three new factors of fermat numbers we report the discovery of a new factor for each of the fermat numbers f 13 f 15 f 16 these new factors have 27 33 and 27 decimal digits respectively each factor was found by the elliptic curve method after division by the new factors and previously known factors the remaining cofactors are seen to be composite numbers with 2391 9808 and 19694 decimal digits respectively

zero one survival behavior of cyclically competing species the coexistence of competing species is due to unavoidable fluctuations always transient in this letter we investigate the ultimate survival probabilities characterizing different species in cyclic competition we show that they often obey a surprisingly simple though nontrivial behavior within a model where coexistence is neutrally stable we demonstrate a robust zero one law when the interactions between the three species are generically asymmetric the weakest species survives at a probability that tends to one for large population sizes while the other two are guaranteed to extinction we rationalize our findings from stochastic simulations by an analytic approach

saddles arrows and spirals deterministic trajectories in cyclic competition of four species population dynamics in systems composed of cyclically competing species has been of increasing interest recently here we investigate a system with four or more species using mean field theory we study in detail the trajectories in configuration space of the population fractions we discover a variety of orbits shaped like saddles spirals and straight lines many of their properties are found explicitly most remarkably we identify a collective variable that evolves simply as an exponential q e t where is a function of the reaction rates it provides information on the state of the system for late times as well as for t we discuss implications of these results for the evolution of a finite stochastic system a generalization to an arbitrary number of cyclically competing species yields valuable insights into universal properties of such systems

coexistence versus extinction in the stochastic cyclic lotka volterra model cyclic dominance of species has been identified as a potential mechanism to maintain biodiversity see e g b kerr m a riley m w feldman and b j m bohannan nature 418 171 2002 and b kirkup and m a riley nature 428 412 2004 through analytical methods supported by numerical simulations we address this issue by studying the properties of a paradigmatic non spatial three species stochastic system namely the rock paper scissors or cyclic lotka volterra model while the deterministic approach rate equations predicts the coexistence of the species resulting in regular yet neutrally stable oscillations of the population densities we demonstrate that fluctuations arising in the system with a finite number of agents drastically alter this picture and are responsible for extinction after long enough time two of the three species die out as main findings we provide analytic estimates and numerical computation of the extinction probability at a given time we also discuss the implications of our results for a broad class of competing population systems

locally conformally kahler manifolds admitting a holomorphic conformal flow a manifold m is locally conformally kahler lck if it admits a kahler covering tilde m with monodromy acting by holomorphic homotheties let m be an lck manifold admitting a holomorphic conformal flow of diffeomorphisms lifted to a non isometric homothetic flow on tilde m we show that m admits an automorphic potential and the monodromy group of its conformal weight bundle is mathbb z

the factorization of the ninth fermat number in this paper we exhibit the full prime factorization of the ninth fermat number f9 2 512 1 it is the product of three prime factors that have 7 49 and 99 decimal digits we found the two largest prime factors by means of the number field sieve which is a factoring algorithm that depends on arithmetic in an algebraic number field in the present case the number field used was q fifth root 2 the calculations were done on approximately 700 workstations scattered around the world and in one of the final stages a supercomputer was used the entire factorization took four months

pfaffian orientations and perfect matchings of scale free networks counting perfect matchings is of great interest in theoretical computer science mathematics among other disciplines however explicitly determining the number of perfect matchings of general graphs is a challenge in this paper we present a first analytical study of the enumeration problem of perfect matchings on scale free networks which are ubiquitous in real life systems we focus on two iteratively growing scale free networks one is fractal while the other is non fractal for each network we construct an orientation and prove that it is pfaffian then based on the connection between the number of perfect matchings and determinant of the skew adjacency matrix corresponding to a pfaffian orientation we derive the recursive expressions for the number of perfect matchings of the networks at two successive iterations furthermore using the recursive relations we show that for both scale free networks their entropy is zero which is in sharp contrast to the previously obtained results for other networks without scale free behavior but having the same average degree as the studied networks such as square lattice and honeycomb lattice whose entropies are larger than zero our results indicate from another angle that scale free networks have a distinct architecture in comparison with those lattices lacking the power law property

factors of fermat numbers and large primes of the form 2ⁿ 1 a new factor is given for each of the fermat numbers f52 f931 f6835 and f9448 in addition a factor of f75 discovered by gary gostin is presented the current status for all f is shown in a table primes of the form k 2 1 k odd are listed for 31 k 149 1500 ii 4000 and for 151 k 199 1000 ii 4000 some primes for even larger values of ii are included the largest one being 5 213165 1 also a survey of several related questions is given in particular values of k such that k 2 i is composite for every ii are considered as well as odd values of h such that 3h 2 1 never yields a twin prime pair

coexistence and survival in conservative lotka volterra networks analyzing coexistence and survival scenarios of lotka volterra lv networks in which the total biomass is conserved is of vital importance for the characterization of long term dynamics of ecological communities here we introduce a classification scheme for coexistence scenarios in these conservative lv models and quantify the extinction process by employing the pfaffian of the network s interaction matrix we illustrate our findings on global stability properties for general systems of four and five species and find a generalized scaling law for the extinction time

kahler parabolicity and the euler number of compact manifolds of non positive sectional curvature let m 2n be a compact riemannian manifold of non positive sectional curvature it is shown that if m 2n is homeomorphic to a kahler manifold then its euler number satisfies the inequality 1 n chi m 2n geq 0

optimal strategies for node selection games on oriented graphs skew matrices and symmetric games abstract in the node selection game d each of the two players simultaneously selects a node from the oriented graph d if there is an arc between the selected nodes then there is a payoff from the dominated player to the dominating player we investigate the set of optimal strategies for the players in the node selection game d we point out that a classical theorem from game theory relates the dimension of the polytope of optimal strategies for d to the nullity of certain skew submatrix of the payoff matrix for d we show that if d is bipartite with at least two nodes in each partite set then an optimal strategy for the node selection game d is never unique our work also implies that if d is a tournament then there is a unique optimal strategy for each player a result obtained by fisher and ryan optimal strategies for a generalized scissors paper and stone game amer math monthly 99 1992 935 942 and independently by laffond laslier and le breton the bipartisan set of a tournament game games econom behav 5 1993 182 201

factoring integers with elliptic curves this paper is devoted to the description and analysis of a new algorithm to factor positive integers it depends on the use of elliptic curves the new method is obtained from pollard s p 1 method proc cambridge philos soc 76 1974 521 528 by replacing the multiplicative group by the group of points on a random elliptic curve it is conjectured that the algorithm determines a non trivial divisor of a composite number n in expected time at most k p log n 2 where p is the least prime dividing n and k is a function for which log k x 2 o 1 log x log log x for x x o in the worst case when n is the product of two primes of the same order of magnitude this is

the twenty second fermat number is composite we have shown by machine proof that f 22 2 222 1 is composite in addition we reenacted young and buell s 1988 resolution of f 20 as composite finding agreement with their final selfridge hurwitz residues we also resolved the character of all extant cofactors of f n n 22 finding no new primes and ruling out prime powers

a golden point rule in rock paper scissors lizard spock game abstract we study a novel five species system on two dimensional lattices when each species have two superior and two inferior partners here we simplify the huge parameter space of predation probability to only two parameters both of monte carlo simulation and mean field theory reveal that two of strategies may die out when the ratio of the two parameters is close to the golden point 0 618 and the remaining three strategies are provided a cyclic dominance system

dimer statistics on the klein bottle abstract the problem of enumerating close packed dimers or perfect matchings on a quadratic lattice embedded on the klein bottle is considered thomassen c thomassen tilings of the torus and the klein bottle and vertex transitive graphs on a fixed surface trans amer math soc 323 2 1991 605 635 characterized that there are six quadrilateral lattices embedded on the klein bottle lu and wu w t lu and f y wu close packed dimers on nonorientable surfaces phys lett a 293 2002 235 246 had obtained a expression for the number of close packed dimers on one of them in this paper we investigate four other embeddings and obtain explicit expressions of the numbers of close packed dimers and free energy per dimer by enumerating pfaffians

evolutionary game theory theoretical concepts and applications to microbial communities ecological systems are complex assemblies of large numbers of individuals interacting competitively under multifaceted environmental conditions recent studies using microbial laboratory communities have revealed some of the self organization principles underneath the complexity of these systems a major role of the inherent stochasticity of its dynamics and the spatial segregation of different interacting species into distinct patterns has thereby been established it ensures the viability of microbial colonies by allowing for species diversity cooperative behavior and other kinds of social behavior a synthesis of evolutionary game theory nonlinear dynamics and the theory of stochastic processes provides the mathematical tools and a conceptual framework for a deeper understanding of these ecological systems we give an introduction into the modern formulation of these theories and illustrate their effectiveness focussing on selected examples of microbial systems intrinsic fluctuations stemming from the discreteness of individuals are ubiquitous and can have an important impact on the stability of ecosystems in the absence of speciation extinction of species is unavoidable it may however take very long times we provide a general concept for defining survival and extinction on ecological time scales spatial degrees of freedom come with a certain mobility of individuals when the latter is sufficiently high bacterial community structures can be understood through mapping individual based models in a continuum approach onto stochastic partial differential equations these allow progress using methods of nonlinear dynamics such as bifurcation analysis and invariant manifolds we conclude with a perspective on the current challenges in quantifying bacterial pattern formation and how this might have an impact on fundamental research in non equilibrium physics

speeding the pollard and elliptic curve methods of factorization since 1974 several algorithms have been developed that attempt to factor a large number n by doing extensive computations module n and occasionally taking gcds with n these began with pollard s p 1 and monte carlo methods more recently williams published a p 1 method and lenstra discovered an elliptic curve method ecm we present ways to speed all of these one improvement uses two tables during the second phases of p 1 and ecm looking for a match polynomial preconditioning lets us search a fixed table of size n with n 2 o n multiplications a parametrization of elliptic curves lets step 1 of ecm compute the x coordinate of np from that of p in about 9 3 1og2 n multiplications for arbitrary p

factorization of the tenth fermat number we describe the complete factorization of the tenth fermat number f10 by the elliptic curve method ecm f10 is a product of four prime factors with 8 10 40 and 252 decimal digits the 40 digit factor was found after about 140 mflop years of computation we also discuss the complete factorization of other fermat numbers by ecm and summarize the factorizations

a practical analysis of the elliptic curve factoring algorithm much asymptotic analysis has been devoted to factoring algorithms we present a practical analysis of the complexity of the elliptic curve algorithm suggesting optimal parameter selection and run time guidelines the parameter selection is aided by a novel use of bayesian statistical decision techniques as applied to random algorithms we discuss how frequently the elliptic curve algorithm succeeds in practice and compare it with the quadratic sieve algorithm

dimer problem on the cylinder and torus institute of mathematics academia sinica nsfc 10771086 10671162 fujian province university nsc nsc96 2115 m 001 005

dimers on the 33 42 lattice in this work we obtain explicit expression of the number of close packed dimers perfect matchings of the 33 42 lattice with cylindrical boundary condition particularly we show that the entropy of 33 42 lattice is the same for cylindrical and toroidal boundary conditions

dimers on the kagome lattice i finite lattices abstract we report exact results on the enumeration of close packed dimers on a finite kagome lattice with general asymmetric dimer weights under periodic and cylindrical boundary conditions for symmetric dimer weights the resulting dimer generating functions reduce to very simple expressions and we show how the simple expressions can be obtained from the consideration of a spin variable mapping

compact kahler manifolds homotopic to negatively curved riemannian manifolds in this paper we show that any compact kahler manifold homotopic to a compact riemannian manifold with negative sectional curvature admits a kahler einstein metric of general type moreover we prove that on a compact symplectic manifold x homotopic to a compact riemannian manifold with negative sectional curvature for any almost complex structure j compatible with the symplectic form there is no non constant j holomorphic entire curve f mathbb c rightarrow x

topologically robust zero sum games and pfaffian orientation how network topology determines the long time dynamics of the antisymmetric lotka volterra equation to explore how the topology of interaction networks determines the robustness of dynamical systems we study the antisymmetric lotka volterra equation alve the alve is the replicator equation of zero sum games in evolutionary game theory in which the strengths of pairwise interactions between strategies are defined by an antisymmetric matrix such that typically some strategies go extinct over time here we show that there also exist topologically robust zero sum games such as the rock paper scissors game for which all strategies coexist for all choices of interaction strengths we refer to such zero sum games as coexistence networks and construct coexistence networks with an arbitrary number of strategies by mapping the long time dynamics of the alve to the algebra of antisymmetric matrices we identify simple graph theoretical rules by which coexistence networks are constructed examples are triangulations of cycles characterized by the golden ratio phi 1 6180 cycles with complete subnetworks and non hamiltonian networks in graph theoretical terms we extend the concept of a pfaffian orientation from even sized to odd sized networks our results show that the topology of interaction networks alone can determine the long time behavior of nonlinear dynamical systems and may help to identify robust network motifs arising for example in ecology

maximum matchings in scale free networks with identical degree distribution the size and number of maximum matchings in a network have found a large variety of applications in many fields as a ubiquitous property of diverse real systems power law degree distribution was shown to have a profound influence on size of maximum matchings in scale free networks where the size of maximum matchings is small and a perfect matching often does not exist in this paper we study analytically the maximum matchings in two scale free networks with identical degree sequence and show that the first network has no perfect matchings while the second one has many for the first network we determine explicitly the size of maximum matchings and provide an exact recursive solution for the number of maximum matchings for the second one we design an orientation and prove that it is pfaffian based on which we derive a closed form expression for the number of perfect matchings moreover we demonstrate that the entropy for perfect matchings is equal to that corresponding to the extended sierpiski graph with the same average degree as both studied scale free networks our results indicate that power law degree distribution alone is not sufficient to characterize the size and number of maximum matchings in scale free networks

a note on euler number of locally conformally kahler manifolds let m 2n be a compact riemannian manifold of non positive resp negative sectional curvature we call m j theta a d bounded locally conformally kahler manifold if the lifted lee form tilde theta on the universal covering space of m is d bounded we show that if m 2n is homeomorphic to a d bounded lck manifold then its euler number satisfies the inequality 1 n chi m 2n ge resp 0

synchronization and extinction in cyclic games with mixed strategies we consider cyclic lotka volterra models with three and four strategies where at every interaction agents play a strategy using a time dependent probability distribution agents learn from a loss by reducing the probability to play a losing strategy at the next interaction for that an agent is described as an urn containing balls of three and four types respectively where after a loss one of the balls corresponding to the losing strategy is replaced by a ball representing the winning strategy using both mean field rate equations and numerical simulations we investigate a range of quantities that allows us to characterize the properties of these cyclic models with time dependent probability distributions for the three strategy case in a spatial setting we observe a transition from neutrally stable to stable when changing the level of discretization of the probability distribution for large values of yielding a good approximation to a continuous distribution spatially synchronized temporal oscillations dominate the system for the four strategy game the system is always neutrally stable but different regimes emerge depending on the size of the system and the level of discretization

extinction in neutrally stable stochastic lotka volterra models populations of competing biological species exhibit a fascinating interplay between the nonlinear dynamics of evolutionary selection forces and random fluctuations arising from the stochastic nature of the interactions the processes leading to extinction of species whose understanding is a key component in the study of evolution and biodiversity are influenced by both of these factors here we investigate a class of stochastic population dynamics models based on generalized lotka volterra systems in the case of neutral stability of the underlying deterministic model the impact of intrinsic noise on the survival of species is dramatic it destroys coexistence of interacting species on a time scale proportional to the population size we introduce a new method based on stochastic averaging which allows one to understand this extinction process quantitatively by reduction to a lower dimensional effective dynamics this is performed analytically for two highly symmetrical models and can be generalized numerically to more complex situations the extinction probability distributions and other quantities of interest we obtain show excellent agreement with simulations

a contribution to von neumann s theory of games ii abstract let a be the skew symmetric matrix attached to a symmetric two person zero sum game g then g is completely mixed i e both players must make some use of every strategy if and only if the principal pfaffians of a alternate in sign

exact solution of close packed dimers on the kagome lattice it is well known that exact enumerations of close packed dimers can be carried out for two dimensional lattices while details of results are now known for most lattices due to the unique nature of the lattice structure there has been no complete analysis for the kagom e lattice here we derive the close form expression 1 3 mathrm ln 4xyz for the free energy of close packed dimers on the kagom e lattice where x y and z are dimer weights we use two different approaches the kasteleyn method of evaluating a pfaffian and an alternative vertex model formulation both methods lead to the same final expression the correlation function between two dimers at a distance equal or greater than two lattice spacings is found to vanish identically