symmetries of lagrangian fibrations abstract we construct fiber preserving anti symplectic involutions for a large class of symplectic manifolds with lagrangian torus fibrations in particular we treat the k3 surface and the six dimensional examples constructed by castano bernard and matessi 2009 8 which include a six dimensional symplectic manifold homeomorphic to the quintic threefold we interpret our results as corroboration of the view that in homological mirror symmetry an anti symplectic involution is the mirror of duality in the same setting we construct fiber preserving symplectomorphisms that can be interpreted as the mirror to twisting by a holomorphic line bundle

on the reconstruction problem in mirror symmetry abstract let m b be a lagrangian torus fibration with singularities such that the fibers are of maslov index zero and unobstructed the paper constructs a rigid analytic space m 0 over the novikov field which is a deformation of the semi flat complex structure of the dual torus fibration over the smooth locus b 0 b of transition functions of m 0 are obtained via a homomorphisms which capture the wall crossing phenomenon of moduli spaces of holomorphic disks

smoothing of stokes discontinuities when the asymptotics of an analytic function f z are given in terms of two asymptotic forms s 1 z s 2 z alternately dominant in alternating sectors of the z plane stokes observed that the coefficients of s 1 s 2 while constant in any one sector can change from sector to sector and that to a first approximation the change appears to be discontinuous recently berry has given an interesting formal argument which shows in more detail how the change takes place and the present paper gives a rigorous treatment of berry s result

smoothing of the stokes phenomenon using mellin barnes integrals abstract recently berry 1989 has put forward a formal theory of the smoothing of the stokes phenomenon based upon optimal truncation of asymptotic expansions and borel summation of the resulting exponentially small remainder terms for certain classes of functions defined by laplace and stieltjes transforms olver jones and boyd 1990 have obtained rigorous uniform exponentially improved asymptotic expansions which describe the smooth transition of a stokes multiplier across a stokes line in this paper we present an alternative theory for functions defined by mellin barnes integrals this approach enables uniform exponentially improved expansions to be established in a direct manner including estimates for the remainder term in the exponentially small series the theory is illustrated by application to the confluent hypergeometric functions considered by olver 1990 1991

error bounds for the method of steepest descents new error bounds are found for the asymptotic expansions which arise in the method of steepest descents these bounds are found by exploiting the reformulation of the method of steepest descents given in recent work by berry howls a major difference between this and previous work is that the global properties of the integrand and not just those on the path of steepest descent play an essential role the bounds are significantly easier to compute than those in the current literature applications of the error bounds are given for asymptotic expansions arising from the pearcey integral the airy function the modified bessel function and the integral of the airy function

overlapping stokes smoothings survival of the error function and canonical catastrophe integrals we derive doubly uniform approximations for the remainder in the optimally truncated saddle point expansion for an integral containing a large parameter double uniformity means that the formulae remain valid while distant saddles responsible for the divergence of the expansion coalesce and separate as described by catastrophe theory and while the subdominant exponentials they contribute switch on and off as described by the error function smoothing of the stokes phenomenon two sorts of asymptotic singularity are thereby united in a common framework the formula for the remainder incorporates both the stokes error function and the canonical catastrophe integrals a numerical illustration is given in which the distant cluster contains two saddles the asymptotic theory gives an accurate description of the details of the fractional remainder even when this is of order exp 36

stieltjes transforms and the stokes phenomenon recently berry olver and jones have found uniform asymptotic expansions for the exponentially small remainder terms that result when asymptotic expansions are optimally truncated these uniform expansions describe the rapid change in the behaviour of the remainders as a stokes line is crossed we show how such uniform expansions may be found when a function can be expressed as a stieltjes transform such an approach has the following advantages the uniform expansion is calculated directly non uniform expansions away from the stokes line are readily found and explicit error bounds may be established we illustrate the method by application to the modified bessel function k v z

smoothing of the stokes phenomenon for high order differential equations we extend the class of functions for which the smooth transition of a stokes multiplier across a stokes line can be rigorously established to functions satisfying a certain differential equation of arbitrary order n the equation chosen admits solutions of hypergeometric function type which in the case n 2 are related to the parabolic cylinder functions in general the solutions of this equation involve compound asymptotic expansions valid in certain sectors of the complex z plane with more than one dominant and subdominant series the functional form of the stokes multipliers expressed in terms of an appropriately scaled variable describing transition across a stokes line is found to obey the error function smoothing law derived by berry

integration theory on the zero sets of polyfold fredholm sections we construct an integration theory for sc differential forms on oriented branched ep subgroupoid for which stokes theorem holds true the construction is compatible with equivalences between ep groupoids and so gives rise to an integration theory for branched suborbifolds of polyfolds examples are the solutions sets of proper oriented fredholm sections of strong polyfold bundles for which we obtain invariants this way

spinor states of real rational curves in real algebraic convex 3 manifolds and enumerative invariants let x be a real algebraic convex 3 manifold whose real part is equipped with a pin i structure we show that every irreducible real rational curve with non empty real part has a canonical spinor state belonging to f 1g the main result is then that the algebraic count of the number of real irreducible rational curves in a given numerical equivalence class passing through the appropriate number of points does not depend on the choice of the real conflguration of points provided that these curves are counted with respect to their spinor states these invariants provide lower bounds for the total number of such real rational curves independantly of the choice of the real conflguration of points

exponentially improved asymptotics for the gamma function by expressing the error term in truncation of the asymptotic expansion in terms of a mellin barnes integral we obtain an exponentially improved asymptotic expansion for z as z in arg z we describe the need for such an expansion arising in applications the result is compared with formal results of dingle obtained via borel summation and rigorous methods of olver using laplace integrals for a different class of functions

hyperasymptotics for integrals with saddles integrals involving exp kf z where k is a large parameter and the contour passes through a saddle of f z are approximated by refining the method of steepest descent to include exponentially small contributions from the other saddles through which the contour does not pass these contributions are responsible for the divergence of the asymptotic expansion generated by the method of steepest descent the refinement is achieved by means of an exact resurgence relation expressing the original integral as its truncated saddle point asymptotic expansion plus a remainder involving the integrals through certain adjacent saddles determined by a topological rule iteration of the resurgence relation and choice of truncation near the least term of the original series leads to a representation of the integral as a sum of contributions associated with multiple scattering paths among the saddles no resummation of divergent series is involved each path gives a hyperseries depending on the terms in the asymptotic expansions for each saddle these depend on the particular integral being studied and so are non universal and certain hyperterminant functions defined by integrals these are always the same and hence universal successive hyperseries get shorter so the scheme naturally halts for two saddles the ultimate error is approximately varepsilon 2 386 where varepsilon proportional to exp a k where a is a positive constant is the error in optimal truncation of the original series as a numerical example an integral with three saddles is computed hyperasymptotically

dispersive hyperasymptotics and the anharmonic oscillator hyperasymptotic summation of steepest descent asymptotic expansions of integrals is extended to functions that satisfy a dispersion relation we apply the method to energy eigenvalues of the anharmonic oscillator for which there is no known integral representation but for which there is a dispersion relation hyperasymptotic summation exploits the rich analytic structure underlying the asymptotics and is a practical alternative to borel summation of the rayleigh schrodinger perturbation series

open gromov witten disk invariants in the presence of an anti symplectic involution abstract for a symplectic manifold with an anti symplectic involution having non empty fixed locus we construct a model of the moduli space of real sphere maps out of moduli spaces of decorated disk maps and give an explicit expression for its first stiefel whitney class as a corollary we obtain a large number of examples which include all odd dimensional projective spaces and many complete intersections for which many types of real moduli spaces are orientable for these manifolds we define open gromov witten invariants with no restriction on the dimension of the manifolds or the type of the constraints if there are no boundary marked points a wdvv type recursion obtained in a sequel computes these invariants for many real symplectic manifolds if there are boundary marked points we define the invariants under some restrictions on the allowed boundary constraints even though the moduli spaces are not orientable in these cases

simultaneous low pass filtering and total variation denoising this paper seeks to combine linear time invariant lti filtering and sparsity based denoising in a principled way in order to effectively filter denoise a wider class of signals lti filtering is most suitable for signals restricted to a known frequency band while sparsity based denoising is suitable for signals admitting a sparse representation with respect to a known transform however some signals cannot be accurately categorized as either band limited or sparse this paper addresses the problem of filtering noisy data for the particular case where the underlying signal comprises a low frequency component and a sparse or sparse derivative component a convex optimization approach is presented and two algorithms derived one based on majorization minimization mm and the other based on the alternating direction method of multipliers admm it is shown that a particular choice of discrete time filter namely zero phase noncausal recursive filters for finite length data formulated in terms of banded matrices makes the algorithms effective and computationally efficient the efficiency stems from the use of fast algorithms for solving banded systems of linear equations the method is illustrated using data from a physiological measurement technique i e near infrared spectroscopic time series imaging that in many cases yields data that is well approximated as the sum of low frequency sparse or sparse derivative and noise components

uniform asymptotic expansions of integrals a selection of problems abstract on the occasion of the conference we mention examples of stieltjes work on asymptotics of special functions the remaining part of the paper gives a selection of asymptotic methods for integrals in particular on uniform approximations we discuss several standard problems and examples in which known special functions error functions airy functions bessel functions etc are needed to construct uniform approximations finally we discuss the recent interest and new insights in the stokes phenomenon an extensive bibliography on uniform asymptotic methods for integrals is given together with references to recent papers on the stokes phenomenon for integrals and related topics

a riemannian approach to reduced plate shell and rod theories abstract we derive a dimensionally reduced limit theory for an n dimensional nonlinear elastic body that is slender along k dimensions the starting point is to view an elastic body as an n dimensional riemannian manifold together with a not necessarily isometric w 1 2 immersion in n dimensional euclidean space the equilibrium configuration is the immersion that minimizes the average discrepancy between the induced and intrinsic metrics the dimensionally reduced limit theory views the elastic body as a k dimensional riemannian manifold along with an isometric w 2 2 immersion in n dimensional euclidean space and linear data in the normal directions the equilibrium configuration minimizes a functional depending on the average covariant derivatives of the linear data the dimensionally reduced limit is obtained using a convergence approach the limit includes as particular cases plate shell and rod theories it applies equally to standard elasticity and to incompatible elasticity thus including as particular cases so called non euclidean plate shell and rod theories

histories of adiabatic quantum transitions the way in which the transition amplitude to an initially unoccupied state increases to its exponentially small final value is studied in detail in the adiabatic approximation for a 2 state quantum system by transforming to a series of superadiabatic bases clinging ever closer to the exact evolving state it is shown that transition histories renormalize onto a universal one in which the amplitude grows to its final value as an error function rather than via large oscillations as in the ordinary adiabatic basis the time for the universal transition is of order surd hslash delta where delta is the small adiabatic slowness parameter in perturbation theory the pre exponential factor of the final amplitude renormalizes superadiabatically from the incorrect value textstyle frac 1 3 pi for the ordinary adiabatic basis to the correct value unity the various histories could be observed in spin experiments

stokes s phenomenon for superfactorial asymptotic series superfactorial series depending on a parameter z are those whose terms a n z grow faster than any power of n if the terms get smaller before they increase the function f z represented by sum 0 infty a n z will exhibit a stokes phenomenon similar to that occurring in asymptotic series whose divergence is merely factorial across stokes lines in the z plane where the late terms all have the same phase a small exponential switches on in the remainder when the series is truncated near its least term the jump is smooth and described by an error function whose argument has a slightly more general form than in the factorial case this result is obtained by a method which is heuristic but applies to superfactorial series where borel summation fails several examples are given including an analytical interpretation of the sum and a numerical test of the error function formula for the function represented by f z sum 0 infty exp n 2 a 2nz where a gg 1

mirror duality and string theoretic hodge numbers we prove in full generality the mirror duality conjecture for string theoretic hodge numbers of calabi yau complete intersections in gorenstein toric fano varieties the proof is based on properties of intersection cohomology

gevrey asymptotics and stieltjes transforms of algebraically decaying functions the development of asymptotic expansions of stieltjes transforms of exponentially decaying functions has been well established in this paper we are concerned with the more difficult case in which the functions decay only algebraically at infinity by using a gevrey type condition we obtain an exponentially improved asymptotic expansion and give three representation theorems to show that the stieltjes transform of algebraically decaying functions can be written as the difference of two integral transforms with exponentially decaying kernels thus making the asymptotic theory developed for integral transforms with exponentially decaying kernels relevant to stieltjes transforms of algebraically decaying functions including the smoothing of the stokes phenomenon

exponentially improved asymptotic solutions of ordinary differential equations ii irregular singularities of rank one re expansions are found for the optimal remainder terms in the well known asymptotic series solutions of homogeneous linear differential equations of the second order in the neighbourhood of an irregular singularity of rank one the re expansions are in terms of generalized exponential integrals and have greater regions of validity than the original expansions as well being considerably more accurate and providing a smooth interpretation of the stokes phenomenon they are also of strikingly simple form in addition explicit asymptotic expansions for the higher coefficients of the original asymptotic solutions are obtained

uniform asymptotic smoothing of stokes s discontinuities across a stokes line where one exponential in an asymptotic expansion maximally dominates another the multiplier of the small exponential changes rapidly if the expansion is truncated near its least term the change is not discontinuous but smooth and moreover universal in form in terms of the singulant f the difference between the larger and smaller exponents and real on the stokes line the change in the multiplier is the error function pi frac 1 2 int sigma infty dt exp t 2 text where sigma imf 2ref frac 1 2 the derivation requires control of exponentially small terms in the dominant series this is achieved with dingle s method of borel summation of late terms starting with the least term in numerical illustrations the multiplier is extracted from dawson s integral erfi and the airy function of the second kind bi the small exponential emerges in the predicted universal manner from the dominant one which can be 10 10 times larger

involutions obstructions and mirror symmetry abstract consider a maslov zero lagrangian submanifold diffeomorphic to a lie group on which an anti symplectic involution acts by the inverse map of the group we show that the fukaya a endomorphism algebra of such a lagrangian is quasi isomorphic to its de rham cohomology tensored with the novikov field in particular it is unobstructed formal and its floer and de rham cohomologies coincide our result implies that the smooth fibers of a large class of singular lagrangian fibrations are unobstructed and their floer and de rham cohomologies coincide this is a step in the syz and family floer cohomology approaches to mirror symmetry more generally our result continues to hold if the lagrangian has cohomology the free graded algebra on a graded vector space v concentrated in odd degree and the anti symplectic involution acts on the cohomology of the lagrangian by the induced map of negative the identity on v it suffices for the maslov class to vanish modulo 4

error bounds and exponential improvements for the asymptotic expansions of the gamma function and its reciprocal in 1994 boyd derived a resurgence representation for the gamma function exploiting the 1991 reformulation of the method of steepest descents by berry and howls using this representation he was able to derive a number of properties of the asymptotic expansion for the gamma function including explicit and realistic error bounds the smooth transition of the stokes discontinuities and asymptotics for the late coefficients the main aim of this paper is to modify boyd s resurgence formula making it suitable for deriving better error estimates for the asymptotic expansions of the gamma function and its reciprocal we also prove the exponentially improved versions of these expansions complete with error terms finally we provide new formal asymptotic expansions for the coefficients appearing in the asymptotic series and compare their numerical efficacy with the results of earlier authors

hyperasymptotics for integrals with finite endpoints berry howls 1991 hereinafter called bh refined the method of steepest descent to study exponentially accurate asymptotics of a general class of integrals involving exp kf z along doubly infinite contours in the complex plane passing over saddlepoints of f z here we derive analogous results for integrals with integrands of a similar form but whose local expansions in powers of 1 k are made about the finite endpoints of semi infinite contours of integration we treat endpoints where f z behaves locally linearly or quadratically generically local endpoint expansions made by the method of steepest descent diverge because of the presence of saddles of f z we derive resurgence relations which express the original integral exactly as a truncated endpoint expansion plus a remainder involving the global saddle structure of f z via integrals through certain adjacent saddles the saddles adjacent to the endpoint are determined by a topological rule if the least term of the endpoint expansion is the n 0 k th summing to here calculates the endpoint integral up to an error of approximately exp n 0 k we develop a scheme involving iteration of the new resurgence relations with a similar one derived in bh which can reduce this error down to exp 2 386n 0 k this hyperasymptotic formalism parallels that of bh and incorporates automatically any change in the complete asymptotic expansion as the endpoint moves in the complex plane provided that it does not coincide with other saddles we illustrate the analytical and numerical use of endpoint hyperasymptotics by application to the complementary error function erfc x and a constructed incomplete airy function

smoothing of stokes s discontinuity for the generalized bessel function ii this is a continuation of an earlier paper in which we investigated the superasymptotics and hyperasymptotics of the generalized bessel function z l 0 z l l 1 l where 0 x and g may be real or complex in this paper we consider the case 1 0 the analysis in the two cases is not quite the same here we shall see that contributions towards hyperasymptotic expansions not only come from adjacent saddles but also from curves that are not even steepest descent paths through any saddles

gamma function asymptotics by an extension of the method of steepest descents recent work of berry howls which reformulated the method of steepest de scents is exploited to derive a new representation for the gamma function it is shown how this representation can be used to derive a number of properties of the asymptotic expansion of the gamma function including explicit and realistic error bounds the berry transition between different asymptotic representations across a stokes line and asymptotic estimates for the late coefficients

invariants of real symplectic 4 manifolds and lower bounds in real enumerative geometry we first build the moduli spaces of real rational pseudo holomorphic curves in a given real symplectic 4 manifold then following the approach of gromov and witten 3 19 11 we define invariants under deformation of real symplectic 4 manifolds these invariants provide lower bounds for the number of real rational j holomorphic curves which realize a given homology class and pass through a given real configuration of points

opening mirror symmetry on the quintic aided by mirror symmetry we determine the number of holomorphic disks ending on the real lagrangian in the quintic threefold we hypothesize that the tension of the domainwall between the two vacua on the brane which is the generating function for the open gromov witten invariants satisfies a certain extension of the picard fuchs differential equation governing periods of the mirror quintic we verify consistency of the monodromies under analytic continuation of the superpotential over the entire moduli space we further check the conjecture by reproducing the first few instanton numbers by a localization computation directly in the a model and verifying ooguri vafa integrality this is the first exact result on open string mirror symmetry for a compact calabi yau manifold