prisoner s dilemma on a stochastic nongrowth network evolution model we investigate the evolution of cooperation on a nongrowth dynamic network model with a death birth dynamics based on tournament selection in the limit of large population size the equilibrium cooperator density is well described by our mean field approximations and inversely related to the average degree in the system small populations are also examined and found to deviate considerably from their expected mean field behavior an expanded replicator equation incorporating gaussian fluctuations in the strategy densities is then constructed with its output agreeing well with our simulation data for all sizes we also briefly comment on the role of strategy mutation in sustaining polymorphic populations in small systems

compatibility of local and global langlands correspondences we prove the compatibility of local and global langlands correspondences for gln which was proved up to semisimplification in ht more precisely for the ndimensional l adic representation rl of the galois group of an imaginary cm field l attached to a conjugate self dual regular algebraic cuspidal automorphic representation of gln al which is square integrable at some finite place we show that frobenius semisimplification of the restriction of rl to the decomposition group of a place v of l not dividing l corresponds to v by the local langlands correspondence if v is square integrable for some finite place v 6 l we deduce that rl is irreducible we also obtain conditional results in the case v l

similarity based cooperation and spatial segregation we analyze a cooperative game where the cooperative act is not based on the previous behavior of the coplayer but on the similarity between the players this system has been studied in a mean field description recently phys rev e 68 046129 2003 here the spatial extension to a two dimensional lattice is studied where each player interacts with eight players in a moore neighborhood the system shows a strong segregation independent of parameters the introduction of a local conversion mechanism towards tolerance allows for four state cycles and the emergence of spiral waves in the spatial game in the case of asymmetric costs of cooperation a rich variety of complex behavior is observed depending on both cooperation costs finally we study the stabilization of a cooperative fixed point of a forecast rule in the symmetric game which corresponds to cooperation across segregation borders this fixed point becomes unstable for high cooperation costs but can be stabilized by a linear feedback mechanism

evolutionary dynamics of spatial games abstract a lattice model of coevolution of strategies for two person 2 2 matrix games is introduced the model allows evolution in an unbounded space of strategies in particular we explore the region of parameter space corresponding to the prisoner s dilemma and classify the types of dynamical and evolutionary behavior that appear for certain restricted strategy spaces we explore the complete space of lattice games

stochastic stability in spatial games we discuss similarities and differences between systems of interacting players maximizing their individual payoffs and particles minimizing their interaction energy long run behavior of stochastic dynamics of spatial games with multiple nash equilibria is analyzed in particular we construct an example of a spatial game with three strategies where stochastic stability of nash equilibria depends on the number of players and the kind of dynamics

conformal blocks and generalized theta functions let yp be the moduli space of rank r vector bundles with trivial determinant on a riemann surface x this space carries a natural line bundle the determinant line bundle s we describe a canonical isomorphism of the space of global sections of sk with the space of conformal blocks defined in terms of representations of the lie algebra 5 c z it follows in particular that the dimension of h yk is given by the verlinde formula

evolutionary games on minimally structured populations population structure induced by both spatial embedding and more general networks of interaction such as model social networks have been shown to have a fundamental effect on the dynamics and outcome of evolutionary games these effects have however proved to be sensitive to the details of the underlying topology and dynamics here we introduce a minimal population structure that is described by two distinct hierarchical levels of interaction similar to the structured metapopulation concept of ecology and island models in population genetics we believe this model is able to identify effects of spatial structure that do not depend on the details of the topology while effects depending on such details clearly lie outside the scope of our approach we expect that those we are able to reproduce should be generally applicable to a wide range of models we derive the dynamics governing the evolution of a system starting from fundamental individual level stochastic processes through two successive mean field approximations in our model of population structure the topology of interactions is described by only two parameters the effective population size at the local scale and the relative strength of local dynamics to global mixing we demonstrate for example the existence of a continuous transition leading to the dominance of cooperation in populations with hierarchical levels of unstructured mixing as the benefit to cost ratio becomes smaller then the local population size applying our model of spatial structure to the repeated prisoner s dilemma we uncover a counterintuitive mechanism by which the constant influx of defectors sustains cooperation further exploring the phase space of the repeated prisoner s dilemma and also of the rock paper scissor game we find indications of rich structure and are able to reproduce several effects observed in other models with explicit spatial embedding such as the maintenance of biodiversity and the emergence of global oscillations

killer geometries in competing species dynamics we discuss a cellular automata model to study the competition between an emergent better fitted species against an existing majority species the model implements local fights among small group of individuals and a synchronous random walk on a 2d lattice the faith of the system i e the spreading or disappearance of the species is determined by their initial density and fight frequency the initial density of the emergent species has to be higher than a critical threshold for total spreading but this value depends in a non trivial way on the fight frequency below the threshold any better adapted species disappears showing that a qualitative advantage is not enough for a minority to win no strategy is involved but spatial organization turns out to be crucial for instance at minority densities of zero measure some very rare local geometries which occur by chance are found to be killer geometries once set they lead with high probability to the total destruction of the preexisting majority species the occurrence rate of these killer geometries is a function of the system size this model may apply to a large spectrum of competing groups like smoker non smoker opinion forming diffusion of innovation setting of industrial standards species evolution epidemic spreading and cancer growth

cooperation in spatial prisoner s dilemma with two types of players for increasing number of neighbors we study a spatial two strategy cooperation and defection prisoner s dilemma game with two types a and b of players located on the sites of a square lattice the evolution of strategy distribution is governed by iterated strategy adoption from a randomly selected neighbor with a probability depending on the payoff difference and also on the type of the neighbor the strategy adoption probability is reduced by a prefactor w 1 from the players of type b we consider the competition between two opposite effects when increasing the number of neighbors k 4 8 and 24 within a range of the portion of influential players type a the inhomogeneous activity in strategy transfer yields a relevant increase dependent on w in the density of cooperators the noise dependence of this phenomenon is also discussed by evaluating phase diagrams

complete branching rules for specht modules abstract we give a combinatorial description for when the specht module of an arbitrary diagram admits a complete branching rule this description given in terms of the maximal rectangles of the diagram generalizes all previously known branching rules for specht modules such as those given by reiner and shimozono for northwest diagrams and by the present author for forest diagrams

effects of strategy migration direction and noise in the evolutionary spatial prisoner s dilemma spatial games are crucial for understanding patterns of cooperation in nature and to some extent society they are known to be more sensitive to local symmetries than e g spin models this paper concerns the evolution of the prisoner s dilemma game on regular lattices with three different types of neighborhoods char22 the von neumann moore and kagom e types we investigate two kinds of dynamics for the players to update their strategies that can be unconditional cooperator or defector depending on the payoff difference an individual can adopt the strategy of a random neighbor a voter model like dynamics vmld or impose its strategy on a random neighbor i e invasion process like dynamics ipld in particular we focus on the effects of noise in combination with the strategy dynamics on the evolution of cooperation we find that vmld compared to ipld better supports the spreading and sustaining of cooperation we see that noise has nontrivial effects on the evolution of cooperation maximum cooperation density can be realized either at a medium noise level in the limit of zero noise or in both these regions the temptation to defect and the local interaction structure determine the outcome especially in the low noise limit the local interaction plays a crucial role in determining the fate of cooperators we elucidate these both by numerical simulations and mean field cluster approximation methods

on the flatness of models of certain shimura varieties of pel type consider a pel shimura variety associated to a unitary group that splits over an unramified extension of q p rapoport and zink have defined a model of the shimura variety over the ring of integers of the completion of the reflex field at a place lying over p with parahoric level structures at p we show that this model is flat as conjectured by rapoport and zink and that its special fibre is reduced

a smooth space of tetrahedra abstract we construct a smooth symmetric compactification of the space of all labeled tetrahedra in p 3

frobenius splitting and cohomology vanishing for schubert varieties let k be an algebraically closed field of characteristic p 0 and x be a projective variety over k we then have the absolute frobenius f x x and an injection ox f 6x given by f fp f e o9x if this makes ox a direct summand in f ox as an x module we call x a frobenius split variety for such a variety the vanishing theorem for ample line bundles follows trivially from serre s vanishing theorem for tensoring ox f ox by an ample line bundle l and noting that l f x f f l f lp projection formula we get that the map in cohomology h x l h x f lp h x lp is an injection iterating this we see that h x l injects into h x lp for every p but for large v the latter is zero thus frobenius split varieties have quite pleasant properties it also turns out that using duality for the frobenius morphism or equivalently the cartier operator one can give a manageable criterion for frobenius splitting the point here is the local nature of duality the relevance of the compatibility of local and global duality was suggested to us by grothendieck s proof of h x y 0 for a noncomplete variety x of dimension n 4 theorem 6 9 and by kempf s paper 10 by this criterion and the bott samelson demazure desingularisation of schubert varieties it follows very easily that schubert varieties in characteristic p are frobenius split the following vanishing theorem is then an immediate consequence let g be a reductive group over the field k of arbitrary characteristic zero or positive let q be a parabolic subgroup and x c g q a schubert variety let l be an ample line bundle on g q then hi x l 0 for i 0 and the restriction map h0 g q l h0 x l is surjective if char k 0 this is a consequence of the compatible frobenius splitting of x in g q and the char k 0 case is handled by semicontinuity the above result for special schubert varieties has been proved by seshadrimusili lakshmibai 12 13 and by kempf 9 by using characteristic free methods for x g q andersen 1 and haboush 5 have given simple proofs using characteristic p methods

topological flatness of local models in the ramified case local models are schemes defined in terms of linear algebra which can be used to study the local structure of integral models of certain shimura varieties with parahoric level structure we investigate the local models for groups of the form open image in new window is a totally ramified extension as defined by pappas and rapoport and show that they are topologically flat in the linear case flatness can be deduced from this

geometry of the tetrahedron space abstract let x be the space of all labeled tetrahedra in p 3 in e babson p e gunnells r scott a smooth space of tetrahedra adv math 165 2 2002 285 312 we constructed a smooth symmetric compactification x of x in this article we show that the complement x x is a divisor with normal crossings and we compute the cohomology ring h x q

spatial evolutionary prisoner s dilemma game with three strategies and external constraints the emergency of mutual cooperation is studied in a spatially extended evolutionary prisoner s dilemma game in which the players are located on the sites of cubic lattices for dimensions d51 2 and 3 each player can choose one of the three following strategies cooperation c defection d or tit for tat t during the evolutionary process the randomly chosen players adopt one of their neighboring strategies if the chosen neighbor has a higher payoff moreover an external constraint imposes that the players always cooperate with probability p the stationary state phase diagram is computed by both using generalized mean field approximations and monte carlo simulations nonequilibrium second order phase transitions associated with the extinction of one of the possible strategies are found and the corresponding critical exponents belong to the directed percolation universality class it is shown that externally forcing the collaboration does not always produce the desired result

evolutionary prisoner s dilemma in random graphs abstract we study an evolutionary version of the spatial prisoner s dilemma game spd where the agents are placed in a random graph for graphs with fixed connectivity we show that for low values of the final density of cooperating agents c depends on the initial conditions however if the graphs have large connectivities c is independent of the initial conditions we characterize the phase diagram of the system using both extensive numerical simulations and analytical computations it is shown that two well defined behaviors are present a nash equilibrium where the final density of cooperating agents c is constant and a non stationary region where c fluctuates in time moreover we study the spd in poisson random graphs and find that the phase diagram previously developed looses its meaning in fact only one regime may be defined this regime is characterized by a non stationary final state where the density of cooperating agents varies in time

on support varieties for lie superalgebras and finite supergroup schemes abstract we study the spectrum of the cohomology rings of cocommutative hopf superalgebras restricted and non restricted lie superalgebras and finite supergroup schemes we also investigate support varieties in these settings and demonstrate that they have the desirable properties of such a theory we completely characterize support varieties for finite supergroup schemes over algebraically closed fields of characteristic zero while for non restricted lie superalgebras in positive characteristic we obtain descriptions that are suggestive of varieties previously defined by duflo and serganova in characteristic zero but which have no known cohomological interpretation our computations for restricted lie superalgebras and infinitesimal supergroup schemes provide natural generalizations of foundational results of friedlander and parshall and of bendel friedlander and suslin in the classical setting

towards optimal control of evolutionary games on networks we investigate the control of evolutionary games on networks in which each edge represents a two player repeating game between neighboring agents after each round of games agents can imitate the strategies of better performing neighbors while a subset of agents can be assigned strategies and thus serve as control inputs we seek here the smallest set of control agents needed to drive the network to a desired uniform strategy state after presenting exact solutions for complete and star networks and describing a general solution approach that is computationally practical only for small networks we design a fast algorithm for approximating the solution on arbitrary networks using a weighted minimum spanning tree and strategy propagation algorithm the resulting approximation is exact for certain classes of games on complete and star networks and simulations suggest that the algorithm performs well in more general cases

evolutionary markovian strategies in 2 2 spatial games abstract evolutionary spatial 2 2 games between heterogeneous agents are analyzed using different variants of cellular automata ca agents play repeatedly against their nearest neighbors 2 2 games specified by a rescaled payoff matrix with two parameters each agent is governed by a binary markovian strategy bms specified by four conditional probabilities p r p s p t p p that take values 0 or 1 the initial configuration consists in a random assignment of strategists among the 2 4 16 possible bms the system then evolves within strategy space according to the simple standard rule each agent copies the strategy of the neighbor who got the highest payoff besides on the payoff matrix the dominant strategy and the degree of cooperation depend on i the type of the neighborhood von neumann or moore ii the way the cooperation state is actualized deterministically or stochastically and iii the amount of noise measured by a parameter e however a robust winner strategy is 1 0 1 1

topological flatness of orthogonal local models in the split even case i local models are schemes defined in terms of linear algebra that were introduced by rapoport and zink to study the etale local structure of integral models of certain pel shimura varieties over p adic fields a basic requirement for the integral models or equivalently for the local models is that they be flat in the case of local models for even orthogonal groups genestier observed that the original definition of the local model does not yield a flat scheme in a recent article pappas and rapoport introduced a new condition to the moduli problem defining the local model the so called spin condition and conjectured that the resulting spin local model is flat we prove a preliminary form of their conjecture in the split iwahori case namely that the spin local model is topologically flat an essential combinatorial ingredient is the equivalence of admissibility and permissibility for two minuscule cocharacters in root systems of type d

stochastic stability in spatial three player games we discuss long run behavior of stochastic dynamics of many interacting agents in particular three player spatial games are studied the effect of the number of players and the noise level on the stochastic stability of nash equilibria is investigated

coincidences among skew schur functions new sufficient conditions and necessary conditions are developed for two skew diagrams to give rise to the same skew schur function the sufficient conditions come from a variety of new operations related to ribbons also known as border strips or rim hooks the necessary conditions relate to the extent of overlap among the rows or among the columns of the skew diagram

a straightening algorithm for row convex tableaux abstract we produce a new basis for the schur and weyl modules associated to a row convex shape d the basis is indexed by a new class of straight tableaux which we introduce by weakening the usual requirements for standard tableaux spanning is proved via a new straightening algorithm for expanding elements of the representation into this basis for skew shapes this algorithm specializes to the classical straightening law the new straight basis is used to produce bases for flagged schur and weyl modules to provide groebner and sagbi bases for the homogeneous coordinate rings of some configuration varieties and to produce a flagged branching rule for row convex representations systematic use of supersymmetric letterplace techniques enables the representation theoretic results to be applied to representations of the general linear lie superalgebra as well as to the general linear group

phase diagrams for an evolutionary prisoner s dilemma game on two dimensional lattices the effects of payoffs and noise on the maintenance of cooperative behavior are studied in an evolutionary prisoner s dilemma game with players located on the sites of different two dimensional lattices this system exhibits a phase transition from a mixed state of cooperators and defectors to a homogeneous one where only the defectors remain alive using monte carlo simulations and the generalized mean field approximations we have determined the phase boundaries critical points separating the two phases on the plane of the temperature noise and temptation to choose defection in the zero temperature limit the cooperation can be sustained only for those connectivity structures where three site clique percolation occurs

bursts in three strategy evolutionary ordinal potential games on a square lattice abstract evolutionary potential games provide an analogy for equilibrium statistical physics under suitable conditions that is broken if cyclic components are added using monte carlo simulations we study the effect of weak cyclic perturbations on the equilibrium in a system of players located on a square lattice while the dynamics is controlled by a logit rule the pair interaction is composed of a symmetric three strategy coordination game of unit strength a weak self dependent term and a cyclic rock paper scissors component the self dependent component favors the first rock strategy which dominates the system behavior at low noises if the strength of the cyclic component is below a threshold value in the opposite case the predator of the first strategy paper rules the behavior at low noises while the first strategy can occasionally form growing domains bursts diminished after the appearance of the third one the variation of noise modifies the invasion velocities as well as the rate of nucleation processes and yields drastic changes in the strategy frequencies fluctuations average payoffs and spatio temporal patterns

bott s formula and enumerative geometry we outline a strategy for computing intersection numbers on smooth varieties with torus actions using a residue formula of bott as an example gromov witten numbers of twisted cubic and elliptic quartic curves on some general complete intersection in projective space are computed the results are consistent with predictions made from mirror symmetry compu tations we also compute degrees of some loci in the linear system of plane curves of degrees less than 10 like those corresponding to sums of powers of linear forms and curves carrying inscribed polygons mathematical institute university of oslo p 0 box 1053 n 0316 oslo norway e mail address ellingsr0math uio no mathematical institute university of bergen alleg 55 n 5007 bergen norway e mail address strommeti i uib no this content downloaded from 157 55 39 29 on tue 12 apr 2016 10 23 14 utc all use subject to http about jstor org terms

cooperation in the noisy case prisoner s dilemma game on two types of regular random graphs we have studied an evolutionary prisoner s dilemma game with players located on two types of random regular graphs with a degree of 4 the analysis is focused on the effects of payoffs and noise temperature on the maintenance of cooperation when varying the noise level and or the highest payoff the system exhibits a second order phase transition from a mixed state of cooperators and defectors to an absorbing state where only defectors remain alive for the random regular graph and bethe lattice the behavior of the system is similar to those found previously on the square lattice with nearest neighbor interactions although the measure of cooperation is enhanced by the absence of loops in the connectivity structure for low noise the optimal connectivity structure is built up from randomly connected triangles

borel weil theorem for configuration varieties and schur modules we present a generalization of the classical schur modules ofgl n exhibiting the same interplay among algebra geometry and combinatorics a generalized young diagramdis an arbitrary finite subset ofn n for eachd we define the schur modulesdofgl n we introduce a projective variety fdand a line bundle ld and describe the schur module in terms of sections of ld for diagrams with the northeast property i1 j1 i2 j2 d min i1 i2 max j1 j2 d which includes the skew diagrams we resolve the singularities of fdand show analogues of bott s and kempf s vanishing theorems finally we apply the atiyah bott fixed point theorem to establish a weyl type character formula of the form charsd x txwt t i j 1 xix 1j dij t wheretruns over certain standard tableaux ofd our results are valid over fields of arbitrary characteristic

specht series for column convex diagrams abstract we give a specht series for the specht module s d associated to a diagram d in two cases 1 when d is column convex 2 when d is the complement of a column convex diagram within a rectangular box

the gorenstein locus of minuscule schubert varieties in this article we explicitly describe the gorenstein locus of all minuscule schubert varieties this proves a special case of the conjecture of a woo and a yong alexander woo alexander yong governing singularities of schubert varieties j algebra 320 2 2008 495 520 regarding the gorenstein locus of schubert varieties

self organization in the spatial battle of the sexes with probabilistic updating abstract the dynamics of a spatial formulation of the iterated battle of the sexes with probabilistic updating is assessed in this work the game is played in the cellular automata manner i e with local and synchronous interaction the effect of memory of past encounters is also taken into account it is concluded that the spatial structure enables the emergence of clusters of coincident choices leading to the mean payoff per encounter to values that are accessible only in the cooperative two person game scenario which constitutes a notable case of self organization with probabilistic updating of choices both kinds of players reach mean payoffs per encounter that are notably higher than those reached with a deterministic updating mechanism albeit the evolutionary dynamics does not stabilize and one of the two possible choices tends to prevail for both kinds of players memory of past iterations induces an inertial effect that moderates this tendency so that intermediate levels of the memory charge tend to favor fairly stable high egalitarian payoffs without impeding the necessary recovering from their initial plummeting

convergence of linear threshold decision making dynamics in finite heterogeneous populations abstract linear threshold models have been studied extensively in structured populations however less attention is paid to perception differences among the individuals of the population to focus on this effect we exclude structure and consider a well mixed population of heterogeneous agents each associated with a threshold in the form of a fixed ratio within zero and one that can be unique to this agent the agents are initialized with a choice of strategy a or b and at each time step one agent becomes active to update if the ratio of agents playing a is higher resp lower than her threshold she updates to a resp b we show that for any given initial condition after a finite number of time steps the population reaches an equilibrium where no agent s threshold is violated however the equilibrium is not necessarily uniquely determined by the initial condition but depends on the agents activation sequence we find all those possible equilibria that the dynamics may reach from a given initial condition and show that in contrast to the case of homogeneous populations heterogeneity in the agents thresholds gives rise to several equilibria where both a playing and b playing agents coexist perception heterogeneity can hence preserve decision diversity even in the absence of population structure we also investigate the asymptotic stability of the equilibria and show how to calculate the contagion probability for a population with two thresholds

statistical mechanics of spatial evolutionary games we discuss the long run behaviour of stochastic dynamics of many interacting players in spatial evolutionary games in particular we investigate the effect of the number of players and the noise level on the stochastic stability of nash equilibria we discuss similarities and differences between systems of interacting players maximizing their individual payoffs and particles minimizing their interaction energy we use concepts and techniques of statistical mechanics to study game theoretic models in order to obtain results in the case of the so called potential games we analyse the thermodynamic limit of the appropriate models of interacting particles

coevolutionary dynamics in large but finite populations coevolving and competing species or game theoretic strategies exhibit rich and complex dynamics for which a general theoretical framework based on finite populations is still lacking recently an explicit mean field description in the form of a fokker planck equation was derived for frequency dependent selection with two strategies in finite populations based on microscopic processes a traulsen j c claussen and c hauert phys rev lett 95 238701 2005 here we generalize this approach in a twofold way first we extend the framework to an arbitrary number of strategies and second we allow for mutations in the evolutionary process the deterministic limit of infinite population size of the frequency dependent moran process yields the adjusted replicator mutator equation which describes the combined effect of selection and mutation for finite populations we provide an extension taking random drift into account in the limit of neutral selection i e whenever the process is determined by random drift and mutations the stationary strategy distribution is derived this distribution forms the background for the coevolutionary process in particular a critical mutation rate uc is obtained separating two scenarios above uc the population predominantly consists of a mixture of strategies whereas below uc the population tends to be in homogeneous states for one of the fundamental problems in evolutionary biology the evolution of cooperation under darwinian selection we demonstrate that the analytical framework provides excellent approximations to individual based simulations even for rather small population sizes this approach complements simulation results and provides a deeper systematic understanding of coevolutionary dynamics

networks of conforming or nonconforming individuals tend to reach satisfactory decisions binary decisions of agents coupled in networks can often be classified into two types coordination where an agent takes an action if enough neighbors are using that action as in the spread of social norms innovations and viral epidemics and anticoordination where too many neighbors taking a particular action causes an agent to take the opposite action as in traffic congestion crowd dispersion and division of labor both of these cases can be modeled using linear threshold based dynamics and a fundamental question is whether the individuals in such networks are likely to reach decisions with which they are satisfied we show that in the coordination case and perhaps more surprisingly also in the anticoordination case the agents will indeed always tend to reach satisfactory decisions that is the network will almost surely reach an equilibrium state this holds for every network topology and every distribution of thresholds for both asynchronous and partially synchronous decision making updates these results reveal that irregular network topology population heterogeneity and partial synchrony are not sufficient to cause cycles or nonconvergence in linear threshold dynamics rather other factors such as imitation or the coexistence of coordinating and anticoordinating agents must play a role

computational complexity of ecological and evolutionary spatial dynamics there are deep yet largely unexplored connections between computer science and biology both disciplines examine how information proliferates in time and space central results in computer science describe the complexity of algorithms that solve certain classes of problems an algorithm is deemed efficient if it can solve a problem in polynomial time which means the running time of the algorithm is a polynomial function of the length of the input there are classes of harder problems for which the fastest possible algorithm requires exponential time another criterion is the space requirement of the algorithm there is a crucial distinction between algorithms that can find a solution verify a solution or list several distinct solutions in given time and space the complexity hierarchy that is generated in this way is the foundation of theoretical computer science precise complexity results can be notoriously difficult the famous question whether polynomial time equals nondeterministic polynomial time i e p np is one of the hardest open problems in computer science and all of mathematics here we consider simple processes of ecological and evolutionary spatial dynamics the basic question is what is the probability that a new invader or a new mutant will take over a resident population we derive precise complexity results for a variety of scenarios we therefore show that some fundamental questions in this area cannot be answered by simple equations assuming that p is not equal to np

on the conormal bundle of the determinantal variety call w such variety if we choose bases b v a a b u v i b for each module we can identify w with the variety of pairs of matrices y x with entries in r x being an m x n matrix y an ii x m matrix such that x y omx y x onxm now let a be the coordinate ring of the affine space a of dimension 2 m x n i e the polynomial ring r x yji i l m j i n and

anisotropic invasion and its consequences in two strategy evolutionary games on a square lattice we have studied invasion processes in two strategy evolutionary games on a square lattice for imitation rule when the players interact with their nearest neighbors monte carlo simulations are performed for systems where the pair interactions are composed of a unit strength coordination game when varying the strengths of the self dependent and cross dependent components at a fixed noise level the visualization of strategy distributions has clearly indicated that circular homogeneous domains evolve into squares with an orientation dependent on the composition this phenomenon is related to the anisotropy of invasion velocities along the interfaces separating the two homogeneous regions the quantified invasion velocities indicate the existence of a parameter region in which the invasions are opposite for the horizontal or vertical and the tilted interfaces in this parameter region faceted islands of both strategies shrink and the system evolves from a random initial state into the homogeneous state that first percolated

mean field approximation for two and three person prisoner s dilemmas abstract we consider the mean field approximation mfa to the system of interacting agents playing two and three person prisoner s dilemma games 2 pd and 3 pd the agents have three available strategies all c all d and a third one which we choose in various ways long time distribution of the strategies is compared for both games the latter game admits coexistence of larger number of strategies in the long run in the case of external pressure for cooperation more pressure is in general necessary for 3 pd than for 2 pd to guarantee cooperation for almost all initial compositions of the strategies results for different types of tit for tat strategies for 3 pd are discussed unlike the 2 pd case the more forgiving tit for tat strategy can survive in the long run in 3 pd

on a grauert riemenschneider vanishing theorem for frobenius split varieties in characteristic p it is known that the grauert riemenschneider vanishing theorem is not valid in characteristic p 1 here we show that it may be restored in the presence of a suitable frobenius splitting the proof uses interchanging two projective limits one involving iterated frobenius maps cf 2 and 4 the other coming from grothendieck s theorem on formal functions that leads to the following general vanishing theorem which we then apply in the situation of the grauert riemenschneider theorem

stochastic stability analysis of evolutionary two player games on regular graphs abstract we study evolutionary two player games and identify stochastically stable equilibria of the network games restricted to infinite populations on regular graphs the players update their strategies according to four different rules birth death death birth imitation and pairwise comparison for prisoner s dilemma and snowdrift games respectively for two player games on regular graphs we show that there is a unique stochastically stable equilibrium for infinite populations for the prisoner s dilemma game if the benefit to cost ratio is larger than k 2 k is the degree of a regular graph the networked game has a higher fraction of cooperators than that for a well mixed population for the snowdrift game the fraction of cooperators in a regular graph would be higher than that of the well mixed population if the benefit to cost ratio is larger than 1 5 under certain conditions the lower graph connectivity can lead to the emergence of more cooperators finally some numerical simulation examples are given to demonstrate the theoretical results

coevolutionary games on networks we study agents on a network playing an iterated prisoner s dilemma against their neighbors the resulting spatially extended coevolutionary game exhibits stationary states which are nash equilibria after perturbation of these equilibria avalanches of mutations reestablish a stationary state scale free avalanche distributions are observed that are in accordance with calculations from the nash equilibria and a confined branching process the transition from subcritical to critical avalanche dynamics can be traced to a change in the degeneracy of the cooperative macrostate and is observed for many variants of this game

the space of triangles vanishing theorems and combinatorics abstract we consider compactifications of pn 3 ij the space of triples of distinct points in projective space one such space is a singular variety of configurations of points and lines another is the smooth compactification of fulton and macpherson and a third is the triangle space of schubert and semple we compute the sections of line bundles on these spaces and show that they are equal as gl n representations to the generalized schur modules associated to bad generalized young diagrams with three rows borel weil theorem on the one hand this yields weyl type character and dimension formulas for the schur modules on the other a combinatorial picture of the space of sections cohomology vanishing theorems play a key role in our analysis

geometry of graph varieties first published in transactions of the american mathematical society in volume 355 2003 4151 4169 published by the american mathematical society

schur functors and schur complexes the study of schur functors has a relatively long history its main impetus derived from representation theory originally in characteristic zero over the years however with the development of modular representations and algebraic geometry over fields of positive characteristic the need for a theory of universal polynomial functors increased and since the mid 1960s approaches to a characteristic free treatment of schur functors have been developing see for instance the recent book of green 1 l in which the treatments by carter and lusztig 6 higman 12 and towber 20 among others are discussed our own interest in such a treatment was awakened by the work of lascoux 141 on resolutions of determinantal ideals although his thesis treated only the characteristic zero case it suggested that a general and elementary theory of schur functors could be developed using only the rudiments of multilinear algebra involving the hopf algebra structures of the symmetric exterior and divided power algebras moreover this elementary development admitted of a natural generalization to the idea of schur complexes whose usefulness was demonstrated for instance in our construction of a universal minimal

phase diagrams for three strategy evolutionary prisoner s dilemma games on regular graphs evolutionary prisoner s dilemma games are studied with players located on square lattice and random regular graph defining four neighbors for each one the players follow one of the three strategies tit for tat unconditional cooperation and defection the simplified payoff matrix is characterized by two parameters the temptation b to choose defection and the cost c of inspection reducing the income of tit for tat the strategy imitation from one of the neighbors is controlled by pairwise comparison at a fixed level of noise using monte carlo simulations and the extended versions of pair approximation we have evaluated the b c phase diagrams indicating a rich plethora of phase transitions between stationary coexistence absorbing and oscillatory states including continuous and discontinuous phase transitions by reasonable costs the tit for tat strategy prevents extinction of cooperators across the whole span of b determining the prisoner s dilemma game irrespective of the connectivity structure we also demonstrate that the system can exhibit a repetitive succession of oscillatory and stationary states upon changing a single payoff value which highlights the remarkable sensitivity of cyclical interactions on the parameters that define the strength of dominance

rock scissors paper game on regular small world networks the spatial rock scissors paper game or cyclic lotka volterra system is extended to study how the spatiotemporal patterns are affected by the rewired host lattice providing uniform number of neighbours degree at each site on the square lattice this system exhibits a self organizing pattern with equal concentration of the competing strategies species if the quenched background is constructed by substituting random links for the nearest neighbour bonds of a square lattice then a limit cycle occurs when the portion of random links exceeds a threshold value this transition can also be observed if the standard link is replaced temporarily by a random one with a probability p at each step of iteration above a second threshold value of p the amplitude of global oscillation increases with time and finally the system reaches one of the homogeneous absorbing states in this case the results of monte carlo simulations are compared with the predictions of the dynamical cluster technique evaluating all the configuration probabilities on one two four and six site clusters

time evolution of the rule 150 cellular automaton activity from a fibonacci iteration the rule 150 cellular automaton is a remarkable discrete dynamical system as it shows 1 f spectra if started from a single seed j nagler and j c claussen phys rev e 71 067103 2005 despite its simplicity a feasible solution for its time behavior is not obvious its self similarity does not follow a one step iteration like other elementary cellular automata here it is shown how its time behavior can be solved as a two step vectorial or string iteration which can be viewed as a generalization of fibonacci iteration generating the time series from a sequence of vectors of increasing length this allows us to compute the total activity time series more efficiently than by simulating the whole spatiotemporal process or even by using the closed expression the results are further extended to the generalization of rule 150 to the two dimensional case and to bethe lattices and the relation to corresponding integer sequences is discussed