critical behavior and synchronization of discrete stochastic phase coupled oscillators synchronization of stochastic phase coupled oscillators is known to occur but difficult to characterize because sufficiently complete analytic work is not yet within our reach and thorough numerical description usually defies all resources we present a discrete model that is sufficiently simple to be characterized in meaningful detail in the mean field limit the model exhibits a supercritical hopf bifurcation and global oscillatory behavior as coupling crosses a critical value when coupling between units is strictly local the model undergoes a continuous phase transition that we characterize numerically using finite size scaling analysis in particular we explicitly rule out multistability and show that the onset of global synchrony is marked by signatures of the xy universality class our numerical results cover dimensions d 2 3 4 and 5 and lead to the appropriate xy classical exponents beta and nu a lower critical dimension dlc 2 and an upper critical dimension duc 4

infinite period bifurcation and global bifurcation branches branches of periodic solutions which exhibit the alternatives of the global hopf bifurcation theorem are calculated for two general systems of differential equations in the first a branch of solu

conformal theory of curves with tractors abstract we present the general theory of curves in conformal geometry using tractor calculus this primarily involves a specification of distinguished parametrizations and relative and absolute conformal invariants of generic curves the absolute conformal invariants are defined via a tractor analogue of the classical frenet frame construction and then expressed in terms of relative ones our approach applies likewise to conformal structures of any signature in the case of indefinite signature we focus especially on the null curves it also provides a conceptual tool for handling distinguished families of curves conformal circles and conformal null helices and conserved quantities along them

stochastic turing patterns in the brusselator model a stochastic version of the brusselator model is proposed and studied via the system size expansion the mean field equations are derived and shown to yield to organized turing patterns within a specific parameters region when determining the turing condition for instability we pay particular attention to the role of cross diffusive terms often neglected in the heuristic derivation of reaction diffusion schemes stochastic fluctuations are shown to give rise to spatially ordered solutions sharing the same quantitative characteristic of the mean field based turing scenario in term of excited wavelengths interestingly the region of parameter yielding to the stochastic self organization is wider than that determined via the conventional turing approach suggesting that the condition for spatial order to appear can be less stringent than customarily believed

conformal geodesics in general relativity conformal geodesics space time curves which are related to conformal structures in a similar way as geodesics are related to metric structures are discussed conformal normal coordinates conformal gauss systems and their associated normal connections normal frames and normal metrics are introduced and used to study i asymptotically simple solutions of ric g lambda g near conformal infinity ii asymptotically simple solutions of ric g 0 with a past null infinity which can be represented as the future null cone of a point i past time like infinity in the first case we define an infty parameter family of physical gauss systems near conformal infinity in the second case a ten parameter family of physical gauss systems covering a neighbourhood of i the behaviour of physical geodesics can be analysed in a particularly simple way in these coordinate systems each of these systems allows an extremely simple transition from the conformal analysis to the physical description of space time for lambda eta 00 0 de sitter type solutions all solutions are characterized in terms of the physical space time by their data on past time like infinity for lambda 0 the conserved quantities of newman and penrose are characterized as the first non trivial coefficient given by the value of the rescaled weyl tensor at i in an expansion of the physical field in a gauss system of the type considered before

cooperative dynamics in a network of stochastic elements with delayed feedback networks of globally coupled noise activated bistable elements with connection time delays are considered the dynamics of these systems is studied numerically using a langevin description and analytically using 1 a gaussian approximation as well as 2 a dichotomous model the system demonstrates ordering phase transitions and multistability that is for a strong enough feedback it exhibits nontrivial stationary states and oscillatory states whose frequencies depend only on the mean of the time delay distribution function other observed dynamical phenomena include coherence resonance and in the case of nonuniform coupling strengths amplitude death and chaos furthermore an increase of the stability of the trivial equilibrium with increasing nonuniformity of the time delays is observed

bases for primitive permutation groups and a conjecture of babai abstract a base of a permutation group g is a sequence b of points from the permutation domain such that only the identity of g fixes b pointwise we show that primitive permutation groups with no alternating composition factors of degree greater than d and no classical composition factors of rank greater than d have a base of size bounded above by a function of d this confirms a conjecture of babai

base sizes of primitive groups bounds with explicit constants abstract we show that the minimal base size b g of a finite primitive permutation group g of degree n is at most 2 log g log n 24 this bound is asymptotically best possible since there exists a sequence of primitive permutation groups g of degrees n such that b g 2 log g log n 2 and b g is unbounded as a corollary we show that a primitive permutation group of degree n that does not contain the alternating group alt n has a base of size at most max n 25

globally coupled stochastic two state oscillators synchronization of infinite and finite arrays we consider arrays of the simplest two state on off stochastic units the units are markovian that is the transitions between the two states occur at a given rate we construct arrays of n globally coupled binary units and observe a remarkable richness of behavior as the control parameter that measures the coupling strength is increased in the mean field limit as we consider the four simplest polynomial forms of coupling that lead to bifurcations and characterize the associated phase transitions of the arrays when n is finite there are fluctuations about the well defined steady states of the infinite arrays we study the nature of these fluctuations and their effects on the bifurcations in all cases by constructing the appropriate langevin equations and the associated fokker planck equations

ensembles of excitable two state units with delayed feedback a two state unit is considered as an abstract modification for an excitable system each state is characterized by a different waiting time distribution this non markovian approach allows for a renewal process description of the system dynamics exact formulas for the interspike interval distribution and power spectral density are found in the limit of an infinity ensemble of globally coupled units the mean field equations for the populations of both states are derived depending on the coupling strength and on the noise intensity the ensemble undergoes saddle node bifurcations and demonstrates bistability while a pitchfork bifurcation emerges on a cusp point the ensemble undergoes hopf bifurcations and bulk oscillations emerge in the onset of coherent activation events only when the feedback affects individual units with a certain time delay

on pyber s base size conjecture let g be a permutation group on a finite set a subset of is a base for g if its pointwise stabilizer in g is trivial the base size of g denoted b g is the smallest size of a base a well known conjecture of pyber from the early 1990s asserts that there exists an absolute constant c such that b g c log g logn for any primitive permutation group g of degree n several special cases have been verified in recent years including the almost simple and diagonal cases in this paper we prove pyber s conjecture for all non affine primitive groups

the base size of a primitive diagonal group abstract a base b for a finite permutation group g acting on a set is a subset of with the property that only the identity of g can fix every point of b we prove that a primitive diagonal group g has a base of size 2 unless the top group of g is the alternating or symmetric group acting naturally in which case the minimal base size of g is determined up to two possible values we also prove that the minimal base size of g satisfies a well known conjecture of pyber moreover we prove that if the top group of g does not contain the alternating group then the proportion of pairs of points that are bases for g tends to 1 as g tends to infinity a similar result for the case when the degree of the top group is fixed is given

synchronization and fluctuations coupling a finite number of stochastic units it is well established that ensembles of globally coupled stochastic oscillators may exhibit a nonequilibrium phase transition to synchronization in the thermodynamic limit infinite number of elements in fact since the early work of kuramoto mean field theory has been used to analyze this transition in contrast work that directly deals with finite arrays is relatively scarce in the context of synchronization and yet it is worth noting that finite number effects should be seriously taken into account since in general the limits n where n is the number of units and t where t is time do not commute mean field theory implements the particular choice first n and then t here we analyze an ensemble of three state coupled stochastic units which has been widely studied in the thermodynamic limit we formally address the finite n problem by deducing a fokker planck equation that describes the system we compute the steady state solution of this fokker planck equation that is finite n but t we use this steady state to analyze the synchronic properties of the system in the framework of the different order parameters that have been proposed in the literature to study nonequilibrium transitions

a theory of semiprimitive groups abstract a transitive permutation group is semiprimitive if each of its normal subgroups is transitive or semiregular interest in this class of groups is motivated by two sources problems arising in universal algebra related to collapsing monoids and the graph restrictive problem for permutation groups here we develop a theory of semiprimitive groups which encompasses their structure their quotient actions and a method by which all finite semiprimitive groups are constructed we also extend some results from the theory of primitive groups to semiprimitive groups and conclude with open problems of a similar nature

bases of primitive linear groups abstract we correct and improve a result in 3 giving the structure of finite primitive linear groups of unbounded base size this confirms a well known conjecture of pyber on base sizes of primitive permutation groups in the case of affine groups whose associated linear group is primitive

a proof of pyber s base size conjecture abstract building on earlier papers of several authors we establish that there exists a universal constant c 0 such that the minimal base size b g of a primitive permutation group g of degree n satisfies log g log n b g 45 log g log n c this finishes the proof of pyber s base size conjecture the main part of our paper is to prove this statement for affine permutation groups g v h where h g l v is an imprimitive linear group an ingredient of the proof is that for the distinguishing number d g in the sense of albertson and collins of a transitive permutation group g of degree n 1 we have the estimates g n d g 48 g n

non markovian approach to globally coupled excitable systems we consider stochastic excitable units with three discrete states each state is characterized by a waiting time density function this approach allows for a non markovian description of the dynamics of separate excitable units and of ensembles of such units we discuss the emergence of oscillations in a globally coupled ensemble with excitatory coupling in the limit of a large ensemble we derive the non markovian mean field equations nonlinear integral equations for the populations of the three states we analyze the stability of their steady solutions collective oscillations are shown to persist in a large parameter region beyond supercritical and subcritical hopf bifurcations we compare the results with simulations of discrete units as well as of coupled fitzhugh nagumo systems

robust ecological pattern formation induced by demographic noise we demonstrate that demographic noise can induce persistent spatial pattern formation and temporal oscillations in the levin segel predator prey model for plankton herbivore population dynamics although the model exhibits a turing instability in mean field theory demographic noise greatly enlarges the region of parameter space where pattern formation occurs to distinguish between patterns generated by fluctuations and those present at the mean field level in real ecosystems we calculate the power spectrum in the noise driven case and predict the presence of fat tails not present in the mean field case these results may account for the prevalence of large scale ecological patterns beyond that expected from traditional nonstochastic approaches

base sizes for simple groups and a conjecture of cameron let g be a permutation group on a finite set a base for g is a subset b with pointwise stabilizer in g that is trivial we write b g for the smallest size of a base for g in this paper we prove that b g 6 if g is an almost simple group of exceptional lie type and is a primitive faithful g set an important consequence of this result when combined with other recent work is that b g 7 for any almost simple group g in a non standard action proving a conjecture of cameron the proof is probabilistic and uses bounds on fixed point ratios

finite size scaling dynamic fluctuations and hyperscaling relation in the kuramoto model we revisit the kuramoto model to explore the finite size scaling fss of the order parameter and its dynamic fluctuations near the onset of the synchronization transition paying particular attention to effects induced by the randomness of the intrinsic frequencies of oscillators for a population of size n we study two ways of sampling the intrinsic frequencies according to the same given unimodal distribution g in the random case frequencies are generated independently in accordance with g which gives rise to oscillator number fluctuation within any given frequency interval in the regular case the n frequencies are generated in a deterministic manner that minimizes the oscillator number fluctuations leading to quasiuniformly spaced frequencies in the population we find that the two samplings yield substantially different finite size properties with clearly distinct scaling exponents moreover the hyperscaling relation between the order parameter and its fluctuations is valid in the regular case but it is violated in the random case in this last case a self consistent mean field theory that completely ignores dynamic fluctuations correctly predicts the fss exponent of the order parameter but not its critical amplitude

laplacian operators and q curvature on conformally einstein manifolds a new definition of canonical conformal differential operators p k k 1 2 with leading term a kth power of the laplacian is given for conformally einstein manifolds of any signature these act between density bundles and more generally between weighted tractor bundles of any rank by construction these factor into a power of a fundamental laplacian associated to einstein metrics there are natural conformal laplacian operators on density bundles due to graham jenne mason sparling gjms it is shown that on conformally einstein manifolds these agree with the p k operators and hence on einstein manifolds the gjms operators factor into a product of second order laplacian type operators in even dimension n the gjms operators are defined only for 1 k n 2 and so on conformally einstein manifolds the p k give an extension of this family of operators to operators of all even orders for n even and k n 2 the operators p k are each given by a natural formula in terms of an einstein metric but they are not natural conformally invariant operators in the usual sense they are shown to be nevertheless canonical objects on conformally einstein structures there are generalisations of these results to operators between weighted tractor bundles it is shown that on einstein manifolds the branson q curvature is constant and an explicit formula for the constant is given in terms of the scalar curvature as part of development conformally invariant tractor equations equivalent to the conformal killing equation are presented

thermodynamic limit of the first order phase transition in the kuramoto model in the kuramoto model a uniform distribution of the natural frequencies leads to a first order i e discontinuous phase transition from incoherence to synchronization at the critical coupling parameter k c we obtain the asymptotic dependence of the order parameter above criticality r r c alpha k k c 2 3 for a finite population we demonstrate that the population size n may be included into a self consistency equation relating r and k in the synchronized state we analyze the convergence to the thermodynamic limit of two alternative schemes to set the natural frequencies other frequency distributions different from the uniform one are also considered

coupled three state oscillators abstract we investigate globally coupled stochastic three state oscillators which we consider as general models of stochastic excitable systems we compare two situations in the first case the transitions between the three states of each unit 1 2 3 1 are determined by poissonian waiting time distributions in the second case only transition 1 2 is poissonian whereas the others are deterministic with a fixed delay when coupled the second system shows coherent oscillations whereas the first remains in a stable stationary state we show that the coherent oscillations are due to a hopf bifurcation in the dynamics of the occupation probabilities of the discrete states and discuss the bifurcation diagram