a polynomial time approximation algorithm for the permanent of a matrix with nonnegative entries we present a polynomial time randomized algorithm for estimating the permanent of an arbitrary n n matrix with nonnegative entries this algorithm technically a fully polynomial randomized approximation scheme computes an approximation that is with high probability within arbitrarily small specified relative error of the true value of the permanent

the bethe permanent of a nonnegative matrix it has recently been observed that the permanent of a nonnegative square matrix i e of a square matrix containing only nonnegative real entries can very well be approximated by solving a certain bethe free energy function minimization problem with the help of the sum product algorithm we call the resulting approximation of the permanent the bethe permanent in this paper we give reasons why this approach to approximating the permanent works well namely we show that the bethe free energy function is convex and that the sum product algorithm finds its minimum efficiently we then discuss the fact that the permanent is lower bounded by the bethe permanent and we comment on potential upper bounds on the permanent based on the bethe permanent we also present a combinatorial characterization of the bethe permanent in terms of permanents of so called lifted versions of the matrix under consideration moreover we comment on possibilities to modify the bethe permanent so that it approximates the permanent even better and we conclude the paper with some observations and conjectures about permanent based pseudocodewords and permanent based kernels

finite lattice expansion for the ising model on the penrose tiling low temperature series are calculated for the free energy magnetization susceptibility and field derivatives of the susceptibility in the ising model on the quasiperiodic penrose lattice the series are computed to an order of 25 and estimates of the critical exponents and are obtained from pade approximants

exact and operator rational approximate solutions of the helmholtz weyl composition equation in underwater acoustics the quadratic profile the application of phase space and path integral methods to both mathematical and computational direct and inverse wave propagation modeling at the level of the scalar one way helmholtz equation is briefly reviewed the role of operator symbols is stressed and their properties briefly discussed the construction of the operator symbol requires the exact or approximate solution of the helmholtz weyl composition equation in the weyl pseudodifferential operator calculus the exact symbols for several quadratic profiles are presented and briefly analyzed the symbols exactly corresponding to a family of operator rational approximations are also presented for the quadratic case these results are used to illustrate several points pertinent to wide angle propagation modeling and the refractive index profile reconstruction problem in underwater acoustics

numerical solution of the coupled mode equations in duct acoustics we present a fast solver for the coupled mode equations in duct acoustics it is based on a partitioning of the resulting system of ordinary differential equations into separate subsystems the number of which increases by the separability of the problem this structure is obtained by transformations of the dependent variable such that weakly interacting modes are separated the decoupling process requires the numerical solution of algebraic riccati equations however these computations can be done on a spatial scale much larger than the characteristic wavelength of the problem 23 refs 7 figs 1 tab

on the number of matrices and a random matrix with prescribed row and column sums and 0 1 entries abstract we consider the set r c of all m n matrices having 0 1 entries and prescribed row sums r r 1 r m and column sums c c 1 c n we prove an asymptotic estimate for the cardinality r c via the solution to a convex optimization problem we show that if r c is sufficiently large then a random matrix d r c sampled from the uniform probability measure in r c with high probability is close to a particular matrix z z r c that maximizes the sum of entropies of entries among all matrices with row sums r column sums c and entries between 0 and 1 similar results are obtained for 0 1 matrices with prescribed row and column sums and assigned zeros in some positions

exact constructions of square root helmholtz operator symbols the focusing quadratic profile operator symbols play a pivotal role in both the exact well posed one way reformulation of solving the elliptic helmholtz equation and the construction of the generalized bremmer coupling series the inverse square root and square root helmholtz operator symbols are the initial quantities of interest in both formulations in addition to providing the theoretical framework for the development and implementation of the parabolic equation pe method in wave propagation modeling exact standard left and weyl symbol constructions are presented for both the inverse square root and square root helmholtz operators in the case of the focusing quadratic profile in one transverse spatial dimension extending and ultimately unifying the previously published corresponding results for the defocusing quadratic case j math phys 33 1887 1914 1992 both i spectral modal summation representations and ii contour integral representations exploiting the underlying periodicity of the associated quantum mechanical harmonic oscillator problem are derived and ultimately related through the propagating and nonpropagating contributions to the operator symbol high and low frequency asymptotic operator symbol expansions are given along with the exact symbol representations for the corresponding operator rational approximations which provide the basis for the practical computational realization of the pe method moreover while the focusing quadratic profile is in some respects nonphysical the corresponding helmholtz operator symbols nevertheless establish canonical symbol features for more general profiles containing locally quadratic wells

bootstrapping on undirected binary networks via statistical mechanics we propose a new method inspired from statistical mechanics for extracting geometric information from undirected binary networks and generating random networks that conform to this geometry in this method an undirected binary network is perceived as a thermodynamic system with a collection of permuted adjacency matrices as its states the task of extracting information from the network is then reformulated as a discrete combinatorial optimization problem of searching for its ground state to solve this problem we apply multiple ensembles of temperature regulated markov chains to establish an ultrametric geometry on the network this geometry is equipped with a tree hierarchy that captures the multiscale community structure of the network we translate this geometry into a parisi adjacency matrix which has a relative low energy level and is in the vicinity of the ground state the parisi adjacency matrix is then further optimized by making block permutations subject to the ultrametric geometry the optimal matrix corresponds to the macrostate of the original network an ensemble of random networks is then generated such that each of these networks conforms to this macrostate the corresponding algorithm also provides an estimate of the size of this ensemble by repeating this procedure at different scales of the ultrametric geometry of the network it is possible to compute its evolution entropy i e to estimate the evolution of its complexity as we move from a coarse to a fine description of its geometric structure we demonstrate the performance of this method on simulated as well as real data networks

matrices with prescribed row and column sums abstract this is a survey of the recent progress and open questions on the structure of the sets of 0 1 and non negative integer matrices with prescribed row and column sums we discuss cardinality estimates the structure of a random matrix from the set discrete versions of the brunn minkowski inequality and the statistical dependence between row and column sums

sequential monte carlo methods for statistical analysis of tables we describe a sequential importance sampling sis procedure for analyzing two way zero one or contingency tables with fixed marginal sums an essential feature of the new method is that it samples the columns of the table progressively according to certain special distributions our method produces monte carlo samples that are remarkably close to the uniform distribution enabling one to approximate closely the null distributions of various test statistics about these tables our method compares favorably with other existing monte carlo based algorithms and sometimes is a few orders of magnitude more efficient in particular compared with markov chain monte carlo mcmc based approaches our importance sampling method not only is more efficient in terms of absolute running time and frees one from pondering over the mixing issue but also provides an easy and accurate estimate of the total number of tables with fixed marginal sums which is far more difficult for an mcmc method to achieve

one way large range step methods for helmholtz waveguides a useful approach for long range computation of the helmholtz equation in a waveguide is to re formulate it as the operator differential riccati equation for the dirichlet to neumann map for waveguides with slow range dependence the piecewise range independent approximation is used to derive a second order range stepping method for this one way re formulation the range marching formulas are exact for each range independent piece and a large range step is possible if the range dependence is gradual based on a fourth order conservative exponential method for linear evolution equations a fourth order method that admits even larger range steps is developed for the one way re formulation numerical examples are used to demonstrate the improved accuracy of the fourth order method

factorization and path integration of the helmholtz equation numerical algorithms the propagator for the reduced scalar helmholtz equation plays a significant role in both analytical and computational studies of acoustic direct wave propagation path functional integrals are taken to provide the principal representation of the propagator and are computed directly the path integral is the primary tool in extending the classical fourier methods so appropriate for wave propagation in homogeneous media to inhomogeneous media for transversely inhomogeneous environments the n dimensional helmholtz equation can be exactly factored into separate forward and backward one way wave equations a parabolic based one way phase space path integral construction provides the generalization of the tappert hardin split step fft algorithm to the full one way factored helmholtz wave equation these extended marching algorithms can readily accommodate density profiles and range updating and further in conjunction with imbedding methods provide the basis for incorporating backscatter effects in a complementary manner for general range dependent environments elliptic based two way path integral constructions lead to an approximate representation of the propagator feynman garrod and a natural statistical monte carlo means of evaluation taken together the path integrals provide the basis for a global analysis in addition to providing a unifying framework for dynamical approximations resolution of the square root operator and the concept of an underlying stochastic process the one way marching algorithms are applied to ocean acoustic environments seismological environments and extreme model environments designed to establish their range of validity and manner of breakdown

the maximal length of a chain in the bruhat order for a class of binary matrices abstract we answer a question by brualdi and deaett about the maximal length of a chain in the bruhat order for an interesting combinatorial class of binary matrices

on the number of interference alignment solutions for the k user mimo channel with constant coefficients in this paper we study the number of different interference alignment ia solutions in a k user multiple input multiple output mimo interference channel when the alignment is performed via beamforming and no symbol extensions are allowed we focus on the case where the number of ia equations matches the number of variables in this situation the number of ia solutions is finite and constant for any channel realization out of a zero measure set and as we prove in this paper it is given by an integral formula that can be numerically approximated using monte carlo integration methods more precisely the number of alignment solutions is the scaled average of the determinant of a certain hermitian matrix related to the geometry of the problem interestingly while the value of this determinant at an arbitrary point can be used to check the feasibility of the ia problem its average properly scaled gives the number of solutions for single beam systems the asymptotic growth rate of the number of solutions is analyzed and some connections with classical combinatorial problems are presented nonetheless our results can be applied to arbitrary interference mimo networks with any number of users antennas and streams per user

a modified schwarz christoffel mapping for regions with piecewise smooth boundaries a method where polygon corners in schwarz christoffel mappings are rounded is used to construct mappings from the upper half plane to regions bounded by arbitrary piecewise smooth curves from a given curve a polygon is constructed by taking tangents to the curve in a number of carefully chosen so called tangent points the schwarz christoffel mapping for that polygon is then constructed and modified to round the corners since such a modification causes effects on the polygon outside the rounded corners the parameters in the mapping have to be redetermined this is done by comparing side lengths in tangent polygons to the given curve and the curve produced by the modified schwarz christoffel mapping the set of equations that this comparison gives can normally be solved using a quasi newton method the resulting function maps the upper half plane on a region bounded by a curve that apart from possible vertices is smooth i e one time continuously differentiable that passes through the tangent points on the given curve has the same direction as the given curve in these points and changes direction monotonically between them furthermore where the original curve has a vertex the constructed curve has a vertex with the same inner angle the method is especially useful for unbounded regions with smooth boundary curves that pass infinity as straight lines such as channels with parallel walls at the ends these properties are kept in the region produced by the constructed mapping

acoustic transmission in curved ducts with varying cross sections methods for predicting the acoustic propagation properties of ducts are developed and analysed for ducts that are asymptotically straight with constant crosssection with application to splitter si

diffraction pattern calculations for a certain class of n fold quasilattices using the cut and project method we obtain seven and nine fold quasiperiodic lattices for such quasilattices the structure factors are calculated analytically through integration in higher dimensional spaces the obtained formulae are compared with the direct numerical calculations of fourier transforms

self similarity of quasilattices in two dimensions i the n gonal quasilattice the self similarity of a two dimensional quasilattice with an n gonal point symmetry has been investigated for even n n or 8 it is shown that for any n an n gonal quasilattice has a self similarity characterised by a complex number tau on inflation the quasilattice is scaled by mod tau mod and subsequently rotated by arg tau tau is a pv unit of the n cyclotomic field q zeta zeta exp 2 pi i n tau satisfies i tau and tau 1 are both algebraic integers in q zeta ii mod tau mod 1 and iii mod tau mod 1 for any conjugate tau but tau the complex conjugate of tau in q zeta pv units are calculated for every n gonal quasilattice whose multiplicity m phi n 2 with phi being the eulerian function is less than 5 m 2 for n 8 10 12 m 3 for n 14 18 and m 4 for n 16 20 24 30 it is found that a quasilattice has two or more independent scales of self similarity if its multiplicity is larger than 2

two algorithmic results for the traveling salesman problem for any norm in a euclidean space and for any number 0 we present a polynomial time algorithm which computes a hamiltonian circuit with the given vertices in the space whose length approximates with relative error less than the largest length of a hamiltonian circuit with these vertices we also present an algorithm which for a given graph with n vertices computes the number of hamiltonian circuits in this graph with 2n olog n time complexity and a polynomial in n space complexity as a by product of our approach we prove that the permanent of a matrix can be computed in polynomial time provided the rank of the matrix is fixed

fast uniform generation of regular graphs abstract an algorithm is presented which randomly selects a labelled graph with specified vertex degrees from a distribution which is arbitrarily close to uniform the algorithm is based on simulation of a rapidly convergent stochastic process and runs in polynomial time for a wide class of degree sequences including all regular sequences and all n vertex sequences with no degree exceeding n 2 the algorithm can be extended to cover the selection of a graph with given degree sequence which avoids a specified set of edges one consequence of this extension is the existence of a polynomial time algorithm for selecting an f factor in a sufficiently dense graph a companion algorithm for counting degree constrained graphs is also presented this algorithm has exactly the same range of validity as the one for selection

block regularized repeated learning testing for estimating generalization error abstract repeated learning testing rlt is a popular cross validation method that is usually adopted in estimation and comparison of algorithm performance in machine learning however the variance of the estimator of the generalization error in the rlt method is easily affected by random partitions poor data partitioning may cause a large overlap between any two training sets of rlt and enlarge the variance in the rlt estimator of the generalization error thus in this study we constrain numbers of overlapping samples between any two training sets and construct a novel data partitioning schema in which rlt is called the block regularized rlt brlt we theoretically prove that the brlt estimator has a smaller variance than does the rlt estimator specifically the variance of the rlt estimator reaches its minimum when samples in a data set equally occur in all training sets and all the numbers of overlapping samples are equal furthermore we provide easy to use construction algorithms of brlt s partition set for several training set size and partition count settings by adopting two level orthogonal arrays we also illustrate brlt s optimal properties with several simulated and real world examples

boolean matrices with prescribed row column sums and stable homogeneous polynomials we prove a new efficiently computable lower bound on the coefficients of stable homogeneous polynomials and present its algorithmic and combinatorial applications our main application is the first poly time deterministic algorithm which approximates the partition functions associated with boolean matrices with prescribed row and column sums within simply exponential multiplicative factor this new algorithm is a particular instance of new polynomial time deterministic algorithms related to the multiple partial differentiation of polynomials given by evaluation oracles