multiresolution analysis generated by a seed function in this paper we use the equivalence result originally proved by the author which relates a multiresolution analysis mra of l2 r and an orthonormal set of single electron wave functions in the lowest landau level to build up a procedure which produces starting with a certain square integrable function a mra of l2 r

curves in calabi yau threefolds and topological quantum field theory we continue our study of the local gromov witten invariants of curves in calabi yau threefolds we define relative invariants for local theory which give rise to a 1 1 dimensional topological quantum field theory tqft taking values in the ring mathbb q t the associated frobenius algebra over mathbb q t is semisimple consequently we obtain a structure result for the local invariants as an easy consequence of our structure formula we recover the closed formulas for the local invariants in the case where either the target genus or the degree equals 1

local gromov witten invariants are log invariants abstract we prove a simple equivalence between the virtual count of rational curves in the total space of an anti nef line bundle and the virtual count of rational curves maximally tangent to a smooth section of the dual line bundle we conjecture a generalization to direct sums of line 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

some results on discrete gabor transforms for finite periodic sequences in this correspondence using linear system theory and frame theory we address computation methods for analysis and synthesis sequences of the discrete gabor transform dgt where all sequences are periodic with the same period l and show that the minimum energy solution to the wexler and raz 1990 biorthogonal condition and the solution to the frame operator are the same via linear system theory and frame theory

multi resolution analysis and fractional quantum hall effect more results in a previous paper we have proved that any multi resolution analysis of 2 produces for even values of the inverse filling factor and for a square lattice a single electron wavefunction of the lowest landau level lll which together with its magnetic translate gives rise to an orthonormal set in the lll we have also discussed the inverse construction in this paper we simplify the procedure clarifying the role of the kq representation moreover we extend our previous results to the more physically relevant case of a triangular lattice and to odd values of the inverse filling factor we also comment on other possible shapes of the lattice as well as on the extension to other landau levels finally just as a first application of our technique we compute an approximation of the coulomb energy for the haar wavefunction for a filling 1 3

oversampling in the gabor scheme a method for calculating the coefficients of the gabor expansion in the case of oversampling is presented the method is based on the concept of frames and utilizes the zak transform as such the zak transform highlights the meaning and importance of frames in the context of oversampling and other aspects of signal representation by means of nonorthogonal bases

the generalized gabor transform the generalized gabor transform for image representation is discussed for a given function f t t spl isin r the generalized gabor transform finds a set of coefficients a sub mr such that f t spl sigma sub m spl infin sup spl infin spl sigma sub r spl infin sup spl infin spl alpha sub mr g t mt exp i2 spl pi rt t the original gabor transform proposed by d gabor 1946 is the special case of t t the computation of the generalized gabor transform with biorthogonal functions is discussed the optimal biorthogonal functions are discussed a relation between a window function and its optimal biorthogonal function is presented based on the zak 1967 transform when t t is rational the finite discrete generalized gabor transform is also derived methods of computation for the biorthogonal function are discussed the relation between a window function and its optimal biorthogonal function derived for the continuous variable generalized gabor transform can be extended to the finite discrete case efficient algorithms for the optimal biorthogonal function and generalized gabor transform for the finite discrete case are proposed

log mirror symmetry and local mirror symmetry we study mirror symmetry of log calabi yau surfaces on one hand we consider the number of affine lines of each degree in 2 b where b is a smooth cubic on the other hand we consider coefficients of a certain expansion of a function obtained from the integrals of dx x dy y over 2 chains whose boundaries lie on b where b is a family of smooth cubics then for small degrees they coincide we discuss the relation between this phenomenon and local mirror symmetry for 2 in a calabi yau 3 fold ckyz

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 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

continuous and discrete wavelet transforms this paper is an expository survey of results on integral representations and discrete sum expansions of functions in l 2 bf r in terms of coherent states two types of coherent states are considered weyl heisenberg coherent states which arise from translations and modulations of a single function and affine coherent states called wavelets which arise as translations and dilations of a single function in each case it is shown how to represent any function in l 2 bf r as a sum or integral of these states most of the paper is a survey of literature most notably the work of i daubechies a grossmann and j morlet a few results of the authors are included

multi resolution analysis and fractional quantum hall effect an equivalence result in this article we prove that any multi resolution analysis of l2 r produces for some values of the filling factor a single electron wave function of the lowest landau level lll which together with its magnetic translation gives rise to an orthonormal set in the lll we also discuss the inverse construction moreover we extend this procedure to the higher landau levels and we discuss the analogies and the differences between this procedure and the one previously proposed by j p antoine and the author

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

the wavelet transform time frequency localization and signal analysis two different procedures for effecting a frequency analysis of a time dependent signal locally in time are studied the first procedure is the short time or windowed fourier transform the second is the wavelet transform in which high frequency components are studied with sharper time resolution than low frequency components the similarities and the differences between these two methods are discussed for both schemes a detailed study is made of the reconstruction method and its stability as a function of the chosen time frequency density finally the notion of time frequency localization is made precise within this framework by two localization theorems

stable logarithmic maps to deligne faltings pairs i we introduce a new compactication of the space of relative stable maps this approach uses logarithmic geoemetry in the sense of kato fontaineillusie without taking expansions of the target the underlying structures of the stable logarithmic maps are stable in the usual sense

multiwindow real valued discrete gabor transform and its fast algorithms based on the biorthogonal analysis approach a multiwindow real valued discrete gabor transform m rdgt for periodic sequences is presented to efficiently analyze the dynamic time frequency content of a signal containing components with multiple and or time varying frequencies the m rdgt offers a computationally efficient implementation as well as a real valued formulation of the multiwindow complex valued discrete gabor transform m cdgt the completeness condition of the m rdgt is proved to be equivalent to its biorthogonality constraint between analysis windows and synthesis windows the m rdgt can utilize the fast discrete hartley transform algorithms for fast computation and has a simple relationship with the m cdgt such that its coefficients can be directly computed from the m rdgt coefficients therefore the m rdgt offers an efficient method to compute the m cdgt since the analyzed sequence analysis and synthesis windows in the existing m cdgt must have an equal period if the period of a sequence is very long solving its windows requires a huge amount of computation and memory and could lead to numerical instability to overcome this problem a modified m rdgt for long periodic or even infinite sequences is presented and its corresponding biorthogonality constraint between analysis windows and synthesis windows is modified in which the period of the analysis and synthesis windows is independent of the period of a analyzed sequence so that one can apply short windows to process any long periodic or even in finite sequence finally the multirate based parallel implementation of the m rdgt is presented which has shown to be effective and fast for time frequency analysis

higher genus gromov witten invariants as genus zero invariants of symmetric products i prove a formula expressing the descendent genus g gromov witten invariants of a projective variety x in terms of genus 0 invariants of its symmetric product stack sg 1 x when x is a point the latter are structure constants of the symmetric group and we obtain a new way of calculating the gromov witten invariants of a point

relative gromov witten invariants and the mirror formula let x be a smooth complex projective variety and let y subset x be a smooth very ample hypersurface such that k y is nef using the technique of relative gromov witten invariants we give a new short and geometric proof of a version of the mirror formula i e we show that the generating function of the genus zero 1 point gromov witten invariants of y can be obtained from that of x by a certain change of variables the so called mirror transformation moreover we use the same techniques to give a similar expression for the virtual numbers of degree d plane rational curves meeting a smooth cubic at one point with multiplicity 3d which play a role in local mirror symmetry