finite sums of products of functions in single variables abstract a characterization is given of smooth functions h of k variables that admit decompositions into a finite sum of products of k functions of single variables only this sufficient and necessary condition is given in terms of certain ordinary linear differential equations formed from a given h which also serve for the construction of the functions in single variables occuring in such a decomposition conditions for the existence of special decompositions are also given including the case of three variables when h x y t n i 1 i x g i y h i t

hodge integrals and degenerate contributions hodge integral techniques are used to compute the degree 1 degenerate contributions of curves of arbitrary genus in the gromov witten theory of 3 folds in the calabi yau case the contributions are compared to related m theoretic calculations in the fano case the contributions suggest new integrality conditions

counting bps states on the enriques calabi yau we study topological string amplitudes for the fhsv model using various techniques this model has a type ii realization involving a calabi yau threefold with enriques fibres which we call the enriques calabi yau by applying heterotic type iia duality we compute the topological amplitudes in the fibre to all genera it turns out that there are two different ways to do the computation that lead to topological couplings with different bps content one of them gives the standard d0 d2 counting amplitudes and from the other one we obtain information about bound states of d0 d4 d2 branes on the enriques fibre we also study the model using mirror symmetry and the holomorphic anomaly equations we verify in this way the heterotic results for the d0 d2 generating functional for low genera and find closed expressions for the topological amplitudes on the total space in terms of modular forms and up to genus three this model turns out to be much simpler than the generic b model and might be exactly solvable

gopakumar vafa invariants via vanishing cycles in this paper we propose an ansatz for defining gopakumar vafa invariants of calabi yau threefolds using perverse sheaves of vanishing cycles our proposal is a modification of a recent approach of kiem li which is itself based on earlier ideas of hosono saito takahashi we conjecture that these invariants are equivalent to other curve counting theories such as gromov witten theory and pandharipande thomas theory our main theorem is that for local surfaces our invariants agree with pt invariants for irreducible one cycles we also give a counter example to the kiem li conjectures where our invariants match the predicted answer finally we give examples where our invariant matches the expected answer in cases where the cycle is non reduced non planar or non primitive

multiple instance content based image retrieval employing isometric embedded similarity measure in image based retrieval global or local features sufficiently discriminative to summarize the image content are commonly extracted first traditional features such as color texture shape or corner characterizing image content are not reliable in terms of similarity measure a good match in the feature domain does not necessarily map to image pairs with similar relationship applying these features as search keys may retrieve dissimilar false positive images or leave similar false negative ones behind moreover images are inherently ambiguous since they contain a great amount of information that justifies many different facets of interpretation using a single image to query a database might employ features that do not match user s expectation and retrieve results with low precision recall ratios how to automatically extract reliable image features as a query key that matches user s expectation in a content based image retrieval cbir system is an important topic the objective of the present work is to propose a multiple instance learning image retrieval system by incorporating an isometric embedded similarity measure multiple instance learning is a way of modeling ambiguity in supervised learning given multiple examples from a small collection of positive and negative example images semantically relevant concepts can be derived automatically and employed to retrieve images from an image database each positive and negative example images are represented by a linear combination of fractal orthonormal basis vectors the mapping coefficients of an image projected onto each orthonormal basis constitute a feature vector the euclidean distance similarity measure is proved to remain consistent i e isometric embedded between any image pairs before and after the projection onto orthonormal axes not only similar images generate points close to each other in the feature space but also dissimilar ones produce feature points far apart the utilization of an isometric embedded fractal based technique to extract reliable image features combined with a multiple instance learning paradigm to derive relevant concepts can produce desirable retrieval results that better match user s expectation in order to demonstrate the feasibility of the proposed approach two sets of test for querying an image database are performed namely the fractal based feature extraction algorithm vs three other feature extractors and single instance vs multiple instance learning both the retrieval results execution time and precision recall curves show favorably for the proposed multiple instance fractal based approach

gromov witten invariants in algebraic geometry gromov witten invariants for arbitrary non singular projective vari eties and arbitrary genus are constructed using the techniques from k behrend b fantechi the intrinsic normal cone

geometry of pdes i integral bordism groups in pdes we improve some our previous theorems on the calculation of integral bordism groups of formally integrable and completely integrable pdes emphasizing the role played by singular solutions and weak solutions some applications to interesting pdes defined on finite dimensional manifolds are also considered

quantum cohomology of the hilbert scheme of points on an resolutions we determine the two point invariants of the equivariant quantum cohomology of the hilbert scheme of points of surface resolutions associated to type an singularities the operators encoding these invariants are expressed in terms of the action of the the affine lie algebra b gl n 1 on its basic representation assuming a certain nondegeneracy conjecture these operators determine the full structure of the quantum cohomology ring a relationship is proven between the quantum cohomology and gromov witten donaldsonthomas theories of an p 1 we close with a discussion of the monodromy properties of the associated quantum differential equation and a generalization to singularities of type d and e

knots links and 4 manifolds in this paper we investigate the relationship between isotopy classes of knots and links in s and the di eomorphism types of homeomorphic smooth 4 manifolds as a corollary of this initial investigation we begin to uncover the surprisingly rich structure of di eomorphism types of manifolds homeomorphic to the k3 surface in order to state our theorems we need to view the seiberg witten invariant of a smooth 4 manifold as a multivariable laurent polynomial to do this recall that the seiberg witten invariant of a smooth closed oriented 4 manifold x with b2 x 1 is an integer valued function which is de ned on the set of spinc structures over x cf w km ko1 t1 in case h1 x z has no 2 torsion which will be the situation in this paper there is a natural identi cation of the spinc structures of x with the characteristic elements of h2 x z in this case we view the seiberg witten invariant as

a multi manifold discriminant analysis method for image feature extraction in this paper we propose a multi manifold discriminant analysis mmda method for an image feature extraction and pattern recognition based on graph embedded learning and under the fisher discriminant analysis framework in an mmda the within class graph and between class graph are respectively designed to characterize the within class compactness and the between class separability seeking for the discriminant matrix to simultaneously maximize the between class scatter and minimize the within class scatter in addition in an mmda the within class graph can represent the sub manifold information while the between class graph can represent the multi manifold information the proposed mmda is extensively examined by using the feret ar and orl face databases and the polyu finger knuckle print databases the experimental results demonstrate that an mmda is effective in feature extraction leading to promising image recognition performance

a comparison theorem for gromov witten invariants in the symplectic category abstract we exploit the geometric approach to the virtual fundamental class due to fukaya ono and li tian to compare gromov witten invariants of a symplectic manifold and a symplectic submanifold whenever all constrained stable maps to the former are contained in the latter to first order various special cases of the comparison theorem in this paper have long been used in the algebraic category some of them have also appeared in the symplectic setting combined with the inherent flexibility of the symplectic category the main theorem leads to a confirmation of pandharipande s gopakumar vafa prediction for gw invariants of fano classes in 6 dimensional symplectic manifolds the proof of the main theorem uses deformations of the cauchy riemann equation that respect the submanifold and carleman similarity principle for solutions of perturbed cauchy riemann equations in a forthcoming paper we apply a similar approach to relative gromov witten invariants and the absolute relative correspondence in genus 0

a geometric approach of the generalized d alembert equation abstract by using a geometric approach we prove that the set of solutions of the generalized d alembert equation n log f x 1 x n 0 considered in domain of the x1 xn space r n is larger than the set of the functions that can be represented in the form as f x1 xn f1 x2 xn fn x1 xn 1

on the extended crystal pde s stability i the n d alembert extended crystal pde s abstract our recent results on extended crystal pde s and geometric theory on pde s stability are applied to the generalized n d alembert pde s d a n n 2 we prove that these are extended crystal pde s for any n 2 for suitable n d a n becomes an extended 0 crystal pde and also a 0 crystal pde an equation having all the same smooth solutions of d a n but without unstabilities at finite time is obtained for each n 2

fractal uncertainty principle with explicit exponent we prove an explicit formula for the dependence of the exponent beta in the fractal uncertainty principle of bourgain dyatlov ann math 187 1 43 2018 on the dimension delta and on the regularity constant c r for the regular set in particular this implies an explicit essential spectral gap for convex co compact hyperbolic surfaces when the hausdorff dimension of the limit set is close to 1

curve counting via stable pairs in the derived category for a nonsingular projective 3 fold x we define integer invariants virtually enumerating pairs c d where c x is an embedded curve and d c is a divisor a virtual class is constructed on the associated moduli space by viewing a pair as an object in the derived category of x the resulting invariants are conjecturally equivalent after universal transformations to both the gromov witten and dt theories of x for calabi yau 3 folds the latter equivalence should be viewed as a wall crossing formula in the derived category

generalized hyers ulam rassias stability of n sesquilinear quadratic mappings on banach modules over c algebras assume that x is a left banach module over a unital c algebra a it is shown that almost every n sesquilinear quadratic mapping h xxxxx n a is an n sesquilinear quadratic mapping when h rx y z 1 z n h x ry z 1 z n h x y rz 1 z 2 z n h x y z 1 z 2 rz n rh x y z 1 z 2 z n r 0 r 1 holds for all x y z 1 z email protected x moreover we prove the generalized hyers ulam rassias stability of an n sesquilinear quadratic mapping on a left banach module over a unital c algebra

learning theory for distribution regression we focus on the distribution regression problem regressing to vector valued outputs from probability measures many important machine learning and statistical tasks fit into this framework including multi instance learning and point estimation problems without analytical solution such as hyperparameter or entropy estimation despite the large number of available heuristics in the literature the inherent two stage sampled nature of the problem makes the theoretical analysis quite challenging since in practice only samples from sampled distributions are observable and the estimates have to rely on similarities computed between sets of points to the best of our knowledge the only existing technique with consistency guarantees for distribution regression requires kernel density estimation as an intermediate step which often performs poorly in practice and the domain of the distributions to be compact euclidean in this paper we study a simple analytically computable ridge regression based alternative to distribution regression where we embed the distributions to a reproducing kernel hilbert space and learn the regressor from the embeddings to the outputs our main contribution is to prove that this scheme is consistent in the two stage sampled setup under mild conditions on separable topological domains enriched with kernels we present an exact computational statistical efficiency trade off analysis showing that our estimator is able to match the one stage sampled minimax optimal rate caponnetto and de vito 2007 steinwart et al 2009 this result answers a 17 year old open question establishing the consistency of the classical set kernel haussler 1999 gartner et al 2002 in regression we also cover consistency for more recent kernels on distributions including those due to christmann and steinwart 2010

an obstruction bundle relating gromov witten invariants of curves and kahler surfaces in a previous paper the authors defined symplectic local gromov witten invariants associated to spin curves and showed that the gw invariants of a k ahler surface x with p g 0 are a sum of such local gw invariants this paper describes how the local gw invariants arise from an obstruction bundle in the sense of taubes over the space of stable maps into curves together with the results of our earlier paper this reduces the calculation of the gw invariants of elliptic and general type complex surfaces to computations in the gw theory of curves with additional classes the euler classes of the real obstruction bundles

the symplectic sum formula for gromov witten invariants in the symplectic category there is a connect sum operation that glues symplectic manifolds by identifying neighborhoods of embedded codimension two submanifolds this paper establishes a formula for the gromov witten invariants of a symplectic sum z x y in terms of the relative gw invariants of x and y several applications to enumerative geometry are given gromov witten invariants are counts of holomorphic maps into symplectic manifolds to define them on a symplectic manifold x w one introduces an almost complex structure j compatible with the symplectic form w and

trace ratio problem revisited dimensionality reduction is an important issue in many machine learning and pattern recognition applications and the trace ratio tr problem is an optimization problem involved in many dimensionality reduction algorithms conventionally the solution is approximated via generalized eigenvalue decomposition due to the difficulty of the original problem however prior works have indicated that it is more reasonable to solve it directly than via the conventional way in this brief we propose a theoretical overview of the global optimum solution to the tr problem via the equivalent trace difference problem eigenvalue perturbation theory is introduced to derive an efficient algorithm based on the newton raphson method theoretical issues on the convergence and efficiency of our algorithm compared with prior literature are proposed and are further supported by extensive empirical results

topology of surface diagrams of smooth 4 manifolds surface diagrams are a new way to specify any smooth closed orientable 4 manifold by an orientable surface decorated with simple closed curves these curves are cyclically indexed and each curve has a unique transverse intersection with the next the aim of this paper is to announce a uniqueness theorem for these objects within a fixed homotopy class that turns out to be similar to the reidemeister singer theorem for heegaard splittings of 3 manifolds

relative gromov witten invariants we define relative gromov witten invariants of a symplectic manifold relative to a codimension two symplectic submanifold these invariants are the key ingredients in the symplectic sum formula of ip4 the main step is the construction of a compact space of v stable maps simple special cases include the hurwitz numbers for algebraic curves and the enumerative invariants of caporaso and harris

generalized re weighting local sampling mean discriminant analysis despite the general success in the pattern recognition community linear discriminant analysis lda has four intrinsic drawbacks in this paper we propose a new feature extraction algorithm namely local sampling mean discriminant analysis lsmda to make up for the first three drawbacks and a generalized re weighting grw framework to make up for the fourth drawback extensive experiments are conducted on both synthetic and real world datasets to evaluate the classification performance of our work the experimental results demonstrate the effectiveness of both lsmda and the grw framework in classifications

virasoro constraints for target curves we prove generalized virasoro constraints for the relative gromov witten theories of all nonsingular target curves descendents of the even cohomology classes are studied first by localization degeneration and completed cycle methods descendents of the odd cohomology are then controlled by monodromy and geometric vanishing relations as an outcome of our results the relative theories of target curves are completely and explicitly determined

multiple instance learning as a classifier combining problem in multiple instance learning mil an object is represented as a bag consisting of a set of feature vectors called instances in the training set the labels of bags are given while the uncertainty comes from the unknown labels of instances in the bags in this paper we study mil with the assumption that instances are drawn from a mixture distribution of the concept and the non concept which leads to a convenient way to solve mil as a classifier combining problem it is shown that instances can be classified with any standard supervised classifier by re weighting the classification posteriors given the instance labels the label of a bag can be obtained as a classifier combining problem an optimal decision rule is derived that determines the threshold on the fraction of instances in a bag that is assigned to the concept class we provide estimators for the two parameters in the model the method is tested on a toy data set and various benchmark data sets and shown to provide results comparable to state of the art mil methods

the topological vertex we construct a cubic field theory which provides all genus amplitudes of the topological a model for all non compact toric calabi yau threefolds the topology of a given feynman diagram encodes the topology of a fixed calabi yau with schwinger parameters playing the role of kahler classes of the threefold we interpret this result as an operatorial computation of the amplitudes in the b model mirror which is the quantum kodaira spencer theory the only degree of freedom of this theory is an unconventional chiral scalar on a riemann surface in this setup we identify the b branes on the mirror riemann surface as fermions related to the chiral boson by bosonization

a structure theorem for the gromov witten invariants of kahler surfaces we prove a structure theorem for the gromov witten invariants of compact kahler surfaces with geometric genus pg 0 under the technical assumption that there is a canonical divisor that is a disjoint union of smooth components the theorem shows that the gw invariants are universal functions determined by the genus of this canonical divisor components and the holomorphic euler characteristic of the surface we compute special cases of these universal functions much of the work on the gromov witten invariants of kahler surfaces has focused on rational and ruled surfaces which have geometric genus pg 0 this paper focuses on surfaces with pg 0 a class that includes most elliptic surfaces and most surfaces of general type in this context we prove a general structure theorem that shows with one technical assumption how the gw invariants are completely determined by the local geometry of a generic canonical divisor the structure theorem is a consequence of a simple fact the image localization lemma of section 3 given a kahler surface x and a canonical divisor d kx this lemma shows that the complex structure j on x can be perturbed to a non integrable almost complex structure jd with the property the image of all jd holomorphic maps lies in the support of d this immediately gives some striking vanishing theorems for the gw invariants of kahler surfaces see section 3 more importantly it implies that the gromov witten invariant of x for genus g and n marked points is a sum gwg n x a gw loc g n dk ak over the connected components dk of d of local invariants that count the contribution of maps whose image lies in or after perturbing to a generic moduli space near dk these local invariants have not been previously defined the proof of their existence relies on using non integrable structures and geometric analysis techniques our structure theorem characterizes the local invariants by expressing them in terms of usual gw invariants of certain standard surfaces for this we make the mild assumption that one can deform the kahler structure on x and choose a canonical divisor d so that all components of d are smooth with this assumption the restriction of kx to each component dk of multiplicity mk is the normal bundle nk to dk and nk k kdk partially supported by the n s f

standard surfaces and nodal curves in symplectic 4 manifolds continuing the program of ds and u1 we introduce refinements of the donaldson smith standard surface count which are designed to count nodal pseudoholomorphic curves and curves with a prescribed decomposition into reducible components in cases where a corresponding analogue of the gromov taubes invariant is easy to define our invariants agree with those analogues we also prove a vanishing result for some of the invariants that count nodal curves

feature extraction based on laplacian bidirectional maximum margin criterion maximum margin criterion mmc based feature extraction is more efficient than linear discriminant analysis lda for calculating the discriminant vectors since it does not need to calculate the inverse within class scatter matrix however mmc ignores the discriminative information within the local structures of samples and the structural information embedding in the images in this paper we develop a novel criterion namely laplacian bidirectional maximum margin criterion lbmmc to address the issue we formulate the image total laplacian matrix image within class laplacian matrix and image between class laplacian matrix using the sample similar weight that is widely used in machine learning the proposed lbmmc based feature extraction computes the discriminant vectors by maximizing the difference between image between class laplacian matrix and image within class laplacian matrix in both row and column directions experiments on the feret and yale face databases show the effectiveness of the proposed lbmmc based feature extraction method

dimensionality reduction of multimodal labeled data by local fisher discriminant analysis reducing the dimensionality of data without losing intrinsic information is an important preprocessing step in high dimensional data analysis fisher discriminant analysis fda is a traditional technique for supervised dimensionality reduction but it tends to give undesired results if samples in a class are multimodal an unsupervised dimensionality reduction method called locality preserving projection lpp can work well with multimodal data due to its locality preserving property however since lpp does not take the label information into account it is not necessarily useful in supervised learning scenarios in this paper we propose a new linear supervised dimensionality reduction method called local fisher discriminant analysis lfda which effectively combines the ideas of fda and lpp lfda has an analytic form of the embedding transformation and the solution can be easily computed just by solving a generalized eigenvalue problem we demonstrate the practical usefulness and high scalability of the lfda method in data visualization and classification tasks through extensive simulation studies we also show that lfda can be extended to non linear dimensionality reduction scenarios by applying the kernel trick

3 dimensional seiberg witten invariants and non kahlerian geometry the paper is concerned with seiberg witten invariants of closed 3 manifolds and their relation to the 4 dimensional invariants the main points are 1 the introduction of parameter dependent seiberg witten invariants for 3 manifolds with b1 0 these invariants depend on a complicated chamber structure defined by a differential geometric brill noether locus in the parameter space 2 a comparison theorem which shows that the seiberg witten invariants of 3 manifolds m with b 1 m 1 are simply the restrictions of the 4 dimensional invariants of the product s 1 m to the set of pull back spinc structures as an immediate consequence one gets the wall crossing formula for the invariants of 3 dimensional manifolds with b1 1 from the 4 dimensional wall crossing formula for manifolds with b 1 3 a novel method to compute the seiberg witten invariants of 3 manifolds m using non kahlerian complex geometry on s 1 m provided this product has such a structure which is compatible with a product metric and orientation induced from m this method relies essentially on a kobayashi hitchin correspondence which is new in the non kahler case 4 a complete description of those 3 manifolds to which our method applies these are riemannian manifolds which admit an oriented riemannian foliation by geodesics such manifolds are always diffeomorphic to seifert manifolds but the foliation has not to be seifert it can have non closed leaves 5 explicit computations of seiberg witten invariants for the 3 sphere s 3 using non seifert riemannian foliations we show that the chamber structure induced by the brill noether locus is very complicated there exist countably many dif ferent chambers and the seiberg witten invariant takes on all integer values

quantum cohomology of the hilbert scheme of points in the plane we determine the ring structure of the equivariant quantum cohomology of the hilbert scheme of points of 2 the operator of quantum multiplication by the divisor class is a nonstationary deformation of the quantum calogero sutherland many body system a relationship between the quantum cohomology of the hilbert scheme and the gromov witten donaldson thomas correspondence for local curves is proven

multiple instance discriminant analysis multiple instance discriminant analysis mida is proposed to cope with the feature extraction problem in multiple instance learning similar to midlabs mida is also derived from linear discriminant analysis lda and both algorithms can be treated as multiple instance extensions of lda different from midlabs which learns from the bag level mida is designed from the instance level mida consists of two versions i e binary class mida b mida and multi class mida m mida which are utilized to cope with binary class standard and multi class multiple instance learning tasks respectively the block coordinate ascent approach by which we seek positive prototypes the most positive instance in a positive bag is termed as the positive prototype of this bag and projection vectors alternatively and iteratively is proposed to optimize b mida and m mida to obtain lower dimensional transformation subspaces extensive experiments empirically demonstrate the effectiveness of b mida and m mida in extracting discriminative components and weakening class label ambiguities for instances in positive bags

gromov witten pairs correspondence for the quintic 3 fold we use the gromov witten pairs descendent correspondence for toric 3 folds and degeneration arguments to establish the gw p correspondence for several compact calabi yau 3 folds including all cy complete intersections in products of projective spaces a crucial aspect of the proof is the study of the gw p correspondence for descendents in relative geometries projective bundles over surfaces relative to a section play a special role the gw p correspondence for calabi yau complete intersections provides a structure result for the gromov witten invariants in a fixed curve class after change of variables the gromov witten series is a rational function in the variable q eiu invariant under q q 1

solving the multiple instance problem with axis parallel rectangles the multiple instance problem arises in tasks where the training examples are ambiguous a single example object may have many alternative feature vectors instances that describe it and yet only one of those feature vectors may be responsible for the observed classification of the object this paper describes and compares three kinds of algorithms that learn axis parallel rectangles to solve the multiple instance problem algorithms that ignore the multiple instance problem perform very poorly an algorithm that directly confronts the multiple instance problem by attempting to identify which feature vectors are responsible for the observed classifications performs best giving 89 correct predictions on a musk odor prediction task the paper also illustrates the use of artificial data to debug and compare these algorithms

reduced open gromov witten invariants on k3 surfaces and multiple cover formula abstract in the paper we study the wall crossing phenomenon of reduced open gromov witten invariants on k3 surfaces with rigid special lagrangian boundary condition as a corollary we derived the multiple cover formula for the reduced open gromov witten invariants which is an open analog of the gopakumar vafa conjecture of genus zero

the local gromov witten theory of curves the local gromov witten theory of curves is solved by localization and degeneration methods localization is used for the exact evalu ation of basic integrals in the local gromov witten theory of p 1 a tqft formalism is defined via degeneration to capture higher genus curves together the results provide a compete and effective solution the local gromov witten theory of curves is equivalent to the lo cal donaldson thomas theory of curves the quantum cohomology of the hilbert scheme points of c 2 and the orbifold quantum cohomol ogy the symmetric product of c 2 the results of the paper provide the local gromov witten calculations required for the proofs of these equivalences

the homotopy lie algebra of symplectomorphism groups of 3 fold blow ups of the projective plane by a result of kedra and pinsonnault we know that the topology of groups of symplectomorphisms of symplectic 4 manifolds is complicated in general however in all known very specific examples the rational cohomology rings of symplectomorphism groups are finitely generated in this paper we compute the rational homotopy lie algebra of symplectomorphism groups of the 3 point blow up of the projective plane with an arbitrary symplectic form and show that in some cases depending on the sizes of the blow ups it is infinite dimensional moreover we explain how the topology is generated by the toric structures one can put on the manifold our method involve the study of the space of almost complex structures compatible with the symplectic structure and it depends on the inflation technique of lalonde mcduff

the intrinsic normal cone let x be an algebraic stack in the sense of deligne mumford we construct a purely 0 dimensional algebraic stack over x in the sense of artin the intrinsic normal cone frak c x the notion of perfect obstruction theory for x is introduced and it is shown how to construct given a perfect obstruction theory for x a pure dimensional virtual fundamental class in the chow group of x we then prove some properties of such classes both in the absolute and in the relative context via a deformation theory interpretation of obstruction theories we prove that several kinds of moduli spaces carry a natural obstruction theory and sometimes a perfect one

enumerative geometry of calabi yau 4 folds gromov witten theory is used to define an enumerative geometry of curves in calabi yau 4 folds the main technique is to find exact solutions to moving multiple cover integrals the resulting invariants are analogous to the bps counts of gopakumar and vafa for calabi yau 3 folds we conjecture the 4 fold invariants to be integers and expect a sheaf theoretic explanation

a degeneration formula of gw invariants this is the sequel to the paper 23 in this paper we construct the virtual moduli cycles of the degeneration of the moduli of stable morphisms constructed in 23 we also construct the virtual moduli cycles of the moduli of relative stable morphisms of a pair of a smooth divisor in a smooth variety based on these we prove a degeneration formula of the gromov witten invariants

gopakumar vafa type invariants on calabi yau 4 folds via descendent insertions the gopakumar vafa type invariants on calabi yau 4 folds which are non trivial only for genus zero and one are defined by klemm pandharipande from gromov witten theory and their integrality is conjectured in a previous work of cao maulik toda mathop mathrm dt nolimits 4 invariants with primary insertions on moduli spaces of one dimensional stable sheaves are used to give a sheaf theoretical interpretation of the genus zero gv type invariants in this paper we propose a sheaf theoretical interpretation of the genus one gv type invariants using descendent insertions on the above moduli spaces the conjectural formula in particular implies nontrivial constraints on genus zero gv type equivalently gw invariants of cy 4 folds which can be proved by the wdvv equation

gromov witten donaldson thomas correspondence for toric 3 folds we prove the equivariant gromov witten theory of a nonsingular toric 3 fold x with primary insertions is equivalent to the equivariant donaldson thomas theory of x as a corollary the topological vertex calculations by agangic klemm marino and vafa of the gromov witten theory of local calabi yau toric 3 folds are proven to be correct in the full 3 leg setting

multivariate online kernel density estimation with gaussian kernels we propose a novel approach to online estimation of probability density functions which is based on kernel density estimation kde the method maintains and updates a non parametric model of the observed data from which the kde can be calculated we propose an online bandwidth estimation approach and a compression revitalization scheme which maintains the kde s complexity low we compare the proposed online kde to the state of the art approaches on examples of estimating stationary and non stationary distributions and on examples of classification the results show that the online kde outperforms or achieves a comparable performance to the state of the art and produces models with a significantly lower complexity while allowing online adaptation

the topology of the space of symplectic balls in rational 4 manifolds we study in this paper the rational homotopy type of the space of symplectic embeddings of the standard ball b4 c r4 into 4 dimensional rational symplectic manifolds where c r2 is the capacity of the standard ball of radius r we compute the rational homotopy groups of that space when the 4 manifold has the form m s2 s2 0 0 where 0 is the area form on the sphere with total area 1 and belongs to the interval 1 2 we show that when is 1 this space retracts to the space of symplectic frames for any value of c however for any given 1 2 the rational homotopy type of that space changes as c crosses the critical parameter 1 which is the difference of areas between the two s2 factors we prove moreover that the full homotopy type of that space changes only at that value that is that the restriction map between these spaces is a homotopy equivalence as long as these values of c remain either below or above that critical value the same methods apply to all other values of and other rational 4 manifolds as well the methods rely on two different tools the study of the action of symplectic groups on the stratified space of almost complex structures developed by gromov abreu and mcduff and the analysis of the relations between the group corresponding to a manifold m the group corresponding to its blow up m and the space of symplectic embedded balls in m

sums of products of functions of x times functions of y we study functions u of the form u x y nk 1ak x bk y mostly by means of two functional determinants wn u and dn u a sample result is that if u has the form on an open subset of a region g on which u is real analytic then u must have this form with possibly different ak and bk on all of g we also study the representability of u x y in the form on a region g in which u is either n times differentiable or is merely continuous the question whether the limit of functions of the form is again of the form is treated moreover we study those functions of the form that are solutions of some classical partial differential equations

m4l maximum margin multi instance multi cluster learning for scene modeling automatically learning and grouping key motion patterns in a traffic scene captured by a static camera is a fundamental and challenging task for intelligent video surveillance to learn motion patterns trajectory obtained by object tracking is parameterized and scene image is spatially and evenly divided into multiple regular cell blocks which potentially contain several primary motion patterns then for each block gaussian mixture model gmm is adopted to learn its motion patterns based on the parameters of trajectories grouping motion pattern can be done by clustering blocks indirectly and each cluster of blocks corresponds to a certain motion pattern for one particular block each of its motion pattern gaussian component can be viewed as an instance and all motion patterns gaussian components constitute a bag which can correspond to multiple semantic clusters therefore blocks can be grouped as a multi instance multi cluster learning mimcl problem and a novel maximum margin multi instance multi cluster learning m 4l algorithm is proposed to avoid processing a difficult optimization problem m 4l is further relaxed and solved by making use of a combination of the cutting plane method and constrained concave convex procedure cccp extensive experiments are conducted on multiple real world video sequences containing various patterns and the results validate the effectiveness of our proposed approach