almost sure well posedness of the cubic nonlinear schrodinger equation below l 2 mathbb t we consider the cauchy problem for the one dimensional periodic cubic non linear schrodinger equation nls with initial data below l 2 in particular we exhibit nonlinear smoothing when the initial data are randomized then we prove local well posedness of nls almost surely for the initial data in the support of the canonical gaussian measures on h s t for each s 1 and global well posedness for each s 1 12

periodic nonlinear schrodinger equation and invariant measures in this paper we continue some investigations on the periodic nlseiu u iu xx u u p 2 p 6 started in lrs we prove that the equation is globally wellposed for a set of data of full normalized gibbs measrue e beta h phi hd phi x h phi tfrac 1 2 int left phi right 2 tfrac 1 p int left phi right p after suitablel 2 truncation the set and the measure are invariant under the flow the proof of a similar result for the kdv and modified kdv equations is outlined the main ingredients used are some estimates from b1 on periodic nls and kdv type equations

on the approximability of numerical taxonomy fitting distances by tree metrics we consider the problem of fitting an n n distance matrix d by a tree metric t let varepsilon be the distance to the closest tree metric under the l infty norm that is varepsilon min t parallel t d parallel infty first we present an o n2 algorithm for finding a tree metric t such that parallel t d parallel infty leq 3 varepsilon second we show that it is cal np hard to find a tree metric t such that parallel t d parallel infty frac 9 8 varepsilon this paper presents the first algorithm for this problem with a performance guarantee

invariant measures for nls in infinite volume we study gibbs measures and invariant dynamics for 1d defocusing nls iut uxx u u u p 2 0 when the period l for all p weak solutions are typically holder continuous on bounded space time regions if p 4 uniqueness and continuous dependence on the initial data may be shown

on the hardness of inferring phylogenies from triplet dissimilarities this work considers the problem of reconstructing a phylogenetic tree from triplet dissimilarities which are dissimilarities defined over taxon triplets triplet dissimilarities are possibly the simplest generalization of pairwise dissimilarities and were used for phylogenetic reconstructions in the past few years we study the hardness of finding a tree best fitting a given triplet dissimilarity table under the norm we show that the corresponding decision problem is np hard and that the corresponding optimization problem cannot be approximated in polynomial time within a constant multiplicative factor smaller than 1 4 on the positive side we present a polynomial time constant rate approximation algorithm for this problem we also address the issue of best fit under maximal distortion which corresponds to the largest ratio between matching entries in two triplet dissimilarity tables we show that it is np hard to approximate the corresponding optimization problem within any constant multiplicative factor

a constructive algorithm for realizing a distance matrix the natural metric of a weighted graph is the length of the shortest paths between all pairs of vertices the investigated problem consists in a representation of a given metric by a graph such that the total length of the graph is minimized for that purpose we give a constructive algorithm based on a technique of reduction fusion and deletion we then show some results on a set of various distance matrices whose optimal realization is known

efficient algorithms for inverting evolution evolution can be mathematically modelled by a stochastic process that operates on the dna of species such models are based on the established theory that the dna sequences or genomes of all extant species have been derived from the genome of the common ancestor of all species by a process of random mutation and natural selection a stochastic model of evolution can be used to construct phylogenies or evolutionary trees for a set of species maximum likelihood estimation mle methods seek the evolutionary tree which is most likely to have produced the dna under consideration while these methods are intellectually satisfying they have not been widely accepted because of their computational intractability in this paper we address the intractability of mle methods as follows we introduce a metric on stochastic process models of evolution we show that this metric is meaningful by proving that in order for any algorithm to distinguish between two stochastic models that are close according to this metric it needs to be given many observations we complement this result with a simple and efficient algorithm for inverting the stochastic process of evolution that is for building a tree from observations on two state characters we will use the same techniques in a subsequent paper to solve the problem for multistate characters and hence for building a tree from dna sequence data the tree we build is provably close in our metric to the tree generating the data and gets closer as more observations become available though there have been many heuristics suggested for the problem of finding good approximations to the most likely tree our algorithm is the first one with a guaranteed convergence rate and further this rate is within a polynomial of the lower bound rate we establish ours is also the first polynomial time algorithm that is proven to converge at all to the correct tree

strichartz inequalities and the nonlinear schrodinger equation on compact manifolds kklwe prove strichartz estimates with fractional loss of derivatives for the schrodinger equation on any riemannian compact manifold as a consequence we infer low regularity local well posedness results in any dimension as well as global existence results for the cauchy problem of nonlinear schrodinger equations on surfaces in the case of defocusing polynomial nonlinearities and on three manifolds in the case of cubic defocusing nonlinearities we also discuss the optimality of these strichartz estimates on spheres

skeletal shape abstraction from examples learning a class prototype from a set of exemplars is an important challenge facing researchers in object categorization although the problem is receiving growing interest most approaches assume a one to one correspondence among local features restricting their ability to learn true abstractions of a shape in this paper we present a new technique for learning an abstract shape prototype from a set of exemplars whose features are in many to many correspondence focusing on the domain of 2d shape we represent a silhouette as a medial axis graph whose nodes correspond to partsrdquo defined by medial branches and whose edges connect adjacent parts given a pair of medial axis graphs we establish a many to many correspondence between their nodes to find correspondences among articulating parts based on these correspondences we recover the abstracted medial axis graph along with the positional and radial attributes associated with its nodes we evaluate the abstracted prototypes in the context of a recognition task

axial symmetric elasticity analysis in non homogeneous bodies under gravitational load by triple reciprocity boundary element method in general internal cells are required to solve elasticity problems by involving a gravitational load in non homogeneous bodies with variable mass density when using a conventional boundary element method bem then the effect of mesh reduction is not achieved and one of the main merits of the bem which is the simplicity of data preparation is lost in this study it is shown that the domain cells can be avoided by using the triple reciprocity bem formulation where the density of domain integral is expressed in terms of other fields that are represented by boundary densities and or source densities at isolated interior points utilizing the rotational symmetry the triple reciprocity bem formulation is developed for axially symmetric elasticity problems in non homogeneous bodies under gravitational force a new computer program was developed and applied to solve several test problems

object recognition as many to many feature matching object recognition can be formulated as matching image features to model features when recognition is exemplar based feature correspondence is one to one however segmentation errors articulation scale difference and within class deformation can yield image and model features which don t match one to one but rather many to many adopting a graph based representation of a set of features we present a matching algorithm that establishes many to many correspondences between the nodes of two noisy vertex labeled weighted graphs our approach reduces the problem of many to many matching of weighted graphs to that of many to many matching of weighted point sets in a normed vector space this is accomplished by embedding the initial weighted graphs into a normed vector space with low distortion using a novel embedding technique based on a spherical encoding of graph structure many to many vector correspondences established by the earth mover s distance framework are mapped back into many to many correspondences between graph nodes empirical evaluation of the algorithm on an extensive set of recognition trials including a comparison with two competing graph matching approaches demonstrates both the robustness and efficacy of the overall approach

a few logs suffice to build almost all trees part ii abstract inferring evolutionary trees is an interesting and important problem in biology but one that is computationally difficult as most associated optimization problems are np hard although many methods are provably statistically consistent i e the probability of recovering the correct tree converges to 1 as the sequence length increases the actual rate of convergence for different methods has not been well understood in a recent paper we introduced a new method for reconstructing evolutionary trees called the dyadic closure method dcm and we showed that dcm has a very fast convergence rate dcm runs in o n 5 log n time where n is the number of sequences and so although polynomial the computational requirements are potentially too large to be of use in practice in this paper we present another tree reconstruction method the witness antiwitness method wam wam is faster than dcm especially on random trees and converges to the true tree topology at the same rate as dcm we also compare wam to other methods used to reconstruct trees including neighbor joining possibly the most popular method among molecular biologists and new methods introduced in the computer science literature

statistical mechanics of the nonlinear schrodinger equation we investigate the statistical mechanics of a complex fieldo whose dynamics is governed by the nonlinear schrodinger equation such fields describe in suitable idealizations langmuir waves in a plasma a propagating laser field in a nonlinear medium and other phenomena their hamiltonian h phi int omega frac 1 2 nabla phi 2 1 p phi p dx is unbounded below and the system will under certain conditions develop self focusing singularities in a finite time we show that when is the circle and thel 2 norm of the field which is conserved by the dynamics is bounded byn the gibbs measure obtained is absolutely continuous with respect to wiener measure and normalizable if and only ifp andn are such that classical solutions exist for all time no collapse of the solitons this measure is essentially the same as that of a one dimensional version of the more realisitc zakharov model of coupled langmuir and ion acoustic waves in a plasma we also obtain some properties of the gibbs state by both analytic and numerical methods asn and the temperature are varied

upgma and the normalized equidistant minimum evolution problem upgma unweighted pair group method with arithmetic mean is a widely used clustering method here we show that upgma is a greedy heuristic for the normalized equidistant minimum evolution neme problem that is finding a rooted tree that minimizes the minimum evolution score relative to the dissimilarity matrix among all rooted trees with the same leaf set in which all leaves have the same distance to the root we prove that the neme problem is np hard in addition we present some heuristic and approximation algorithms for solving the neme problem including a polynomial time algorithm that yields a binary rooted tree whose neme score is within o log2n of the optimum

random data cauchy theory for supercritical wave equations ii a global existence result we prove that the subquartic wave equation on the three dimensional ball with dirichlet boundary conditions admits global strong solutions for a large set of random supercritical initial data in bigcap s 1 2 h s theta times h s 1 theta we obtain this result as a consequence of a general random data cauchy theory for supercritical wave equations developed in our previous work 6 and invariant measure considerations inspired by earlier works by bourgain 2 3 on the non linear schrodinger equation which allow us to obtain also precise large time dynamical informations on our solutions

random data cauchy theory for supercritical wave equations i local theory we study the local existence of strong solutions for the cubic nonlinear wave equation with data in hs m s 1 2 where m is a three dimensional compact riemannian manifold this problem is supercritical and can be shown to be strongly ill posed in the hadamard sense however after a suitable randomization we are able to construct local strong solution for a large set of initial data in hs m where s 1 4 in the case of a boundary less manifold and s 8 21 in the case of a manifold with boundary

provably fast and accurate recovery of evolutionary trees through harmonic greedy triplets we give a greedy learning algorithm for reconstructing an evolutionary tree based on a certain harmonic average on triplets of terminal taxa after the pairwise distances between terminal taxa are estimated from sequence data the algorithm runs in smallbigo numtaxa 2 time using smallbigo numtaxa work space where numtaxa is the number of terminal taxa these time and space complexities are optimal in the sense that the size of an input distance matrix is numtaxa 2 and the size of an output tree is numtaxa moreover in the jukes cantor model of evolution the algorithm recovers the correct tree topology with high probability using sample sequences of length polynomial in 1 numtaxa 2 the logarithm of the error probability and 3 the inverses of two small parameters

improving skeletal shape abstraction using multiple optimal solutions shape abstraction is an important problem faced by researchers in many fields such as pattern recognition computer vision and industrial design a recently developed previous shape abstraction framework demirci et al 20 generates an abstracted shape based on the correspondences between the features of the input shapes where the correspondences are obtained using the first optimal solution of a well known transportation problem as the size of the feature space grows the possibility of having more than one optimal solution for the same problem increases considering the case where multiple optimal solutions exist for the same transportation problem we first rank all optimal solutions based on how much they preserve the local neighborhood relations in this paper instead of creating the abstracted shape using the first optimal solution as done by the previous work we create the abstracted shape using the highest ranked optimal solution with this new property more effective abstracted shapes are generated experimental evaluation of the framework demonstrates that the proposed approach compares favorably with the previous technique in a set of shape retrieval experiments for different datasets highlightsskeletal shape abstraction is reformulated as the transportation problem a distortion free graph embedding is used for a better representation all optimal solutions for the transportation problem are considered the optimal solution which most resembles the neighborhood relations is used experiments demonstrate the improved performance

invariant measure for the schrodinger equation on the real line abstract in this paper we build a gibbs measure for the cubic defocusing schrodinger equation on the real line with a decreasing interaction potential in the sense that the non linearity u 2 u is multiplied by a function which we assume integrable and smooth enough we prove that this equation is globally well posed in the support of this measure and that the measure is invariant under the flow of the equation what is more the support of the measure the set of initial data is disjoint from l 2