phase space networks of the six vertex model under different boundary conditions the six vertex model is mapped to three dimensional sphere stacks and different boundary conditions corresponding to different containers the shape of the container provides a qualitative visualization of the boundary effect based on the sphere stacking picture we map the phase spaces of the six vertex models to discrete networks a node in the network represents a state of the system and an edge between two nodes represents a zero energy spin flip which corresponds to adding or removing a sphere the network analysis shows that the phase spaces of systems with different boundary conditions share some common features we derived a few formulas for the number and the sizes of the disconnected phase space subnetworks under the periodic boundary conditions the sphere stacking provides new challenges in combinatorics and may cast light on some two dimensional models

internal diffusion limited aggregation parallel algorithms and complexity the computational complexity of internal diffusion limited aggregation dla is examined from both a theoretical and a practical point of view we show that for two or more dimensions the problem of predicting the cluster from a given set of paths is complete for the complexity class cc the subset of p characterized by circuits composed of comparator gates cc completeness is believed to imply that in the worst case growing a cluster of size n requires polynomial time in n even on a parallel computer a parallel relaxation algorithm is presented that uses the fact that clusters are nearly spherical to guess the cluster from a given set of paths and then corrects defects in the guessed cluster through a nonlocal annihilation process the parallel running time of the relaxation algorithm for two dimensional internal dla is studied by simulating it on a serial computer the numerical results are compatible with a running time that is either polylogarithmic in n or a small power of n thus the computational resources needed to grow large clusters are significantly less on average than the worst case analysis would suggest for a parallel machine with k processors we show that random clusters in d dimensions can be generated in mathcal o n k logk n 2 d steps this is a significant speedup over explicit sequential simulation which takes mathcal o n 1 2 d time on average finally we show that in one dimension internal dla can be predicted in mathcal o logn parallel time and so is in the complexity class nc

height representation critical exponents and ergodicity in the four state triangular potts antiferromagnet we study the four state antiferromagnetic potts model on the triangular lattice we show that the model has six types of defects which diffuse and annihilate according to certain conservation laws consistent with their having a vector valued topological charge using the properties of these defects we deduce a 2 2 dimensional height representation for the model and hence show that the model is equivalent to the three state potts antiferromagnet on the kagome lattice and to bond coloring models on the triangular and honeycomb lattices we also calculate critical exponents for the ground state ensemble of the model we find that the exponents governing the spin spin correlation function and spin fluctuations violate the fisher scaling law because of constraints on path length which increase the effective wavelength of the spin operator on the height lattice we confirm our predictions by extensive monte carlo simulations of the model using the wang swendsen koteck cluster algorithm although this algorithm is not ergodic on lattices with toroidal boundary conditions we prove that it is ergodic on lattices whose topology has no noncontractible loops of infinite order such as the projective plane to guard against biases introduced by lack of ergodicity we perform our simulations on both the torus and the projective plane

a quantitative central limit theorem for the excursion area of random spherical harmonics over subdomains of s2 in recent years considerable interest has been drawn by the analysis of geometric functionals for the excursion sets of random eigenfunctions on the unit sphere spherical harmonics in this paper we extend those results to proper subsets of the sphere s2 i e spherical caps focussing in particular on the excursion area precisely we show that the asymptotic behaviour of the excursion area is dominated by the so called second order chaos component and we exploit this result to establish a quantitative central limit theorem in the high energy limit these results generalize analogous findings for the full sphere their proofs however require more sophisticated techniques in particular a careful analysis of some independent interest for smooth approximations of the indicator function for spherical cap subsets

on the area of excursion sets of spherical gaussian eigenfunctions the high frequency behaviour for random eigenfunctions of the spherical laplacian has been recently the object of considerable interest also because of strong motivation arising from physics and cosmology in this paper we are concerned with the high frequency behaviour of excursion sets in particular we establish a uniform central limit theorem for the empirical measure i e the proportion of spherical surface where spherical gaussian eigenfunctions lie below a level z our proofs borrow some techniques from the literature on stationary long memory processes in particular we expand the empirical measure into hermite polynomials and establish a uniform weak reduction principle entailing that the asymptotic behaviour is asymptotically dominated by a single term in the expansion as a result we establish a functional central limit theorem the limiting process is fully degenerate

on the computational complexity of ising spin glass models in a spin glass with ising spins the problems of computing the magnetic partition function and finding a ground state are studied in a finite two dimensional lattice these problems can be solved by algorithms that require a number of steps bounded by a polynomial function of the size of the lattice in contrast to this fact the same problems are shown to belong to the class of np hard problems both in the two dimensional case within a magnetic field and in the three dimensional case np hardness of a problem suggests that it is very unlikely that a polynomial algorithm could exist to solve it

on the dynamics of lotka volterra equations having an invariant hyperplane the dynamics of n dimensional lotka volterra equations having an invariant hyperplane is investigated herein there exists a fundamental difference between the even and the odd dimensional case for even dimensions the dynamics are generated by a hamiltonian system with respect to an appropriately chosen poisson structure for odd dimensions the dynamics are hamiltonian if there is a continuum of interior fixed points and gradient like if the interior fixed point is unique in this case the invariant hyperplane is part of the omega limit set and the dynamics restricted to this set are those of a hamiltonian system furthermore an example of a completely integrable four dimensional hamiltonian lotka volterra system is presented

geometric approach for global asymptotic stability of three dimensional lotka volterra systems abstract in this paper by applying the geometric criterion and time average property to lotka volterra systems some results for the global asymptotic stability of the systems are obtained furthermore we consider li wang conjecture for a three dimensional system which is transformed from a lotka volterra system

decomposition of besov and triebel lizorkin spaces on the sphere abstract a discrete system of almost exponentially localized elements needlets on the n dimensional unit sphere s n is constructed it shown that the needlet system can be used for decomposition of besov and triebel lizorkin spaces on the sphere as an application of besov spaces on s n a jackson estimate for nonlinear m term approximation from the needlet system is obtained

from logarithmic to subdiffusive polynomial fluctuations for internal dla and related growth models we consider a cluster growth model on zd called internal diffusion limited aggregation internal dla in this model random walks start at the origin one at a time and stop moving when reaching a site not occupied by previous walks it is known that the asymptotic shape of the cluster is spherical when dimension is 2 or more we prove that fluctuations with respect to a sphere are at most a power of the logarithm of its radius in dimension d 2 in so doing we introduce a closely related cluster growth model that we call the flashing process whose fluctuations are controlled easily and accurately this process is coupled to internal dla to yield the desired bound part of our proof adapts the approach of lawler bramson and griffeath on another space scale and uses a sharp estimate written by blachere in our appendix on the expected time spent by a random walk inside an annulus

phase space networks of geometrically frustrated systems we illustrate a network approach to the phase space study by using two geometrical frustration models antiferromagnet on triangular lattice and square ice their highly degenerated ground states are mapped as discrete networks such that the quantitative network analysis can be applied to phase space studies the resulting phase spaces share some comon features and establish a class of complex networks with unique gaussian spectral densities although phase space networks are heterogeneously connected the systems are still ergodic due to the random poisson processes this network approach can be generalized to phase spaces of some other complex systems

on nonlinear functionals of random spherical eigenfunctions we prove central limit theorems and stein like bounds for the asymptotic behaviour of nonlinear functionals of spherical gaussian eigenfunctions our investigation combines asymptotic analysis of higher order moments for legendre polynomials and in addition recent results on malliavin calculus and total variation bounds for gaussian subordinated fields we discuss applications to geometric functionals like the defect and invariant statistics e g polyspectra of isotropic spherical random fields both of these have relevance for applications especially in an astrophysical environment

on the distribution of the nodal sets of random spherical harmonics we study the volume of the nodal set of eigenfunctions of the laplacian on the m dimensional sphere it is well known that the eigenspaces corresponding to en n n m 1 are the spaces en of spherical harmonics of degree n of dimension n we use the multiplicity of the eigenvalues to endow en with the gaussian probability measure and study the distribution of the m dimensional volume of the nodal sets of a randomly chosen function the expected volume is proportional to en one of our main results is bounding the variance of the volume to be o en n in addition to the volume of the nodal set we study its leray measure we find that its expected value is n independent we are able to determine that the asymptotic form of the variance is const n

tilings with trichromatic colored edges triangles this paper studies the tilings with colored edges triangles constructed on a triangulation of a simply connected orientable surface such that the degree of each interior vertex is even such as for fundamental example a part of the triangular lattice of the plane the constraints are that we only use three colors all the colors appear in each tile and two tile can share an edge only if this edge has the same color in both tiles using previous results on lozenge tilings we give a linear algorithm of coloration for triangulations of the sphere or of planar regions with the constraint that the boundary is monochromatic we define a flip as a shift of colors on a cycle of edges using only two colors we prove flip connectivity of the set of solutions for the cases seen above i e two tiling are mutually accessible by a sequence of flips and prove that there is no flip accessibility in the general case where the boundary is not assumed to be monochromatic nevertheless using flips we obtain a tiling invariant even in the general case we finish relaxing the condition allowing monochromatic triangles with this hypothesis some local flips are sufficient for connectivity we give a linear algorithm of coloration and strong structural results on the set of solutions

the c1 property of carrying simplices for a class of competitive systems of odes abstract in this paper we consider a class of dissipative competitive systems of odes on the nonneptive orthant c we assume among others that the system restricted to any face of c is weakly persistent it is proved that the carrying simplex is a c1 manifold with corners in the proof we analyze the asymptotic behavior of the linearization of the original system

limit cycles for competitor competitor mutualist lotka volterra systems abstract it is known that a limit cycle or periodic coexistence can occur in a competitor competitor mutualist lotka volterra system x 1 x 1 r 1 a 11 x 1 a 12 x 2 a 13 x 3 x 2 x 2 r 2 a 21 x 1 a 22 x 2 a 23 x 3 x 3 x 3 r 3 a 31 x 1 a 32 x 2 a 33 x 3 where r i a i j are positive real constants x liang j jiang the dynamical behavior of type k competitive kolmogorov systems and its applications to 3 dimensional type k competitive lotka volterra systems nonlinearity 16 2003 785 801 in this paper we shall construct an example with at least two limit cycles and furthermore we will show that the number of periodic orbits and hence a fortiori of limit cycles is finite it is also shown that contrary to three dimensional competitive lotka volterra systems nontrivial periodic coexistence does occur even if none of the three species can resist invasion from either of the others in this case new amenable conditions are given on the coefficients under which the system has no nontrivial periodic orbits these conditions imply that the positive equilibrium if it exists is globally asymptotically stable

ribbon tilings and multidimensional height functions we fix n and say a square in the two dimensional grid indexed by x y has color c if x y c mod n a ribbon tile of order n is a connected polyomino containing exactly one square of each color we show that the set of order n ribbon tilings of a simply connected region r is in one to one correspondence with a set of height functions from the vertices of r to z n satisfying certain difference restrictions it is also in one to one correspondence with the set of acyclic orientations of a certain partially oriented graph using these facts we describe a linear in the area of r algorithm for determining whether r can be tiled with ribbon tiles of order n and producing such a tiling when one exists we also resolve a conjecture of pak by showing that any pair of order n ribbon tilings of r can be connected by a sequence of local replacement moves some of our results are generalizations of known results for order 2 ribbon tilings a k a domino tilings we also discuss applications of multidimensional height functions to a broader class of polyomino tiling problems

predicting nonlinear cellular automata quickly by decomposing them into linear ones we show that a wide variety of nonlinear cellular automata cas can be decomposed into a quasidirect product of linear ones these cas can be predicted by parallel circuits of depth o log2 t using gates with binary inputs or o log t depth if sum mod p gates with an unbounded number of inputs are allowed thus these cas can be predicted by idealized parallel computers much faster than by explicit simulation even though they are nonlinear this class includes any ca whose rule when written as an algebra is a solvable group we also show that cas based on nilpotent groups can be predicted in depth o log t or o 1 by circuits with binary or sum mod p gates respectively we use these techniques to give an efficient algorithm for a ca rule which like elementary ca rule 18 has diffusing defects that annihilate in pairs this can be used to predict the motion of defects in rule 18 in o log2 t parallel time

limit cycles for the competitive three dimensional lotka volterra system abstract in the first part of this paper it is proved that the number of limit cycles of the competitive three dimensional lotka volterra system in r 3 is finite if this system has not any heteroclinic polycycles in r 3 in the second part of this paper a 3d competitive lotka volterra system with two small parameters is discussed this system always has a heteroclinic polycycle with three saddles it is proved that there exists one parameter range in which the system is persistence and has at least two limit cycles and there exists other parameter ranges in which the system is not persistence and has at least one limit cycle

a new kempe invariant and the non ergodicity of the wang swendsen kotecký algorithm we prove that for the class of three colorable triangulations of a closed oriented surface the degree of a four coloring modulo 12 is an invariant under kempe changes we use this general result to prove that for all triangulations t 3l 3m of the torus with 3 l m there are at least two kempe equivalence classes this result implies in particular that the wang swendsen koteck algorithm for the zero temperature 4 state potts antiferromagnet on these triangulations t 3l 3m of the torus is not ergodic

nodal length fluctuations for arithmetic random waves using the spectral multiplicities of the standard torus we endow the laplace eigenspaces with gaussian probability measures this induces a notion of random gaussian laplace eigenfunctions on the torus arithmetic random waves we study the distribution of the nodal length of random eigenfunctions for large eigenvalues and our primary result is that the asymptotics for the variance is nonuniversal our result is intimately related to the arithmetic of lattice points lying on a circle with radius corresponding to the energy

non deterministic approach to anisotropic growth patterns with continuously tunable morphology the fractal properties of some real snowflakes the authors demonstrate that a tunable family of patterns resembling a range of experimentally observed snowflakes is obtained from a simple but somewhat ad hoc non deterministic growth model an extension of the niemeyer pietronero wiesmann eta model in which they i use noise reduction to enable the asymptotic behaviour to become more readily apparent from simulations of reasonable size ii replace the eta model boundary conditions by the boundary conditions of the diffusion limited aggregation dla model and iii include a surface tension parameter they measure the fractal dimension of some real snowflakes and obtain quantitative agreement with certain cases of the model they conclude that the model although somewhat ad hoc gives patterns that are in both qualitative and quantitative agreement with a range of real snowflake patterns

carrying simplices in nonautonomous and random competitive kolmogorov systems abstract the purpose of this paper is to investigate the asymptotic behavior of positive solutions of nonautonomous and random competitive kolmogorov systems via the skew product flows approach it is shown that there exists an unordered carrying simplex which attracts all nontrivial positive orbits of the skew product flow associated with a nonautonomous random competitive kolmogorov system

generalized quotients in coxeter groups for w s a coxeter group we study sets of the form w v w e w i l wv 1 w i v for all v e v where v c w such sets w v here called generalized quotients are shown to have much of the rich combinatorial structure under bruhat order that has previously been known only for the case when v c s i e for minimal coset representatives modulo a parabolic subgroup we show that bruhat intervals in w v for general v c w are lexicographically shellable the mobius function on w v under bruhat order takes values in 1 0 1 for finite groups w generalized quotients are the same thing as lower intervals in the weak order this is however in general not true connections with the weak order are explored and it is shown that w v is always a complete meet semilattice and a convex order ideal as a subset of w under weak order descent classes di w e w il1 ws 1 w xt s e i for all s e s i c s are also analyzed using generalized quotients it is shown that each descent class as a poset under bruhat order or weak order is isomorphic to a generalized quotient under the corresponding ordering the latter half of the paper is devoted to the symmetric group and to the study of some specific examples of generalized quotients which arise in combinatorics for instance the set of standard young tableaux of a fixed shape or the set of linear extensions of a rooted forest suitably interpreted form generalized quotients we prove a factorization result for the quotients that come from rooted forests which shows that algebraically these quotients behave as a system of minimal coset representatives of a subset which is in general not a subgroup we also study the rank generating function for certain quotients in the symmetric group

the computational complexity of pattern formation the computational complexity of diffusion limited aggregation and fluid invasion in porous media is studied the time requirements on an idealized parallel computer for simulating the patterns formed by these models are investigated it is shown that these growth models are p complete these results provide strong evidence that such pattern formation processes are inherently sequential and cannot be simulated efficiently in parallel

gaussian random measures generated by berry s nodal sets we consider vectors of random variables obtained by restricting the length of the nodal set of berry s random wave model to a finite collection of possibly overlapping smooth compact subsets of mathbb r 2 our main result shows that as the energy diverges to infinity and after an adequate normalisation these random elements converge in distribution to a gaussian vector whose covariance structure reproduces that of a homogeneous independently scattered random measure a by product of our analysis is that when restricted to rectangles the dominant chaotic projection of the nodal length field weakly converges to a standard wiener sheet in the banach space of real valued continuous mappings over a fixed compact set an analogous study is performed for complex valued random waves in which case the nodal set is a locally finite collection of random points

the computational complexity of generating random fractals we examine a number of models that generate random fractals the models are studied using the tools of computational complexity theory from the perspective of parallel computation diffusion limited aggregation and several widely used algorithms for equilibrating the ising model are shown to be highly sequential it is unlikely they can be simulated efficiently in parallel this is in contrast to mandelbrot percolation which can be simulated in constant parallel time our research helps shed light on the intrinsic complexity of these models relative to each other and to different growth processes that have been recently studied using complexity theory in addition the results may serve as a guide to simulation physics

exponential stability of a competitive lotka volterra system with delays abstract in this paper the exponential stability of a two species competitive lotka volterra system with discrete delays has been studied by constructing appropriate lyapunov functional and using the linear matrix inequality lmi optimization approach a series of sufficient criteria is obtained which ensure the exponential stability of the positive equilibrium of the given lotka volterra system the results are new and improve some earlier work an illustrative example is given to demonstrate the effectiveness of the obtained results

asymptotics for spherical needlets we investigate invariant random fields on the sphere using a new type of spherical wavelets called needlets these are compactly supported in frequency and enjoy excellent localization properties in real space with quasi exponentially decaying tails we show that for random fields on the sphere the needlet coefficients are asymptotically uncorrelated for any fixed angular distance this property is used to derive clt and functional clt convergence results for polynomial functionals of the needlet coefficients here the asymptotictheory is considered in the high frequency sense our proposals emerge from strong empirical motivations especially in connection with the analysis of cosmological data sets 1 introduction over the last two decades wavelets have emerged as one of the most interesting tools of statistical investigation in this paper we give an application to the statistical analysis of data sets indexed by the unit sphere s 2 this is motivated mostly by the analysis of the cosmic

quasilinear cellular automata abstract simulating a cellular automaton ca for t time steps into the future requires t 2 serial computation steps or t parallel ones however certain cas based on an abelian group such as addition mod 2 are termed linear because they obey a principle of superposition this allows them to be predicted efficiently in serial time o t robinson 1987 or o log t in parallel in this paper we generalize this by looking at cas with a variety of algebraic structures including quasigroups non abelian groups steiner systems and other structures we show that in many cases an efficient algorithm exists even though these cas are not linear in the previous sense we term them quasilinear we find examples which can be predicted in serial time proportional to t t log t t log 2 t and t for t log t log log t and log 2 t we also discuss what algebraic properties are required or implied by the existence of scaling relations and principles of superposition and exhibit several novel vector valued cas

an o n log n algorithm for finding a domino tiling of a plane picture whose number of holes is bounded we consider the problem of tiling a plane picture with dominoes this picture can be with holes or without holes we give a necessary and sufficient condition for the existence of such a tiling and then we deduce an algorithm to decide whether a picture is tileable or not and if it is to determine a tiling this algorithm runs in o kn n log n time where n and k are the respective numbers of cells and holes we also propose a simplified version of this algorithm which is linear in the area enclosed by the outer boundary provided the picture has a bounded number of holes

domino tiling in planar graphs with regular and bipartite dual abstract we prove that thurston s linear time tiling algorithm can be extended to all planar graphs that have a regular bipartite dual and that it does not depend on group theory

dynamics of a family of lotka volterra systems in r3 abstract we provide the phase portraits of the 3 dimensional lotka volterra systems x x y a z y y x z z b z a x y for all the values of the parameters a and b in the finite region and in the infinity region through the poincare compactification we also study the integrability of the system

factoring polynomials with rational coefficients in this paper we present a polynomial time algorithm to solve the following problem given a non zero polynomial fe q x in one variable with rational coefficients find the decomposition of f into irreducible factors in q x it is well known that this is equivalent to factoring primitive polynomials fez x into irreducible factors in z x here we call f z x primitive if the greatest common divisor of its coefficients the content of f is 1 our algorithm performs well in practice cf 8 its running time measured in bit operations is o nl2 n9 log fd3

parallel algorithm and dynamic exponent for diffusion limited aggregation a parallel algorithm for diffusion limited aggregation dla is described and analyzed from the perspective of computational complexity the dynamic exponent z of the algorithm is defined with respect to the probabilistic parallel random access machine pram model of parallel computation according to t sim l z where l is the cluster size t is the running time and the algorithm uses a number of processors polynomial in l it is argued that z d d 2 2 where d is the fractal dimension and d 2 is the second generalized dimension simulations of dla are carried out to measure d 2 and to test scaling assumptions employed in the complexity analysis of the parallel algorithm it is plausible that the parallel algorithm attains the minimum possible value of the dynamic exponent in which case z characterizes the intrinsic history dependence of dla

nodal intersections for random eigenfunctions on the torus we investigate the number of nodal intersections of random gaussian laplace eigenfunctions on the standard two dimensional flat torus arithmetic random waves with a fixed smooth reference curve with nonvanishing curvature the expected intersection number is universally proportional to the length of the reference curve times the wavenumber independent of the geometry our main result prescribes the asymptotic behavior of the nodal intersections variance for smooth curves in the high energy limit remarkably it is dependent on both the angular distribution of lattice points lying on the circle with radius corresponding to the given wavenumber and the geometry of the given curve in particular this implies that the nodal intersection number admits a universal asymptotic law with arbitrarily high probability

the computational complexity of sandpiles given an initial distribution of sand in an abelian sandpile what final state does it relax to after all possible avalanches have taken place in d 3 we show that this problem is p complete so that explicit simulation of the system is almost certainly necessary we also show that the problem of determining whether a sandpile state is recurrent is p complete in d 3 and briefly discuss the problem of constructing the identity in d 1 we give two algorithms for predicting the sandpile on a lattice of size n both faster than explicit simulation a serial one that runs in time mathcal o n log n and a parallel one that runs in time mathcal o log3n i e the class nc3 the latter is based on a more general problem we call additive ranked generability this leaves the two dimensional case as an interesting open problem

global dynamics of a classical lotka volterra competition diffusion advection system abstract in this paper we study a classical two species lotka volterra competition diffusion advection system where the diffusion and advection rates of two competitors are supposed to be proportional by employing the principal spectral theory we first establish a key a priori estimate on the co existence positive steady state which is a powerful tool to link the local and global dynamics we then further present a complete classification on all possible long time dynamical behaviors by appealing to the theory of monotone dynamical systems lastly we apply these results to a special situation where two species are competing for the same resources and obtain a sharp criteria in term of certain variable parameters for all kinds of global dynamics this work gives a positive answer to the conjecture proposed by lou et al in 34 by considering a more general model under certain conditions and also can be seen as a further development of he and ni 19 for competition diffusion system where we bring new ingredients in the arguments to overcome the difficulty caused by the involvement of advection

logarithmic fluctuations for internal dla let each of n particles starting at the origin in z 2 perform simple random walk until reaching a site with no other particles lawler bramson and griffeath proved that the resulting random set a n of n occupied sites is with high probability close to a disk br of radius r p n we show that the discrepancy between a n and the disk is at most logarithmic in the radius i e there is an absolute constant c such that with probability 1 br c log r a r 2 br c log r for all suciently large r

three dimensional competitive lotka volterra systems with no periodic orbits the following conjecture of m l zeeman is proved if three interacting species modeled by a competitive lotka volterra system can each resist invasion at carrying capacity then there can be no coexistence of the species indeed two of the species are driven to extinction it is also proved that in the other extreme if none of the species can resist invasion from either of the others then there is stable coexistence of at least two of the species in this case if the system has a fixed point in the interior of the positive cone in r3 then that fixed point is globally asymptotically stable representing stable coexistence of all three species otherwise there is a globally asymptotically stable fixed point in one of the coordinate planes of r3 representing stable coexistence of two of the species

from local to global behavior in competitive lotka volterra systems in this paper we exploit the linear quadratic monotone and geometric structures of competitive lotka volterra systems of arbitrary dimension to give geometric algebraic and computational hypotheses for ruling out nontrivial recurrence we thus deduce the global dynamics of a system from its local dynamics the geometric hypotheses rely on the introduction of a split liapunov function we show that if a system has a fixed point p int r n and the carrying simplex of the system lies to one side of its tangent hyperplane at p then there is no nontrivial recurrence and the global dynamics are known we translate the geometric hypotheses into algebraic hypotheses in terms of the definiteness of a certain quadratic function on the tangent hyperplane finally we derive a computational algorithm for checking the algebraic hypotheses and we compare this algorithm with the classical volterra liapunov stability theorem for lotka volterra systems

structural and computational depth of diffusion limited aggregation diffusion limited aggregation dla is studied from the perspective of computational complexity a parallel algorithm is exhibited that requires a number of steps that scales as the depth of the tree defined by the cluster the existence of this algorithm suggests a connection between a fundamental computational and structural property of dla

tilings of rectangles with t tetrominoes we prove that any two tilings of a rectangular region by t tetrominoes are connected by moves involving only 2 and 4 tiles we also show that the number of such tilings is an evaluation of the tutte polynomial the results are extended to a more general class of regions

a central limit theorem and higher order results for the angular bispectrum the angular bispectrum of spherical random fields has recently gained an enormous importance especially in connection with statistical inference on cosmological data in this paper we analyze its moments and cumulants of arbitrary order and we use these results to establish a multivariate central limit theorem and higher order approximations the results rely upon combinatorial methods from graph theory and a detailed investigation for the asymptotic behavior of coefficients arising in matrix representation theory for the group of rotations so 3

ribbon tile invariants let t be a finite set of tiles and b a set of regions f tileable by t we introduce a tile counting group g t b as a group of all linear relations for the number of times each tile t c t can occur in a tiling of a region r c b we compute the tile counting group for a large set of ribbon tiles also known as rim hooks in a context of representation theory of the symmetric group the tile counting group is presented by its set of generators which consists of certain new tile invariants in a special case these invariants generalize the conway lagarias invariant for tromino tilings and a height invariant which is related to computation of characters of the symmetric group the heart of the proof is the known bijection between rim hook tableaux and certain standard skew young tableaux we also discuss signed tilings by the ribbon tiles and apply our results to the tileability problem

chaotic attractors in the four dimensional leslie gower competition model abstract we study the occurrence of the chaotic attractor in the four dimensional classical leslie gower competition model we find that chaos can be generated by a cascade of quasiperiod doubling bifurcations starting from a supercritical neimark sacker bifurcation of the positive fixed point in this model the chaotic attractor is contained in the three dimensional carrying simplex that is a globally attracting invariant manifold biologically the result implies that the invasion attempts by an invader into a trimorphic population under the leslie gower dynamics can lead to chaos

limiting the complexity of limit sets in self regulating systems in the analysis of models of ecosystems one seeks to discover conditions that limit the complexity of the possible behavior of solutions since intuitively one feels that the full range of possible complex behaviors in systems of order three or more ought not to occur for simple ecological interactions it has been shown by hirsch 6 7 that the solutions of competitive and cooperative systems have limit sets which cannot be more complicated than invariant sets of systems of one lower dimension in particular autonomous 2 dimensional systems of these types have only trivial dynamics in the sense that all bounded solutions approach equilibrium asymptotically in planar systems for example the absence of limit cycles makes the dynamics trivial in the sense that bounded solutions can have only critical points or orbits connecting critical points in their omega limit sets 3 in this note we prove a theorem which appears to be useful in limiting the complexity of limit sets for a class of biologically important equations

dynamics integrability and topology for some classes of kolmogorov hamiltonian systems in r 4 abstract in this paper we first give the sufficient and necessary conditions in order that two classes of polynomial kolmogorov systems in r 4 are hamiltonian systems then we study the integrability of these hamiltonian systems in the liouville sense finally we investigate the global dynamics of the completely integrable lotka volterra hamiltonian systems in r 4 as an application of the invariant subsets of these systems we obtain topological classifications of the 3 submanifolds in r 4 defined by the hypersurfaces a x y b z w c x 2 y d x y 2 e z 2 w f z w 2 h where a b c d e f w and h are real constants

internal dla and the stefan problem generalized internal diffusion limited aggregation is a stochastic growth model on the lattice in which a finite number of sites act as poisson sources of particles which then perform symmetric random walks with an attractive zero range interaction until they reach the first site which has been visited by fewer than alpha particles at which point they stop sites on which particles are frozen constitute the occupied set we prove that in appropriate regimes the particle density has a hydrodynamic limit which is the one phase stefan problem this is then used to study the asymptotic behavior of the occupied set in two dimensions when the walks are independent with one source at the origin and alpha 1 we obtain in particular that the occupied set is asymptotically a disc of radius k sqrt t where k is the solution of exp k 2 4 pi k 2 settling a conjecture of lawler bramson and griffeath

majority vote cellular automata ising dynamics and p completeness we study cellular automata where the state at each site is decided by a majority vote of the sites in its neighborhood these are equivalent for a restricted set of initial conditions to nonzero probability transitions in single spin flip dynamics of the ising model at zero temperature we show that in three or more dimensions these systems can simulate boolean circuits of and and or gates and are therefore p complete that is predicting their state t time steps in the future is at least as hard as any other problem that takes polynomial time on a serial computer therefore unless a widely believed conjecture in computer science is false it is impossible even with parallel computation to predict majority vote cellular automata or zero temperature single spin flip ising dynamics qualitatively faster than by explicit simulation

asymptotic behavior of densities for two particle annihilating random walks consider the system of particles on d where particles are of two types a andb and execute simple random walks in continuous time particles do not interact with their own type but when ana particle meets ab particle both disappear i e are annihilated this system serves as a model for the chemical reactiona b inert we analyze the limiting behavior of the densities a t and b t when the initial state is given by homogeneous poisson random fields we prove that for equal initial densities a 0 b 0 there is a change in behavior fromd 4 where a t b t c t d 4 tod 4 where a t b t c tast for unequal initial densities a 0 b 0 a t e c l ind 1 a t e ct logt ind 2 and a t e ct ind 3 the termc depends on the initial densities and changes withd techniques are from interacting particle systems the behavior for this two particle annihilation process has similarities to those for coalescing random walks a a a and annihilating random walks a a inert the analysis of the present process is made considerably more difficult by the lack of comparison with an attractive particle system

subdiffusive fluctuations for internal diffusion limited aggregation internal diffusion limited aggregation internal dla is a cluster model in z d where new points are added by starting random walkers at the origin and letting them run until they have found a new point to add to the cluster it has been shown that the limiting shape of internal dla clusters is spherical here we show that for d 2 the fluctuations are subdiffusive in fact that they are of order at most n 1 3 at least up to logarithmic corrections more precisely we show that for all sufficiently large n the cluster after m d n d steps covers all points in the ball of radius n n 1 3 in n 2 and is contained in the ball of radius n n 1 3 in n 4

triangular sos models and cubic crystal shapes a solid on solid sos model in a field h conjugate to the orientation of the surface is exactly solved with the aid of pfaffians the free energy h directly gives the equilibrium shape of a finite crystal the phase diagram exhibits rough and smooth phases corresponding to rounded and flat portions of the crystal surface the solid on solid model undergoes transitions of the pokrovsky talapov type 1979 characterised by a specific heat exponent alpha 1 2 one special point of the phase diagram corresponds to the appearance of a facet via an alpha 0 transition height height correlations are derived along a special line in the phase diagram with the aid of the known equivalence of this sos model with an ising model several exponents can be translated from one model to the other this enables one to derive the topology of the phase diagram of the antiferromagnetic triangular ising model with first and second neighbour couplings in a field

upper and lower bounds for the ground state entropy of antiferromagnetic potts models we derive rigorous upper and lower bounds for the ground state entropy of the q state potts antiferromagnet on the honeycomb and triangular lattices these bounds are quite restrictive especially for large q copyright ital 1997 ital the american physical society

stein malliavin approximations for nonlinear functionals of random eigenfunctions on s d abstract we investigate stein malliavin approximations for nonlinear functionals of geometric interest for random eigenfunctions on the unit d dimensional sphere s d d 2 all our results are established in the high energy limit i e as the corresponding eigenvalues diverge in particular we prove a quantitative central limit theorem for the excursion volume of gaussian eigenfunctions this goal is achieved by means of several results of independent interest concerning the asymptotic analysis for the variance of moments of gaussian eigenfunctions the rates of convergence in various probability metrics for hermite subordinated processes and quantitative central limit theorems for arbitrary polynomials of finite order or general square integrable nonlinear transforms some related issues were already considered in the literature for the 2 dimensional case s 2 our results are new or improve the existing bounds even in these special circumstances proofs are based on the asymptotic analysis of moments of all order for gegenbauer polynomials and make extensive use of the recent literature on so called fourth moment theorems by nourdin and peccati

antiferromagnetic potts models on the square lattice a high precision monte carlo study we study the antiferromagnetic q state potts model on the square lattice for q 3 and q 4 using the wang swendsen koteck wsk monte carlo algorithm and a powerful finite size scaling extrapolation method for q 3 we obtain good control up to correlation length 5000 the data are consistent with ae2 p 1 a1e as with p 1 the staggered susceptibility behaves as stagg 5 3 for q 4 the model is disordered 2 even at zero temperature in appendices we prove a correlation inequality for potts antiferromagnets on a bipartite lattice and we prove ergodicity of the wsk algorithm at zero temperature for potts antiferromagnets on a bipartite lattice

types of markov fields and tilings the method of types is one of the most popular techniques in information theory and combinatorics however thus far the method has been mostly applied to 1 d markov processes and it has not been thoroughly studied for general markov fields markov fields over a finite alphabet of size m ge 2 can be viewed as models for multidimensional systems with local interactions the locality of these interactions is represented by a shape mathcal s while its marking by symbols of the underlying alphabet is called a tile two assignments in a markov field have the same type if they have the same empirical distribution i e if they have the same number of tiles of a given type our goal is to study the growth of the number of possible markov field types in either a d dimensional box of lengths n 1 ldots n d or its cyclic counterpart a d dimensional torus we relate this question to the enumeration of nonnegative integer solutions of a large system of diophantine linear equations called the conservation laws we view a markov type as a vector in a d m mathcal s dimensional space and count the number of such vectors satisfying the conservation laws which turns out to be the number of integer points in a certain polytope for the torus this polytope is of dimension mu d 1 mathrm rk mathrm mathrm c where mathrm rk mathrm mathrm c is the number of linearly independent conservation laws mathrm mathrm c this provides an upper bound on the number of types then we construct a matching lower bound leading to the conclusion that the number of types in the torus markov field is theta n mu where n n 1 ldots n d these results are derived by geometric tools including ideas of discrete and convex multidimensional geometry

parallel dynamics and computational complexity of network growth models the parallel computational complexity or depth of growing network models is investigated the networks considered are generated by preferential attachment rules where the probability of attaching a new node to an existing node is given by a power alpha of the connectivity of the existing node algorithms for generating growing networks very quickly in parallel are described and studied the sublinear and superlinear cases require distinct algorithms as a result there is a discontinuous transition in the parallel complexity of sampling these networks corresponding to the discontinuous structural transition at alpha 1 where the networks become scale free for alpha 1 networks can be generated in constant time while for 0 alpha 1 logarithmic parallel time is required the results show that these networks have little depth and embody very little history dependence despite being defined by sequential growth rules

tile invariants new horizons let t be a finite set of tiles the group of invariants g t introduced by pak trans ams 352 2000 5525 is a group of linear relations between the number of copies of tiles in tilings of the same region we survey known results about g the height function approach the local move property various applications and special cases

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

dynamics of competitive lotka volterra systems that can be projected to a line abstract it is known that for a three species autonomous competitive lotka volterra system where axial fixed points are local attractors almost all trajectories converge to the axial fixed points for an n species autonomous competitive lotka volterra system we prove that almost all trajectories converge to the axial fixed points if the axial fixed points are local attractors and the system can be radially projected to a line

a combinatorial formula for the character of the diagonal coinvariants author s haglund j haiman m loehr n remmel j b ulyanov a abstract let rn be the ring of coinvariants for the diagonal action of the symmetric group sn it is known that the character of rn as a doubly graded s module can be expressed using the frobenius characteristic map as nabla en where en is the n th elementary symmetric function and nabla is an operator from the theory of macdonald polynomials we conjecture a combinatorial formula for nabla en and prove that it has many desirable properties that support our conjecture in particular we prove that our formula is a symmetric function which is not obvious and that it is schur positive these results make use of the theory of ribbon tableau generating functions of lascoux leclerc and thibon we also show that a variety of earlier conjectures and theorems on nabla en are special cases of our conjecture finally we extend our conjectures on nabla en and several on the results supporting them to higher powers nablam en

dynamical behaviors determined by the lyapunov function in competitive lotka volterra systems dynamical behaviors of the competitive lotka volterra system even for 3 species are not fully understood in this paper we study this problem from the perspective of the lyapunov function we construct explicitly the lyapunov function using three examples of the competitive lotka volterra system for the whole state space 1 the general 2 species case 2 a 3 species model and 3 the model of may leonard the basins of attraction for these examples are demonstrated including cases with bistability and cyclical behavior the first two examples are the generalized gradient system where the energy dissipation may not follow the gradient of the lyapunov function in addition under a new type of stochastic interpretation the lyapunov function also leads to the boltzmann gibbs distribution on the final steady state when multiplicative noise is added

linearization and invariant manifolds on the carrying simplex for competitive maps abstract a result due to m w hirsch states that most competitive maps admit a carrying simplex i e an invariant hypersurface of codimension one which attracts all nontrivial orbits the common approach in the study of these maps is to focus on the dynamical behavior on the carrying simplex however this manifold is normally non smooth therefore not every tool coming from differential geometry can be applied in this paper we prove that the restriction of the map to the carrying simplex in a neighborhood of an interior fixed point is topologically conjugate to the restriction of the map to its pseudo unstable manifold by an invariant foliation this implies that the linearization techniques are applicable for studying the local dynamics of the interior fixed points on the carrying simplex we further construct the stable and unstable manifolds on the carrying simplex our results give partial responses to hirsch s problem regarding the smoothness of the carrying simplex we discuss some applications in classical models of population dynamics

geometric method for a global repellor in competitive lotka volterra systems abstract the geometric method of involving the relative positions of the nullcline planes is used to analyze the global asymptotic behavior of solutions of autonomous lotka volterra competitive systems some observations about limit sets are made and based on these observations new criteria are established for the system to have a single point global repellor clearly these criteria can be used to preclude the existence of nonconstant periodic solutions

high resolution asymptotics for the angular bispectrum of spherical random fields in this paper we study the asymptotic behavior of the angular bispectrum of spherical random fields here the asymptotic theory is developed in the framework of fixed radius fields which are observed with increasing resolution as the sample size grows the results we present are then exploited in a set of procedures aimed at testing non gaussianity for these statistics we are able to show convergence to functionals of standard brownian motion under the null hypothesis analytic results are also presented on the behavior of the tests in the presence of a broad class of non gaussian alternatives the issue of testing for non gaussianity on spherical random fields has recently gained enormous empirical importance especially in connection with the statistical analysis of cosmic microwave background radiation