asymptotic normality of a nonparametric estimator of sample coverage this paper establishes a necessary and sufficient condition for the asymptotic normality of the nonparametric estimator of sample coverage proposed by good biometrica 40 1953 237 264 this new necessary and sufficient condition extends the validity of the asymptotic normality beyond the previously proven cases

out of plane vibrations of curved non uniform beams of constant radius the governing differential equations for out of plane vibrations of curved non uniform beams of constant radius are derived two physical parameters are introduced to simplify the analysis the explicit relations between the flexural displacement its first three order derivatives and the torsional displacement are derived with these explicit relations the two coupled governing characteristic differential equations can be decoupled and reduced to a sixth order ordinary differential equation with variable coefficients in the torsional displacement it is shown that if the material and geometric properties of the beam are in arbitrary polynomial forms of spatial variable then exact solutions for the out of plane vibrations of the beam can be obtained the derived explicit relations can also be used to reduce the difficulty in experimental measurements finally the influence of taper ratio center angle and arc length on the first two natural frequencies of the beams is illustrated

analytical and numerical approaches to nonlinear galloping of internally resonant suspended cables a study is carried out on nonlinear multimodal galloping of suspended cables a consistent model of a curved cable beam geometrically nonlinear and able to torque recently formulated by the authors is used the model accounts for quasi steady aerodynamic forces including the effect of static swing of the cable and dynamic twist of the cross section complementary solution methods are employed namely finite difference and galerkin spatial discretization followed by numerical time integration or galerkin spatial discretization in conjunction with multiple scale perturbation analysis the different techniques are applied to a cable close to the first cross over point at which a number of internal resonances exist branches of periodic solutions and their stability are evaluated as functions of wind velocity the existence of branches of quasi periodic solutions originating from narrow unstable intervals and propagating elsewhere is also proved qualitative and quantitative results furnished by the different investigation tools are compared among them and the importance of the various components of motion accounted or neglected in the reduced models is discussed

on classification of n 2 supersymmetric theories we find a relation between the spectrum of solitons of massiven 2 quantum field theories ind 2 and the scaling dimensions of chiral fields at the conformal point the condition that the scaling dimensions be real imposes restrictions on the soliton numbers and leads to a classification program for symmetricn 2 conformal theories and their massive deformations in terms of a suitable generalization of dynkin diagrams which coincides with the a d e dynkin diagrams for minimal models the landau ginzburg theories are a proper subset of this classification in the particular case of lg theories we relate the soliton numbers with intersection of vanishing cycles of the corresponding singularity the relation between soliton numbers and the scaling dimensions in this particular case is a well known application of picard lefschetz theory

moduli of sheaves supported on curves of genus four contained in a quadric surface abstract we study the moduli space of stable sheaves of euler characteristic one supported on curves of degree 3 3 contained in a smooth quadric surface we compute its hodge numbers by investigating the variation of the moduli spaces of semi stable coherent systems we find the classification of the semi stable sheaves by means of extensions or resolutions

on the homology of the hilbert scheme of points in the plane geir ellingsrud 1 and stein arild stromme 2 i matematisk institutt universitetet i oslo blindern n oslo 3 norway 2 matematisk institutt universitetet i bergen n 5014 bergen norway although several authors have been interested in the hilbert scheme hilba ip z parametrizing finite subschemes of length d in the projective plane i1 i2 f 1 f2 br among others not much is known about the topological properties of this space the picard group has been calculated i f2 and the homology groups of hilb3 ip 2 have been computed hi in this paper we give a precise description of the additive structure of the homology of hilba ip2 applying the results of birula bialynicki b1 b2 on the cellular decompositions de fined by a torus action to the natural action of a maximal torus of sl 3 on hilbe ipz a rather easy consequence of the fact that this action has finitely many fixpoints is that the cycle maps between the chow groups and the homology groups are isomorphisms in particular there is no odd homology and the homology groups are all free the main objective of this work is to compute their ranks the betti numbers of hilba ipz as a byproduct of our method we get similar results on the homology of the punctual hilbert scheme and of the hilbert scheme of points in the affine plane it seems natural to generalize our results to any toric smooth surface however we give the results only for the rational ruled surfaces if with an indication of the necessary changes in the proofs for simplicity we work over the field of complex numbers but with an appropriate interpretation of the word homology our results remain valid over any base field

multiple d3 instantons and mock modular forms i we study d3 instanton corrections to the hypermultiplet moduli space in type iib string theory compactified on a calabi yau threefold in a previous work consistency of d3 instantons with s duality was established at first order in the instanton expansion using the modular properties of the m5 brane elliptic genus we extend this analysis to the two instanton level where wall crossing phenomena start playing a role we focus on the contact potential an analogue of the kahler potential which must transform as a modular form under s duality we show that it can be expressed in terms of a suitable modification of the partition function of d4 d2 d0 bps black holes constructed out of the generating function of msw invariants the latter coincide with donaldson thomas invariants in a particular chamber modular invariance of the contact potential then requires that in case where the d3 brane wraps a reducible divisor the generating function of msw invariants must transform as a vector valued mock modular form with a specific modular completion built from the msw invariants of the constituents physically this gives a powerful constraint on the degeneracies of bps black holes mathematically our result gives a universal prediction for the modular properties of donaldson thomas invariants of pure two dimensional sheaves

three dimensional free vibrations of a circular arch using the theory of a cosserat point small deformation three dimensional free vibrations of a circular arch with uniform rectangular cross section have been investigated by using different theoretical approaches and by experimental verification special emphasis has been focused on a numerical formulation which models each element of the arch using the theory of a cosserat point comparison has been made with accurate three dimensional numerical modeling of the arch using the computer program ansys also three dimensional beam elements in the computer programs ansys and i deas have been considered along with an exact solution of approximate model equations of the arch the results indicate that appropriate modeling of rotary inertia is necessary to predict accurate frequencies associated with torsional out of plane modes moreover in its general form the theory of a cosserat point is a fully nonlinear theory that has already been tested for buckling of beams and arches therefore the success of the cosserat theory for the dynamic problem considered in this paper suggests that the theory of a cosserat point can be used for more complicated nonlinear dynamic problems of thin rod like structures

kodaira spencer theory of gravity and exact results for quantum string amplitudes we develop techniques to compute higher loop string amplitudes for twistedn 2 theories with 3 i e the critical case an important ingredient is the discovery of an anomaly at every genus in decoupling of brst trivial states captured to all orders by a master anomaly equation in a particular realization of then 2 theories the resulting string field theory is equivalent to a topological theory in six dimensions the kodaira spencer theory which may be viewed as the closed string analog of the chern simons theory using the mirror map this leads to computation of the number of holomorphic curves of higher genus curves in calabi yau manifolds it is shown that topological amplitudes can also be reinterpreted as computing corrections to superpotential terms appearing in the effective 4d theory resulting from compactification of standard 10d superstrings on the correspondingn 2 theory relations withc 1 strings are also pointed out

asymptotic bias in information estimates and the exponential bell polynomials we present a new derivation of the asymptotic correction for bias in the estimate of information from a finite sample the new derivation reveals a relationship between information estimates and a sequence of polynomials with combinatorial significance the exponential bell polynomials and helps to provide an understanding of the form and behavior of the asymptotic correction for bias

virtual refinements of the vafa witten formula we conjecture a formula for the generating function of virtual chi y genera of moduli spaces of rank 2 sheaves on arbitrary surfaces with holomorphic 2 form specializing the conjecture to minimal surfaces of general type and to virtual euler characteristics we recover part of a formula of c vafa and e witten these virtual chi y genera can be written in terms of descendent donaldson invariants using t mochizuki s formula the latter can be expressed in terms of seiberg witten invariants and certain explicit integrals over hilbert schemes of points these integrals are governed by seven universal functions which are determined by their values on mathbb p 2 and mathbb p 1 times mathbb p 1 using localization we calculate these functions up to some order which allows us to check our conjecture in many cases in an appendix by h nakajima and the first named author the virtual euler characteristic specialization of our conjecture is extended to include mu classes thereby interpolating between vafa witten s formula and witten s conjecture for donaldson invariants

elliptic genera of symmetric products and second quantized strings in this note we prove an identity that equates the elliptic genus partition function of a supersymmetric sigma model on the n fold symmetric product m n sn of a manifold m to the partition function of a second quantized string theory on the space m s 1 the generating function of these elliptic genera is shown to be almost an automorphic form for o 3 2 z in the context of d brane dynamics this result gives a precise computation of the free energy of a gas of d strings inside a higher dimensional brane

from sheaves on p 2 to a generalization of the rademacher expansion moduli spaces of stable coherent sheaves on a surface are of much interest for both mathe matics and physics yoshioka computed generating functions of poincare polynomials of such moduli spaces if the surface is p 2 and the rank of the sheaves is 2 motivated by physical arguments this pa per investigates the modular properties of these generating functions it is shown that these functions can be written in terms of the lerch sum and theta function based on this we prove a conjecture by vafa and witten which expresses the generating functions of euler numbers as a mixed mock mod ular form moreover we derive an exact formula of rademacher type for the fourier coefficients of this function this formula requires a generalization of the classical circle method this is the first example of an exact formula for the fourier coefficients of mixed mock modular forms which is of independent mathematical interest 1 introduction and statement of results in the past interactions between physics and mathematics have led to many interesting results motivated by strong weak coupling duality or s duality in physics this article considers various gen erating functions which appear in the study of moduli spaces of stable coherent sheaves on the projective plane p 2 we express the generating functions of poincare polynomials of moduli spaces of rank 2 sheaves in terms of the lerch sum and theta function which we will recall later using these expressions we prove a conjec ture by vafa and witten 45 for the generating functions of euler numbers these functions appear to be related to ramanujan s mock theta functions and therefore transform almost as weakly holomorphic modular forms i e meromorphic mod ular forms whose poles if there are any may only lie in cusps our second main result is an exact formula for the fourier coefficients of these generating functions that formally resembles the rademacher expansion for the coefficients of weakly holomorphic modular forms moduli spaces of coherent sheaves on a complex surface s receive much attention see for example 28 for an extensive work on such moduli spaces more specifically one is interested in the moduli space m r c1 c 2 of semi stable sheaves of rank r with first chern class c1 and second chern class c2 w e will consider topological invariants of m in particular the poincare polynomial p m s 2d imc m i 0 bi m s i and the euler number m p m 1 w here

coverage adjusted entropy estimation summary data on neural coding have frequently been analyzed using information theoretic measures these formulations involve the fundamental and generally difficult statistical problem of estimating entropy we review briefly several methods that have been advanced to estimate entropy and highlight a method the coverage adjusted entropy estimator cae due to chao and shen that appeared recently in the environmental statistics literature this method begins with the elementary horvitz thompson estimator developed for sampling from a finite population and adjusts for the potential new species that have not yet been observed in the sample these become the new patterns or words in a spike train that have not yet been observed the adjustment is due to i j good and is called the good turing coverage estimate we provide a new empirical regularization derivation of the coverage adjusted probability estimator which shrinks the maximum likelihood estimate we prove that the cae is consistent and first order optimal with rate op 1 logn in the class of distributions with finite entropy variance and that within the class of distributions with finite qth moment of the log likelihood the good turing coverage estimate and the total probability of unobserved words converge at rate op 1 logn q we then provide a simulation study of the estimator with standard distributions and examples from neuronal data where observations are dependent the results show that with a minor modification the cae performs much better than the mle and is better than the best upper bound estimator due to paninski when the number of possible words m is unknown or infinite copyright q 2007 john wiley sons ltd

in plane and out of plane free vibrations of curved beams with variable sections we present an approximate method to study the analysis for both the in plane and out of plane free vibration of horizontally curved beams with arbitrary shapes and variable cross sections the characteristic equation for free vibration can be derived by applying the green function which is obtained as a discrete type solution of differential equations governing the flexural behaviour of the curved beam under the action of a concentrated load

bayesian entropy estimation for countable discrete distributions we consider the problem of estimating shannon s entropy h from discrete data in cases where the number of possible symbols is unknown or even countably infinite the pitman yor process a generalization of dirichlet process provides a tractable prior distribution over the space of countably infinite discrete distributions and has found major applications in bayesian non parametric statistics and machine learning here we show that it provides a natural family of priors for bayesian entropy estimation due to the fact that moments of the induced posterior distribution over h can be computed analytically we derive formulas for the posterior mean bayes least squares estimate and variance under dirichlet and pitman yor process priors moreover we show that a fixed dirichlet or pitman yor process prior implies a narrow prior distribution over h meaning the prior strongly determines the entropy estimate in the under sampled regime we derive a family of continuous measures for mixing pitman yor processes to produce an approximately flat prior over h we show that the resulting pitman yor mixture pym entropy estimator is consistent for a large class of distributions finally we explore the theoretical properties of the resulting estimator and show that it performs well both in simulation and in application to real data

s duality and refined bps indices whenever available refined bps indices provide considerably more information on the spectrum of bps states than their unrefined version extending earlier work on the modularity of generalized donaldson thomas invariants counting d4 d2 d0 brane bound states in type iia strings on a calabi yau threefold mathfrak y we construct the modular completion of generating functions of refined bps indices supported on a divisor class although for compact mathfrak y the refined indices are not protected switching on the refinement considerably simplifies the construction of the modular completion furthermore it leads to a non commutative analogue of the tba equations which suggests a quantization of the moduli space consistent with s duality in contrast for a local cy threefold given by the canonical bundle over a complex surface s refined dt invariants are well defined and equal to vafa witten invariants of s our construction provides a modular completion of the generating function of these refined invariants for arbitrary rank in cases where all reducible components of the divisor class are collinear which occurs e g when b 2 mathfrak y 1 or in the local case we show that the holomorphic anomaly equation satisfied by the completed generating function truncates at quadratic order in the local case it agrees with an earlier proposal by minahan et al for unrefined invariants and extends it to the refined level using the afore mentioned non commutative structure finally we show that these general predictions reproduce known results for u 2 and u 3 vafa witten theory on mathrm p 2 and make them explicit for u 4

a note on entropy estimation we compare an entropy estimator recently discussed by zhang 2012 with two estimators and introduced by grassberger 2003 and schurmann 2004 we prove the identity which has not been taken into account by zhang 2012 then we prove that the systematic error bias of is less than or equal to the bias of the ordinary likelihood or plug in estimator of entropy finally by numerical simulation we verify that for the most interesting regime of small sample estimation and large event spaces the estimator has a significantly smaller statistical error than

nonparametric estimation of kullback leibler divergence in this letter we introduce an estimator of kullback leibler divergence based on two independent samples we show that on any finite alphabet this estimator has an exponentially decaying bias and that it is consistent and asymptotically normal to explain the importance of this estimator we provide a thorough analysis of the more standard plug in estimator we show that it is consistent and asymptotically normal but with an infinite bias moreover if we modify the plug in estimator to remove the rare events that cause the bias to become infinite the bias still decays at a rate no faster than further we extend our results to estimating the symmetrized kullback leibler divergence we conclude by providing simulation results which show that the asymptotic properties of these estimators hold even for relatively small sample sizes

configurations in abelian categories iv invariants and changing stability conditions this is the last in a series on configurations in an abelian category a given a finite poset i an i configuration is a finite collection of objects j and morphisms j k or j k j k in a satisfying some axioms where j k are subsets of i configurations describe how an object x in a decomposes into subobjects the first paper defined configurations and studied moduli spaces of configurations in a using artin stacks it showed well behaved moduli stacks obja m i a of objects and configurations in a exist when a is the abelian category coh p of coherent sheaves on a projective scheme p or mod kq of representations of a quiver q the second studied algebras of constructible functions and stack functions on obja the third introduced stability conditions t on a and showed the moduli space objss of semistable objects in class is a constructible subset in obja so its characteristic function ss is a constructible function it formed algebras h pa h to h pa h to of constructible and stack functions on obja and proved many identities in them in this paper if t and t are stability conditions on a we write ss in terms of the ss and deduce the algebras h pa h to are independent of t we study invariantsiss or iss i counting semistable objects or configurations in a which satisfy additive and multiplicative identities we compute them completely when a mod kq or a coh p for p a smooth curve we also find invariants with special properties when a coh p for p a smooth surface with kp 1 nef or a calabi yau 3 fold

euler characteristics of su 2 instanton moduli spaces on rational elliptic surfaces recently minahan nemeschansky vafa and warner computed partition functions for n 4 topological yang mills theory on rational elliptic surfaces in particular they computed generating functions of euler characteristics of su 2 instanton moduli spaces in mathematics they are expected to coincide with those of gieseker compactifications in this paper we compute euler characteristics of these spaces and show that our results coincide with theirs we also check the modular property of z su 2 and z so 3 conjectured by vafa and witten

topological strings and almost modular forms the b model topological string theory on a calabi yau threefold x has a symmetry group generated by monodromies of the periods of x this acts on the topological string wave function in a natural way governed by the quantum mechanics of the phase space h3 x we show that depending on the choice of polarization the genus g topological string amplitude is either a holomorphic quasi modular form or an almost holomorphic modular form of weight 0 under moreover at each genus certain combinations of genus g amplitudes are both modular and holomorphic we illustrate this for the local calabi yau manifolds giving rise to seiberg witten gauge theories in four dimensions and local ip2 and ip1 ip1 as a byproduct we also obtain a simple way of relating the topological string amplitudes near different points in the moduli space which we use to give predictions for gromov witten invariants of the orbifold mathbb c 3 mathbb z 3

partial desingularisations of quotients of nonsingular varieties and their betti numbers when a reductive group g acts linearly on a nonsingular complex projective variety x one can define a projective quotient variety x g using mumford s geometric invariant theory if the condition that every semistable point be properly stable is satisfied this quotient is the ordinary topological quotient of an open subset xss of the variety by the group in k a formula is obtained for the rational cohomology of x g under this condition the formula involves the rational cohomology of x and various linear sections of x together with the rational cohomology of the classifying spaces of g and certain reductive subgroups of g in many interesting examples the condition required in k is not satisfied thus the question arises as to what information we can obtain in general the quotient x g can now have serious singularities in contrast to the good case considered in k where the only singularities are those caused by finite isotropy groups it will be shown in this paper that there is a systematic way of blowing up x along a sequence of nonsingular subvarieties to obtain a variety x with a linear action of g such that every semistable point of x is stable the only assumption that we have to make is that there exists at least one stable point of x then x g is almost a resolution of singularities of x1 g in the sense that the most serious singularities have been resolved moreover there is a formula for the rational cohomology of x g again involving the rational cohomology of x and certain linear sections of x together with the rational cohomology of the classifying spaces of g and some reductive subgroups of g see theorem 8 14 for convenience we shall assume throughout that g is connected however the construction of x works in general and it is straightforward to modify the cohomological formulas to apply to the general case the construction of x g can also be modified to apply in some cases when x has no stable points the layout of the paper is as follows section 1 is a review of the basic facts of geometric invariant theory which will be needed and section 2 describes the relationship of geometric invariant theory with symplectic geometry and the moment map in section 3 semistability and stability in a blow up of x along a nonsingular c invariant subvariety are related to semistability and stability in x

s duality for surfaces with a_n type singularities we show that the generating series of euler characteristics of hilbert schemes of points on any algebraic surface with at worst a n type singularities is described by the theta series determined by integer valued positive definite quadratic forms and the dedekind eta function in particular it is a fourier development of a meromorphic modular form with possibly half integer weight the key ingredient is to apply the flop transformation formula of donaldson thomas type invariants counting two dimensional torsion sheaves on threefolds proved in the author s previous paper

atiyah class and sheaf counting on local calabi yau fourfolds abstract we discuss donaldson thomas dt invariants of torsion sheaves with 2 dimensional support on a smooth projective surface in an ambient non compact calabi yau fourfold given by the total space of a rank 2 bundle on the surface we prove that in certain cases when the rank 2 bundle is chosen appropriately the universal truncated atiyah class of these codimension 2 sheaves reduces to one defined over the moduli space of such sheaves realized as torsion codimension 1 sheaves in a noncompact divisor threefold embedded in the ambient fourfold such reduction property of universal atiyah class enables us to relate our fourfold dt theory to a reduced dt theory of a threefold and subsequently then to the moduli spaces of sheaves on the base surface using results in 15 16 we finally make predictions about modularity of such fourfold invariants when the base surface is an elliptic k3

a normal limit law for a nonparametric estimator of the coverage of a random sample the coverage of a multinomial random sample is the sum of the probabilities of the observed classes a normal limit law is rigorously proved for good s 1953 coverage estimator the result is valid under very general conditions and all terms except the coverage itself are observable nevertheless the implied confidence intervals are not much wider than those developed under restrictive assumptions such as in the classical occupancy problem the asymptotic variance is somewhat unexpected the proof utilizes a method of holst 1979

noether lefschetz theory and the yau zaslow conjecture the yau zaslow conjecture predicts the genus 0 curve counts of k3 surfaces in terms of the dedekindfunction the classical intersection theory of curves in the moduli of k3 surfaces with noether lefschetz divisors is related to 3 fold gromov witten invariants via the k3 curve counts results by borcherds and kudla millson determine these classical intersections in terms of vector valued modular forms proven mirror transformations can often be used to calculate the 3 fold invariants which arise via a detailed study of the stu model determining special curves in the moduli of k3 surfaces we prove the yau zaslow conjecture for all curve classes on k3 surfaces two modular form identities are required the first the klemm lerche mayr identity relating hypergeometric series to modular forms after mirror trans formation is proven here the second the harvey moore identity is proven by d zagier and presented in the paper

in plane and out of plane motions of an extensible semi circular pipe conveying fluid abstract in plane and out of plane motions of a semi circular pipe conveying fluid are analyzed in this paper assuming that the centerline of the semi circular pipe is extensible nonlinear equations of in plane and out of plane motions are derived according to the extended hamilton principle the lagrange nonlinear strain theory and the euler bernoulli beam theory are used to derive the equations the derived equations of motion are discretized by applying the galerkin method linearized equations around the equilibrium position are obtained from the discretized equations and then the dynamic characteristics of the pipe are investigated in addition some modelling issues which are related to the nonlinearity of the circumferential strain and stress are discussed this study finds that a semi circular pipe conveying fluid does not lose stability even at a high fluid velocity although a model using the lagrange nonlinear strain and the corresponding nonlinear stress yields the most accurate computational results of the natural frequencies a model using the lagrange strain and a linearized stress is recommended to compute the natural frequencies efficiently while still maintaining accuracy

vertical d4 d2 d0 bound states on k3 fibrations and modularity an explicit formula is derived for the generating function of vertical d4 d2 d0 bound states on smooth k3 fibered calabi yau threefolds generalizing previous results of gholampour and sheshmani it is also shown that this formula satisfies strong modularity properties as predicted by string theory this leads to a new construction of vector valued modular forms which exhibit some of the features of a generalized hecke transform

integral operators and integral cohomology classes of hilbert schemes the methods of integral operators on the cohomology of hilbert schemes of points on surfaces are developed they are used to establish integral bases for the cohomology groups of hilbert schemes of points on a class of surfaces and conjecturally for all simply connected surfaces

vafa witten theory and iterated integrals of modular forms vafa witten vw theory is a topologically twisted version of mathcal n 4 supersymmetric yang mills theory s duality suggests that the partition function of vw theory with gauge group su n transforms as a modular form under duality transformations interestingly vafa and witten demonstrated the presence of a modular anomaly when the theory has gauge group su 2 and is considered on the complex projective plane mathbb p 2 this modular anomaly could be expressed as an integral of a modular form and also be traded for a holomorphic anomaly we demonstrate that the modular anomaly for gauge group su 3 involves an iterated integral of modular forms moreover the modular anomaly for su 3 can be traded for a holomorphic anomaly which is shown to factor into a product of the partition functions for lower rank gauge groups the su 3 partition function is mathematically an example of a mock modular form of depth two

asymptotic normality of an entropy estimator with exponentially decaying bias this paper establishes the asymptotic normality of an entropy estimator with an exponentially decaying bias on any finite alphabet furthermore it is shown that the nonparametric estimator is asymptotically efficient

in plane free vibration of a single crystal silicon ring in this paper the natural frequencies and the associated mode shapes of in plane free vibration of a single crystal silicon ring are analyzed it is found that the si 1 1 1 ring is two dimensionally isotropic in the 1 1 1 plane for elastic constants but three dimensionally anisotropic while the si 1 0 0 ring is fully anisotropic hamilton s principle is used to derive the equations of vibration which is a set of partial differential equations with coefficients being periodic in polar variable expressing the radial and tangential displacements in sinusoidal form with non predetermined amplitudes and through the integration with respect to the circumferential variable the original governing equations in partial differential form can be converted into the amplitude equations in ordinary differential form the exact expressions for frequencies and mode shapes are obtained it is found that for si 1 0 0 rings the frequencies of a pair of modes which are equal for an isotropic ring split due to the anisotropic effect only for the second in plane vibration mode the phenomena of frequency splitting and degenerate modes can be proved either based on the conservation of averaged mechanical energy or by the concept of crystallographic symmetry groups when the single crystal silicon is replaced by the polycrystalline silicon which is isotropic in elastic constants the derived equations for frequencies correctly predict the vanishing of the phenomenon of frequency splitting

twisted cubics on cubic fourfolds we construct a new twenty dimensional family of projective eight dimen sional irreducible holomorphic symplectic manifolds the compactified moduli space m3 y of twisted cubics on a smooth cubic fourfold y that does not contain a plane is shown to be smooth and to admit a contractionm3 y z y to a projective eight dimensional symplec tic manifoldz y the construction is based on results on linear determinantal representations of singular cubic surfaces

on the existence of a crepant resolution of some moduli spaces of sheaves on an abelian surface let j be an abelian surface with a generic ample line bundle open image in new window for n 1 the moduli space mj 2 0 2n of open image in new window 1 semistable sheaves f of rank 2 with chern classes c1 f 0 c2 f 2n is a singular projective variety endowed with a holomorphic symplectic structure on the smooth locus in this paper we show that there does not exist a crepant resolution of mj 2 0 2n for n 2 this certainly implies that there is no symplectic desingularization of mj 2 0 2n for n 2

free non linear vibration of a rotating thin ring with the in plane and out of plane motions free non linear vibration of a rotating thin ring with a constant speed is analyzed when the ring has both the in plane and out of plane motions the geometric non linearity of displacements is considered by adopting the lagrange strain theory for the circumferential strain instead of the infinitesimal strain theory by using hamilton s principle the coupled non linear partial differential equations are derived which describe the out of plane bending and torsional motions as well as the in plane bending and extensional motions during deriving the equations of motion we discuss how to model the circumferential stress and strain in order to consider the geometric non linearity four models are established three non linear models and one linear model for the four models the linearized equations of motion are obtained in the neighbourhood of the steady state equilibrium position based on the linearized equations of the four cases the natural frequencies are computed at various rotational speeds and then they are compared through the comparison this study recommends which model is appropriate to describe the non linear behaviour more precisely

vertex operators grassmannians and hilbert schemes we approximate the infinite grassmannian by finite dimensional cutoffs and define a family of fermionic vertex operators as the limit of geometric correspondences on the equivariant cohomology groups with respect to a one dimensional torus action we prove that in the localization basis these are the well known fermionic vertex operators on the infinite wedge representation furthermore the boson fermion correspondence locality and intertwining properties with the virasoro algebra are the limits of relations on the finite dimensional cutoff spaces which are true for geometric reasons we then show that these operators are also almost by definition the vertex operators defined by okounkov and the author in carlsson and okounkov http arxiv org abs 0801 2565v2 math ag 2009 on the equivariant cohomology groups of the hilbert scheme of points on mathbb c 2 with respect to a special torus action

free vibrations of circular arches a review abstract in this paper a critical brief review of the free dynamics of circular arches is presented under the classical hypotheses that 1 the dimensions of the cross section are small in comparison with the radius 2 the axis is inextensible and 3 shear forces and rotatory inertia are neglected a simple yet efficient method is proposed which allows one to take into account quite naturally the cross section variations and the presence of flexible supports a number of other approaches are also briefly illustrated from the ritz and galerkin methods to the finite element techniques and their numerical performances are compared for a wide range of examples uniform arches with rotationally flexible and axially flexible supports have been extensively examined together with three different kinds of stepped arches and two different arches with linearly varying cross section whenever possible a comparison with available exact solutions is also performed the agreement among all these methods is generally good even if there are some theoretical discrepancies to be pointed out

ideals of the cohomology rings of hilbert schemes and their applications we study the ideals of the rational cohomology ring of the hilbert scheme x n of n points on a smooth projective surface x as an application for a large class of smooth quasi projective surfaces x we show that every cup product structure constant of h x n is independent of n moreover we obtain two sets of ring generators for the cohomology ring h x n similar results are established for the chen ruan orbifold cohomology ring of the symmetric product in particular we prove a ring isomorphism between h x n c and h orb x n sn c for a large class of smooth quasi projective surfaces with numerically trivial canonical class

genus zero gopakumar vafa type invariants for calabi yau 4 folds abstract in analogy with the gopakumar vafa conjecture on cy 3 folds klemm and pandharipande defined gv type invariants on calabi yau 4 folds using gromov witten theory and conjectured their integrality in this paper we propose a sheaf theoretic interpretation of their genus zero invariants using donaldson thomas theory on cy 4 folds more specifically we conjecture genus zero gv type invariants are d t 4 invariants for one dimensional stable sheaves on cy 4 folds some examples are computed for both compact and non compact cy 4 folds to support our conjectures we also propose an equivariant version of the conjectures for local curves and verify them in certain cases

entropy estimation in turing s perspective a new nonparametric estimator of shannon s entropy on a countable alphabet is proposed and analyzed against the well known plug in estimator the proposed estimator is developed based on turing s formula which recovers distributional characteristics on the subset of the alphabet not covered by a size n sample the fundamental switch in perspective brings about substantial gain in estimation accuracy for every distribution with finite entropy in general a uniform variance upper bound is established for the entire class of distributions with finite entropy that decays at a rate of o ln n n compared to o ln n 2 n for the plug in in a wide range of subclasses the variance of the proposed estimator converges at a rate of o 1 n and this rate of convergence carries over to the convergence rates in mean squared errors in many subclasses specifically for any finite alphabet the proposed estimator has a bias decaying exponentially in n several new bias adjusted estimators are also discussed

wall crossing holomorphic anomaly and mock modularity of multiple m5 branes using wall crossing formulae and the theory of mock modular forms we derive a holomorphic anomaly equation for the modified elliptic genus of two m5 branes wrapping a rigid divisor inside a calabi yau manifold the anomaly originates from restoring modularity of an indefinite theta function capturing the wall crossing of bps invariants associated to d4 d2 d0 brane systems we show the compatibility of this equation with anomaly equations previously observed in the context of mathcal n 4 topological yang mills theory on mathbbm p 2 and e strings obtained from wrapping m5 branes on a del pezzo surface the non holomorphic part is related to the contribution originating from bound states of singly wrapped m5 branes on the divisor we show in examples that the information provided by the anomaly is enough to compute the bps degeneracies for certain charges we further speculate on a natural extension of the anomaly to higher d4 brane charge

the cup product of hilbert schemes for k3 surfaces to any graded frobenius algebra a we associate a sequence of graded frobenius algebras a n so that there is canonical isomorphism of rings h x 2 n h x n 2n for the hilbert scheme x n of generalised n tuples of any smooth projective surface x with numerically trivial canonical bundle

black holes and higher depth mock modular forms by enforcing invariance under s duality in type iib string theory compactified on a calabi yau threefold we derive modular properties of the gene rating function of bps degeneracies of d4 d2 d0 black holes in type iia string theory compactified on the same space mathematically these bps degeneracies are the generalized donaldson thomas invariants counting coherent sheaves with support on a divisor cal d at the large volume attractor point for cal d irreducible this function is closely related to the elliptic genus of the superconformal field theory obtained by wrapping m5 brane on cal d and is therefore known to be modular instead when cal d is the sum of n irreducible divisors cal d i we show that the generating function acquires a modular anomaly we characterize this anomaly for arbitrary n by providing an explicit expression for a non holomorphic modular completion in terms of generalized error functions as a result the generating function turns out to be a mixed mock modular form of depth n 1

on dyson s crank conjecture and the uniform asymptotic behavior of certain inverse theta functions in this paper we prove a longstanding conjecture by freeman dyson concerning the limiting shape of the crank generating function we fit this function in a more general family of inverse theta functions which play a key role in physics

superconformal algebras and mock theta functions it is known that characters of bps representations of extended superconformal algebras do not have good modular properties due to extra singular vectors coming from the bps condition in order to improve their modular properties we apply the method of zwegers which has recently been developed to analyze modular properties of mock theta functions we consider the case of the superconformal algebra at general levels and obtain the decomposition of characters of bps representations into a sum of simple jacobi forms and an infinite series of non bps representations we apply our method to study elliptic genera of hyper kahler manifolds in higher dimensions in particular we determine the elliptic genera in the case of complex four dimensions of the hilbert scheme of points on k3 surfaces k 2 and complex tori a 3

motivic degree zero donaldson thomas invariants given a smooth complex threefold x we define the virtual motive operatorname hilb n x operatorname vir of the hilbert scheme of n points on x in the case when x is calabi yau operatorname hilb n x operatorname vir gives a motivic refinement of the n point degree zero donaldson thomas invariant of x the key example is x 3 where the hilbert scheme can be expressed as the critical locus of a regular function on a smooth variety and its virtual motive is defined in terms of the denef loeser motivic nearby fiber a crucial technical result asserts that if a function is equivariant with respect to a suitable torus action its motivic nearby fiber is simply given by the motivic class of a general fiber this allows us to compute the generating function of the virtual motives operatorname hilb n mathbb c 3 operatorname vir via a direct computation involving the motivic class of the commuting variety we then give a formula for the generating function for arbitrary x as a motivic exponential generalizing known results in lower dimensions the weight polynomial specialization leads to a product formula in terms of deformed macmahon functions analogous to gottsche s formula for the poincare polynomials of the hilbert schemes of points on surfaces

counting rational curves on text k3 surfaces the aim of these notes is to explain the remarkable formula found by yau and zaslow y z to express the number of rational curves on a k3 surface projective k3 surfaces fall into countably many families fg g 1 a surface in fg admits a gdimensional linear system of curves of genus g a naive count of constants suggests that such a system will contain a positive number say n g of rational highly singular curves the formula is

bias analysis in entropy estimation we consider the problem of finite sample corrections for entropy estimation new estimates of the shannon entropy are proposed and their systematic error the bias is computed analytically we find that our results cover correction formulae of current entropy estimates recently discussed in the literature the trade off between bias reduction and the increase of the corresponding statistical error is analysed

estimation of entropy and mutual information we present some new results on the nonparametric estimation of entropy and mutual information first we use an exact local expansion of the entropy function to prove almost sure consistency and central limit theorems for three of the most commonly used discretized information estimators the setup is related to grenander s method of sieves and places no assumptions on the underlying probability measure generating the data second we prove a converse to these consistency theorems demonstrating that a misapplication of the most common estimation techniques leads to an arbitrarily poor estimate of the true information even given unlimited data this inconsistency theorem leads to an analytical approximation of the bias valid in surprisingly small sample regimes and more accurate than the usual 1 n formula of miller and madow over a large region of parameter space the two most practical implications of these results are negative 1 information estimates in a certain data regime are likely contaminated by bias even if bias corrected estimators are used and 2 confidence intervals calculated by standard techniques drastically underestimate the error of the most common estimation methods finally we note a very useful connection between the bias of entropy estimators and a certain polynomial approximation problem by casting bias calculation problems in this approximation theory framework we obtain the best possible generalization of known asymptotic bias results more interesting this framework leads to an estimator with some nice properties the estimator comes equipped with rigorous bounds on the maximum error over all possible underlying probability distributions and this maximum error turns out to be surprisingly small we demonstrate the application of this new estimator on both real and simulated data

d branes categories and n 1 supersymmetry we show that boundary conditions in topological open string theory on calabi yau cy manifolds are objects in the derived category of coherent sheaves as foreseen in the homological mirror symmetry proposal of kontsevich together with conformal field theory considerations this leads to a precise criterion determining the supersymmetry preserving branes at any point in cy moduli space completing the proposal of ii stability

theta functions and hodge numbers of moduli spaces of sheaves on rational surfaces we compute generating functions for the hodge numbers of the moduli spaces of h stable rank 2 sheaves on a rational surface s in terms of theta functions for indefinite lattices if h lies in the closure of the ample cone and has self intersection 0 it follows that the generating functions are jacobi forms in particular the generating functions for the euler numbers can be expressed in terms of modular forms and their transformation behaviour is compatible with the predictions of s duality we also express the generating functions for the signatures in terms of modular forms it turns out that these generating functions are also with respect to another developing parameter the generating function for the donaldson invariants of s evaluated on all powers of the point class

minimax estimation of functionals of discrete distributions we propose a general methodology for the construction and analysis of essentially minimax estimators for a wide class of functionals of finite dimensional parameters and elaborate on the case of discrete distributions where the support size s is unknown and may be comparable with or even much larger than the number of observations n we treat the respective regions where the functional is nonsmooth and smooth separately in the nonsmooth regime we apply an unbiased estimator for the best polynomial approximation of the functional whereas in the smooth regime we apply a bias corrected version of the maximum likelihood estimator mle we illustrate the merit of this approach by thoroughly analyzing the performance of the resulting schemes for estimating two important information measures 1 the entropy h p sum i 1 s p i ln p i and 2 f alpha p sum i 1 s p i alpha alpha 0 we obtain the minimax l 2 rates for estimating these functionals in particular we demonstrate that our estimator achieves the optimal sample complexity n asymp s ln s for entropy estimation we also demonstrate that the sample complexity for estimating f alpha p 0 is n asymp s 1 alpha ln s which can be achieved by our estimator but not the mle for 1 we show the minimax l 2 rate for estimating f alpha p is n ln n 2 alpha 1 for infinite support size while the maximum l 2 rate for the mle is n 2 alpha 1 for all the above cases the behavior of the minimax rate optimal estimators with n samples is essentially that of the mle plug in rule with n ln n samples which we term effective sample size enlargement we highlight the practical advantages of our schemes for the estimation of entropy and mutual information we compare our performance with various existing approaches and demonstrate that our approach reduces running time and boosts the accuracy moreover we show that the minimax rate optimal mutual information estimator yielded by our framework leads to significant performance boosts over the chow liu algorithm in learning graphical models the wide use of information measure estimation suggests that the insights and estimators obtained in this paper could be broadly applicable