modeling and analysis of timed petri nets using heaps of pieces the authors show that safe timed petri nets can be represented by special automata over the max semiring which compute the height of heaps of pieces this extends to the timed case the classical representation a la mazurkiewicz of the behavior of safe petri nets by trace monoids and trace languages for a subclass including all safe free choice petri nets we obtain reduced heap realizations using structural properties of the net covering by safe state machine components the authors illustrate the heap based modeling by the typical case of safe jobshops for a periodic schedule the authors obtain a heap based throughput formula which is simpler to compute than its traditional timed event graph version particularly if one is interested in the successive evaluation of a large number of possible schedules

complexity of stability and controllability of elementary hybrid systems in this paper we consider simple classes of nonlinear systems and prove that basic questions related to their stability and controllability are either undecidable or computationally intractable np hard as a special case we consider a class of hybrid systems in which the state space is partitioned into two halfspaces and the dynamics in each halfspace correspond to a different linear system

on the limit products of a family of matrices in this paper we direct attention at bounded families of complex n n matrices in order to study their asymptotic behaviour we recall from linear algebra appl 322 2001 162 the concept of limit spectrum maximizing product and show that nondefective families always admit such limit products then we consider defective families in loc cite we proved that for finite families of 2 2 matrices defectivity is equivalent to the existence of defective such limit products this result led us to conjecture the validity of this property also for higher dimensions n 3 here instead by making use of the results obtained by bousch and mairesse j am math soc 15 2002 77 that disproved the well known finiteness conjecture we find some counterexamples to our conjecture in loc cite for all n 3 2002 elsevier science inc all rights reserved

convergence analysis of c2 hermite interpolatory subdivision schemes by explicit joint spectral radius formulas abstract subdivision schemes are popular iterative processes to build graphs of functions curves and surfaces we analyze the 2 point hermite c 2 subdivision scheme introduced by merrien in 26 for the analysis of its convergence and its smoothness properties we are concerned with the computation of the joint spectral radius of a family of 2 matrices associated with the scheme in this paper by an explicit computation of the joint spectral radius of such pairs of matrices we determine necessary and sufficient conditions for the scheme to be c 2 convergent whenever it reproduces cubic polynomials in addition we present two one parameter families of convergent subdivision schemes belonging to the class in 26 possessing interesting properties from the shape control point of view

stability of discrete linear inclusion let m ai be a set of linear operators on rn the discrete linear inclusion dli m is the set of possible trajectories xi i 0 such that xn ainain 1 ai1x0 where x0 rn and aij m we study several notions of stability for dli m including absolute asymptotic stability aas which is that all products ain ai1 0 as n we mainly study the case that m is a finite set we give criteria for the various forms of stability two new approaches are taken one relates the question of aas of dli m to formal language theory and finite automata while the second connects the aas property to the structure of a lie algebra associated to the elements of m more generally the discrete linear inclusion dli m makes sense for m contained in a banach algebra b we prove some results for aas in this case and give counterexamples showing that some results valid for finite sets of operators on rn are not true for finite sets m in a general banach algebras b

a generalization of sourour s theorem abstract let x be an invertible n n matrix n 1 with entries in some field k assume x diag a a for any a e k then for every sequence a 1 a n 1 a i e k there is a matrix y with entries in k and det y 1 such that the n 1 principal minors of yxy 1 have the values a 1 a n 1 respectively this generalizes sourour theorem where a i 0 is assumed for all i

on the asymptotic regularity of a family of matrices abstract in this paper we consider bounded families f of complex n n matrices we give sufficient conditions under which the sequence k f 1 k k 1 where k f is the supremum of the spectral radii of all possible products of k matrices chosen in f is convergent to its supremum f the so called generalized spectral radius of f we also illustrate a possible practical application

asymptotic stability and stabilizability of special classes of discrete time positive switched systems abstract in this paper we consider discrete time positive switched systems switching among autonomous subsystems characterized either by monomial matrices or by circulant matrices necessary and sufficient conditions are provided guaranteeing either global uniform asymptotic stability or stabilizability i e the possibility of driving to zero the state trajectory corresponding to any initial state by resorting to some switching sequence such conditions lead to simple algorithms that allow to easily detect under suitable conditions whether a given positive switched system is not stabilizable

some criteria for spectral finiteness of a finite subset of the real matrix space rd d abstract in this paper we present some checkable criteria for the spectral finiteness of a finite subset of the real d d matrix space r d d where 2 d

on the complexity of computing the capacity of codes that avoid forbidden difference patterns some questions related to the computation of the capacity of codes that avoid forbidden difference patterns are analysed the maximal number of n bit sequences whose pairwise differences do not contain some given forbidden difference patterns is known to increase exponentially with n the coefficient of the exponent is the capacity of the forbidden patterns in this paper new inequalities for the capacity are given that allow for the approximation of the capacity with arbitrary high accuracy the computational cost of the algorithm derived from these inequalities is fixed once the desired accuracy is given subsequently a polynomial time algorithm is given for determining if the capacity of a set is positive while the same problem is shown to be np hard when the sets of forbidden patterns are defined over an extended set of symbols finally the existence of extremal norms is proved for any set of matrices arising in the capacity computation based on this result a second capacity approximating algorithm is proposed the usefulness of this algorithm is illustrated by computing exactly the capacity of particular codes that were only known approximately

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

powers of 1 cyclic conjugacy classes in the groups gln f and sln f abstract let a gl n f where f is a field we say that a is 1 cyclic if a is similar to a matrix of the form a diag a 1 a 2 a k where a i gl l i f is cyclic for 1 i k l 1 0 1 and l i 2 for 2 i k it is shown that if a gl n f is 1 cyclic where n 2 and f 4 then every nonscalar matrix m gl n f whose determinant equals det a 4 is the product of four matrices which are similar to a under matrices of sl n f the problem of expressing a scalar matrix as a product of similar 1 cyclic matrices is also discussed the above result is applied to problems of factorizing matrices in the group sl n f into products of unipotent matrices of index 2 and into products of matrices of fixed finite order

optimal periodic orbits of chaotic systems occur at low period invariant sets embedded in a chaotic attractor can generate time averages that differ from the average generated by typical orbits on the attractor motivated by two different topics namely controlling chaos and riddled basins of attraction we consider the question of which invariant set yields the largest optimal value of an average of a given smooth function of the system state we present numerical evidence and analysis that indicate that the optimal average is typically achieved by a low period unstable periodic orbit embedded in the chaotic attractor in particular our results indicate that if we consider that the function to be optimized depends on a parameter gamma then the lebesgue measure in gamma corresponding to optimal periodic orbits of period ital p or greater decreases exponentially with increasing ital p furthermore the set of parameter values for which optimal orbits are nonperiodic typically has zero lebesgue measure copyright ital 1996 the american physical society

sums of orbits of algebraic groups i abstract let g be a simple algebraic group which is defined and split over a field k and let l be a corresponding lie algebra further let r be the corresponding root system we prove here that if char k 0 or a very good prime for r and k r then there exists an orbit o l l k with respect to the adjoint action of g g k such that l o o this is an analogue of the corresponding result for chevalley groups the thompson problem see e w ellers n gordeev on the conjecture of j thompson and o ore trans amer math soc 350 1998 3657 3671 we also prove that if char k 2 for r b r c r f 4 and char k 3 for r g 2 then the zariski closure of a sum of any r r b r r 3 d r e r f 4 r 1 r a r b 3 g 2 2r r c r g orbits of elements of l z l coincides with l for the group g sl 2 k where k is an algebraically closed field of characteristic zero we list all cases of rational k g modules v which have an orbit o v such that o o v

a tree based approach to joint spectral radius determination abstract we suggest a novel method to determine the joint spectral radius of finite sets of matrices by validating the finiteness property it is based on finding a certain finite tree with nodes representing sets of matrix products our approach accounts for cases where one or several matrix products satisfy the finiteness property moreover is potentially functional even for reducible sets of matrices

the finite step realizability of the joint spectral radius of a pair of d d matrices one of which being rank one abstract we study the finite step realizability of the joint generalized spectral radius of a pair of real square matrices s 1 and s 2 one of which has rank 1 where 2 d let a denote the spectral radius of a square matrix a then we prove that there always exists a finite length word i 1 i m 1 2 m for some finite m 1 such that s i 1 s i m m sup n 1 max i 1 i n 1 2 n s i 1 s i n n in other words there holds the spectral finiteness property for s 1 s 2 explicit formula for computation of the joint spectral radius is derived this implies that the stability of the switched system induced by s 1 s 2 is algorithmically decidable in this case

the boundedness of all products of a pair of matrices is undecidable we show that the boundedness of the set of all products of a given pair sigma of rational matrices is undecidable furthermore we show that the joint or generalized spectral radius rho sigma is not computable because testing whether rho sigma less than or equal to1 is an undecidable problem as a consequence the robust stability of linear systems under time varying perturbations is undecidable and the same is true for the stability of a simple class of hybrid systems we also discuss some connections with the so called finiteness conjecture our results are based on a simple reduction from the emptiness problem for probabilistic finite automata which is known to be undecidable c 2000 elsevier science b v all rights reserved

criteria for the stability of the finiteness property and for the uniqueness of barabanov norms a set of matrices is said to have the finiteness property if the maximal rate of exponential growth of long products of matrices drawn from that set is realised by a periodic product the extent to which the finiteness property is prevalent among finite sets of matrices is the subject of ongoing research in this article we give a condition on a finite irreducible set of matrices which guarantees that the finiteness property holds not only for that set but also for all sufficiently nearby sets of equal cardinality we also prove a theorem giving conditions under which the barabanov norm associated to a finite irreducible set of matrices is unique up to multiplication by a scalar and show that in certain cases these conditions are also persistent under small perturbations

the normed finiteness property of compact contraction operators we study the joint spectral radius given by a finite set of compact operators on a hilbert space it is shown that the normed finiteness property holds in this case that is if all the compact operators are contractions and the joint spectral radius is equal to 1 then there exists a finite product that has a spectral radius equal to 1 we prove an additional statement in that the requirement that the joint spectral radius be equal to 1 can be relaxed to the asking that the maximum norm of finite products of a length norm is equal to 1 the length of this product is related to the dimension of the subspace on which the set of operators is norm preserving

density dependent limits for a nonlinear reaction diffusion model a reaction diffusion model constructed on a grid by linking nonlinear density dependent birth and death processes through particle diffusion is studied by proving two laws of large numbers and a fluctuation theorem under varying assumptions on the density parameter

contractibility of compact contractions in hilbert space for a finite set of compact contractions in a complex hilbert space h it is shown that r a 1 for all a in the multiplicative semigroup generated by if and only if there exists a positive integer n such that a 1 for all a in the multiplicative semigroup generated by with length greater than n here r a denotes the spectral radius of a as an application an answer is given to an infinite dimensional case of the finiteness conjecture for the generalized spectral radius attributed to j c lagarias and y wang linear algebra appl 214 1995 17

efficient algorithms for deciding the type of growth of products of integer matrices for a given finite set sigma of matrices with nonnegative integer entries we study the growth with t of max parallel to a 1 a t parallel to a i epsilon sigma we show how to determine in polynomial time whether this growth is bounded polynomial or exponential and we characterize all possible behaviors c 2007 elsevier inc all rights reserved

comparison of stochastic and deterministic models of a linear chemical reaction with diffusion particles placed in n cells on the unit interval give birth or die according to linear rates adjacent cells are coupled by diffusion with a rate proportional to n2 cell numbers are divided by a density parameter to represent concentrations and the resulting space time markov process is compared to a corresponding deterministic model the solution to a partial differential equation the models are viewed as hilbert space valued processes and compared by means of a law of large numbers and central limit theorem new and nearly optimal results are obtained by exploiting the ornstein uhlenbeck type structure of the stochastic model

sets of matrices all infinite products of which converge an infinite product iit lmi of matrices converges on the right if limi m mi exists a set z ai i l of n x n matrices is called an rcp set rightconvergent product set if all infinite products with each element drawn from z converge such sets of matrices arise in constructing self similar objects like von koch s snowflake curve in various interpolation schemes in constructing wavelets of compact support and in studying nonhomogeneous markov chains this paper gives necessary conditions and also some sufficient conditions for a set x to be an rcp set these are conditions on the eigenvalues and left eigenspaces of matrices in 2 and finite products of these matrices necessary and sufficient conditions are given for a finite set z to be an rcp set having a limit function m d rit lad where d d d which is a continuous function on the space of all sequences d with the sequence topology finite rcp sets of column stochastic matrices are completely characterized some results are given on the problem of algorithmically deciding if a given set x is an rcp set

covering gl v and products of blockcyclic matrices let v be an n dimensional vector space over a field k where n n n 3 k 4 a linear transformation v v is called blockcyclic if the minimum rank satisfies n 2 we study products of blockcyclic conjugacy classes in gl v example if and are products of blockcyclic conjugacy classes in gl v then contains a cyclic transformation the results are used in our study on products of singular similarity classes knuppel and nielsen in press 8 a further application of our methods is the following theorem let k n n 1 a non central conjugacy class of gl v and gl v such that det det k then k a similar result was proved in lev s article lev 1996 10

brief paper explicit construction of a barabanov norm for a class of positive planar discrete time linear switched systems we consider the stability under arbitrary switching of a discrete time linear switched system a powerful approach for addressing this problem is based on studying the most unstable switching law musl if the solution of the switched system corresponding to the musl converges to the origin then the switched system is stable for any switching law the musl can be characterized using optimal control techniques this variational approach leads to a hamilton jacobi bellman equation describing the behavior of the switched system under the musl the solution of this equation is sometimes referred to as a barabanov norm of the switched system although the barabanov norm was studied extensively it seems that there are few examples where it was actually computed in closed form in this paper we consider a special class of positive planar discrete time linear switched systems and provide a closed form expression for a corresponding barabanov norm and a musl the unit circle in this norm is a parallelogram

bounded normal generation and invariant automatic continuity abstract we study the question of how quickly products of a fixed conjugacy class in the projective unitary group of a ii1 factor von neumann algebra cover the entire group our result is that the number of factors that are needed is essentially as small as permitted by the 1 norm in analogy to results of liebeck and shalev for non abelian finite simple groups as an application of the techniques we prove that every homomorphism from the projective unitary group of a finite factor to a polish sin group is continuous a result which is even new for pu n moreover we show that the projective unitary group of a ii1 factor carries a unique polish group topology

corrigendum addendum to sets of matrices all infinite products of which converge abstract this corrigendum addendum supplies corrected statements and proofs of some results in our paper appearing in linear algebra appl 161 1992 227 263 these results concern special kinds of bounded semigroups of matrices it also reports on progress on the topics of this paper made in the last eight years

hourglass alternative and the finiteness conjecture for the spectral characteristics of sets of non negative matrices abstract recently blondel nesterov and protasov proved 1 2 that the finiteness conjecture holds for the generalized and the lower spectral radii of the sets of non negative matrices with independent row column uncertainty we show that this result can be obtained as a simple consequence of the so called hourglass alternative used in 3 by the author and his companions to analyze the minimax relations between the spectral radii of matrix products axiomatization of the statements that constitute the hourglass alternative makes it possible to define a new class of sets of positive matrices having the finiteness property which includes the sets of non negative matrices with independent row uncertainty this class of matrices supplemented by the zero and identity matrices forms a semiring with the minkowski operations of addition and multiplication of matrix sets which gives means to construct new sets of non negative matrices possessing the finiteness property for the generalized and the lower spectral radii

brief paper periodically switched stability induces exponential stability of discrete time linear switched systems in the sense of markovian probabilities the conjecture that periodically switched stability implies absolute asymptotic stability of random infinite products of a finite set of square matrices has recently been disproved under the guise of the finiteness conjecture in this paper we show that this conjecture holds in terms of markovian probabilities more specifically let s k c n x n 1 k k be arbitrarily given k matrices and s k k j j 1 1 k j k for each j 1 where n k 2 then we study the exponential stability of the following discrete time switched dynamics s x j s k j s k 1x 0 j 1 and x 0 c n where k j j 1 s k can be an arbitrary switching sequence for a probability row vector p p 1 p k r k and an irreducible markov transition matrix p r k x k with pp p we denote by m p p the markovian probability on s k corresponding to p p by using symbolic dynamics and ergodic theoretic approaches we show that if s possesses the periodically switched stability then i it is exponentially stable m p p almost surely ii the set of stable switching sequences k j j 1 s k has the same hausdorff dimension as s k thus the periodically switched stability of a discrete time linear switched dynamics implies that the system is exponentially stable for almost all switching sequences

the covering number of the grouppsln f abstract let g be the group psl n f where n 3 f is a field and f 4 assume further that if n 3 then f is either finite or algebraically closed given an integer k and a subset a g denote a k a 1 a 2 a k a 1 a 2 a k a denote by cn g the minimal value of k such that c k g for every nontrivial conjugacy class c of g it is shown that cn g n related results on factorizations of matrices in gl n f and sl n f are also discussed

stability and stabilizability criteria for discrete time positive switched systems in this paper we consider the class of discrete time switched systems switching between p autonomous positive subsystems first sufficient conditions for testing stability based on the existence of special classes of common lyapunov functions are investigated and these conditions are mutually related thus proving that if a linear copositive common lyapunov function can be found then a quadratic positive definite common function can be found too and this latter in turn ensures the existence of a quadratic copositive common function secondly stabilizability is introduced and characterized it is shown that if these systems are stabilizable they can be stabilized by means of a periodic switching sequence which asymptotically drives to zero every positive initial state conditions for the existence of state dependent stabilizing switching laws based on the values of a copositive linear quadratic lyapunov function are investigated and mutually related too finally some properties of the patterns of the stabilizing switching sequences are investigated and the relationship between a sufficient condition for stabilizability the existence of a schur convex combination of the subsystem matrices and an equivalent condition for stabilizability the existence of a schur matrix product of the subsystem matrices is explored

the generalized spectral radius and extremal norms the generalized spectral radius also known under the name of joint spectral radius or after taking logarithms maximal lyapunov exponent of a discrete inclusion is examined we present a new proof for a result of barabanov which states that for irreducible sets of matrices an extremal norm always exists this approach lends itself easily to the analysis of further properties of the generalized spectral radius we prove that the generalized spectral radius is locally lipschitz continuous on the space of compact irreducible sets of matrices and show a strict monotonicity property of the generalized spectral radius sufficient conditions for the existence of extremal norms are obtained

the berger wang formula for the markovian joint spectral radius abstract the berger wang formula establishes equality between the joint and generalized spectral radii of a set of matrices for matrix products whose multipliers are applied not arbitrarily but in accordance with some markovian law there are also known analogs of the joint and generalized spectral radii however the known proofs of the berger wang formula hardly can be directly applied in the case of markovian products of matrices since they essentially rely on the arbitrariness of appearance of different matrices in the related matrix products nevertheless as has been shown by x dai 1 the berger wang formula is valid for the case of markovian analogs of the joint and the generalized spectral radii too although the proof in this case heavily exploits the more involved techniques of multiplicative ergodic theory in the paper we propose a matrix theory construction allowing to deduce the markovian analog of the berger wang formula from the classical berger wang formula

the finiteness conjecture for the generalized spectral radius of a set of matrices the generalized spectral radius g9 of a set of n n matrices is g9 lim supk g9k 1k where g9k sup a1a2 ak each ai the joint spectral radius g9 is g9 lim supk g9k 1k where g9k sup a1 ak each ai it is known that g9 g9 holds for any finite set of n n matrices the finiteness conjecture asserts that for any finite set of real n n matrices there exists a finite k such that g9 g9 g9k 1k the normed finiteness conjecture for a given operator norm asserts that for any finite set a1 am having all ai op 1 either g9 1 or g9 g9 g9k 1k 1 for some finite k it is shown that the finiteness conjecture is true if and only if the normed finiteness conjecture is true for all operator norms the normed finiteness conjecture is proved for a large class of operator norms extending results of gurvits in particular for polytope norms and for the euclidean norm explicit upper bounds are given for the least k having g9 g9k 1k these results imply upper bounds for generalized critical exponents for these norms

the extended covering number of sln is n 1 abstract for a special linear group sln k k a field we want to find the minimal integer k extended covering number such that 1 c1 ck for arbitrary non central conjugacy classes ci of sln k using methods from chevalley groups length theorems on products of simple mappings and theorems on products of cyclic mappings we find that k n 1 provided n and the field are not too small

the eigenvalue problem for linear and affine iterated function systems abstract the eigenvalue problem for a linear function l centers on solving the eigen equation lx x this paper generalizes the eigenvalue problem from a single linear function to an iterated function system f consisting of possibly an infinite number of linear or affine functions the eigen equation becomes f x x where 0 is real x is a compact set and f x f f f x the main result is that an irreducible linear iterated function system f has a unique eigenvalue equal to the joint spectral radius of the functions in f and a corresponding eigenset s that is centrally symmetric star shaped and full dimensional results of barabanov and of dranisnikov konyagin protasov on the joint spectral radius follow as corollaries

performance evaluation of max automata automata with multiplicities over the max semiring can be used to represent the behavior of timed discrete event systems this formalism which extends both conventional automata and max linear representations covers a class of systems with synchronization phenomena and variable schedules performance evaluation is considered in the worst mean and optimal cases a simple algebraic reduction is provided for the worst case the last two cases are solved for the subclass of deterministic series recognized by deterministic automata deterministic series frequently arise due to the finiteness properties of max linear projective semigroups the mean performance is given by the kolmogorov equation of a markov chain the optimal performance is given by a hamilton jacobi bellman equation

kazhdan constants for conjugacy classes of compact groups kazhdan constants relative to conjugacy classes of compact groups are computed they depend on the nontrivial irreducible characters of the respective group the result is applied in particular to finite groups of lie type symmetric groups and the group su n

a stochastic model of a chemical reaction with diffusion stochastic systems of brownian motions with multiple deletion of particles are introduced to model a chemical reaction with diffusion convergence to the solution of a deterministic nonlinear reaction diffusion equation is proved without high density assumptions

bounded semigroups of matrices in this note are proved two conjectures of daubechies and lagarias the first asserts that if z is a bounded set of matrices such that all left infinite products converge then 8 generates a bounded semigroup the second asserts the equality of two differently defined joint spectral radii for a bounded set of matrices one definition involves the conventional spectral radius and one involves the operator norm

optimal norms and the computation of joint spectral radius of matrices the notion of spectral radius of a set of matrices is a natural extension of spectral radius of a single matrix the finiteness conjecture fc claims that among the infinite products made from the elements of a given finite set of matrices there is a certain periodic product made from the repetition of the optimal product whose rate of growth is maximal fc has been disproved in this paper it is conjectured that fc is almost always true and an algorithm is presented to verify the optimality of a given product the algorithm uses optimal norms as a special subset of extremal norms several conjectures related to optimal norms and non decomposable sets of matrices are presented the algorithm has successfully calculated the spectral radius of several parametric families of pairs of matrices associated with compactly supported multi resolution analyses and wavelets the results of related numerical experiments are presented

asymptotic stability and generalized gelfand spectral radius formula let be a set of n n complex matrices for m 1 2 let m be the set of all products of matrices in of length m denote by the multiplicative semigroup generated by is said to be asymptotically stable in the sense of dynamical systems if there is 0 1 such that there are bounded neighborhoods u v cn of the origin for which av mu for all a m m 1 2 for a bounded set of n n complex matrices it is shown that the following conditions are mutually equivalent i is asymptotically stable ii g9 lim supm supa m a 1 m 1 iii lim supm supa m a 1 m 1 where a stands for the spectral radius of a and iv there exists a positive number such that a 1 for all a this fact answers an open question raised by brayton and tong the generalized gelfand spectral radius formula that is g9 conjectured by daubechies and lagarias and proved by berger and wang using advanced tools from ring theory and then by elsner using analytic geometric tools follows immediately form the above asymptotic stability theorem

asymptotic height optimization for topical ifs tetris heaps and the finiteness conjecture given an iterated function system ifs of topical maps verifying some conditions we prove that the asymptotic height optimization problems are equivalent to finding the extrema of a continuous functional the average height on some compact space of measures we give general results to determine these extrema and then apply them to two concrete problems first we give a new proof of the theorem that the densest heaps of two tetris pieces are sturmian second we construct an explicit counterexample to the lagarias wang finiteness conjecture

high density limit theorems for nonlinear chemical reactions with diffusion the solutionx of a nonlinear reaction diffusion equation on then dimensional unit cubes is approximated by a space time jump markov processx v n law of large numbers lln x v n is constructed on a grids n ons ofn cells wherev is proportional to the initial number of particles in each cell the deviation ofx v n fromx is computed by a central limit theorem clt the assumptions on the parametersv n are for the lln asn and for the clt frac n upsilon to 0 asn the limity y x in the clt which is a generalized ornstein uhlenbeck process is represented as the mild solution of a linear stochastic partial differential equation spde and its best possible state spaces are described the problem of stationary solutions ofy x in dependence ofx is also investigated