the zero locus of an admissible normal function we prove that the zero locus of an admissible normal function over an algebraic parameter space s is algebraic in the case where s is a curve

pathwise approximation of stochastic differential equations on domains higher order convergence rates without global lipschitz coefficients we study the approximation of stochastic differential equations on domains for this we introduce modified ito taylor schemes which preserve approximately the boundary domain of the equation under consideration assuming the existence of a unique non exploding solution we show that the modified ito taylor scheme of order has pathwise convergence order e for arbitrary e 0 as long as the coefficients of the equation are sufficiently differentiable in particular no global lipschitz conditions for the coefficients and their derivatives are required this applies for example to the so called square root diffusions

information cascade with marginal stability in a network of chaotic elements abstract a newly discovered cascade process of clusters is studied in a network of chaotic elements it is shown that the splitting of clusters and the synchronization of elements are balanced in a class of partially ordered states where marginal stability is sustained over an interval of the bifurcation parameters the partition information creation in bit space shows an avalanche process of information which leads to the anomalous behavior of power spectra roughly fitted by a power law form of the wavenumber lyapunov spectra have accumulation at null exponents analogous with those studied in fluid turbulence models

the euler scheme with irregular coefficients weak convergence of the euler scheme for stochastic differential equations is established when coefficients are discontinuous on a set of lebesgue measure zero the rate of convergence is presented when coefficients are holder continuous monte carlo simulations are also discussed

minimal asymptotic error for one point approximation of sdes with time irregular coefficients we consider strong one point approximation of solutions of scalar stochastic differential equations sdes with irregular coefficients the drift coefficient a 0 t i r r is assumed to be lipschitz continuous with respect to the space variable but only measurable with respect to the time variable for the diffusion coefficient b 0 t r we assume that it is only piecewise holder continuous with holder exponent 0 1 we show that roughly speaking the error of any algorithm which uses n values of the diffusion coefficient cannot converge to zero faster than n min 1 2 as n this best speed of convergence is achieved by the randomized euler scheme

singularities of admissible normal functions in a recent paper m green and p griffiths used r thomas work on nodal hypersurfaces to sketch a proof of the equivalence of the hodge conjecture and the existence of certain singular admissible normal functions inspired by their work we study normal functions using morihiko saito s mixed hodge modules and prove that the existence of singularities of the type considered by griffiths and green is equivalent to the hodge conjecture several of the intermediate results including a relative version of the weak lefschetz theorem for perverse sheaves are of independent interest

relative monodromy weight filtrations in this paper we establish the existence of relative weight filtrations of cones of monodromy logarithms and identities between relative weight filtrations in a context more general than m kashiwara s infinitesimal mixed hodge modules this generalizes a theorem of m kashiwara and adds conceptual clarity to the problem our approach to working with relative weight filtrations w n w is to split the weight filtrations w with canonical splittings introduced by p deligne these splittings agree with the splittings that occur in the mathfrak sl 2 theory of variations of pure hodge structure of w schmid and of e cattani a kaplan and w schmid but are more general as they don t require filtrations w to be monodromy weight filtrations

on the geometry of classifying spaces and horizontal slices we study the local properties of the moduli space of a polarized calabi yau manifold let u be a neighborhood of the moduli space then we know the universal covering space v of u is a smooth manifold suppose d is the classifying space of a polarized calabi yau manifold with the automorphism group g then we prove that the map from v to d induced by the period map is a pluriharmonic map we also give a kahler metric on v which is called the hodge metric we prove that the ricci curvature of the hodge metric is negative away from zero we also proved the nonexistence of the kahler metric on the classifying space of a calabi yau threefold which is invariant under a cocompact lattice of g 1 introduction let x be a polarized simply connected calabi yau manifold that is x is a simply connected compact kahler manifold of dimension n with zero first chern class and is akform of x such that h 2 x z in this paper we study the local properties of the moduli space of the polarized calabi yau manifold x by definition is the parameter space of the complex structures over x for the fixed polarization is a quasi projective variety by a theorem of viehweg 17 suppose x is a calabi yau manifold let n d im h 1 x tx 0

asymptotic error distributions for the euler method for stochastic differential equations we are interested in the rate of convergence of the euler scheme approximation of the solution to a stochastic differential equation driven by a general possibly discontinuous semimartingale and by the asymptotic behavior of the associated normalized error it is well known that for ito s equations the rate is 1 sqrt n we provide a necessary and sufficient condition for this rate to be 1 sqrt n when the driving semimartingale is a continuous martingale or a continuous semimartingale under a mild additional assumption we also prove that in these cases the normalized error processes converge in law the rate can also differ from 1 sqrt n this is the case for instance if the driving process is deterministic or if it is a levy process without a brownian component it is again 1 sqrt n when the driving process is levy with a nonvanishing brownian component but then the normalized error processes converge in law in the finite dimensional sense only while the discretized normalized error processes converge in law in the skorohod sense and the limit is given an explicit form

globally coupled circle maps abstract a network of circle maps is investigated successive transitions are found among coherent ordered partially ordered po and turbulent phases in the ordered phase two cluster attractors with similarity have a large basin volume which are reduced to a unique circle map through a scaling transformation with two parameters the po phase is characterized by the increase of partition complexity in the phase chaotic itinerancy is observed whose dynamics consists of quasistationary high dimensional chaotic motion low dimensional attractor ruins and switching among them dependence of the diffusion constant on the size and the nonlinearity clearly deviates from that by a random phase approximation suggesting remaining correlation among elements even in the turbulent phase

chaotic itinerancy and power law residence time distribution in stochastic dynamical systems chaotic itinerant motion among varieties of ordered states is described by a stochastic model based on the mechanism of chaotic itinerancy the model consists of a random walk on a half line and a markov chain with a transition probability matrix the stability of attractor ruin in the model is investigated by analyzing the residence time distribution of orbits at attractor ruins it is shown that the residence time distribution averaged over all attractor ruins can be described by the superposition of truncated power law distributions if the basin of attraction for each attractor ruin has a zero measure this result is confirmed by simulation of models exhibiting chaotic itinerancy chaotic itinerancy is also shown to be absent in coupled milnor attractor systems if the transition probability among attractor ruins can be represented as a markov chain

variation of mixed hodge structure i around 1970 griffiths introduced the notion of a variation of hodge structure on a complex manifold s see 17 w 2 it constitutes the axiomatization of the features possessed by the local systems of cohomology associated to a family of compact k ihler manifolds esp smooth projective varieties parametrized by s these are for a variation of hodge structure u of weight m 0 a locally constant sheaf u of q vector spaces or u of real vector spaces 1 hodge filtration horizontality a decreasing filtration o r of gs by holomorphic subbundles such that the evident connection v on takes o p into q 2 hodge decompositions if gs denotes the sheaf of functions on s and if

optimal global approximation of sdes with time irregular coefficients in asymptotic setting we investigate strong approximation of solutions of scalar stochastic differential equations sdes with irregular coefficients in przybylowicz 2015 23 an approximation of solutions of sdes at a single point is considered such kind of approximation is also called a one point approximation comparing to that article we are interested here in a global reconstruction of trajectories of the solutions of sdes in a whole interval of existence we assume that a drift coefficient a 0 t r r is globally lipschitz continuous with respect to a space variable but only measurable with respect to a time variable a diffusion coefficient b 0 t r is only piecewise holder continuous with holder exponent 0 1 the algorithm and results concerning lower bounds from przybylowicz 2015 23 cannot be applied for this problem and therefore we develop a suitable new technique in order to approximate solutions of sdes under such assumptions we define a discrete type randomized euler scheme we provide the error analysis of the algorithm showing that its error is o n min 1 2 moreover we prove that roughly speaking the error of an arbitrary algorithm for fixed a and b that uses n values of the diffusion coefficient cannot converge to zero faster than n min 1 2 as n hence the proposed version of the randomized euler scheme achieves the established best rate of convergence

strong and weak divergence in finite time of euler s method for stochastic differential equations with non globally lipschitz continuous coefficients the stochastic euler scheme is known to converge to the exact solution of a stochastic differential equation sde with globally lipschitz continuous drift and diffusion coefficients recent results extend this convergence to coefficients that grow at most linearly for superlinearly growing coefficients finite time convergence in the strong mean square sense remains in this article we answer this question to the negative and prove for a large class of sdes with non globally lipschitz continuous coefficients that euler s approximation converges neither in the strong mean square sense nor in the numerically weak sense to the exact solution at a finite time point even worse the difference of the exact solution and of the numerical approximation at a finite time point diverges to infinity in the strong mean square sense and in the numerically weak sense

asymptotics of degenerations of mixed hodge structures abstract we construct a hermitian metric on the classifying spaces of graded polarized mixed hodge structures and prove analogs of the strong distance estimate 6 between an admissible period map and the approximating nilpotent orbit we also consider the asymptotic behavior of the biextension metric introduced by hain 12 analogs of the norm estimates of 19 and the asymptotics of the naive limit hodge filtration considered in 21

the law of the euler scheme for stochastic differential equations i convergence rate of the distribution function we study the approximation problem ofef x t byef x t n where x t is the solution of a stochastic differential equation x t n is defined by the euler discretization scheme with stept n andf is a given function for smoothf s talay and tubaro have shown that the erroref x t f x t n can be expanded in powers of 1 n which permits to construct romberg extrapolation precedures to accelerate the convergence rate here we prove that the expansion exists also whenf is only supposed measurable and bounded under an additional nondegeneracy condition of hormander type for the infinitesimal generator of x t to obtain this result we use the stochastic variations calculus in the second part of this work we will consider the density of the law ofx t n and compare it to the density of the law ofx t

variance reduction for simulated diffusions this article develops some variance reduction techniques for the monte carlo integration of functionals of the solutions of ito stochastic differential equations sdes the monte carlo method for sdes offers a means of calculating solutions to certain types of parabolic partial differential equation and so has applications in various fields including stochastic control particle physics and econometrics it involves the representation of the required integrals as means of random variables defined on infinite dimensional wiener spaces which cannot be simulated directly sthey must at some stage be approximated by variables defined on high but finite dimensional spaces the approach taken here is to construct variance reduced random variables on the infinite dimensional spaces which can subsequently be approximated by any of a number of known finite difference methods the methods of control variates and importance sampling are developed in both cases a perfect variate i e one which is unbiased and h

on the strength of attractors in a high dimensional system milnor attractor network robust global attraction and noise induced selection abstract strength of attractor is studied by the return rate to itself after perturbations for a multi attractor state of a globally coupled map it is found that fragile milnor attractors have a large basin volume at the partially ordered phase such dominance of fragile attractors is understood by robustness of global attraction in the phase space change of the attractor strength and basin volume against the parameter and size are studied in the partially ordered phase the dynamics is often described as milnor attractor network which leads to a new interpretation of chaotic itinerancy noise induced selection of fragile attractors is found that has a sharp dependence on the noise amplitude relevance of the observed results to neural dynamics and cell differentiation is also discussed

the euler scheme for levy driven stochastic differential equations in relation with monte carlo methods to solve some integro differential equations we study the approximation problem of ee g x t by ee g bar x t n where x t 0 leq t leq t is the solution of a stochastic differential equation governed by a levy process z t bar x n t is defined by the euler discretization scheme with step fractn with appropriate assumptions we show that the error ee g x t ee g bar x t n can be expanded in powers of frac1n if the levy measure of z has finite moments of order high enough otherwise the rate of convergence is slower and its speed depends on the behavior of the tails of the levy measure the simulation of the increments of z t is also discussed

an euler maruyama method for diffusion equations with discontinuous coefficients and a family of interface conditions abstract numerical methods are developed for linear parabolic equations in one spatial dimension having piecewise constant diffusion coefficients along with a one parameter family of interface conditions at the discontinuity we construct an euler maruyama numerical method for the stochastic differential equation sde corresponding to the alternative divergence formulation of these equations our main result is the construction of an euler scheme that can accommodate specification of any one of a family of interface conditions considered we then prove convergence estimates for the euler scheme to illustrate our method and its theoretical analysis we implement it for the stochastic formulation of the parabolic system

weak rate of convergence of the euler maruyama scheme for stochastic differential equations with non regular drift we consider an euler maruyama type approximation method for a stochastic differential equation sde with a non regular drift and regular diffusion coefficient the method regu larizes the drift coefficient within a certain class of functions and then the euler maruyama scheme for the regularized scheme is used as an approximation this methodology gives two errors the first one is the error of regularization of the drift coefficient within a given class of parametrized functions the second one is the error of the regularized euler maruyama scheme after an optimization procedure with respect to the parameters we obtain various rates which improve other known results

clustering coding switching hierarchical ordering and control in a network of chaotic elements abstract a network of chaotic elements is investigated with the use of globally coupled maps a simple coding of many attractors with clustering is shown through the coding the attractors are organized so that their change exhibits bifurcation like phenomena a precision dependent tree is constructed which leads to the similarity of our attractor with those of spin glasses hierarchical dynamics is constructed on the tree which leads to the dynamical change of trees and the temporal change of effective degrees of freedom by a simple input on a site we can switch among attractors and tune the strength of chaos a threshold on a cluster size is found beyond which a peculiar posi nega switch occurs possible application to biological information processing is discussed with the emphasis on the fuzzy switch chaotic search and hierarchical code categorization

numerical approximation of irregular sdes via skorokhod embeddings abstract we provide a new algorithm for approximating the law of a one dimensional diffusion m solving a stochastic differential equation with possibly irregular coefficients the algorithm is based on the construction of markov chains whose laws can be embedded into the diffusion m with a sequence of stopping times the algorithm does not require any regularity or growth assumption in particular it applies to sdes with coefficients that are nowhere continuous and that grow superlinearly we show that if the diffusion coefficient is bounded and bounded away from 0 then our algorithm has a weak convergence rate of order 1 4 finally we illustrate the algorithm s performance with several examples

on the concept of attractor this note proposes a definition for the concept of attractor based on the probable asymptotic behavior of orbits the definition is sufficiently broad so that every smooth compact dynamical system has at least one attractor