tropical linear algebra with the łukasiewicz t norm the max ukasiewicz semiring is defined as the unit interval 0 1 equipped with the arithmetics a b max a b and a b 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 ukasiewicz 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 ukasiewicz semiring

x simple image eigencones of tropical matrices abstract we investigate max algebraic tropical one sided systems a 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 if any eigenvector x associated with and belonging to the interval x is the unique solution of the system a y x 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

semigroup of matrices acting on the max plus projective space abstract we investigate the action of semigroups of d d matrices with entries in the max plus semifield on the max plus projective space recall that semigroups generated by one element with projectively bounded image are projectively finite and thus contain idempotent elements in terms of orbits our main result states that the image of a minimal orbit by an idempotent element of the semigroup with minimal rank has at most d elements moreover each idempotent element with minimal rank maps at least one orbit onto a singleton this allows us to deduce the central limit theorem for stochastic recurrent sequences driven by independent random matrices that take countably many values as soon as the semigroup generated by the values contains an element with projectively bounded image

max algebraic attraction cones of nonnegative irreducible matrices abstract it is known that the max algebraic powers ar of a nonnegative irreducible matrix are ultimately periodic this leads to the concept of attraction cone attr a t by which we mean the solution set of a two sided system t a ar x ar t x where r is any integer after the periodicity transient t a and a is the maximum cycle geometric mean of a a question which this paper answers is how to describe attr a t by a concise system of equations without knowing t a this study requires knowledge of certain structures and symmetries of periodic max algebraic powers which are also described we also consider extremals of attraction cones in a special case and address the complexity of computing the coefficients of the system which describes attraction cone

reachability of eigenspaces for interval circulant matrices in max algebra abstract a nonnegative matrix a is said to be strongly robust if its max algebraic eigencone is universally reachable i e if the orbit of any initial vector ends up with a max algebraic eigenvector of a consider the case when the initial vector is restricted to an interval and a can be any matrix from a given interval of nonnegative circulant matrices the main aim of this paper is to classify and characterize the six types of interval robustness in this situation this naturally leads us also to study the max algebraic spectral theory of circulant matrices and the relation of inclusion between attraction cones of circulant matrices in max algebra

generators extremals and bases of max cones abstract max cones are max algebraic analogs of convex cones in the present paper we develop a theory of generating sets and extremals of max cones in r n this theory is based on the observation that extremals are minimal elements of max cones under suitable scalings of vectors we give new proofs of existing results suitably generalizing restating and refining them of these it is important that any set of generators may be partitioned into the set of extremals and the set of redundant elements we include results on properties of open and closed cones on properties of totally dependent sets and on computational bounds for the problem of finding the essentially unique basis of a finitely generated cone

harmonic maps with defects two problems concerning maps with point singularities from a domain c 3 tos2 are solved the first is to determine the minimum energy of when the location and topological degree of the singularities are prescribed in the second problem is the unit ball and g is given on we show that the only cases in whichg x x minimizes the energy isg const org x rx withr a rotation extensions of these problems are also solved e g points are replaced by holes 3 s2 is replaced by n sn 1 or by n pn 1 the latter being appropriate for the theory of liquid crystals

a linear system theoretic view of discrete event processes and its use for performance evaluation in manufacturing a discrete event system is a system whose behavior can be described by means of a set of time consuming activities performed according to a prescribed ordering events correspond to starting or ending some activity an analogy between linear systems and a class of discrete event systems is developed following this analogy such discrete event systems can be viewed as linear in the sense of an appropriate algebra the periodical behavior of closed discrete event systems i e involving a set of repeatedly performed activities can be totally characterized by solving an eigenvalue and eigenvector equation in this algebra this problem is numerically solved by an efficient algorithm which basically consists of finding the shortest paths from one node to all other nodes in a graph the potentiality of this approach for the performance evaluation of flexible manufacturing systems is emphasized the case of a flowshop like production process is analyzed in detail

modifying the power method in max algebra in the max algebra system the eigenequation for an n n irreducible nonnegative matrix a aij is a x a x here a x i maxjaijxj and a is the maximum circuit geometric mean the complexity of the power method given in l elsner p van den driessche linear algebra appl to appear to compute a and x is considered under some assumptions on the critical matrix it is shown that the algorithm may have time complexity o n4 a modified power method based on karp s formula is presented for this new algorithm with no assumptions on the critical matrix a and x can be computed in o n3 time furthermore this algorithm can be used to compute all linearly independent eigenvectors corresponding to a

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

a graphical approach of the spectral theory in the max algebra in this paper the author studies matrices in the max algebra the author introduces a new tool for describing the deterministic spectral behavior of matrices of size 3 spl times 3 it consists of a graphical representation of eigenvectors and domains of attraction in the projective space

elastic energy of liquid crystals in convex polyhedra we consider nematic liquid crystals in a bounded convex polyhedron described by a director field n r subject to tangent boundary conditions we derive lower bounds for the one constant elastic energy in terms of topological invariants for a right rectangular prism and a large class of topologies we derive upper bounds by introducing test configurations constructed from local conformal solutions of the euler lagrange equation the ratio of the upper and lower bounds depends only on the aspect ratios of the prism as the aspect ratios are varied the minimum energy conformal state undergoes a sharp transition from being smooth to having singularities on the edges

elastic energy for reflection symmetric topologies nematic liquid crystals in a polyhedral domain a prototype for bistable displays may be described by a unit vector field subject to tangent boundary conditions here we consider the case of a rectangular prism for configurations with reflection symmetric topologies we derive a new lower bound for the one constant elastic energy for certain topologies called conformal and anticonformal the lower bound agrees with a previous result for the remaining topologies called nonconformal the new bound is an improvement for nonconformal topologies we derive an upper bound which differs from the lower bound by a factor depending only on the aspect ratios of the prism

generalized matrix period in max plus algebra the matrix power sequences and their periodic properties in max plus algebra are studied the notion of generalized periodicity is introduced for which both periodicity and linear periodicity are special cases every matrix over a max plus algebra is shown to be almost generally periodic

x robustness of interval circulant matrices in fuzzy algebra abstract fuzzy algebra is an algebraic structure in which classical addition and multiplication are replaced by and where a b max a b a b min a b a matrix is x robust if its eigenspace is achieved starting at each vector from a given interval vector x the possible and universal x robustness of interval matrices over fuzzy algebra are defined polynomial algorithms for checking the possible and universal x robustness of interval circulant matrices are described

on matrix powers in max algebra abstract let a a ij r n n n 1 n and da be the digraph n i j a ij the matrix a is called irreducible if da is strongly connected and strongly irreducible if every max algebraic power of a is irreducible a is called robust if for every x with at least one finite component a k x is an eigenvector of a for some natural number k we study the eigenvalue eigenvector problem for powers of irreducible matrices this enables us to characterise robust irreducible matrices in particular robust strongly irreducible matrices are described in terms of eigenspaces of matrix powers

reachability of eigenspaces for interval matrices in max min algebra abstract in max min algebra the standard pair of operations plus and times is substituted by pair of operations and where a b max a b a b min a b a square matrix a is x robust if its eigenspace is reached starting at each vector of a given interval vector x x x x x various versions of the reachability for an interval vector x and an interval matrix a a a a a depending on the used quantifiers and their order are studied the universal possible tolerance and weak tolerance a robustness of x and x robustness of a over max min algebra are defined and polynomial algorithms for its checking in the case of circulant matrices are described

orbits in max min algebra abstract properties of orbits in max min algebra are described mainly the properties of periodic orbits an o n 3 algorithm computing the period of a periodic orbit is presented as a consequence an o n 3 log n algorithm computing the period of arbitrary orbit is obtained as the pre periodic part of the orbit has length at most n 1 2 1

linear matrix period in max plus algebra abstract linear periodicity of matrices in max plus algebra is studied it is proved that the linear period of a matrix can be computed in o n 3 time if the digraph of the matrix is strongly connected the general problem of deciding whether a given matrix is almost linear periodic is shown to be np complete

subeigenvectors and supereigenvectors of fuzzy matrices abstract a fuzzy algebra is a triple b where b is a nonempty bounded linearly ordered set and a b max a b a b min a b for a b b a vector x is said to be a eigenvector of a square matrix a if a x x for some b the aim of the paper is to solve subeigenproblem and supereigenproblem for some b that is to find a solution x of a x x and a x x x is called subeigenvector and supereigenvector respectively the problems are related to and motivated by the similar problems of tropical linear algebra in this paper the properties of subeigenvectors and supereigenvectors are described and the values associated with subeigenvectors and supereigenvectors are characterized as a consequence efficient algorithms for checking all equivalent conditions are introduced

spectrum transformations of the laurent biorthogonal polynomials i q appel polynomials we study the simplest closure conditions of chains of spectral transformations of the laurent biorthogonal polynomials lbp it is shown that the 1 periodic q closure condition leads to the q appel lbp which are a special case of the lbp introduced by pastro

the robustness of interval fuzzy matrices abstract robust interval matrices over max min algebra fuzzy matrices are studied and equivalent conditions for interval fuzzy matrices to be robust are presented polynomial algorithms for checking the necessary and sufficient conditions for robustness of interval fuzzy matrices are introduced in addition more efficient algorithms for verifying the robustness of interval circulant matrices are described

on the ultimate behavior of the sequence of consecutive powers of a matrix in the max plus algebra abstract we study the sequence of consecutive powers of a matrix in the max plus algebra which has maximum and addition as its basic operations if the matrix is irreducible then it is well known that the ultimate behavior of the sequence is cyclic for reducible matrices the ultimate behavior is more complex but it is also cyclic in nature we will give a detailed characterization of the rates and periods of the ultimate behavior for a general matrix

max balancing weighted directed graphs and matrix scaling a weighted directed graph g is a triple v a g where v a is a directed graph and g is an arbitrary real valued function defined on the arc set a let g be a strongly connected simple weighted directed graph we say that g is max balanced if for every nontrivial subset of the vertices w the maximum weight over arcs leaving w equals the maximum weight over arcs entering w we show that there exists a up to an additive constant unique potential pv for v v such that v a gp is max balanced where gap pu ga pv for a u v a we describe an o v 2 a algorithm for computing p using an algorithm for computing the maximum cycle mean of g finally we apply our principal result to the similarity scaling of nonnegative matrices

the minkowski theorem for max plus convex sets abstract we establish the following max plus analogue of minkowski s theorem any point of a compact max plus convex subset of r n can be written as the max plus convex combination of at most n 1 of the extreme points of this subset we establish related results for closed max plus cones and closed unbounded max plus convex sets in particular we show that a closed max plus convex set can be decomposed as a max plus sum of its recession cone and of the max plus convex hull of its extreme points

on visualization scaling subeigenvectors and kleene stars in max algebra abstract the purpose of this paper is to investigate the interplay arising between max algebra convexity and scaling problems the latter which have been studied in nonnegative matrix theory are strongly related to max algebra one problem is that of strict visualization scaling defined as for a given nonnegative matrix a a diagonal matrix x such that all elements of x 1 ax are less than or equal to the maximum cycle geometric mean of a with strict inequality for the entries which do not lie on critical cycles in this paper such scalings are described by means of the max algebraic subeigenvectors and kleene stars of nonnegative matrices as well as by some concepts of convex geometry

max algebraic powers of irreducible matrices in the periodic regime an application of cyclic classes abstract in max algebra it is well known that the sequence of max algebraic powers a k with a an irreducible square matrix becomes periodic after a finite transient time t a and the ultimate period is equal to the cyclicity of the critical graph of a in this connection we study computational complexity of the following problems 1 for a given k compute a periodic power a r with r k mod and r t a 2 for a given x find the ultimate period of a l x we show that both problems can be solved by matrix squaring in o n 3 log n operations the main idea is to apply an appropriate diagonal similarity scaling a x 1 ax called visualization scaling and to study the role of cyclic classes of the critical graph

orbits and critical components of matrices in max min algebra abstract a speed up of a known o n 3 algorithm computing the period of a periodic orbit in max min algebra is presented if the critical components or the transitive closure a of the transition matrix a are known the computational complexity of the algorithm is o n 2 this is achieved by using only those coordinates of the orbit that are related to the critical components on the other hand no critical component can be omitted as the critical components are pairwise disjoint the new formula is helpful also in solving the converse problem generating an orbit with a prescribed period this is demonstrated by examples all parts of the paper are connected with the fact that the periodic regime of an orbit is encoded in its critical coordinates we show that the non critical coordinates of the state vector after n 1 steps can be ignored how to generate a known periodic regime by using a small number of coordinates of the state vector and that the difference between the defect of an orbit and the quasidefect of its critical coordinates is small the final part deals with the situation when the matrix is reduced after the orbit has reached its periodic regime

ae and ea robustness of interval circulant matrices in max min algebra abstract max min algebra is defined as a linearly ordered set with two binary operations classical addition and multiplication are replaced by maximum and minimum respectively a square matrix is robust if its eigenspace is achieved starting at arbitrary vector and it is x robust if its eigenspace is achieved starting at each vector from a given interval vector x the ea and ae robustness and x robustness of interval circulant matrices over max min algebra are defined polynomial algorithms for checking these types of robustness and x robustness are given

on the power method in max algebra abstract the eigenvalue problem for an irreducible nonnegative matrix a a ij in the max algebra system is a otimes x lambda x where a otimes x i mathop max j a ij x j and lambda turns out to be the maximum circuit geometric mean mu a a power method algorithm is given to compute mu a and eigenvector x the algorithm is developed by using results on the convergence of max powers of a which are proved using nonnegative matrix theory in contrast to an algorithm developed in 4 this new method works for any irreducible nonnegative a and calculates eigenvectors in a simpler and more efficient way some asymptotic formulas relating mu a the spectral radius and norms are also given

the robustness of interval matrices in max plus algebra abstract robust interval matrices in max plus algebra are studied and equivalent conditions for interval matrices to be robust are presented the necessary and sufficient conditions for robustness of interval matrices are introduced in addition efficient algorithms for verifying the robustness of interval circulant and interval zero one matrices are described