on the asymptotic convergence of b spline wavelets to gabor functions a family of nonorthogonal polynomial spline wavelet transforms is considered these transforms are fully reversible and can be implemented efficiently the corresponding wavelet functions have a compact support it is proven that these b spline wavelets converge to gabor functions modulated gaussian pointwise and in all l sub p norms with 1

preliminary study on helical ct algorithms for patient motion estimation and compensation helical computed tomography helical spiral ct has replaced conventional ct in many clinical applications in current helical ct a patient is assumed to be rigid and motionless during scanning and planar projection sets are produced from raw data via longitudinal interpolation however rigid patient motion is a problem in some cases such as in the skull base and temporal bone imaging motion artifacts thus generated in reconstructed images can prevent accurate diagnosis modeling a uniform translational movement the authors address how patient motion is ascertained and how it may be compensated first mismatch between adjacent fan beam projections of the same orientation is determined via classical correlation which is approximately proportional to the patient displacement projected onto an axis orthogonal to the central ray of the involved fan beam then the patient motion vector the patient displacement per gantry rotation is estimated from its projections using a least square root method to suppress motion artifacts adaptive interpolation algorithms are developed that synthesize full scan and half scan planar projection data sets respectively in the adaptive scheme the interpolation is performed along inclined paths dependent upon the patient motion vector the simulation results show that the patient motion vector can be accurately and reliably estimated using the authors correlation and least square root algorithm patient motion artifacts can be effectively suppressed via adaptive interpolation and adaptive half scan interpolation is advantageous compared with its full scan counterpart in terms of high contrast image resolution

a wavelet shrinkage approach to tomographic image reconstruction abstract a method is proposed for reconstructing images from tomographic data with respect to a two dimensional wavelet basis the wavelet vaguelette decomposition wvd is used as a framework within which expressions for the necessary wavelet coefficients may be derived these coefficients are calculated using a version of the filtered back projection algorithm as a computational tool in a multiresolution fashion the necessary filters are defined in terms of the underlying wavelets denoising is accomplished through an adaptation of the wavelet shrinkage ws approach of donoho et al and amounts to a form of regularization combining these two steps yields the proposed wvd ws reconstruction algorithm which is compared to the traditional filtered backprojection method in a small study

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

discretization of the radon transform and of its inverse by spline convolutions we present an explicit formula for b spline convolution kernels these are defined as the convolution of several b splines of variable widths h sub i and degrees n sub i we apply our results to derive spline convolution based algorithms for two closely related problems the computation of the radon transform and of its inverse first we present an efficient discrete implementation of the radon transform that is optimal in the least squares sense we then consider the reverse problem and introduce a new spline convolution version of the filtered back projection algorithm for tomographic reconstruction in both cases our explicit kernel formula allows for the use of high degree splines these offer better approximation performance than the conventional lower degree formulations e g piecewise constant or piecewise linear models we present multiple experiments to validate our approach and to find the parameters that give the best tradeoff between image quality and computational complexity in particular we find that it can be computationally more efficient to increase the approximation degree than to increase the sampling rate

on board selection of relevant images an application to linear feature recognition we propose an on board selection scheme for aerial and space images based on linear feature detection in a feature hyperspace the detection task is performed by means of the radon transform rt and the wavelet transform a fast algorithm for the rt computation is described and counteractions against the discretization errors are proposed a new wavelet based algorithm is introduced which performs a fine analysis of the waveforms of the rt peaks yielding a possibly error free detection in images corrupted by a high level of noise a technique based on the feature hyperspace is proposed able to significantly exploit all the available pieces of information on these peaks results of the tests on synthetic and real images are reported which show that this method achieves satisfactory results making the detection task highly reliable in the presence of both noise and clutter

dynamical irreducibility of polynomials modulo primes for a class of polynomials f in mathbb z x which in particular includes all quadratic polynomials and also trinomials of some special form we show that under some natural conditions necessary for quadratic polynomials the set of primes p such that all iterations of f are irreducible modulo p is of relative density zero furthermore we give an explicit bound on the rate of the decay of the density of such primes in an interval 1 q as q rightarrow infty for this class of polynomials this gives a more precise version of a recent result of ferraguti proc am math soc 146 2773 2784 2018 which applies to arbitrary polynomials but requires a certain assumption about their galois group furthermore under the generalised riemann hypothesis we obtain a stronger bound on this density

the interior radon transform the interior radon transform arises from a limited data problem in computerized tomography when only rays travelling through a specified region of interest are measured this problem occurs due to technical restrictions of the sampling apparatus or in an endeavour to reduce the x ray dose the corresponding operator mathcal r i is investigated as a mapping between weighted l 2 spaces the main result is a singular value decomposition svd for this operator for functions of unbounded support in mathbb r 2 the proof is based on the construction of intertwining differential operators the techniques used are unified in the sense that svds for other radon transforms with rotational symmetry can easily be derived in the same way consistency conditions are also obtained for mathcal r i for functions of bounded and unbounded support many of the results generalize to higher dimensions they are stated whenever they follow directly from the two dimensional case

regularization in tomographic reconstruction using thresholding estimators in tomographic medical devices such as single photon emission computed tomography or positron emission tomography cameras image reconstruction is an unstable inverse problem due to the presence of additive noise a new family of regularization methods for reconstruction based on a thresholding procedure in wavelet and wavelet packet wp decompositions is studied this approach is based on the fact that the decompositions provide a near diagonalization of the inverse radon transform and of prior information in medical images a wp decomposition is adaptively chosen for the specific image to be restored corresponding algorithms have been developed for both two dimensional and full three dimensional reconstruction these procedures are fast noniterative and flexible numerical results suggest that they outperform filtered back projection and iterative procedures such as ordered subset expectation maximization

nombre de points de hauteur bornee sur les surfaces de del pezzo de degre 5 we state manin s conjecture in the particular case of the split del pezzo s surfaces of degree 5 over q we show that for an open set u v nu q b card p u q h p b c b log b 4 b the constant c is the one conjectured by e peyre sommaire

multiresolution tomographic reconstruction using wavelets shows how the separable two dimensional wavelet representation leads naturally to an efficient multiresolution tomographic reconstruction algorithm this algorithm is similar to the conventional filtered backprojection algorithm except that the filters are now angle dependent and the backprojection gives the wavelet coefficients of the reconstruction which are then used to synthesize the reconstruction at various resolution levels by reconstructing only a small localized region at high resolution the authors show how radiation exposure and computation can be significantly reduced compared to a standard reconstruction

wavelet sampling and localization schemes for the radon transform in two dimensions two theorems are presented for wavelet decompositions of the two dimensional radon transform the rst theorem establishes an upper error bound in l 2 norm between the radon transform and its wavelet approximation whose coecients at dierent scales are estimated from radon data acquired at corresponding sampling rates the second theorem gives an estimate of the accuracy of a local image reconstructed from localized radon data at multiple levels these results show how to design a multilevel sampling and localization strategy for parallel beam scanning by using wavelet regularity and vanishing moment characteristics clarify the interaction between the wavelet structure and the essential bandwidth of an object image and provide guidelines for wavelet local tomography

density of châtelet surfaces failing the hasse principle ch telet surfaces provide a rich source of geometrically rational surfaces which do not always satisfy the hasse principle restricting attention to a special class of ch telet surfaces we investigate the frequency that such counter examples arise over the rational numbers

mesures et equidistribution sur les espaces de berkovich the proof by ullmo and zhang of bogomolov s conjecture about points of small height in abelian varieties made a crucial use of an equidistribution property for small points in the associated complex abelian variety we study the analogous equidistribution property at p adic places our results can be conveniently stated within the framework of the analytic spaces defined by berkovich the first one is valid in any dimension but is restricted to algebraic metrics the second one is valid for curves but allows for more general metrics in particular to the normalized heights with respect to dynamical systems

equal sums of three powers in a region maxxi b here d will be a fixed positive integer there are 6b o b trivial solutions in which x4 x5 x6 are a permutation of x1 x2 x3 we write n 0 d b for the number of non trivial solutions and our task is to estimate this quantity when d 2 it is a simple exercise with the circle method to show that nd b cb for an appropriate positive constant c so that the non trivial solutions dominate the trivial ones for larger values of d it follows from hua s inequality 11 that

an arithmetic bernstein kusnirenko inequality we present an upper bound for the height of the isolated zeros in the torus of a system of laurent polynomials over an adelic field satisfying the product formula this upper bound is expressed in terms of the mixed integrals of the local roof functions associated to the chosen height function and to the system of laurent polynomials we also show that this bound is close to optimal in some families of examples this result is an arithmetic analogue of the classical bernstein kusnirenko theorem its proof is based on arithmetic intersection theory on toric varieties

new sampling scheme for region of interest tomography the wavelet localization technique was previously applied to the study of region of interest roi tomography it achieves a significant saving in the required projections if only a small region of a tomographic image is of interest in this paper we first show that with the same sampling scheme a simple interpolation applied to the samples can give a result at least as good as that using the original wavelet localization approach it implies that the use of the wavelet transform is not the key to the reduction of the sampling requirement in fact the quality of the reconstructed roi is largely determined by the structure of the sampling scheme rather than directly reducing the projection number the use of the wavelet theory permits a clear understanding of how to achieve a good sampling pattern based on an error analysis using the wavelet theory we further suggest a new sampling scheme such that the number of required projections in each angle is reduced in a multiresolution form a new multiresolution interpolation algorithm is then used to interpolate the missing samples to obtain the full projections as a result more than 84 of projections are saved as compared with the traditional approach in reconstructing an roi of 32 spl times 32 pixels in an image of 256 spl times 256 pixels a series of simulations was performed to reconstruct different sizes of the roi all results show that the signal to error ratios of the reconstructed roi are comparable with that using full projection data set

wavelet based multiresolution local tomography we develop an algorithm to reconstruct the wavelet coefficients of an image from the radon transform data the proposed method uses the properties of wavelets to localize the radon transform and can be used to reconstruct a local region of the cross section of a body using almost completely local data that significantly reduces the amount of exposure and computations in x ray tomography the property that distinguishes our algorithm from the previous algorithms is based on the observation that for some wavelet bases with sufficiently many vanishing moments the ramp filtered version of the scaling function as well as the wavelet function has extremely rapid decay we show that the variance of the elements of the null space is negligible in the locally reconstructed image also we find an upper bound for the reconstruction error in terms of the amount of data used in the algorithm to reconstruct a local region 16 pixels in radius in a 256 spl times 256 image we require 22 of full exposure data

counting rational points on hypersurfaces for any n geq 2 let f in mathbb z x 1 ldots x n be a form of degree d geq 2 which produces a geometrically irreducible hypersurface in mathbb p n 1 this paper is concerned with the number n f b of rational points on f 0 which have height at most b for any varepsilon 0 we establish the estimate n f b o b n 2 varepsilon whenever either n leq 5 or the hypersurface is not a union of lines here the implied constant depends at most upon d n and varepsilon

on compactly supported spline wavelets and a duality principle let v 1 v 0 v 1 be a multiresolution analysis of l 2 generated by the mth order b spline n m x we exhibit a compactly supported basic wavelet m x that generates the corresponding orthogonal complementary wavelet subspaces w 1 w 0 w 1 consequently the two finite sequences that describe the two scale relations of n m x and m x in terms of n m 2x j j z yield an efficient reconstruction algorithm to give an efficient wavelet decomposition algorithm based on these two finite sequences we derive a duality principle which also happens to yield the dual bases n m x j and 59 m x j relative to n m x j and m x j respectively

the curvelet transform for image denoising we describe approximate digital implementations of two new mathematical transforms namely the ridgelet transform and the curvelet transform our implementations offer exact reconstruction stability against perturbations ease of implementation and low computational complexity a central tool is fourier domain computation of an approximate digital radon transform we introduce a very simple interpolation in the fourier space which takes cartesian samples and yields samples on a rectopolar grid which is a pseudo polar sampling set based on a concentric squares geometry despite the crudeness of our interpolation the visual performance is surprisingly good our ridgelet transform applies to the radon transform a special overcomplete wavelet pyramid whose wavelets have compact support in the frequency domain our curvelet transform uses our ridgelet transform as a component step and implements curvelet subbands using a filter bank of a spl grave trous wavelet filters our philosophy throughout is that transforms should be overcomplete rather than critically sampled we apply these digital transforms to the denoising of some standard images embedded in white noise in the tests reported here simple thresholding of the curvelet coefficients is very competitive with state of the art techniques based on wavelets including thresholding of decimated or undecimated wavelet transforms and also including tree based bayesian posterior mean methods moreover the curvelet reconstructions exhibit higher perceptual quality than wavelet based reconstructions offering visually sharper images and in particular higher quality recovery of edges and of faint linear and curvilinear features existing theory for curvelet and ridgelet transforms suggests that these new approaches can outperform wavelet methods in certain image reconstruction problems the empirical results reported here are in encouraging agreement

on codimension one submersions of euclidean spaces it is shown that when n 1 first integrals of a dynamical systemx inr n are known and they are independent at any point ofr n then one can have n 3 that certain orbits ofx are diffeomorphic to circles and others diffeomorphic to straight lines an analytical criterion is also given involving only the first derivatives of the first integrals in order thatall the orbits be diffeomorphic to straight lines therefore the criterion is sufficient in order to avoid the presence of geometrical chaos among the orbits of the dynamical systemx

density of rational points on a quadric bundle in mathbb p 3 times mathbb p 3 we establish an asymptotic formula for the number of rational points of bounded anticanonical height which lie on a certain zariski dense subset of the biprojective hypersurface x1y12 x4y42 0 in p3 p3 this confirms the modified manin conjecture for this variety in which the removal of a thin set of rational points is allowed

a kalman filtering approach to stochastic global and region of interest tomography we define two forms of stochastic tomography in global tomography the goal is to reconstruct an object from noisy observations of all of its projections in region of interest roi tomography the goal is to reconstruct a small portion of an object an roi from noisy observations of its projections densely sampled in and near the roi and sparsely sampled away from the roi we solve both problems by expanding the object and its projections in a circular harmonic fourier series in the angular variable so that the radon transform becomes abel transforms of integer orders applied to the harmonics the algorithm has three major components first we fit state space models to each order of abel transform and thus represent the radon transform operation as a parallel bank of systems each of which computes the appropriate abel transform of a circular harmonic a variable transformation here allows either the global or roi problem to be solved second the object harmonics are modeled as a brownian branch this is a two point boundary value system which is markovianized into a form suitable for the kalman filter finally a parallel bank of kalman smoothing filters independently estimates each circular harmonic from the noisy projection data numerical examples illustrate the proposed procedure

incomplete data problems in x ray computerized tomography in the present paper we study truncated projections for the fanbeam geometry in computerized tomography first we derive consistency conditions for the divergent beam transform then we study a singular value decomposition for the case where only the interior rays in the fan are provided as for example in region of interest tomography we show that the high angular frequency components of the searched for densities are well determined and we present reconstructions from real data where the missing information is approximated based on the singular value decomposition

the hasse norm principle for abelian extensions we study the distribution of abelian extensions of bounded discriminant of a number field k which fail the hasse norm principle for example we classify those finite abelian groups g for which a positive proportion of g extensions of k fail the hasse norm principle we obtain a similar classification for the failure of weak approximation for the associated norm one tori these results involve counting abelian extensions of bounded discriminant with infinitely many local conditions imposed which we achieve using tools from harmonic analysis building on work of wright

rational points of bounded height on general conic bundle surfaces a conjecture of manin predicts the asymptotic distribution of rational points of bounded height on fano varieties in this paper we use conic bundles to obtain correct lower bounds for a wide class of surfaces over number fields for which the conjecture is still far from being proved for example we obtain the conjectured lower bound of manin s conjecture for any del pezzo surface whose picard rank is sufficiently large or for arbitrary del pezzo surfaces after possibly an extension of the ground field of small degree

escape to infinity under the action of a potential and a constant electromagnetic field escape to infinity is proved for a great variety of potentials including the potential created by an infinite number of sources relativistic escape is studied escape in the presence of a constant electromagnetic field and a potential is also considered

manin s conjecture for quartic del pezzo surfaces with a conic fibration an asymptotic formula is established for the number of q rational points of bounded height on a non singular quartic del pezzo surface with a conic bundle structure

quadratic polynomials at prime arguments for a fixed quadratic irreducible polynomial f with no fixed prime factors at prime arguments we prove that there exist infinitely many primes p such that f p has at most 4 prime factors improving a classical result of richert who requires 5 in place of 4 denoting by p n the greatest prime factor of n it is also proved that p f p p 0 847 infinitely often

frobenius fields for elliptic curves let e bbb q be an elliptic curve over the field of rational numbers with mathop rm end bar bbb q e bbb z let k be a fixed imaginary quadratic field over bbb q and x a positive real number for each prime p of good reduction for e let pi p e be a root of the characteristic polynomial of the frobenius endomorphism of e over the finite field bbb f p let pi e k x be the number of primes p leq x such that the field extension bbb q pi p e is the fixed imaginary quadratic field k we present upper bounds for pi e k x obtained using two different approaches the first one inspired from work of serre is to consider the image of frobenius in a mixed galois representation associated to k and to the elliptic curve e the second one inspired from work of cojocaru fouvry and murty is based on an application of the square sieve the bounds obtained using the first approach are better pi e k x ll x 4 5 log x 1 5 and are the best known so far the bounds obtained using the second approach are weaker but are independent of the number field k a property which is essential for other applications all results are conditional upon grh

random diophantine equations i abstract we consider additive diophantine equations of degree k in s variables and establish that whenever s 3 k 2 then almost all such equations satisfy the hasse principle the equations that are soluble form a set of positive density and among the soluble ones almost all equations admit a small solution our bound for the smallest solution is nearly best possible

the density of rational points on curves and surfaces let c be an irreducible projective curve of degree d in mathbb p 3 defined over overline mathbb q it is shown that c has o varepsilon d b 2 d varepsilon rational points of height at most b for any varepsilon 0 uniformly for all curves c this result extends an estimate of bombieri and pila duke math j 59 1989 337 357 to projective curves for a projective surface s in mathbb p 3 of degree d ge 3 it is shown that there are o varepsilon d b 2 varepsilon rational points of height at most b of which at most o varepsilon d b 52 27 varepsilon do not lie on a rational line in s for non singular surfaces one may reduce the exponent to 4 3 16 9d for d 4 or 5 or max 1 3 sqrt d 2 d 1 for d ge 6 even for the surface x 1 d x 2 d x 3 d x 4 d this last result improves on the previous best known as a further application it is shown that almost all integers represented by an irreducible binary form f x y in mathbb z x y have essentially only one such representation this extends a result of hooley j reine angew math 226 1967 30 87 which concerned cubic forms only the results are not restricted to projective surfaces and as an application of other results in the paper it is shown that x 1 x 2 x 3 in mathbb n 3 x 1 d x 2 d x 3 d n ll varepsilon d n theta d varepsilon with theta frac 2 sqrt d frac 2 d 1 when d ge 8 this provides the first non trivial bound for the number of representations as a sum of three d th powers

wavelet methods for inverting the radon transform with noisy data because the radon transform is a smoothing transform any noise in the radon data becomes magnified when the inverse radon transform is applied among the methods used to deal with this problem is the wavelet vaguelette decomposition wvd coupled with wavelet shrinkage as introduced by donoho 1995 we extend several results of donoho and others here first we introduce a new sufficient condition on wavelets to generate a wvd for a general homogeneous operator whose class includes the radon transform we show that a variant of donoho s method for solving inverse problems can be derived as the exact minimizer of a variational problem that uses a besov norm as the smoothing functional we give a new proof of the rate of convergence of wavelet shrinkage that allows us to estimate rather sharply the best shrinkage parameter needed to recover an image from noise corrupted data we conduct tomographic reconstruction computations that support the hypothesis that near optimal shrinkage parameters can be derived if one can estimate only two besov space parameters about an image f both theoretical and experimental results indicate that our choice of shrinkage parameters yields uniformly better results than kolaczyk s 1996 variant of donoho s method and the classical filtered backprojection method

on the distribution of squarefree integers in arithmetic progressions we investigate the error term of the asymptotic formula for the number of squarefree integers up to some bound and lying in some arithmetic progression in particular we prove an upper bound for its variance as a varies over mathbb z q mathbb z times which considerably improves upon earlier work of blomer

escape to infinity in a newtonian potential escape to infinity in the presence of a newtonian potential is examined in the classical and relativistic cases

on the hilbert transform of wavelets a wavelet is a localized function having a prescribed number of vanishing moments in this correspondence we provide precise arguments as to why the hilbert transform of a wavelet is again a wavelet in particular we provide sharp estimates of the localization vanishing moments and smoothness of the transformed wavelet we work in the general setting of non compactly supported wavelets our main result is that in the presence of some minimal smoothness and decay the hilbert transform of a wavelet is again as smooth and oscillating as the original wavelet whereas its localization is controlled by the number of vanishing moments of the original wavelet we motivate our results using concrete examples

wavelet localization of the radon transform the authors develop an algorithm which significantly reduces radiation exposure in x ray tomography when a local region of the body is to be imaged the algorithm uses the properties of wavelets to essentially localize the radon transform this algorithm differs from previous algorithms for doing local tomography because it recovers an approximation to the original image not the image module the nullspace of the local tomography operator or the lambda transform of the image this is possible because the authors do not truly invert the interior radon transform but rather sample the radon transform sparsely away from the local region of interest much attention in the field has been directed towards localized tomography the authors believe that this technique represents a significant contribution towards this effort

orthonormal bases of compactly supported wavelets we construct orthonormal bases of compactly supported wavelets with arbitrarily high regularity the order of regularity increases linearly with the support width we start by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction the construction then follows from a synthesis of these different approaches

tomographic reconstruction using nonseparable wavelets in this paper the use of nonseparable wavelets for tomographic reconstruction is investigated local tomography is also presented the algorithm computes both the quincunx approximation and detail coefficients of a function from its projections simulation results showed that nonseparable wavelets provide a reconstruction improvement versus separable wavelets

on the special role of symmetric periodic orbits in a chaotic system abstract after early work of henon it has become folk knowledge that symmetric periodic orbits are of particular importance we reinforce this belief by additional studies and we further find that invariant closed symplectic submanifolds caused by discrete symmetries prove to be an important complement to the long known role of the orbits the latter have particular importance in semi classics based on the structural stability of hyperbolic horseshoes we give an argument that opens an avenue to the understanding of these facts

wada basins and chaotic invariant sets in the henon heiles system the h enon heiles hamiltonian is investigated in the context of chaotic scattering in the range of energies where escaping from the scattering region is possible special attention is paid to the analysis of the different nature of the orbits and the the invariant sets such as the stable and unstable manifolds and the chaotic saddle furthermore a discussion on the average decay time associated to the typical chaotic transients which are present in this problem is presented the main goal of this paper is to show by using various computational methods that the corresponding exit basins of this open hamiltonian are not only fractal but they also verify the more restrictive property of wada we argue that this property is verified by typical open hamiltonian systems with three or more escapes

detection of edges from projections in a number of applications of computerized tomography the ultimate goal is to detect and characterize objects within a cross section detection of edges of different contrast regions yields the required information the problem of detecting edges from projection data is addressed it is shown that the class of linear edge detection operators used on images can be used for detection of edges directly from projection data this not only reduces the computational burden but also avoids the difficulties of postprocessing a reconstructed image this is accomplished by a convolution backprojection operation for example with the marr hildreth edge detection operator the filtering function that is to be used on the projection data is the radon transform of the laplacian of the 2 d gaussian function which is combined with the reconstruction filter simulation results showing the efficacy of the proposed method and a comparison with edges detected from the reconstructed image are presented

feldkamp type cone beam tomography in the wavelet framework x ray computed tomography ct is in transition from fan beam to cone beam geometry for cone beam volumetric imaging reduction of radiation exposure remains an important issue because the wavelet approach was shown to be effective and flexible for two dimensional 2 d local region reconstruction the authors are motivated to perform wavelet local ct in cone beam geometry in this paper they formulate the feldkamp cone beam reconstruction from the wavelet perspective derive both full scan and half scan feldkamp type formulas for either global or local reconstruction and demonstrate the feasibility and utility in synthetic and real data it is found that using the wavelet feldkamp approach a three dimensional 3 d region of interest roi can be reconstructed with neither severe image artifacts nor any significant constant bias in the authors simulation and experiments

on scattering trajectories of dynamical systems dynamical systems on rn presenting geometric chaos i e open domains where bounded and unbounded orbits are intermingled have been constructed the opposite situation open scattering has been studied for integrable hamiltonian and non hamiltonian vector fields and for equations of type x v x when the potential has the form v a r b r f in spherical coordinates

variants of equidistribution in arithmetic progression and the twin prime conjecture let mathcal h h 1 ldots h k be a fixed set of k distinct non negative integers we show that bombieri vinogradov type theorems for a certain class of functions f in arithmetic progressions can be extended to the product f n mu 2 n h 1 cdots mu 2 n h k up to almost the same level of distribution as that for f as an application of this result we show equidistribution of tuples of squarefree integers in arithmetic progressions with a level of distribution up to 2 3 this generalizes a result of orr j number theory 3 474 497 1971 we also formulate arithmetic progression analogues of the chowla conjecture on correlations of the mobius function we are able to prove this for some special cases thereby generalizing a result of siebert and wolke math z 122 4 327 341 1971 we also show the relevance of such conjectures to the twin prime problem

rational points of bounded height on fano varieties a prime pe7z let v be an algebraic variety defined over f and li a metrized line bundle on v i e a system l consisting of a line bundle l and a family of banach v adic metrics on l f for all v s satisfying well known conditions cf 1 chap vi then we can define a height function hl x f i r by the formula h x is x 1 where s is a local section of l non vanishing

a refinement of the bernstein kusnirenko estimate a theorem of ku nirenko and bern tein also known as the bkk theorem shows that the number of isolated solutions in a torus to a system of polynomial equations is bounded above by the mixed volume of the newton polytopes of the given polynomials and this upper bound is generically exact we improve on this result by introducing refined combinatorial invariants of polynomials and a generalization of the mixed volume of convex bodies the mixed integral of concave functions the proof is based on new techniques and results from relative toric geometry

a hankel transform approach to tomographic image reconstruction a relatively unexplored algorithm is developed for reconstructing a two dimensional image from a finite set of its sampled projections the algorithm referred to as the hankel transform reconstruction htr algorithm is polar coordinate based the algorithm expands the polar form fourier transform f r theta of an image into a fourier series in theta calculates the appropriately ordered hankel transform of the coefficients of this series giving the coefficients for the fourier series of the polar form image f p phi resolves this series giving a polar form reconstruction and interpolates this reconstruction to a rectilinear grid the htr algorithm is outlined and it is shown that its performance compares favorably to the popular convolution backprojection algorithm

bifurcation to chaotic scattering abstract we investigate a novel type of bifurcation to chaos which occurs in the context of chaotic scattering in chaotic scattering the deflection angle versus impact parameter is singular on a set of impact parameters which is fractal this behavior is caused by the presence of a chaotic invariant set of unstable bounded orbits in the bifurcation considered here the chaotic set arises abruptly in a sense to be discussed as the particle energy e decreases from above a critical value em to below we call this transition an abrupt bifurcation to fully developed chaotic scattering numerical computation of the dimension d of the chaotic set shows that in agreement with theoretical prediction d 1 in em e 1 near the abrupt bifurcatio

lagrange s four squares theorem with one prime and threealmost prime variables it is conjectured that every sufficiently large integer n equiv 4 pmod 24 should be a sum of the squares of 4 primes the best approximation to this in the literature is the result of brudern and fouvry j reine angew math 454 1994 59 96 who showed that every sufficiently large integer n equiv 4 pmod 24 is a sum of the squares of 4 almost primes each of which has at most 34 prime factors the present paper proves such a result with the square of one prime and 3 almost primes which in this case have at most 101 prime factors each the work of brudern and fouvry was based on kloosterman s approach to representations by quaternary forms but this does not lend itself to situations in which one of the variables is restricted to be a prime instead the present paper works with an almost all result for the representation of numbers m as sums of 3 squares to use this approach one has to take m of the form n p 2 and such numbers are too sparse for the standard theory it is therefore necessary to use an amplification procedure which emphasizes those integers m for which n m is a square all this machinery is coupled with kloosterman s version of the circle method there are considerable technical complications in which bounds for the kloosterman sum play a key role at one point in the argument a saving has to be extracted from a non trivial averaging over the denominators of the farey arcs this is an instance of the second kloosterman refinement

tomographic reconstruction and estimation based on multiscale natural pixel bases we use a natural pixel type representation of an object originally developed for incomplete data tomography problems to construct nearly orthonormal multiscale basis functions the nearly orthonormal behavior of the multiscale basis functions results in a system matrix relating the input the object coefficients and the output the projection data which is extremely sparse in addition the coarsest scale elements of this matrix capture any ill conditioning in the system matrix arising from the geometry of the imaging system we exploit this feature to partition the system matrix by scales and obtain a reconstruction procedure that requires inversion of only a well conditioned and sparse matrix this enables us to formulate a tomographic reconstruction technique from incomplete data wherein the object is reconstructed at multiple scales or resolutions in case of noisy projection data we extend our multiscale reconstruction technique to explicitly account for noise by calculating maximum a posteriori probability map multiscale reconstruction estimates based on a certain self similar prior on the multiscale object coefficients the framework for multiscale reconstruction presented can find application in regularization of imaging problems where the projection data are incomplete irregular and noisy and in object feature recognition directly from projection data

region of interest tomography using exponential radial sampling the authors combine several ideas including nonuniform sampling and circular harmonic expansions into a new procedure for reconstructing a small region of interest roi of an image from a set of its projections that are densely sampled in the roi and coarsely sampled outside the roi specifically the radial sampling density of both the projections and the reconstructed image decreases exponentially with increasing distance from the roi the problem and data are reminiscent of the recently formulated local tomography problem however the authors algorithm reconstructs the roi of the image itself not the filtered version of it obtained using local tomography the new algorithm has the added advantages of speed it can be implemented entirely using the fft and parallelizability each image harmonic is computed independently numerical examples compare the new algorithm to filtered backprojection

counting rational points on quartic del pezzo surfaces with a rational conic upper and lower bounds of the expected order of magnitude are obtained for the number of rational points of bounded height on any quartic del pezzo surface over mathbb q that contains a conic defined over mathbb q

size of regulators and consecutive square free numbers we prove that for almost all d the fundamental solution x 0 y 0 sqrt d of the associated pell equation x2 dy2 1 is greater than d1 749 we also show a strong link between this question and the error term in the asymptotic formula of the number of pairs of consecutive square free numbers

wavelet based reconstruction for limited angle x ray tomography the aim of x ray tomography is to reconstruct an unknown physical body from a collection of projection images when the projection images are only available from a limited angle of view the reconstruction problem is a severely ill posed inverse problem statistical inversion allows stable solution of the limited angle tomography problem by complementing the measurement data by a priori information in this work the unknown attenuation distribution inside the body is represented as a wavelet expansion and a besov space prior distribution together with positivity constraint is used the wavelet expansion is thresholded before reconstruction to reduce the dimension of the computational problem feasibility of the method is demonstrated by numerical examples using in vitro data from mammography and dental radiology