from real affine geometry to complex geometry we construct from a real ane manifold with singularities a tropical manifold a degeneration of calabi yau manifolds this solves a fundamental problem in mirror symmetry furthermore a striking feature of our approach is that it yields an explicit and canonical order by order description of the degeneration via families of tropical trees this gives complete control of the b model side of mirror symmetry in terms of tropical geometry for example we expect that our deformation parameter is a canonical coordinate and expect period calculations to be expressible in terms of tropical curves we anticipate this will lead to a proof of mirror symmetry via tropical methods

multi peak solutions for the henon equation with slightly subcritical growth we study the dirichlet problem for the henon equation left begin array ll delta u x alpha u frac n 2 n 2 varepsilon hbox in omega u 0 hbox in omega u 0 hbox on partial omega end array right where is the unit ball in mathbb r n with n 3 the power is positive and varepsilon is a small positive parameter we prove that for every integer k 1 the above problem has a solution which blows up at k different points of as varepsilon goes to zero we also show that the ground state solution which blows up at one point is unique

a note on the radial solutions for the supercritical henon equation we prove the existence of a positive radial solution for the henon equation with arbitrary growth the solution is found by means of a shooting method and turns out to be an increasing function of the radial variable some numerical experiments suggest the existence of many positive oscillating solutions

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

tropical counting from asymptotic analysis on maurer cartan equations let be a toric surface and let be its landau ginzburg lg mirror where is the hori vafa potential as shown in their preprint we apply asymptotic analysis to study the extended deformation theory of the lg model and prove that semi classical limits of fourier modes of a specific class of maurer cartan solutions naturally give rise to tropical disks in with maslov index 0 or 2 the latter of which produces a universal unfolding of for our construction reproduces gross perturbed potential adv math 224 2010 pp 169 245 which was proven to be the universal unfolding of written in canonical coordinates we also explain how the extended deformation theory can be used to reinterpret the jumping phenomenon of across walls of the scattering diagram formed by maslov index 0 tropical disks originally observed by gross in the same work in the case of

tropicalization of the moduli space of stable maps let x be an algebraic variety and let s be a tropical variety associated to x we study the tropicalization map from the moduli space of stable maps into x to the moduli space of tropical curves in s we prove that it is a continuous map and that its image is compact and polyhedral loosely speaking when we deform algebraic curves in x the associated tropical curves in s deform continuously moreover the locus of realizable tropical curves inside the space of all tropical curves is compact and polyhedral our main tools are berkovich spaces formal models balancing conditions vanishing cycles and quantifier elimination for rigid subanalytic sets

functorial tropicalization of logarithmic schemes the case of constant coefficients the purpose of this article is to develop foundational techniques from logarithmic geometry in order to define a functorial tropicalization map for fine and saturated logarithmic schemes in the case of constant coefficients our approach crucially uses the theory of fans in the sense of kato and generalizes thuillier s retraction map onto the non archimedean skeleton in the toroidal case for the convenience of the reader many examples as well as an introductory treatment of the theory of kato fans are included

enhanced adaptive volterra filtering by automatic attenuation of memory regions and its application to acoustic echo cancellation this paper presents a novel scheme for nonlinear acoustic echo cancellation based on adaptive volterra filters with linear and quadratic kernels which automatically prefers those diagonals contributing most to the output of the quadratic kernel with the goal of minimizing the overall mean square error in typical echo cancellation scenarios not all coefficients will be equally relevant for the modeling of the nonlinear echo but coefficients close to the main diagonal of the second order kernel will describe most of the nonlinear echo distortions such that not all diagonals need to be implemented however it is difficult to decide the most appropriate number of diagonals a priori since there are many factors that influence this decision such as the energy of the nonlinear echo the shape of the room impulse response or the step size used for the adaptation of kernel coefficients our proposed scheme incorporates adaptive scaling factors that control the influence of each group of adjacent diagonals contributing to the quadratic kernel output an appropriate selection of these factors serves to emphasize or neglect diagonals of the model as required by the present situation we provide adaptation rules for these factors based on previous works on combination of adaptive filters and comprehensive simulations showing the reduced gradient noise reached by the new echo canceller

infinitely many spike solutions for the henon equation with critical growth following the constructive method of wei and yan 22 with new ingredients to take care of n 3 n 3 we prove the existence of infinitely many solutions of the henon equation u x u n 2 n 2 in the unit ball of r n r n n 3 n 3 0 0 with the dirichlet boundary condition

recursive even mirror fourier nonlinear filters and simplified structures in this paper a novel class of recursive nonlinear filters is presented as an extension of finite memory even mirror fourier nonlinear filters these filters are characterized by two relevant properties i they are able to arbitrarily well approximate any discrete time time invariant causal infinite memory continuous nonlinear system and ii they are always stable according to the bounded input bounded output criterion even though recursive models can represent many systems with fewer coefficients than their finite memory counterparts it is still possible to further reduce their computational complexity in fact while in general simplified structures lead to a loss of performance it is pointed out in the paper that in various common real world situations they are able of giving remarkable complexity reductions without negatively affecting the modeling capabilities

enumerative geometry of singular plane curves general points in the plane the answer is easy for 6 1 namely 3 d l 2 but otherwise was to my knowledge unknown until now we are going to develop a recursive procedure for solving such problems to make the recursion possible it will be necessary to consider a generalization of independent interest of the above problem let s be the blow up of ip 2 at a point p with exceptional divisor e1 and a fixed line section eo we call s an irreducible fan a curve c c s is said to be of type d e if it is the proper transform of a curve of degree d in z having multiplicity e at p one then asks how many curves c c s of given type d e having only smooth branches of given configuration along e0 w e1 and otherwise having a given number of nodes as their only singularities pass through the appropriate number of points of s some of which may be on e o w e we will solve this problem by recursion on d e the recursion is based on degenerating the surface s to a surface called a 2 fan of the form so 1 w s 2 with si s i i 2 using and suitably enhancing methods of 3 we can describe a suitable part of the limit on so of the relevant families of curves of type d e on s d e 1 in terms of similar families of curves of type d e and e e e e d on 1 and 2 respectively

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

syz mirror symmetry for toric calabi yau manifolds we investigate mirror symmetry for toric calabi yau manifolds from the perspective of the syz conjecture starting with a non toric special lagrangian torus fibration on a toric calabi yau manifold x we construct a complex manifold check x using t duality modified by quantum corrections these corrections are encoded by fourier transforms of generating functions of certain open gromov witten invariants we conjecture that this complex manifold check x which belongs to the hori iqbal vafa mirror family is inherently written in canonical flat coordinates in particular we obtain an enumerative meaning for the inverse mirror maps and this gives a geometric reason for why their taylor series expansions in terms of the kahler parameters of x have integral coefficients applying the results in a formula equating open and closed gromov witten invariants and its applications to mirror symmetry to appear in pacific j math and a relation for gromov witten invariants of local calabi yau threefolds to appear in math res lett we compute the open gromov witten invariants in terms of local bps invariants and give evidences of our conjecture for several 3 dimensional examples including k mathbb p 2 and k mathbb p 1

fast and robust adaptation of dft domain volterra filters in diagonal coordinates using iterated coefficient updates this paper presents a novel data reusing technique for implementing adaptive discrete fourier transform dft domain volterra filters in diagonal coordinates having arbitrary nonlinear order in general a major drawback of such nonlinear filters is the large number of parameters thus a weak excitation of higher order kernels results in a slow convergence for system identification tasks in order to exploit the available innovation of the signals more effectively each data frame is processed for a specified number of iterations in an overlap save scheme thereby enhancing the convergence performance due to inherent recursions of the repeated filtering and updating steps this concept also lends itself to a fast algorithm requiring only a moderate increase in algorithmic complexity over a baseline implementation a detailed comparison of the required number of multiplications is given for several different algorithm versions despite the relatively simple concept the outlined iteration algorithm exhibits significant gains in both adaptation speed and steady state convergence this is demonstrated by various experiments for both stationary noise and speech input considering an application of nonlinear acoustic echo cancellation finally investigations on the tracking behavior for time variant nonlinearities and robustness against misdetection of double talk situations confirm the promising benefits and good trade off of the proposed iterated coefficient updates volterra filters approach

laurent determinants and arrangements of hyperplane amoebas abstract we study amoebas associated with laurent polynomials and obtain new results regarding the number and structure of the connected components of the complement of the amoeba we also investigate the associated laurent determinant in the case of a hyperplane arrangement we perform explicit computations leading to a closed formula for the laurent determinant

baseband volterra filters for implementing carrier based nonlinearities a diagonal coordinate representation for volterra filters is developed and exploited to derive efficient volterra filter implementations for processing carrier based input signals in the diagonal coordinate representation the output is expressed as a sum of linear filters applied to modified input signals hence linear filtering methods are employed to implement the nonlinear filter on a baseband version of the input downsampling is then used to reduce computational complexity the same approach is employed to develop efficient implementations for processing continuous time carrier based signals pulse amplitude modulated signals and frequency division multiplexed input signals

adaptive volterra filters with evolutionary quadratic kernels using a combination scheme for memory control this paper proposes a new paradigm for adaptive volterra filtering using a time variant size of the quadratic kernel memory in order to optimally identify any unknown transversal second order nonlinear system to this end competing volterra structures of different sizes are employed in a hierarchical combination scheme so as to find the best configuration of the second order kernel memory using the already known diagonal coordinate representation the length and number of required quadratic kernel diagonals can be concurrently estimated by monitoring the combination performance subsequently the memory size of the involved models is dynamically increased or decreased following a set of intuitive rules since this automatic memory adaptation is performed along with the coefficient updates an efficient volterra filter is realized offering great flexibility and minimizing the risk of under or overmodeling any given quadratic nonlinearity besides the straightforward scheme a simplified version is presented greatly reducing the algorithmic demands this efficient version is based on a virtualization of the competing volterra filters by jointly using common coefficients and hence exhibits a computational complexity suitable for practical implementations the robust estimation performance of the approach is demonstrated by various examples for a nonlinear acoustic echo cancellation scenario involving stationary noise real speech signals and realistic volterra kernels

constructing integrable systems of semitoric type let m be a connected symplectic 4 manifold a semitoric integrable system on m essentially consists of a pair of independent real valued smooth functions j and h on m for which j generates a hamiltonian circle action under which h is invariant in this paper we give a general method to construct starting from a collection of five ingredients a symplectic 4 manifold equipped a semitoric integrable system then we show that every semitoric integrable system on a symplectic 4 manifold is obtained in this fashion in conjunction with the uniqueness theorem proved recently by the authors this gives a classification of semitoric integrable systems on 4 manifolds in terms of five invariants

real algebraic curves the moment map and amoebas in this paper we prove the topological uniqueness of maximal arrangements of a real plane algebraic curve with respect to three lines more generally we prove the topological uniqueness of a maximally arranged algebraic curve on a real toric surface we use the moment map as a tool for studying the topology of real algebraic curves and their complexiflcations

localized mirror functor for lagrangian immersions and homological mirrorsymmetry for mathbb p 1_ a b c this paper gives a new way of constructing landau ginzburg mirrors using deformation theory of lagrangian immersions motivated by the works of seidel strominger yau zaslow and fukaya oh ohta ono moreover we construct a canonical functor from the fukaya category to the mirror category of matrix factorizations this functor derives homological mirror symmetry under some explicit assumptions as an application the construction is applied to spheres with three orbifold points to produce their quantum corrected mirrors and derive homological mirror symmetry furthermore we discover an enumerative meaning of the inverse mirror map for elliptic curve quotients

enumerative meaning of mirror maps for toric calabi yau manifolds abstract we prove that the inverse of a mirror map for a toric calabi yau manifold of the form k y where y is a compact toric fano manifold can be expressed in terms of generating functions of genus 0 open gromov witten invariants defined by fukaya oh ohta ono 2010 15 such a relation between mirror maps and disk counting invariants was first conjectured by gross and siebert 2011 24 conjecture 0 2 and remark 5 1 as part of their program and was later formulated in terms of fukaya oh ohta ono s invariants in the toric calabi yau case in chan et al 2012 8 conjecture 1 1

mirror symmetry via logarithmic degeneration data i this paper lays the foundations of a program to study mirror symmetry by studying the log structures of illusie fontaine and kato on degenerations of calabi yau manifolds the basic idea is that one can associate to certain sorts of degenerations of calabi yau manifolds a log calabi yau space which is a log structure on the degenerate fibre the log cy space captures essentially all the information of the degeneration and hence all mirror statements for the large complex structure limit given by the degeneration can already be derived from the log cy space in this paper we begin by discussing affine manifolds with singularities given such an affine manifold along with a polyhedral decomposition we show how to construct a scheme consisting of a union of toric varieties in certain non degenerate cases we can also construct log structures on these schemes conversely given certain sorts of degenerations one can build an affine manifold with singularities structure on the dual intersection complex of the degeneration mirror symmetry is then obtained as a discrete legendre transform on these affine manifolds thus providing an algebro geometrization of the strominger yau zaslow conjecture the deepest result of this paper shows an isomorphism between log complex moduli of a log cy space and log kahler moduli of its mirror

complete moduli in the presence of semiabelian group action i prove the existence and describe the structure of moduli space of pairs p consisting of a projective variety p with semiabelian group action and an ample cartier divisor on it satisfying a few simple conditions every connected component of this moduli space is proper a component containing a projective toric variety is described by a configuration of several polytopes the main one of which is the secondary polytope on the other hand the component containing a principally polarized abelian variety provides a moduli compactification of a g the main irreducible component of this compactification is described by an infinite periodic analog of the secondary polytope and coincides with the toroidal compactification of a g for the second voronoi decomposition

adaptive volterra filters for active control of nonlinear noise processes this paper presents a volterra filtered x least mean square lms algorithm for feedforward active noise control the research has demonstrated that linear active noise control anc systems can be successfully applied to reduce the broadband noise and narrowband noise specifically such linear anc systems are very efficient in reduction of low frequency noise however in some situations the noise that comes from a dynamic system may he a nonlinear and deterministic noise process rather than a stochastic white or tonal noise process and the primary noise at the canceling point may exhibit nonlinear distortion furthermore the secondary path estimate in the anc system which denotes the transfer function between the secondary source secondary speaker and the error microphone may have nonminimum phase and hence the causality constraint is violated if such situations exist the linear anc system will suffer performance degradation an implementation of a volterra filtered x lms vfxlms algorithm based on a multichannel structure is described for feedforward active noise control numerical simulation results show that the developed algorithm achieves performance improvement over the standard filtered x lms algorithm for the following two situations 1 the reference noise is a nonlinear noise process and at the same time the secondary path estimate is of nonminimum phase 2 the primary path exhibits the nonlinear behavior in addition the developed vfxlms algorithm can also be employed as an alternative in the case where the standard filtered x lms algorithm does not perform well

a sparse interpolated scheme for implementing adaptive volterra filters in most practical applications the major drawback for using adaptive volterra filters is the large number of coefficients to cope with several research works discussing strategies to reduce the computational burden of these structures have been presented in the open literature for such a common approach has been the use of some type of sparseness in volterra filter kernels in this work a sparse interpolated approach with the interpolation having the purpose of recreating in an approximate way the elements disregarded for obtaining sparse kernels is presented and discussed thus for the adaptive sparse interpolated volterra filter coefficient update expressions considering both least mean square lms and normalized lms nlms algorithms are derived by using a constrained approach in general the proposed strategy outperforms other sparse schemes in terms of the tradeoff between computational complexity and mean square error mse performance as shown through numerical simulations

existence of infinitely many spike solutions for a critical henon type biharmonic equation abstract following the idea of wei and yan 32 with new ingredients to take care of the critical dimension n 5 we construct infinitely many solutions of the biharmonic equation 2 v x v n 4 n 4 in the unit ball of r n n 5 0 with the navier conditions v v 0 on the boundary

syz mirror symmetry for hypertoric varieties we construct a lagrangian torus fibration on a smooth hypertoric variety and a corresponding syz mirror variety using t duality and generating functions of open gromov witten invariants the variety is singular in general we construct a resolution of the variety using the wall and chamber structure on the base of the syz fibration

enumeration of holomorphic cylinders in log calabi yau surfaces i we define the counting of holomorphic cylinders in log calabi yau surfaces although we start with a complex log calabi yau surface the counting is achieved by applying methods from non archimedean geometry this gives rise to new geometric invariants moreover we prove that the counting satisfies a property of symmetry explicit calculations are given for a del pezzo surface in detail which verify the conjectured wall crossing formula for the focus focus singularity our holomorphic cylinders are expected to give a geometric understanding of the combinatorial notion of broken line by gross hacking keel and siebert our tools include berkovich spaces tropical geometry gromov witten theory and the gaga theorem for non archimedean analytic stacks