients chosen from tropical support sets are described. An application to a model of economic complementarityand weak links is developed. It is shown that weak links do not necessary bound total factor productivity from below but ...

Added: February 1, 2015

Efimov A., International Mathematics Research Notices 2018 No. 12 P. 3834-3869

In this article, we will show that for a smooth quasi-projective variety X over complex numbers, and regular function W on X, the periodic cyclic homology of the DG category of matrix factorizations of W on X is identified (under the Riemann–Hilbert correspondence) with the vanishing cohomology, with the monodromy twisted by a sign. Also, ...

Added: October 14, 2018

Shitov Y., Linear Algebra and its Applications 2016 Vol. 499 P. 26-30

Let χ(A) denote the characteristic polynomial of a matrix A over a field; a standard result of linear algebra states that χ(A−1) is the reciprocal polynomial of χ(A). More formally, the condition χn(A)χk(A−1) = χn−k(A) holds for any invertible n × n matrix A over a field, where χi(A) denotes the coefficient of λn−i in the characteristic polynomial ...

Added: March 9, 2016

Katzarkov L. V., Deliu D., Favero D. et al., Advances in Mathematics 2016 Vol. 295 P. 195-249

Building upon ideas of Eisenbud, Buchweitz, Positselski, and others, we introduce the notion of a factorization category. We then develop some essential tools for working with factorization categories, including constructions of resolutions of factorizations from resolutions of their components and derived functors. Using these resolutions, we lift fully-faithfulness and equivalence statements from derived categories of ...

Added: October 23, 2017

191574970, Functional Analysis and Its Applications 2006 Vol. 40 No. 2 P. 81-90

It is well known that every module M over the algebra ℒ(X) of operators on a finite-dimensional space X can be represented as the tensor product of X by some vector space E, M ≅ = E ⊗ X. We generalize this assertion to the case of topological modules by proving that if X is a stereotype space with the stereotype approximation property, then for each stereotype module M over the ...

Added: September 23, 2016

Losev A. S., Slizovskiy S., JETP Letters 2010 Vol. 91 P. 620-624

Added: February 27, 2013