fiedler ptak scaling in max algebra abstract this is essentially the text of my talk on fiedler ptak scaling in max algebra delivered in the invited minisymposium in honor of miroslav fiedler at the 17th ilas conference in braunschweig germany

centre manifold reduction and the stuart landau equation for fluid motions the centre manifold reduction to derive the stuart landau equation is examined a double expansion in terms of the fourier series and linear eigenfunctions is introduced in hydrodynamic equations a centre manifold reduction scheme is then applied to reduce the resultant system of ordinary differential equations to the stuart landau equation through a formal expansion in linear eigenfunctions the latter equation is shown to be equivalent with the one derived by the method of multiple scales numerical coefficients involved in the quintic stuart landau equation are evaluated for plane poiseuille flow convection in a vertical slot and rayleigh benard convection in all the cases the coefficients converge as a dimension of the ode system increases and approach the numerical values obtained by the method of multiple scales

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

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

bifurcating solutions at the onset of convection in the begard problem for two fluids abstract we consider two fluids with different thermal and mechanical properties arranged in parallel layers between two infinite horizontal plates the bottom plate is kept at a higher temperature than the top plate in the unbounded directions we impose periodic boundary conditions with the periods chosen such that the problem has hexagonal symmetry in contrast to the benard problem for one fluid the onset of convection in the two fluid benard problem considering here can be oscillatory the oscillations are essentially due to the competition between the destabilizing temperature gradient and a stable interface between the two fluids the hexagonal symmetry of the problem causes a sixfold degeneracy of the critical eigenvalues on the abstract level hopf bifurcations with this type of symmetry induced degeneracy were investigated by roberts swift and wagner they showed that there are eleven qualitatively different types of bifurcating solutions and they identified the parameters which determine the stability of these solutions in this paper we apply their results to the two fluid benard problem since the eigenfunctions at criticality are not explicitly known in this problem we shall use a combination of analysis and numerical computation in the examples we study we find that most branches are subcritical and none are stable near the bifurcation point

the 1 2 hopf steady state mode interaction in three dimensional magnetoconvection abstract the first analysis of a properly three dimensional mode interaction between steady and oscillatory forms of convection with different preferred wavenumbers is presented by varying the fluid parameters for boussinesq magnetoconvection we locate a point where the conduction state is unstable to both steady and oscillatory motion simultaneously we then construct and analyse the normal form the complex transition between steady and oscillatory convection near onset can be explained this extends and completes the work of clune and knobloch pattern selection in three dimensional magnetoconvection physica d 74 1994 151 176 selecting the most marginal wavenumbers in the problem ensures that the analysis is relevant to the behaviour which would be observed in an unbounded plane layer the symmetries of the resulting d 4 t 2 equivariant bifurcation problem play a large role in determining the bifurcation structure and explain the appearance of interesting phenomena such as drifting solutions a m rucklidge m silber bifurcations of periodic orbits with spatio temporal symmetries nonlinearity 11 1998 1435 1455 we also find new phenomena in the normal form for a hopf bifurcation with d 4 t 2 symmetry m silber e knobloch hopf bifurcation on a square lattice nonlinearity 4 1991 1063 1106 the introduction of weakly non boussinesq effects leads to qualitative changes in the dynamics near onset different convection planforms are stabilised and chaotic heteroclinic cycling behaviour is observed

tropical planar networks abstract we show that every tropical totally positive matrix can be uniquely represented as the transfer matrix of a canonical totally connected weighted planar network we deduce a uniqueness theorem for the factorization of a tropical totally positive in terms of elementary jacobi matrices

normal forms for driven surface waves boundary conditions symmetry and genericity abstract the parametric excitation of standing waves in a vertically oscillated fluid can be studied by analyzing the period doubling bifurcation in the associated stroboscopic map when the fluid container has symmetry the stroboscopic map will exhibit this symmetry and this in turn significantly affects the structure of the bifurcation problem for both square and circular containers the normal forms appropriate to the reduced maps for the critical mode amplitudes are discussed the experiments of simonelli and gollub with square containers strongly suggest that the effective symmetry of the problem is larger than the obvious geometric symmetry of the boundaries for the ideal fluid model of benjamin and ursell with neumann boundary conditions nbc one can show that the problem does indeed have a larger symmetry because it embeds in a model on a larger domain with periodic boundary conditions pbc the original nbc are then realized as a symmetry constraint on the solutions of this extended model the additional symmetries introduced in this way lead to a large number of effects that would otherwise be non generic including critical eigenspaces carrying reducible representations of and reduced maps on these eigenspaces that have symmetries larger than these extra symmetries relate in phase and out of phase mixed modes and lead to new branches to pure modes the case of circular geometry is also discussed as it does not allow this extension to pbc and thus provides an instructive contrast with the subtleties found for square geometry

max plus definite matrix closures and their eigenspaces 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

symmetries and pattern selection in rayleigh benard convection abstract this paper describes the process of pattern selection between rolls and hexagons in rayleigh benard convection with reflectional symmetry in the horizontal midplane this symmetry is a consequence of the boussinesq approximation provided the boundary conditions are the same on the top and bottom plates all possible local bifurcation diagrams assuming certain non degeneracy conditions are found using only group theory the results are therefore applicable to other systems with the same symmetries rolls hexagons or a new solution regular triangles can be stable depending on the physical system rolls are stable in ordinary rayleigh benard convection the results are compared to those of buzano and golubitsky 1 without the midplane reflection symmetry the bifurcation behavior of the two cases is quite different and a connection between them is established by considering the effects of breaking the reflectional symmetry finally the relevant experimental results are described

the 1 2 spatial resonance on a hexagonal lattice in two layered rayleigh benard problems this paper investigates pattern formation resulting from the interaction between steady modes with wavenumbers in the ratio 1 2 a two layered rayleigh benard problem with infinite extent is examined two layers are separated by a non deformable and thin dividing plate this physical set up is known to provide an exact 1 2 resonance between critical modes restricting the wavevectors of interacting modes on a hexagonal lattice we derive 12 dimensional amplitude equations up to cubic order among steady solutions of the equations the stable ones are hexagons and mixed hexagons travelling waves corresponding to relative equilibria are also found to be stable depending on parameter values

limit theorems for products of positive random matrices let s be the set of q times q matrices with positive entries such that each column and each row contains a strictly positive element and denote by s circ the subset of these matrices all entries of which are strictly positive consider a random ergodic sequence x n n geq1 in s the aim of this paper is to describe the asymptotic behavior of the random products x n x n ldots x 1 n geq 1 under the main hypothesis p bigcup n geq 1 x n in s circ 0 we first study the behavior in direction of row and column vectors of x n then adding a moment condition we prove a law of large numbers for the entries and lengths of these vectors and also for the spectral radius of x n under the mixing hypotheses that are usual in the case of sums of real random variables we get a central limit theorem for the previous quantities the variance of the gaussian limit law is strictly positive except when x n n geq 1 is tight this tightness property is fully studied when the x n n geq 1 are independent

computational geometry an introduction from the reviews this book offers a coherent treatment at the graduate textbook level of the field that has come to be known in the last decade or so as computational geometry the book is well organized and lucidly written a timely contribution by two founders of the field it clearly demonstrates that computational geometry in the plane is now a fairly well understood branch of computer science and mathematics it also points the way to the solution of the more challenging problems in dimensions higher than two

bifurcation analysis of the eckhaus instability the eckhaus instability 1 is the only purely two dimensional instability occurring in convection 2 taylor couette flow 3 liquid crystals 4 and other pattern forming systems despite its widespread occurrence the eckhaus instability has not been analyzed in bifurcation theoretic terms it can be studied analytically by averaging over the depth 5 leading to the classic one dimensional ginzburg landau equation frac partial a partial t mu a frac partial 2 a partial x 2 left a right 2 a 1

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

sil nikov saddle node interaction near a codimension 2 bifurcation laser with injected signal abstract we describe a sil nikov saddle node interaction in the proximity of a hopf saddle node bifurcation both from numerical and theoretical viewpoints inspired by numerical observations on a three dimensional model of a laser with injected signal we describe the interaction near the codimension 2 point using a return map controlled by the parameters of the bifurcation the theoretical predictions compare well with numerical experiments finally we discuss the reasons for the existence of the sil nikov saddle node bifurcation in the laser with injected signal and correspondingly the characteristics of other systems that might display this bifurcation

identities in upper triangular tropical matrix semigroups and the bicyclic monoid abstract we establish necessary and sufficient conditions for a semigroup identity to hold in the monoid of n n upper triangular tropical matrices in terms of equivalence of certain tropical polynomials this leads to an algorithm for checking whether such an identity holds in time polynomial in the length of the identity and size of the alphabet it also allows us to answer a question of izhakian and margolis by showing that the identities which hold in the monoid of 2 2 upper triangular tropical matrices are exactly the same as those which hold in the bicyclic monoid our results extend to a broader class of chain structured tropical matrix semigroups we exhibit a faithful representation of the free monogenic inverse semigroup within such a semigroup which leads also to a representation by 3 3 upper triangular tropical matrices

complex dynamics in the 1 3 spatial resonance abstract the interaction between two steady state bifurcations with spatial wave numbers in the ratio 1 3 is considered periodic boundary conditions are assumed the resulting o 2 equivariant normal form truncated at third order exhibits a number of global bifurcations that may result in complex dynamics the origin of this behavior is elucidated with the help of careful numerical simulations and analysis of appropriate return maps the results generalize to other 1 n resonances

on the spectrum in max algebra abstract we give new proofs of several fundamental results of spectral theory in max algebra this includes the description of the spectrum in max algebra of a given non negative matrix via local spectral radii the spectral theorem and the spectral mapping theorem in max algebra the latter result is also generalized to the setting of power series in max algebra by applying certain continuity properties of the spectrum in max algebra our methods enable us to obtain some related results for the usual spectrum of complex matrices and distinguished spectrum for non negative matrices

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

extremals of the supereigenvector cone in max algebra a combinatorial description abstract we give a combinatorial description of extremal generators of the supereigenvector cone x a x x in max algebra

the ultimate rank of tropical matrices a tropical matrix is a matrix defined over the max plus semiring for such matrices there exist several non coinciding notions of rank the row rank the column rank the schein barvinok rank the kapranov rank or the tropical rank among others in the present paper we show that there exists a natural notion of ultimate rank for the powers of a tropical matrix which does not depend on the underlying notion of rank furthermore we provide a simple formula for the ultimate rank of a matrix which can therefore be computed in polynomial time then we turn our attention to finitely generated semigroups of matrices for which our notion of ultimate rank is generalized naturally we provide both combinatorial and geometric characterizations of semigroups having maximal ultimate rank as a consequence we obtain a polynomial algorithm to decide if the ultimate rank of a finitely generated semigroup is maximal

tropicalization of facets of polytopes it is known that any tropical polytope is the image under the valuation map of ordinary polytopes over the puiseux series field the latter polytopes are called lifts of the tropical polytope we prove that any pure tropical polytope is the intersection of the tropical half spaces given by the images under the valuation map of the facet defining half spaces of a certain lift we construct this lift explicitly taking into account geometric properties of the given polytope moreover when the generators of the tropical polytope are in general position we prove that the above property is satisfied for any lift this solves a conjecture of develin and yu

pure dimension and projectivity of tropical polytopes abstract we study how geometric properties of tropical convex sets and polytopes which are of interest in many application areas manifest themselves in their algebraic structure as modules over the tropical semiring our main results establish a close connection between pure dimension of tropical convex sets and projectivity in the sense of ring theory these results lead to a geometric understanding of idempotency for tropical matrices as well as their direct interest our results suggest that there is substantial scope to apply ideas and techniques from abstract algebra in particular ring theory in tropical geometry

on tropical supereigenvectors abstract the task of finding tropical eigenvectors and subeigenvectors that is non trivial solutions to a x x and a x x in the max plus algebra has been studied by many authors since the 1960s in contrast the task of finding supereigenvectors that is solutions to a x x has attracted attention only recently we present a number of properties of supereigenvectors focusing on a complete characterization of the values of associated with supereigenvectors and in particular finite supereigenvectors the proof of the main statement is constructive and enables us to find a non trivial subspace of finite supereigenvectors we also present an overview of key related results on eigenvectors and subeigenvectors

on common eigenvectors for semigroups of matrices in tropical and traditional linear algebra we prove the existence of a common eigenvector for commutative nilpotent and quasinilpotent semigroups of matrices with complex or real nonnegative entries both in the conventional and tropical linear algebra

dynamic response of a rotating beam subjected to an accelerating distributed surface force abstract a method is given by which the response of a rotating or non rotating timoshenko beam can be determined subjected to an accelerating fixed direction distributed surface force the beam model includes the gyroscopically induced displacement transverse to the direction of the load the solution for pinned supports is set up using multi integral transforms and the inversion is expressed in terms of convulution integrals these are numerically integrated for a uniformly distributed load having an exponentially varying velocity function results are presented for the displacement under the load s center as a function of position and for the displacement of every point on the beam at an instant in time comparisons are made between the beam response to a constant velocity load and its response to a load which accelerates to the same velocity

the whitney embedding theorem for tropical torsion modules classification of tropical modules abstract we prove here a tropical version of the well known whitney embedding theorem 32 stating that a smooth connected m dimensional compact differential manifold can be embedded into r 2 m 1 the tropical version of this theorem states that a tropical torsion module with m generators can always be embedded into the free tropical module r p where p equals to 2 for m 2 and 3 p m m 1 otherwise is the number of rows supporting the torsion when the generators are given by the independent columns of a matrix of size n m as a corollary we get that tropical m dimensional torsion modules are classified by a m 1 m m 1 1 parameter family

duality between invariant spaces for max plus linear discrete event systems we extend the notions of conditioned and controlled invariant spaces to linear dynamical systems over the max plus or tropical semiring we establish a duality theorem relating both notions which we use to construct dynamic observers these are useful in situations in which some of the system coefficients may vary within certain intervals the results are illustrated by an application to a manufacturing system

the non transverse shil nikov hopf bifurcation uncoupling of homoclinic orbits and homoclinic tangencies abstract it is known that in a neighbourhood of a codimension two shil nikov hopf bifurcation primary periodic orbits lie on a single wiggly curve in period parameter space with accumulation points at parameter values of a pair of homoclinic tangencies to a periodic orbit in contrast it has recently been shown by hirschberg and laing that primary periodic orbits lie on an infinity of isolas in a neighbourhood of certain degenerate homoclinic tangency to a periodic orbit this paper analyses the codimension three bifurcation caused by a non transverse i e degenerately parametrically unfolded shil nikov hopf bifurcation which contains nearby dynamics akin to both degeneracies two cases are classified as being downward pointing or upward pointing depending on whether the variation of a third parameter causes either the annihilation of a locus of saddle focus homoclinic orbits to equilibria or the uncoupling of this locus from the locus of hopf bifurcations we undertake a heuristic analysis of the unfolding showing that in both cases it contains codimension two non transversal homoclinic orbits to equilibria and non transversal homoclinic tangencies to periodic orbits unfolding the former non transverse orbit is shown to cause two wiggly curves to coalesce and leave finitely many isolas of periodic orbits unfolding the latter causes two wiggly curves to coalesce into first infinitely many and then finitely many isolas asymptotic expressions are given for the accumulation of two types of isola forming bifurcations the implications of z 2 equivariance on the unfolding is discussed finally numerical evidence is presented for both upward and downward pointing non transverse shil nikov hopf bifurcations occuring in a model of an autonomous non linear electronic circuit both with respect to a z 2 equivariant and a z 2 non equivariant equilibrium numerical computation of curves of periodic orbits homoclinic orbits and homoclinic tangencies to periodic orbits are shown to agree broadly with the theory but to uncover extra complications

the 1 2 mode interaction and heteroclinic networks in boussinesq convection abstract methods of equivariant bifurcation theory are applied to boussinesq convection in a plane layer with stress free horizontal boundaries and an imposed square lattice periodicity in the horizontal directions we consider the problem near the onset of instability of the uniform conducting state where spatial roll patterns with two different wavelengths in the ratio 1 2 become simultaneously unstable at a mode interaction center manifold reduction yields a normal form on c 4 with very rich dynamical behavior for a fixed prandtl number p the mode interaction occurs at an isolated point in the parameter plane r l where r is the rayleigh number and l the length of the horizontal periodicities and acts as an organizing center for many nearby bifurcations the normal form predicts the appearance of many steady states and travelling waves which are classified by their symmetries it also predicts the appearance of robust heteroclinic networks involving steady states with several different symmetries and robust attractors of generalized heteroclinic type that include connections from equilibria to subcycles this is the first example of a heteroclinic network in a fluid dynamical system that has depth greater than one the normal form dynamics is in good correspondence both quantitatively and qualitatively with direct numerical simulations of the full convection equations

modal analysis of rotating shafts a body fixed axis formulation approach abstract a spinning timoshenko beam subjected to a constant moving load is analyzed using a modal expansion technique the formulation is based on a body fixed axis reference system and thus should be applicable to handle any general cross sections using this alternative approach dynamic quantities such as natural frequencies mode shapes and system responses are easily computable it is shown that simply supported spinning timoshenko beams posses two pairs of natural frequencies corresponding to each mode shape through appropriate simplifications the coupled eighth order differential equations governing the problem are degenerated into a set of uncoupled fourth order equations with this simplified theory closed form expressions for natural frequencies and system transient response are derived however one pair of the natural frequencies is lost in the process but this is acceptable as the frequencies of this pair are much higher than those of the retained pair in addition a linearized approximation of the natural frequency similar to the one given for euler bernoulli beams is also proposed for a timoshenko shaft numerical simulations are performed to demonstrate the characteristics of the response

on commuting matrices in max algebra and in classical nonnegative algebra abstract this paper studies commuting matrices in max algebra and nonnegative linear algebra our starting point is the existence of a common eigenvector which directly leads to max analogs and nonnegative analogs of some classical results for complex matrices we also investigate frobenius normal forms of commuting matrices particularly when the perron roots of the components are distinct for the case of max algebra we show how the intersection of eigencones of commuting matrices can be described and we consider connections with boolean algebra which enables us to prove that two commuting irreducible matrices in max algebra have a common eigennode

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

matrices commuting with a given normal tropical matrix abstract consider the space m n nor of square normal matrices x x i j over r i e x i j 0 and x i i 0 endow m n nor with the tropical sum and multiplication fix a real matrix a m n nor and consider the set a of matrices in m n nor which commute with a we prove that a is a finite union of alcoved polytopes in particular a is a finite union of convex sets the set a a of x such that a x x a a is also a finite union of alcoved polytopes the same is true for the set a of x such that a x x a x a topology is given to m n nor then the set a a is a neighborhood of the identity matrix i if a is strictly normal then a is a neighborhood of the zero matrix in one case a is a neighborhood of a we give an upper bound for the dimension of a we explore the relationship between the polyhedral complexes span a span x and span a x when a and x commute two matrices denoted a and a arise from a in connection with a the geometric meaning of them is given in detail for one example we produce examples of matrices which commute in any dimension

multiplicative structure of 2x2 tropical matrices we study the algebraic structure of the semigroup of all 2x2 tropical matrices under multiplication using ideas from tropical geometry we give a complete description of green s relations and the idempotents and maximal subgroups of this semigroup

0 2 symmetric bifurcation theory for convection rolls abstract the interaction of two steady state modes for the two dimensional benard problem with lateral periodic boundary conditions p b c and free horizontal boundary conditions is considered the p b c generate an action of o 2 on the navier stokes equations the boussinesq approximation induces an additional z 2 symmetry the o 2 z 2 equivariant bifurcation equations are determined and two fifth order terms are identified two types of mixed mode solutions and a travelling wave solution are found using macsyma the fifth order terms are calculated and the stability of the solutions to phase perturbations depending on the prandtl number is determined

memory loss property for products of random matrices in the max plus algebra products of random matrices in the max plus algebra are used as models of a wide range of discrete event systems including train or queueing networks job shops timed digital circuits or parallel processing systems several mathematical models such as timed event graph or task resources models also lead to max plus products of matrices some stability and computability results such as convergence of waiting times to a unique stationary regime or limit theorems for the throughput have been proved under the so called memory loss property mlp when the random matrices are i i d we prove that this property is generic in the following sense if it is not fulfilled the support of the common law of the random matrices is included in a union of finitely many affine hyperplanes

noncommuting graphs of matrices over semirings we calculate diameters and girths of commuting graphs of the set of all nilpotent matrices over a semiring the group of all invertible matrices over a semiring and the full matrix semiring

new stability results for patterns in a model of long wavelength convection abstract we consider the transition from a spatially uniform state to a steady spatially periodic pattern in a partial differential equation describing long wavelength convection e knobloch pattern selection in long wavelength convection physica d 41 1990 450 479 this both extends existing work on the study of rolls squares and hexagons and demonstrates how recent generic results for the stability of spatially periodic patterns may be applied in practice we find that squares even if stable to roll perturbations are often unstable when a wider class of perturbations is considered we also find scenarios where transitions from hexagons to rectangles can occur in some cases we find that near onset more exotic spatially periodic planforms are preferred over the usual rolls squares and hexagons

least squares residuals and minimal residual methods we study krylov subspace methods for solving unsymmetric linear algebraic systems that minimize the norm of the residual at each step minimal residual mr methods mr methods are often formulated in terms of a sequence of least squares ls problems of increasing dimension we present several basic identities and bounds for the ls residual these results are interesting in the general context of solving ls problems when applied to mr methods they show that the size of the mr residual is strongly related to the conditioning of different bases of the same krylov subspace using different bases is useful in theory because relating convergence to the characteristics of different bases offers new insight into the behavior of mr methods different bases also lead to different implementations which are mathematically equivalent but can differ numerically our theoretical results are used for a finite precision analysis of implementations of the gmres method y saad and m h schultz siam j sci statist comput 7 1986 pp 856 869 we explain that the choice of the basis is fundamental for the numerical stability of the implementation as demonstrated in the case of simpler gmres h f walker and l zhou numer linear algebra appl 1 1994 pp 571 581 the best orthogonalization technique used for computing the basis does not compensate for the loss of accuracy due to an inappropriate choice of the basis in particular we prove that simpler gmres is inherently less numerically stable than the classical gmres implementation due to saad and schultz siam j sci statist comput 7 1986 pp 856 869

dynamics in the 1 2 spatial resonance with broken reflection symmetry the dynamics near the 1 2 spatial resonance in systems with broken o 2 symmetry are described the system is assumed to remain so 2 invariant and attention is focused on the behavior that replaces the structurally stable heteroclinic cycles present in the o 2 symmetric system the symmetry breaking destroys these cycles and introduces a variety of new bifurcations into the system some which are associated with complex behavior the possible scenarios with increasing symmetry breaking are described and possible applications of the results are discussed

the geometric structure of max plus hemispaces abstract given a set s endowed with a convexity structure a hemispace is a convex subset of s which has convex complement we recall that r max n is a semimodule over the max plus semifield r max r max a convexity structure of current interest is provided by r max n naturally endowed with the max plus or tropical convexity in this paper we provide a geometric description of a max plus hemispace in r max n we show that a max plus hemispace has a conical decomposition as a finite union of disjoint max plus cones these cones can be interpreted as faces of several max plus hyperplanes briec and horvath proved that the closure of a max plus hemispace is bounded by a max plus hyperplane given a hyperplane we give a simple condition for the assignment of the faces between a pair of complementary max plus hemispaces our result allows for counting and enumeration of the associated max plus hemispaces we recall that an n dimensional max plus hyperplane is called strictly affine and nondegenerate if it has a linear equation that contains all variables x 1 x 2 x n and a free term we prove that the number of max plus hemispaces in r max n supported by strictly affine nondegenerate hyperplanes centered in the origin is twice the n th ordered bell number our work can be viewed as a complement to the recent results of katz nitica and sergeev who described generating sets for max plus hemispaces and the results of briec and horvath who proved that closed open max plus hemispaces are max plus closed open halfspaces

pattern selection in three dimensional magnetoconvection abstract weakly nonlinear boussinesq convection in an imposed vertical magnetic field is studied stress free boundary conditions are used at top and bottom of the layer where the magnetic field is assumed to remain vertical both steady and oscillatory convection are considered among all the spatially periodic patterns steady rolls are the preferred pattern when the onset of convection is steady state in the overstable regime a variety of spatially and temporally periodic patterns may be stable depending on the parameters these include in particular a pattern called alternating rolls the location of stable alternating rolls appears to be consistent with recent numerical simulations of compressible magnetoconvection where such patterns have been found in certain cases the conduction solution loses stability directly to spatially periodic patterns with nonperiodic time dependence the analysis of the possible patterns and their stability takes advantage of existing results from equivariant bifurcation theory and uses the critical wavenumber selected by the linear stability calculation as a function of the imposed magnetic field

rank functions of tropical matrices abstract we consider rank functions important in tropical linear algebra the tropical rank equal to the topological dimension of the tropical linear span of the columns of a given matrix the factor rank equal to the smallest number of vectors containing these columns in their span the kapranov rank related to problems of tropical algebraic geometry and the determinantal and gondran minoux ranks which generalize the classical linear algebraic notion of rank we are interested in studying the arithmetic properties of the rank functions their mutual behavior and the related computational problems we discuss different open problems related to rank functions of tropical matrices and illustrate them by the number of examples

period doubling mode interactions with circular symmetry abstract a two parameter analysis of the interaction between two period doubling modes with azimuthal wavenumbers k and l l k 1 is carried out for systems with circular symmetry the problem is formulated in terms of an o 2 equivariant map on c 2 in the generic case all primary secondary and tertiary solution branches and their stability properties are classified the results depend on whether k l is odd or even when one mode bifurcates subcritically and the other supercritically the pure mode branches lose stability to a branch of reflection symmetric mixed modes which in turn can undergo a tertiary hopf bifurcation to a quasiperiodic reflection symmetric pattern we conjecture that this invariant circle can break down with increasing amplitude to produce reflection symmetric chaos additional phase instabilities may occur in which the reflection symmetry is broken and the resulting pattern drifts either clockwise or counterclockwise the results explain a number of observations by ciliberto and gollub on parametrically excited surface waves in circular geometry and imply several new predictions for such experiments in an appendix the theory is compared with previous attempts to model the experiments

simple and superlattice turing patterns in reaction diffusion systems bifurcation bistability and parameter collapse abstract we use equivariant bifurcation theory to investigate pattern selection at the onset of a turing instability in a general two component reaction diffusion system the analysis is restricted to patterns that periodically tile the plane in either a square or hexagonal fashion both simple periodic patterns stripes squares hexagons and rhombs and superlattice patterns are considered the latter correspond to patterns that have structure on two disparate length scales the short length scale is dictated by the critical wave number from linear theory while the periodicity of the pattern is on a larger scale analytic expressions for the coefficients of the leading nonlinear terms in the bifurcation equations are computed from the general reaction diffusion system using perturbation theory we show that no matter how complicated the reaction kinetics might be the nonlinear reaction terms enter the analysis through just four parameters moreover for hexagonal problems all patterns bifurcate unstably unless a particular degeneracy condition is satisfied and at this degeneracy we find that the number of effective system parameters drops to two allowing a complete characterization of the possible bifurcation results at this degeneracy for example we find that rhombs squares and superlattice patterns always bifurcate unstably we apply these general results to some specific model equations including the lengyel epstein cima model to investigate the relative stability of patterns as a function of system parameters and to numerically test the analytical predictions

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

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

the tropical analogue of polar cones we study the max plus or tropical analogue of the notion of polar the polar of a cone represents the set of linear inequalities satisfied by its elements we establish an analogue of the bipolar theorem which characterizes all the inequalities satisfied by the elements of a tropical convex cone we derive this characterization from a new separation theorem we also establish variants of these results concerning systems of linear equalities

max plus control design for temporal constraints meeting in timed event graphs the aim of the presented work is the control of timed event graph to meet tight temporal constraints the problem of temporal constraints is formulated in terms of control of linear max plus models first the synthesis of a control law that ensures the satisfaction of a single constraint for a single input system is presented then the single input multi constraints problem is tackled and finally the method is extended to the multi inputs multi constraints problem the proposed method is illustrated on the example of a simple production process

max algebra the linear algebra of combinatorics abstract let a b max a b a b a b for a b r r by max algebra we understand the analogue of linear algebra developed for the pair of operations extended to matrices and vectors max algebra which has been studied for more than 40 years is an attractive way of describing a class of nonlinear problems appearing for instance in machine scheduling information technology and discrete event dynamic systems this paper focuses on presenting a number of links between basic max algebraic problems like systems of linear equations eigenvalue eigenvector problem linear independence regularity and characteristic polynomial on one hand and combinatorial or combinatorial optimisation problems on the other hand this indicates that max algebra may be regarded as a linear algebraic encoding of a class of combinatorial problems the paper is intended for wider readership including researchers not familiar with max algebra

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

bases in max algebra for n tuples over the algebraic system r r max concepts such as linear dependence space and basis may be defined by analogy with classical linear algebra whenever a space is finitely generated it possesses a basis and all its bases are trivially related and therefore have the same cardinality however for any given n 2 spaces with bases of arbitrary cardinality may be constructed as well as spaces with no basis

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

subadditivity generalized products of random matrices and operations research an elementary theorem on subadditive sequences provides the key to a far reaching theory of subadditive processes one important instance of this theory is the limit theory for products of stationary random matrices this paper shows that the subadditive inequality that governs the log of the norm of ordinary matrix products also governs other functions of several generalized matrix products these generalized matrix products are used to calculate minimal cost transportation routes schedules in manufacturing and minimal and maximal probabilities of multistage processes the application of subadditive ergodic theory to generalized products of stationary random matrices yields new information about the limiting behavior of generalized products exact calculations of the asymptotic behavior are possible in some examples

dynamic response of a rotating timoshenko shaft subject to axial forces and moving loads abstract the dynamic response of a rotating shaft subject to axial force and moving loads is analyzed by using timoshenko beam theory and the assumed mode method the deformations of the shaft are expressed in terms of an inertial reference frame the kinetic and potential energy are then expressed in matrix form by using the assumed mode method the resulting equations of motion in matrix form are derived by using hamilton s principle the influences of the rotational speed of the shaft the axial speed of the loads and the rayleigh coefficient are investigated and compared with the available reported results the effects of compressive axial forces and perturbation of the axial velocity of the moving loads are also included in the analysis

secondary turing type instabilities due to strong spatial resonance we investigate the dynamics of pattern forming systems in large domains near a codimension two point corresponding to a strong spatial resonance where competing instabilities with wavenumbers in th

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

resonant mode interactions in rayleigh benard convection department of physics university of california berkeley california 94720 received 5 february 1998 in systems with midplane re ection symmetry the dominant spatial resonance is the 1 3 resonance numeri cal continuation is used to study this and other 1 2k11 resonances in two dimensional convection betweenno slip perfectly conducting horizontal plates periodic boundary conditions are imposed in the horizontal these resonances in uence the process of wave number selection at moderate rayleigh numbers through thegeneration of hybrid solutions and thereby modify the eckhaus picture of wave number selection unlike thebetter known mixed modes the hybrid solutions have the same symmetry as a pair of primary rolls both hybridand symmetry breaking mixed modes are computed and their linear stability properties with respect to pertur bations preserving different spatial periods are determined a complete description of the effects of midplanere ection on wave number selection emerges only steady solutions are considered and the prandtl number is xed ats510 s1063 651x 98 05709 2 pacs number s 47 20 ky 05 45 1bi introduction

tropical polar cones hypergraph transversals and mean payoff games we discuss the tropical analogues of several basic questions of convex duality in particular the polar of a tropical polyhedral cone represents the set of linear inequalities that its elements satisfy we characterize the extreme rays of the polar in terms of certain minimal set covers which may be thought of as weighted generalizations of minimal transversals in hypergraphs we also give a tropical analogue of farkas lemma which allows one to check whether a linear inequality is implied by a finite family of linear inequalities here the certificate is a strategy of a mean payoff game we discuss examples showing that the number of extreme rays of the polar of the tropical cyclic polyhedral cone is polynomially bounded and that there is no unique minimal system of inequalities defining a given tropical polyhedral cone

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 polynomial algorithm for solving system of inequalities in max plus algebra abstract this paper deals with the system of inequalities a x x in which a is a square matrix x is a column vector and is a max plus composition it provides two algorithms which can be used to find a finite proper supereigenvectors of a corresponding to the eigenvalue 0 in a case of irreducible matrix a with the maximum cycle mean a 0 and a finite solution of the system in a case of reducible matrix a respectively in a polynomial number of steps

homoclinic orbits and mixed mode oscillations in far from equilibrium systems nonlinear autonomous dynamical systems with ahomoclinic tangency to a periodic orbit are investigated we study the bifurcation sequences of the mixed mode oscillations generated by the homoclinicity which are shown to belong to two different types depending on the nature of the liapunov numbers of the basic periodic orbit a detailed numerical analysis is carried out to show how the existence of a tangent homoclinic orbit allows us to understand in a quantitative way a particular and regular sequence of cool flame ignition oscillations observed in a thermokinetic model of hydrocarbon oxidation chaotic cool flame oscillations are also observed in the same model when the control parameter crosses a critical value this chaotic set of trajectories becomes globally unstable and forms a cantor like hyperbolic repellor and the ignition mechanism generates ahomoclinic tangency to the cantor set of trajectories the complex bifurcation diagram may be globally reconstructed from a one dimensional dynamical system thanks to the strong contractivity of thermokinetics it is found that a symbolic dynamics with three symbols is necessary to classify the periodic windows of the complex bifurcation sequence observed numerically in this system

2 1 resonance in non boussinesq convection abstract numerical and laboratory experiments on non boussinesq convection have revealed a possibly stable pattern with square symmetry but with upflow in isolated plumes and downflow in connected sheets a topology normally associated with a hexagonal pattern the appearance of this pattern can be interpreted as the sum of two different pairs of orthogonal rolls aligned at 45 and with wave numbers in the ratio 2 1 we examine the interaction between these modes by expanding about the multiple bifurcation point at which they are both marginal the relevant amplitude equations are written down and are shown to account for several of the features found in the experiments the new pattern can be the only stable solution near onset and can occur subcritically these features of the solution depend crucially on the lack of boussinesq symmetry which induces quadratic resonance between the two sets of rolls in close analogy with the 2 1 resonance problem in non boussinesq convection with o 2 symmetry

heteroclinic cycles and modulated travelling waves in a system with d 4 symmetry we study a system of two complex ordinary differential equations with d4 symmetry describing the effects of the symmetry breaking o 2 to d4 on an interaction between fourier modes with wavenumbers in the ratio 1 2 this change from continuous to discrete symmetry is relevant to the effect of introducing riblets on a wall to reduce boundary layer drag numerical simulations planar analysis and melnikov s method reveal the persistence under this symmetry breaking of just two of the continuous group orbit of heteroclinic cycles of the o 2 system it is further shown that one of these cycles destabilizes to regular modulated travellin waves while the other loses stability to a new type of solution modulated travelling waves which quasi periodically reverse their direction of propagation a bifurcation diagram for this behavior and the subsequent destabilization of modulated travelling waves to travelling waves is established the method of averaging is used to show that some of the quasiperiodic solutions of the o 2 system persist under the symmetry breaking while others break into sets of periodic orbits

basic solutions of systems with two max linear inequalities abstract we give an explicit description of the basic solutions of max linear systems a x b x with two inequalities

tropical linear maps on the plane abstract in this paper we fully describe all tropical linear maps in the tropical projective plane tp 2 that is maps from the tropical plane to itself given by tropical multiplication by a real 3 3 matrix a the map f a is continuous and piecewise linear in the classical sense in some particular cases the map f a is a parallel projection onto the set spanned by the columns of a in the general case after a change of coordinates the map collapses at most three regions of the plane onto certain segments called antennas and is a parallel projection elsewhere theorem 3 in order to study f a we may assume that a is normal i e i a 0 up to changes of coordinates a given matrix a admits infinitely many normalizations our approach is to define and compute a unique normalization for a which we call lower canonical normalization theorem 1 and then always work with it due both to its algebraic simplicity and its geometrical meaning on r n any n n some aspects of tropical linear maps have been studied in 6 we work in tp 2 adding a geometric view and doing everything explicitly we give precise pictures inspiration for this paper comes from 3 5 6 8 12 26 we have tried to make it self contained our preparatory results present noticeable relationships between the algebraic properties of a given matrix a idempotent normal matrix permutation matrix etc and classical geometric properties of the points spanned by the columns of a classical convexity and others see theorem 2 and corollary 1 as a by product we compute all the tropical square roots of normal matrices of a certain type see corollary 4 this is perhaps a curious result in tropical algebra our final aim is however to give a precise description of the map f a tp 2 tp 2 this is particularly easy when two tropical triangles arising from a denoted t a and t a fit as much as possible then the action of f a is easily described on the closure of each cell of the cell decomposition c a see theorem 3 normal matrices play a crucial role in this paper the tropical powers of normal matrices of size n n satisfy a n 1 a n a n 1 this statement can be traced back at least to 26 and appears later many times such as 1 2 6 9 10 in lemma 1 we give a direct proof of this fact for n 3 but now the equality a 2 a 3 means that the columns of a 2 are three fixed points of f a and in fact any point spanned by the columns of a 2 is fixed by f a among 3 3 normal matrices the idempotent ones i e those satisfying a a 2 are particularly nice we prove that the columns of such a matrix tropically span a set which is classically compact connected and convex lemma 2 and corollary 1 in our terminology it is a good tropical triangle

a characterization of bases of tropical kernels in terms of cramer s rule abstract the tropical kernel of a min plus matrix is a linear space it has a finite basis that is uniquely determined up to scalar multiplication in this paper we give a characterization of vectors in a basis of the kernel using the tropical analogue of cramer s rule we first show that each finite vector in a basis can be obtained by applying cramer s rule to a submatrix consisting of certain rows of the original matrix we prove the main results by presenting an algorithm for choosing such rows further we extend the result for kernel vectors that contain infinity in its entries

tropical plactic algebra the cloaktic monoid and semigroup representations abstract a new tropical plactic algebra is introduced in which the knuth relations are inferred from the underlying semiring arithmetic encapsulating the ubiquitous plactic monoid p n this algebra manifests a natural framework for accommodating representations of p n and equivalently of young tableaux and its moderate coarsening the cloaktic monoid k n and the co cloaktic monoid the faithful linear representations of k n and by tropical matrices which constitute a tropical plactic algebra are shown to provide linear representations of the plactic monoid to this end the paper develops a special type of configuration tableaux corresponding bijectively to semi standard young tableaux these special tableaux allow a systematic encoding of combinatorial properties in numerical algebraic ways including algorithmic benefits the interplay between these algebraic combinatorial structures establishes a profound machinery for exploring semigroup attributes in particular satisfying of semigroup identities this machinery is utilized here to prove that k n and admit all the semigroup identities satisfied by n n triangular tropical matrices which holds also for p 3

applications of entropy of product systems higher rank graphs abstract we consider c algebras of finite higher rank graphs along with their rotational action we show how the entropy theory of product systems with finite frames applies to identify the phase transitions of the dynamics we compute the positive inverse temperatures where symmetry breaks and in particular we identify the subharmonic parts of the gauge invariant equilibrium states our analysis applies to positively weighted rotational actions through a recalibration of the entropies

characterization of tropical hemispaces by p r decompositions abstract 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

the tropical commuting variety we study tropical commuting matrices from two viewpoints linear algebra and algebraic geometry in classical linear algebra there exist various criteria to test whether two square matrices commut

new type of complex dynamics in the 1 2 spatial resonance abstract the amplitude equations describing the 1 2 spatial resonance in o 2 symmetric systems are re examined complex dynamical behavior associated with a new class of global connections is identified and analyzed these connections form further from the mode interaction point where the structurally stable heteroclinic cycles described by armbruster et al physica d 29 1988 257 no longer exist the resulting behavior is characteristic of systems with o 2 z 2 symmetry in which the symmetry z 2 is weakly broken

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

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

dynamic response of a rotating multi span shaft with general boundary conditions subjected to a moving load abstract this paper investigates the dynamic response of a rotating multi span shaft with general boundary conditions subjected to an axially moving load the system equations of motion are derived based on the global assumed mode method and a modified transformation matrix is proposed to deal with the multi span geometric constraints the numerical results are solved by the runge kutta method to ensure the validity of the method the convergence test for dynamic response and the comparison of natural frequencies obtained by both the present method and finite element method are demonstrated numerical simulations are performed to study the effects of the moving speed of the load the number of spans and the rotational speed of the shaft on the deflections under the moving load in addition the effects on the occurring position of maximum deflection are presented the numerical results show that the maximum deflection in the direction of the load may occur at any span and near the midpoint of one span however the maximum deflection induced by the gyroscopic effect in the direction perpendicular to the load always occurs at the first span moreover the occurring position of maximum deflection is independent of the rotational speed

convexity of tropical polytopes abstract we study the relationship between min plus max plus and euclidean convexity for subsets of r n we introduce a construction which associates to any max plus convex set with compact projectivisation a canonical matrix called its dominator the dominator is a kleene star whose max plus column space is the min plus convex hull of the original set we apply this to show that a set which is any two of i a max plus polytope ii a min plus polytope and iii a euclidean polytope must also be the third in particular these results answer a question of sergeev schneider and butkovic 16 and show that row spaces of tropical kleene star matrices are exactly the polytropes studied by joswig and kulas 13

hexagons and triangles in the rayleigh benard problem quintic order equations on a hexagonal lattice on a weakly nonlinear basis we revisit the pattern formation problem in the boussinesq convection for which nonlinear terms of the quadratic order are known to vanish from amplitude equations it