on witten s proof of the positive energy theorem this paper gives a mathematically rigorous proof of the positive energy theorem using spinors this completes and simplifies the original argument presented by edward witten we clarify the geometric aspects of this argument and prove the necessary analytic theorems concerning the relevant dirac operator

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

stationary sets for the wave equation in crystallographic domains let w be a crystallographic group in r n generated by reflections and let be the fundamental domain of w we characterize stationary sets for the wave equation in when the initial data is supported in the interior of the stationary sets are the sets of time invariant zeros of nontrivial solutions that are identically zero at t 0 lie show that for these initial data the n 1 dimensional part of the stationary sets consists of hyperplanes that are mirrors of a crystallographic group w w w this part comes from a corresponding odd symmetry of the initial data in physical language the result is that if the initial source is localized strictly inside of the crystalline then unmovable interference hypersurfaces can only be faces of a crystalline substructure of the original one

thermoacoustic tomography for an integro differential wave equation modeling attenuation abstract in this article we study the inverse problem of thermoacoustic tomography tat on a medium with attenuation represented by a time convolution or memory term and whose consideration is motivated by the modeling of ultrasound waves in heterogeneous tissue via fractional derivatives with spatially dependent parameters under the assumption of being able to measure data on the whole boundary we prove uniqueness and stability and propose a convergent reconstruction method for a class of smooth variable sound speeds by a suitable modification of the time reversal technique we obtain a neumann series reconstruction formula

analytic explanation of spatial resolution related to bandwidth and detector aperture size in thermoacoustic or photoacoustic reconstruction an analytic explanation of the spatial resolution in thermoacoustic or photoacoustic reconstruction is presented three types of specific recording geometries including spherical planar and cylindrical surface as well as other general cases are investigated analytic expressions of the point spread functions psf s as a function of the bandwidth of the measurement system and the finite size of the detector aperture are derived based on rigorous reconstruction formulas the analyses clearly reveal that the dependence of the psf s on the bandwidth of all recording geometries shares the same space invariant expression while the dependence on the aperture size of the detector differs the bandwidth affects both axial and lateral resolutions in contrast the detector aperture blurs the lateral resolution greatly but the axial resolution only slightly

injectivity sets for the radon transform over circles and complete systems of radial functions abstract a necessary and sufficient characterization is given that specifies which sets of sums of translations of radial functions are dense in the set of continuous functions in the plane this problem is shown to be equivalent to inversion for the radon transform on circles centered on restricted subsets of the plane the proofs rest on the geometry of zero sets for harmonic polynomials and the microlocal analysis of this circular radon transform a characterization of nodal sets for the heat and wave equation in the plane are consequences of our theorems and questions of pinkus and ehrenpreis are answered

ranges and inversion formulas for spherical mean operator and its dual abstract we consider the spherical mean operator r and its dual t r we establish some results from harmonic analysis for the generalized fourier transform associated with the operator r we deduce that the operators r and t r are transmutation operators on some spaces of functions next we give inversion formulas for these operators and a plancherel theorem for the operator t r

mathematical modeling in photoacoustic imaging of small absorbers this paper is devoted to mathematical modeling in photoacoustic imaging of small absorbers we propose a new method for reconstructing small absorbing regions inside a bounded domain from boundary measurements of the induced acoustic signal we also show the focusing property of the back propagated acoustic signal indeed we provide two different methods for locating a targeted optical absorber from boundary measurements of the induced acoustic signal the first method consists of a multiple signal classification music type algorithm and the second one uses a multifrequency approach we also show results of computational experiments to demonstrate efficiency of the algorithms

inversion of spherical means and the wave equation in even dimensions we establish inversion formulas of the so called filtered back projection type to recover a function supported in the ball in even dimensions from its spherical means over spheres centered on the boundary of the ball we also find several formulas to recover initial data of the form f 0 or 0 g for the free space wave equation in even dimensions from the trace of the solution on the boundary of the ball provided that the initial data has support in the ball

recovering functions from the spherical mean transform with limited radii data by expansion into spherical harmonics abstract the aim of the article is to generalize the method presented in 3 theorem 1 by g ambartsoumian r gouia zarrad and m lewis for recovering functions from their spherical mean transform with limited radii data from the two dimensional case to the general n dimensional case the idea behind the method is to expand each function in question into spherical harmonics and then obtain for each term in the expansion an integral equation of volterra s type that can be solved iteratively we show also how this method can be modified for the spherical case of recovering functions from the spherical transform with limited radii data lastly we solve the analogous problem for the case of the funk transform by again using expansion into spherical harmonics and then obtain an abel type integral equation which can be inverted by a method introduced in 14

thermodynamics of the general diffusion process time reversibility and entropy production we introduce an axiomatic thermodynamic theory for the general diffusion process and prove a theorem concerning entropy and irreversibility the equivalence among time reversibility zero entropy production symmetricity of the stationary diffusion process and a potential condition

interpolation of stochastic and deterministic reduced dynamics abstract there are roughly speaking the two extreme types of dynamical density functional theories one is deterministic without noise and another is fully stochastic with noise we attempt to interpolate these extreme limiting theories for general reduced description of time evolution by making use of a series of projection operators which generalize the so called kawasaki gunton operator this helps to clarify though in a formal way the controversies or confusions surrounding the current dynamical density functional theories

recovery of a source term or a speed with one measurement and applications we study the problem of recovery the source a t x f x in the wave equation in anisotropic medium with a known so that a 0 x 6d 0 with a single measurement we use carleman estimates combined with geometric arguments and give sharp conditions for uniqueness we also study the non linear problem of recovery the sound speed in the equation utt c x u d 0with one measurement we give sharp conditions for stability as well an application to thermoacoustic tomography is also presented

the exponentially convergent trapezoidal rule it is well known that the trapezoidal rule converges geometrically when applied to analytic functions on periodic intervals or the real line the mathematics and history of this phenomenon are reviewed and it is shown that far from being a curiosity it is linked with computational methods all across scientific computing including algorithms related to inverse laplace transforms special functions complex analysis rational approximation integral equations and the computation of functions and eigenvalues of matrices and operators

the effect of interference of coloured additive and multiplicative white noises on escape rate in this paper we have calculated the escape rate from a meta stable state for coloured and correlated noise driven open systems based on the fokker planck description of the stochastic process we consider the effect of two correlation times due to the additive coloured noise and the correlation between additive coloured and multiplicative white noises the effect of the noise correlation strength on the rate has also been investigated

to recovering the moments from the spherical mean radon transform abstract this article deals with characterizations of a function in terms of its circular mean radon transform we present a new approach the consistency method showing how to describe the class of real valued planar functions f which have the given circular mean radon transform m f over circles centered on the unit circle also expressions are derived for the geometric moments of an unknown function in terms of its circular mean radon transform

modified szabo s wave equation models for lossy media obeying frequency power law szabo s models of acoustic attenuation szabo j acoust soc am 96 1 491 500 1994 comply well with the empirical frequency power law involving noninteger and odd integer exponent coefficients while guaranteeing causality but nevertheless encounter the troublesome issues of hypersingular improper integral and obscurity in implementing initial conditions the purpose of this paper is to ease or remove these drawbacks of the szabo s models via the caputo fractional derivative concept the positive time fractional derivative is also introduced to include the positivity of the attenuation processes

total mean curvature scalar curvature and a variational analog of brown york mass we study the supremum of the total mean curvature on the boundary of compact mean convex 3 manifolds with nonnegative scalar curvature and a prescribed boundary metric we establish an additivity property for this supremum and exhibit rigidity for maximizers assuming the supremum is attained when the boundary consists of 2 spheres we demonstrate that the finiteness of the supremum follows from the previous work of shi tam and wang yau on the quasi local mass problem in general relativity in turn we define a variational analog of brown york quasi local mass without assuming that the boundary 2 sphere has positive gauss curvature

a mollification approach for inverting the spherical mean radon transform the filtered backprojection algorithm is still one of the most important reconstruction methods in computed tomography it is based on explicit formulas for recovering mollified versions of the original unknown function in this paper we derive such formulas for the spherical mean radon transform with centers restricted to the boundary of a ball in two spatial dimensions we present an efficient implementation strategy that leads to a stable reconstruction algorithm of filtered backprojection type

embedded minimal surfaces exotic spheres and manifolds with positive ricci curvature let n be a three dimensional riemannian manifold let e be a closed embedded surface in n then it is a question of basic interest to see whether one can deform in its isotopy class to some canonical embedded surface from the point of view of geometry a natural canonical surface will be the extremal surface of some functional defined on the space of embedded surfaces the simplest functional is the area functional the extremal surface of the area functional is called the minimal surface such minimal surfaces were used extensively by meeks yau my21 in studying group actions on three dimensional manifolds in my2 the theory of minimal surfaces was used to simplify and strengthen the classical dehn s lemma loop theorem and the sphere theorem in the setting there one minimizes area among all immersed surfaces and proves that the extremal object is embedded in this paper we minimize area among all embedded surfaces isotopic to a fixed embedded surface in the category of these surfaces we prove a general existence theorem theorem 1 a particular consequence of this theorem is that for irreducible manifolds an embedded incompressible surface is isotopic to an embedded incompressible surface with minimal area we also prove that there exists an embedded sphere of least area enclosing a fake cell provided the complementary volume is not a standard ball and provided there exists no embedded one sided rp2 by making use of the last result and a cutting and pasting argument we are able to settle a well known problem in the theory of three dimensional manifolds we prove that the covering space of any irreducible orientable three dimensional manifold is irreducible it is possible to exploit our existence theorem to study

a reconstruction algorithm for photoacoustic imaging based on the nonuniform fft fourier reconstruction algorithms significantly outperform conventional backprojection algorithms in terms of computation time in photoacoustic imaging these methods require interpolation in the fourier space domain which creates artifacts in reconstructed images we propose a novel reconstruction algorithm that applies the one dimensional nonuniform fast fourier transform to photoacoustic imaging it is shown theoretically and numerically that our algorithm avoids artifacts while preserving the computational effectiveness of fourier reconstruction

inversion of circular means and the wave equation on convex planar domains we study the problem of recovering the initial data of the two dimensional wave equation from values of its solution on the boundary w of a smooth convex bounded domain w r 2 as a main result we establish back projection type inversion formulas that recover any initial data with support in w modulo an explicitly computed smoothing integral operator k w for circular and elliptical domains the operator k w is shown to vanish identically and hence we establish exact inversion formulas of the back projection type in these cases similar results are obtained for recovering a function from its mean values over circles with centers on w both reconstruction problems are amongst others essential for the hybrid imaging modalities photoacoustic and thermoacoustic tomography

sampling conditions for the circular radon transform recovering a function from circular or spherical mean values is the basis of many modern imaging technologies such as photo and thermoacoustic computed tomography or ultrasound reflection tomography recently much progress has been made concerning the problem of recovering a function from its circular mean values its circular radon transform in particular theoretically exact inversion formulas of the back projection type have been discovered using continuously sampled data in practical applications however only a discrete number of circular mean values can be collected in this paper we address this issue in the context of the shannon sampling theory we derive sharp sampling conditions for the number of angular and radial samples such that any essentially b0 bandlimited function can be recovered from a finite number of such circular mean values

nonlinear langevin equations and the time dependent density functional method to study the time dependent density functional method tddfm two streaming velocity reversible terms are reformulated in the nonlinear langevin equation mori s prog theor phys 33 423 1965 projection operator method shows a variety of nonlinear langevin equations this is because the equations depend on the choice of phase space functions employed in the projection if phase space functions include particular functions however the streaming velocity term has an invariable form the form is independent of the choice of other phase space functions since the invariable streaming velocity term does not introduce the tddfm the second viewpoint is presented in this the linearization of the streaming velocity term agrees with the frequency term in the linear langevin equation since only the second streaming velocity term introduces the tddfm one needs to be cautious in the derivation of the tddfm

the universal back projection formula for spherical means and the wave equation on certain quadric hypersurfaces abstract in many tomographic applications and elsewhere there arises a need to reconstruct a function from the data on the boundary of some domain given either by the spherical means of the function or by the corresponding solution of the free space wave equation in this paper we show that the so called universal back projection formulas provide exact recovery of the unknown function with compact support for data on any quadric hypersurfaces that can be approximated by elliptic hypersurfaces these quadric hypersurfaces include elliptic paraboloid as well as parabolic and elliptic cylinders

the inversion problem and applications of the generalized radon transform we prove that under certain conditions the inversion problem for the generalized radon transform reduces to solving a fredholm integral equation and we obtain the asymptotic expansion of the symbol of the integral operator in this equation we consider applications of the generalized radon transform to partial differential equations with variable coefficients and provide a solution to the inversion problem for the attenuated and exponential radon transforms

the mass of an asymptotically flat manifold we show that the mass of an asymptotically flat n manifold is a geometric invariant the proof is based on harmonic coordinates and to develop a suitable existence theory results about elliptic operators with rough coefficients on weighted sobolev spaces are summarised some relations between the mass scalar curvature and harmonic maps are described and the positive mass theorem for n dimensional spin manifolds is proved

positive mass theorem and the boundary behaviors of compact manifolds with nonnegative scalar curvature in this paper we study the boundary behaviors of compact manifolds with nonnegative scalar curvature and nonempty boundary using a general version of positive mass theorem of schoen yau and witten we prove the following theorem for any compact manifold with boundary and nonnegative scalar curvature if it is spin and its boundary can be isometrically embedded into euclidean space as a strictly convex hypersurface then the integral of mean curvature of the boundary of the manifold cannot be greater than the integral of mean curvature of the embedded image as a hypersurface in euclidean space moreover equality holds if and only if the manifold is isometric with a domain in the euclidean space conversely under the assumption that the theorem is true then one can prove the adm mass of an asymptotically flat manifold is nonnegative which is part of the positive mass theorem

levy flights in confining potentials we analyze confining mechanisms for levy flights when they evolve in suitable external potentials their variance may exist and show signatures of a superdiffusive transport two classes of stochastic jump type processes are considered those driven by langevin equation with levy noise and those named topological levy processes occurring in systems with topological complexity such as folded polymers or complex networks whose langevin representation is unknown and possibly nonexistent our major finding is that both above classes of processes stay in affinity and may share common stationary probability density even if their detailed dynamical behavior look different this near equilibrium observation seems to be generic to a broad class of levy noise driven processes such as e g superdiffusion on folded polymers geophysical flows and even climatic changes

entropy production in nonequilibrium statistical mechanics we consider systems of nonequilibrium statistical mechanics driven by nonconservative forces and in contact with an ideal thermostat these are smooth dynamical systems for which one can define natural stationary states srb in the simplest case and entropy production e minus the sum of the lyapunov exponents in the simplest case we give exact and explicit definitions of the entropy production e for the various situations of physical interest we prove that e 0 and indicate cases where e 0 the novelty of the approach is that we do not try to compute entropy production directly but make it depend on the identification of a natural stationary state for the system

sampling in fan beam tomography in fan beam tomography functions in mathbb r 2 are constructed from integrals along straight lines emanating from a finite number of sources sitting on a circle around the reconstruction region using a sampling theorem for periodic functions and asymptotic estimates for the fourier transform of the fan beam transform the exact sampling conditions are found for standard fan beam scanning necessary to obtain a certain resolution new efficient sampling schemes and corresponding reconstruction algorithms are also found these sampling schemes need significantly less data to obtain a certain resolution than a standard fan beam geometry the correctness of the sampling conditions for standard fan beam scanning and the superiority of the new schemes are shown by computer simulations

principle of maximum entropy and reduced dynamics a new method to obtain a series of reduced dynamics at various stages of coarse graining is proposed this ranges from the most coarse grained one which agrees with the deterministic time evolution equation for averages of the relevant variables to the least coarse grained one which is the generalized fokker planck equation for the probability distribution function of the relevant variables the method is based on the extention of the kawasaki gunton operator with the help of the principle of maximum entropy

inversion formulas and stability estimates of the wave operator on the hyperplane abstract one of the mathematical problems arising in photoacoustic tomography pat a novel and promising technology in medical imaging is to recover the initial function from the solution of the wave equation on a surface where the detectors of pat are located we define the wave operator as the transform assigning to a given function f the solution of the wave equation on the detector surface where the detectors are located with the initial function f here we study many properties of this wave operator including the inversion formulas and stability estimates assuming that the detector surface is a hyperplane

universal back projection algorithm for photoacoustic computed tomography we report results of a reconstruction algorithm for three dimensional photoacoustic computed tomography a universal back projection formula is presented for three types of imaging geometries planar spherical and cylindrical surfaces a solid angle weighting factor is introduced in the back projection formula to compensate for the variations of detection views a method for implementing this algorithm is described numerical simulation is used to demonstrate the performance of the algorithm

the approximate inverse in action with an application to computerized tomography the approximate inverse is a scheme to obtain stable numerical inversion formulae for linear operator equations of the first kind yet in some applications the computation of a crucial ingredient the reconstruction kernel is time consuming and instable it may even happen that the kernel does not exist for a particular semidiscrete system to cure this dilemma we propose and analyze a technique that is based on a singular value decomposition of the underlying operator the results are applied to the reconstruction problem in 2d computerized tomography where they enable the design of reconstruction filters and lead to a novel error analysis of the filtered backprojection algorithm

exact and approximative imaging methods for photoacoustic tomography using an arbitrary detection surface two universal reconstruction methods for photoacoustic also called optoacoustic or thermoacoustic computed tomography are derived applicable to an arbitrarily shaped detection surface in photoacoustic tomography acoustic pressure waves are induced by illuminating a semitransparent sample with pulsed electromagnetic radiation and are measured on a detection surface outside the sample the imaging problem consists in reconstructing the initial pressure sources from those measurements the first solution to this problem is based on the time reversal of the acoustic pressure field with a second order embedded boundary method the pressure on the arbitrarily shaped detection surface is set to coincide with the measured data in reversed temporal order in the second approach the reconstruction problem is solved by calculating the far field approximation a concept well known in physics where the generated acoustic wave is approximated by an outgoing spherical wave with the reconstruction point as center numerical simulations are used to compare the proposed universal reconstruction methods with existing algorithms

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

langevin equation for the density of a system of interacting langevin processes we present a simple derivation of the stochastic equation obeyed by the density function for a system of langevin processes interacting via a pairwise potential the resulting equation is considerably different from the phenomenological equations usually used to describe the dynamics of non conserved model a and conserved model b particle systems the major feature is that the spatial white noise for this system appears not additively but multiplicatively this simply expresses the fact that the density cannot fluctuate in regions devoid of particles the steady state for the density function may however still be recovered formally as a functional integral over the coursed grained free energy of the system as in models a and b

is a curved flight path in sar better than a straight one in the plane we study the transform r gamma f of integrating an unknown function f over circles centered at a given curve gamma this is a simplified model of synthetic aperture radar sar when the radar is not directed but has other applications like thermoacoustic tomography for example we study the problem of recovering the wave front set wf f if the visible singularities of f hit gamma once we show that wf f cannot be recovered i e the artifacts cannot be resolved if gamma partial omega is the boundary of a strictly convex domain omega we show that this is still true on the other hand in the latter case if f is known a priori to have singularities in a compact set then we show that one can recover wf f omega and moreover this can be done in a simple explicit way using backpropagation for the wave equation

proof of the positive mass theorem ii the positive mass theorem states that for a nontrivial isolated physical system the total energy which includes contributions from both matter and gravitation is positive this assertion was demonstrated in our previous paper in the important case when the space time admits a maximal slice here this assumption is removed and the general theorem is demonstrated abstracts of the results of this paper appeared in 11 and 13

stochastic operators information and entropy for a stochastic operatoru on andl1 space i eu is linear positive and norm preserving on the positive cone ofl1 it is shown thatu decreases relative information between two nonnegativel1 functions furthermore it is shown that the following properties ofu are closely related u is energy decreasing energy preserving u ish decreasing whereh is boltzmann sh functional and the maxwell distributions are fixed points ofu

time domain wave equations for lossy media obeying a frequency power law for attenuation described by a slowly varying power law function of frequency 0 y classical lossy time domain wave equations exist only for the restricted cases where y 0 or y 2 for the frequently occurring practical situation in which attenuation is much smaller than the wave number a lossy dispersion characteristic is derived that has the desired attenuation general power law dependence in order to obtain the corresponding time domain lossy wave equation time domain loss operators similar in function to existing derivative operators are developed through the use of generalized functions three forms of lossy wave equations are found depending on whether y is an even or odd integer or a noninteger a time domain expression of causality analogous in function to the kramers kronig relations in the frequency domain is used to derive the causal wave equations final causal versions of the time domain wave equations are obtained even for the cases where y 1 which according to the paley wiener th

discrete radon transform this paper describes the discrete radon transform drt and the exact inversion algorithm for it similar to the discrete fourier transform dft the drt is defined for periodic vector sequences and studied as a transform in its own right casting the forward transform as a matrix vector multiplication the key observation is that the matrix although very large has a block circulant structure this observation allows construction of fast direct and inverse transforms moreover we show that the drt can be used to compute various generalizations of the classical radon transform rt and in particular the generalization where straight lines are replaced by curves and weight functions are introduced into the integrals along these curves in fact we describe not a single transform but a class of transforms representatives of which correspond in one way or another to discrete versions of the rt and its generalizations an interesting observation is that the exact inversion algorithm cannot be obtained directly from radon s inversion formula given the fact that the rt has no nontrivial one dimensional analog exact invertibility makes the drt a useful tool geared specifically for multidimensional digital signal processing exact invertibility of the drt flexibility in its definition and fast computational algorithm affect present applications and open possibilities for new ones some of these applications are discussed in the paper

an inversion formula for the spherical mean transform with data on an ellipsoid in two and three dimensions abstract in the articles 1 and 2 of d finch m haltmeier s patch and d rakesh inversion formulas were found in any dimension greater than or equal to 2 for recovering a smooth function with compact support in the unit ball from spherical means centered on the unit sphere the aim of this article is to show that the methods used in 1 2 can be modified in order to get similar inversion formulas from spherical means centered on an ellipsoid in two and three dimensional spaces

to local reconstruction from the spherical mean radon transform abstract the article suggests a new approach using the so called consistency method for the inversion of the spherical mean radon transform in 2d with detectors on a line we present a new iterative formula which gives an algorithm to recover an unknown function supported completely on one side of a line l from its spherical means over circles centered on the line l our reconstruction formula has the benefit of being local such an inversion is required in problems of thermo and photo acoustic tomography ultrasound reflection tomography radar imaging and a few other

on the trend to equilibrium for some dissipative systems with slowly increasing a priori bounds we prove convergence to equilibrium with explicit rates for various kinetic equations with relatively bad control of the distribution tails in particular boltzmann type equations with smoothed soft potentials we compensate the lack of uniform in time estimates by the use of precise logarithmic sobolev type inequalities and the assumption that the initial datum decays rapidly at large velocities our method not only gives explicit results on the times of convergence but is also able to cover situations in which compactness arguments apparently do not apply even mere convergence to equilibrium was an open problem for soft potentials

isometric embeddings into the minkowski space and new quasi local mass the definition of quasi local mass for a bounded space like region in space time is essential in several major unsettled problems in general relativity the quasi local mass is expected to be a type of flux integral on the boundary two surface sigma partial omega and should be independent of whichever space like region sigma bounds an important idea which is related to the hamiltonian formulation of general relativity is to consider a reference surface in a flat ambient space with the same first fundamental form and derive the quasi local mass from the difference of the extrinsic geometries this approach has been taken by brown york 4 5 and liu yau 16 17 see also related works 3 6 9 12 14 15 28 32 to define such notions using the isometric embedding theorem into the euclidean three space however there exist surfaces in the minkowski space whose quasilocal mass is strictly positive 19 it appears that the momentum information needs to be accounted for to reconcile the difference in order to fully capture this information we use isometric embeddings into the minkowski space as references in this article we first prove an existence and uniqueness theorem for such isometric embeddings we then solve the boundary value problem for jang s 13 equation as a procedure to recognize such a surface in the minkowski space in doing so we discover a new expression of quasi local mass for a large class of admissible surfaces see theorem a and remark 1 1 the new mass is positive when the ambient space time satisfies the dominant energy condition and vanishes on surfaces in the minkowski space it also has the nice asymptotic behavior at spatial infinity and null infinity some of these results were announced in 29

the spherical radon transform with centers on cylindrical surfaces abstract recovering a function from its spherical radon transform with centers of spheres of integration restricted to a hypersurface is at the heart of several modern imaging technologies including sar ultrasound imaging and photo and thermoacoustic tomography in this paper we study an inversion of the spherical radon transform with centers of integration restricted to cylindrical surfaces of the form r m where is a hypersurface in r n we show that this transform can be decomposed into two lower dimensional spherical radon transforms one with centers on and one with a planar center set in r m 1 together with explicit inversion formulas for the spherical radon transform with a planar center set and existing algorithms for inverting the spherical radon transform with a center set this yields reconstruction procedures for general cylindrical domains in the special case of spherical or elliptical cylinders we obtain novel explicit inversion formulas for three spatial dimensions these inversion formulas can be implemented efficiently by backprojection type algorithms only requiring o n 4 3 floating point operations where n is the total number of unknowns to be recovered we present numerical results demonstrating the efficiency of the derived algorithms

modeling power law absorption and dispersion for acoustic propagation using the fractional laplacian the efficient simulation of wave propagation through lossy media in which the absorption follows a frequency power law has many important applications in biomedical ultrasonics previous wave equations which use time domain fractional operators require the storage of the complete pressure field at previous time steps such operators are convolution based this makes them unsuitable for many three dimensional problems of interest here a wave equation that utilizes two lossy derivative operators based on the fractional laplacian is derived these operators account separately for the required power law absorption and dispersion and can be efficiently incorporated into fourier based pseudospectral and k space methods without the increase in memory required by their time domain fractional counterparts a framework for encoding the developed wave equation using three coupled first order constitutive equations is discussed and the model is demonstrated through several one two and three dimensional simulations

inversion of n dimensional spherical averages in this paper we give a simple inversion formula for determining the even part of a function from its averages over the surfaces of spheres in mathbb r n centered on a hyperplane y 0 we emphasize the relation between the backprojection approximation to the function and the full inversion result a simple necessary and sufficient condition for data to correspond to the spherical averages of a function is given

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 galerkin least squares approach for photoacoustic tomography the development of fast and accurate image reconstruction algorithms is a central aspect of computed tomography in this paper we address this issue for photoacoustic computed tomography in circular geometry we investigate the galerkin least squares method for that purpose for approximating the function to be recovered we use subspaces of translation invariant spaces generated by a single function this includes many systems that have previously been employed in photoacoustic tomography such as generalized kaiser bessel basis functions or the natural pixel basis by exploiting an isometry property of the forward problem we are able to efficiently set up the galerkin equation for a wide class of generating functions and devise efficient algorithms for its solution we establish a convergence analysis and present numerical simulations that demonstrate the efficiency and accuracy of the derived algorithm

noise and conditional entropy evolution abstract we study the convergence properties of the conditional kullback leibler entropy in stochastic systems we have proved general results showing that asymptotic stability is a necessary and sufficient condition for the monotone convergence of the conditional entropy to its maximal value of zero additionally we have made specific calculations of the rate of convergence of this entropy to zero in a one dimensional situation illustrated by ornstein uhlenbeck and rayleigh processes higher dimensional situations and a two dimensional ornstein uhlenbeck process with a stochastically perturbed harmonic oscillator and colored noise as examples we also apply our general results to the problem of conditional entropy convergence in the presence of dichotomous noise in both the one dimensional and multidimensional cases we show that the convergence of the conditional entropy to zero is monotone and at least exponential in the specific cases of the ornstein uhlenbeck and rayleigh processes as well as the stochastically perturbed harmonic oscillator and colored noise examples we obtain exact formulae for the temporal evolution of the conditional entropy starting from a concrete initial distribution the rather surprising result in this case is that the rate of convergence of the entropy to zero is independent of the noise amplitude

fluctuations and non equilibrium thermodynamics abstract a closed macroscopic system with fluctuating local properties is treated as a thermodynamic system with internal degrees of freedom gibb s entropy postulate is used to define the systems entropy as a functional of the probability density in internal coordinate space it is shown that application of the scheme of thermodynamics of irreversible processes then leads directly to the theory of fluctuations as markov processes described by a multivariate fokker planck equation in this perspective fluctuation theory may be said to have become integrated into non equilibrium thermodynamics

optimized kaiser bessel window functions for computed tomography kaiser bessel window functions are frequently used to discretize tomographic problems because they have two desirable properties 1 their short support leads to a low computational cost and 2 their rotational symmetry makes their imaging transform independent of the direction in this paper we aim at optimizing the parameters of these basis functions we present a formalism based on the theory of approximation and point out the importance of the partition of unity condition while we prove that for compact support functions this condition is incompatible with isotropy we show that minimizing the deviation from the partition of unity condition is highly beneficial the numerical results confirm that the proposed tuning of the kaiser bessel window functions yields the best performance

singular values of the attenuated photoacoustic imaging operator abstract we analyse the ill posedness of the photoacoustic imaging problem in the case of an attenuating medium to this end we introduce an attenuated photoacoustic operator and determine the asymptotic behaviour of its singular values dividing the known attenuation models into strong and weak attenuation classes we show that for strong attenuation the singular values of the attenuated photoacoustic operator decay exponentially and in the weak attenuation case the singular values of the attenuated photoacoustic operator decay with the same rate as the singular values of the non attenuated photoacoustic operator

dynamical density functional theory for interacting brownian particles stochastic or deterministic we aim to clarify confusions in the literature as to whether or not dynamical density functional theories for the one body density of a classical brownian fluid should contain a stochastic noise term we point out that a stochastic as well as a deterministic equation of motion for the density distribution can be justified depending on how the fluid one body density is defined i e whether it is an ensemble averaged density distribution or a spatially and or temporally coarse grained density distribution

a class of singular fourier integral operators in synthetic aperture radar imaging in this article we analyze the microlocal properties of the linearized forward scattering operator f and the normal operator f f where f is the l 2 adjoint of f which arises in synthetic aperture radar imaging for the common midpoint acquisition geometry when f is applied to the scattered data artifacts appear we show that f f can be decomposed as a sum of four operators each belonging to a class of distributions associated to two cleanly intersecting lagrangians i p l 0 1 thereby explaining the latter artifacts

on the positive mass theorem for manifolds with corners we study the positive mass theorem for certain non smooth metrics following p miao s work our approach is to smooth the metric using the ricci flow as well as improving some previous results on the behaviour of the adm mass under the ricci flow we extend the analysis of the zero mass case to higher dimensions

microscopic derivation of time dependent density functional methods time dependent density functional methods tddfm are studied from the microscopic viewpoint using projection operator methods in classical liquids a density field is defined without averaging so that a time evolution equation of the density field is derived with a random force the derived equation includes a free energy functional which is different from that defined in the tddfm the projection operator method provides the exact expression of the free energy functional another definition of the density field by an average leads to the equation of the tddfm in addition an equation describing fluctuations is also derived

the approximate inverse in action iii 3d doppler tomography the approximate inverse is a powerful tool for solving first kind operator equations in a stable way its abstract convergence and stability theory developed in our articles siam j numer anal 37 1909 1929 2000 and math comp 72 1399 1415 2003 is applied to the reconstruction problem of 3d vector field tomography resulting in a reconstruction algorithm of filtered backprojection type for an analytically computed reconstruction filter reconstruction kernel convergence with rates as well as the regularization property are established

boundary rigidity with partial data we study the boundary rigidity problem with partial data consisting of determining locally the riemannian metric of a riemannian manifold with boundary from the distance function measured at pairs of points near a fixed point on the boundary we show that one can recover uniquely and in a stable way a conformal factor near a strictly convex point where we have the information in particular this implies that we can determine locally the isotropic sound speed of a medium by measuring the travel times of waves joining points close to a convex point on the boundary the local results lead to a global lens rigidity uniqueness and stability result assuming that the manifold is foliated by strictly convex hypersurfaces

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

temporal behavior of the conditional and gibbs entropies we study the temporal approach to equilibrium of the gibbs and conditional entropies for stochastic systems in the presence of white noise the conditional entropy will either remain constant or monotonically increase to its maximum of zero however the gibbs entropy may have a variety of patterns of approach to its final value ranging from a monotone increase or decrease to an oscillatory approach we have illustrated all of these behaviors using examples in which both entropy dynamics can be determined analytically

positive scalar curvature with skeleton singularities we study positive scalar curvature on the regular part of riemannian manifolds with singular uniformly euclidean l infty metrics that consolidate gromov s scalar curvature polyhedral comparison theory and edge metrics that appear in the study of einstein manifolds we show that in all dimensions edge singularities with cone angles le 2 pi along codimension 2 submanifolds do not affect the yamabe type in three dimensions we prove the same for more general singular sets which are allowed to stratify along 1 skeletons exhibiting edge singularities angles le 2 pi and arbitrary l infty isolated point singularities we derive as an application of our techniques positive mass theorems for asymptotically flat manifolds with analogous singularities

effective discretization of direct reconstruction schemes for photoacoustic imaging in spherical geometries recently kernel methods for the recovery of a function from its spherical means in spherical acquisition geometry have been proposed we present efficient algorithms for these formulas in the two and three dimensional cases our scheme applies fourier techniques for certain convolution type integrals and discretizes physical space on a polar and a spherical grid respectively