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

relating h sub 2 and h sub spl infin norm bounds for sampled data systems we relate the h sub spl infin and h sub 2 norms for multi input multi output sampled data feedback control systems where a continuous time plant is controlled by a digital compensator with hold and sampler upper bounds on both h sub 2 and h sub spl infin norms are obtained based on fundamental relations derived by two different approaches namely the hybrid state space approach and the fast sampling and lifting approach

theoretical approach to biological aging abstract we present a model for biological aging that considers the number of individuals whose inherited genotype determines the maximum age for death each individual may die before that age due to some external factor but never after that limit the genotype of the offspring is inherited from the parent with some mutations described by a transition matrix the model can describe different strategies of reproduction and it is exactly soluble we applied our method to the bit string model for aging and the results are in perfect agreement with numerical simulations

comparison of peak and rms gains for discrete time systems abstract a convolution system can have a frequency response which is small for all frequencies yet still greatly amplify the peaks of signals passing though it for finite dimensional systems however we establish the simple bound h 1 2 n 1 h h h where h 1 is the peak gain of the system h h is the maximum frequency response of the system and n its dimension the same result for continuous time systems is due to gohberg and doyle and it is mentioned in 6 the bound implies that h optimal controllers which minimize the maximum of some disturbance to error transfer function cannot have very large peak gains from the disturbance to error

weighted hankel norm approximation calculation of bounds abstract frequency domain bounds are computed for the approximation error involved in the optimum hankel norm approximation procedure the bounds differ from those arising in the unweighted optimal hankel norm approximation procedure by a multiplicative factor depending solely on the weighting function this factor becomes unity when the weighting function is constant

analytical solution of a generalized penna model in 1995 penna introduced a simple model of biological aging a modified penna model has been demonstrated to exhibit behavior of real life systems including catastrophic senescence in salmon and a mortality plateau at advanced ages we present a general steady state analytic solution to the penna model which is able to deal with arbitrary birth and survivability functions this solution is employed to solve standard variant penna models studied by simulation different verhulst factors regulating both the birth rate and external death rate are considered

compensation of some time dependent deformations in tomography this work concerns 2d t dynamic tomography we show that a much larger class of deformations than the affine transforms can be compensated analytically within filtered back projection algorithms in 2d parallel beam and fan beam dynamic tomography we present numerical experiments on the shepp and logan phantom showing that nonaffine deformations can be compensated a generalization to 3d cone beam tomography is proposed

a problem of integral geometry abstract a new analytic method for finding a function from the knowledge of its integrals over the spheres passing through a fixed point is given this problem has been earlier studied by a cormack and e quinto

new relations between norms of system transfer functions it is well known that the h2 norm and h norm of a transfer function can differ arbitrarily since both norms reflect fundamentally different properties however if the pole structure of the transfer function is known it is possible to bound the h norm from above by a constant multiple of the h2 norm it is desirable to compute this constant as tightly as possible in this article we derive a tight bound for the h norm given knowledge of the h2 norm and the poles of a transfer function we compute the bound in closed form for multiple input multiple output transfer functions in continuous and discrete time furthermore we derive a general procedure to compute the bound given a weighted l2 norm

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

all optimal hankel norm approximations of linear multivariable systems and their l error bounds the problem of approximating a multivariable transfer function g s of mcmillan degree n by s of mcmillan degree k is considered a complete characterization of all approximations that minimize the hankel norm is derived the solution involves a characterization of all rational functions s f s that minimize where s has mcmillan degree k and f s is anticavisal the solution to the latter problem is via results on balanced realizations all pass functions and the inertia of matrices all in terms of the solutions to lyapunov equations it is then shown that where k 1 g s is the k l st hankel singular value of g s and for one class of optimal hankel norm approximations the method is not computationally demanding and is applied to a 12 state model

phase transition in a generalized eden growth model on a tree we study analytically the late time statistics of the number of particles in an eden growth model on a tree in this model a cluster grows in continuous time on a binary cayley tree starting from the root by absorbing new particles at the empty perimeter sites at a rate proportional to c l where c is a positive parameter and l is the distance of the perimeter site from the root for c 1 this model corresponds to random binary search trees and for c 2 it corresponds to digital search trees in computer science by introducing a backward fokker planck approach we calculate the mean and the variance of the number of particles at large times and show that the variance undergoes a phase transition at a critical value c sqrt 2 while for c sqrt 2 the variance is proportional to the mean and the distribution is normal for c sqrt 2 the variance is anomalously large and the distribution is non gaussian due to the appearance of extreme fluctuations the model is generalized to one where growth occurs on a tree with m branches and in this more general case we show that the critical point occurs at c sqrt m

a characterization of the radon transform s range by a system of pdes abstract let g be a compactly supported function of d planes in r n we prove that then g is in the range of the radon transform if and only if g satisfies an ultrahyperbolic system of pdes we parameterize the d planes by d 1 points x0 x1 xd on them and get the pde 2 x i k x j 1 2 x i 1 x j k g x 0 x 1 x d vol x i x 0 i 1 d 0 where xik denotes the kth coordinate of x1 at the end we analyze in detail the case of d 1

relating h 2 and h bounds for finite dimensional systems for a linear time invariant system the infinity norm of the transfer function can be used as a measure of the gain of the system this notion of system gain is ideally suited to the frequency domain design techniques such as h infinity optimal control another measure of the gain of a system is the h 2 norm which is often associated with the lqg optimal control problem the only known connection between these two norms is that for discrete time transfer functions the h 2 norm is bounded by the h infinity norm it is shown in this paper that given precise or certain partial knowledge of the poles of the transfer function it is possible to obtain an upper bound of the h infinity norm as a function of the h 2 norm both in the continuous and discrete time cases it is also shown that in continuous time the h 2 norm can be bounded by a function of the h infinity norm and the bandwidth of the system

exact reconstruction filters to invert radon transforms with finite elements abstract the new reconstruction filters introduced in this paper make possible various schemes to invert the radon transform r in r n and the general integral transform over k dimensional planes in r m potential applications include nuclear magnetic resonance and projection pursuit techniques in statistics for the exploration of large data sets the first filter presented here solves a mathematical problem in computed tomography posed by z h cho and restated by l a shepp and j b kruskal the construction of a transfer function whose dual radon transform r reproduces exactly cho s cylindrical approximate dirac measure in r 2 this function then provides the basis for a finite element galerkin rayleigh ritz method to invert the radon transform with least squares estimates of accuracy the generalization of to r n exists but requires singular distributions a second reconstruction filter real analytic avoids this difficulty and its dual radon transform r equals a gaussian approximate dirac measure in r n the smooth function unifies the dirac and fourier approaches with a common convolution kernel and allows a rigorous control of pointwise accuracy and random noise with both a priori and a posteriori estimates of error in both the quadratic and uniform norms finally the same numerical methods extend to invert the k dimensional integral transform in r m

invariant radon transforms on a symmetric space injectivity and support theorems are proved for a class of radon transforms r for u a smooth family of measures defined on a certain space of affine planes in x0 where x0 is the tangent space of a riemannian symmetric space of rank one the transforms are defined by integrating against 8 over these planes we show that if r i f is supported inside a ball of radius r then so is f this is true for f e l2 x0 or f e x x0 furthermore 1 is invertible on either of these domains the main technique is to use facts about spherical harmonics to reduce the problem to a one dimensional integral equation 0 introduction radon transfortns have been extensively studied see for example guillemin 2 helgason 6 and gelfand 1 in this paper we will generalize and unify several results on radon transforms if a compact group k acts on a real vector space x0 then one has the following notion of a k invariant radon transform the ingredients are 0 the space of affine hyperplanes f x 4 ix e c x 60 and a smooth function ul on f the radon transform is defined for f e lc x0 by r lf 4 fxef x u x 4 dx dx is lebesgue measure if rp intertwines the actions of k it is called k invariant this is easily seen to be equivalent to the condition u k x k x for all k e k and x e 4 it is natural to ask if r is invertible and to look for support theorems quinto 11 covers the case when k 0 n and x0 rdn he proves a support theorem and invertibility for k invariant radon transforms also helgason 3 proves support and injectivity theorems in the case u 1 helgason 5 proves a support theorem for the classical radon transform on euclidean space if x0 has a natural complex or quaternionic structure one can play the same game with ewf the space of complex or quaternionic hyperplanes f is either the complex numbers c or the quaternions h quinto 10 covers the case when k u n and x0 cn once again he proves injectivity and a support theorem in this paper we will take x0 to be the tangent space of a rank one riereceived by the editors july 1 1988 1980 mathematics subject classification 1985 revision primary 44a15 secondary 43a85 1990 american mathematical society 0002 9947 90 1 00 25 per page

null spaces and ranges for the classical and spherical radon transforms abstract let r be the classical radon transform that integrates a function over hyperplanes in r n and let sm be the transform that integrates a function over spheres containing the origin in r n we prove continuity results for both transforms and explicitly give the null space of r for a class of square integrable functions on the exterior of a ball in r n as well as the null space of sm for square integrable functions on a ball we show sm l 2 r n l 2 r n is one one and we characterize the range of sm on classes of smooth functions and square integrable functions by certain moment conditions if g x is a schwartz function on r n that is zero to infinite order at x 0 we prove moment conditions sufficient for g to be in the range of sm c r n we apply our results on sm to existence and uniqueness theorems for solutions to a characteristic initial value problem for the darboux partial differential equation

dynamics of the bit string model of age structured populations abstract the dynamics of age structured populations in the framework of the bit string model developed by t j p penna and collaborators is analyzed in terms of the spectral properties of a time evolution generator this is a functional of inclusive properties of the current state of the population which is closely related to the well known leslie operators the analysis of the generator allows for the identification of propagation modes associated with invariant subspaces of the propagator matrix to which in general different gain factors are associated restrictions on the reproductive age span translate into the number structure and relative dominance of the propagation modes the time evolution of semelparous populations is found to be associated in general with oscillatory steady regimes through the appearance of a dominant family of propagation modes with identical gain factors in complete analogy with the semelparous regime of the leslie model explicit results are given and discussed for the case in which one deleterious trait is lethal

a markovian growth dynamics on rooted binary trees evolving according to the gompertz curve inspired by biological dynamics we consider a growth markov process taking values on the space of rooted binary trees similar to the aldous shields probab theory relat fields 79 4 509 542 1988 model fix n 1 and 0 we start at time 0 with the tree composed of a root only at any time each node with no descendants independently from the other nodes produces two successors at rate n k n where k is the distance from the node to the root denote by z n t the number of nodes with no descendants at time t and let t n 1 nln n ln4 ln2 2 we prove that 2 n z n t n n converges to the gompertz curve exp ln2 e we also prove a central limit theorem for the martingale associated to z n t

a variational method using fractional order hilbert spaces for tomographic reconstruction of blurred and noised binary images we provide in this article a refined functional analysis of the radon operator restricted to axisymmetric functions and show that it enjoys strong regularity properties in fractional order hilbert spaces this study is motivated by a problem of tomographic reconstruction of binary axially symmetric objects for which we have available one single blurred and noised snapshot we propose a variational approach to handle this problem consisting in solving a minimization problem settled in adapted fractional order hilbert spaces we show the existence of solutions and then derive first order necessary conditions for optimality in the form of optimality systems

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

gompertz mortality law and scaling behavior of the penna model the penna model is a model of evolutionary ageing through mutation accumulation where traditionally time and the age of an organism are treated as discrete variables and an organism s genome is represented by a binary bit string we reformulate the asexual penna model and show that a universal scale invariance emerges as we increase the number of discrete genome bits to the limit of a continuum the continuum model introduced by almeida and thomas int j mod phys c 11 1209 2000 can be recovered from the discrete model in the limit of infinite bits coupled with a vanishing mutation rate per bit finally we show that scale invariant properties may lead to the ubiquitous gompertz law for mortality rates for early ages which is generally regarded as being empirical

approximative model reconstruction of cascade systems this letter considers how to approximately reconstruct a cascade system from a given unstructured system estimate many system identification methods including subspace methods provide reliable but generally unstructured black box models the problem we consider is how to find cascade systems that are close to such black box models for this we use model matching techniques and optimal weighted hankel norm approximation to obtain accurate low order cascade systems we show that it is possible to bound the reconstruction error in terms of an error tolerance parameter and weighted hankel singular values the suggested methods are illustrated on both a numerical example and a real double tank system with experimental data

singular value decompositions and inversion methods for the exterior radon transform and a spherical transform abstract the classical radon transform r maps an integrable function in rn to its integrals over all n 1 dimensional hyperplanes and the exterior radon transform is the transform r restricted to hyperplanes that do not intersect a given disc a singular value decomposition for the exterior transform is given for spaces of square integrable functions on the exterior of the disc this decomposition in orthogonal functions explicitly produces the null space and range of the exterior transform and gives a new method for inverting the transform modulo the null space a modification of this method is given that will exactly invert functions of compact support these results generalize theorems of r m perry and the author a singular value decomposition for the radon transform that integrates over spheres in rn containing the origin is also given this follows from the singular value decomposition for r and yields the null space and a new inversion method for this transform

radon transforms on affine grassmannians we develop an analytic approach to the radon transform f f f where f is a flinction on the affine grassmann manifold g n k of k dimensional planes in r n and is a k dimensional plane in the similar manifold g n k k k for f e l p g n k we prove that this transform is finite almost everywhere on g n k if and only if 1 n 1 and interpreted as a direct quasi orthogonal radon transform for another pair of affine grassmannians as a consequence we obtain that the radon transform and the dual radon transform are injective simultaneously if and only if k k n 1 the investigation is carried out for locally integrable and continuous functions satisfying natural weak conditions at infinity

range descriptions for the spherical mean radon transform abstract the transform considered in the paper averages a function supported in a ball in r n over all spheres centered at the boundary of the ball this radon type transform arises in several contemporary applications e g in thermoacoustic tomography and sonar and radar imaging range descriptions for such transforms are important in all these areas for instance when dealing with incomplete data error correction and other issues four different types of complete range descriptions are provided some of which also suggest inversion procedures necessity of three of these appropriately formulated conditions holds also in general domains while the complete discussion of the case of general domains would require another publication

mortality curves using a bit string aging model abstract in this work we studied the shape of mortality curves for a population under different conditions using a bit string model for biological aging more specifically we studied how the model parameters affect the shape of the mortality curves although these curves are in reasonable agreement with the gompertz law for the increase of mortality for a extended range of ages other parameters play an important role to determine the overall behavior of the mortality s curves

a diffusion limit for a class of randomly growing binary trees binary trees are grown by adding one node at a time an available node at height i being added with probability proportional to c i c 1 we establish both a strong law of large numbers and a central limit theorem for the vector x t x i t where x i t is the proportion of nodes of height i that are available at time t we show in fact that there is a deterministic process x i t such that sum x i t x i t text converges to 0 a text s text and such that if c2 tfrac 1 2 z i n t 2 n 2 x n 1 tc n x n 1 tc n and z n t z i n t then z n t converges weakly to a gaussian diffusion z t the results are applied to establish asymptotic normality in the unbiased coin tossing case for an entropy estimation procedure due to j ziv and to obtain results on the growth of the maximum height of the tree

numerical solution of the penna model of biological aging with age modified mutation rate in this paper we present results of numerical calculation of the penna bit string model of biological aging modified for the case of a dependent mutation rate m a where a is the parent s age the mutation rate m a is the probability per bit of an extra bad mutation introduced in offspring inherited genome we assume that m a increases with age a as compared with the reference case of the standard penna model based on a constant mutation rate m the dynamics of the population growth shows distinct changes in age distribution of the population here we concentrate on mortality q a a fraction of items eliminated from the population when we go from age a to a 1 in simulated transition from time t to next time t 1 the experimentally observed q a dependence essentially follows the gompertz exponential law for a above the minimum reproduction age deviation from the gompertz law is however observed for the very old items close to the maximal age this effect may also result from an increase in mutation rate m with age a discussed in this paper the numerical calculations are based on analytical solution of the penna model presented in a series of papers by coe et al j b coe y mao and m e cates phys rev lett 89 288103 2002 results of the numerical calculations are supported by the data obtained from computer simulation based on the solution by coe et al