a method for solving ill posed linear operator equations an algorithm is presented to compute least squares solutions of the linear equation tx y where t is a bounded pperator from one hilbert space into another the algorithm is a generalization of marti s method the convergence to a solution minimizing the seminorm lx is proved where the operator l is a closed densely defined linear operator with closed range the method is applied to integral equations and three examples are treated numerically

a biological algorithm to solve the assignment problem based on dna molecules computation abstract the assignment problem ap is to optimally solve the task of assigning n jobs to n individuals such that minimum cost or maximum profit can be obtained it is a vitally important np complete problem in operation management and applied mathematics having a wide range of applications in this paper we present a new parallel dna algorithm to solve the assignment problem using dna molecular operations meanwhile we design flexible length dna strands to represent different jobs and individuals take appropriate steps and get the solutions of the ap in proper length range and with o n 2 time complexity we extend the application of dna molecular operations and simultaneity reduce the complexity of the computation

a fourier approach to nonparametric deconvolution of a density estimate we consider the problem of constructing a nonparametric estimate of a probability density function h from independent random samples of observations from densities a and f when a represents the convolution of h and f our approach is based on truncated fourier inversion in which the truncation point plays the role of a smoothing parameter we derive the asymptotic mean integrated squared error of the estimate and use this formula to suggest a simple practical method for choosing the truncation point from the data strikingly when the smoothing parameter is chosen in this way then in many circumstances the estimator behaves to first order as though the true f were known

the multivariate krawtchouk polynomials as matrix elements of the rotation group representations on oscillator states an algebraic interpretation of the bivariate krawtchouk polynomials is provided in the framework of the three dimensional isotropic harmonic oscillator model these polynomials in two discrete variables are shown to arise as matrix elements of unitary reducible representations of the rotation group in three dimensions many of their properties are derived by exploiting the group theoretic setting the bivariate tratnik polynomials of krawtchouk type are seen to be special cases of the general polynomials that correspond to particular rotations involving only two parameters it is explained how the approach generalizes naturally to d 1 dimensions and allows us to interpret multivariate krawtchouk polynomials as matrix elements of so d 1 unitary representations indications are given on the connection with other algebraic models for these polynomials

solving the minimum bisection problem using a biologically inspired computational model the traditional trend of dna computing aims at solving computationally intractable problems the minimum bisection problem mbp is a well known np hard problem which is intended to partition the vertices of a given graph into two equal halves so as to minimize the number of those edges with exactly one end in each half based on a biologically inspired computational model this paper describes a novel algorithm for the minimum bisection problem which requires a time cost and a dna strand length that are linearly proportional to the instance size

numerical determination of hitting time distributions from their laplace transforms simple cases abstract an important problem in probabilistic modeling consists of the determination of the distribution function of a positive random variable in particular a hitting time from its laplace transform in this note we use two versions of the method of maximum entropy to invert the laplace transform using a few of its values on the real line or equivalently a few fractional moments of the exponential of the variable to compare the performance of the methods we consider some simple one dimensional examples in which the laplace transform can be computed by solving a differential equation exactly and comparisons can be made

adaptive wavelet estimator for nonparametric density deconvolution 1999 institute of mathematical statistics the electronic version of this article is the complete one and can be found online at http projecteuclid org euclid aos 1017939249

connection coefficients for classical orthogonal polynomials of several variables abstract connection coefficients between different orthonormal bases satisfy two discrete orthogonal relations themselves for classical orthogonal polynomials whose weights are invariant under the action of the symmetric group connection coefficients between a basis consisting of products of hypergeometric functions and another basis obtained from the first one by applying a permutation are studied for the jacobi polynomials on the simplex it is shown that the connection coefficients can be expressed in terms of tratnik s multivariable racah polynomials and their weights this gives in particular a new interpretation of the hidden duality between the variables and the degree indices of the racah polynomials which lies at the heart of their bispectral properties these techniques also lead to explicit formulas for connection coefficients of hahn and krawtchouk polynomials of several variables as well as for orthogonal polynomials on balls and spheres

solving traveling salesman problem in the adleman lipton model abstract the traveling salesman problem is to find a minimum cost weight path for a given set of cities vertices and roads edges the path must start at a specified city and end there after going through all the other given cites only once it is a classical np complete problem in graph theory in this paper we consider a dna procedure for solving the traveling salesman problem in the adleman lipton model the procedure works in o n steps for the traveling salesman of an edge weighted graph with n vertices

3nj coefficients of su 1 1 as connection coefficients between orthogonal polynomials in n variables in the tensor product of n 1 positive discrete series representations of su 1 1 a coupled basis vector can be described by a certain binary coupling tree to every such binary coupling tree polynomials rl k x and rl k x are associated these polynomials are n variable jacobi and continuous hahn polynomials and are orthogonal with respect to a weight function the connection coefficients expressing such a polynomial associated with a given binary coupling tree in terms of those polynomials associated with another binary coupling tree are proportional to 3nj coefficients of su 1 1

hahn polynomials on polyhedra and quantum integrability abstract orthogonal polynomials with respect to the hypergeometric distribution on lattices in polyhedral domains in r d which include hexagons in r 2 and truncated tetrahedrons in r 3 are defined and studied the polynomials are given explicitly in terms of the classical one dimensional hahn polynomials they are also characterized as common eigenfunctions of a family of commuting partial difference operators these operators provide symmetries for a system that can be regarded as a discrete extension of the generic quantum superintegrable system on the d sphere moreover the discrete system is proved to possess all essential properties of the continuous system in particular the symmetry operators for the discrete hamiltonian define a representation of the kohno drinfeld lie algebra on the space of orthogonal polynomials and an explicit set of 2 d 1 generators for the symmetry algebra is constructed furthermore other discrete quantum superintegrable systems which extend the quantum harmonic oscillator are obtained by considering appropriate limits of the parameters

adaptive estimation of a quadratic functional by model selection we consider the problem of estimating s 2 when s belongs to some separable hilbert space and one observes the gaussian process y t langles t rangle sigmal t for all t epsilon mathbb h where l is some gaussian isonormal process this framework allows us in particular to consider the classical gaussian sequence model for which mathbb h l 2 mathbb n and l t sum lambda geq1 t lambda varepsilon lambda where varepsilon lambda lambda geq1 is a sequence of i i d standard normal variables our approach consists in considering some at most countable families of finite dimensional linear subspaces of mathbb h the models and then using model selection via some conveniently penalized least squares criterion to build new estimators of s 2 we prove a general nonasymptotic risk bound which allows us to show that such penalized estimators are adaptive on a variety of collections of sets for the parameter s depending on the family of models from which they are built in particular in the context of the gaussian sequence model a convenient choice of the family of models allows defining estimators which are adaptive over collections of hyperrectangles ellipsoids l p bodies or besov bodies we take special care to describe the conditions under which the penalized estimator is efficient when the level of noise sigma tends to zero our construction is an alternative to the one by efroimovich and low for hyperrectangles and provides new results otherwise

a regularization method for the numerical inversion of the laplace transform a method for numerical inversion of the laplace transform based on tikhonov regularization is described the method requires values of u s only for real s and appears relatively reliable in practice since the ill posedness of this problem has been considered the precondition to use the algorithm is that the exact solution of the inversion exists and it is lipschitz continuous with finite and known support

laguerre polynomials and the inverse laplace transform using discrete data abstract we consider the problem of finding a function defined on 0 from a countable set of values of its laplace transform the problem is severely ill posed we shall use the expansion of the function in a series of laguerre polynomials to convert the problem in an analytic interpolation problem then using the coefficients of lagrange polynomials we shall construct a stable approximation solution error estimate is given numerical results are produced

discrete orthogonal polynomials and difference equations of several variables the goal of this work is to characterize all second order difference operators of several variables that have discrete orthogonal polynomials as eigenfunctions under some mild assumptions we give a complete solution of the problem

some multivariable orthogonal polynomials of the askey tableau discrete families a multivariable generalization is presented for all the discrete families of the askey tableau this significantly extends the multivariable hahn polynomials introduced by karlin and mcgregor the latter are recovered as a limit case from a family of multivariable racah polynomials

the multivariate hahn polynomials and the singular oscillator karlin and mcgregor s d variable hahn polynomials are shown to arise in the dimensional singular oscillator model as the overlap coefficients between bases associated with the separation of variables in cartesian and hyperspherical coordinates these polynomials in d discrete variables depend on real parameters and are orthogonal with respect to the multidimensional hypergeometric distribution the focus is put on the d 2 case for which the connection with the three dimensional singular oscillator is used to derive the main properties of the polynomials forward backward shift operators orthogonality relation generating function recurrence relations bispectrality difference equations and explicit expression in terms of the univariate hahn polynomials the extension of these results to an arbitrary number of variables is presented at the end of the paper

the inverse laplace transform and analytic pseudo differential operators abstract by comparing the laplace transform l with the differential operatord we obtain a formula for the inverse laplace transform l 1 1 v 1 cos d v l wherevis a unitary transformation operator this helps us obtain an explicit spectral representation of l some applications of the above relation are discussed

dna ternary addition abstract in this paper we first show a dna representation of a ternary number with digit set 1 0 1 which features its address and each bit position based on this dna representation operations for value assigning and bit position shifting are proposed the algorithm of dna computing for adding two ternary integers is presented the algorithm works in o m steps for adding two ternary integers of m bits

another proof of the triple sum formula for wigner 9j symbols we give a simple proof of the alisauskas jucys triple sum formula for wigner 9j symbols

complex variable and regularization methods of inversion of the laplace transform in this paper three methods are derived for approximating f given its laplace transform g on 0 oo i e fo f t exp st dt g s assuming that g e l 0 oo the first method is based on a sinc like rational approximation of 9 the second on a sinc solution of the integral equation fo0 f t exp st dt g s via standard regularization and the third method is based on first converting fo0 f t exp st dt g s to a convolution integral over r and then finding a sinc approximation to f via the application of a special regularization procedure to solve the fourier transform problem we also obtain bounds on the error of approximation which depend on both the method of approximation and the regularization parameter

laplace deconvolution on the basis of time domain data and its application to dynamic contrast enhanced imaging in the present paper we consider the problem of laplace deconvolution with noisy discrete non equally spaced observations on a finite time interval which appears in many different contexts we propose a new method for laplace deconvolution which is based on expansions of the convolution kernel the unknown function and the observed signal over laguerre functions basis which acts as a surrogate eigenfunction basis using regression setting the expansion results in a small system of linear equations with the matrix of the system being triangular and toeplitz the number m of the terms in the expansion of the estimator is controlled via complexity penalty the advantage of this methodology is that it leads to very fast computations produces no boundary effects due to extension at zero and cut off at t and provides an estimator with the risk within a logarithmic factor of m of the oracle risk under no assumptions on the model and within a constant factor of the oracle risk under mild assumptions we emphasize that in the present paper we consider the true observational model with possibly non equispaced observations which are available on a finite interval of length t and account for the bias associated with this model the study is motivated by perfusion imaging using a short injection of contrast agent a procedure which is widely applied for medical assessment of blood flows within tissues such as cancerous tumors presence of a tuning parameter a allows to choose the most advantageous time units so that both the kernel and the unknown right hand side of the equation are well represented for the further deconvolution in addition absence of boundary effects allows for an accurate estimator of the tissue blood flow parameter which is of great interest to medical doctors the methodology is illustrated by an extensive simulation study and a real data example which confirms that the proposed technique is fast efficient accurate usable from a practical point of view and competitive

summary of sinc numerical methods sinc approximation methods excel for problems whose solutions may have singularities or infinite domains or boundary layers this article summarizes results obtained to date on sinc numerical methods of computation sinc methods provide procedures for function approximation over bounded or unbounded regions encompassing interpolation approximation of derivatives approximate definite and indefinite integration solving initial value ordinary differential equation problems approximation and inversion of fourier and laplace transforms approximation of hilbert transforms and approximation of indefinite convolutions the approximate solution of partial differential equations and the approximate solution of integral equations methods for constructing conformal maps and methods for analytic continuation indeed sinc are ubiquitous for approximating every operation of calculus

on monotone solutions of an integrodifferential equation in linear viscoelasticity we consider the integrodifferential equation u t x int 0 t a t s u xx s x ds with initial and boundary conditions corresponding to the rayleigh problem the kernel has the form a t a 0 a infty t int 0 t a 1 s ds where a 0 geq 0 a infty geq 0 and a 1 in l loc 1 bf r is of positive type and satisfies the condition int 0 infty e epsilon t ert a 1 t ert dt 0 by solving the equation numerically and performing a careful error analysis we show that the solution u t x need not be nondecreasing in t geq 0 for fixed x 0 if a 1 is nonnegative nonincreasing and convex the same result is shown to hold under the assumption that a 1 is completely positive this answers a question that remained unsolved in j pru ss it math ann 279 1987 p 330 in the case where a 1 is convex piecewise linear the solution is shown to be almost everywhere equal to a function which is discontinuous across infinitely many parallel lines

explicit nearly optimal linear rational approximation with preassigned poles this paper gives explicit rational functions for interpolating and approximating functions on the intervals 1 1 0 oc and oc oc the rational functions are linear in the functions to be approximated and they have preassigned poles the error of approximation of these rationals is nearly as small as the error of best rational approximation with numerator and denominator polynomials of the same degrees regions of analyticity are described which make it possible to tell a priori the accuracy which we can expect from this type of rational approximation 1 introduction and summary in this paper we attempt to give a constructive affirmative answer to each of the following questions

numerical methods based on whittaker cardinal or sinc functions this paper summarizes the results known to date for using sine functions composed with other functions as bases for approximations in numerical analysis described in this paper are methods of interpolation and approximation of functions and their derivatives quadrature the approximate evaluation of transforms hilbert fourier laplace hankel and mellin and the approximate solution of differential and integral equations the methods have many advantages over classical methods which use polynomials as bases in addition all of the methods converge at an optimal rate if singularities on the boundary of approximation are ignored

a polynomial time dna computing solution for the bin packing problem it is suggested here an algorithm based on stickers for the dna computing model s roweis e winfree r burgoyne n chelyapov m goodman p rothemund l adleman a sticker based model for dna computation journal of computational biology 5 1998 615 629 that solves the well known bin packing problem bpp that belongs to the class np hard in the strong sense in a time bounded by o n 2q where n is the quantity of items and q the space requirements expressed in bits to the best of the authors knowledge this is the first polynomial time algorithmic solution for the bpp in such a model

the bad truth about laplace s transform inverting the laplace transform is a paradigm for exponentially ill posed problems for a class of operators including the laplace transform we give forward and inverse formulae that have fast implementations using the fast fourier transform these formulae lead easily to regularized inverses whose effects on noise and filtered data can be precisely described our results give cogent reasons for the general sense of dread most mathematicians feel about inverting the laplace transform