reconstructing algebraic functions from mixed data we consider a variant of the traditional task of explicitly reconstructing algebraic functions from black box representations in the traditional setting for such problems one is given access to an unknown function f that is represented by a black box or an oracle which can be queried for the value of f at any input given a guarantee that this unknown function f is some nice algebraic function say a polynomial in its input of degree bound d the goal of the reconstruction problem is to explicitly determine the coefficients of the unknown polynomial all work on polynomial interpolation especially sparse ones are or may be presented in such a setting the work of kaltofen and trager computing with polynomials given by black boxes for their evaluations greatest common divisors factorization separation of numerators and denominators in proc 29th ann ieee symp on foundations of computer science 1988 pp 296 305 for instance highlights the utility of this setting by performing numerous manipulations on polynomials presented as black boxes the variant considered in this paper differs from the traditional setting in that our black boxes represent several algebraic functions f1 fk where at each input x the box arbitrarily chooses a subset of f1 x fk x to output and we do not know which subset it outputs we show how to reconstruct the functions f1 fk from the black box provided the black box outputs according to these functions often this allows us to group the sample points into sets such that for each set all outputs to points in the set are from the same algebraic function our methods are robust in the presence of a small fraction of arbitrary errors in the black box our model and techniques can be applied in the areas of computer vision machine learning curve fitting and polynomial approximation self correcting programs and bivariate polynomial factorization

global strong solutions for nonlinear schro dinger equations on considere l existence globale unique des solutions fortes pour les equations de schrodinger non lineaire a interaction en loi de puissance i u t u u p 1 u t r x r n u t 0 x u 0 x x r n ou t 0 r

analytic smoothing effect for nonlinear schrodinger equation with quintic nonlinearity abstract we prove the global existence of analytic solutions to nonlinear schrodinger equation with quintic nonlinearity in n space dimensions for sufficiently small cauchy data with exponential decay the smallness assumption on the data is imposed in terms of the critical sobolev space h n 2 1 2 moreover a characterization of some class of analytic functions is given

remarks on the semilinear schrodinger equation abstract we investigate the initial value problem for some semi linear schrodinger equations in two space dimensions with exponential growth

direct fourier reconstruction in fan beam tomography we consider the problem of reconstructing tomographic imagery from fan beam projections using the direct fourier method dfm previous dfm reconstructions from parallel beam projections produced images of quality comparable to that of filtered convolution back projection moreover the number of operations using dfm in the parallel beam case is proportional to n2 log n versus n3 for back projection 3 the fan beam case is more complicated because additional interpolation of the nonuniformly spaced rebinned data is required we derive bounds on the detector spacing in fan beam ct that enable direct fourier reconstruction and describe the full algorithm necessary for processing the fan beam data the feasibility of the method is demonstrated with an example a key result of this paper is that high quality imagery can be reconstructed from fan beam data using the dfm in 0 n2 log n operations

a high speed reconstruction from projections using direct fourier method with optimized parameters an experimental analysis some issues of the direct fourier method dfm implementation are discussed a hybrid spline linear interpolation for the dfm is proposed the results of comprehensive simulation research are presented the following reconstruction problems and parameters are emphasized interpolation increasing the radial density of the polar raster filtering the 2 d inverse fourier transformation dimension and considering the cases of noiseless and noisy input data for the a priori prescribed resolution of the reconstructed image values of reconstruction parameters have been determined which are optimal with regard to reconstruction quality and computation cost the computational requirements of the dfm algorithm which correspond to distinct interpolation schemes are compared to one another for ct and mr tomography respectively the estimations obtained are compared to computational characteristics of the convolution backprojection method

asymptotic behavior and decay estimates of the solutions for a nonlinear parabolic equation with exponential nonlinearity we consider a nonlinear parabolic equation with an exponential nonlinearity which is critical with respect to the growth of the nonlinearity and the regularity of the initial data after showing the equivalence of the notions of weak and mild solutions we derive decay estimates and the asymptotic behavior for small global in time solutions

multigrid tomographic inversion with variable resolution data and image spaces a multigrid inversion approach that uses variable resolutions of both the data space and the image space is proposed since the computational complexity of inverse problems typically increases with a larger number of unknown image pixels and a larger number of measurements the proposed algorithm further reduces the computation relative to conventional multigrid approaches which change only the image space resolution at coarse scales the advantage is particularly important for data rich applications where data resolutions may differ for different scales applications of the approach to bayesian reconstruction algorithms in transmission and emission tomography with a generalized gaussian markov random field image prior are presented both with a poisson noise model and with a quadratic data term simulation results indicate that the proposed multigrid approach results in significant improvement in convergence speed compared to the fixed grid iterative coordinate descent method and a multigrid method with fixed data resolution

on zero testing and interpolation of k sparse multivariate polynomials over finite fields abstract given a black box which will produce the value of a k sparse multivariate polynomial for any given specific argument one may ask for optimal strategies 1 to distinguish such a polynomial from the zero polynomial 2 to distinguish any two such polynomials from one other and 3 to uniformly reconstruct the polynomial from such an information source while such strategies are known already for polynomials over fields of characteristic zero the equally important but considerably more complicated case of a finite field k of small characteristic is studied in the present paper the result is that the time complexity of such strategies depends critically on the degree m of the extension field of k from which the arguments are to be chosen e g if m equals the number n of variables then 1 can be solved by k 1 and 2 as well as 3 by 2 k 1 queries while in case m 1 essentially 2 log n log k queries are needed

the global cauchy problem for the nls and nlkg with small rough data abstract by using the unit cube decomposition to the frequency spaces we study the cauchy problem for the nonlinear schrodinger equation and the nonlinear klein gordon equation some global well posedness results are obtained for the small cauchy data in some modulation spaces m p 1 s

asymptotic behavior of solutions to schrodinger equations with a subcritical dissipative nonlinearity abstract we study the asymptotic behavior in time of solutions to the initial value problem of the nonlinear schrodinger equation with a subcritical dissipative nonlinearity u p 1 u where 1 p 1 2 n n is the space dimension and is a complex constant satisfying im 0 we show the time decay estimates and the large time asymptotics of the solution when the space dimension n 3 p is sufficiently close to 1 2 n and the initial data is sufficiently small

the cauchy problem for heat equations with exponential nonlinearity abstract the cauchy problem for the semilinear heat equations is studied in the orlicz space exp l 2 r n where any power behavior of interaction works as a subcritical nonlinearity we prove the existence of global solutions for the semilinear heat equations with the exponential nonlinearity under the smallness condition on the initial data in exp l 2 r n

a multilevel domain decomposition algorithm for fast o n sup 2 logn reprojection of tomographic images a novel algorithm for fast computation of tomographic image projections is presented the method comprises a decomposition of an image into subimages followed by an aggregation of projections computed for the subimages the multilevel domain decomposition algorithm is formulated as a recursive procedure the computational cost of the proposed algorithm is comparable to that of fft based techniques while it appears to be more flexible than the latter numerical results demonstrate the effectiveness of the method

interpolation of polynomials given by straight line programs we give an algorithm for the interpolation of a polynomial a given by a straight line program its complexity is polynomial in t log d l n where t is an input bound on the number of terms in a d is a bound on its partial degree in all variables l is the length of the given straight line program and n is the number of variables

analyticity of solutions to a quadratic system of schrodinger equations without mass resonance condition abstract we consider the cauchy problem for a quadratic system of schrodinger equations in space dimensions n 4 without mass resonance condition analytic smoothing effect for the quadratic system of schrodinger equations under the mass resonance condition for data which satisfy exponentially decaying condition has been studied in 14 in this study we construct global solutions in the analytic hardy space via the phase modulation operator based on the scale critical sobolev space h n 2 2 with data like the test functions of fourier hyperfunctions under the condition m 1 m 2 1 4 n m 1 between two masses m 1 and m 2

accelerating x ray data collection using pyramid beam ray casting geometries image reconstruction from its projections is a necessity in many applications such as medical ct security inspection and others this paper extends the 2 d fan beam method in to 3 d the algorithm called pyramid beam pb is based upon the parallel reconstruction algorithm in it allows fast capturing of the scanned data and in 3 d the reconstructions are based upon the discrete x ray transform the pb geometries are reordered to fit parallel projection geometry the underlying idea is to use the algorithm in by porting the proposed pb geometries to fit the algorithm in the complexity of the algorithm is comparable with the 3 d fft the results show excellent reconstruction qualities while being simple for practical use

on learning functions from noise free and noisy samples via occam s razor an occam approximation is an algorithm that takes as input a set of samples of a function and a tolerance epsilon and produces as output a compact representation of a function that is within epsilon of the given samples we show that the existence of an occam approximation is sufficient to guarantee the probably approximate learnability of classes of functions on the reals even in the presence of arbitrarily large but random additive noise one consequence of our results is a general technique for the design and analysis of nonlinear filters in digital signal processing

solvability of the cauchy problem of non isotropic schrodinger equations in sobolev spaces abstract we study the existence of solutions for the cauchy problem of the non isotropically perturbed nonlinear schrodinger equation i u t u u u a u x 1 x 1 x 1 x 1 b u x 1 x 1 x 1 x 1 x 1 x 1 0 where a b are not simultaneously vanishing real constants is a positive constant and x x 1 x 2 r 2 by using kato s method we establish some local existence results for initial data belonging to h s r 2 where s 0 if either b 0 0 3 or b 0 a 0 0 8 3 s 1 3 if b 0 3 and s 1 3 8 if b 0 a 0 8 3 global existence is also established under some additional conditions

hardness randomness tradeoffs for bounded depth arithmetic circuits in this paper we show that lower bounds for bounded depth arithmetic circuits imply derandomization of polynomial identity testing for bounded depth arithmetic circuits more formally if there exists an explicit polynomial f that cannot be computed by a depth d arithmetic circuit of small size then there exists an efficient deterministic black box algorithm to test whether a given depth d 5 circuit that computes a polynomial of relatively small individual degrees is identically zero or not in particular if we are guaranteed that the tested circuit computes a multilinear polynomial then we can perform the identity test efficiently to the best of our knowledge this is the first hardness randomness tradeoff for bounded depth arithmetic circuits the above results are obtained using the arithmetic nisan wigderson generator of kabanets and impagliazzo together with a new theorem on bounded depth circuits which is the main technical contribution of our work this theorem deals with polynomial equations of the form p x 1 dots x n y equiv0 and shows that if p has a circuit of depth d and size s and if the polynomial f x 1 dots x n satisfies p x 1 dots x n f equiv0 then f has a circuit of depth d 3 and size mathrm poly s m r where m is the total degree of f and r is the degree of y in p this circuit for f can be found probabilistically in time mathrm poly s m r in the other direction we observe that the methods of kabanets and impagliazzo can be used to show that derandomizing identity testing for bounded depth circuits implies lower bounds for the same class of circuits more formally if we can derandomize polynomial identity testing for bounded depth circuits then nexp does not have bounded depth arithmetic circuits that is either mathrm nexp not subseteq p mathrm poly or the permanent is not computable by polynomial size bounded depth arithmetic circuits

line detection in images through regularized hough transform the problem of determining the location and orientation of straight lines in images is of great importance in the fields of computer vision and image processing traditionally the hough transform a special case of the radon transform has been widely used to solve this problem for binary images in this paper we pose the problem of detecting straight lines in gray scale images as an inverse problem our formulation is based on use of the inverse radon operator which relates the parameters determining the location and orientation of the lines in the image to the noisy input image the advantage of this formulation is that we can then approach the problem of line detection within a regularization framework and enhance the performance of the hough based line detector through the incorporation of prior information in the form of regularization we discuss the type of regularizers that are useful for this problem and derive efficient computational schemes to solve the resulting optimization problems enabling their use in large applications finally we show how our new approach can be alternatively viewed as one of finding an optimal representation of the noisy image in terms of elements chosen from a dictionary of lines this interpretation relates the problem of hough based line finding to the body of work on adaptive signal representation

fast calculation of multiple line integrals a line integral is defined as the integral of two dimensional data along a one dimensional straight line of given length and orientation line integrals are used in various forms of edge and line detectors in images and in the computation of the radon transform we present a recursive algorithm which enables approximation of discretized line integrals at all lengths orientations and locations to within a prescribed error bound in at most o n log n log log n operations where n is the number of data points furthermore for most applications in particular where even small amounts of noise are present in the data all of these integrals can be computed to the desired accuracy in about 24n log n operations

fast inversion of the radon transform using log polar coordinates and partial back projections in this paper a novel filtered back projection algorithm for inversion of a discretized radon transform is presented it makes use of invariance properties possessed by both the radon transform and its dual by switching to log polar coordinates both operators can be expressed in a displacement invariant manner explicit expressions for the corresponding transfer functions are calculated furthermore by dividing the back projection into several partial back projections inversion can be performed by means of finite convolutions and hence implemented by an fft algorithm in this way a fast and accurate reconstruction method is obtained

well posedness of cauchy problem of fourth order schrodinger equations in anisotropic sobolev space abstract we study the existence of solutions for the cauchy problem of the non isotropically perturbed nonlinear schrodinger equation iu t u a i 1 d u x i x i x i x i u u 0 where a is a real number a 0 is a positive constant 1 d n and x x 1 x 2 x n r n t r by using kato s method we establish local solvability results for initial data belonging to w 2 2 1 r n

the global cauchy problem for the non linear klein gordon equation on demontre l unicite de solutions faibles et l existence et l unicite de solutions globales fortement continues dans l espace energie

a fast digital radon transform an efficient means for evaluating the hough transform a fast digital radon transform based on recursively defined digital straight lines is described which has the sequential complexity of n 2 log n additions for an n x n image this transform can be used to evaluate the hough transform to detect straight lines in a digital image whilst a parallel implementation of the hough transform algorithm is difficult because of global memory access requirements the fast digital radon transform is vectorizable and therefore well suited for parallel computation the structure of the fast algorithm is shown to be quite similar to the fft algorithm for decimation in frequency it is demonstrated that even for sequential computation the fast radon transform is an attractive alternative to the classical hough transform algorithm

a framework for discrete integral transformations ii the 2d discrete radon transform although naturally at the heart of many fundamental physical computations and potentially useful in many important image processing tasks the radon transform lacks a coherent discrete definition for two dimensional 2d discrete images which is algebraically exact invertible and rapidly computable we define a notion of 2d discrete radon transforms for 2d discrete images which is based on summation along lines of absolute slope less than 1 values at nongrid locations are defined using trigonometric interpolation on a zero padded grid our definition is shown to be geometrically faithful the summation avoids wrap around effects our proposal uses a special collection of lines in mathbb r 2 for which the transform is rapidly computable and invertible we describe a fast algorithm using o n log n operations where n n 2 is the number of pixels in the image the fast algorithm exploits a discrete projection slice theorem which associates the discrete radon transform with the pseudopolar fourier transform a averbuch et al siam j sci comput 30 2008 pp 764 784 our definition for discrete images converges to a natural continuous counterpart with increasing refinement

primality and identity testing via chinese remaindering we give a simple and new randomized primality testing algorithm by reducing primality testing for number n to testing if a specific univariate identity over zn holds we also give new randomized algorithms for testing if a multivariate polynomial over a finite field or over rationals is identically zero the first of these algorithms also works over zn for any n the running time of the algorithms is polynomial in the size of arithmetic circuit representing the input polynomial and the error parameter these algorithms use fewer random bits and work for a larger class of polynomials than all the previously known methods for example the schwartz zippel test schwartz 1980 zippel 1979 chen kao and lewin vadhan tests chen and kao 1997 lewin and vadhan 1998

fast tomographic reconstruction via rotation based hierarchical backprojection we introduce a family of fast algorithms for tomographic backprojection in the parallel beam geometry the algorithms reduce the computational cost of backprojecting p projections onto an n times n pixel image from the conventional o n 2 p to o n 2 log p the new algorithms aggregate the projections in a hierarchical structure with images in the hierarchy formed by the rotation and addition of other images made up of fewer projections while these algorithms are related to existing fast algorithms this work places them within the signal processing framework providing a systematic means to optimize and adjust the trade off between computational cost and accuracy rotations are performed separably in order that higher order interpolators may be used with low computational cost the same ideas are applied to create a tomographic projection algorithm which computes projections of an n times n pixel image onto p view angles at a cost of o n 2 log p

computational complexity of sparse rational interpolation the authors analyze the computational complexity of sparse rational interpolation and give the first deterministic algorithm for this problem with singly exponential bounds on the number of arithmetic operations

shear based fast hierarchical backprojection for parallel beam tomography we introduce a family of fast algorithms for 2 d parallel beam tomographic backprojection they aggregate the projections in a hierarchical structure involving the shearing and addition of sparsely sampled images the algorithms achieve a computational cost of o n2 logp when backprojecting an ntimesn pixel image from p projections the algorithms provide a systematic means guided by a fourier domain interpretation to adjust and optimize the tradeoff between computational cost and accuracy in an example with n 512 and p 1458 the algorithms provide high accuracy with more than an order of magnitude reduction in operation counts

a fast multilevel domain decomposition algorithm for radar imaging a novel algorithm for radar imaging is presented the method comprises two steps first a decomposition of the radar data domain into subdomains and computation of pertinent low resolution images second interpolation phase correction and aggregation of the low resolution images into the final high resolution one a multilevel algorithm is formulated via a recursive application of the domain decomposition and image aggregation steps the computational cost of the proposed algorithm is comparable to that of the fast fourier transform fft based techniques while it appears to be considerably more flexible than the latter

comments and corrections on the use of polar sampling theorems in ct in two recent papers an exact polar interpolation formula was used to reconstruct computer tomography ct imagery by a procedure known as direct fourier transform inversion the resulting imagery compared favorably to ct imagery reconstructed by filtered convolution back projection strictly speaking however the interpolation formula was inappropriately used since it is valid only for an odd number of azimuthal samples while in ct one uses an even number of samples when the number of samples n is large say 200 as in ct the error is not noticeable and the appropriateness of the formula has no practical significance however when n is small a large error can result we derive in this paper exact azimuthal interpolation formulas for n even and arbitrary n

authors reply to comments on direct fourier reconstruction in computer tomography in their comments fan and sanz discuss an interpolation formula used by us in our papers on direct fourier transform tomography 1 2 they raise questions relating to 1 the relation between our work and others and 2 the optimality of the interpolation procedure in direct fourier reconstruction in ct we show that 1 most of their points are irrelevant and 2 other points follow from an error which was already corrected by the authors in 3 prior to the publication of the comments

o n sup 2 log sub 2 n filtered backprojection reconstruction algorithm for tomography we present a new fast reconstruction algorithm for parallel beam tomography the new algorithm is an accelerated version of the standard filtered backprojection fbp reconstruction and uses a hierarchical decomposition of the backprojection operation to reduce the computational cost from o n sup 3 to o n sup 2 log sub 2 n we discuss the choice of the various parameters that affect the behavior of the algorithm and present numerical studies that demonstrate the cost versus distortion tradeoff comparisons with fourier reconstruction algorithms and a multilevel inversion algorithm by brandt et al both of which also have o n sup 2 log sub 2 n cost suggest that the proposed hierarchical scheme has a superior cost versus distortion performance it offers rms reconstruction errors comparable to the fbp with considerable speedup for an example with a 512 spl times 512 pixel image and 1024 views the speedup achieved with a particular implementation is over 40 fold with reconstructions visually indistinguishable from the fbp

on some approximation problems concerning sparse polynomials over finite fields abstract we obtain new lower bounds on the number of non zeros of sparse polynomials and give a fully polynomial time e approximation algorithm for the number of non zeros of multivariate sparse polynomials over a finite field of q elements and degree less than q 1 this partially answers an open problem of d grigoriev and m karpinski also probabilistic and deterministic algorithms for testing identity to zero of a sparse polynomial given by a black box are given finally we propose an algorithm to estimate the size of the image of a univariate sparse polynomial

a fast and accurate multilevel inversion of the radon transform a number of imaging technologies reconstruct an image function from its radon projection using the convolution backprojection method the convolution is an o n2 log n algorithm where the image consists of n x n pixels while the backprojection is an o n3 algorithm thus constituting the major computational burden of the convolution backprojection method an o n2 log n multilevel backprojection method is presented here when implemented with a fourier domain postprocessing technique also presented here the resulting image quality is similar or superior to the image quality of the classical backprojection technique

a graph based unified technique for computing and representing coefficients over finite fields this paper presents the generalized theory and an efficient graph based technique for the calculation and representation of coefficients of multivariate canonic polynomials over arbitrary finite fields in any polarity the technique presented for computing coefficients is unlike polynomial interpolation or matrix based techniques and takes into consideration efficient graph based forms which can be available as an existing resource during synthesis verification or simulation of digital systems techniques for optimization of the graph based forms for representing the coefficients are also presented the efficiency of the algorithm increases for larger fields as a test case the proposed technique has been applied to benchmark circuits over gf 2m the experimental results show that the proposed technique can significantly speed up execution time

nonlinear scattering theory for a class of wave equations in hs abstract for the semi linear higher order wave equation and the nonlinear higher order schrodinger equation we show that the scattering operators map a band in h s into h s if the nonlinearities have sub critical powers in h s the smoothness of the scattering operators and the uniform boundedness of strong solutions for the defocusing nls equation are also shown provided that the nonlinearities have subcritical growth in h 1 moreover the spatial decaying behavior of solutions in energy space for the defocusing nls equation are obtained

on critical cases of sobolev s inequalities abstract we present a new form of the trudinger type inequality which shows an explicit dependence moreover we give an alternative proof of the brezis gallouet wainger inequality

isometric decomposition operators function spaces ep qλ and applications to nonlinear evolution equations abstract by using the isometric decomposition to the frequency spaces we will introduce a new class of function spaces e p q which is a subspace of gevrey 1 class g 1 r n c r n for 0 and we will study the cauchy problem for the nonlinear schrodinger equation the complex ginzburg landau equation and the navier stokes equation some well posed results are obtained for the cauchy data in e 2 1 0 and the regularity behavior in e 2 1 ct g 1 r n for the complex ginzburg landau equation and the navier stokes equation is also obtained as time t 0

discrete radon transform has an exact fast inverse and generalizes to operations other than sums along lines gotz druckmuller and independently brady have defined a discrete radon transform drt that sums an image9s pixel values along a set of aptly chosen discrete lines complete in slope and intercept the transform is fast o n 2 log n for an n n image it uses only addition not multiplication or interpolation and it admits a fast exact algorithm for the adjoint operation namely backprojection this paper shows that the transform additionally has a fast exact although iterative inverse the inverse reproduces to machine accuracy the pixel by pixel values of the original image from its drt without artifacts or a finite point spread function fourier or fast fourier transform methods are not used the inverse can also be calculated from sampled sinograms and is well conditioned in the presence of noise also introduced are generalizations of the drt that combine pixel values along lines by operations other than addition for example there is a fast transform that calculates median values along all discrete lines and is able to detect linear features at low signal to noise ratios in the presence of pointlike clutter features of arbitrarily large amplitude

nonlinear schrodinger equations in the sobolev space of critical order the cauchy problem for the nonlinear schrodinger equations is considered in the sobolev spacehn 2 rn of critical ordern 2 where the embedding intol rn breaks down and any power behavior of interaction works as a subcritical nonlinearity under the interaction of exponential type the existence and uniqueness is proved for globalhn 2 solutions with small cauchy data

fast fully 3 d image reconstruction in pet using planograms we present a method of performing fast and accurate three dimensional 3 d backprojection using only fourier transform operations for line integral data acquired by planar detector arrays in positron emission tomography this approach is a 3 d extension of the two dimensional 2 d linogram technique of edholm by using a special choice of parameters to index a line of response lor for a pair of planar detectors rather than the conventional parameters used to index a lor for a circular tomograph all the lors passing through a point in the field of view fov lie on a 2 d plane in the four dimensional 4 d data space thus backprojection of all the lors passing through a point in the fov corresponds to integration of a 2 d plane through the 4 d planogram the key step is that the integration along a set of parallel 2 d planes through the planogram that is backprojection of a plane of points can be replaced by a 2 d section through the origin of the 4 d fourier transform of the data backprojection can be performed as a sequence of fourier transform operations for faster implementation in addition we derive the central section theorem for planogram format data and also derive a reconstruction filter for both backprojection filtering and filtered backprojection reconstruction algorithms with software based fourier transform calculations we provide preliminary comparisons of planogram backprojection to standard 3 d backprojection and demonstrate a reduction in computation time by a factor of approximately 15

sparse interpolation of multivariate rational functions consider the black box interpolation of a sparse n variate rational function f where is the maximum number of terms in either numerator or denominator when numerator and denominator are at most of degree d then the number of possible terms in f is o dn and explodes exponentially as the number of variables increases the complexity of our sparse rational interpolation algorithm does not depend exponentially on n anymore it still depends on d because we densely interpolate univariate auxiliary rational functions of the same degree we remove the exponent n and introduce the sparsity in the complexity by reconstructing the auxiliary function s coefficients via sparse multivariate interpolation the approach is new and builds on the normalization of the rational function s representation our method can be combined with probabilistic and deterministic components from sparse polynomial black box interpolation to suit either an exact or a finite precision computational environment the latter is illustrated with several examples running from exact finite field arithmetic to noisy floating point evaluations in general the performance of our sparse rational black box interpolation depends on the choice of the employed sparse polynomial black box interpolation if the early termination ben or tiwari algorithm is used our method achieves rational interpolation in o d black box evaluations and thus is sensitive to the sparsity of the multivariate f

a computational study of reconstruction algorithms for diffraction tomography interpolation versus filtered backpropagation from the standpoint of reporting a new contribution this paper shows that by using bilinear interpolation followed by direct two dimensional fourier inversion one can obtain reconstructions of quality which is comparable to that produced by the filtered backpropagation algorithm proposed recently by devaney for an n n image reconstructed from n diffracted projections the former approach requires approximately 4n fft s whereas the backpropagation technique requires approximately n2fft s we have also taken this opportunity to present the reader with a tutorial introduction to diffraction tomography an area that is becoming increasingly important not only in medical imaging but also in underwater and seismic mapping with microwaves and sound the main feature of the tutorial part is the statement of the fourier diffraction projection theorem which is an extension of the traditional fourier slice theorem to the case of image formation with diffracting illumination

the solutions for nonlinear schrodinger equations abstract we consider the nonlinear schrodinger equation i u t u f u 0 f u u 2 m u n m n n x t r n r u 0 x u 0 x we give a formal solution of the cauchy problem by this formal solution we obtain that there exists a constant b independent of p n such that for any initial data u 0 h s b 0 s n 2 2 m n 1 4 n 2 s the schrodinger equation has a unique global solution u l r b p 2 s moreover when m 0 we give a unique weak solution of the cauchy problem in d distribution

the gridding method for image reconstruction by fourier transformation the authors explore a computational method for reconstructing an n dimensional signal f from a sampled version of its fourier transform f spl circ the method involves a window function w spl circ and proceeds in three steps first the convolution g spl circ w spl circ f spl circ is computed numerically on a cartesian grid using the available samples of f spl circ then g wf is computed via the inverse discrete fourier transform and finally f is obtained as g w due to the smoothing effect of the convolution evaluating w spl circ f spl circ is much less error prone than merely interpolating f spl circ the method was originally devised for image reconstruction in radio astronomy but is actually applicable to a broad range of reconstructive imaging methods including magnetic resonance imaging and computed tomography in particular it provides a fast and accurate alternative to the filtered backprojection the basic method has several variants with other applications such as the equidistant resampling of arbitrarily sampled signals or the fast computation of the radon hough transform

a deterministic multivariate interpolation algorithm for small finite fields we present a new multivariate interpolation algorithm over arbitrary fields which is primarily suited for small finite fields given function values at arbitrary t points we show that it is possible to find an n variable interpolating polynomial with at most t terms using the number of field operations that is polynomial in t and n the algorithm exploits the structure of the multivariate generalized vandermonde matrix associated with the problem relative to the univariate interpolation only the minimal degree selection of terms cannot be guaranteed and several term selection heuristics are investigated toward obtaining low degree polynomials the algorithms were applied to obtain reed muller and related transforms for incompletely specified functions

on the cauchy problem for the nls equations in r1 1 abstract in this paper we consider the local and global solutions for the nonlinear schrodinger equation with data in the homogeneous and nonhomogeneous besov space and the scattering result for small data the techniques to be used are adapted from the littlewood paley trichotomy the strichartz type estimate kato s smoothing effect and the maximal function in time estimate for the schrodinger equation

comments on direct fourier reconstruction in computer tomography in this correspondence several theoretical aspects of direct fourier reconstruction in cat as presented in 1 3 are revisited specifically the following points are discussed 1 a lack of appropriate references to analogous previous work in the theory of the proposed interpolation technique and its application in physics and engineering areas 1 2 some extensions and remarks on well known trigonometric interpolation formulas 1 3 and 3 some ambiguities related to the optimality of the reconstruction technique and the exactness of the interpolation procedure 1

direct fourier reconstruction in computer tomography direct fourier transform ft reconstruction of images in computerized tomography ct is not widely used because of the difficulty of precisely interpolating from polar to cartesian samples in this paper an exact interpolation scheme is proposed which in practice can be approached with arbitrary accuracy using well conditioned algorithms several features of the direct ft method are discussed a method that allows angular band limiting of the data before processing to avoid angular aliasing artifacts in the reconstructed image is discussed and experimentally verified the experimental results demonstrate the feasibility of direct ft reconstruction of ct data

learning read once formulas with queries a read once formula is a boolean formula in which each variable occurs at most once such formulas are also called formulas or boolean trees this paper treats the problem of exactly identifying an unknown read once formula using specific kinds of queries the main results are a polynomial time algorithm for exact identification of monotone read once formulas using only membership queries and a polynomial time algorithm for exact identification of general read once formulas using equivalence and membership queries a protocol based on the notion of a minimally adequate teacher 1 the results of the authors improve on valiant s previous results for read once formulas 26 it is also shown that no polynomial time algorithm using only membership queries or only equivalence queries can exactly identify all read once formulas

line detection as an inverse problem application to lung ultrasound imaging this paper presents a novel method for line restoration in speckle images we address this as a sparse estimation problem using both convex and non convex optimization techniques based on the radon transform and sparsity regularization this breaks into subproblems which are solved using the alternating direction method of multipliers thereby achieving line detection and deconvolution simultaneously we include an additional deblurring step in the radon domain via a total variation blind deconvolution to enhance line visualization and to improve line recognition we evaluate our approach on a real clinical application the identification of b lines in lung ultrasound images thus an automatic b line identification method is proposed using a simple local maxima technique in the radon transform domain associated with known clinical definitions of line artefacts using all initially detected lines as a starting point our approach then differentiates between b lines and other lines of no clinical significance including z lines and a lines we evaluated our techniques using as ground truth lines identified visually by clinical experts the proposed approach achieves the best b line detection performance as measured by the f score when a non convex ell text p regularization is employed for both line detection and deconvolution the f scores as well as the receiver operating characteristic roc curves show that the proposed approach outperforms the state of the art methods with improvements in b line detection performance of 54 40 and 33 for text f 0 5 text f 1 and text f 2 respectively and of 24 based on roc curve evaluations

fast parallel algorithms for sparse multivariate polynomial interpolation over finite fields the authors consider the problem of reconstructing i e interpolating a t sparse multivariate polynomial given a black box which will produce the value of the polynomial for any value of the arguments it is shown that if the polynomial has coefficients in a finite field gf q and the black box can evaluate the polynomial in the field gf q ulcorner 2 log q nt 3 urcorner where n is the number of variables then there is an algorithm to interpolate the polynomial in o log 3 nt boolean parallel time and o n 2 t 6 log 2 nt processors this algorithm yields the first efficient deterministic polynomial time algorithm and moreover boolean nc algorithm for interpolating t sparse polynomials over finite fields and should be contrasted with the fact that efficient interpolation using a black box that only evaluates the polynomial at points in gf q is not possible cf m clausen a dress j grabmeier and m karpinski theoret comput sci 1990 to appear this algorithm tog

zero testing of p adic and modular polynomials we obtain new algorithms for testing whether a given by a black box multivariate polynomial over p adic fields given by a black box is identical to zero we also remark on the zero testing of polynomials in residue rings our results complement a known results on the zero testing of polynomials over the integers the rationals and over finite fields

existence of solutions for schrodinger evolution equations we study the existence uniqueness and regularity of the solution of the initial value problem for the time dependent schrodinger equationi u t 1 2 u v t x u u 0 u0 we provide sufficient conditions onv t x such that the equation generates a unique unitary propagatoru t s and such thatu t s u0ec1 l2 c0 h2 n foru0eh2 n the conditions are general enough to accommodate moving singularities of type x 2 n 4 or x n 2 n 3

learning functions represented as multiplicity automata we study the learnability of multiplicity automata in angluin s exact learning model and we investigate its applications our starting point is a known theorem from automata theory relating the number of states in a minimal multiplicity automaton for a function to the rank of its hankel matrix with this theorem in hand we present a new simple algorithm for learning multiplicity automata with improved time and query complexity and we prove the learnability of various concept classes these include among others the class of disjoint dnf and more generally satisfy o 1 dnf the class of polynomials over finite fields the class of bounded degree polynomials over infinite fields the class of xor of terms certain classes of boxes in high dimensions in addition we obtain the best query complexity for several classes known to be learnable by other methods such as decision trees and polynomials over gf 2 while multiplicity automata are shown to be useful to prove the learnability of some subclasses of dnf formulae and various other classes we study the limitations of this method we prove that this method cannot be used to resolve the learnability of some other open problems such as the learnability of general dnf formulas or even k term dnf for k log n or satisfy s dnf formulas for s 1 these results are proven by exhibiting functions in the above classes that require multiplicity automata with super polynomial number of states

well posedness of the cauchy problem of high dimension non isotropic fourth order schrodinger equations in sobolev spaces abstract we study the well posedness for the cauchy problem of the non isotropically perturbed nonlinear schrodinger equation i u t u u u a i 1 d u x i x i x i x i 0 where a is a real constant 1 d n is an integer is a positive constant and x x 1 x 2 x n r n by using kato s method we establish some local and global existence results for initial data belonging to h s r n where s 0 if 0 8 2 n d s n 2 1 8 2 n d if 8 2 n d

ct reconstruction from parallel and fan beam projections by a 2 d discrete radon transform the discrete radon transform drt was defined by abervuch et al as an analog of the continuous radon transform for discrete data both the drt and its inverse are computable in o n2logn operations for images of size n n in this paper we demonstrate the applicability of the inverse drt for the reconstruction of a 2 d object from its continuous projections the drt and its inverse are shown to model accurately the continuum as the number of samples increases numerical results for the reconstruction from parallel projections are presented we also show that the inverse drt can be used for reconstruction from fan beam projections with equispaced detectors

continuous dependence of cauchy problem for nonlinear schrodinger equation in hs abstract we consider the cauchy problem for the nonlinear schrodinger equation i t u u 0 u 1 u u in r n where 0 1 c in h s subcritical and critical case 0 4 n 2 s when 1 s n 2 and 0 when s n 2 we show that the solution depends continuously on the initial value in the standard sense in h s r n if satisfies certain assumptions

multilevel matrix multiplication and fast solution of integral equations abstract a fast multigrid approach is described for the task of calculating k x y u y dy for each x r d discretizing by an equidistant grid with n points and meshsize h and approximating the integrations to o h2s accuracy it is shown that the complexity of this calculation can be reduced from o n2 to o sn provided the kernel k is sufficiently smooth for potential type kernels the complexity is reduced to o sn log n corresponding integral equations can be solved to a similar accuracy in basically the same amount of work using a special kind of distributed relaxation in a multigrid algorithm one and two dimensional numerical tests and theoretical derivations of optimal strategies are reported the method is applicable to the task of multiplying by any matrix with appropriate smoothness properties including most types of many body interactions