maximal unipotent monodromy for complete intersection cy manifolds the computations that are suggested by string theory in the b model requires the existence of degenerations of cy manifolds with maximum unipotent monodromy in string theory such a point in the moduli space is called a large radius limit or large complex structure limit in this paper we are going to construct one parameter families of n dimensional calabi yau manifolds which are complete intersections in toric varieties and which have a monodromy operator t such that t n id n 1 0 but t n id n 0 i e the monodromy operator is maximal unipotent

mirror symmetry on kummer type k3 surfaces we investigate both geometric and conformal field theoretic aspects of mirror symmetry on n 4 4 superconformal field theories with central charge c 6 our approach enables us to determine the action of mirror symmetry on non stable singular fibers in elliptic fibrations of n orbifold limits of k3 the resulting map gives an automorphism of order 4 8 or 12 respectively on the smooth universal covering space of the moduli space we explicitly derive the geometric counterparts of the twist fields in our orbifold conformal field theories the classical mckay correspondence allows for a natural interpretation of our results

solutions of some monge ampere equations with isolated and line singularities in this paper we study existence regularity classification and asymptotic behaviors of solutions of some monge ampere equations with isolated and line singularities we classify all solutions of det 2u 1 in r n with one puncture point this can be applied to characterize ellipsoids in the same spirit of serrin s overdetermined problem for the laplace operator in the case of having k non removable singular points for k 1 modulo affine equivalence the set of all generalized solutions can be identified as an explicit orbifold we also establish existence of global solutions with general singular sets regularity properties and optimal estimates of the second order derivatives of generalized solutions near the singularity consisting of a point or a straight line the geometric motivation comes from singular semi flat calabi yau metrics

sublinear perturbations of the differential equation y n 0 and of the analogous difference equation abstract according to a result of a ghizzetti for any solution y t of the differential equation where y n t i 0 n 1 g i t y i t 0 t 1 1 g i x x n i 1 dx 0 i n 1 either y t 0 for t 1 or there is an integer r with 0 r n 1 such that lim t y t t r exists and 0 related results are obtained for difference and differential inequalities a special case of the former has interesting applications in the study of orthogonal polynomials

affine manifolds syz geometry and the y vertex we study the real monge ampere equation in two and three dimensions both from the point of view of the syz conjecture where solutions give rise to semi flat calabi yau s and in affine differential geometry where solutions yield parabolic affine sphere hypersurfaces we find explicit examples connect the holomorphic function representation to hitchin s description of special lagrangian moduli space and construct the developing map explicitly for a singularity corresponding to the type in elliptic fiber after hyper kahler rotation following baues and cortes we show that various types of metric cones over two dimensional elliptic affine spheres generate solutions of the monge ampere equation in three dimensions we then prove a local and global existence theorem for an elliptic affine two sphere metric with prescribed singularities the metric cone over the two sphere minus three points yields a parabolic affine sphere with singularities along a y shaped locus this gives a semi flat calabi yau metric in a neighborhood of the y vertex

mirror symmetry and noncommutative geometry of a categories homological mirror symmetry aims to explain the phenomenon of mirror symmetry in the language of a categories and their deformation theory in these notes i discuss various aspects of this approach from the point of view of noncommutative algebraic geometry in the tensor category of graded vector spaces

geometry of twisted kahler einstein metrics and collapsing we prove that the twisted kahler einstein metrics that arise on the base of certain holomorphic fiber space with calabi yau fibers have conical type singularities along the discriminant locus these fiber spaces arise naturally when studying the collapsing of ricci flat kahler metrics on calabi yau manifolds and of the kahler ricci flow on compact kahler manifolds with semiample canonical bundle and intermediate kodaira dimension our results allow us to understand their collapsed gromov hausdorff limits when the base is smooth and the discriminant has simple normal crossings

collapsing of abelian fibered calabi yau manifolds we study the collapsing behavior of ricci flat kahler metrics on a projective calabi yau manifold which admits an abelian fibration when the volume of the fibers approaches zero we show that away from the critical locus of the fibration the metrics collapse with locally bounded curvature and along the fibers the rescaled metrics become flat in the limit the limit metric on the base minus the critical locus is locally isometric to an open dense subset of any gromov hausdorff limit space of the ricci flat metrics we then apply these results to study metric degenerations of families of polarized hyperkahler manifolds in the large complex structure limit in this setting we prove an analogue of a result of gross and wilson for k3 surfaces which is motivated by the strominger yau zaslow picture of mirror symmetry

on k3 surfaces with large complex structure abstract we discuss a notion of large complex structure for elliptic k3 surfaces with section inspired by the eight dimensional f theory heterotic duality in string theory this concept is naturally associated with the type ii mumford partial compactification of the moduli space of periods for these structures the paper provides an explicit hodge theoretic condition for the complex structure of an elliptic k3 surface with section to be large we also establish certain geometric consequences of this large complex structure condition in terms of the kodaira types of the singular fibers of the elliptic fibration

open gromov witten invariants on elliptic k3 surfaces and wall crossing in this paper we study holomorphic discs in k3 surfaces and defined the open gromov witten invariants using this new invariant we can establish a version of correspondence between tropical discs and holomorphic discs with non trivial invariants we give an example of wall crossing phenomenon of the invariant and expect it satisfies kontsevich soibelman wall crossing formula

a special lagrangian type equation for holomorphic line bundles let l be a holomorphic line bundle over a compact kahler manifold x motivated by mirror symmetry we study the deformed hermitian yang mills equation on l which is the line bundle analogue of the special lagrangian equation in the case that x is calabi yau we show that this equation is the euler lagrange equation for a positive functional and that solutions are unique global minimizers we provide a necessary and sufficient criterion for existence in the case that x is a kahler surface for the higher dimensional cases we introduce a line bundle version of the lagrangian mean curvature flow and prove convergence when l is ample and x has non negative orthogonal bisectional curvature

some new finite difference inequalities abstract in this paper we establish some new finite difference inequalities related to a certain integral inequality arising in the theory of differential equations the inequalities obtained here can be used as handy tools in the theory of some new classes of finite difference and sum difference equations

convergence of calabi yau manifolds abstract in this paper we study the convergence of calabi yau manifolds under kahler degeneration to orbifold singularities and complex degeneration to canonical singularities including the conifold singularities and the collapsing of a family of calabi yau manifolds

a liouville theorem on the pde det f_ i bar jmath 1 det fiȷ 1 let f be a smooth plurisubharmonic function which solves begin aligned det f i bar jmath 1 text on mathbb c n end aligned suppose that the metric omega f sqrt 1 f i bar jmath dz i wedge d bar z j is complete and f satisfies the growth condition begin aligned mathsf n 0 1 1 z 2 le f le mathsf n 0 1 z 2 text as z rightarrow infty end aligned for some mathsf n 0 0 then f is a quadratic polynomial

velocity probability density functions of high reynolds number turbulence abstract this paper deals with the probability density function pdf of velocity differences between two points separated by distance r measurements of pdfs were made for r lying in the inertial range for two different flows in a jet with r 852 and in a wind tunnel with r 2720 these pdfs have a characteristic non gaussian shape with exponential tails following kolmogorov s general ideas of log normality a new model for the pdf is developed which contains two parameters determined by experiments this empirical model agrees with the experimental results that the tails of the pdf deviate from a truly exponential behaviour in particular for small r in addition the model leads to the general scaling law ln e 2 r r 0 differenr from kolmogorov s third hypothesis ln e 2 ln r r 0 restricted to the inertial range only x is x x we develop also a formalism based on an extremum principle which is consistent with both the log normality of e and the above mentioned power law in this formalism can be interpreted as the codimension of dissipative structures and asymptotically varies as 1 ln r

multifractality in stock indexes fact or fiction abstract multifractal analysis and extensive statistical tests are performed upon intraday minutely data within individual trading days for four stock market indexes including hsi szsc s p 500 and nasdaq to check whether the indexes instead of the returns possess multifractality we find that the mass exponent q is linear and the singularity q is close to 1 for all trading days and all indexes furthermore we find strong evidence showing that the scaling behaviors of the original data sets cannot be distinguished from those of shuffled time series hence the so called multifractality in the intraday stock market indexes is merely an illusion

on some finite difference inequalities in two independent variables in this article we establish some new finite difference inequalities in two independent variables which can be used as tools in the study of certain classes of finite difference and sum difference equations

a new complete calabi yau metric on mathbb c 3 c 3 motivated by the study of collapsing calabi yau 3 folds with a lefschetz k3 fibration we construct a complete calabi yau metric on mathbb c 3 with maximal volume growth which in the appropriate scale is expected to model the collapsing metric near the nodal point this new calabi yau metric has singular tangent cone at infinity mathbb c 2 mathbb z 2 times mathbb c and its riemannian geometry has certain non standard features near the singularity of the tangent cone which are more typical of adiabatic limit problems the proof uses an existence result in h j hein s ph d thesis to perturb an asymptotic approximate solution into an actual solution and the main difficulty lies in correcting the slowly decaying error terms

the kahler ricci flow on surfaces of positive kodaira dimension 4 estimates 9 4 1 the zeroth order and volume estimates 9 4 2 a partial second order estimate 13 4 3 gradient estimates 14 4 4 the second order estimates 18

on a conjecture of candelas and de la ossa we prove that the metric completion of a canonical ricci flat kahler metric on the nonsingular part of a projective calabi yau variety x with ordinary double point singularities is a compact metric length space homeomorphic to the projective variety x itself as an application we prove a conjecture of candelas and de la ossa for conifold flops and transitions

treatment of kurtosis in financial markets abstract since mandelbrot 1963 2 highlighted the fact that data on the yield of financial assets exhibit leptokurtosis different distributions have been presented as alternatives to the normal distribution so far little consideration has been given to the capacity that these distributions have to recover the kurtosis of the sample data our work aims to present distributions which given the broad range of their kurtosis have the capacity to perform adjustment on many occasions where other distributions fail while also being capable of recovering the peakedness of the empirical data another key characteristic of these distributions is that they are defined within a bounded domain in the same way as the sample data an empirical application of these distributions is presented within the financial field by using daily returns

complete ricci flat kahler manifolds of infinite topological type we display an infinite dimensional family of complete ricci flat kahler manifolds of complex dimension 2 for which the second homology is infinitely generated these are obtained from the gibbons hawking ansatz 2 by using infinitely many sparsely distributed centers

isolated singularities of affine special kahler metrics in two dimensions we prove that there are just two types of isolated singularities of special kahler metrics in real dimension two provided the associated holomorphic cubic form does not have essential singularities we also construct examples of such metrics

scaling laws and dissipation scale of a passive scalar in fully developed turbulence abstract the scaling laws of the temperature structure functions and their relation with those of velocity have been experimentally studied the relationship between the dissipative scales for velocity and the temperature is first investigated in agreement with recent numerical simulations of pumir 1994 it is found that for prandtl number close to 1 the dissipation scale for a scalar is smaller than that of velocity thus temperature structure functions present a larger scaling interval than that of the velocity the intermittent corrections of scaling are then analyzed it is shown that as proposed in literature r benzi et al 1992 the second order structure function is affected only by the velocity intermittency this structure function is then used as the reference for testing the applicability of the extended self similarity ess to the passive scalar case ess holds but in a narrower interval than that observed in velocity statistics finally a hierarchy for the temperature structure functions similar to that proposed by she and leveque 1994 for the velocity field is introduced and experimentally tested

scalar intermittency in the variational theory of turbulence abstract the evolution of the probability density function pdf of temperature differences passive or active scalar t x r t x versus the scale r is considered on the point of view of the variational theory the pdf at scale r appears as a superposition of rescaled large scale pdfs with a log normal distribution of the scaling factors as in the velocity case the variance of this log normal distribution behaves as a power law versus r the exponent going to zero as the reynolds or grasshof numbers go to infinity

empirical distributions of chinese stock returns at different microscopic timescales abstract we study the distributions of event time returns and clock time returns at different microscopic timescales using ultra high frequency data extracted from the limit order books of 23 stocks traded in the chinese stock market in 2003 we find that the returns at the one trade timescale obey the inverse cubic law for larger timescales 2 32 trades and 1 5 min the returns follow the student distribution with power law tails with the decrease in timescale the tail becomes fatter which is consistent with the variational theory in turbulence