determining the inertia of a matrix pencil as a function of the parameter abstract we determine the inertia of a linear real symmetric matrix pencil a t a tb of order n as a function of the parameter t finding the critical values where the inertia of a t changes is reduced to determining the real eigenvalues of a not necessarily symmetric matrix the order of this matrix is at most r s r 1 where r s and r 1 are the ranks of a b and a b respectively we illustrate the method by means of a numerical example then we reduce determining the inertia of a quadratic real symmetric matrix pencil a t a tb 1 t 2 b 2 to the linear case our results are extensions of those by caron and gould

stochastic output feedback control convex lifting approach abstract this paper presents a method to design output feedback control for discrete time linear systems affected by bounded additive state output disturbances and subject to chance constraints on the state and hard constraints on the control input this method relies on a so called convex lifting which is a nonnegative convex piecewise affine function equal to 0 over a given stochastic positively invariant set and strictly positive outside this set accordingly it is shown that this function is strictly decreasing along the closed loop dynamics outside this invariant set and convergent to 0 as time tends to infinity consequently the state is convergent to the given invariant set while the method only requires solving a linear program at each sampling instant

on the tail of the branching random walk local time consider a critical branching random walk on mathbb z d d ge 1 started with a single particle at the origin and let l x be the total number of particles that ever visit a vertex x we study the tail of l x under suitable conditions on the offspring distribution in particular our results hold if the offspring distribution has an exponential moment

universality of high dimensional spanning forests and sandpiles we prove that the wired uniform spanning forest exhibits mean field behaviour on a very large class of graphs including every transitive graph of at least quintic volume growth and every bounded degree nonamenable graph several of our results are new even in the case of mathbb z d d ge 5 in particular we prove that every tree in the forest has spectral dimension 4 3 and walk dimension 3 almost surely and that the critical exponents governing the intrinsic diameter and volume of the past of a vertex in the forest are 1 and 1 2 respectively the past of a vertex in the uniform spanning forest is the union of the vertex and the finite components that are disconnected from infinity when that vertex is deleted from the forest we obtain as a corollary that the critical exponent governing the extrinsic diameter of the past is 2 on any transitive graph of at least five dimensional polynomial growth and is 1 on any bounded degree nonamenable graph we deduce that the critical exponents describing the diameter and total number of topplings in an avalanche in the abelian sandpile model are 2 and 1 2 respectively for any transitive graph with polynomial growth of dimension at least five and are 1 and 1 2 respectively for any bounded degree nonamenable graph in the case of mathbb z d d ge 5 some of our results regarding critical exponents recover earlier results of bhupatiraju et al electron j probab 22 85 51 2017 in this case we improve upon their results by showing that the tail probabilities in question are described by the appropriate power laws to within constant order multiplicative errors rather than the polylogarithmic order multiplicative errors present in that work

on the segre weyr characteristics of right left regular matrix pencils the classical segre weyr characteristic theory of the standard eigenvalue eigenvector problem defined on a n n is extended to the case of right left regular pencils sf g f g m n the notions of toeplitz matrices and of the right left f g sequences j r f g j 1 f g respectively are introduced for right left regular pairs it is shown that j r f g j 1 f g are piecewise arithmetic progression sequences paps the singular points of those paps define the possible degrees of the elementary divisors e d at whereas the deviations or gaps from the arithmetic progression sequence define the multiplicity of the corresponding degree e d s the links between the algebraic notion of the segre characteristic and the geometric notion of the weyr characteristic are established for such families of pencils two different procedures for the computation of the segre characteristic are given the first is based on the analysis of singular points of paps and the second is based o

infinite volume limit of the abelian sandpile model in dimensions d 3 we study the abelian sandpile model on mathbb z d in d 3 we prove existence of the infinite volume addition operator almost surely with respect to the infinite volume limit of the uniform measures on recurrent configurations we prove the existence of a markov process with stationary measure and study ergodic properties of this process the main techniques we use are a connection between the statistics of waves and uniform two component spanning trees and results on the uniform spanning forest measure on mathbb z d

the infinite volume limit of dissipative abelian sandpiles we construct the thermodynamic limit of the stationary measures of the bak tang wiesenfeld sandpile model with a dissipative toppling matrix sand grains may disappear at each toppling we prove uniqueness and mixing properties of this measure and we obtain an infinite volume ergodic markov process leaving it invariant we show how to extend the dhar formalism of the abelian group of toppling operators to infinite volume in order to obtain a compact abelian group with a unique haar measure representing the uniform distribution over the recurrent configurations that create finite avalanches

an improved algorithm for the computation of kronecker s canonical form of a singular pencil we give an 0 t algorithm for computing the kronecker structure of an arbitrary m x n pencil xe a the algorithm is shown to he numerically stable because only unitary transformations are used the improved speed over earlier unitary methods is due to the efficient use of condensed forms which are maintained throughout the recursions of the algorithm

waves of topplings in an abelian sandpile abstract we study the structure of the avalanche process in the 2d abelian sandpile model an avalanche may be represented as a sequence of waves each consisting of sites that toppled only once in that wave it is shown that the waves are in one to one correspondence with the set of two rooted spanning trees and can be described in terms of green functions we argue that the probability distribution of waves of size s varies as 1 s for large s we prove also that the avalanches started at the open boundary consist of only one wave and find the asymptotic distribution of their sizes which varies as s 3 2 we establish the equivalence between waves and inverse avalanches introduced by dhar and manna and reproduce their result 11 8 for the critical exponent of the size distribution of the first inverse avalanche or in our terms the last wave

infinite volume limit for the stationary distribution of abelian sandpile models we study the stationary distribution of the standard abelian sandpile model in the box n n n d d for d 2 we show that as n the finite volume stationary distributions weakly converge to a translation invariant measure on allowed sandpile configurations in d this allows us to define infinite volume versions of the avalanche size distribution and related quantities the proof is based on a mapping of the sandpile model to the uniform spanning tree due to majumdar and dhar and the existence of the wired uniform spanning forest measure on d in the case d 4 we also make use of wilson s method

output feedback robust mpc with one free control move for the linear polytopic uncertain system with bounded disturbance in this paper a previous approach for the robust model predictive control mpc for a linear polytopic uncertain system is extended to the case with bounded disturbance and unmeasurable state the controller on line optimizes a free control move followed by an output feedback control law based on the pre specified state estimator a key technique for this controller is an appropriate formulation of the estimation error bound which accounts for recursive feasibility of the optimization problem the quadratic boundedness qb of the augmented state is guaranteed by the proposed approach a numerical example is given to illustrate the effectiveness of the proposed controller

abelian sandpiles and the harmonic model we present a construction of an entropy preserving equivariant surjective map from the d dimensional critical sandpile model to a certain closed shift invariant subgroup of mathbb t mathbb z d the harmonic model a similar map is constructed for the dissipative abelian sandpile model and is used to prove uniqueness and the bernoulli property of the measure of maximal entropy for that model

the computation of the kronecker canonical form of an arbitrary symmetric pencil abstract we present an algorithm for the computation of the kronecker structure of a symmetric pencil a m this method which preserves the property of symmetry is somewhat different from van dooren s algorithm for the determination of the kronecker structure of an arbitrary pencil we show how to use this method to determine the structure of the infinite elementary divisors of a m and its kronecker blocks which may occur for the case of a singular pencil the cost of computations of our algorithm is cheaper than van dooren s algorithm and the symmetry of the reduced pencils can be preserved the algorithm is fairly stable though we use some nonunitary transformation matrices but the norms of these matrices are bounded under a tolerance the present procedure can also be used to separate from a symmetric pencil a smaller symmetric regular pencil which contains only the finite eigenvalues of the original one so this method can be used as a preprocessing for a qz or hr algorithm in order to get rid of singularities which can be troublesome for those algorithms

linear systems with state and control constraints the theory and application of maximal output admissible sets the initial state of an unforced linear system is output admissible with respect to a constraint set y if the resulting output function satisfies the pointwise in time condition y t in y t or 0 the set of all possible such initial conditions is the maximal output admissible set o sub infinity the properties of o sub infinity and its characterization are investigated in the discrete time case it is generally possible to represent o sub infinity or a close approximation of it by a finite number of functional inequalities practical algorithms for generating the functions are described in the continuous time case simple representations of the maximal output admissible set are not available however it is shown that the discrete time results may be used to obtain approximate representations

the abelian sandpile model on an infinite tree we consider the standard abelian sandpile process on the bethe lattice we show the existence of the thermodynamic limit for the finite volume stationary measures and the existence of a unique infinite volume markov process exhibiting features of self organized criticality

pairs of matrices with quadratic minimal polynomials abstract let a b be n n matrices over a field f and suppose a b have quadratic minimal polynomials then the algebra generated by a b has dimension at most 2n 1 if n is odd and 2n if n is even these bounds are exact

a counterexample to kippenhahn s conjecture on hermitian pencils abstract a counterexample is constructed to a conjecture of kippenhahn math nachr 6 193 288 1951 52 a pair of hermitian 8 8 matrices h k is found such that 1 h k generate m 8 c and 2 the minimal polynomial of the pencil xh yk has degree 4 recent work of h shapiro e g linear algebra appl 43 201 221 1982 has established the conjecture for n n matrices n 5

self organized critical state of sandpile automaton models we study a general bak tang wiesenfeld type automaton model of self organized criticality in which the toppling conditions depend on local height but not on its gradient we characterize the critical state and determine its entropy for an arbitrary finite lattice in any dimension the two point correlation function is shown to satisfy a linear equation the spectrum of relaxation times describing the approach to the critical state is also determined exactly

a note on stewart s theorem for definite matrix pairs abstract let a and b be n n hermitian matrices the matrix pair a b is called definite pair and the corresponding eigenvalue problem ax bx is definite if c a b inf x 1 h a ib x 0 in this note we develop a uniform upper bound for differences of corresponding eigenvalues of two definite pairs and so improve a result which is obtained by g w stewart 2 moreover we prove that this upper bound is a projective metric in the set of n n definite pairs

robust constrained model predictive control using linear matrix inequalities abstract the primary disadvantage of current design techniques for model predictive control mpc is their inability to deal explicitly with plant model uncertainty in this paper we present a new approach for robust mpc synthesis that allows explicit incorporation of the description of plant uncertainty in the problem formulation the uncertainty is expressed in both the time and frequency domains the goal is to design at each time step a state feedback control law that minimizes a worst case infinite horizon objective function subject to constraints on the control input and plant output using standard techniques the problem of minimizing an upper bound on the worst case objective function subject to input and output constraints is reduced to a convex optimization involving linear matrix inequalities lmis it is shown that the feasible receding horizon state feedback control design robustly stabilizes the set of uncertain plants several extensions such as application to systems with time delays problems involving constant set point tracking trajectory tracking and disturbance rejection which follow naturally from our formulation are discussed the controller design is illustrated with two examples

approaching criticality via the zero dissipation limit in the abelian avalanche model the discrete height abelian sandpile model was introduced by bak tang wiesenfeld and dhar as an example for the concept of self organized criticality when the model is modified to allow grains to disappear on each toppling it is called bulk dissipative we provide a detailed study of a continuous height version of the abelian sandpile model called the abelian avalanche model which allows an arbitrarily small amount of dissipation to take place on every toppling we prove that for non zero dissipation the infinite volume limit of the stationary measure of the abelian avalanche model exists and can be obtained via a weighted spanning tree measure we show that in the whole non zero dissipation regime the model is not critical i e spatial covariances of local observables decay exponentially we then study the zero dissipation limit and prove that the self organized critical model is recovered both for the stationary measure and for the dynamics we obtain rigorous bounds on toppling probabilities and introduce an exponent describing their scaling at criticality we rigorously establish the mean field value of this exponent for d 4

worst case formulations of model predictive control for systems with bounded parameters two different predictive control formulations are developed based on minimization of the worst case quadratic cost for systems with bounded parameters the two formulations differ on the assumptions made about the future inputs in optimizing the current input one assumes open loop control while the other considers closed loop control their closed loop properties such as asymptotic stability are examined we then focus on a moving average model with an integrator and derive computationally simpler suboptimal algorithms

the upper critical dimension of the abelian sandpile model the existing estimation of the upper critical dimension of the abelian sandpile model is based on a qualitative consideration of avalanches as self avoiding branching processes we find an exact representation of an avalanche as a sequence of spanning subtrees of two component spanning trees using equivalence between chemical paths on the spanning tree and loop erased random walks we reduce the problem to determination of the fractal dimension of spanning subtrees then the upper critical dimension du 4 follows from lawler s theorems for intersection probabilities of random walks and loop erased random walks

relations between strict equivalence invariants and structure at infinity of matrix pencils the structure at infinity of matrix pencils is investigated via their smith macmillan forms at infinity 1 relations between some strict equivalence invariants and pole zero structure at infinity are established

pencils of complex and real symmetric and skew matrices abstract this expository paper establishes the canonical forms under congruence for pairs of complex or real symmetric or skew matrices the treatment is in the spirit of the well known book of gantmacher on matrix theory and may be regarded as a supplement to gantmacher s chapters on pencils of matrices

control sharing and merging control lyapunov functions given two control lyapunov functions clfs a merging is a new clf whose gradient is a positive combination of the gradients of the two parents clfs the merging function is an important trade off since this new function may for instance approximate one of the two parents functions close to the origin while being close to the other far away for nonlinear control affine systems some equivalence properties are shown between the control sharing property i e the existence of a single control law which makes simultaneously negative the lyapunov derivatives of the two given clfs and the existence of merging clfs it is shown that even for linear time invariant systems the control sharing property does not always hold with the remarkable exception of planar systems the class of linear differential inclusions is also discussed and similar equivalence results are presented for this class of systems linear matrix inequalities conditions are provided to guarantee the control sharing property finally a constructive procedure based on the recently considered r functions is defined to merge two smooth positively homogeneous clfs

implicit improved vertex control for uncertain time varying linear discrete time systems with state and control constraints the problem of regulating an uncertain and or time varying linear discrete time system with state and control constraints to the origin is addressed it is shown that feasibility and a robustly asymptotically stable closed loop can be achieved using an interpolation technique the design method can be seen as an alternative to optimization based control schemes such as robust model predictive control especially for problems requiring complex calculations to find the optimal solution the present method can provide a straightforward suboptimal solution a simulation demonstrates the performance of this class of constrained controllers

explicit robustness and fragility margins for linear discrete systems with piecewise affine control law in this paper we focus on the robustness and fragility problem for piecewise affine pwa control laws for discrete time linear system dynamics in the presence of parametric uncertainty of the state space model a generic geometrical approach will be used to obtain robustness fragility margins with respect to the positive invariance properties for pwa control laws defined over a bounded region in the state space it is shown that these margins can be described in terms of polyhedral sets in parameter space the methodology is further extended to the fragility problem with respect to the partition defining the controller finally several computational aspects are presented regarding the transformation from the theoretical formulations to explicit representations vertex halfspace representation of polytopes of these sets

the u n free rigid body integrability and stability analysis of the equilibria a natural extension of the free rigid body dynamics to the unitary group u n is considered the dynamics is described by the euler equation on the lie algebra u n which has a bi hamiltonian structure and it can be reduced onto the adjoint orbits as in the case of the so n the complete integrability and the stability of the isolated equilibria on the generic orbits are considered by using the method of bolsinov and oshemkov in particular it is shown that all the isolated equilibria on generic orbits are lyapunov stable c 2015 elsevier inc all rights reserved

singularities of bi hamiltonian systems we study the relationship between singularities of bi hamiltonian systems and algebraic properties of compatible poisson brackets as the main tool we introduce the notion of linearization of a poisson pencil from the algebraic viewpoint a linearized poisson pencil can be understood as a lie algebra with a fixed 2 cocycle in terms of such linearizations we give a criterion for non degeneracy of singular points of bi hamiltonian systems and describe their types

on the reduction of pairs of hermitian or symmetric matrices to diagonal form by congruence abstract let a 1 a 2 be given n by n hermitian or symmetric matrices and consider the simultaneous transformations a i sa i s if a i is hermitian or a i sa i s t if a i is symmetric we give necessary and sufficient conditions for the existence of a unitary s which reduces both a 1 and a 2 to diagonal form in this way when at least one of a 1 or a 2 is nonsingular we give necessary and sufficient conditions for a reduction of this sort with a nonsingular s these results are a generalization of the classical theorem from mechanics that a positive definite matrix and a hermitian matrix can always be diagonalized simultaneously by a nonsingular congruence

critical exponents for boundary avalanches in two dimensional abelian sandpile we investigate the properties of boundary avalanches in 2d abelian sandpile model asm we construct the one to one correspondence of boundary avalanches and two rooted spanning trees using the connection between the obtained graph representation and lattice green functions we calculate the exact values of critical exponents for size and lifetime distributions of avalanches starting at the open boundary that forms an angle alpha we find that the probability of a boundary avalanche of the size s varies as s 1 pi 2 alpha for large s and the probability of an avalanche of lifetime t varies as t 1 4 pi 5 alpha for large t the obtained values are verified by numerical simulations

an algorithm for finding generalized eigenpairs of a symmetric definite matrix pencil abstract a new method for finding eigenpairs of any symmetric definite matrix pencil is proposed it is a modification of the well known rayleigh quotient iteration it circumvents a drawback of that method i e its nonconvergence or slow convergence for poor initial vectors the global convergence and cubic convergence rate of the new method are proved its efficiency is also demonstrated by numerical results

on a deflation method for the symmetric generalized eigenvalue problem abstract a generalization of the fix heiberger reduction is used to deflate the infinite and the singular structure from a symmetric matrix pencil a b the finite eigenvalues can be determined from the remaining symmetric problem with the aid of this deflation method it is shown that the kronecker canonical form of a b is very special if b is positive semidefinite

some remarks about mishchenko fomenko subalgebras abstract we discuss and compare two different approaches to the notion of mishchenko fomenko subalgebras in poisson lie algebras of finite dimensional lie algebras one of them commonly accepted by the algebraic community uses polynomial ad invariants the other is based on formal ad invariants and allows one to deal with arbitrary lie algebras not necessarily algebraic in this sense the latter is more universal

webs veronese curves and bihamiltonian systems abstract we define a special kind of multidimensional webs connected with the veronese curve for these webs the foliations in question depend not on a discrete parameter but on the point on a projective line for each bihamiltonian system of odd dimension in general position we construct such a web and show how to reconstruct the original bihamiltonian system based on these data

mean field avalanche size exponent for sandpiles on galton watson trees we show that in abelian sandpiles on infinite galton watson trees the probability that the total avalanche has more than t topplings decays as t 1 2 we prove both quenched and annealed bounds under suitable moment conditions our proofs are based on an analysis of the conductance martingale of morris probab theory relat fields 125 259 265 2003 that was previously used by lyons et al electron j probab 13 58 1702 1725 2008 to study uniform spanning forests on mathbb z d d ge 3 and other transient graphs

the condition number of equivalence transformations that block diagonalize matrix pencils how ill conditioned must a matrix s be if its columns are constrained to span certain subspaces we answer this question in order to find nearly best conditioned matrices sr and sl that block diagonalize a given matrix pencil t a b i e s l 1 tsr is bloc diagonal we show that the best conditioned sr has a condition number approximately equal to the cosecant of the smallest angle between right subspaces belonging to different diagonal blocks of thus the more nearly the right subspaces overlap the more ill conditioned sr must be the same is true of sl and the left subspaces for the standard eigenproblem t a i sl sr and the cosecant of the angle between subspaces turns out equal to an earlier estimate of the smallest condition number namely the norm of the projection matrix associated with one of the subspaces we apply this result to bound the error in an algorithm to compute analytic functions of matrices for instance exp t

choosing a spanning tree for the integer lattice uniformly consider the nearest neighbor graph for the integer lattice zd in d dimensions for a large finite piece of it consider choosing a spanning tree for that piece uniformly among all possible subgraphs that are spanning trees as the piece gets larger this approaches a limiting measure on the set of spanning graphs for zd this is shown to be a tree if and only if d 4 in this case the tree has only one topological end that is there are no doubly infinite paths when d 2 5 the spanning forest has infinitely many components almost surely with each component having one or two topological ends

uniform spanning forests we study uniform spanning forest measures on infinite graphs which are weak limits of uniform spanning tree measures from finite subgraphs these limits can be taken with free fsf or wired wsf boundary conditions pemantle proved that the free and wired spanning forests coincide in z d and that they give a single tree iff d 4 in the present work we extend pemantle s alternative to general graphs and exhibit further connections of uniform spanning forests to random walks potential theory invariant percolation and amenability the uniform spanning forest model is related to random cluster models in statistical physics but because of the preceding connections its analysis can be carried further among our results are the following the fsf and wsf in a graph g coincide iff all harmonic dirichlet functions on g are constant the tail fields of the wsf and the fsf are trivial on any graph on any cayley graph that is not a finite extension of z all component trees of the wsf have one end this is new in z d for d 5 on any tree as well as on any graph with spectral radius less than 1 a s all components of the wsf are recurrent the basic topology of the free and the wired uniform spanning forest measures on lattices in hyperbolic space h d is analyzed a cayley graph is amenable iff for all e 0 the union of the wsf and bernoulli percolation with parameter e is connected harmonic measure from infinity is shown to exist on any recurrent proper planar graph with finite codegrees we also present numerous open problems and conjectures

min max control of constrained uncertain discrete time linear systems for discrete time uncertain linear systems with constraints on inputs and states we develop an approach to determine state feedback controllers based on a min max control formulation robustness is achieved against additive norm bounded input disturbances and or polyhedral parametric uncertainties in the state space matrices we show that the finite horizon robust optimal control law is a continuous piecewise affine function of the state vector and can be calculated by solving a sequence of multiparametric linear programs when the optimal control law is implemented in a receding horizon scheme only a piecewise affine function needs to be evaluated on line at each time step the technique computes the robust optimal feedback controller for a rather general class of systems with modest computational effort without needing to resort to gridding of the state space

equivalence between the abelian sandpile model and the q 0 limit of the potts model abstract we establish an equivalence between the undirected abelian sandpile model and the q 0 limit of the q state potts model the equivalence is valid for arbitrary finite graphs two dimensional abelian sandpile models thus correspond to a conformal field theory with central charge c 2 the equivalence also gives a monte carlo algorithm to generate random spanning trees we study the growth process of the spread of fire under the burning algorithm in the background of a random recurrent configuration of the abelian sandpile model the average number of sites burnt upto time t varies at ta in two dimensions our numerically determined value of a agrees with the theoretical prediction a 8 5 we relate this exponent to the conventional exponents characterizing the distributions of avalanche sizes

admissible sets and feedback control for discrete time linear dynamical systems with bounded controls and states for linear discrete time dynamical systems necessary and sufficient conditions are given for a control sequence belonging to a polyhedral constraint set to stabilize the system under the condition that the state is restricted to a polyhedral state constraint set a variable structure linear state feedback controller given in terms of the controls at the vertices of the polyhedral state constraint set is presented

indistinguishability of trees in uniform spanning forests we prove that in both the free and the wired uniform spanning forest fusf and wusf of any unimodular random rooted network in particular of any cayley graph it is impossible to distinguish the connected components of the forest from each other by invariantly defined graph properties almost surely this confirms a conjecture of benjamini et al ann probab 29 1 1 65 2001 we also answer positively two additional questions of benjamini et al ann probab 29 1 1 65 2001 under the assumption of unimodularity we prove that on any unimodular random rooted network the fusf is either connected or has infinitely many connected components almost surely and if the fusf and wusf are distinct then every component of the fusf is transient and infinitely ended almost surely all of these results are new even for cayley graphs

on perturbations and the equivalence orbit of a matrix pencil abstract two theorems on the canonical kronecker form of a perturbed matrix pencil and the characterization of the closure of the set of all matrix pencils with a fixed kronecker canonical form are given

cotangent models for integrable systems we associate cotangent models to a neighbourhood of a liouville torus in symplectic and poisson manifolds focusing on b poisson b symplectic manifolds the semilocal equivalence with such models uses the corresponding action angle theorems in these settings the theorem of liouville mineur arnold for symplectic manifolds and an action angle theorem for regular liouville tori in poisson manifolds laurent gengoux et al intmath res notices imrn 8 1839 1869 2011 our models comprise regular liouville tori of poisson manifolds but also consider the liouville tori on the singular locus of a b poisson manifold for this latter class of poisson structures we define a twisted cotangent model the equivalence with this twisted cotangent model is given by an action angle theorem recently proved by the authors and scott math pures appl 9 105 1 66 85 2016 this viewpoint of cotangent models provides a new machinery to construct examples of integrable systems which are especially valuable in the b symplectic case where not many sources of examples are known at the end of the paper we introduce non degenerate singularities as lifted cotangent models on b symplectic manifolds and discuss some generalizations of these models to general poisson manifolds

perturbation theorems for the generalized eigenvalue problem abstract given a matrix pair z a b the perturbation of its eigenvalues is studied considering two pairs z w as points of the grassman manifold g n 2 n and its eigenvalues as points in g 1 2 the projective complex plane the distance of the spectra measured in the chordal metric in g 1 2 is bounded by some distance of the matrix pairs in g n 2 n analogs of the bauer fike theorem henrici s theorem and the hoffman weilandt theorem are obtained from which the classical results can be derived

simultaneous diagonalization of rectangular matrices abstract a matrix d is said to be diagonal if its i j th element is null whenever i and j are unequal for a set a of matrices a of the same order the paper gives necessary and sufficient conditions for nonsingular matrices s and t to exist such that sa t d is diagonal for each matrix a in the set

finding a positive definite linear combination of two hermitian matrices abstract let a and b be n by n hermitian matrices over the complex field a result of au yeung 1 and stewart 8 states that if x a ib x 0 for all nonzero n vectors x then there is a linear combination of a and b which is positive definite in this article we present an algorithm which finds such a linear combination in a finite number of steps we also discuss the implementation of the algorithm in case a and b are real symmetric sparse matrices

note on pencils of matrices let a and 8 be hermitian matrices of order n if x is a real parameter then the adjoint of a xb can be written as xa0 a a 2 an it where the ai are hermitian it is shown that if a and b are positive definite non negative definite then so are the ai the ranks of the ai are also considered the results of kocaoglan and demirekler 1984 are proved in a more concise and clear manner

the perturbation bounds for eigenspaces of a definite matrix pair leta andb be hermitian matrices the matrix pair a b is called definite pair and the corresponding eigenvalue problem ax bx is definite if c a b equiv mathop min limits left x right 1 x h a ib x 0 the perturbation bounds for eigenspaces of a definite pair on every unitary invariant matrix norm were obtained by imposing additional restrictions on the location of the generalized eigenvalues thus it gives a positive answer for an open question proposed by stewart 7 the famous davis kahan sin theorems and sin 2 theorem 2 can also be deduced from the present results

the abelian sandpile and related models abstract the abelian sandpile model is the simplest analytically tractable model of self organized criticality this paper presents a brief review of known results about the model the abelian group structure of the algebra of operators allows an exact calculation of many of its properties in particular when there is a preferred direction one can calculate all the critical exponents characterizing the distribution of avalanche sizes in all dimensions for the undirected case the model is related to q 0 state potts model this enables exact calculation of some exponents in two dimensions and there are some conjectures about others we also discuss a generalization of the model to a network of communicating reactive processors this includes sandpile models with stochastic toppling rules as a special case we also consider a nondashabelian stochastic variant which lies in a different universality class related to directed percolation

lower eigenvalue bounds for pencils of matrices abstract eigenvalue bounds are obtained for pencils of matrices a vb where a is a stieltjes matrix and b is positive definite under assumptions suitable for the estimation of asymptotic convergence rates of factorization iterative methods where b represents the approximate factorization of a the upper bounds obtained depend on the connectivity structure of the matrices involved which enters through matrix graph considerations in addition a more classical argument is used to obtain a lower bound potential applications of these results include a partial confirmation of gustafsson s conjecture concerning the nonnecessity of axelsson s perturbations

some interlacing results for indefinite hermitian matrices abstract a classical theorem of cauchy states that the eigenvalues of a principal submatrix a 0 of a hermitian matrix a interlace the eigenvalues of a we consider the case of a matrix a which is hermitian with respect to an indefinite inner product

the inertia of a hermitian pencil abstract let a and b be hermitian matrices and p a b where r 2 using parametric dependence of the eigenvalues we study the inertia of p under variation of and under small hermitian perturbations in particular we give semicontinuous dependence results for the set of where inertia of p is discontinuous

right left characteristic sequences and column row minimal indices of a singular pencil for a general singular pencil sf g m n s the right left characteristic sequences r f g l f g are defined and they are shown to be of the piecewise arithmetic progression type the sets of column row minimal indices c f g r f g are defined by a singular points analysis of r f g l f g respectively thus c f g r f g emerge as numerical invariants of the ordered pair f g and they may be computed by rank tests on toeplitz matrices defined on f g

introduction to the sandpile model abstract this article is based on a talk given by one of us evi at the conference statphys taipei 1997 it overviews the exact results in the theory of the sandpile model and discusses briefly the yet unsolved problem of calculation of avalanche distribution exponents the key ingredients include the analogy with the critical reaction diffusion system the spanning tree representation of height configurations and the decomposition of the avalanche process into waves of topplings

theoretical studies of self organized criticality these notes are intended to provide a pedagogical introduction to the abelian sandpile model of self organized criticality and its related models the abelian group the algebra of particle addition operators the burning test for recurrent states equivalence to the spanning trees problem are described the exact solution of the directed version of the model in any dimension is explained the model s equivalence to scheidegger s model of river basins takayasu s aggregation model and the voter model is discussed for the undirected case the solution for one dimensional lattices and the bethe lattice is briefly described known results about the two dimensional case are summarized generalization to the abelian distributed processors model is discussed time dependent properties and the universality of critical behavior in sandpiles are briefly discussed i conclude by listing some still unsolved problems

structure of two dimensional sandpile i height probabilities the height probabilities of the two dimensional abelian sandpile model are the fractionial numbers of lattice sites having heights 1 2 3 4 a combinatorial method for evaluation of these quantities is proposed the method is based on mapping the set of allowed sandpile configurations onto the set of spanning trees covering a given lattice exact analytical expressions for all probabilities are obtained

hermitian pencils with a cubic minimal polynomial let a be an n n complex matrix and write a h ik where h and k are hermitian matrices we show that if the minimal polynomial of the pencil xh yk has degree 3 then there is a unitary matrix u such that u 1au is block diagonal with blocks of size 3 3 or smaller this is a special case of a conjecture made by kippenhahn in 1951

r matrix theory formal casimirs and the periodic toda lattice the nonunitary r matrix theory and the associated bi and trihamiltonian schemes are considered the language of poisson pencils and of their formal casimirs is applied in this framework to characterize the bihamiltonian chains of integrals of motion pointing out the role of the schur polynomials in these constructions this formalism is subsequently applied to the periodic toda lattice some different algebraic settings and lax formulations proposed in the literature for this system are analyzed in detail and their full equivalence is exploited in particular the equivalence between the loop algebra approach and the method of differential difference operators is illustrated moreover two alternative lax formulations are considered and appropriate reduction algorithms are found in both cases allowing us to derive the multihamiltonian formalism from r matrix theory the systems of integrals for the periodic toda lattice known after flaschka and henon and their functional relations are recovered through systematic application of the previously outlined schemes