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

toric hyperkahler manifolds with quaternionic kahler bases and supergravity solutions in the present work some examples of toric hyperkahler metrics in eight dimensions are constructed first it is described how the calderbank pedersen metrics arise as a consequence of the joyce description of selfdual structures in four dimensions the jones tod correspondence and a result due to tod and przanowski it is also shown that any quaternionic kahler metric with t2 isometry is locally isometric to a calderbank pedersen one the swann construction of hyperkahler metrics in eight dimensions is applied to them to find hyperkahler examples with u 1 u 1 isometry the connection with the pedersen poon toric hyperkahler metrics is explained and it is shown that there is a class of solutions of the generalized monopole equation related to eigenfunctions of a certain linear equation these hyperkahler examples are lifted to solutions of the d 11 supergravity and type iia and iib backgrounds are found by use of dualities as before all the description is achieved in terms of a single eigenfunction f some explicit f are found together with the toda structure corresponding to the trajectories of the killing vectors of the calderbank pedersen bases

monopoles and the gibbons manton metric we show that in the region where monopoles are well separated the l 2 metric on the moduli space of n monopoles is exponentially close to the l 2 invariant hyperkahler metric proposed by gibbons and manton the proof is based on a description of the gibbons manton metric as a metric on a certain moduli space of solutions to nahm s equations and on twistor methods in particular we show how the twistor description of monopole metrics determines the asymptotic metric the construction of the gibbons manton metric in terms of nahm s equations yields a class of interesting pseudo hyperkahler metrics for example we show for each semisimple lie group g and a maximal torus t g the existence of a g t invariant pseudo hyperkahler manifold whose hyperkahler quotients by t are precisely kronheimer s hyperkahler metrics on g t a similar result holds for kronheimer s ale spaces

hodge cohomology of gravitational instantons we study the space of l2 harmonic forms on complete manifolds with metrics of fibred boundary or fibred cusp type these metrics generalize the geometric structures at infinity of several different well known classes of metrics including asymptotically locally euclidean manifolds the known types of gravitational instantons and also poincare metrics on q rank 1 ends of locally symmetric spaces and on the complements of smooth divisors in kahler manifolds the answer in all cases is given in terms of intersection cohomology of a stratified compactification of the manifold the l2 signature formula implied by our result is closely related to the one proved by dai 25 and more generally by vaillant 67 and identifies dai s invariant directly in terms of intersection cohomology of differing perversities this work is also closely related to a recent paper of carron 12 and the forthcoming paper of cheeger and dai 17 we apply our results to a number of examples gravitational instantons among them arising in predictions about l2 harmonic forms in duality theories in string theory

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

finite number fluctuations in a three state model with growing population abstract we study the role of the finite number of units in a three state model with a growing population the model presents two attractors namely a fixed point and a limit cycle in the mean field approximation it has been shown that there is a bistability region for which both attractors are stable we observe that contrary to constant population models the bistability is preserved in the presence of finite number fluctuations

instantons monopoles and toric hyperkahler manifolds in this paper the metric on the moduli space of the k 1 su n periodic instanton or caloron with arbitrary gauge holonomy at spatial infinity is explicitly constructed the metric is toric hyperkahler and of the form conjectured by lee and yi the torus coordinates describe the residual u 1 n 1 gauge invariance and the temporal position of the caloron and can also be viewed as the phases of n monopoles that constitute the caloron the 1 1 1 monopole is obtained as a limit of the caloron the calculation is performed on the space of nahm data which is justified by proving the isometric property of the nahm construction for the cases considered an alternative construction using the hyperkahler quotient is also presented the effect of massless monopoles is briefly discussed

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

moduli spaces of instantons on the taub nut space we present adhm nahm data for instantons on the taub nut space and encode these data in terms of bow diagrams we study the moduli spaces of the instantons and present these spaces as finite hyperkahler quotients as an example we find an explicit expression for the metric on the moduli space of one su 2 instanton

hyperkahler quotient construction of bps monopole moduli spaces we use the hyperkahler quotient of flat space to obtain some monopole moduli space metrics in explicit form using this new description we discuss their topology completeness and isometries we construct the moduli space metrics in the limit when some monopoles become massless which corresponds to non maximal symmetry breaking of the gauge group we also introduce a new family of hyperkahler metrics which depending on the mass parameter being positive or negative tend to either the asymptotic metric on the moduli space of many su 2 monopoles or to previously unknown metrics these new metrics are singular or complete depending on the particular choice of the level set of the moment map the singular metrics are of relevance to the moduli spaces of vacua of three dimensional gauge theories for higher rank gauge groups finally we make a few comments concerning the existence of closed or bound orbits on some of these manifolds and the integrability of the geodesic flow

hybrid impotent incompatible eigenstrain based homogenization summary we present a constitutive framework for a periodic heterogeneous medium with minimal number of internal variables the method is based on a variant of the transformation field analysis tfa where eigenstrains are discretized using c 0 continuous approximation in matrix dominated mode of deformation hereafter referred to as impotent eigenstrain mode whereas in multiphase mode of deformation the eigenstrains are approximated using the usual c 1 approximation the delay in the onset of inelastic response and the eigenstrain induced anisotropy in a microphase both characteristic to averaging methods are alleviated by introducing an eigenstrain upwinding scheme and by enhancing constitutive laws of microphases the proposed formulation has been verified against a direct numerical simulation the method has been found to be very accurate in predicting an overall material response at a computational cost comparable with the phenomenological modeling of a periodic heterogeneous medium copyright 2013 john wiley sons ltd

continuous and discontinuous phase transitions and partial synchronization in stochastic three state oscillators we investigate both continuous second order and discontinuous first order transitions to macroscopic synchronization within a single class of discrete stochastic globally phase coupled oscillators we provide analytical and numerical evidence that the continuity of the transition depends on the coupling coefficients and in some nonuniform populations on the degree of quenched disorder hence in a relatively simple setting this class of models exhibits the qualitative behaviors characteristic of a variety of considerably more complicated models in addition we study the microscopic basis of synchronization above threshold and detail the counterintuitive subtleties relating measurements of time averaged frequencies and mean field oscillations most notably we observe a state of suprathreshold partial synchronization in which time averaged frequency measurements from individual oscillators do not correspond to the frequency of macroscopic oscillations observed in the population

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

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

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

integrating over higgs branches we develop some useful techniques for integrating over higgs branches in supersymmetric theories with 4 and 8 supercharges in particular we define a regularized volume for hyperkahler quotients we evaluate this volume for certain ale and alf spaces in terms of the hyperkahler periods we also reduce these volumes for a large class of hyperkahler quotients to simpler integrals these quotients include complex coadjoint orbits instanton moduli spaces on 4 and ale manifolds hitchin spaces and moduli spaces of parabolic higgs bundles on riemann surfaces in the case of hitchin spaces the evaluation of the volume reduces to a summation over solutions of bethe ansatz equations for the non linear schrodinger system we discuss some applications of our results

effects of disorder on synchronization of discrete phase coupled oscillators we study synchronization in populations of phase coupled stochastic three state oscillators characterized by a distribution of transition rates we present results on an exactly solvable dimer as well as a systematic characterization of globally connected arrays of n types of oscillators n 2 3 4 by exploring the linear stability of the nonsynchronous fixed point we also provide results for globally coupled arrays where the transition rate of each unit is drawn from a uniform distribution of finite width even in the presence of transition rate disorder numerical and analytical results point to a single phase transition to macroscopic synchrony at a critical value of the coupling strength numerical simulations make possible further characterization of the synchronized arrays

reduction of hkt structures kt geometry is the geometry of a hermitian connection whose torsion is a 3 form hkt geometry is the geometry of a hyper hermitian connection whose torsion is a 3 form we identify nontrivial conditions for a reduction theory for these types of geometry

hyperkahler metrics and supersymmetry we describe two constructions of hyperkahler manifolds one based on a legendre transform and one on a sympletic quotient these constructions arose in the context of supersymmetric nonlinear models but can be described entirely geometrically in this general setting we attempt to clarify the relation between supersymmetry and aspects of modern differential geometry along the way reviewing many basic and well known ideas in the hope of making them accessible to a new audience

asymptotic metrics for su n monopoles with maximal symmetry breaking we compute the asymptotic metrics for moduli spaces of su n monopoles with maximal symmetry breaking these metrics are exponentially close to the exact monopole metric1 as soon as for each simple root the individual monopoles corresponding to that root are well separated we also show that the estimates can be differentiated term by term in natural coordinates which is a new result even for su 2 monopoles

bounds for the effective constitutive relation of a nonlinear composite for a nonlinear composite a bound on its effective energy density does not induce a corresponding bound on its constitutive relation because differentiating a bound on a function does not automatically bound its derivative in this work a method introduced by milton and serkov for bounding directly the constitutive relation is refined by employing a linear comparison material in a way similar to that employed by the present authors to obtain bounds of hashin shtrikma type for the effective energy of a nonlinear composite the original milton serkov approach produces bounds with a close relationship to the classical energy bounds of voigt and reuss type the bounds produced in the present implementation are closely related to bounds of hashin shtrikman type for the composite indeed in the case of a linear composite the present method delivers the hashin shtrikman bounds in their generalized form valid for any two point statistics it is demonstrated for nonlinear examples that the approximate constitutive relation that is obtained by differentiating the energy bound can be on the boundary of the bounding set for the exact constitutive relation but a simple counterexample is presented to show that this is not always the case

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