legendre polynomial solutions of high order linear fredholm integro differential equations in this study a legendre collocation matrix method is presented to solve high order linear fredholm integro differential equations under the mixed conditions in terms of legendre polynomials the proposed method converts the equation and conditions to matrix equations by means of collocation points on the interval 1 1 which corresponding to systems of linear algebraic equations with legendre coefficients thus by solving the matrix equation legendre coefficients and polynomial approach are obtained also examples that illustrate the pertinent features of the method are presented and by using the error analysis the results are discussed

lattice coding and decoding achieve the optimal diversity multiplexing tradeoff of mimo channels this paper considers communication over coherent multiple input multiple output mimo flat fading channels where the channel is only known at the receiver for this setting we introduce the class of lattice space time last codes we show that these codes achieve the optimal diversity multiplexing tradeoff defined by zheng and tse under generalized minimum euclidean distance lattice decoding our scheme is based on a generalization of erez and zamir mod lambda scheme to the mimo case in our construction the scalar scaling of erez zamir and costa gaussian dirty paper schemes is replaced by the minimum mean square error generalized decision feedback equalizer mmse gdfe this result settles the open problem posed by zheng and tse on the construction of explicit coding and decoding schemes that achieve the optimal diversity multiplexing tradeoff moreover our results shed more light on the structure of optimal coding decoding techniques in delay limited mimo channels and hence open the door for novel approaches for space time code constructions in particular 1 we show that mmse gdfe plays a fundamental role in approaching the limits of delay limited mimo channels in the high signal to noise ratio snr regime unlike the additive white gaussian noise awgn channel case and 2 our random coding arguments represent a major departure from traditional space time code designs based on the rank and or mutual information design criteria

linear time nearest point algorithms for coxeter lattices the coxeter lattices are a family of lattices containing many of the important lattices in low dimensions this includes an e 7 e 8 and their duals an e 7 and e 8 we consider the problem of finding a nearest point in a coxeter lattice we describe two new algorithms one with worst case arithmetic complexity o nlogn and the other with worst case complexity o n where n is the dimension of the lattice we show that for the particular lattices an and an the algorithms are equivalent to nearest point algorithms that already exist in the literature

worst case to average case reductions based on gaussian measures we show that finding small solutions to random modular linear equations is at least as hard as approximating several lattice problems in the worst case within a factor almost linear in the dimension of the lattice the lattice problems we consider are the shortest vector problem the shortest independent vectors problem the covering radius problem and the guaranteed distance decoding problem a variant of the well known closest vector problem the approximation factor we obtain is n log o 1 n for all four problems this greatly improves on all previous work on the subject starting from ajtai s seminal paper generating hard instances of lattice problems in complexity of computations and proofs quad mat 13 dept math seconda univ napoli caserta italy 2004 pp 1 32 up to the strongest previously known results by micciancio siam j comput 34 2004 pp 118 169 our results also bring us closer to the limit where the problems are no longer known to be in np intersect conp our main tools are gaussian measures on lattices and the high dimensional fourier transform we start by defining a new lattice parameter which determines the amount of gaussian noise that one has to add to a lattice in order to get close to a uniform distribution in addition to yielding quantitatively much stronger results the use of this parameter allows us to simplify many of the complications in previous work our technical contributions are twofold first we show tight connections between this new parameter and existing lattice parameters one such important connection is between this parameter and the length of the shortest set of linearly independent vectors second we prove that the distribution that one obtains after adding gaussian noise to the lattice has the following interesting property the distribution of the noise vector when conditioning on the final value behaves in many respects like the original gaussian noise vector in particular its moments remain essentially unchanged

new bounds in some transference theorems in the geometry of numbers the aim of this paper is to give new bounds in certain inequalities concerning mutually reciprocal lattices in r to formulate the problem we have to introduce some notation and terminology we shall treat ir as an n dimensional euclidean space with the norm ii ii and metric d the inner product of vectors u v will be denoted by uv we shall usually write u 2 instead of uu the closed and open unit balls in f will be denoted by b and b respectively if a c ir then span a and a z denote the linear subspace spanned over a and the orthogonal complement of a in p a lattice in ir is an additive subgroup of ir generated by n linearly independent vectors the family of all lattices in ir will be denoted by a given a lattice l e a we define the polar dual reciprocal lattice l in the usual way l u u v z for each v l

lattice based signatures optimization and implementation on reconfigurable hardware nearly all of the currently used signature schemes such as rsa or dsa are based either on the factoring assumption or the presumed intractability of the discrete logarithm problem as a consequence the appearance of quantum computers or algorithmic advances on these problems may lead to the unpleasant situation that a large number of today s schemes will most likely need to be replaced with more secure alternatives in this work we present such an alternative an efficient signature scheme whose security is derived from the hardness of lattice problems it is based on recent theoretical advances in lattice based cryptography and is highly optimized for practicability and use in embedded systems the public and secret keys are roughly 1 5 kb and 0 3 kb long while the signature size is approximately 1 1 kb for a security level of around 80 bits we provide implementation results on reconfigurable hardware spartan virtex 6 and demonstrate that the scheme is scalable has low area consumption and even outperforms classical schemes

stable systolic inequalities and cohomology products multiplicative relations in the cohomology ring of a manifold impose constraints upon its stable systoles given a compact riemannian manifold x g its real homology h x is naturally endowed with the stable norm briefly if h hk x then the stable norm of h is the infimum of the riemannian k volumes of real cycles representing h the stable k systole is the minimum of the stable norm over nonzero elements in the lattice of integral classes in hk x relying on results from the geometry of numbers due to w banaszczyk and extending work by m gromov and j hebda we prove metric independent inequalities for products of stable systoles where the product can be as long as the real cup length of x 2003 wiley periodicals inc

pseudospectral method for numerical solution of daes with an error estimation in e babolian m m hosseini reducing index and pseudospectral methods for differential algebraic equations appl math comput 140 2003 77 90 numerical solution of linear differential algebraic equations daes has been presented by pseudospectral method in this paper a new error estimation technique is proposed to pseudospectral method such that is well done for linear semi explicit daes when the daes has index 2 the obtained approximate solution can be improved by using the proposed error estimation technique furthermore with providing some examples the proposed error estimation method is dealt with numerically

achieving awgn channel capacity with lattice gaussian coding we propose a new coding scheme using only one lattice that achieves the 1 2 log 1 snr capacity of the additive white gaussian noise awgn channel with lattice decoding which is provable for signal to noise ratio snr e at present the scheme applies a discrete gaussian distribution over an awgn good lattice but otherwise does not require a shaping lattice or dither thus it significantly simplifies the default lattice coding scheme of erez and zamir which involves a quantization good lattice as well as an awgn good lattice using the flatness factor we show that the error probability of the proposed scheme under minimum mean square error lattice decoding is almost the same as that of erez and zamir for any rate up to the awgn channel capacity we introduce the notion of good constellations which carry almost the same mutual information as that of continuous gaussian inputs we also address the implementation of gaussian shaping for the proposed lattice gaussian coding scheme

mixing time and eigenvalues of the abelian sandpile markov chain the abelian sandpile model defines a markov chain whose states are integer valued functions on the vertices of a simple connected graph g by viewing this chain as a nonreversible random walk on an abelian group we give a formula for its eigenvalues and eigenvectors in terms of multiplicative harmonic functions on the vertices of g we show that the spectral gap of the sandpile chain is within a constant factor of the length of the shortest noninteger vector in the dual laplacian lattice while the mixing time is at most a constant times the smoothing parameter of the laplacian lattice we find a surprising inverse relationship between the spectral gap of the sandpile chain and that of simple random walk on g if the latter has a sufficiently large spectral gap then the former has a small gap in the case where g is the complete graph on n vertices we show that the sandpile chain exhibits cutoff at time 1 4 2 n logn

awgn goodness is enough capacity achieving lattice codes based on dithered probabilistic shaping in this paper we show that any sequence of infinite lattice constellations which is good for the unconstrained gaussian channel can be shaped into a capacity achieving sequence of codes for the power constrained gaussian channel under lattice decoding and non uniform signaling unlike previous results in the literature our scheme holds with no extra condition on the lattices e g quantization goodness or vanishing flatness factor thus establishing a direct implication between awgn goodness in the sense of poltyrev and capacity achieving codes our analysis uses properties of the discrete gaussian distribution in order to obtain precise bounds on the probability of error and achievable rates in particular we obtain a simple characterization of the finite blocklength behavior of the scheme showing that it approaches the optimal dispersion coefficient for high signal to noise ratio we further show that for low signal to noise ratio the discrete gaussian over centered lattice constellations cannot achieve capacity and thus a shift or dither is essentially necessary

hardness of approximating the shortest vector problem in lattices let p 1 be any fixed real we show that assuming np n rp there is no polynomial time algorithm that approximates the shortest vector problem svp in e p norm within a constant factor under the stronger assumption np n rtime 2poly log n we show that there is no polynomial time algorithm with approximation ratio 2 log n 1 2 e where n is the dimension of the lattice and e 0 is an arbitrarily small constant we first give a new randomized reduction from closest vector problem cvp to svp that achieves some constant factor hardness the reduction is based on bch codes its advantage is that the svp instances produced by the reduction behave well under the augmented tensor product a new variant of tensor product that we introduce this enables us to boost the hardness factor to 2 log n 1 2 e

on one approach to solve the linear boundary value problems for fredholm integro differential equations necessary and sufficient conditions for the well posedness of linear boundary value problems for fredholm integro differential equations are established algorithms for finding their solutions are offered

numerical solution of volterra integro differential equations by the tau method with the chebyshev and legendre bases in this paper we present a solution to the volterra integro differential equations vide by the tau method with arbitrary polynomial bases we apply some simple matrices in extension of the tau method for the numerical solution of vide in particular the chebyshev and legendre polynomials are applied as the bases polynomials the results of examples are illustrated that the tau method bases are very useful in computing a numerical solution of vide

joint physical layer coding and network coding for bidirectional relaying we consider a communication system where two transmitters wish to exchange information through a central relay the transmitter and relay nodes exchange data over synchronized average power constrained additive white gaussian noise channels with a real input with signal to noise ratio snr of snr an upper bound on the capacity is 1 2 log 1 snr bits per transmitter per use of the multiple access phase and broadcast phase of the bidirectional relay channel we show that using lattice codes and lattice decoding we can obtain a rate of 1 2 log 1 2 snr bits per transmitter which is essentially optimal at high snr the main idea is to decode the sum of the codewords modulo a lattice at the relay followed by a broadcast phase which performs slepian wolf coding we also show that if the two transmitters use identical lattices with minimum angle decoding we can achieve the same rate of 1 2 log 1 2 snr the proposed scheme can be thought of as a joint physical layer network layer code which outperforms other recently proposed analog network coding schemes

evolutionary computational intelligence in solving a class of nonlinear volterra fredholm integro differential equations in this paper a stochastic computational intelligence technique for solving a class of nonlinear volterra fredholm integro differential equations with mixed conditions is presented the strength of feed forward artificial neural networks is used to accurately model the integro equation comparisons with the exact solution and other numerical techniques are presented to show the efficiency of the proposed method theoretical and numerical results are presented analysis for the presented method is given

a nonparametric clustering algorithm with a quantile based likelihood estimator clustering is a representative of unsupervised learning and one of the important approaches in exploratory data analysis by its very nature clustering without strong assumption on data distribution is desirable information theoretic clustering is a class of clustering methods that optimize information theoretic quantities such as entropy and mutual information these quantities can be estimated in a nonparametric manner and information theoretic clustering algorithms are capable of capturing various intrinsic data structures it is also possible to estimate information theoretic quantities using a data set with sampling weight for each datum assuming the data set is sampled from a certain cluster and assigning different sampling weights depending on the clusters the cluster conditional information theoretic quantities are estimated in this letter a simple iterative clustering algorithm is proposed based on a nonparametric estimator of the log likelihood for weighted data sets the clustering algorithm is also derived from the principle of conditional entropy minimization with maximum entropy regularization the proposed algorithm does not contain a tuning parameter the algorithm is experimentally shown to be comparable to or outperform conventional nonparametric clustering methods

lattices which are good for almost everything we define an ensemble of lattices and show that for asymptotically high dimension most of its members are simultaneously good as sphere packings sphere coverings additive white gaussian noise awgn channel codes and mean squared error mse quantization codes these lattices are generated by applying construction a to a random linear code over a prime field of growing size i e by lifting the code to spl ropf sup n

the hardness of the closest vector problem with preprocessing we give a new simple proof of the np hardness of the closest vector problem in addition to being much simpler than all previously known proofs the new proof yields new interesting results about the complexity of the closest vector problem with preprocessing this is a variant of the closest vector problem in which the lattice is specified in advance and can be preprocessed for an arbitrarily long amount of time before the target vector is revealed we show that there are lattices for which the closest vector problem remains hard regardless of the amount of preprocessing

a novel semi analytical approach for solving nonlinear volterra integro differential equations in this work we present an efficient semi analytical method based on the taylor series for solving nonlinear volterra integro differential equations namely the differential transform method dtm the dtm provides a recursive relation for the coefficients of the taylor series that is derived from the given equations we provide a new recursive relation for the nonlinear volterra integro differential equations with complex nonlinear kernels since the dtm is based on the taylor series it is difficult to obtain accurate approximate solutions in a large domain to overcome this difficulty the standard dtm is applied in each subdomain called the multistage differential transform method msdtm we also present an convergence analysis for the proposed method to demonstrate the efficiency of the proposed method several numerical examples are performed and support the results in our analysis

information cut for clustering using a gradient descent approach we introduce a new graph cut for clustering which we call the information cut it is derived using parzen windowing to estimate an information theoretic distance measure between probability density functions we propose to optimize the information cut using a gradient descent based approach our algorithm has several advantages compared to many other graph based methods in terms of determining an appropriate affinity measure computational complexity memory requirements and coping with different data scales we show that our method may produce clustering and image segmentation results comparable or better than the state of the art graph based methods

markov chain monte carlo methods for lattice gaussian sampling convergence analysis and enhancement sampling from lattice gaussian distribution has emerged as an important problem in coding decoding and cryptography in this paper the classic gibbs algorithm from markov chain monte carlo mcmc methods is demonstrated to be geometrically ergodic for lattice gaussian sampling which means that the markov chain arising from it converges exponentially fast to the stationary distribution meanwhile the exponential convergence rate of the markov chain is also derived through the spectral radius of the forward operator then a comprehensive analysis of the convergence rate is carried out and two sampling schemes are proposed to further enhance the convergence performance the first one referred to as a metropolis within gibbs mwg algorithm improves the convergence by refining the state space of the univariate sampling the second is a blocked strategy of the gibbs algorithm which performs sampling over multivariates at each markov move and is shown to yield a better convergence rate than the traditional univariate sampling in order to perform blocked sampling efficiently the gibbs klein gk algorithm is proposed which samples block by block using the kleins algorithm furthermore the validity of the gk algorithm is demonstrated by showing its ergodicity simulation results based on mimo detections are presented to confirm the convergence gain brought by the proposed gibbs sampling schemes

on the asymptotics of solving the lwe problem using coded bkw with sieving the learning with errors problem lwe has become a central topic in recent cryptographic research in this paper we present a new solving algorithm combining important ideas from previous work on improving the blum kalai wasserman bkw algorithm and ideas from sieving in lattices the new algorithm is analyzed and demonstrates an improved asymptotic performance for the regev parameters q n 2 and noise level sigma n 1 5 sqrt 2 pi log 2 2 n the asymptotic complexity is 2 0 893n in the standard setting improving the previously best known complexity of roughly 2 0 930n the newly proposed algorithm also provides asymptotic improvements when a quantum computer is assumed or when the number of samples is limited

random ensembles of lattices from generalized reductions we propose a general framework to study constructions of euclidean lattices from linear codes over finite fields in particular we prove general conditions for an ensemble constructed using linear codes to contain dense lattices i e with packing density comparable to the minkowski hlawka lower bound specializing to number field lattices we obtain a number of interesting corollaries for instance the best known packing density of ideal lattices and an elementary coding theoretic construction of asymptotically dense hurwitz lattices all results are algorithmically effective in the sense that for any dimension a finite family containing dense lattices is exhibited for suitable constructions based on craig s lattices this family is smaller in terms of alphabet size than previous ensembles in the literature

superconvergence of collocation methods for volterra and abel integral equations of the second kind this paper deals with the question of the attainable order of convergence in the numerical solution of volterra and abel integral equations by collocation methods in certain piecewise polynomial spaces and which are based on suitable interpolatory quadrature for the resulting moment integrals the use of a nonlinear variation of constants formula for the representation of the error function in terms of the defect allows for a unified treatment of equations with continuous and weakly singular kernels

almost universal codes for mimo wiretap channels despite several works on secrecy coding for fading and mimo wiretap channels from an error probability perspective the construction of information theoretically secure codes over such channels remains an open problem in this paper we consider a fading wiretap channel model where the transmitter has only partial statistical channel state information our channel model includes static channels i i d block fading channels and ergodic stationary fading with fast decay of large deviations for the eavesdropper s channel we extend the flatness factor criterion from the gaussian wiretap channel to fading and mimo wiretap channels and establish a simple design criterion where the normalized product distance minimum determinant of the lattice and its dual should be maximized simultaneously moreover we propose concrete lattice codes satisfying this design criterion which are built from algebraic number fields with constant root discriminant in the single antenna case and from division algebras centered at such number fields in the multiple antenna case the proposed lattice codes achieve strong secrecy and semantic security for all rates r where c b and c e are bob and eve s channel capacities respectively and kappa is an explicit constant gap furthermore these codes are almost universal in the sense that a fixed code is good for secrecy for a wide range of fading models finally we consider a compound wiretap model with a more restricted uncertainty set and show that rates r are achievable where bar c b is a lower bound for bob s capacity and bar c e is an upper bound for eve s capacity for all the channels in the set

successive minima of line bundles abstract we introduce and study the successive minima of line bundles on proper algebraic varieties the first resp last minima are the width resp seshadri constant of the line bundle at very general points the volume of the line bundle is equivalent to the product of the successive minima for line bundles on toric varieties the successive minima are equivalent to the reciprocal of usual successive minima of the difference of the moment polytope

on lattices learning with errors random linear codes and cryptography our main result is a reduction from worst case lattice problems such as gapsvp and sivp to a certain learning problem this learning problem is a natural extension of the learning from parity with error problem to higher moduli it can also be viewed as the problem of decoding from a random linear code this we believe gives a strong indication that these problems are hard our reduction however is quantum hence an efficient solution to the learning problem implies a quantum algorithm for gapsvp and sivp a main open question is whether this reduction can be made classical i e nonquantum we also present a classical public key cryptosystem whose security is based on the hardness of the learning problem by the main result its security is also based on the worst case quantum hardness of gapsvp and sivp the new cryptosystem is much more efficient than previous lattice based cryptosystems the public key is of size o n2 and encrypting a message increases its size by a factor of o n in previous cryptosystems these values are o n4 and o n2 respectively in fact under the assumption that all parties share a random bit string of length o n2 the size of the public key can be reduced to o n

estimating mutual information we present two classes of improved estimators for mutual information m x y from samples of random points distributed according to some joint probability density mu x y in contrast to conventional estimators based on binnings they are based on entropy estimates from k nearest neighbor distances this means that they are data efficient with k 1 we resolve structures down to the smallest possible scales adaptive the resolution is higher where data are more numerous and have minimal bias indeed the bias of the underlying entropy estimates is mainly due to nonuniformity of the density at the smallest resolved scale giving typically systematic errors which scale as functions of k n for n points numerically we find that both families become exact for independent distributions i e the estimator m x y vanishes up to statistical fluctuations if mu x y mu x mu y this holds for all tested marginal distributions and for all dimensions of x and y in addition we give estimators for redundancies between more than two random variables we compare our algorithms in detail with existing algorithms finally we demonstrate the usefulness of our estimators for assessing the actual independence of components obtained from independent component analysis ica for improving ica and for estimating the reliability of blind source separation

iterative numerical solution of non linear integro differential equations by the tau method abstract in this paper we try to give an iterative application of the operational tau method e l ortiz h samara an operational approach to the tau method for the numerical solution of non linear differential equations computing 27 1981 15 25 for the numerical solution of non linear integro differential nide a simple method convert non linear equations to linear equations then we use the operational tau method for the numerical solution of linearized equation iteratively various numerical examples illustrate the accuracy of the method

stability results for one step discretized collocation methods in the numerical treatment of volterra integral equations this paper is concerned with the stability analysis of the discretized collocation method for the second kind volterra integral equation with degenerate kernel a fixed order recurrence relation with variable coefficients is derived and local stability conditions are given independent of the discretization local stability and stability with respect to an isolated perturbation of some methods are proved the reliability of the derived stability conditions is shown by numerical experiments

nonlinear component analysis as a kernel eigenvalue problem a new method for performing a nonlinear form of principal component analysis is proposed by the use of integral operator kernel functions one can efficiently compute principal components in high dimensional feature spaces related to input space by some nonlinear map for instance the space of all possible five pixel products in 16 16 images we give the derivation of the method and present experimental results on polynomial feature extraction for pattern recognition

kernel entropy component analysis we introduce kernel entropy component analysis kernel eca as a new method for data transformation and dimensionality reduction kernel eca reveals structure relating to the renyi entropy of the input space data set estimated via a kernel matrix using parzen windowing this is achieved by projections onto a subset of entropy preserving kernel principal component analysis kernel pca axes this subset does not need in general to correspond to the top eigenvalues of the kernel matrix in contrast to the dimensionality reduction using kernel pca we show that kernel eca may produce strikingly different transformed data sets compared to kernel pca with a distinct angle based structure a new spectral clustering algorithm utilizing this structure is developed with positive results furthermore kernel eca is shown to be an useful alternative for pattern denoising

sphere bound achieving coset codes and multilevel coset codes a simple sphere bound gives the best possible tradeoff between the volume per point of an infinite array l and its error probability on an additive white gaussian noise awgn channel it is shown that the sphere bound can be approached by a large class of coset codes or multilevel coset codes with multistage decoding including certain binary lattices these codes have structure of the kind that has been found to be useful in practice capacity curves and design guidance for practical codes are given exponential error bounds for coset codes are developed generalizing poltyrev s 1994 bounds for lattices these results are based on the channel coding theorems of information theory rather than the minkowski hlawka theorem of lattice theory

lattice closures of polyhedra given p subset mathbb r n a mixed integer set p i p cap mathbb z t times mathbb r n t and a k tuple of n dimensional integral vectors pi 1 ldots pi k where the last n t entries of each vector is zero we consider the relaxation of p i obtained by taking the convex hull of points x in p for which pi 1 tx ldots pi t kx are integral we then define the k dimensional lattice closure of p i to be the intersection of all such relaxations obtained from k tuples of n dimensional vectors when p is a rational polyhedron we show that given any collection of such k tuples there is a finite subcollection that gives the same closure more generally we show that any k tuple is dominated by another k tuple coming from the finite subcollection the k dimensional lattice closure contains the convex hull of p i and is equal to the split closure when k 1 therefore a result of cook et al math program 47 155 174 1990 implies that when p is a rational polyhedron the k dimensional lattice closure is a polyhedron for k 1 and our finiteness result extends this to all k ge 2 we also construct a polyhedral mixed integer set with n integer variables and one continuous variable such that for any k n finitely many iterations of the k dimensional lattice closure do not give the convex hull of the set our result implies that t branch split cuts cannot give the convex hull of the set nor can valid inequalities from unbounded full dimensional convex lattice free sets

a grouped ranking model for item preference parameter given a set of rating data for a set of items determining preference levels of items is a matter of importance various probability models have been proposed to solve this task one such model is the plackett luce model which parameterizes the preference level of each item by a real value in this letter the plackett luce model is generalized to cope with grouped ranking observations such as movie or restaurant ratings since it is difficult to maximize the likelihood of the proposed model directly a feasible approximation is derived and the em algorithm is adopted to find the model parameter by maximizing the approximate likelihood which is easily evaluated the proposed model is extended to a mixture model and two applications are proposed to show the effectiveness of the proposed model numerical experiments with real world data are carried out

geodesic entropic graphs for dimension and entropy estimation in manifold learning in the manifold learning problem one seeks to discover a smooth low dimensional surface i e a manifold embedded in a higher dimensional linear vector space based on a set of measured sample points on the surface in this paper we consider the closely related problem of estimating the manifold s intrinsic dimension and the intrinsic entropy of the sample points specifically we view the sample points as realizations of an unknown multivariate density supported on an unknown smooth manifold we introduce a novel geometric approach based on entropic graph methods although the theory presented applies to this general class of graphs we focus on the geodesic minimal spanning tree gmst to obtaining asymptotically consistent estimates of the manifold dimension and the re spl acute nyi spl alpha entropy of the sample density on the manifold the gmst approach is striking in its simplicity and does not require reconstruction of the manifold or estimation of the multivariate density of the samples the gmst method simply constructs a minimal spanning tree mst sequence using a geodesic edge matrix and uses the overall lengths of the msts to simultaneously estimate manifold dimension and entropy we illustrate the gmst approach on standard synthetic manifolds as well as on real data sets consisting of images of faces

new lattice based cryptographic constructions we introduce the use of fourier analysis on lattices as an integral part of a lattice based construction the tools we develop provide an elegant description of certain gaussian distributions around lattice points our results include two cryptographic constructions that are based on the worst case hardness of the unique shortest vector problem the main result is a new public key cryptosystem whose security guarantee is considerably stronger than previous results o n1 5 instead of o n7 this provides the first alternative to ajtai and dwork s original 1996 cryptosystem our second result is a family of collision resistant hash functions with an improved security guarantee in terms of the unique shortest vector problem surprisingly both results are derived from one theorem that presents two indistinguishable distributions on the segment 0 1 it seems that this theorem can have further applications as an example we use it to solve an open problem in quantum computation related to the dihedral hidden subgroup problem

lattice reduction aided gibbs algorithm for lattice gaussian sampling convergence enhancement and decoding optimization sampling from the lattice gaussian distribution has emerged as an important problem in coding decoding and cryptography in this paper lattice reduction technique is adopted to gibbs sampler for lattice gaussian sampling first with respect to lattice gaussian distribution we show the convergence rate of systematic scan gibbs sampling is characterized by the hirschfeld gebelein renyi maximal correlation among the multivariate of being sampled then lattice reduction aided gibbs algorithm is proposed to sample from an equivalent lattice gaussian distribution but with less correlated multivariate thus leading to a better markov mixing after that we extend the proposed lattice reduction aided gibbs sampling to lattice decoding where the choice of the standard deviation for the sampling is fully investigated a customized solution that suits for each specific decoding case by euclidean distance is given which results in a better tradeoff between markov mixing and sampler decoding based on it a startup mechanism is also proposed for gibbs sampler decoding where decoding complexity can be reduced without performance loss moreover the recycling gibbs sampling that exploits the potential of samples is also considered to improve the decoding performance in lattice decoding simulation results based on large scale uncoded multiple input multiple output detection are presented to confirm the performance gain and complexity reduction

computing the ehrhart quasi polynomial of a rational simplex we present a polynomial time algorithm to compute any fixed number of the highest coefficients of the ehrhart quasi polynomial of a rational simplex previously such algorithms were known for integer simplices and for rational polytopes of a fixed dimension the algorithm is based on the formula relating the kth coefficient of the ehrhart quasi polynomial of a rational polytope to volumes of sections of the polytope by affine lattice subspaces parallel to k dimensional faces of the polytope we discuss possible extensions and open questions

on ideal lattices and learning with errors over rings the learning with errors lwe problem is to distinguish random linear equations which have been perturbed by a small amount of noise from truly uniform ones the problem has been shown to be as hard as worst case lattice problems and in recent years it has served as the foundation for a plethora of cryptographic applications unfortunately these applications are rather inefficient due to an inherent quadratic overhead in the use of lwe a main open question was whether lwe and its applications could be made truly efficient by exploiting extra algebraic structure as was done for lattice based hash functions and related primitives we resolve this question in the affirmative by introducing an algebraic variant of lwe called ring lwe and proving that it too enjoys very strong hardness guarantees specifically we show that the ring lwe distribution is pseudorandom assuming that worst case problems on ideal lattices are hard for polynomial time quantum algorithms applications include the first truly practical lattice based public key cryptosystem with an efficient security reduction moreover many of the other applications of lwe can be made much more efficient through the use of ring lwe

lattice problems in np conp we show that the problems of approximating the shortest and closest vector in a lattice to within a factor of n lie in np intersect conp the result almost subsumes the three mutually incomparable previous results regarding these lattice problems banaszczyk 1993 goldreich and goldwasser 2000 and aharonov and regev 2003 our technique is based on a simple fact regarding succinct approximation of functions using their fourier series over the lattice this technique might be useful elsewhere we demonstrate this by giving a simple and efficient algorithm for one other lattice problem cvpp improving on a previous result of regev 2003 an interesting fact is that our result emerged from a dequantization of our previous quantum result in aharonov and regev 2003 this route to proving purely classical results might be beneficial elsewhere

on the geometric ergodicity of metropolis hastings algorithms for lattice gaussian sampling sampling from the lattice gaussian distribution has emerged as an important problem in coding decoding and cryptography in this paper the classic metropolis hastings mh algorithm in markov chain monte carlo methods is adopted for lattice gaussian sampling two mh based algorithms are proposed which overcome the limitation of klein s algorithm the first one referred to as the independent metropolis hastings klein mhk algorithm establishes a markov chain via an independent proposal distribution we show that the markov chain arising from this independent mhk algorithm is uniformly ergodic namely it converges to the stationary distribution exponentially fast regardless of the initial state moreover the rate of convergence is analyzed in terms of the theta series leading to predictable mixing time a symmetric metropolis klein algorithm is also proposed which is proven to be geometrically ergodic

a semi analytical approach to solve integro differential equations in this work we present a semi analytical method for solving integro differential equations under multi point or mixed boundary conditions the proposed method solves linear and nonlinear fredholmvolterra integro differential equations a convergence analysis of the proposed method is directly examined numerical examples are worked out to demonstrate the main results moreover proper graphs are provided to confirm the efficiency and the accuracy of the proposed scheme we show that with a few number of obtained approximating terms we achieve a high accuracy level of the obtained results however increasing the number of approximating terms yields a significant decrease of the error of the approximation the proposed method is very useful reliable and flexible for solving different kinds of integro differential equations

compute and forward harnessing interference through structured codes interference is usually viewed as an obstacle to communication in wireless networks this paper proposes a new strategy compute and forward that exploits interference to obtain significantly higher rates between users in a network the key idea is that relays should decode linear functions of transmitted messages according to their observed channel coefficients rather than ignoring the interference as noise after decoding these linear equations the relays simply send them towards the destinations which given enough equations can recover their desired messages the underlying codes are based on nested lattices whose algebraic structure ensures that integer combinations of codewords can be decoded reliably encoders map messages from a finite field to a lattice and decoders recover equations of lattice points which are then mapped back to equations over the finite field this scheme is applicable even if the transmitters lack channel state information

iterated solutions of linear operator equations with the tau method the tau method produces polynomial approximations of solutions of differential equations the purpose of this paper is i to extend the recursive formulation of this method to general linear operator equations defined in a separable hilbert space and ii to develop an iterative refinement procedure which improves on the accuracy of tau approximations applications to fredholm integral equations demonstrate the effectiveness of this technique

hermitian vector bundles and extension groups on arithmetic schemes i geometry of numbers we define and investigate extension groups in the context of arakelov geometry the arithmetic extension groups ext xi f g we introduce are extensions by groups of analytic types of the usual extension groups extxi f g attached to ox modules f and g over an arithmetic scheme x in this paper we focus on the first arithmetic extension group ext x1 f g the elements of which may be described in terms of admissible short exact sequences of hermitian vector bundles over x and we especially consider the case when x is an arithmetic curve namely the spectrum specok of the ring of integers in some number field k then the study of arithmetic extensions over x is related to old and new problems concerning lattices and the geometry of numbers namely for any two hermitian vector bundles f and g over x specok we attach a logarithmic sizesf g to any element of ext x1 f g and we give an upper bound on sf g in terms of slope invariants of f and g we further illustrate this notion by relating the sizes of restrictions to points in p1 z of the universal extension over pz1 to the geometry of psl2 z acting on poincare s upper half plane and by deducing some quantitative results in reduction theory from our previous upper bound on sizes finally we investigate the behaviour of size by base change i e under extension of the ground field k to a larger number field k when the base field k is q we establish that the size which cannot increase under base change is actually invariant when the field k is an abelian extension of k or when f g is a direct sum of root lattices and of lattices of voronoi s first kind the appendices contain results concerning extensions in categories of sheaves on ringed spaces and lattices of voronoi s first kind which might also be of independent interest

a relation of primal dual lattices and the complexity of shortest lattice vector problem abstract we give a simplified proof of a theorem of lagarias lenstra and schnorr 17 that the problem of approximating the length of the shortest lattice vector within a factor of cn for an appropriate constant c cannot be np hard unless np conp we also prove that the problem of finding a n 1 4 unique shortest lattice vector is not np hard under polynomial time many one reductions unless the polynomial time hierarchy collapses 1998 elsevier science b v all rights reserved

the adaptive operational tau method for systems of odes as the tau method like many other numerical methods has the limitation of using a fixed step size with some high degree order of approximation for solving initial value problems over long intervals we introduce here the adaptive operational tau method this limitation is very much problem dependent and in such case the fixed step size application of the tau method loses the true track of the solution but when we apply this new adaptive method the true solution is recovered with a reasonable number of steps to illustrate the effectiveness of this method we apply it to some stiff systems of ordinary differential equations odes the numerical results confirm the efficiency of the method

cdt based gaussian sampling from multi to double precision the renyi divergence is a measure of closeness of two probability distributions which has found several applications over the last years as an alternative to the statistical distance in lattice based cryptography a tight bound has recently been presented for the renyi divergence of distributions that have a bounded relative error we show that it can be used to bound the precision requirement in gaussian sampling to the ieee 754 floating point standard double precision for usual lattice based signature parameters by using a modified cumulative distribution table cdt which reduces the memory needed by cdt based algorithms and makes their constant time implementation faster and simpler then we apply this approach to a variable center variant of the cdt algorithm which occasionally requires the online computation of the cumulative distribution function as a result the amount of costly floating point operations is drastically decreased which makes the constant time and cache resistant variants of this algorithm viable and efficient finally we provide some experimental results indicating that comparing to rejection sampling our approach increases the gpv signature rate by a factor 4 to 8 depending on the security parameter

bessel polynomial solutions of high order linear volterra integro differential equations abstract in this study a practical matrix method which is based on collocation points is presented to find approximate solutions of high order linear volterra integro differential equations vides under the mixed conditions in terms of bessel polynomials numerical examples are included to demonstrate the validity and applicability of the technique and comparisons are made with the existing results the results show the efficiency and accuracy of the present work all of the numerical computations have been performed on the computer using a program written in matlab v7 6 0 r2008a

improvements in closest point search based on dual hkz bases in this paper we review the technique to solve the cvp based on dual hkz bases by j blomer 4 the technique is based on the transference theorems given by banaszczyk 3 which imply some necessary conditions on the coefficients of the closest vectors with respect to a basis whose dual is hkz reduced recursively starting with the last coefficient intervals of length i can be derived for the i th coefficient of any closest vector this leads to n candidates for closest vectors in this paper we refine the necessary conditions derived from the transference theorems giving an exponential reduction of the number of candidates the improvement is due to the fact that the lengths of the intervals are not independent in the original algorithm the candidates for a coefficient pair v i v i 1 correspond to the integer points in a rectangle of volume email protected i 1 in our analysis we show that the candidates for v i v i 1 in fact lie in an ellipse with transverse and conjugate diameter i 1 respectively i this reduces the expected number of points to be enumerated by an exponential factor of about 0 886 n we further show how a choice of the coefficients v n v i 1 influences the interval from which v i can be chosen numerical computations show that these considerations allow to bound the expected number of points to be enumerated by n 0 7 5 n for 10

iterated chvatal gomory cuts and the geometry of numbers chvatal gomory cutting planes cg cuts for short are a fundamental tool in integer programming given any single cg cut one can derive an entire family of cg cuts by iterating its multiplier vector modulo one this leads naturally to two questions first which iterates correspond to the strongest cuts and second can we find such strong cuts efficiently we answer the first question empirically by showing that one specific approach for selecting the iterate tends to perform much better than several others the approach essentially consists of solving a nonlinear optimization problem over a special lattice associated with the cg cut we then provide a partial answer to the second question by presenting a polynomial time algorithm that yields an iterate that is strong in a certain well defined sense the algorithm is based on results from the algorithmic geometry of numbers

numerical solution of the nonlinear volterra integro differential equations by the tau method in this paper we use the operational approach to the tau method for solving nonlinear volterra integro differential equation vides with analytic function coefficients with initial or boundary conditions we do this without linearizing nonlinear terms we introduce an error estimation of the method we give some examples to clarify the efficiency and high accuracy of the method

a new collocation method for solution of mixed linear integro differential difference equations numerical solution of mixed linear integro differential difference equation is presented using chebyshev collocation method the aim of this article is to present an efficient numerical procedure for solving mixed linear integro differential difference equations our method depends mainly on a chebyshev expansion approach this method transforms mixed linear integro differential difference equations and the given conditions into matrix equation which corresponds to a system of linear algebraic equation the reliability and efficiency of the proposed scheme are demonstrated by some numerical experiments and performed on the computer algebraic system maple10

a modified pseudospectral method for numerical solution of ordinary differential equations systems in this paper numerical solution of first order systems of ordinary differential equations odes is considered and an error estimation method is proposed by this method the efficiency of pseudospectral method to solve the systems is determined and a modified pseudospectral method is presented furthermore with providing some examples the aforementioned cases are dealt with numerically

on the proximity factors of lattice reduction aided decoding lattice reduction aided decoding lrad features reduced decoding complexity and near optimum performance in multiinput multioutput communications in this paper a quantitative error rate analysis of lrad is presented to this aim the proximity factors are defined to measure the worst case losses in decoding distances associated with the decision region relative to infinite lattice decoding ild namely closest point search in an infinite lattice upper bounds on the proximity factors are derived which are functions of the dimension n of the lattice alone the study is then extended to the dual basis reduction it is found that the bounds for dual basis reduction may be smaller reasonably good bounds are derived in many cases proximity factors not only imply the same diversity order in fading channels but also relate the error probabilities of ild and lrad

decoding by sampling part ii derandomization and soft output decoding in this paper a derandomized algorithm for sampling decoding is proposed to achieve near optimal performance in lattice decoding by setting a probability threshold to sample candidates the whole sampling procedure becomes deterministic which brings considerable performance improvement and complexity reduction over to the randomized sampling moreover the upper bound on the sample size k which corresponds to near maximum likelihood ml performance is derived we also find that the proposed algorithm can be used as an efficient tool to implement soft output decoding in multiple input multiple output mimo systems an upper bound of the sphere radius r in list sphere decoding lsd is derived based on it we demonstrate that the derandomized sampling algorithm is capable of achieving near maximum a posteriori map performance simulation results show that near optimum performance can be achieved by a moderate size k in both lattice decoding and soft output decoding

numerical solution of the system of fredholm integro differential equations by the tau method the tau method produces approximate polynomial solution of differential integral and integro differential equations see e l ortiz the tau method siam j numer anal 6 3 1969 480 492 e l ortiz h samara an operational approach to the tau method for the numerical solution of non linear differential equations computing 27 1981 15 25 s m hosseini s shahmorad a matrix formulation of the tau for fredholm and volterra linear integro differential equations the korean j comput appl math 9 2 2002 497 507 s m hosseini s shahmorad numerical solution of a class of integro differential equations by the tau method with an error estimation appl math comput 136 2003 559 570 in this paper we extend the tau method for the numerical solution of integro differential equations systems ides we also give a brief description of the structure of the tau program by the maple software an efficient error estimation of the numerical solution of the method is also introduced some examples are given to clarify the efficiency and high accuracy of the method

tau approximate solution of fractional partial differential equations in this study we improve the algebraic formulation of the fractional partial differential equations fpdes by using the matrix vector multiplication representation of the problem this representation allows us to investigate an operational approach of the tau method for the numerical solution of fpdes we introduce a converter matrix for the construction of converted chebyshev and legendre polynomials which is applied in the operational approach of the tau method we present the advantages of using the method and compare it with several other methods some experiments are applied to solve fpdes including linear and nonlinear terms by comparing the numerical results obtained from the other methods we demonstrate the high accuracy and efficiency of the proposed method

an upper bound on the number of perfect quadratic forms abstract in a recent publication roland bacher showed that the number p d of non similar perfect d dimensional quadratic forms satisfies e d p d e o d 3 log d we improve the upper bound to e o d 2 log d by a volumetric argument based on voronoi s first reduction theory

improved inapproximability of lattice and coding problems with preprocessing we show that the closest vector problem with preprocessing cvpp is np hard to approximate to within spl radic 3 spl epsi for any spl epsi 0 in addition we show that the nearest codeword problem with preprocessing ncpp is np hard to approximate to within 3 spl epsi these results improve previous results of feige and micciancio we also present the first inapproximability result for the relatively nearest codeword problem with preprocessing rncp finally we describe an n approximation algorithm to cvpp

efficient decoding algorithms for the compute and forward strategy we address in this paper decoding aspects of the compute and forward cf physical layer network coding strategy under the cf framework encoders use a special class of nested lattice codes and decoders are based on suboptimal minimum distance decoding of unknown performance gap with respect to optimal decoders in this work we develop and assess the performance of novel decoding algorithms for cf operating in the multiple access channel starting with the gaussian channel we investigate the maximum a posteriori map decoder we derive a novel map decoding metric and develop practical decoding algorithms shown numerically to outperform the original one for the fading channel we analyze the ml decoder for integer valued lattices and develop a novel diophantine approximation based near ml decoding algorithm shown numerically to outperform the original cf decoder in the 1 d case using mathbb z lattices

blind compute and forward compute and forward c f is a promising new approach to interference management enjoying several advantages over other information theoretic schemes c f usually requires channel state information csi at the receivers so that an optimal scaling factor can be computed for the purposes of decoding in this paper a blind c f scheme i e one not requiring csi is developed rather than attempting to compute the optimal scaling factor this new scheme seeks one or more good scalars i e scalars that allow correct decoding despite possibly being suboptimal the region of all such good scalars is characterized to find a good scalar a computationally efficient scheme is proposed which involves error detection a hierarchically organized list as well as a use of the smoothing lemma from lattice theory simulation results show that our blind c f scheme achieves for a class of nested lattice codes the same throughput as its csi enabled counterpart at the expense of approximately a two fold increase in computational complexity in the high throughput region moreover our blind c f scheme can be applied to multisource multirelay networks with a good performance complexity tradeoff

precoded integer forcing universally achieves the mimo capacity to within a constant gap an open loop single user multiple input multiple output mimo communication scheme is considered where a transmitter equipped with multiple antennas encodes the data into independent streams all taken from the same linear code the coded streams are then linearly precoded using the encoding matrix of a perfect linear dispersion space time code at the receiver side integer forcing equalization is applied followed by standard single stream decoding it is shown that this communication architecture achieves the capacity of any gaussian mimo channel up to a gap that depends only on the number of transmit antennas

numerical solution of a class of integro differential equations by the tau method with an error estimation the tau method by construction produces approximate polynomial solutions of differential equations the purpose of this paper is to extend the tau method to the integro differential equations an efficient error estimation for the tau method is also introduced details of this method are presented and some numerical results along with estimated errors are given to clarify the method and its error estimator

improvements in the analysis of kannan s cvp algorithm finding a closest lattice vector to a given target vector is a widely studied problem in computational mathematics and computer science the classical enumeration algorithm was given by kannan in 1983 together with an algorithm for the shortest vector problem the time complexity of the closest vector enumeration algorithm based on hkz reduced bases was proven to within 2 o n n n 2 by hanrot and stehle in 2007 using their point counting method the constant hidden in o n is strictly bigger than 1 9 in this paper we give a new counting method which we call sequence counting method to bound the number of loop iterations and show that the time complexity is actually upper bounded by 2 0 401 n o n n n 2 our paper revisits kannan s cvp algorithm and points out a new method sequence counting to upper bound the time complexity by 2 0 401 n o n n n 2 we connects the loops of the enumeration algorithm with lattice points by a one to one mapping using the sequence as a tool we use different methods to upper bound the number of lattice points near a target vector according to the dimension of the lattice and then combine them to conclude our result

information theoretic clustering clustering is an important topic in pattern recognition since only the structure of the data dictates the grouping unsupervised learning information theory is an obvious criteria to establish the clustering rule the paper describes a novel valley seeking clustering algorithm using an information theoretic measure to estimate the cost of partitioning the data set the information theoretic criteria developed here evolved from a renyi entropy estimator a renyi 1960 that was proposed recently and has been successfully applied to other machine learning applications j c principe et al 2000 an improved version of the k change algorithm is used in optimization because of the stepwise nature of the cost function and existence of local minima even when applied to nonlinearly separable data the new algorithm performs well and was able to find nonlinear boundaries between clusters the algorithm is also applied to the segmentation of magnetic resonance imaging data mri with very promising results

a universal lattice code decoder for fading channels we present a maximum likelihood decoding algorithm for an arbitrary lattice code when used over an independent fading channel with perfect channel state information at the receiver the decoder is based on a bounded distance search among the lattice points falling inside a sphere centered at the received point by judicious choice of the decoding radius we show that this decoder can be practically used to decode lattice codes of dimension up to 32 in a fading environment

estimating the number of clusters in a data set via the gap statistic we propose a method the gap statistic for estimating the number of clusters groups in a set of data the technique uses the output of any clustering algorithm e g k means or hierarchical comparing the change in within cluster dispersion with that expected under an appropriate reference null distribution some theory is developed for the proposal and a simulation study shows that the gap statistic usually outperforms other methods that have been proposed in the literature

information based clustering in an age of increasingly large data sets investigators in many different disciplines have turned to clustering as a tool for data analysis and exploration existing clustering methods however typically depend on several nontrivial assumptions about the structure of data here we reformulate the clustering problem from an information theoretic perspective that avoids many of these assumptions in particular our formulation obviates the need for defining a cluster prototype does not require an a priori similarity metric is invariant to changes in the representation of the data and naturally captures nonlinear relations we apply this approach to different domains and find that it consistently produces clusters that are more coherent than those extracted by existing algorithms finally our approach provides a way of clustering based on collective notions of similarity rather than the traditional pairwise measures

dimension reduction techniques for the minimization of theta functions on lattices we consider the minimization of theta functions theta lambda alpha sum p in lambda e pi alpha p 2 amongst lattices lambda subset mathbb r d by reducing the dimension of the problem following as a motivation the case d 3 where minimizers are supposed to be either the bcc or the fcc lattices a first way to reduce dimension is by considering layered lattices and minimize either among competitors presenting different sequences of repetitions of the layers or among competitors presenting different shifts of the layers with respect to each other the second case presents the problem of minimizing theta functions also on translated lattices namely minimizing l u mapsto theta l u alpha another way to reduce dimension is by considering lattices with a product structure or by successively minimizing over concentric layers the first direction leads to the question of minimization amongst orthorhombic lattices whereas the second is relevant for asymptotics questions which we study in detail in two dimensions