Searcharxiv⌕ Search

arXiv subjects

Sergei Sergeev

Publications and source records attributed to Sergei Sergeev.

At least 37 records · Page 2Linked to original sources
arXiv2016

Tropical convexity over max-min semiring

This is a survey on an analogue of tropical convexity developed over the max-min semiring, starting with the descriptions of max-min segments, semispaces, hyperplanes and an account of separation and non-separation results based on semispaces. There are some new results. In particular, we give new "colorful" extensions of the max-min Caratheodory theorem. In the end of the paper, we list some consequences of the topological Radon and Tverberg theorems (like Helly and Centerpoint theorems), valid over a more general class of max-T semirings, where multiplication is a triangular norm.

math.MG↗
arXiv2016

X-simple image eigencones of tropical matrices

We investigate max-algebraic (tropical) one-sided systems $A\otimes x=b$ where $b$ is an eigenvector and $x$ lies in an interval $X$. A matrix $A$ is said to have $X$-simple image eigencone associated with an eigenvalue $λ$, if any eigenvector $x$ associated with $λ$ and belonging to the interval $X$ is the unique solution of the system $A\otimes y=λx$ in $X$. We characterize matrices with $X$-simple image eigencone geometrically and combinatorially, and for some special cases, derive criteria that can be efficiently checked in practice.

math.RA↗
arXiv2014

Tropical linear algebra with the Lukasiewicz T-norm

The max-Lukasiewicz semiring is defined as the unit interval [0,1] equipped with the arithmetics "a+b"=max(a,b) and "ab"=max(0,a+b-1). Linear algebra over this semiring can be developed in the usual way. We observe that any problem of the max-Lukasiewicz linear algebra can be equivalently formulated as a problem of the tropical (max-plus) linear algebra. Based on this equivalence, we develop a theory of the matrix powers and the eigenproblem over the max-Lukasiewicz semiring.

math.CO↗
arXiv2014

On the dimension of max-min convex sets

We introduce a notion of dimension of max-min convex sets, following the approach of tropical convexity. We introduce a max-min analogue of the tropical rank of a matrix and show that it is equal to the dimension of the associated polytope. We describe the relation between this rank and the notion of strong regularity in max-min algebra, which is traditionally defined in terms of unique solvability of linear systems and trapezoidal property.

math.MG↗
arXiv2014

Characterizing matrices with $X$-simple image eigenspace in max-min semiring

A matrix $A$ is said to have $X$-simple image eigenspace if any eigenvector $x$ belonging to the interval $X=\{x\colon \underline{x}\leq x\leq\overline{x}\}$ is the unique solution of the system $A\otimes y=x$ in $X$. The main result of this paper is a combinatorial characterization of such matrices in the linear algebra over max-min (fuzzy) semiring. The characterized property is related to and motivated by the general development of tropical linear algebra and interval analysis, as well as the notions of simple image set and weak robustness (or weak stability) that have been studied in max-min and max-plus algebras.

math.RA↗
arXiv2013

On sets of eigenvalues of matrices with prescribed row sums and prescribed graph

Motivated by a work of Boros, Brualdi, Crama and Hoffman, we consider the sets of (i) possible Perron roots of nonnegative matrices with prescribed row sums and associated graph, and (ii) possible eigenvalues of complex matrices with prescribed associated graph and row sums of the moduli of their entries. To characterize the set of Perron roots or possible eigenvalues of matrices in these classes we introduce, following an idea of Al'pin, Elsner and van den Driessche, the concept of row uniform matrix, which is a nonnegative matrix where all nonzero entries in every row are equal. Furthermore, we completely characterize the sets of possible Perron roots of the class of nonnegative matrices and the set of possible eigenvalues of the class of complex matrices under study. Extending known results to the reducible case, we derive new sharp bounds on the set of eigenvalues or Perron roots of matrices when the only information available is the graph of the matrix and the row sums of the moduli of its entries. In the last section of the paper a new constructive proof of the Camion-Hoffman theorem is given.

math.RA↗
arXiv2013

Characterization of tropical hemispaces by (P,R)-decompositions

We consider tropical hemispaces, defined as tropically convex sets whose complements are also tropically convex, and tropical semispaces, defined as maximal tropically convex sets not containing a given point. We introduce the concept of $(P,R)$-decomposition. This yields (to our knowledge) a new kind of representation of tropically convex sets extending the classical idea of representing convex sets by means of extreme points and rays. We characterize tropical hemispaces as tropically convex sets that admit a (P,R)-decomposition of certain kind. In this characterization, with each tropical hemispace we associate a matrix with coefficients in the completed tropical semifield, satisfying an extended rank-one condition. Our proof techniques are based on homogenization (lifting a convex set to a cone), and the relation between tropical hemispaces and semispaces.

math.MG↗
arXiv2013

Weak CSR expansions and transience bounds in max-plus algebra

This paper aims to unify and extend existing techniques for deriving upper bounds on the transient of max-plus matrix powers. To this aim, we introduce the concept of weak CSR expansions: A^t=CS^tR + B^t. We observe that most of the known bounds (implicitly) take the maximum of (i) a bound for the weak CSR expansion to hold, which does not depend on the values of the entries of the matrix but only on its pattern, and (ii) a bound for the CS^tR term to dominate. To improve and analyze (i), we consider various cycle replacement techniques and show that some of the known bounds for indices and exponents of digraphs apply here. We also show how to make use of various parameters of digraphs. To improve and analyze (ii), we introduce three different kinds of weak CSR expansions (named after Nachtigall, Hartman-Arguelles, and Cycle Threshold). As a result, we obtain a collection of bounds, in general incomparable to one another, but better than the bounds found in the literature.

math.CO↗
arXiv2013

On the max-algebraic core of a nonnegative matrix

The max-algebraic core of a nonnegative matrix is the intersection of column spans of all max-algebraic matrix powers. Here we investigate the action of a matrix on its core. Being closely related to ultimate periodicity of matrix powers, this study leads us to new modifications and geometric characterizations of robust, orbit periodic and weakly stable matrices.

math.RA↗
arXiv2013

Two cores of a nonnegative matrix

We prove that the sequence of eigencones (i.e., cones of nonnegative eigenvectors) of positive powers A^k of a nonnegative square matrix A is periodic both in max algebra and in nonnegative linear algebra. Using an argument of Pullman, we also show that the Minkowski sum of the eigencones of powers of A is equal to the core of A defined as the intersection of nonnegative column spans of matrix powers, also in max algebra. Based on this, we describe the set of extremal rays of the core. The spectral theory of matrix powers and the theory of matrix core is developed in max algebra and in nonnegative linear algebra simultaneously wherever possible, in order to unify and compare both versions of the same theory.

math.RA↗
arXiv2012

The Analytic Hierarchy Process, Max Algebra and Multi-objective Optimisation

The Analytic Hierarchy Process (AHP) is widely used for decision making involving multiple criteria. Elsner and van den Driessche introduced a max-algebraic approach to the single criterion AHP. We extend this to the multi-criteria AHP, by considering multi-objective generalisations of the single objective optimisation problem solved in these earlier papers. We relate the existence of globally optimal solutions to the commutativity properties of the associated matrices; we relate min-max optimal solutions to the generalised spectral radius; and we prove that Pareto optimal solutions are guaranteed to exist.

math.RA↗
arXiv2012

An application of the max-plus spectral theory to an ultradiscrete analogue of the Lax pair

We study the ultradiscrete analogue of Lax pair proposed by Willox et al. This "pair" is a max-plus linear system comprising four equations. Our starting point is to treat this system as a combination of two max-plus eigenproblems, with two additional constraints. Though infinite-dimensional, these two eigenproblems can be treated by means of the "standard" max-plus spectral theory. In particular, any solution to the system can be described as a max-linear combination of fundamental eigenvectors associated with each soliton. We then describe the operation of undressing using pairs of fundamental eigenvectors. We also study the solvability of the complete system of four equations as proposed by Willox et al.

math.RA↗
arXiv2012

Z-matrix equations in max algebra, nonnegative linear algebra and other semirings

We study the max-algebraic analogue of equations involving Z-matrices and M-matrices, with an outlook to a more general algebraic setting. We show that these equations can be solved using the Frobenius trace down method in a way similar to that in non-negative linear algebra, characterizing the solvability in terms of supports and access relations. We give a description of the solution set as combination of the least solution and the eigenspace of the matrix, and provide a general algebraic setting in which this result holds.

math.RA↗