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

max algebra and pairwise comparison matrices abstract this paper is a continuation of our 2004 paper max algebra and pairwise comparison matrices in which the max eigenvector of a symmetrically reciprocal matrix was used to approximate such a matrix by a transitive matrix this approximation was based on minimizing the maximal relative error in a later paper by dahl a different error measure was used and led to a slightly different approximating transitive matrix here some geometric properties of this approximation problem are discussed these lead among other results to a new characterization of a max eigenvector of an irreducible nonnegative matrix the case of toeplitz matrices is discussed in detail and an application to music theory that uses toeplitz symmetrically reciprocal matrices is given

limit eigenvalues of nonnegative matrices abstract in this paper we give a simple closed formula for a certain limit eigenvalue of a nonnegative matrix we bound the spectral radius of a nonnegative matrix in terms of this limit eigenvalue

occupation time large deviations of two dimensional symmetric simple exclusion process we prove a large deviations principle for the occupation time of a site in the two dimensional symmetric simple exclusion process the decay probability rate is of order t logt and the rate function is given by 2 sin 1 2 1 sin 1 2 1 2 the proof relies on a large deviations principle for the polar empirical measure which contains an interesting log scale spatial average a contraction principle permits us to deduce the occupation time large deviations from the large deviations for the polar empirical measure

barabanov norms lipschitz continuity and monotonicity for the max algebraic joint spectral radius abstract we present several results describing the interplay between the max algebraic joint spectral radius jsr for compact sets of matrices and suitably defined matrix norms in particular we extend a classical result for the conventional algebra showing that the max algebraic jsr can be described in terms of induced norms of the matrices in the set we also show that for a set generating an irreducible semigroup in a cone theoretic sense a monotone barabanov norm always exists this fact is then used to show that the max algebraic jsr is locally lipschitz continuous on the space of compact irreducible sets of matrices with respect to the hausdorff distance we then prove that the max algebraic jsr is locally hoelder continuous on the space of compact sets of nonnegative matrices finally we prove a strict monotonicity property for the max algebraic jsr that echoes a fact for the classical jsr the single matrix characterisation of the max algebraic jsr plays a vital role in our proofs

functional inequalities for max eigenvalues of nonnegative matrices let pn denote the set of all n n nonnegative matrices for a function f r m r and matrices ak aijk i j 1n k 1 m define f a1 am f aij1 aijm ij 1n for each a pn we denote its spectral radius by a and its max eigenvalue by a in a previous paper all functions f which satisfy f a1 am f a1 am n n a1 am pn and some functions which satisfy f a1 am f a1 am n n a1 am pn were characterized here for an interval i in r we characterize those functions f satisfying f a1 am f a1 am n n a1 am inn as well as the functions satisfying f 0 0 and f a1 am f a1 am n n a1 am inn

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

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 2004 2010 10 11 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

polynomial convolutions in max plus algebra abstract recently in a work that grew out of their exploration of interlacing polynomials marcus spielman and srivastava 21 and marcus 20 studied certain combinatorial polynomial convolutions these convolutions preserve real rootedness and capture expectations of characteristic polynomials of unitarily invariant random matrices thus providing a link to free probability we explore analogues of these types of convolutions in the setting of max plus algebra in this setting the max permanent replaces the determinant the maximum is the analogue of the expected value and real rootedness is replaced by full canonical form our results resemble those of marcus et al however in contrast to the classical setting we obtain an exact and simple description of all roots of the convolution of p x and q x in terms of the roots of p x and q x

bounds for the perron root using max eigenvalues abstract using the techniques of max algebra a new proof of al pin s lower and upper bounds for the perron root of a nonnegative matrix is given the bounds depend on the row sums of the matrix and its directed graph if the matrix has zero main diagonal entries then these bounds may improve the classical row sum bounds this is illustrated by a generalized tournament matrix

an exact general remeshing scheme applied to physically conservative voxelization we present an exact general remeshing scheme to compute analytic integrals of polynomial functions over the intersections between convex polyhedral cells of old and new meshes in physics applications this allows one to ensure global mass momentum and energy conservation while applying higher order polynomial interpolation we elaborate on applications of our algorithm arising in the analysis of cosmological n body data computer graphics and continuum mechanics problems we focus on the particular case of remeshing tetrahedral cells onto a cartesian grid such that the volume integral of the polynomial density function given on the input mesh is guaranteed to equal the corresponding integral over the output mesh we refer to this as physically conservative voxelization at the core of our method is an algorithm for intersecting two convex polyhedra by successively clipping one against the faces of the other this algorithm is an implementation of the ideas presented abstractly by sugihara 48 who suggests using the planar graph representations of convex polyhedra to ensure topological consistency of the output this makes our implementation robust to geometric degeneracy in the input we employ a simplicial decomposition to calculate moment integrals up to quadratic order over the resulting intersection domain we also address practical issues arising in a software implementation including numerical stability in geometric calculations management of cancellation errors and extension to two dimensions in a comparison to recent work we show substantial performance gains we provide a c implementation intended to be a fast accurate and robust tool for geometric calculations on polyhedral mesh elements

a generalized eigenvalue problem in the max algebra abstract we consider the generalized eigenvalue problem a x b x x 0 x 0 where a and b are entrywise nonnegative n n matrices and the max product satisfies a x i max m 1 n a im x m the case b i has been studied by several authors and for irreducible e g positive a there is exactly one eigenvalue in the above max sense the generalized problem is different and for example neither existence nor uniqueness of eigenvalues is guaranteed even for 2 2 positive matrices a and b which can be analysed by graphical methods for general n existence and uniqueness are discussed by different methods including degree theory

permanents max algebra and optimal assignment abstract the max algebra consists of the set of real numbers together with equipped with two binary operations maximization and addition for a square matrix its permanent over the max algebra is simply the maximum diagonal sum of the matrix several results are proved for the permanent over the max algebra which are analogs of the corresponding results for the permanent of a nonnegative matrix these include alexandroff inequality bregman s inequality cauchy binet formula and a bebianotype expansion

a max version of the generalized spectral radius theorem abstract let be a bounded set of n n nonnegative matrices in max algebra in this paper we propose the notions of the max algebra version of the generalized spectral radius of and the max algebra version of the joint spectral radius of the max algebra version of the generalized spectral radius theorem is established we propose the relationship between the generalized spectral radius of in the sense of daubechies and lagarias and its max algebra version moreover a generalization of elsner and van den driessche s lemma is presented as well

the influence of the marked reduced graph of a nonnegative matrix on the jordan form and on related properties a survey abstract for a nonnegative matrix p we discuss the relation of its marked reduced graph to that part of the jordan form that is associated with the perron frobenius root to the nonnegativity of the eigenvectors and generalized eigenvectors to the nonnegativity of solutions of linear equations and to the asymptotic growth of powers of the matrix results are often stated in terms of m matrices and standard results on irreducible matrices are assumed we give examples to illustrate the theorems surveyed

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

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

on the max version of the generalized spectral radius theorem abstract let be a bounded set of n n non negative matrices recently the max algebra version of the generalized spectral radius of was introduced we show that lim t i t 1 t where denotes the generalized spectral radius and t the hadamard power of this provides a description of that uses no max terminology as an application we give a short proof of the max version of the generalized spectral radius theorem

a max version of the perron frobenius theorem abstract if a an n n nonnegative irreducible matrix then there exists a 0 and a positive vector x such that maxjaijxj a xi i 1 2 n furthermore a is the maximum geometric mean of a circuit in the weighted directed graph corresponding to a this theorem which we refer to as the max version of the perron frobenius theorem is well known in the context of matrices over the max algebra and also in the context of matrix scalings in the present work which is partly expository we bring out the intimate connection between this result and the perron frobenius theory we present several proofs of the result some of which use the perron frobenius theorem structure of max eigenvalues and max eigenvectors is described possible ways to unify the perron frobenius theorem and its max version are indicated some inequalities for a are proved

some monotonicity properties of schur powers of matrices and related inequalities abstract a series of inequalities are developed relating the spectral radius a b of the schur product a b of two nonnegative matrices a and b with those of a a and b b yielding a b a a b b 1 2 as a corollary it is proved that the spectral radius of the schur powers r a r a r a a a r factors satisfies 1 r log r is decreasing while 1 r 1 log r is increasing the latter provided a is a stochastic matrix the entropy of a finite stationary markov chain is identified with d r dr r 1 a number of majorization comparisons for the spectral radius of schur powers is given

the spectra of irreducible matrices over completed idempotent semifields motivated by some spectral results in the characterization of concept lattices we investigate the spectra of reducible matrices over complete idempotent semifields in the framework of naturally ordered semirings or dioids we find non null eigenvectors for every non null element in the semifield and conclude that the notion of spectrum has to be refined to encompass that of the incomplete semifield case so as to include only those eigenvalues with eigenvectors that have finite coordinates considering special sets of eigenvectors brings out finite complete lattices in the picture and we contend that such structure may be more important than standard eigenspaces for matrices over completed idempotent semifields

the markov chain tree theorem in commutative semirings and the state reduction algorithm in commutative semifields we extend the markov chain tree theorem to general commutative semirings and we generalize the state reduction algorithm to general commutative semifields this leads to a new universal algorithm whose prototype is the state reduction algorithm which computes the markov chain tree vector of a stochastic matrix

generalized spectral radius and its max algebra version abstract let c n n and r n n be bounded subsets and let and denote the generalized spectral radius of and the max algebra version of the generalized spectral radius of respectively we apply a single matrix description of to give a new elementary and straightforward proof of the berger wang formula in max algebra and consequently a new short proof of the original berger wang formula in the case of bounded subsets of n n non negative matrices we also obtain a new description of in terms of the schur hadamard product and prove new trace and max trace descriptions of and in particular we show that limsup m sup a m tr a 1 m limsup m sup a m tr a 1 m and limsup m sup b m tr b 1 m limsup m sup b m tr b 1 m where tr a max i 1 n a ii and b b ij