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

the lp electrostatic q capacitary minkowski problem for general measures abstract existence of solutions to the l p minkowski problem of electrostatic q capacity for discrete measures was very recently given by g xiong j xiong and l xu jfa 2019 when 0 p 1 and 1 q 2 in this paper an existence result of solutions to this problem for general measures is proved when 0 p 1 and 1 q n which is a supplement to the previous result

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

the minkowski problem for polytopes abstract the traditional solution to the minkowski problem for polytopes involves two steps first the existence of a polytope satisfying given boundary data is demonstrated in the second step the uniqueness of that polytope up to translation is then shown to follow from the equality conditions of minkowski s inequality a generalized isoperimetric inequality for mixed volumes that is typically proved in a separate context in this article we adapt the classical argument to prove both the existence theorem of minkowski and his mixed volume inequality simultaneously thereby providing a new proof of minkowski s inequality that demonstrates the equiprimordial relationship between these two fundamental theorems of convex geometry

the logarithmic minkowski problem in analogy with the classical minkowski problem necessary and sufficient conditions are given to assure that a given measure on the unit sphere is the cone volume measure of the unit ball of a finite dimensional banach space

fast decoding of expander codes expander codes are tanner codes defined on sparse graphs that have good expansion properties sipser and spielman 1996 showed that there is a linear time decoding algorithm for expander codes when the vertex expansion is at least 3 4 and the number of errors corrected is a constant fraction of the code length later feldman et al 2007 gave a decoding algorithm that allows the expansion to be 2 3 1 3c where c is the left degree of the underlying bipartite graph at the expense of polynomial time decoding complexity recently viderman 2013 further improved the expansion parameter to 2 3 1 6c and the decoding algorithm runs in linear time these results are for expander codes whose inner codes are parity check codes by using stronger inner codes chilappagari et al 2010 showed that there is a linear time decoding algorithm for every vertex expansion greater than 1 2 in this paper it is shown that for every vertex expansion there is a linear time decoding algorithm for expander codes using inner codes with minimum distance depending on the vertex expansion and that the number of errors corrected is a constant fraction of the code length

on the case of equality in the brunn minkowski inequality for capacity suppose that and 1 are convex open subsets of rn denote their convex combination by the brunn minkowski inequality says that vol t 1 t vol 0 1 n t vol for 0 t l moreover if there is equality for some t other than an endpoint then the domains 1 and 0 are translates and dilates of each other borell proved an analogue of the brunn minkowski inequality with capacity defined below in place of volume borel s theorem b says theorem a let t t 1 1 t 0 be a convex combination of two convex subsets of rn n 3 then cap the main purpose of this note is to prove

on the range of the first two dirichlet and neumann eigenvalues of the laplacian in this paper we study the set of points in the plane defined by x y lambda 1 omega lambda 2 omega omega 1 where lambda 1 omega lambda 2 omega are either the two first eigenvalues of the dirichlet laplacian or the two first non trivial eigenvalues of the neumann laplacian we consider the case of general open sets together with the case of convex open domains we give some qualitative properties of these sets show some pictures obtained through numerical computations and state several open problems

a solution of the problem of van den essen and shpilrain ii abstract let k be a field of characteristic zero we show that every endomorphism of k x 1 x n taking any coordinate polynomial to a coordinate one is an automorphism

braided block codes a new class of binary iteratively decodable codes with good decoding performance is presented these codes called braided block codes bbcs operate on continuous data streams and are constructed by interconnection of two component block codes bbcs can be considered as convolutional or sliding version of either elias product codes or expander codes in this paper we define bbcs describe methods of their construction analyze code properties and study asymptotic iterative decoding performance

on solvability of two dimensional lp minkowski problem abstract given p 0 and a positive continuous function g with g x t g x for some 0 0 the functional f u 0 t u x 2 u 2 x dx c 0 t g x u p x dx 2 p has a minimizer in the class of positive functions u c1 r for which u x t u x for all x r this minimizer is used to prove the existence of a positive periodic solution y c2 r of two dimensional lp minkowski problem y1 p x y x y x g x where p 0 2

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

convexity of solutions and brunn minkowski inequalities for hessian equations in r3 by using minkowski addition of convex functions we prove convexity and rearrangement properties of solutions to some hessian equations in r 3 and brunn minkowski and isoperimetric inequalities for related functionals

ironing sweeping and multidimensional screening we provide existence proofs and characterization results for the multidimensional version of the multiproduct monopolist problem of mussa and rosen 1978 these results are also directly applicable to the multidimensional nonlinear pricing problems studied by wilson 1993 and armstrong 1996 we establish that bunching is robust in these multidimensional screening problems even with very regular distributions of types this comes from a strong conflict between participation constraints and second order incentive compatibility conditions we consequently design a new technique the sweeping procedure for dealing with bunching in multidimensional contexts this technique extends the ironing procedure of mussa and rosen 1978 to several dimensions we illustrate it on two examples we first solve a linear quadratic version of the bidimensional nonlinear pricing problem where consumers types are exponentially distributed the solution involves pure bundling for consumers with low demands the second example is the bidimensional version of the mussa and rosen problem when consumers types are uniformly distributed on a square the solution is such that the seller offers a full differentiation of products in the upper part of the qualities spectrum but only limited choice for lower qualities this seems to be a quite general pattern for multidimensional screening problems the sweeping procedure is potentially applicable to other multidimensional screening problems

variational formula and overdetermined problem for the principle eigenvalue of k hessian operator abstract we first deduce the first variational formula and some overdetermined problems for the principle eigenvalue of the k hessian operator and then prove serrin type symmetry result for our overdetermined problems

linear time encodable decodable codes with near optimal rate we present an explicit construction of linear time encodable and decodable codes of rate r which can correct a fraction 1 r spl epsiv 2 of errors over an alphabet of constant size depending only on spl epsiv for every 0 0 the error correction performance of these codes is optimal as seen by the singleton bound these are near mds codes such near mds linear time codes were known for the decoding from erasures our construction generalizes this to handle errors as well concatenating these codes with good constant sized binary codes gives a construction of linear time binary codes which meet the zyablov bound and also the more general blokh zyablov bound by resorting to multilevel concatenation our work also yields linear time encodable decodable codes which match forney s error exponent for concatenated codes for communication over the binary symmetric channel the encoding decoding complexity was quadratic in forney s result and forney s bound has remained the best constructive error exponent for almost 40 years now in summary our results match the performance of the previously known explicit constructions of codes that had polynomial time encoding and decoding but in addition have linear time encoding and decoding algorithms

expander codes using expander graphs we construct a new family of asymptotically good linear error correcting codes these codes have linear time sequential decoding algorithms and logarithmic time parallel decoding algorithms that use a linear number of processors we present both randomized and explicit constructions of these codes experimental results demonstrate the good performance of the randomly chosen codes

approximating optimization problems over convex functions many problems of theoretical and practical interest involve finding an optimum over a family of convex functions for instance finding the projection on the convex functions in hk and optimizing functionals arising from some problems in economics in the continuous setting and assuming smoothness the convexity constraints may be given locally by asking the hessian matrix to be positive semidefinite but in making discrete approximations two difficulties arise the continuous solutions may be not smooth and functions with positive semidefinite discrete hessian need not be convex in a discrete sense previous work has concentrated on non local descriptions of convexity making the number of constraints to grow super linearly with the number of nodes even in dimension 2 and these descriptions are very difficult to extend to higher dimensions in this paper we propose a finite difference approximation using positive semidefinite programs and discrete hessians and prove convergence under very general conditions even when the continuous solution is not smooth working on any dimension and requiring a linear number of constraints in the number of nodes using semidefinite programming codes we show concrete examples of approximations to problems in two and three dimensions

interior w 2 p estimates for solutions of the monge ampere equation in this work we adapt the techniques developed in c1 to prove interior estimates for solutions of perturbations of the monge ampere equation

regularity of solutions for some variational problems subject to a convexity constraint we first study the minimizers in the class of convex functions of an elliptic functional with nonhomogeneous dirichlet boundary conditions we prove c1 regularity of the minimizers under the assumption that the upper envelope of admissible functions is c1 this condition is optimal at least when the functional depends only on the gradient 3 we then give various extensions of this result in particular we consider a problem without boundary conditions arising in an economic model introduced by rochet and chone in 4

deforming a hypersurface by gauss curvature and support function abstract we study the motion of smooth strictly convex bodies in r n expanding in the direction of their normal vector field with speed depending on gauss curvature and support function

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 trapping sets and guaranteed error correction capability of ldpc codes and gldpc codes the relation between the girth and the guaranteed error correction capability of left regular low density parity check ldpc codes when decoded using the bit flipping serial and parallel algorithms is investigated a lower bound on the size of variable node sets which expand by a factor of at least 3 4 is found based on the moore bound this bound combined with the well known expander based arguments leads to a lower bound on the guaranteed error correction capability the decoding failures of the bit flipping algorithms are characterized using the notions of trapping sets and fixed sets the relation between fixed sets and a class of graphs known as cage graphs is studied upper bounds on the guaranteed error correction capability are then established based on the order of cage graphs the results are extended to left regular and right uniform generalized ldpc codes it is shown that this class of generalized ldpc codes can correct a linear number of worst case errors in the code length under the parallel bit flipping algorithm when the underlying tanner graph is a good expander a lower bound on the size of variable node sets which have the required expansion is established

a brunn minkowski inequality for the monge ampere eigenvalue abstract we prove a brunn minkowski type inequality for the eigenvalue of the monge ampere operator 1 2 n is concave in the class of c 2 domains in r n endowed with minkowski addition the equality case is explicitly described too the main device of the proof is a notion of addition for convex functions called infimal convolution which corresponds to the minkowski addition of the graphs of the involved functions

on an isoperimetric inequality for capacity conjectured by polya and szego abstract we study a conjecture by polya and szego on the approximation of the electrostatic capacity of convex bodies in terms of their surface measure we prove that a local version of this conjecture holds true and we give some results which bring further evidence to its global validity

the logarithmic minkowski problem for polytopes abstract the logarithmic minkowski problem asks for necessary and sufficient conditions for a finite borel measure on the unit sphere so that it is the cone volume measure of a convex body this problem was solved recently by boroczky lutwak yang and zhang for even measures boroczky et al 2013 8 this paper solves the case of discrete measures whose supports are in general position

conformal modulus and its reduction via mean curvatures abstract this note shows that the conformal modulus of a condenser and the reduced conformal modulus of a compact subset in r n 2 are naturally but optimally tied with the corresponding graphic mass and the semi diameter through mean curvatures

decoding of expander codes at rates close to capacity the decoding error probability of codes is studied as a function of their block length it is shown that the existence of codes with a polynomially small decoding error probability implies the existence of codes with an exponentially small decoding error probability specifically it is assumed that there exists a family of codes of length n and rate r 1 epsiv c c is a capacity of a binary symmetric channel whose decoding probability decreases inverse polynomially in n it is shown that if the decoding probability decreases sufficiently fast but still only inverse polynomially fast in n then there exists another such family of codes whose decoding error probability decreases exponentially fast in n moreover if the decoding time complexity of the assumed family of codes is polynomial in n and 1 epsiv then the decoding time complexity of the presented family is linear in n and polynomial in 1 epsiv these codes are compared to the recently presented codes of barg and zemor error exponents of expander codes ieee transactions on information theory 2002 and concatenated codes serial and parallel ieee transactions on information theory 2005 it is shown that the latter families cannot be tuned to have exponentially decaying in n error probability and at the same time to have decoding time complexity linear in n and polynomial in 1 epsiv

regularity and singularities of optimal convex shapes in the plane we focus here on the analysis of the regularity or singularity of solutions om 0 to shape optimization problems among convex planar sets namely j om 0 min j om om textrm convex omega in mathcal s ad where mathcal s ad is a set of 2 dimensional admissible shapes and j mathcal s ad rightarrow r is a shape functional our main goal is to obtain qualitative properties of these optimal shapes by using first and second order optimality conditions including the infinite dimensional lagrange multiplier due to the convexity constraint we prove two types of results begin itemize item i under a suitable convexity property of the functional j we prove that omega 0 is a w 2 p set p in 1 infty this result applies for instance with p infty when the shape functional can be written as j omega r omega p omega where r om f om e f om la 1 om involves the area omega the dirichlet energy e f omega or the first eigenvalue of the laplace dirichlet operator lambda 1 omega and p omega is the perimeter of om item ii under a suitable concavity assumption on the functional j we prove that omega 0 is a polygon this result applies for instance when the functional is now written as j om r omega p omega with the same notations as above end itemize

geometrical logarithmic capacitance abstract this paper is devoted to a novel and nontrivial exploration of eight aspects of the geometrical logarithmic capacitance a very key notion in mathematical physics quasiconformal geometry and variational calculus through 1 identifying with the reduced conformal module 2 evaluating the minimal log potential energy 3 relating to both the volume radius and the surface radius 4 linking with the n harmonic radius and the log capacity of the kevin image of a compact surface 5 finding the minkowski inequality and the general variational formula for the log capacity 6 pinching the log isocapacitary inequality from left and solving the left prescribed problem for the normalized log capacitary curvature measure 7 pinching the log isocapacitary inequality from right and handling the right prescribed problem for the normalized log capacitary curvature measure 8 handling an overdetermination of the n equilibrium potential of a given convex body via the log capacitary concavity

generalized and doubly generalized ldpc codes with random component codes for the binary erasure channel in this paper a method for the asymptotic analysis of generalized low density parity check gldpc codes and doubly generalized low density parity check d gldpc codes over the binary erasure channel bec based on extrinsic information transfer exit chart is described this method overcomes the problem consisting of the impossibility to evaluate the exit function for the check or variable component codes in situations where the information functions or split information functions for component codes are unknown according to the proposed technique gldpc codes and d gldpc codes where the generalized check and variable component codes are random codes with minimum distance at least 2 are considered a technique is then developed which finds the exit chart for the overall gldpc or d gldpc code by evaluating the expected exit function for each check and variable component code this technique is finally combined with the differential evolution algorithm in order to generate some good gldpc and d gldpc edge distributions numerical results of long random codes are presented which confirm the effectiveness of the proposed approach they also reveal that d gldpc codes can outperform standard ldpc codes and gldpc codes in terms of both waterfall performance and error floor

regularity of weak solutions to the monge ampere equation we study the properties of generalized solutions to the monge ampere equation det d 2 u v where the borel measure v satisfies a condition introduced by jerison that is weaker than the doubling property when v f dx this condition which we call d e admits the possibility of f vanishing or becoming infinite our analysis extends the regularity theory due to caffarelli available when 0 a f a which implies that v f dx is doubling the main difference between the d e case and the case when f is bounded between two positive constants is the need to use a variant of the aleksandrov maximum principle due to jerison and some tools from convex geometry in particular the hausdorff metric

a localization property of viscosity solutions to the monge ampere equation and their strict convexity the purpose of this note is to show a localization property of convex viscosity solutions to the monge ampere inequality 0 1 2 n are strictly convex

a brunn minkowski inequality for the hessian eigenvalue in three dimensional convex domain abstract we use the deformation methods to obtain the strictly log concavity of solution of a class hessian equation in bounded convex domain in r 3 as an application we get the brunn minkowski inequality for the hessian eigenvalue and characterize the equality case in bounded strictly convex domain in r 3

the brunn minkowski inequality for p capacity of convex bodies in this paper we prove the brunn minkowski inequality for the p capacity of convex bodies i e convex compact sets with non empty interior in r n for every p 1 n moreover we prove that the equality holds in such inequality if and only if the involved bodies coincide up to a translation and a dilatation

the orlicz minkowski problem for torsional rigidity abstract in this paper we prove a variational formula of torsional rigidity under the orlicz sum and then introduce the orlicz mixed torsional rigidity which yields the orlicz brunn minkowski inequality and orlicz minkowski inequality for torsional rigidity as an application we prove the existence of solution to the orlicz minkowski problem for discrete measure and general measure associated with torsional rigidity

the lp minkowski problem for n p 1 abstract chou and wang s existence result for the l p minkowski problem on s n 1 for p n 1 and an absolutely continuous measure is discussed and extended to more general measures in particular we provide an almost optimal sufficient condition for the case p 0 1

the lp minkowski problem for torsion abstract in the present paper we introduce the so called l p torsion measure l p mixed torsion a hadamard variational formula for torsion under the p sum is established as an application the existence of solutions to a l p minkowski problem for torsion is proved for any fixed p 1 and p n 2 which extends a result of colesanti and fimiani 24 a l p minkowski inequality for torsion is also obtained which implies the uniqueness of the solutions for the l p minkowski problem mentioned above

the hadamard variational formula and the minkowski problem for p capacity abstract a hadamard variational formula for p capacity of convex bodies in r n is established when 1 p n the formula is applied to solve the minkowski problem for p capacity which involves a degenerate monge ampere type equation uniqueness for the minkowski problem for p capacity is established when 1 p n and existence and regularity when 1 p 2 these results are non linear extensions of the now classical solution of jerison of the minkowski problem for electrostatic capacity p 2

polygons as optimal shapes with convexity constraint in this paper we focus on the following general shape optimization problem min j omega omega convex omega in mathcal s ad where mathcal s ad is a set of 2 dimensional admissible shapes and j mathcal s ad rightarrow mathbb r is a shape functional using a specific parameterization of the set of convex domains we derive some extremality conditions first and second order for this kind of problem moreover we use these optimality conditions to prove that for a large class of functionals satisfying a concavity like property any solution to this shape optimization problem is a polygon

capacities surface area and radial sums a dual capacitary brunn minkowski inequality is established for the n 1 capacity of radial sums of star bodies in rn this inequality is a counterpart to the capacitary brunn minkowski inequality for the p capacity of minkowski sums of convex bodies in rn 1 p n proved by borell colesanti and salani when n 3 the dual capacitary brunn minkowski inequality follows from an inequality of bandle and marcus but here a new proof is given that provides an equality condition note that when n 3 the n 1 capacity is the classical electrostatic capacity a proof is also given of both the inequality and a different equality condition when n 2 the latter case requires completely different techniques and an understanding of the behavior of surface area perimeter under the operation of radial sum these results can be viewed as showing that in a sense n 1 capacity has the same status as volume in that it plays the role of its own dual set function in the brunn minkowski and dual brunn minkowski theories

the set of fixed points of a unipotent group abstract let k be an algebraically closed field let g be a non trivial connected unipotent group which acts effectively on an affine variety x then every non empty component r of the set of fixed points of g is a k uniruled variety i e there exist an affine cylinder w k and a dominant generically finite polynomial mapping w k r we show also that if an arbitrary infinite algebraic group g acts effectively on k n and the set of fixed points contains a hypersurface h then this hypersurface is k uniruled

rebuttal of donnelly s paper on the spectral gap the spectral gap conjecture of m van den berg 2 formula 65 asserts that 2 1 3 for all convex euclidean domains of diameter 1 where 1 and 2 denote the first two eigenvalues of the dirichlet laplacian notice that equality holds for the 1 dimensional unit interval which can be regarded also as a degenerate n dimensional rectangular box the gap estimate is conjectured to hold more generally for schrodinger operators with convex potentials under dirichlet boundary conditions see the work of s t yau and collaborators 9 11 this schrodinger gap conjecture was proved some time ago in 1 dimension by r lavine 8 and more recently in all dimensions by b andrews and j clutterbuck 1 the proof in this journal by h donnelly 3 of the original gap conjecture in 2 dimensions for the dirichlet laplacian with zero potential is not correct the editors of mathematische zeitschrift have asked us to describe the flaws in the proof in order to clarify the state of the literature donnelly s approach to the problem is a natural one first perform a shape optimization to rule out a non degenerate minimizing domain and then analyze the spectral gap for a sequence of domains degenerating to an interval with the help of results by d jerison 5 for some history on this approach and on the gap conjecture more generally see the report on the aim meeting low eigenvalues of laplace and schrodinger operators 10 especially page 12 of the open problems list the error lies in the proof of the shape optimization step as we now explain donnelly wishes to prove that no minimizing domain can exist for

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

capacitary inequalities of the brunn minkowski type the main purpose of this paper is to show how so called a concave capacities may be built up from newton capacity c in lr n 3 in much the same way as concave measures from lebesgue measure m n in lr n 1 throughout o f2 o stand for non empty open and convex subsets of ir and j o denotes the class of all convex bodies contained in f2 recall that a positive radon measure in f2 is said to be concave if a and

p capacity vs surface area abstract this paper discovers two relationships between the 1 n p capacity and the surface area via the lebesgue volume and the willmore energy in r n respectively whence showing if r n is a convex compact smooth set with its interior and the mean curvature h 0 of its boundary then n p 1 p n 1 p 1 cap p p 1 n p 1 p n 1 area n 1 n p n 1 h n 1 d n 1 p 1 n 1 and hence c a p 1 area 1 h n 1 d n 1 of particular interest is that if p 2 3 n in the last but one inequality then cap 2 2 1 3 area where 2 1 3 is smaller than 2 2 2 conjectured by polya szego in 25 2 while bigger than 2 2 obtained by polya szego in 27 p 165 4

the brunn minkowski inequality in 1978 osserman 124 wrote an extensive survey on the isoperimetric inequality the brunn minkowski inequality can be proved in a page yet quickly yields the classical isoperimetric inequality for important classes of subsets of rn and deserves to be better known this guide explains the relationship between the brunn minkowski inequality and other inequalities in geometry and analysis and some applications

interior a priori estimates for solutions of fully non linear equations on etend une theorie de perturbations aux solutions d equations uniformement elliptiques d ordre 2 totalement non lineaires

testing sets for properness of polynomial mappings in this paper we characterize testing sets for properness of polynomial mappings f mathbb c n rightarrow mathbb c m we also study the set of points at which such mappings are not proper as the first application we give a proof of the complementary conjecture of mckay and wang conjecture 12 in 16 the second application is an answer to the problem of kraft russell about a characterization of mathbb c n the third application is given in 13 a solution of the problem of van den essen and shpilrain about endomorphism of the ring mathbb c x 1 x n the fourth application is a theorem about extensions of affine varieties

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 lp capacitary minkowski problem for polytopes abstract sufficient conditions are given for the existence of solutions to the discrete l p minkowski problem for p capacity when 0 p 1 and 1 p 2

brunn minkowski inequalities for variational functionals and related problems abstract recently several inequalities of brunn minkowski type have been proved for well known functionals in the calculus of variations e g the first eigenvalue of the laplacian the newton capacity the torsional rigidity and generalizations of these examples in this paper we add new contributions to this topic in particular we establish equality conditions in the case of the first eigenvalue of the laplacian and of the torsional rigidity and we prove a brunn minkowski inequality for another class of variational functionals moreover we describe the links between brunn minkowski type inequalities and the resolution of minkowski type problems

local correctability of expander codes in this work we present the first local decoding algorithm for expander codes this yields a new family of constant rate codes that can recover from a constant fraction of errors in the codeword symbols and where any symbol of the codeword can be recovered with high probability by reading n e symbols from the corrupted codeword where n is the block length of the code expander codes introduced by sipser and spielman are formed from an expander graph g v e of degree d and an inner code of block length d over an alphabet each edge of the expander graph is associated with a symbol in a string in e will be a codeword if for each vertex in v the symbols on the adjacent edges form a codeword in the inner code we show that if the inner code has a smooth reconstruction algorithm in the noiseless setting then the corresponding expander code has an efficient local correction algorithm in the noisy setting instantiating our construction with inner codes based on finite geometries we obtain novel locally decodable codes with rate approaching one this provides an alternative to the multiplicity codes of kopparty saraf and yekhanin stoc 11 and the lifted codes of guo kopparty and sudan itcs 13

estimates of first and second order shape derivatives in nonsmooth multidimensional domains and applications in this paper we investigate continuity properties of first and second order shape derivatives of functionals depending on second order elliptic pde s around nonsmooth domains essentially either lipschitz or convex or satisfying a uniform exterior ball condition we prove rather sharp continuity results for these shape derivatives with respect to sobolev norms of the boundary traces of the displacements with respect to previous results of this kind the approach is quite different and is valid in any dimension n geq 2 it is based on sharp regularity results for poisson type equations in such nonsmooth domains we also enlarge the class of functionals and pdes for which these estimates apply applications are given to qualitative properties of shape optimization problems under convexity constraints for the variable domains or their complement

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