the monogenic signal this paper introduces a two dimensional 2 d generalization of the analytic signal this novel approach is based on the riesz transform which is used instead of the hilbert transform the combination of a 2 d signal with the riesz transformed one yields a sophisticated 2 d analytic signal the monogenic signal the approach is derived analytically from irrotational and solenoidal vector fields based on local amplitude and local phase an appropriate local signal representation that preserves the split of identity i e the invariance equivariance property of signal decomposition is presented this is one of the central properties of the one dimensional 1 d analytic signal that decomposes a signal into structural and energetic information we show that further properties of the analytic signal concerning symmetry energy allpass transfer function and orthogonality are also preserved and we compare this with the behavior of other approaches for a 2 d analytic signal as a central topic of this paper a geometric phase interpretation that is based on the relation between the 1 d analytic signal and the 2 d monogenic signal established by the radon 1986 transform is introduced possible applications of this relationship are sketched and references to other applications of the monogenic signal are given

the invertibility of rotation invariant radon transforms let r denote the radon transform on rn that integrates a function over hyperplanes in given smooth positive measures depending on the hyperplane we characterize the measures for which r is rotation invariant we prove rotation invariant transforms are all one to one and hence invertible on the domain of square integrable functions of compact support l02 rn we prove the hole theorem if f l02rn and r f 0 for hyperplanes not intersecting a ball centered at the origin then f is zero outside of that ball using the theory of fourier integral operators we extend these results to the domain of distributions of compact support on rn our results prove invertibility for a mathematical model of positron emission tomography and imply a hole theorem for the constantly attenuated radon transform as well as invertibility for other radon transforms

moment conditions and support theorems for radon transforms on affine grassmann manifolds abstract let g p n and g q n be the affine grassmann manifolds of p and q planes in r n respectively and let r p q be the radon transform from smooth functions on g p n to smooth functions on g q n arising from the inclusion incidence relation when p q and dim g p n dim g p n we present a range characterization theorem for r p q via moment conditions we then use this range result to prove a support theorem for r p q this complements a previous range characterization theorem for r p q via differential equations when dim g p n dim g p n we also present a support theorem in this latter case

use of radon s projection theory in electromagnetic inverse scattering in a wide variety of electromagnetic profile reconstruction or shape imaging techniques a need often arises to deduce the three two or one dimensional distribution of different physical quantities from their projections e g in radio astronomy structural biology roentgenology geophysics and also in electromagnetic imaging investigation of such problems in various specialized areas resulted in the establishment of the new interdisciplinary subject known as reconstruction from projections the underlying theory which was first rigorously formulated by radon will become of increasing importance to radar target mapping wave imaging and related electromagnetic inverse problems therefore a tutorial exposition is timely and well suited for this issue major emphasis will be placed on showing how ludwig s theorems on support determinacy and self consistency can be used favorably to analyze the data limited reconstruction cases the derived theorems will be applied to the specific case of physical optics far field inverse scattering clearly proving that the bojarski and the kennaugh identities constitute a fourier radon transform pair

reconstruction of a three dimensional object from a limited range of views abstract we define the radon transform r and the back projection r adjoint of r on the space of c rapidly decreasing functions j and on the space of distributions of compact support e we show that r r is a convolution operator the result is true in the case of full views as well as in the case of a limited range of views an application to the medical image reconstruction is given

a survey of the physical optics inverse scattering identity the physical optics approximation is applied to the acoustic and electromagnetic direct scattering integral representation yielding an inverse scattering identity which relates the characteristic function of a scatterer to the three dimensional spatial fourier transform of the augmented far field scattering amplitude this identity requires full scattering information for all frequencies and aspect angles an integral equation for incomplete scattering information for this identity is developed this integral equation is for the unknown characteristic function of the scatterer in terms of the known incomplete scattering information the kernel of this integral equation is the three dimensional spatial fourier transform of the known characteristic function of the scattering information aperture a regularized analytic closed form solution to this integral equation is obtained synthesized numerico experimental results verifying this solution are presented the details of some special cases consisting of a priori knowledge about the scatterer or the scattering information aperture are presented

on the gegenbauer transformation the gegenbauer transformation g k is defined for 1 2 k 0 1 2 by where if being the gegenbauer polynomial of index and degree k and l 0 k is the tchebichef polynomial of degree k the transformation is studied on the spaces l p denned by the norm and its boundedness and range on these spaces is determined and inversion formulae are found

a filter bank for the directional decomposition of images theory and design the authors introduce a directionally oriented 2 d filter bank with the property that the individual channels may be critically sampled without loss of information the passband regions of the component filters are wedge shaped and thus provide directional information it is shown that these filter bank outputs may be maximally decimated to achieve a minimum sample representation in a way that permits the original signal to be exactly reconstructed the authors discuss the theory for directional decomposition and reconstruction in addition implementation issues are addressed where realizations based on both recursive and nonrecursive filters are considered

the non homogeneous biharmonic plate equation fundamental solutions abstract real materials and structural components are often non homogeneous either by design or because of the physical composition and imperfections in the underlying material thus analytical solutions for non homogeneous materials under mechanical loads are of considerable interest to engineers and have widespread applications given the prevalence of these materials in fields as diverse as aerospace construction electronics etc more precisely those are essentially composites with carefully manufactured properties that yield desirable mechanical characteristics and properties such as optimal arrangement of the material minimum weight etc to this end the displacement fundamental solution or green s function corresponding to a point force for the non homogenous biharmonic equation in two dimensions are derived in this work by employing a conformal mapping technique in conjunction with the radon transformation these functions besides being useful in their own right can also be used within the context of integral equation formulations for the solution of boundary value problems finally a series of numerical examples that deal with the non homogeneous plate on elastic foundation problem serve to illustrate the present method

the first suslin homology sheaf of a split simply connected semisimple algebraic group abstract let k be a perfect field and g be a k split simply connected semisimple algebraic group denote k 2 m the second unramified milnor k theory sheaf in the nisnevich topology we prove in this article that the first suslin singular homology sheaf h 1 nis g of the variety g is isomorphic to the direct sum k 2 m d where d is the number of irreducible components of the root system of g

torsion in cohomology of compact lie groups and chow rings of reductive algebraic groups the purpose of the present note is to give a general procedure for calculating the singular cohomology h k ifp with coefficients in an arbitrary finite field ifv of a connected compact lie group k simultaneously we calculate the chow ring a g ifp a g p of the corresponding complex algebraic group g and the degrees of the basic generalized invariants of the weyl group w in arbitrary characteristic p a mysterious connection between the degrees of generators of a g ifp and the kernel of the principal nilpotent element in characteristic p is pointed out the computation of real cohomology of compact lie groups started by e cartan in 1929 was completed for classical groups by r brauer and by l pontryagin in 1935 h hopf and h samelson showed in 1941 that h k f is an exterior algebra on generators of degrees 2 d 1 1 2dr l where l r a n k k but only in 1949 c t yen managed to compute the d for all exceptional groups case by case on the other hand c chevalley and j leray showed that the d i are nothing else but the degrees of the basic invariants over of the weyl group w a simple procedure for calculating the dl was given in the late fifties by a j coleman and by b kostant nowadays this is common knowledge the problem of computing the torsion in h k z proved to be more recalcitrant this was done by several authors c e miller settled so in 1953 23 a borel settled spin ad spin and the quotients of su and sp in 1954 6 p f baum and w browder settled the remaining classical group the semispin group in 1965 4 g 2 and f 4 were settled in 1954 6 and the p 3 torsion for e 6 and p 5tors ion for ev e s in 1961 7 by a borel s araki computed the 2 torsion for e 6 e 7 1 and the 3 torsion for e 7 e s 2 and announced the 2 torsion for e 8 in a joint paper with y shikata in 1961 3 the details of the proof for the last result have appeared only recently 18 there was an error in 1 in the case of ad e 7 which has been fixed in 13

statistical projection completion in x ray ct using consistency conditions projection data incompleteness arises in many situations relevant to x ray computed tomography ct imaging we propose a penalized maximum likelihood statistical sinogram restoration approach that incorporates the helgason ludwig hl consistency conditions to accommodate projection data incompleteness image reconstruction is performed by the filtered backprojection fbp in a second step in our problem formulation the objective function consists of the log likelihood of the x ray ct data and a penalty term the hl condition poses a linear constraint on the restored sinogram and can be implemented efficiently via fast fourier transform fft and inverse fft we derive an iterative algorithm that increases the objective function monotonically the proposed algorithm is applied to both computer simulated data and real patient data we study different factors in the problem formulation that affect the properties of the final fbp reconstructed images including the data truncation level the amount of prior knowledge on the object support as well as different approximations of the statistical distribution of the available projection data we also compare its performance with an analytical truncation artifacts reduction method the proposed method greatly improves both the accuracy and the precision of the reconstructed images within the scan field of view and to a certain extent recovers the truncated peripheral region of the object the proposed method may also be applied in areas such as limited angle tomography metal artifacts reduction and sparse sampling imaging

efficient algorithms for computing the 2 d hexagonal fourier transforms in this paper representations of the two dimensional 2 d signals are presented that reduce the computation of 2 d discrete hexagonal fourier transforms 2 d dhfts the representations are based on the concept of the covering that reveals the mathematical structure of the transforms specifically a set of unitary paired transforms is derived that splits the 2 d dhft into a number of smaller one dimensional 1 d dfts examples for the 8 spl times 4 and 16 spl times 8 hexagonal lattices are described in detail the number of multiplications required for computing the 8 spl times 4 and 16 spl times 8 point dhfts are equal 20 and 136 respectively in the general n spl ges 8 case the number of multiplications required to compute the 2n spl times n point dhft by the paired transforms equals n sup 2 log n 1 n

null spaces of radon transforms we obtain new descriptions of the null spaces of sev eral projectively equivalent transforms in integral geometry the paper deals with the hyperplane radon transform the totally ge odesic transforms on the sphere and the hyperbolic space the spherical slice transform and the cormack quinto spherical mean transform for spheres through the origin the consideration ex tends to the corresponding dual transforms and the relevant exte rior interior modifications the method relies on new results for the gegenbauer chebyshev integrals which generalize abel type fractional integrals on the positive half line

shape reconstruction from po multifrequency scattered fields via the singular value decomposition approach this paper deals with the problem of determining the shape of unknown perfectly conducting infinitely long cylinders starting from the knowledge of the scattered electric far field under the incidence of plane waves with a fixed angle of incidence and varying frequency the problem is formulated as a nonlinear inverse one by searching for a compact support distribution accounting for the objects contour the nonlinear unknown to data mapping is then linearized by means of the kirchhoff approximation which reduces it into a fourier transform relationship then the fourier transform inversion from incomplete data is dealt with by means of the singular value decomposition svd approach and the features of the reconstructable unknowns are investigated finally numerical results confirm the performed analysis

complex nonlinear parameter estimation cnpe and obstacle shape reconstruction abstract the complex nonlinear parameter estimation cnpe formula and a robust algorithm are introduced the formula is simple however it is exact explicit and systematic the algorithm is flexible accurate fast and robust it can be applied to many different complex nonlinear parameter estimation problems as well as complex nonlinear least squares cnls problems the algorithm is applied to an obstacle shape reconstruction problem to show the power of the formula and the algorithm the problem is known to be highly nonlinear and severely ill posed

transgressive loop group extensions a central extension of the loop group of a lie group is called transgressive if it corresponds under transgression to a degree four class in the cohomology of the classifying space of the lie group transgressive loop group extensions are those that can be explored by finite dimensional higher categorical geometry over the lie group we show how transgressive central extensions can be characterized in a loop group theoretical way in terms of loop fusion and thin homotopy equivariance

trees grids and mso decidability from graphs to matroids monadic second order mso logic has proved to be a useful tool in many areas of application reaching flom decidability and complexity to picture processing correctness of programs and parallel processes to characterize the structural borderline between decidability and undecidability is a classical research problem here this problem is related to questions in computational complexity especially to the model checking problem for which many tools developed in the area of decidability have proved to be useful for more than two decades it was conjectured in d seese the structure of the models of decidable monadic theories of graphs ann pure appl logic 53 1991 169 195 that decidability of monadic theories of countable structures implies that the theory can be reduced via interpretability to a theory of trees it is one of the main goals of this article to prove a variant of this conjecture for matroids representable over a finite field matroids can be viewed as a wide generalization of graphs and they seem to capture some second order properties in a more suitable way than graphs themselves cf the recent development in matroid structure theory j f geelen a h m gerards g p whittle branch width and well quasi ordering in matroids and graphs j combin theory ser b 84 2002 270 290 j f geelen a h m gerards n robertson g p whittle excluding a planar graph from a gf q representable matroid manuscript 2003 more exactly we prove for every finite field f that any class of f representable matroids with a decidable mso theory must have uniformly bounded branch width moreover we show that bounding the branch width of all matroids in general is not sufficient to obtain a decidable mso theory our paper gives a rather detailed introduction to these different subjects and shows that a blend of ideas and methods from logic together with structural matroid theory can lead to new tools and algorithms and can shed light on some old open problems

unification of the inverse radon transform in odd and even dimensions it is well known that the inverse radon transform exists in two forms for odd and even dimension functions respectively this note shows that these formulas become zero when the dimension requirement is violated which results in a simple unified formula for the inverse radon transform of arbitrary dimensions

the fourier argand representation an optimal basis of steerable patterns computing the convolution between a 2d signal and a corresponding filter with variable orientations is a basic problem that arises in various tasks ranging from low level image processing e g ridge edge detection to high level computer vision e g pattern recognition through decades of research there still lacks an efficient method for solving this problem in this paper we investigate this problem from the perspective of approximation by considering the following problem what is the optimal basis for approximating all rotated versions of a given bivariate function surprisingly solely minimising the l 2 approximation error leads to a rotation covariant linear expansion which we name fourier argand representation this representation presents two major advantages 1 rotation covariance of the basis which implies a strong steerability rotating by an angle alpha corresponds to multiplying each basis function by a complex scalar e ik alpha 2 optimality of the fourier argand basis which ensures a few number of basis functions suffice to accurately approximate complicated patterns and highly direction selective filters we show the relation between the fourier argand representation and the radon transform leading to an efficient implementation of the decomposition for digital filters we also show how to retrieve accurate orientation of local structures patterns using a fast frequency estimation algorithm

the monadic second order logic of graphs x linear orderings graphs are finite and handled as relational structures we give some answers to the following general questions 1 1 for which classes of graphs c is it possible to specify a linear ordering of the set of vertices of each graph of c by fixed monadic second order formulas 2 2 for which classes of graphs c does there exist an extension l of monadic second order logic such that a subclass l of c is recognizable if and only if it is the class of graphs in c that satisfy a formula of l in this paper recognizability is understood in an algebraic sense relative to a finite set of graph operations and basic graphs that generate all graphs of c 3 3 for which classes of graphs c is it possible to construct in every graph of the class and by fixed formulas of a suitable extension of monadic second order logic its hierarchichal structure i e a finite term written with the operations and basic graphs of 2 that defines the considered graph applications concern dependency graphs of partially commutative words partial k paths cographs and graphs the modular decomposition of which uses prime graphs of bounded size

on the x ray transform of planar symmetric 2 tensors abstract in this paper we study the attenuated x ray transform of 2 tensors supported in convex bounded subsets with sufficiently smooth boundary in the euclidean plane we characterize its range and reconstruct all possible 2 tensors yielding identical x ray data the characterization is in terms of a hilbert transform associated with a analytic maps in the sense of bukhgeim

a fast method for acoustic imaging of multiple three dimensional objects this paper is concerned with the inverse problem for imaging multiple three dimensional objects using the information of the far field pattern of the scattered wave a spatially dependent function which has noticeably different values inside and outside the obstacle is derived a numerical method based on the characterization is developed to obtain a visualization of the obstacle the most remarkable advantage of this method is that it does not need any prior knowledge about the geometry and physical properties of the scatterer and requires only the information of the far field measurements for a finite number of directions of incidence and observation distributed over a limited range furthermore the scheme is very simple and fast since it avoids the use of the iterative procedure and requires only the solution of a linear system some numerical examples with synthetic far field data are given showing the practicality and efficiency of this scheme

wavelet based deconvolution algorithm for time domain near field isar imaging based on correlation receiver in communication theory the measured backscattering waveform of a calibration conducting sphere is adopted as the system impulse response this antenna target system response is then approximated by the derivatives of a gaussian function dgf as well as the coiflets to restore the impulse response of the target and to reconstruct the isar images both the dgf and coiflets are applied to the deconvolution and their results are compared the gaussian and coiflet function parameters are derived by a best fit to the measured time domain system response employing the parametric fits a lowpass filter has been automatically embedded which performs regularization of the conjugate gradient cg optimization and speeds up the cg convergence numerical results show that the image quality from the coiflet system is similar to or better than from the dgf system owing to the continuity smoothness orthogonality and compact support of the coiflets the size of the system matrix has been reduced from 1024 times 1024 for dgf to 261 times 261 and the deconvolution process is accelerated more than 22 fold without sacrificing image characters

gerbal central extensions of reductive groups by k3 we classify central extensions of a reductive group g by mathcal k 3 and b mathcal k 3 the sheaf of third quillen k theory groups and its classifying stack these turn out to be parametrized by the group of weyl invariant quadratic forms on the cocharacter lattice valued in k times and the group of integral weyl invariant cubic forms on the cocharacter lattice respectively

partial radon transforms this article formally defines partial radon transforms for functions of more than two dimensions it shows that a generalized projection slice theorem exists which connects planar and hyperplanar projections of a function to its fourier transform in addition a general theoretical framework is provided for carrying out n dimensional backprojection reconstruction in a multistage fashion through the use of the partial radon transform

propagation estimates for dispersive wave equations application to the stratified wave equation the plane stratified wave equation t2 h 0 with h c y 2 z2 is studied where z x y x rk y r1 and c y c 0 as y solutions to such an equation are solved for the propagation of waves through a layered medium and can include waves which propagate in the x directions only i e trapped modes this leads to a consideration of the pseudo differential wave equation t2 x 0 such that the dispersion relation 2 is analytic and satisfies c1 2 c2 for c 0 uniform propagation estimates like x t e utp 0 dkx c 1 t e 0 dkx are obtained where ut is the evolution group p are projection operators onto the hilbert space of initial conditions h and e is the local energy density in special cases scattering of trapped modes off a local perturbation satisfies the causality estimate p jsp k c for each 1 2 here p j p k are remote outgoing detector incoming transmitter projections for the jth kth trapped mode also r is compact so the projections lo

the monadic second order logic of graphs vii graphs as relational structures abstract relational structures form a unique framework in which various types of graphs and hypergraphs can be formalized and studied we define operations on structures that are compatible with monadic second order logic and that are powerful enough to represent context free graph and hypergraph grammars of various types namely the so called hyperedge replacement c ednce and separated handle rewriting grammars several results concerning monadic second order properties of the generated sets are obtained in a uniform way we also give a logical characterization of the equational sets of structures that generalizes the ones obtained by engelfriet and courcelle for hyperedge replacement and c ednce sets of graphs

on the determination of functions from their integral values along certain manifolds when one integrates a function of two variables x y a point function f p in the plane subject to suitable regularity conditions along an arbitrary straight line g then one obtains in the integral values f g a line function in part a of the present paper the problem which is solved is the inversion of this linear functional transformation that is the following questions are answered can every line function satisfying suitable regularity conditions be regarded as constructed in this way if so is f uniquely known from f and how can f be calculated in part b a solution of the dual problem of calculating a line function f g from its point mean values f p is solved in a certain sense finally in part c certain generalizations are discussed prompted by consideration of non euclidean manifolds as well as higher dimensional spaces the treatment of these problems themselves of interest gains enhanced importance through the numerous relationships that exist between this topic and the theory of logarithmic and newtonian potentials these are mentioned at appropriate places in the text

the monadic second order logic of graphs i recognizable sets of finite graphs abstract the notion of a recognizable set of finite graphs is introduced every set of finite graphs that is definable in monadic second order logic is recognizable but not vice versa the monadic second order theory of a context free set of graphs is decidable

the arason invariant and mod 2 algebraic cycles introduction 73 1 review of the arason invariant 75 2 the special clifford group 76 3 k cohomology of split reductive algebraic groups 78 4 k cohomology of bg 86 5 gl n and cliff n n 92 6 two invariants for clifford bundles 95 7 snaking a bloch ogus differential 100 8 proof of theorem 1 101 9 application to quadratic forms 102 appendix a toral descent 104 appendix b the rost invariant 108 appendix c an amusing example 113 acknowledgements 116 references 116

hierarchical reconstruction using geometry and sinogram restoration the authors describe and demonstrate a hierarchical reconstruction algorithm for use in noisy and limited angle or sparse angle tomography the algorithm estimates an object s mass center of mass and convex hull from the available projections and uses this information along with fundamental mathematical constraints to estimate a full set of smoothed projections the mass and center of mass estimates are made using a least squares estimator derived from the principles of consistency of the radon transform the convex hull estimate is produced by first estimating the positions of support lines of the object from each available projection and then estimating the overall convex hull using prior shape information estimating the position of two support lines from a single projection is accomplished using a generalized likelihood ratio technique for estimating jumps in linear systems results for simulated objects in a variety of measurement situations are shown and several possible extensions to this work are discussed

monadic second order evaluations on tree decomposable graphs abstract every graph generated by a hyperedge replacement graph grammar can be represented by a tree namely the derivation tree of the derivation sequence that produced it certain functions on graphs can be computed recursively on the derivation trees of these graphs by using monadic second order logic and semiring homomorphisms we describe in a single formalism a large class of such functions polynomial and even linear algorithms can be constructed for some of these functions we unify similar results obtained by takamizawa 1982 bern 1987 arnborg 1991 and habel 1989

time domain imaging of radar targets using algorithms for reconstruction from projections a time domain bistatic inverse scattering identity based on the radon transform and the space time magnetic integral equation is derived for real time use in short pulse radar systems it is shown that using a sine modulated exponential pulse siliep and algorithms for reconstruction from projections provides an edge enhanced image a simulation based on stepped frequency multi aspect measurements of aircraft models produces clear images with highly defined edges images using data from a restricted range of aspect angles are also demonstrated

some mathematical considerations in dealing with the inverse problem many problems of mathematical physics can be formulated in terms of the operator equation ax y where a is an integro differential operator given a and x the solution for y is usually straightforward however the inverse problem which consists of the solution for x when given a and y is much more difficult the following questions relative to the inverse problem are explored 1 does specification of the operator a determine the set y for which a solution x is possible 2 does the inverse problem always have a unique solution 3 do small perturbations of the forcing function y always result in small perturbations of the solution 4 what are some of the considerations that enter into the choice of a solution technique for a specific problem the concept of an ill posed problem versus that of a well posed problem is discussed specifically the manner by which an ill posed problem may be regularized to a well posed problem is presented the concepts are illustrated by several examples

the monadic second order logic of graphs xii planar graphs and planar maps we prove that we can specify by formulas of monadic second order logic the unique planar embedding of a 3 connected planar graph if the planar graph is not 3 connected but given with a linear order of its set of edges we can also define a planar embedding by monadic second order formulas we cannot do so in general without the ordering even for 2 connected planar graphs the planar embedding of a graph can be specified by a relational structure called a map which is a graph enriched with a circular ordering of the edges incident with each vertex this circular ordering called a rotation system represents a planar embedding of the neighbourhood of each vertex for each connected map one can define a linear order on its vertices by formulas of monadic second order logic hence we have for planar graphs some kind of equivalence between linear orderings and planar embeddings

shape reconstruction by the spectral data of the far field operator analysis and performances the second version of the linear sampling method is considered with reference to perfectly conducting cylinders and its performances analyzed it is pointed out that for nonconvex objects the shape of the object under test is not retrieved whereas in its place the convex hull is recovered we also point out that in the case of aspect limited data neither the convex hull of the object is reconstructed

a characterization of fourier and radon transforms on euclidean space we show that a continuous operator behaving under rotations positive dilations and translations like the fourier or the radon transform on r must be a constant multiple of one of these transforms we prove this characterization for various function spaces e g we characterize the fourier transform as an operator acting on spaces between 6d r and r on the other hand a counterexample shows that the radon transform is not determined by its behaviour above as an operator from r to s x r but we can characterize the radon trans form as an operator acting between r and s 1 x r the space of integrable distributions on s x r in the special case n our methods sharpen results of j l b cooper and h kober who characterize the fourier transform as an operator from lp r into lp r 1 s p z 2 introduction several authors have been concerned with the question whether an operator on function spaces on r is determined up to a constant multiple by its behaviour under rotations dilations and translations for instance stein 15 showed that the translation invariant hubert and riesz transforms on l2 can be characterized in such a way the fourier and radon transforms on r also have a well known simple be haviour under rotations positive dilations and translations in this article we prove that such a behaviour is characteristic an operator acting under those operations like the fourier or the radon transform is a constant multiple of one of these transforms in the case of the fourier transform for n 1 our methods e g determining homogeneous distributions by euler s equation sharpen results of cooper 1 and kober 10 who characterized by different methods the fourier transform as an operator from lp r into lp r 1 p 2 we also obtain connections to hardy 7 and plancherel 12 who looked for operators on l2 r behaving under dilations like the fourier transform further our characterizations can be used to obtain corresponding ones for operators related to the fourier and radon transforms our result implies the result on hilbert and riesz transforms mentioned above stein 15 we also remark that such characteristic properties can be applied to verify inversion formulas for the corresponding operator in the case of the radon

analytic representations in quantum mechanics various euclidean hyperbolic and elliptic analytic representations are introduced and relations among them are discussed the bargmann analytic representation in the complex plane is considered and its relation to other phase space methods for the harmonic oscillator is reviewed the general theory that relates the growth of analytic functions with the density of their zeros is applied to bargmann functions and it leads to theorems on the completeness of sequences of glauber coherent states two hyperbolic analytic representations in the unit disc based on su 1 1 coherent states and also on phase states are introduced a third analytic representation in the complex plane based on barut girardello states is also considered and transformations which relate it to the other ones are studied in the case of systems with finite dimensional hilbert space an elliptic analytic representation in the extended complex plane and also another analytic representation based on theta functions are introduced the berezin formalism in the euclidean hyperbolic and elliptic cases is discussed contour analytic representations in these three cases are also presented

the monadic second order logic of graphs xiii graph drawings with edge crossings we introduce finite relational structures called sketches that represent edge crossings in drawings of finite graphs we consider the problem of characterizing sketches in monadic second order logic we answer positively the question for framed sketches i e for those representing drawings of graphs consisting of a planar connected spanning subgraph the frame augmented with additional edges that may cross one another and that may cross the edges of the frame we prove the 3 edge theorem stating that a structure of appropriate type with a frame is a sketch if and only if every induced substructure representing the frame and at most 3 edges not in the frame is a sketch

method of paired transforms for reconstruction of images from projections discrete model an effective method of discrete image reconstruction from its projections is introduced the method is based on the vector and paired representations of the two dimensional 2d image with respect to the 2d discrete fourier transform such representations yield algorithms for image reconstruction by a minimal number of attenuation measurements in certain projections the proposed algorithms are described in detail for an n spl times n image when n 2 sup r r 1 the inverse formulas for image reconstruction are given the efficiency of the algorithms is expressed in the fact that they require a minimal number of multiplications or can be implemented without such at all the problem of discrete image reconstruction is also considered in three dimensional 3d space namely on the 3d torus where the reconstruction is performed by means of the nonlinear projections that are integral over 3d spirals on the torus

fredholm integral equation for shape reconstruction of an underwater object using backscattered ramp response signatures inverse scattering problems for identifying underwater objects are of importance for all kinds of applications in seas that occupy more than two thirds of the surface of the earth the physical optics approximation has been applied to the acoustic scattering integral representation yielding an inverse scattering equation which relates the profile function of a submerged object to the fourier transform of the complex backscattered acoustic field this technique is the so called ramp response technique and usually requires the full scattering information for all frequencies to obtain accurate profile function of an object in order to retrieve the shape information of an object from the band limited backscattered frequency data a fredholm integral equation is formulated in this paper for calculating the ramp response of the submerged object this fredholm integral equation is for the unknown ramp response of an object in terms of the known incomplete scattering information the kernel of this integral equ