on expander codes sipser and spielman see ibid vol 42 p 1717 22 nov 1996 have introduced a constructive family of asymptotically good linear error correcting codes expander codes together with a simple parallel algorithm that will always remove a constant fraction of errors we introduce a variation on their decoding algorithm that with no extra cost in complexity provably corrects up to 12 times more errors

transport and instability for perfect fluids incompressible perfect fluids are described by the euler equations we provide a new simple proof for well posedness for velocities in c 1 alpha and linear and nonlinear instability results using transport techniques the results have an important consequence the topology of c 1 alpha is too fine for interesting questions about large time behavior

fast growth of the vorticity gradient in symmetric smooth domains for 2d incompressible ideal flow abstract we construct initial data for the two dimensional euler equation in a bounded smooth symmetric domain such that the gradient of vorticity in l grows as a double exponential in time for all time our construction is based on the recent result by kiselev and sverak 9

improved nearly mds expander codes a construction of expander codes is presented with the following three properties i the codes lie close to the singleton bound ii they can be encoded in time complexity that is linear in their code length and iii they have a linear time bounded distance decoder by using a version of the decoder that corrects also erasures the codes can replace maximum distance separable mds outer codes in concatenated constructions thus resulting in linear time encodable and decodable codes that approach the zyablov bound or the capacity of memoryless channels the presented construction improves on an earlier result by guruswami and indyk in that any rate and relative minimum distance that lies below the singleton bound is attainable for a significantly smaller alphabet size

small scale creation for solutions of the incompressible two dimensional euler equation we construct an initial data for the two dimensional euler equation in a disk for which the gradient of vorticity exhibits double exponential growth in time for all times this estimate is known to be sharp the double exponential growth is the fastest possible growth rate

goodness of fit tests for log linear models in sparse contingency tables abstract the asymptotic normality of the likelihood ratio goodness of fit statistic is demonstrated for testing the fit of log linear models with closed form maximum likelihood estimates in sparse contingency tables unlike the traditional chi squared theory the number of categories in the table increases as the sample size increases but not all of the expected frequencies are required to become large some results of a small monte carlo study are presented the traditional chi squared approximation is reasonably accurate for the pearson statistic for many sparse tables but cases are presented for which it fails the normal approximation can be much more accurate than the chi squared approximation for the likelihood ratio statistic but the bias of estimated moments is a potential problem for very sparse tables

homogeneity tests against central mixture alternatives abstract a test of homogeneity is derived for a general sampling model the alternative hypothesis is a mixture over a parameter of the sampling model centered at the null hypothesis the test statistic is derived through a score test its functional form is independent of the particular mixing distribution suppose yi i 1 n are independent random variables with respective probability density or mass functions fi yi i under the null hypothesis suppose all i homogeneous and equal to the common value 0 under the alternative hypothesis suppose the i behave as random samples taken from a distribution with mean 0 and finite third moment the score statistic for testing these hypotheses rejects the null for large values of where is the maximum likelihood estimate of 0 under the null hypothesis and when the sample size n is large s is normally distributed under both the null hypothesis and a sequence of alternative hypotheses in which the variance of the mixing distribution tends

models for the analysis of association in multivariate contingency tables abstract the present work describes a family of models that can be used for the analysis of association in multiway cross classifications the models are complementary to the usual i e the so called hierarchical log linear models an advantage of this family of models is that it is frequently possible to give a more parsimonious description of the association than is possible with the usual log linear models approach the models will be especially useful when some or all of the variables are ordinal the models are derived from a reparameterization of the saturated model based on singular value decompositions of interaction terms in the log linear parameterization multivariate association models given in goodman 1979 1986 clogg 1982a b agresti and kezouh 1983 gilula and haberman 1988 and becker and clogg 1989 are obtained as special cases the utility of the models developed in this article is demonstrated with the analysis of two examples the first example a cross classification of

methods for exact goodness of fit tests abstract numerous goodness of fit tests with asymptotic chi squared distributions have been proposed for discrete multivariate data and there has been much discussion about using asymptotic results for computing critical values when there are small expected cell values although exact methods would be preferred in these situations it generally is believed that such methods are computationally intractable we propose methods for calculating exact distributions and significance levels for goodness of fit statistics that are computationally feasible over a wide range of models in particular the distribution for a simple multinomial model can be evaluated in polynomial time for composite null hypotheses we calculate the distribution conditional on the sufficient statistics for the nuisance parameters we calculate the characteristic function of a distribution and invert the characteristic function using the fast fourier transform fft our approach emphasizes the relationship between exact methods and

local regularity for the modified sqg patch equation we study the patch dynamics on the whole plane and on the half plane for a family of active scalars called modified surface quasi geostrophic sqg equations these involve a parameter that appears in the power of the kernel in their biot savart laws and describes the degree of regularity of the equation the values 0 and correspond to the two dimensional euler and sqg equations respectively we establish here local in time regularity for these models for all 0 on the whole plane and for all small 0 on the half plane we use the latter result in 16 where we show existence of regular initial data on the half plane that lead to a finite time singularity 2016 wiley periodicals inc

a recursive approach to low complexity codes a method is described for constructing long error correcting codes from one or more shorter error correcting codes referred to as subcodes and a bipartite graph a graph is shown which specifies carefully chosen subsets of the digits of the new codes that must be codewords in one of the shorter subcodes lower bounds to the rate and the minimum distance of the new code are derived in terms of the parameters of the graph and the subeodes both the encoders and decoders proposed are shown to take advantage of the code s explicit decomposition into subcodes to decompose and simplify the associated computational processes bounds on the performance of two specific decoding algorithms are established and the asymptotic growth of the complexity of decoding for two types of codes and decoders is analyzed the proposed decoders are able to make effective use of probabilistic information supplied by the channel receiver e g reliability information without greatly increasing the number of computations required it is shown that choosing a transmission order for the digits that is appropriate for the graph and the subcodes can give the code excellent burst error correction abilities the construction principles

finite time blow up for a 1d model of 2d boussinesq system the 2d conservative boussinesq system describes inviscid incompressible buoyant fluid flow in a gravity field the possibility of finite time blow up for solutions of this system is a classical problem of mathematical hydrodynamics we consider a 1d model of the 2d boussinesq system motivated by a particular finite time blow up scenario we prove that finite time blow up is possible for the solutions to the model system

the sharp corner formation in 2d euler dynamics of patches infinite double exponential rate of merging for the 2d euler dynamics of patches we investigate the convergence to the singular stationary solution in the presence of a regular strain it is proved that the rate of merging can be double exponential infinitely in time and the estimates we obtain are sharp

l infty l ill posedness for a class of equations arising in hydrodynamics we give a new approach to studying norm inflation in some critical spaces for a wide class of equations arising in hydrodynamics as an application we prove strong ill posedness of the n dimensional euler equations in the class c 1 cap l 2 omega and also in c k cap l 2 omega where omega can be the whole space a smooth bounded domain or the torus we also apply our method to the oldroyd b surface quasi geostrophic and boussinesq systems

using linear programming to decode binary linear codes a new method is given for performing approximate maximum likelihood ml decoding of an arbitrary binary linear code based on observations received from any discrete memoryless symmetric channel the decoding algorithm is based on a linear programming lp relaxation that is defined by a factor graph or parity check representation of the code the resulting lp decoder generalizes our previous work on turbo like codes a precise combinatorial characterization of when the lp decoder succeeds is provided based on pseudocodewords associated with the factor graph our definition of a pseudocodeword unifies other such notions known for iterative algorithms including stopping sets irreducible closed walks trellis cycles deviation sets and graph covers the fractional distance d sub frac of a code is introduced which is a lower bound on the classical distance it is shown that the efficient lp decoder will correct up to spl lceil d sub frac 2 spl rceil 1 errors and that there are codes with d sub frac spl omega n sup 1 spl epsi an efficient algorithm to compute the fractional distance is presented experimental evidence shows a similar performance on low density parity check ldpc codes between lp decoding and the min sum and sum product algorithms methods for tightening the lp relaxation to improve performance are also provided

correcting a fraction of errors in nonbinary expander codes with linear programming a linear programming decoder for nonbinary expander codes is presented it is shown that the proposed decoder has the nearest neighbor certificate properties it is also shown that this decoder corrects any pattern of errors of a relative weight up to approximately 1 4 a b where a and b are the relative minimum distances of the constituent codes

on the finite time blowup of a one dimensional model for the three dimensional axisymmetric euler equations in connection with the recent proposal for possible singularity formation at the boundary for solutions of three dimensional axisymmetric incompressible euler s equations luo and hou proc natl acad sci usa 2014 we study models for the dynamics at the boundary and show that they exhibit a finite time blowup from smooth data

blowup with vorticity control for a 2d model of the boussinesq equations abstract we propose a system of equations with nonlocal flux in two space dimensions which is closely modeled after the 2d boussinesq equations in a hyperbolic flow scenario our equations involve a vorticity stretching term and a non local biot savart law and provide insight into the underlying intrinsic mechanisms of singularity formation we prove stable controlled finite time blowup involving upper and lower bounds on the vorticity up to the time of blowup for a wide class of initial data

log linear modeling with the negative multinomial distribution we develop a negative multinomial sampling plan in which observed cell counts are positively correlated we show that maximum likelihood estimates of cell means are the same as those found under independent poisson sampling there is no maximum likelihood estimate for the shape parameter in general we propose an estimate of the shape parameter based on the mean and quantiles of pearson s chi squared statistic these techniques are applied to models of cancer incidence for three cities in ohio and longitudinal health care utilization by a group of senior citizens

exponential growth of the vorticity gradient for the euler equation on the torus abstract we prove that there are solutions to the euler equation on the torus with c 1 vorticity and smooth except at one point such that the vorticity gradient grows in l at least exponentially as t the same result is shown to hold for the vorticity hessian and smooth solutions our proofs use a version of a recent result by kiselev and sverak 6

goodness of fit tests for large sparse multinomial distributions abstract a goodness of fit statistic d 2 is introduced for use in multinomial distributions pearson s x2 and d 2 are both approximately normally distributed when the sample size n is not large relative to the number of multinomial categories k under sequences of local alternative hypotheses the test based on d 2 exhibits moderate power when the x2 test is biased application is made to the analysis of large sparse contingency tables a theorem is presented that describes the likelihood ratio for testing a simple multinomial distribution against a mixture of multinomial distributions a wide variety of mixing distributions is considered and the d 2 statistic is a special case of log when testing for dirichlet mixtures of multinomial distributions in the case where n is large relative to k x2 and d 2 k behave approximately as chi squared random variables and differ by a very small amount under the null hypothesis in this situation x2 and d 2 k will generally yield the same inference with a l

blow up for the 2d euler equation on some bounded domains abstract we find a smooth solution of the 2d euler equation on a bounded domain which exists and is unique in a natural class locally in time but blows up in finite time in the sense of its vorticity losing continuity the domain s boundary is smooth except at two points which are interior cusps

the vanishing viscosity limit in the presence of a porous medium we consider the flow of a viscous incompressible newtonian fluid in a perforated domain in the plane the domain is the exterior of a regular lattice of rigid particles we study the simultaneous limit of vanishing particle size and distance and of vanishing viscosity under suitable conditions on the particle size particle distance and viscosity we prove that solutions of the navier stokes system in the perforated domain converges to solutions of the euler system modeling inviscid incompressible flow in the full plane that is the flow is not disturbed by the porous medium and becomes inviscid in the limit convergence is obtained in the energy norm with explicit rates of convergence

finite time singularity for the modified sqg patch equation it is well known that the incompressible euler equations in two dimensions have globally regular solutions the inviscid surface quasi geostrophic sqg equation has a biot savart law which is one derivative less regular than in the euler case and the question of global regularity for its solutions is still open we study here the patch dynamics in the half plane for a family of active scalars which interpolates between these two equations via a parameter 0 12 appearing in the kernels of their biot savart laws the values 0 and 12 correspond to the 2d euler and sqg cases respectively we prove global in time regularity for the 2d euler patch model even if the patches initially touch the boundary of the half plane on the other hand for any sufficiently small 0 we exhibit initial data which lead to a singularity in finite time thus these results show a phase transition in the behavior of solutions to these equations and provide a rigorous foundation for classifying the 2d euler equations as critical department of mathematics rice university 6100 s main st houston tx 77005 1892 kiselev rice edu department of mathematics stanford university stanford ca 94305 ryzhik math stanford edu school of mathematics georgia institute of technology 686 cherry street atlanta ga 30332 0160 yaoyao math gatech edu department of mathematics university of wisconsin 480 lincoln dr madison wi 53706 1325 zlatos math wisc edu

linear time encodable and decodable error correcting codes we present a new class of asymptotically good linear error correcting codes these codes can be both encoded and decoded in linear time they can also be encoded by logarithmic depth circuits of linear size and decoded by logarithmic depth circuits of size o nlogn we present both randomized and explicit constructions of these codes

one dimensional model equations for hyperbolic fluid flow abstract in this paper we study the singularity formation for two nonlocal 1d active scalar equations focusing on the hyperbolic flow scenario those 1d equations can be regarded as simplified models of some 2d fluid equations

statistical inference for familial disease clusters in many epidemiologic studies the first indication of an environmental or genetic contribution to the disease is the way in which the diseased cases cluster within the same family units the concept of clustering is contrasted with incidence we assume that all individuals are exchangeable except for their disease status this assumption is used to provide an exact test of the initial hypothesis of no familial link with the disease conditional on the number of diseased cases and the distribution of the sizes of the various family units new parametric generalizations of binomial sampling models are described to provide measures of the effect size of the disease clustering we consider models and an example that takes covariates into account ascertainment bias is described and the appropriate sampling distribution is demonstrated four numerical examples with real data illustrate these methods

on decoding irregular tanner codes with local optimality guarantees we consider decoding of binary linear tanner codes using message passing iterative decoding and linear programming lp decoding in memoryless binary input output symmetric mbios channels we present new certificates that are based on a combinatorial characterization for the local optimality of a codeword in irregular tanner codes with respect to any mbios channel this characterization is a generalization of arora proc acm symp theory of computing 2009 and vontobel proc inf theory and appl workshop 2010 and is based on a conical combination of normalized weighted subtrees in the computation trees of the tanner graph these subtrees may have any finite height h even equal or greater than half of the girth of the tanner graph in addition the degrees of local code nodes in these subtrees are not restricted to two i e these subtrees are not restricted to skinny trees we prove that local optimality in this new characterization implies maximum likelihood ml optimality and lp optimality and show that a certificate can be computed efficiently we also present a new message passing iterative decoding algorithm called normalized weighted min sum nwms nwms decoding is a belief propagation bp type algorithm that applies to any irregular binary tanner code with single parity check local codes e g low density and high density parity check codes we prove that if a locally optimal codeword with respect to height parameter h exists whereby notably h is not limited by the girth of the tanner graph then nwms decoding finds this codeword in h iterations the decoding guarantee of the nwms decoding algorithm applies whenever there exists a locally optimal codeword because local optimality of a codeword implies that it is the unique ml codeword the decoding guarantee also provides an ml certificate for this codeword finally we apply the new local optimality characterization to regular tanner codes and prove lower bounds on the noise thresholds of lp decoding in mbios channels when the noise is below these lower bounds the probability that lp decoding fails to decode the transmitted codeword decays doubly exponentially in the girth of the tanner graph

ldpc codes over rings for psk modulation this paper describes the design and analysis of low density parity check ldpc codes over rings and shows how these codes when mapped onto appropriate signal constellations can be used to effect bandwidth efficient modulation specifically ldpc codes are constructed over the integer rings spl zopf sub m and g sub m sup 2 and mapped onto phase shift keying psk type signal sets to yield geometrically uniform signal space codes this paper identifies and addresses the design issues that affect code performance examples of codes over spl zopf sub 8 and g sub 64 mapped onto 8 ary and 64 ary signal sets at a spectral efficiency of 1 5 and 2 0 bits per second per hertz b s hz illustrate the approach simulation of these codes over the additive white gaussian noise awgn channel demonstrates that this approach is a good alternative to bandwidth efficient techniques based on binary ldpc codes e g bit interleaved coded modulation

stability of perturbations near a background magnetic field of the 2d incompressible mhd equations with mixed partial dissipation abstract the stability and large time behavior problem on some partially dissipated systems is not well understood the vorticity gradient of the 2d incompressible euler equation can grow double exponentially in time while the same quantity to the 2d navier stokes equation decays algebraically in time however the stability and large time behavior of the vorticity gradients of the 2d navier stokes equation with only vertical or horizontal dissipation appears to be unknown this paper presents a global stability result on perturbations near a background magnetic field to the 2d incompressible magnetohydrodynamic mhd equations with vertical dissipation and horizontal magnetic diffusion this stability result provides a significant example for the stabilizing effects of the magnetic field on electrically conducting fluids in addition we obtain an explicit decay rate for the solution of this nonlinear system

estimating model discrepancy a quantity 2 is defined as the discrepancy between an actual and an assumed statistical model an estimator is developed for the case in which data are grouped the estimator is based on pearson

cumulant methods for analysing independence of r way contingency tables and goodness of fit frequency data cumulant methods for analysisng the independence of r way contingency tables and goodness of fit frequency data are presented these methods are based on the exact first three cumulants of both the classical pearson chi squared statistic and a recent advantageous modification of this statistic these approximate methods provide major distributional improvements over existing asymptotic methods

lp decoding corrects a constant fraction of errors we show that for low density parity check ldpc codes whose tanner graphs have sufficient expansion the linear programming lp decoder of feldman karger and wainwright can correct a constant fraction of errors a random graph will have sufficient expansion with high probability and recent work shows that such graphs can be constructed efficiently a key element of our method is the use of a dual witness a zero valued dual solution to the decoding linear program whose existence proves decoding success we show that as long as no more than a certain constant fraction of the bits are flipped by the channel we can find a dual witness this new method can be used for proving bounds on the performance of any lp decoder even in a probabilistic setting our result implies that the word error rate of the lp decoder decreases exponentially in the code length under the binary symmetric channel bsc this is the first such error bound for ldpc codes using an analysis based on pseudocodewords recent work by koetter and vontobel shows that lp decoding and min sum decoding of ldpc codes are closely related by the graph cover structure of their pseudocodewords in their terminology our result implies that that there exist families of ldpc codes where the minimum bsc pseudoweight grows linearly in the block length

the exponential map of the group of area preserving diffeomorphisms of a surface with boundary we prove that the riemannian exponential map of the right invariant l2 metric on the group of volume preserving diffeomorphisms of a two dimensional manifold with a nonempty boundary is a nonlinear fredholm map of index zero

linear inviscid damping for a class of monotone shear flow in sobolev spaces in this paper we prove the decay estimates of the velocity and h1 scattering for the two dimensional linearized euler equations around a class of monotone shear flow in a finite channel our result is consistent with the decay rate predicted by case in 1960 2016 wiley periodicals inc

iterative linear programming decoding of nonbinary ldpc codes with linear complexity the problem of low complexity linear programming lp decoding of nonbinary low density parity check ldpc codes is considered and an iterative lp decoding algorithm is presented results that were previously derived for binary ldpc codes are extended to the nonbinary case both simple and generalized nonbinary ldpc codes are considered it is shown how the algorithm can be implemented efficiently using a finite field fast fourier transform then the convergence rate of the algorithm is analyzed the complexity of the algorithm scales linearly in the block length and it can approximate up to an arbitrarily small relative error the objective function of the exact lp solution when applied to a typical code from an appropriate nonbinary ldpc code ensemble the algorithm can correct a constant fraction of errors in linear in the block length computational complexity computer experiments with the new iterative lp decoding algorithm show that in the error floor region it can have better performance compared to belief propagation decoding with similar computational requirements

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

small sample comparisons for the power divergence goodness of fit statistics abstract the small sample properties of the power divergence family of goodness of fit statistics cressie and read in press are examined the asymptotic 2 and three alternative approximations are used to obtain appropriate test levels for comparison with the exact level it is shown that if the 2 approximation is to be used then there are many preferable statistics to the log likelihood ratio g 2 including pearson s 2 furthermore exact power comparisons under specific alternatives to the symmetric null hypothesis show that the power of the well known tests based on g 2 and 2 can be improved by choosing other statistics from the family an explanation is proposed in terms of the effect of relative deviations between observed and expected counts

on the asymptotic stability of stationary solutions of the inviscid incompressible porous medium equation we study the stability of stationary solutions of the two dimensional inviscid incompressible porous medium equation ipm we show that solutions which are near certain stable stationary solutions must converge as t to a stationary solution of the ipm equation it turns out that linearizing the ipm equation about certain stable stationary solutions gives a non local partial damping mechanism on the torus the linearized problem has a very large set of stationary undamped modes this makes the problem of long time behavior more difficult since there is the possibility of a cascading non linear growth along the stationary modes of the linearized problem we solve this by more or less doing a second linearization around the undamped modes exploiting a special non linear structure and showing that the stationary modes can be controlled

no local double exponential gradient growth in hyperbolic flow for the euler equation we consider smooth double odd solutions of the two dimensional euler equation in 1 1 2 with periodic boundary conditions it is tempting to think that the symmetry in the

a warning on the use of chi squared statistics with frequency tables with small expected cell counts abstract when applied to frequency tables with small expected cell counts pearson chi squared test statistics may be asymptotically inconsistent even in cases in which a satisfactory chi squared approximation exists for the distribution under the null hypothesis this problem is particularly important in cases in which the number of cells is large and the expected cell counts are quite variable to illustrate this bias of the chi squared test this article considers the pearson chi squared test of the hypothesis that the cell probabilities for a multinomial frequency table have specified values in this case the expected value and variance of the pearson chi square may be evaluated under both the null and alternative hypotheses when the number of cells is large normal approximations and discrete edgeworth expansions may also be used to assess the size and power of the pearson chi squared test these analyses show that unless all cell probabilities are equal it is possible to select a significance lev

finite time blow up in the hyperbolic boussinesq system abstract in recent work of luo and hou 10 a new scenario for finite time blow up in solutions of 3d euler equation has been proposed the scenario involves a ring of hyperbolic points of the flow located at the boundary of a cylinder in this paper we propose a two dimensional model that we call hyperbolic boussinesq system this model is designed to provide insight into the hyperbolic point blow up scenario the model features an incompressible velocity vector field a simplified biot savart law and a simplified term modeling buoyancy we prove that finite time blow up happens for a natural class of initial data

distance properties of expander codes the minimum distance of some families of expander codes is studied as well as some related families of codes defined on bipartite graphs the weight spectrum and the minimum distance of a random ensemble of such codes are computed and it is shown that it sometimes meets the gilbert varshamov gv bound a lower bound on the minimum distances of constructive families of expander codes is derived the relative minimum distance of the expander code is shown to exceed the product bound i e the quantity spl delta sub 0 spl delta sub 1 where spl delta sub 0 and spl delta sub 1 are the minimum relative distances of the constituent codes as a consequence of this a polynomially constructible family of expander codes is obtained whose relative distance exceeds the zyablov bound on the distance of serial concatenations

local ill posedness of the incompressible euler equations in c 1 and b 1_ infty 1 we show that the 2d euler equations are not locally well posed in the sense of hadamard in the c 1 space and in the besov space b 1 infty 1 our approach relies on the technique of lagrangian deformations of bourgain and li strong ill posedness of the incompressible euler equations in borderline sobolev spaces arxiv 1307 7090 we show that the assumption that the data to solution map is continuous in either c 1 or b 1 infty 1 leads to a contradiction with a result in w 1 p of kato and ponce rev mat iberoam 2 73 88 1986