on the well posedness of the euler equations in the triebel lizorkin spaces we prove local in time unique existence and a blowup criterion for solutions in the triebel lizorkin space for the euler equations of inviscid incompressible fluid flows in n n 2 as a corollary we obtain global persistence of the initial regularity characterized by the triebel lizorkin spaces for the solutions of two dimensional euler equations to prove the results we establish the logarithmic inequality of the beale kato majda type the moser type of inequality as well as the commutator estimate in the triebel lizorkin spaces the key methods of proof used are the littlewood paley decomposition and the paradifferential calculus by j m bony 2002 john wiley sons inc

feature extraction by non parametric mutual information maximization we present a method for learning discriminative feature transforms using as criterion the mutual information between class labels and transformed features instead of a commonly used mutual information measure based on kullback leibler divergence we use a quadratic divergence measure which allows us to make an efficient non parametric implementation and requires no prior assumptions about class densities in addition to linear transforms we also discuss nonlinear transforms that are implemented as radial basis function networks extensions to reduce the computational complexity are also presented and a comparison to greedy feature selection is made

on estimating a density using hellinger distance and some other strange facts on s interesse ici aux possibles vitesses d estimation d une densite a support compact dans m sous des hypotheses de regularite lorsque la perte est mesuree par le carre de la distance de hellinger on regardera aussi le cas connu des normes mathbb l q pour 1 q 2 et le risque est le risque minimax sur la famille on donne une methode generale permettant de traiter les problemes dans le cadre de la theorie de l approximation sous des conditions concernant l entropie metrique et l e capacite des familles a estimer les rapports entre regularite et entropie metrique etant bien connus nous pourrons aussi traiter les cas classiques et d autres qui le sont moins sous des conditions de bornes inferieures les vitesses sont celles observees pour la norme mathbb l q mais elles different dans le cas general on montre aussi que les restrictions sur la compacite du support ou la regularite sont indispensables et que leur absence mene a l impossibilite d obtenir une estimation raisonnable en ce sens que n importe quelle suite d estimateurs sera arbitrairement mauvaise en un point au moins un resultat analogue est vrai sous des conditions de regularite

moduli spaces and macromolecules techniques from moduli spaces are applied to biological macromolecules the first main result provides new a priori constraints on protein geometry discovered empirically and confirmed computationally the second main result identifies up to homotopy the natural moduli space of several interacting rna molecules with the riemann moduli space of a surface with several boundary components in each fixed genus applications to rna folding prediction are discussed the mathematical and biological frameworks are surveyed and presented from first principles

the complexity of finding minimum length generator sequences abstract the computational complexity of the following problem is investigated given a permutation group specified as a set of generators and a single target permutation which is a member of the group what is the shortest expression for the target permutation in terms of the generators the general problem is demonstrated to be psppace complete and indeed is shown to remain so even when the generator set is restricted to contain only two permutations the restriction on generator set cardinality is the best possible as the problem becomes soluble in polynomial time if the generator set cantains only one permutation an interesting feature of this problem is that it does not fall under the headings of two person games or formal languages which cover the great majority of known pspace complete problems some restricted versions of the problem in which the generator set is fixed rather than being part of the problem instance are also investigated and shown to be computationally tractable one result of this kind is that determining the most compact expression of a permutation in terms of cyclicly adjacent transpositions can be achieved in polynomial time thus from an initial arrangement of distinct objects on a circle one can quickly compute the smallest number of interchanges of adjacent objects required to realise any other arrangement surprisingly this problem appears substantially more difficult to solve than the related one for which a solution has been known for some time in which the objects are arranged on a line segment

remarks on the blow up of the euler equations and the related equations we consider the euler system for inviscid incompressible fluid flows and its perturbations in n n 2 we prove global well posedness of this perturbed euler system in the triebel lizorkin spaces for initial vorticity which is small in the critical besov norms comparison type theorems about the blow up of solutions are proved between the euler system and its perturbations we also study the possiblity of the self similar type of blow up of solutions to the equations

bayesian measures of model complexity and fit summary we consider the problem of comparing complex hierarchical models in which the number of parameters is not clearly defined using an information theoretic argument we derive a measure pd for the effective number of parameters in a model as the difference between the posterior mean of the deviance and the deviance at the posterior means of the parameters of interest in general pd approximately corresponds to the trace of the product of fisher s information and the posterior covariance which in normal models is the trace of the hat matrix projecting observations onto fitted values its properties in exponential families are explored the posterior mean deviance is suggested as a bayesian measure of fit or adequacy and the contributions of individual observations to the fit and complexity can give rise to a diagnostic plot of deviance residuals against leverages adding pd to the posterior mean deviance gives a deviance information criterion for comparing models which is related to other information criteria and has an approximate decision theoretic justification the procedure is illustrated in some examples and comparisons are drawn with alternative bayesian and classical proposals throughout it is emphasized that the quantities required are trivial to compute in a markov chain monte carlo analysis

model selection for multivariate regression in small samples we develop a small sample criterion aicc for selecting multivariate regression models this criterion adjusts the akaike information criterion to be an exact unbiased estimator for the expected kullback leibler information a small sample comparison shows that aicc provides better model order choices than other available model selection methods data from an agricultural experiment are analyzed

complexity reduced scheme for feature extraction with linear discriminant analysis owing to the singularity of the within class scatter linear discriminant analysis lda becomes ill posed for small sample size sss problems null space based lda nlda which is an extension of lda provides good discriminant performances for sss problems yet as the original scheme for the feature extractor fe of nlda suffers from a complexity burden a few modified schemes have since been proposed for complexity reduction in this brief by transforming the problem of finding the fe of nlda into a linear equation problem a novel scheme is derived offering a further reduction of the complexity

the three dimensional euler equations where do we stand the three dimensional euler equations have stood for a quarter of a millenium as a challenge to mathematicians and physicists while much has been discovered the nature of solutions is still largely a mystery this paper surveys some of the issues such as singularity formation that have cost so much effort in the last 25 years in this light we review the beale kato majda theorem and its consequences and then list some of the results of numerical experiments that have been attempted a different line of endeavour focuses on work concerning the pressure hessian and how it may be used and modelled the euler equations are finally discussed in terms of their membership of a class of general lagrangian evolution equations using hamilton s quaternions these are reformulated in an elegant manner to describe the motion and rotation of fluid particles c 2007 elsevier b v all rights reserved

some problems on cayley graphs this survey paper presents the historical development of some problems on cayley graphs which are interesting to graph and group theorists such as hamiltonicity or diameter problems to computer scientists and molecular biologists such as pancake problem or sorting by reversals to coding theorists such as the vertex reconstruction problem related to error correcting codes but not related to ulam s problem

on the finite time singularities of the 3d incompressible euler equations we prove the finite time vorticity blowup in the pointwise sense for solutions of the 3d incompressible euler equations assuming some conditions on the initial data and its corresponding solutions near initial time these conditions are represented by the relation between the deformation tensor and the hessian of pressure both coupled with the vorticity directions associated with the initial data and solutions near initial time we also study the possibility of the enstrophy blowup for the 3d euler and the 3d navier stokes equations and prove the finite time enstrophy blowup for initial data satisfying suitable conditions finally we obtain a new blowup criterion that controls the blowup by a quantity containing the hessian of the pressure 2006 wiley periodicals inc

3d euler equations and ideal mhd mapped to regular systems probing the finite time blowup hypothesis abstract we prove by an explicit construction that solutions to incompressible 3d euler equations defined in the periodic cube 0 l 3 can be mapped bijectively to a new system of equations whose solutions are globally regular we establish that the usual beale kato majda criterion for finite time singularity or blowup of a solution to the 3d euler system is equivalent to a condition on the corresponding regular solution of the new system in the hypothetical case of euler finite time singularity we provide an explicit formula for the blowup time in terms of the regular solution of the new system the new system is amenable to being integrated numerically using similar methods as in euler equations we propose a method to simulate numerically the new regular system and describe how to use this to draw robust and reliable conclusions on the finite time singularity problem of euler equations based on the conservation of quantities directly related to energy and circulation the method of mapping to a regular system can be extended to any fluid equation that admits a beale kato majda type of theorem e g 3d navier stokes 2d and 3d magnetohydrodynamics and 1d inviscid burgers we discuss briefly the case of 2d ideal magnetohydrodynamics in order to illustrate the usefulness of the mapping we provide a thorough comparison of the analytical solution versus the numerical solution in the case of 1d inviscid burgers equation

small global solutions to the damped two dimensional boussinesq equations abstract the two dimensional 2d incompressible euler equations have been thoroughly investigated and the resolution of the global in time existence and uniqueness issue is currently in a satisfactory status in contrast the global regularity problem concerning the 2d inviscid boussinesq equations remains widely open in an attempt to understand this problem we examine the damped 2d boussinesq equations and study how damping affects the regularity of solutions since the damping effect is insufficient in overcoming the difficulty due to the vortex stretching we seek unique global small solutions and the efforts have been mainly devoted to minimizing the smallness assumption by positioning the solutions in a suitable functional setting more precisely the homogeneous besov space b 1 1 we are able to obtain a unique global solution under a minimal smallness assumption

sequential vs parallel complexity in simple gene assembly we investigate some differences between the general intramolecular model for gene assembly and its restricted simple model although both models satisfactorily sort all current experimental data we show that the general model offers assembly strategies for a given string that vary in both assembly length and the operations used while the simple model will always use the same number of each type of operation to sort a gene when simple operations are applied in parallel this is given a new twist we prove that for any n 1 there exists a string having maximally parallel assemblies of any length between n and 2n

on the euler equations in the critical triebel lizorkin spaces in this paper we study the initial value problem of the incompressible euler equations in n for initial data belonging to the critical triebel lizorkin spaces i e v0 fn 11 q q 1 we prove the blow up criterion of solutions in fn 11 q for n 2 3 for n 2 in particular we prove global well posedness of the euler equations in f31 q q 1 for the proof of these results we establish a sharp moser type inequality as well as a commutator type estimate in these spaces the key methods are the littlewood paley decomposition and the paradifferential calculus by j m bony

signed genome rearrangement by reversals and transpositions models and approximations an important problem in computational biology is the genome rearrangement using reversals and transpositions analysis of genome evolving by reversals and transpositions leads to a combinatorial optimization problem of sorting by reversals and transpositions i e sorting of a permutation using reversals and transpositions of arbitrary fragments the reversal operation works on a single segment of the genome by reversing the selected segment two kinds of transpositions have been studied in the literature the first kind of transposition operations delete a segment of the genome and insert it into another position in the genome the second kind of transposition operations delete a segment of the genome and insert its inverse into another position in the genome both transposition operations can be viewed as operations working on two consecutive segments in this paper we introduce a third transposition operation which works on two consecutive segments and study sorting of a signed permutation by reversals and transpositions by allowing reversals and the first kind of transpositions or reversals and the first two kinds of transpositions or reversals and all three kinds of transpositions we have three problem models after establishing a common lower bound on the number of operations needed we present a unified 2 approximation algorithm for all these problems finally we present a better 1 75 approximation for the third problem

any discrimination rule can have an arbitrarily bad probability of error for finite sample size consider the basic discrimination problem based on a sample of size n drawn from the distribution of x y on the borel sets of rdx 0 1 if 0 thus any attempt to find a nontrivial distribution free upper bound for rn will fail and any results on the rate of convergence of rn to r must use assumptions about the distribution of x y

a representation theoretic approach to the calculation of evolutionary distance in bacteria in the context of bacteria and models of their evolution under genome rearrangement we explore a novel application of group representation theory to the inference of evolutionary history our contribution is to show in a very general maximum likelihood setting how to use elementary matrix algebra to sidestep intractable combinatorial computations and convert the problem into one of eigenvalue estimation amenable to standard numerical approximation techniques

bayesian model averaging for linear regression models abstract we consider the problem of accounting for model uncertainty in linear regression models conditioning on a single selected model ignores model uncertainty and thus leads to the underestimation of uncertainty when making inferences about quantities of interest a bayesian solution to this problem involves averaging over all possible models i e combinations of predictors when making inferences about quantities of interest this approach is often not practical in this article we offer two alternative approaches first we describe an ad hoc procedure occam s window which indicates a small set of models over which a model average can be computed second we describe a markov chain monte carlo approach that directly approximates the exact solution in the presence of model uncertainty both of these model averaging procedures provide better predictive performance than any single model that might reasonably have been selected in the extreme case where there are many candidate predictors but

on the deformations of the incompressible euler equations we consider systems of deformed system of equations which are obtained by some transformations from the system of incompressible euler equations these have similar properties to the original euler equations including the scaling invariance for one form of deformed system we prove that finite time blow up actually occurs for generic initial data while for the other form of the deformed system we prove the global in time regularity for smooth initial data moreover using the explicit functional relations between the solutions of those deformed systems and that of the original euler system we derive the condition of finite time blow up of the euler system in terms of solutions of one of its deformed systems as another application of those relations we deduce a lower estimate of the possible blow up time of the 3d euler equations

sorting permutations and binary strings by length weighted rearrangements abstract comparative genomics is a line of study which aims at determining dis similarities between genomes one way of doing this is to infer the large scale mutations also called genome rearrangements that are responsible for transforming one genome into another under the parsimony principle numerous efforts have been made over the past years regarding the traditional approach in which all genome rearrangements have the same unit cost however since long rearrangements are more likely to disturb the organism some works considered the length weighted approach in which the cost of each rearrangement is a function of its length the problem is thus modeled as follows given two genomes g 1 and g 2 and a set of allowed genome rearrangements each with a given cost determine their distance i e the total cost of a minimum cost sequence of rearrangements from that transforms g 1 into g 2 in this paper we study length weighted reversals and transpositions mainly considering their prefix variants in which the leftmost part of the genome must always be included we present approximation algorithms and bounds on the distances and diameters i e maximum possible distance over any two genomes of same length for eight problems when genomes are represented as permutations and another eight problems when genomes are represented as binary strings in each case considering two different cost functions

sorting by reversals and the theory of 4 regular graphs abstract we show that the theory of sorting by reversals fits into the well established theory of circuit partitions of 4 regular multigraphs which also involves the combinatorial structures of circle graphs and delta matroids in this way we expose strong connections between the two theories that have not been fully appreciated before we also discuss a generalization of sorting by reversals involving the double cut and join dcj operation finally we also show that the theory of sorting by reversals is closely related to that of gene assembly in ciliates

limiting case of the sobolev inequality in bmo with application to the euler equations we shall prove a logarithmic sobolev inequality by means of the bmo norm in the critical exponents as an application we shall establish a blow up criterion of solutions to the euler equations

approximation algorithms for sorting by length weighted prefix and suffix operations we addressed sorting permutations by prefix and suffix reversals and transpositions we initiated the study of these rearrangements in a length weighted context we presented approximation algorithms for 10 such problems the traditional approach for the problems of sorting permutations by rearrangements is to consider that all operations have the same unitary cost in this case the goal is to find the minimum number of allowed rearrangements that are needed to sort a given permutation and numerous efforts have been made over the past years regarding these problems on the other hand a long rearrangement which is in fact a mutation is more likely to disturb the organism therefore weights based on the length of the segment involved may have an important role in the evolutionary process in this paper we present the first results regarding problems of sorting permutations by length weighted operations that consider rearrangement models with prefix and suffix variations of reversals and transpositions which are the two most common types of genome rearrangements our main results are o lg 2 n approximation algorithms for 10 such problems

geometric linear discriminant analysis for pattern recognition we propose an innovative technique geometric linear discriminant analysis geometric lda to reduce the complexity of pattern recognition systems by using a linear transformation to lower the dimension of the observation space we experimentally compare geometric lda to other dimensionality reduction methods found in the literature we show that geometric lda produces the same and in many cases a significantly better linear transformation than other methods found in the literature

quick greedy computation for minimum common string partition in the minimum common string partition problem one is given two strings s and t with the same character statistics and one seeks for the smallest partition of s into substrings so that t can also be partitioned into the same substring multiset the problem is fundamental in several variants of edit distance with block operations e g signed reversal distance with duplicates and edit distance with moves the minimum common string partition problem is known to be np complete and the best approximation algorithm known has an approximation factor of o lognlog n since the minimum common string partition problem is of utmost practical importance one seeks a heuristic that will 1 usually have a low approximation factor and 2 will run fast a simple greedy algorithm is known which iteratively choose non overlapping longest common substrings of the input strings this algorithm has been well studied from an approximation point of view and it has been shown to have a bad worst case approximation factor however all the bad approximation factors presented so far stem from complicated recursive construction in practice the greedy algorithm seems to have small approximation factors however the best current implementation of greedy runs in quadratic time we propose a novel method to implement greedy in linear time

inference in molecular population genetics full likelihood based inference for modern population genetics data presents methodological and computational challenges the problem is of considerable practical importance and has attracted recent attention with the development of algorithms based on importance sampling is and markov chain monte carlo mcmc sampling here we introduce a new is algorithm the optimal proposal distribution for these problems can be characterized and we exploit a detailed analysis of genealogical processes to develop a practicable approximation to it we compare the new method with existing algorithms on a variety of genetic examples our approach substantially outperforms existing is algorithms with efficiency typically improved by several orders of magnitude the new method also compares favourably with existing mcmc methods in some problems and less favourably in others suggesting that both is and mcmc methods have a continuing role to play in this area we offer insights into the relative advantages of each approach and we discuss diagnostics in the is framework

a 2 approximation algorithm for genome rearrangements by reversals and transpositions recently a new approach to analyze genomes evolving which is based on comparision of gene orders versus traditional comparision of dna sequences was proposed sankoff et al 1992 the approach is based on the global rearrangements e g inversions and transpositions of fragments analysis of genomes evolving by inversions and transpositions leads to a combinatorial problem of sorting by reversals and transpositions i e sorting of a permutation using reversals and transpositions of arbitrary fragments we study sorting of signed permutations by reversals and transpositions a problem which adequately models genome rearrangements as the genes in dna are oriented we establish a lower bound and give a 2 approximation algorithm for the problem

empirically estimable classification bounds based on a nonparametric divergence measure information divergence functions play a critical role in statistics and information theory in this paper we show that a nonparametric f divergence measure can be used to provide improved bounds on the minimum binary classification probability of error for the case when the training and test data are drawn from the same distribution and for the case where there exists some mismatch between training and test distributions we confirm these theoretical results by designing feature selection algorithms using the criteria from these bounds and by evaluating the algorithms on a series of pathological speech classification tasks

asymptotic properties of maximum likelihood estimators and likelihood ratio tests under nonstandard conditions abstract large sample properties of the likelihood function when the true parameter value may be on the boundary of the parameter space are described specifically the asymptotic distribution of maximum likelihood estimators and likelihood ratio statistics are derived these results generalize the work of moran 1971 chant 1974 and chernoff 1954 some of chant s results are shown to be incorrect the approach used in deriving these results follows from comments made by moran and chant the problem is shown to be asymptotically equivalent to the problem of estimating the restricted mean of a multivariate gaussian distribution from a sample of size 1 in this representation the gaussian random variable corresponds to the limit of the normalized score statistic and the estimate of the mean corresponds to the limit of the normalized maximum likelihood estimator thus the limiting distribution of the maximum likelihood estimator is the same as the distribution of the projection of the gaussian random v

a phase transition in the random transposition random walk our work is motivated by bourque pevzner s simulation study of the effectiveness of the parsimony method in studying genome rearrangement and leads to a surprising result about the random transposition walk in continuous time on the group of permutations on n elements starting from the identity let d t be the minimum number of transpositions needed to go back to the identity element from the location at time t d t undergoes a phase transition for 0 1 d cn 2 u c n where u is an explicit function satisfying u x x 2 moreover we describe the fluctuations of d cn 2 about its mean at each of the three stages subcritical critical and supercritical the techniques used involve viewing the cycles in the random permutation as a coagulation fragmentation process and relating the behavior to the erd s renyi random graph model

lower bounds for bayes error estimation we give a short proof of the following result let x y be any distribution on n spl times 0 1 and let x sub 1 y sub 1 x sub n y sub n be an i i d sample drawn from this distribution in discrimination the bayes error l inf sub g p g x spl ne y is of crucial importance here we show that without further conditions on the distribution of x y no rate of convergence results can be obtained let spl phi sub n x sub 1 y sub 1 x sub n y sub n be an estimate of the bayes error and let spl phi sub n be a sequence of such estimates for any sequence a sub n of positive numbers converging to zero a distribution of x y may be found such that e l spl phi sub n x sub 1 y sub 1 x sub n y sub n spl ges a sub n often converges infinitely

note on loss of regularity for solutions of the 3 d incompressible euler and related equations one of the central problems in the mathematical theory of turbulence is that of breakdown of smooth indefinitely differentiable solutions to the equations of motion in 1934 j leray advanced the idea that turbulence may be related to the spontaneous appearance of singularities in solutions of the 3 d incompressible navier stokes equations the problem is still open we show in this report that breakdown of smooth solutions to the 3 d incompressible slightly viscous i e corresponding to high reynolds numbers or highly turbulent navier stokes equations cannot occur without breakdown in the corresponding solution of the incompressible euler ideal fluid equation we prove then that solutions of distorted euler equations which are equations closely related to the euler equations for short term intervals do breakdown

on maximal instances for the original syntenic distance the syntenic distance between two genomes has been introduced by ferretti nadeau and sankoff as an approximation of the evolutionary distance between genomes for which the gene order is not known this distance is the minimum number of fusions fissions and translocations required to transform a genome into another kleinberg and liben nowell as well as pisanti and sagot proved independently that for genomes with n chromosomes the diameter for this distance is 2n 4 pisanti and sagot also generalized this result showing that the maximal distance between a genome with m chromosomes and a genome with n chromosomes is n m 4 kleinberg and liben nowell asked for a characterization of maximal instances for the syntenic distance pairs of n chromosome genomes at a distance of 2n 4 in this paper we give such a characterization and show that we can extend it to pairs of genomes with respectively n and m chromosomes that are at maximal distance

a no panacea theorem for classifier combination we introduce the no panacea theorem npt for multiple classifier combination previously proved only in the case of two classifiers and two classes in this paper we extend the npt to cases of multiple classifiers and multiple classes we prove that if the combination function is continuous and diverse there exists a situation in which the combination algorithm will give very bad performance the proof relies on constructing pathological probability density distributions that have high densities in particular areas such that the combination functions give incorrect classification thus there is no optimal combination algorithm that is suitable in all situations it can be seen from this theorem that the probability density functions pdfs play an important role in the performance of combination algorithms so studying the pdfs becomes the first step of finding a good combination algorithm although devised for classifier combination the npt is also relevant to all supervised classification problems

genome rearrangements and sorting by reversals sequence comparison in molecular biology is in the beginning of a major paradigm shift a shift from gene comparison based on local mutations i e insertions deletions and substitutions of nucleotides to chromosome comparison based on global rearrangements i e inversions and transpositions of fragments the classical methods of sequence comparison do not work for global rearrangements and little is known in computer science about the edit distance between sequences if global rearrangements are allowed in the simplest form the problem of gene rearrangements corresponds to sorting by reversals i e sorting of an array using reversals of arbitrary fragments recently kececioglu and sankoff gave the first approximation algorithm for sorting by reversals with guaranteed error bound 2 and identified open problems related to chromosome rearrangements one of these problems is gollan s conjecture on the reversal diameter of the symmetric group this paper proves the conjecture further the problem of expected reversal distance between two random permutations is investigated the reversal distance between two random permutations is shown to be very close to the reversal diameter thereby indicating that reversal distance provides a good separation between related and nonrelated sequences in molecular evolution studies the gene rearrangement problem forces us to consider reversals of signed permutations as the genes in dna could be positively or negatively oriented an approximation algorithm for signed permutation is presented which provides a performance guarantee of 3 over 2 finally using the signed permutations approach an approximation algorithm for sorting by reversals is described which achieves a performance guarantee of 7 over 4

bayesian analysis of quantitative trait locus data using reversible jump markov chain monte carlo the advent of molecular markers has created a great potential for the understanding of quantitative inheritance in plants as well as in animals taking the newly available data into account biometric models have been constructed for the mapping of quantitative trait loci qtls in current approaches the lack of knowledge on the number and location of the most important qtls contributing to a trait is a major problem in this paper we utilize reversible jump markov chain monte carlo methodology green 1995 biometrika 82 711 732 in order to compute the posterior quantities required for fully bayesian inference it yields posterior densities not only for the parameters given the number of qtl but also for the number of qtl itself as an example the algorithm is applied to simulated data according to a standard design in plant breeding

bayesian analysis of binary and polychotomous response data abstract a vast literature in statistics biometrics and econometrics is concerned with the analysis of binary and polychotomous response data the classical approach fits a categorical response regression model using maximum likelihood and inferences about the model are based on the associated asymptotic theory the accuracy of classical confidence statements is questionable for small sample sizes in this article exact bayesian methods for modeling categorical response data are developed using the idea of data augmentation the general approach can be summarized as follows the probit regression model for binary outcomes is seen to have an underlying normal regression structure on latent continuous data values of the latent data can be simulated from suitable truncated normal distributions if the latent data are known then the posterior distribution of the parameters can be computed using standard results for normal linear models draws from this posterior are used to sample new latent data and t

a simpler and faster 1 5 approximation algorithm for sorting by transpositions an important problem in genome rearrangements is sorting permutations by transpositions the complexity of the problem is still open and two rather complicated 1 5 approximation algorithms for sorting linear permutations are known bafna and pevzner 98 and christie 99 the fastest known algorithm is the quadratic algorithm of bafna and pevzner in this paper we observe that the problem of sorting circular permutations by transpositions is equivalent to the problem of sorting linear permutations by transpositions hence all algorithms for sorting linear permutations by transpositions can be used to sort circular permutations our main result is a new o n3 2 log n 1 5 approximation algorithm which is considerably simpler than the previous ones and whose analysis is significantly less involved

nonexistence of self similar singularities for the 3d incompressible euler equations we prove that there exists no self similar finite time blowing up solution to the 3d incompressible euler equations if the vorticity decays sufficiently fast near infinity in mathbb r 3 by a similar method we also show nonexistence of self similar blowing up solutions to the divergence free transport equation in mathbb r n this result has direct applications to the density dependent euler equations the boussinesq system and the quasi geostrophic equations for which we also show nonexistence of self similar blowing up solutions

the critical sobolev inequalities in besov spaces and regularity criterion to some semi linear evolution equations we show the critical sobolev inequalities in the besov spaces with the logarithmic form such as brezis gallouet wainger and beale kato majda as an application of those inequalities the regularity problem under the critical condition to the navier stokes equations the euler equations in mathbb r n and the gradient flow to the harmonic map to the sphere are discussed namely the serrin ohyama type regularity criteria are improved in the terms of the besov spaces

successor rules for flipping pancakes and burnt pancakes a stack of n pancakes can be rearranged in all n ways by a sequence of n 1 flips and a stack of n burnt pancakes can be rearranged in all 2 n n ways by a sequence of 2 n n 1 flips in both cases a computer program can efficiently generate suitable solutions we approach these tasks instead from a human perspective how can we determine the next flip directly from the current stack how can we flip the minimum or maximum number of burnt pancakes overall what if we are only allowed to flip the top n 2 n 1 or n burnt pancakes we answer the first question with simple successor rules that take worst case o n time and amortized o 1 time then we answer the second question exactly for minimization and provide conjectures for maximization for the third question we prove that solutions almost certainly exist for pancakes and burnt pancakes using only these three flips more broadly we discuss how efficiency and optimality can shape iterative solutions to hamilton cycle problems in highly symmetric graphs

using em to obtain asymptotic variance covariance matrices the sem algorithm abstract the expectation maximization em algorithm is a popular and often remarkably simple method for maximum likelihood estimation in incomplete data problems one criticism of em in practice is that asymptotic variance covariance matrices for parameters e g standard errors are not automatic byproducts as they are when using some other methods such as newton raphson in this article we define and illustrate a procedure that obtains numerically stable asymptotic variance covariance matrices using only the code for computing the complete data variance covariance matrix the code for em itself and code for standard matrix operations the basic idea is to use the fact that the rate of convergence of em is governed by the fractions of missing information to find the increased variability due to missing information to add to the complete data variance covariance matrix we call this supplemented em algorithm the sem algorithm theory and particular examples reinforce the conclusion that the sem alg

discussion on the meeting on statistical modelling and analysis of genetic data david j balding imperial college school of medicine london i extendmy apologies to the authors that an unavoidable commitment arising unexpectedly in the 18 hours before the meeting robbed me of my final preparation so that my comments at the meeting were poorly presented i shall try to do a better job in this written version and to leave enough space i shall omit my introductory comments about the role of statisticians in bioinformatics

pivots determinants and perfect matchings of graphs we consider sequences of local and edge complementations on graphs with loops we allow local complementation only on looped vertices we recall that these operations together form the matrix operation of principal pivot transform or pivot restricted to graphs this fact is not well known and as a consequence various special cases of the properties of pivot have been rediscovered multiple times in this paper we give a gentle overview of various properties of pivot for local and edge complementations on graphs moreover we relate the pivot operation to perfect matchings to obtain a purely graph theoretical characterization of the effect of sequences of pivot operations finally we show that two of the three operations that make up a formal graph model of the biological process of gene assembly in ciliates together form the matrix operation of schur complement restricted to graphs

bias of nearest neighbor error estimates the bias of the finite sample nearest neighbor nn error from its asymptotic value is examined expressions are obtained which relate the bias of the nn and 2 nn errors to sample size dimensionality metric and distributions these expressions isolate the effect of sample size from that of the distributions giving an explicit relation showing how the bias changes as the sample size is increased experimental results are given which suggest that the expressions accurately predict the bias it is shown that when the dimensionality of the data is high it may not be possible to estimate the asymptotic error simply by increasing the sample size a new procedure is suggested to alleviate this problem this procedure involves measuring the mean nn errors at several sample sizes and using our derived relationship between the bias and the sample size to extrapolate an estimate of the asymptotic nn error the results are extended to the multiclass problem the choice of an optimal metric to minimize the bias is also discussed

bayesian approximate kernel regression with variable selection abstractnonlinear kernel regression models are often used in statistics and machine learning because they are more accurate than linear models variable selection for kernel regression models is a challenge partly because unlike the linear regression setting there is no clear concept of an effect size for regression coefficients in this paper we propose a novel framework that provides an effect size analog for each explanatory variable in bayesian kernel regression models when the kernel is shift invariant for example the gaussian kernel we use function analytic properties of shift invariant reproducing kernel hilbert spaces rkhs to define a linear vector space that i captures nonlinear structure and ii can be projected onto the original explanatory variables this projection onto the original explanatory variables serves as an analog of effect sizes the specific function analytic property we use is that shift invariant kernel functions can be approximated via random fourier bases based

measuring heterogeneity in forensic databases using hierarchical bayes models summary as currently defined dna fingerprint profiles do not uniquely identify individuals for criminal cases involving dna evidence forensic scientists evaluate the conditional probability that an unknown but distinct individual matches the crime sample given that the defendant matches estimates of the conditional probability of observing matching profiles are based on reference populations maintained by forensic testing laboratories each of these databases is heterogeneous being composed of subpopulations of different heritages this heterogeneity has an impact on the weight of the evidence a hierarchical bayes model is formulated that incorporates the key physical characteristics inherent in these data with the help of markov chain monte carlo sampling levels of heterogeneity are estimated for three major ethnic groups in the database of lifecodes corporation

model selection and accounting for model uncertainty in graphical models using occam s window abstract we consider the problem of model selection and accounting for model uncertainty in high dimensional contingency tables motivated by expert system applications the approach most used currently is a stepwise strategy guided by tests based on approximate asymptotic p values leading to the selection of a single model inference is then conditional on the selected model the sampling properties of such a strategy are complex and the failure to take account of model uncertainty leads to underestimation of uncertainty about quantities of interest in principle a panacea is provided by the standard bayesian formalism that averages the posterior distributions of the quantity of interest under each of the models weighted by their posterior model probabilities furthermore this approach is optimal in the sense of maximizing predictive ability but this has not been used in practice because computing the posterior model probabilities is hard and the number of models is very large often greater than 1

bounds on the bayes error given moments we show how to compute lower bounds for the supremum bayes error if the class conditional distributions must satisfy moment constraints where the supremum is with respect to the unknown class conditional distributions our approach makes use of curto and fialkow s solutions for the truncated moment problem the lower bound shows that the popular gaussian assumption is not robust in this regard we also construct an upper bound for the supremum bayes error by constraining the decision boundary to be linear

on arbitrarily slow rates of global convergence in density estimation let a density f on rd be estimated by fn x x1 xn where x rd fn is a borel measurable function of its arguments and x1 xn are independent random vectors with common density f let p 1 be a constant one of the main results of this note is that for every sequence fn and for every positive number sequence an satisfying lim an 0 there exists an f such that e left smallint f n left x right f left x right p dx right a n infinitely often

transforming cabbage into turnip polynomial algorithm for sorting signed permutations by reversals genomes frequently evolve by reversals r i j that transform a gene order 1 i i 1 j 1 j n into 1 i j 1 i 1 j n reversal distance between permutations and sis the minimum number of reversals to transform into a analysis of genome rearrangements in molecular biology started in the late 1930 s when dobzhansky and sturtevant published a milestone paper presenting a rearrangement scenario with 17 inversions between the species of drosophilia analysis of genomes evolving by inversions leads to a combinatorial problem of sorting by reversals studied in detail recently we study sorting of signed permutations by reversals a problem that adequately models rearrangements in a small genomes like chloroplast or mitochondrial dna the previously suggested approximation algorithms for sorting signed permutations by reversals compute the reversal distance between permutations with an astonishing accuracy for both simulated and biological data we prove a duality theorem explaining this intriguing performance and show that there exists a hidden parameter that allows one to compute the reversal distance between signed permutations in polynomial time

multivariate bayesian variable selection and prediction the multivariate regression model is considered with p regressors a latent vector with p binary entries serves to identify one of two types of regression coefficients those close to 0 and those not specializing our general distributional setting to the linear model with gaussian errors and using natural conjugate prior distributions we derive the marginal posterior distribution of the binary latent vector fast algorithms aid its direct computation and in high dimensions these are supplemented by a markov chain monte carlo approach to sampling from the known posterior distribution problems with hundreds of regressor variables become quite feasible we give a simple method of assigning the hyperparameters of the prior distribution the posterior predictive distribution is derived and the approach illustrated on compositional analysis of data involving three sugars with 160 near infra red absorbances as regressors

remarks on the breakdown of smooth solutions for the 3 d euler equations the authors prove that the maximum norm of the vorticity controls the breakdown of smooth solutions of the 3 d euler equations in other words if a solution of the euler equations is initially smooth and loses its regularity at some later time then the maximum vorticity necessarily grows without bound as the critical time approaches equivalently if the vorticity remains bounded a smooth solution persists

successful pressing sequences for a bicolored graph and binary matrices abstract we apply matrix theory over f 2 to understand the nature of so called successful pressing sequences of black and white vertex colored graphs these sequences arise in computational phylogenetics where by a celebrated result of hannenhalli and pevzner the space of sortings by reversal of a signed permutation can be described by pressing sequences in particular we offer several alternative linear algebraic and graph theoretic characterizations of successful pressing sequences describe the relation between such sequences and provide bounds on the number of them we also offer several open problems that arose as a result of the present work

variable selection via gibbs sampling abstract a crucial problem in building a multiple regression model is the selection of predictors to include the main thrust of this article is to propose and develop a procedure that uses probabilistic considerations for selecting promising subsets this procedure entails embedding the regression setup in a hierarchical normal mixture model where latent variables are used to identify subset choices in this framework the promising subsets of predictors can be identified as those with higher posterior probability the computational burden is then alleviated by using the gibbs sampler to indirectly sample from this multinomial posterior distribution on the set of possible subset choices those subsets with higher probability the promising ones can then be identified by their more frequent appearance in the gibbs sample

an efficient algorithm for one sided block ordering problem under block interchange distance abstract in this work we study the one sided block ordering problem under block interchange distance given two signed permutations and of size n where represents a partially assembled genome consisting of several blocks i e contigs and represents a completely assembled genome the one sided block ordering problem under block interchange distance is to order i e assemble the blocks of such that the block interchange distance between the assembly of and is minimized the one sided block ordering problem is useful in genome resequencing because its algorithms can be used to assemble the contigs of partially assembled resequencing genomes based on their completely assembled genomes by using permutation groups in algebra we design an efficient algorithm to solve the one sided block ordering problem under block interchange distance in o n log n time moreover we show that the assembly of can be done in o n time and its block interchange distance from can also be calculated in advance in o n time

a faster and simpler algorithm for sorting signed permutations by reversals we give a quadratic time algorithm for finding the minimum number of reversals needed to sort a signed permutation our algorithm is faster than the previous algorithm of hannenhalli and pevzner and its faster implementation by berman and hannenhalli the algorithm is conceptually simple and does not require special data structures our study also considerably simplifies the combinatorial structures used by the analysis

on the complexity of unsigned translocation distance translocation is one of the basic operations for genome rearrangement translocation distance is the minimum number of translocations required to transform one genome into the other in this paper we show that computing the translocation distance for unsigned genomes is np hard moreover we show that approximating the translocation distance for unsigned genomes within ratio 1 00017 is np hard

the evolution of the random reversal graph genomes are typically represented as signed permutations where each integer corresponds to a single gene and its sign indicates the orientation of the gene in this paper we study genome rearrangements via reversals sankoff and blanchette 1999 20 flipping entire segments of genes and changing their orientations this abstraction leads to the notion of the reversal graph in which two signed permutations are neighbors if they differ by one reversal we identify the evolution of multiple genomes within the random reversal graph i e the probability space consisting of subgraphs over signed permutations obtained by selecting edges reversals with independent probability n which is an important framework for the analysis of genome rearrangements in the sense of relating the reversal rate n with the global evolution of genomes our main result shows that the structure of the random reversal graph changes dramatically at n 1 n 1 2 for n 1 n 1 2 the random graph consists of components of size at most o n ln n a s and for 1 n 1 2 there emerges a unique largest component of size 2 n n a s where is the survival probability of a certain branching process this giant component is furthermore dense in the reversal graph

polynomial time sortable stacks of burnt pancakes pancake flipping a famous open problem in computer science can be formalised as the problem of sorting a permutation of positive integers using as few prefix reversals as possible in that context a prefix reversal of length k reverses the order of the first k elements of the permutation the burnt variant of pancake flipping involves permutations of signed integers and reversals in that case not only reverse the order of elements but also invert their signs although three decades have now passed since the first works on these problems neither their computational complexity nor the maximal number of prefix reversals needed to sort a permutation is yet known in this work we prove a new lower bound for sorting burnt pancakes and show that an important class of permutations known as simple permutations can be optimally sorted in polynomial time