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

area minimizers and boundary rigidity of almost hyperbolic metrics this paper is a continuation of our paper about boundary rigidity and filling minimality of metrics close to flat ones we show that compact regions close to a hyperbolic one are boundary distance rigid and strict minimal fillings we also provide a more invariant view on the approach used in the above mentioned paper

minimal sensitivity optimisation of perturbatively approximated excited state wavefunctions of the quartic oscillator the direct point by point application of minimal sensitivity optimisation to the first order perturbative approximation to the ground state wavefunction of the quartic oscillator carried out previously is extended to low lying excited states an analytic formula for the derivative with respect to the redundant parameter is developed and used to obtain the minimally sensitive value for this parameter as a function of configuration space generally more than one such minimally sensitive value is found at each configuration space point these values are categorised into disjoint continuous curves for all the cases studied it has proved possible to piece together sections of such curves in an unambiguous manner so as to obtain optimised wavefunction approximations which are both renormalisable and continuously differentiable comparison of these optimised wavefunction approximations with the exact results revealed satisfactory agreement particularly in the large mod x mod asymptotic region where orthodox perturbation methods fail completely due to the strong coupling

on asymptotic volume of finsler tori minimal surfaces in normed spaces and symplectic filling volume the main unconditional result of this paper theorem 3 states that every two dimensional affine disc in a normed space that is a disc contained in a two dimensional affine subspace is an area minimizing surface among all immersed discs with the same boundary with respect to the symplectic holmes thompson surface area to emphasize that this is not at all obvious it may be worth mentioning that a similar statement with rational chains in place of immersed discs is incorrect theorem 2 and that it is not known for surfaces that may not be topological discs the result still may not sound too exciting to the reader who never looked at the problem before even though it goes back to busemann s works in the 50 s see bes th and references there and the proof heavily relies on asymptotic geometry of tori we believe that it is more important that we embed this problem into a whole area of mostly open problems as well as give some partial results and suggest certain directions of how to attack them we begin with a trivial statement

entropy rigidity of negatively curved manifolds of finite volume we prove the following entropy rigidity result in finite volume if x is a negatively curved manifold with curvature b 2 k x 1 then ent top x n 1 if and only if x is hyperbolic in particular if x has the same length spectrum of a hyperbolic manifold x 0 the it is isometric to x 0 we also give a direct entropy free proof of this fact we compare with the classical theorems holding in the compact case pointing out the main difficulties to extend them to finite volume manifolds

on a pseudo differential equation for stokes waves it is shown that the existence of a smooth solution to a nonlinear pseudo differential equation on the unit circle is equivalent to the existence of a globally injective conformal mapping in the complex plane which gives a smooth solution to the nonlinear elliptic free boundary problem for stokes waves in hydrodynamics

cusp development in hele shaw flow with a free surface free surface flow in a hele shaw cell is considered it has been known for some time that when the fluid region is contracting a finite time blow up can occur in which a cusp forms in the free surface the solution does not exist beyond the time of blow up examples are presented of this blow up and also of a new process in which a cusp of a different type forms in the free surface but after which the solution continues to exist after a transformation of the dependent variable the pressure the new cusps are related to permissible singularities in the free boundary of the obstacle problem

unique recovery of lower order coefficients for hyperbolic equations from data on disjoint sets abstract we consider a restricted dirichlet to neumann map s r t associated with the operator t 2 g a q where g is the laplace beltrami operator of a riemannian manifold m g and a and q are a vector field and a function on m the restriction s r t corresponds to the case where the dirichlet traces are supported on 0 t s and the neumann traces are restricted on 0 t r here s and r are open sets which may be disjoint on the boundary of m we show that s r t determines uniquely up the natural gauge invariance the lower order terms a and q in a neighborhood of the set r assuming that r is strictly convex and that the wave equation is exactly controllable from s in time t 2 we give also a global result under a convex foliation condition the main novelty is the recovery of a and q when the sets r and s are disjoint we allow a and q to be non self adjoint and in particular the corresponding physical system may have dissipation of energy

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

traveling waves in deep water with gravity and surface tension this paper is concerned with the simulation of periodic traveling deep water free surface water waves under the influence of gravity and surface tension in two and three dimensions a variety of techniques is utilized including the numerical simulation of a weakly nonlinear model explicit solutions of low order perturbation theories and the direct numerical simulation of the full water wave equations the weakly nonlinear models which we present are new and extend the work of akers and milewski siam j appl math 70 2010 pp 2390 2408 to arbitrary bond number and fluid depth the numerical scheme for the full water wave problem features a novel extension of the transformed field expansions method of nicholls and reitich euro j mech b fluids 2 5 2006 pp 406 424 to accommodate capillary effects in a stable and rapid fashion the purpose of this paper is apply the new numerical method then compare small amplitude solutions of potential flow with those of the approximate model particular attention is paid to the behavior near quadratic resonances an example of which is the wilton ripple

lens rigidity for manifolds with hyperbolic trapped set for a riemannian manifold m g with strictly convex boundary m the lens data consists in the set of lengths of geodesics with endpoints on m together with their endpoints x x m m and tangent exit vectors v v t x m t x m we show deformation lens rigidity for manifolds with hyperbolic trapped set and no conjugate points a class which contains all manifolds with negative curvature and strictly convex boundary including those with non trivial topology and trapped geodesics for the same class of manifolds in dimension 2 we prove that the set of endpoints and exit vectors of geodesics ie the scattering data determines the riemann surface up to conformal diffeomorphism

travelling times in scattering by obstacles in curved space abstract we consider travelling times of billiard trajectories in the exterior of an obstacle k on a two dimensional riemannian manifold m we prove that given two obstacles with almost the same travelling times the generalised geodesic flows on the non trapping parts of their respective phase spaces will have a time preserving conjugacy moreover if m has non positive sectional curvature we prove that if k and l are two obstacles with strictly convex boundaries and almost the same travelling times then k and l are identical

some analytical properties of solutions to a two dimensional steadily translating inviscid bubble we consider some analytical properties for a steadily translating two dimensional bubble in an inviscid irrotational flow when surface tension effects are included but gravity neglected for a general value of the bernoulli constant we show that a conformal mapping function from a unit circle to the exterior of the bubble has a very specific polar decomposition the pole locations and corresponding residues are determined numerically while in specific limits their values are determined asymptotically

optimisation of parameterised perturbative approximations to wavefunctions the authors have justified a point by point optimisation procedure for parametrised perturbative approximations to wavefunctions by appealing to a principle of minimal sensitivity here it is shown that if the diagonal matrix elements of the perturbation vanish which can always be arranged via a simple redefinition of the unperturbed hamiltonian then the optimisation procedure is also justified by an underlying stationary principle with the nth order perturbative approximations being stationary to that order

rigidity of broken geodesic flow and inverse problems consider broken geodesics alpha 0 l on a compact riemannian manifold m g with boundary of dimension n geq 3 the broken geodesics are unions of two geodesics with the property that they have a common end point assume that for every broken geodesic alpha 0 l starting at and ending to the boundary partial m we know the starting point and direction alpha 0 alpha 0 the end point and direction alpha l alpha l and the length l we show that this data determines uniquely up to an isometry the manifold m g this result has applications in inverse problems on very heterogeneous media for situations where there are many scattering points in the medium and arises in several applications including geophysics and medical imaging as an example we consider the inverse problem for the radiative transfer equation or the linear transport equation with a nonconstant wave speed assuming that the scattering kernel is everywhere positive we show that the boundary measurements determine the wave speed inside the domain up to an isometry

local boundary rigidity of a compact riemannian manifold with curvature bounded above this paper considers the boundary rigidity problem for a compact convex riemannian manifold m g with boundary o9m whose curvature satisfies a general upper bound condition this includes all nonpositively curved manifolds and all sufficiently small convex domains on any given riemannian manifold it is shown that in the space of metrics g on m there is a c3 neighborhood of g such that g is the unique metric with the given boundary distance function i e the function that assigns to any pair of boundary points their distance as measured in m more precisely given any metric g in this neighborhood with the same boundary distance function there is diffeomorphism o which is the identity on o9m such that g g there is also a sharp volume comparison result for metrics in this neighborhood in terms of the boundary distance function 1 statement of the result the general boundary rigidity problem reads to which extent is a riemannian metric on a compact manifold with boundary determined from the distances between boundary points more precisely it can be formulated as follows let m g be a compact riemannian manifold with boundary om let g be another riemannian metric on m we say that g and g have the same boundary distance function if d1 x y d9 x y for arbitrary boundary points x y e om where d1 resp d9 represents distance in m with respect to g resp g it is easy to give examples of pairs of metrics with the same boundary distance function indeed if m m is an arbitrary diffeomorphism of m onto itself which is the identity on the boundary then the metrics g and g g have the same boundary distance function here g og is the pull back of g under i e for arbitrary vectors 1 e txm we have r1 71 x i where p txm t x m is the differential of at x and resp is the inner product with respect to the metric g resp g we say that a compact riemannian manifold is boundary rigid if this is the only type of nonuniqueness more precisely m g is boundary rigid if for every received by the editors august 10 1998 2000 mathematics subject classification primary 53c65 53c22 53c20 secondary 58c25 58c40 the first and third authors were supported by crdf grant rm2 143 the first author was also supported by nsf grants mds 9505175 and mds 9626232 the third author was also supported by intas rfbr grant 95 0763 2000 american matlhematical society 3937 this content downloaded from 207 46 13 167 on thu 21 jul 2016 04 20 00 utc all use subject to http about jstor org terms 3938 c b croke n s dairbekov and v a sharafnjtdinov riemannian metric g on m1with the same boundary distance function as g there is a diffeomorphism o m m which is the identity on the boundary and for which g 0 of in ward and outward vectors here v is the unit outward norn al to om1 given x c qmvi we denote by k x the maximum of the sectional curvatures of all two planes oc t1 m such that c a for m g a cdrm we define the following invariant

singularities in water waves and rayleigh taylor instability waves numerically in a relatively efficient fashion we also show that for twodimensional water waves in a finite depth channel the nearest singularity in the unphysical region has the form as for deep water waves however associated with such a singularity there is a series of image singularities at increasing distances from the physical plane with possibly different behaviour further for the rayleightaylor problem of motion of fluid over vacuum and for the unsteady water wave problem we derive integro differential equations valid in the unphysical region and show how these equations can give information on the nature of singularities for arbitrary initial conditions we give indications to suggest that a one half point singularity on its approach to the physical domain corresponds to a spike observed in rayleigh taylor experiment

a well posed numerical method to track isolated conformal map singularities in hele shaw flow we present a new numerical method for calculating an evolving 2d hele shaw interface when surface tension effects are neglected in the case where the flow is directed from the less viscous fluid into the more viscous fluid the motion of the interface is ill posed small deviations in the initial condition will produce significant changes in the ensuing motion this situation is disastrous for numerical computation as small roundoff errors can quickly lead to large inaccuracies in the computed solution our method of computation is most easily formulated using a conformal map from the fluid domain into a unit disk the method relies on analytically continuing the initial data and equations of motion into the region exterior to the disk where the evolution problem becomes well posed the equations are then numerically solved in the extended domain the presence of singularities in the conformal map outside of the disk introduces specific structures along the fluid interface our method can explicitly track the location of isolated pole and branch point singularities allowing us to draw connections between the development of interfacial patterns and the motion of singularities as they approach the unit disk in particular we are able to relate physical features such as finger shape side branch formation and competition between fingers to the nature and location of the singularities the usefulness of this method in studying the formation of topological singularities self intersections of the interface is also pointed out

santalo s formula and stability of trapping sets of positive measure abstract billiard trajectories broken generalised geodesics are considered in the exterior of an obstacle k with smooth boundary on an arbitrary riemannian manifold we prove a generalisation of the well known santalo s formula as a consequence it is established that if the set of trapped points has positive measure then for all sufficiently small smooth perturbations of the boundary of k the set of trapped points for the new obstacle obtained in this way also has positive measure more generally the measure of the set of trapped points depends continuously on perturbations of the obstacle k some consequences are derived in the case of scattering by an obstacle k in i r n for example it is shown that for a large class of obstacles k the volume of k is uniquely determined by the average travelling times of scattering rays in the exterior of k

rigidity of the scattering length spectrum in this paper we consider properties of obstacles satisfying some non degeneracy conditions that can be recovered from the scattering length spectrum sls clearly the latter tells us whether the obstacle k is trapping or non trapping if the set of trapped points is relatively small then the sls also determines the volume of the obstacle the number of its connected components and whether its boundary is convex everywhere or it has non trivial concavities under the additional assumption that the curvature of the obstacle does not vanish of infinite order it is proved that from the sls one can recover certain information about the number of reflection points of any simply reflecting ray in the exterior of the obstacle finally for some special classes of obstacles e g star shaped ones it is shown that the sls completely determines the obstacle

modeling extreme waves based on equations of potential flow with a free surface a method for numerical investigation of nonlinear wave dynamics based on direct hydrodynamical modeling of 1 d potential periodic surface waves is created the model is a part of an interactive wind wave model using a non stationary conformal mapping the principal equations are rewritten in a surface following coordinate system and reduced to two simple evolutionary equations for the elevation and the velocity potential of the surface fourier expansion is used to approximate these equations high accuracy was confirmed by validation of the non stationary model against known solutions and by comparison between the results obtained with different resolution in the horizontal the method developed is applied to the simulation of waves evolution with different initial conditions numerical experiments with initially monochromatic waves with different steepness show that the model is able to simulate breaking conditions when the surface becomes a multi valued function of the horizontal coordinate an estimate of the critical initial wave height that divides between non breaking and eventually breaking waves is obtained simulations of nonlinear evolution of a wave field is represented initially by two modes with close wave numbers amplitude modulation and a wave field with a phase modulation both runs result in the appearance of large and very steep waves these also break if the initial amplitudes are sufficiently large

boundary integral techniques for multi connected domains abstract several boundary integral techniques are available for the computation of the solution to laplace s equation in multi connected domains however for cases where the domain is changing such as incompressible inviscid fluid flow with free surfaces iterative methods are highly attractive the paper describes one such formulation and tests it on circular and elliptic annuli it is necessary to use interpolated quadrature points to maintain accuracy when regions of the annuli are thin

stability for the lens rigidity problem let g be a riemannian metric for mathbb r d d geqq 3 which differs from the euclidean metric only in a smooth and strictly convex bounded domain m the lens rigidity problem is concerned with recovering the metric g inside m from the corresponding lens relation on the boundary partial m in this paper the stability of the lens rigidity problem is investigated for metrics which are a priori close to a given non trapping metric satisfying the strong fold regular condition a metric g is called strong fold regular if for each point x in m there exists a set of geodesics passing through x whose conormal bundle covers t x m moreover these geodesics contain either no conjugate points or only fold conjugate points with a non degeneracy condition examples of strong fold regular metrics are constructed our main result gives the first stability result for the lens rigidity problem in the case of anisotropic metrics which allow conjugate points the approach is based on the study of the linearized inverse problem of recovering a metric from its induced geodesic flow which is a weighted geodesic x ray transform problem for symmetric 2 tensor fields a key ingredient is to show that the kernel of the x ray transform on symmetric solenoidal 2 tensor fields is of finite dimension it remains open whether the kernel space is trivial or not

new conformal mapping for adaptive resolving of the complex singularities of stokes wave a new highly efficient method is developed for computation of travelling periodic waves stokes waves on the free surface of deep water a convergence of numerical approximation is determined by the complex singularities above the free surface for the analytical continuation of the travelling wave into the complex plane an auxiliary conformal mapping is introduced which moves singularities away from the free surface thus dramatically speeding up numerical convergence by adapting the numerical grid for resolving singularities while being consistent with the fluid dynamics the efficiency of that conformal mapping is demonstrated for the stokes wave approaching the limiting stokes wave the wave of the greatest height which significantly expands the family of numerically accessible solutions it allows us to provide a detailed study of the oscillatory approach of these solutions to the limiting wave generalizations of the conformal mapping to resolve multiple singularities are also introduced

computation of hele shaw flows with free boundaries abstract a novel method for calculation of hele shaw flows with receding free boundaries is presented the method is applied to flows with suction from a point sink and to flow in a channel with parallel walls in each case the unknown fluid region is mapped conformally onto the unit disc the free boundary being mapped onto the unit circle this mapping which is a function of position and time is calculated numerically at points on the unit circle using a version of the boundary integral method the free boundary is thus found without explicit calculation of the pressure at internal points and the computation times are much less than those for other numerical methods for this problem numerical results are compared with explicit analytic solutions for several test problems

gravity perturbed crapper waves crapper waves are a family of exact periodic travelling wave solutions of the free surface irrotational incompressible euler equations these are pure capillary waves meaning that surface tension is accounted for but gravity is neglected for certain parameter values crapper waves are known to have multi valued height using the implicit function theorem we prove that any of the crapper waves can be perturbed by the effect of gravity yielding the existence of gravity capillary waves nearby to the crapper waves this result implies the existence of travelling gravity capillary waves with multi valued height the solutions we prove to exist include waves with both positive and negative values of the gravity coefficient we also compute these gravity perturbed crapper waves by means of a quasi newton iterative scheme again using both positive and negative values of the gravity coefficient a phase diagram is generated which depicts the existence of single valued and multi valued travelling waves in the gravity amplitude plane a new largest water wave is computed which is composed of a string of bubbles at the interface

gaussian effective potential and coleman s normal ordering prescription the functional integral formalism for a class of systems the potential of whose bosonic hamiltonian has a fourier representation in the sense of tempered distributions we calculate the gaussian effective potential within the framework of the functional integral formalism we show that coleman s normal ordering prescription can be formally generalized to the functional integral formalism

the inverse problem for the local geodesic ray transform under a convexity assumption on the boundary we solve a local inverse problem namely we show that the geodesic x ray transform can be inverted locally in a stable manner one even has a reconstruction formula we also show that under an assumption on the existence of a global foliation by strictly convex hypersurfaces the geodesic x ray transform is globally injective in addition we prove stability estimates and propose a layer stripping type algorithm for reconstruction

improved harmonic approximation and the 2d ising model at t tc and h 0 we propose a new method to determine the unknown parameter associated with a self consistent harmonic approximation we check the validity of our technique in the context of the sine gordon model as a non trivial application we consider the scaling regime of the 2d ising model away from the critical point and in the presence of a magnetic field h we derive an expression that relates the approximate correlation length t tc and h

the boundary rigidity problem in the presence of a magnetic field abstract for a compact riemannian manifold with boundary endowed with a magnetic potential we consider the problem of restoring the metric g and the magnetic potential from the values of the mane action potential between boundary points and the associated linearized problem we study simple magnetic systems in this case knowledge of the mane action potential is equivalent to knowledge of the scattering relation on the boundary which maps a starting point and a direction of a magnetic geodesic into its end point and direction this problem can only be solved up to an isometry and a gauge transformation of for the linearized problem we show injectivity up to the natural obstruction under explicit bounds on the curvature and on we also show injectivity and stability for g and in a generic class g including real analytic ones for the nonlinear problem we show rigidity for real analytic simple g rigidity for metrics in a given conformal class and locally near any g g we also show that simple magnetic systems on two dimensional manifolds are always rigid

recovery of material parameters in transversely isotropic media in this paper we show that in anisotropic elasticity in the particular case of transversely isotropic media under appropriate convexity conditions knowledge of the qsh wave travel times determines the tilt of the axis of isotropy as well as some of the elastic material parameters and the knowledge of qp and qsv travel times conditionally determines a subset of the remaining parameters in the sense that if some of the remaining parameters are known the rest are determined or if the remaining parameters satisfy a suitable relation they are all determined under certain non degeneracy conditions furthermore we give a precise description of the additional issues which are a subject of ongoing work that need to be resolved for a full treatment

semiglobal boundary rigidity for riemannian metrics let m g be a riemannian manifold with boundary in this paper we consider the problem of determining the metric g from its associated boundary distance function dg x y dist x y x y m that is the geodesic distance between boundary points this problem arose in rigidity questions in riemannian geometry 12 6 8 for the case in which m is a bounded domain of euclidean space and the metric is conformal to the euclidean one this problem is called the inverse kinematic problem which arose in geophysics and has a long history starting at least in the early part of the 20th century with herglotz 10 he considered the case where m is a ball x r r x r equipped with a spherically symmetric metric ds2 dx2 c2 r where c r is a positive function depending only on the radius r x herglotz found a formula to determine c r from the boundary distance function physically this corresponds to the case of a spherically symmetric earth model with an index of refraction depending only on the radius the boundary distance function corresponds to the travel times of e g acoustic waves going through the earth and measured at the surface the general problem for the case that the sound speed depends on all variables has been extensively studied see for instance 17 and the references given there also this problem has a close connection for other inverse problems related to determining the sound speed from boundary measurements see 20

rigorous analytical approximation of tritronquee solution to painleve i and the first singularity we use a recently developed method 1 2 to determine approximate expression for tritronquee solution for p 1 y 6y 2 x 0 y 6 y 2 x 0 in a domain d with rigorous bounds in particular we rigorously confirm the location of the closest singularity from the origin to be at x 770766 323285 2 3841687675 to within 5 10 6 5 10 6 accuracy in agreement with previous numerical calculations 6

the geodesic ray transform on riemannian surfaces with conjugate points we study the geodesic x ray transform x on compact riemannian surfaces with conjugate points regardless of the type of the conjugate points we show that we cannot recover the singularities and therefore this transform is always unstable ill posed we describe the microlocal kernel of x and relate it to the conjugate locus we present numerical examples illustrating the cancellation of singularities we also show that the attenuated x ray transform is well posed if the attenuation is positive and there are no more than two conjugate points along each geodesic but it is still ill posed if there are three or more conjugate points those results follow from our analysis of the weighted x ray transform

existence and uniqueness of analytic solution for rayleigh taylor problem abstract we study the initial value problem for the classical rayleigh taylor problem if the initial data are analytic in disk r r containing the unit disk it is proved that unique solution which is analytic in r s for s 1 r exists locally in time the analysis is based on a nirenberg theorem on abstract cauchy kowalewski problem in properly chosen banach spaces

integral geometry of tensor fields on a class of non simple riemannian manifolds we study the geodesic x ray transform ig of tensor fields on a compact riemannian manifold m with non necessarily convex boundary and with possible conjugate points we assume that ig is known for geodesics belonging to an open set g with endpoints on the boundary we prove generic s injectivity and a stability estimate under some topological assumptions and under the condition that for any x t m there is a geodesic in g through x normal to without conjugate points

a numerical study on parasitic capillary waves using unsteady conformal mapping this paper describes fully nonlinear computation of unsteady motion of parasitic capillary waves that appear on the front face of steep gravity waves progressing on water of infinite depth within the framework of irrotational plane flow as an alternative to the widely used boundary integral method with mixed eulerian lagrangian mel time updating we focus on a numerical method based on unsteady conformal mapping which will be hereafter referred to as the unsteady hodograph transformation uht method in this method we solve the nonlinear evolution equations to find an unsteady conformal map in a complex plane with which the flow domain is mapped onto the unit disk while the free surface is fixed on the unit circle the aim of this work is to compare the uht method with the mel method and find a more efficient method to compute parasitic capillary waves from linear stability analysis it is found that a critical difference between these two methods arises from the kernel of cotangent function in singular integrals and the uht method can avoid some numerical instability due to it numerical examples demonstrate that the uht method is more suitable than the mel method for not only parasitic capillary waves but also capillary dominated waves in particular the uht method requires no artificial techniques such as filtering to control numerical errors in these examples in addition another major difference between the two methods is observed in terms of the clustering property of sample points on the free surface depending on the restoring force of waves gravity or surface tension

boundary rigidity and filling volume minimality of metrics close to a flat one we say that a riemannian manifold m g with a non empty boundary m is a minimal orientable filling if for every compact orientable m g with m m the inequality d g x y d g x y for all x y m implies vol m g vol m g we show that if a metric g on a region m c r n with a connected boundary is sufficiently c 2 close to a euclidean one then it is a minimal filling by studying the equality case vol m g vol m g we show that if d g x y d g x y for all x y m then m g is isometric to m g this gives the first known open class of boundary rigid manifolds in dimensions higher than two and makes a step towards a proof of michel s conjecture

the geodesic x ray transform with matrix weights author s paternain gp salo m uhlmann g zhou h abstract 2019 by johns hopkins university press consider a compact riemannian manifold of dimension 3 with strictly convex boundary such that the manifold admits a strictly convex function we show that the attenuated ray transform in the presence of an arbitrary connection and higgs field is injective modulo the natural obstruction for functions and one forms we also show that the connection and the higgs field are uniquely determined by the scattering relation modulo gauge transformations the proofs involve a reduction to a local result showing that the geodesic x ray transform with a matrix weight can be inverted locally near a point of strict convexity at the boundary and a detailed analysis of layer stripping arguments based on strictly convex exhaustion functions as a somewhat striking corollary we show that these integral geometry problems can be solved on strictly convex manifolds of dimension 3 having nonnegative sectional curvature similar results were known earlier in negative sectional curvature we also apply our methods to solve some inverse problems in quantum state tomography and polarization tomography

an inverse problem in elastodynamics uniqueness of the wave speeds in the interior abstract we consider the unique determination of internal properties of a nonhomogeneous isotropic elastic object from measurements made at the surface the 3 dimensional object is modelled by solutions of the linear hyperbolic system of equations for elastodynamics whose leading coefficients correspond to the internal properties of the object its density and elasticity we model surface measurements by the dirichlet to neumann map on a finite time interval in a previous paper the author has shown that the density and elastic properties of the surface of the object are uniquely determined by the dirichlet to neumann map here we apply that result to conclude that certain properties of the interior of the object the wave speeds are also determined we then observe that the elastodynamic polarization is determined outside the object by the dirichlet to neumann map we conclude in the case that the wave speeds are constant that the polarization data do not determine the density in the interior this problem and techniques used in its study are closely related to those in for example seismology and medical imaging the techniques used here though from geometric optics integral geometry and microlocal analysis lead to the solution of this fully three dimensional problem

on the existence of stationary splash singularities for the euler equations abstract in this paper we discuss the existence of stationary incompressible fluids with splash singularities specifically we show that there are stationary solutions to the euler equations with two fluids whose interfaces are arbitrarily close to a splash and that there are stationary water waves with splash singularities

a variational perturbation approximation method in tsallis non extensive statistical physics abstract for the generalized statistical mechanics based on the tsallis entropy a variational perturbation approximation method with the principle of minimal sensitivity is developed by calculating the generalized free energy up to the third order in variational perturbation expansion the approximation up to the first order amounts to a variational approach which covers the variational method developed by e k lenzi l c malacarne r s mendes phys rev lett 80 1998 218 and the approximations up to higher orders can systematically improve variational results as an illustrated example the generalized free energy for a classical harmonic oscillator considered in the lenzi s joint work are calculated up to the third order and the resultant approximations up to the first second and third orders are numerically compared with the exact result

minimal sensitivity optimisation of perturbative wavefunctions stevenson s principle 1981 of minimal sensitivity is applied directly to configuration space wavefunctions calculated in first order perturbation theory the redundant parameters which comprise the essential elements of minimal sensitivity calculations actually become functions of configuration space giving the method considerable flexibility and potential accuracy it is shown that the method is exact in an illustrative class of simple cases and for the more realistic case of the ground state wavefunction of the quartic oscillator it is considerably more accurate than the closely related perturbative variational approach particularly in the asymptotic region

inverse scattering problem in anisotropic media a new family of scattering invariants is constructed that differs from the scattering invariants obtained by the geometric optics approximations a local uniqueness modulo a diffeomorphism of the metric with a given hodograph function is proven under some restrictions on the curvature of the metric

on tori without conjugate points in this paper we consider riemannian metrics without conjugate points on an n torus recent work of j heber established that the gradient vector fields of busemann functions on the universal cover of such a manifold induce a natural foliation akin to the weak stable foliation for a riemannian manifold with negative sectional curvature on the unit tangent bundle the main result in the paper is that the metric is flat if this foliation is lipschitz we also prove that this foliation is lipschitz if and only if the metric has bounded asymptotes this confirms a conjecture of e hopf in this case

boundary rigidity and stability for generic simple metrics let m g be a riemannian manifold with boundary denote by pg the distance function in the metric g we consider the inverse problem of whether pg x y known for all x y on 9m determines the metric uniquely this problem arose in geophysics in an attempt to determine the inner structure of the earth by measuring the travel times of seismic waves it goes back to herglotz h and wiechert and zoeppritz wz although the emphasis has been in the case that the medium is isotropic the anisotropic case has been of interest in geophysics since it has been found that the inner core of the earth exhibits anisotropic behavior cr in differential geometry this inverse problem has been studied because of rigidity questions and is known as the boundary rigidity problem it is clear that one cannot determine the metric uniquely any isometry which is the identity at the boundary will give rise to the same measurements furthermore the boundary distance function only takes into account the shortest paths and it is easy to find counterexamples to unique determination so one needs to pose some restrictions on the metric michel mi conjectured that a simple metric g is uniquely determined up to an action of a diffeomorphism fixing the boundary by the boundary distance function pg x y known for all x and y on dm we recall definition 1 1 we say that the riemannian metric g is simple in m if dm is strictly convex w r t g and for any x g m the exponential map exp exp 1 m m is a diffeomorphism

removing the stiffness from interfacial flows with surface tension a new formulation and new methods are presented for computing the motion of fluid interfaces with surface tension in two dimensional irrotational and incompressible fluids through the laplace young condition at the interface surface tension introduces high order terms both nonlinear and nonlocal into the dynamics this leads to severe stability constraints for explicit time integration methods and makes the application of implicit methods difficult this new formulation has all the nice properties for time integration methods that are associated with having a linear constant coefficient highest order term that is using this formulation we give implicit time integration methods that have no high order time step stability constraint associated with surface tension and are explicit in fourier space the approach is based on a boundary integral formulation and applies more generally even to problems beyond the fluid mechanical context here they are applied to computing with high resolution the motion of interfaces in hele shaw flows and the motion of free surfaces in inviscid flows governed by the euler equations one hele shaw computation shows the behavior of an expanding gas bubble over long time as the interface undergoes successive tip splittings and finger competition a second computation shows the formation of a very ramified interface through the interaction of surface tension with an unstable density stratification in euler flows the computation of a vortex sheet shows its roll up through the kelvin helmholtz instability this motion culminates in the late time self intersection of the interface creating trapped bubbles of fluid this is we believe a type of singularity formation previously unobserved for such flows in 2d finally computations of falling plumes in an unstably stratified boussinesq fluid show a very similar behavior

tensor tomography on surfaces we show that on simple surfaces the geodesic ray transform acting on solenoidal symmetric tensor fields of arbitrary order is injective this solves a long standing inverse problem in the two dimensional case

on deformation boundary rigidity and spectral rigidity of riemannian surfaces with no focal points although the main results of the paper are obtained in the two dimensional case the problems under consideration can be posed in an arbitrary dimension let m g be a compact riemannian manifold with boundary the boundary rigidity problem consists of determining the riemannian metric g on m up to isometries which are the identity on the boundary by measuring the geodesic distance dg between boundary points it is easy to see that the answer to this question is no in general we can find metrics g such that there exists x0 m satisfying dist x0 m supx y mdg x y then we can change the metric g in a neighborhood of x0 without changing the boundary distance function a compact riemannian manifold m g with boundary is called boundary rigid if the metric g is uniquely determined by dg up to isometries which are the identity on the boundary there are a few classes of metrics which are known to be boundary rigid see 13 for references in this paper we consider the linearized problem let g be a one parameter family of riemannian metrics on m such that dg dg0 e e for a geodesic a b m of the metric g with a b m it is easy to see that

local lens rigidity with incomplete data for a class of non simple riemannian manifolds let be the scattering relation on a compact riemannian manifold m with nonnecessarily convex boundary that maps initial points of geodesic rays on the boundary and initial directions to the outgoing point on the boundary and the outgoing direction let be the length of that geodesic ray we study the question of whether the metric g is uniquely determined up to an isometry by knowledge of and restricted on some subset d we allow possible conjugate points but we assume that the conormal bundle of the geodesics issued from d covers t m and that those geodesics have no conjugate points under an additional topological assumption we prove that and restricted to d uniquely recover an isometric copy of g locally near generic metrics and in particular near real analytic ones

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

propagation of polarization in elastodynamics with residual stress and travel times we show that knowing the displacement to traction map associated to the equations of isotropic elastodynamics with residual stress we can determine the lens maps of compressional and shear waves we derive several consequences of this for the inverse problem of determining the residual stress and the lame parameters from the displacement to traction map

lengths and volumes in riemannian manifolds we consider the question of when an inequality between lengths of corresponding geodesics implies a corresponding inequality between volumes we prove this in a number of cases for compact manifolds with and without boundary in particular we show that for two riemannian metrics of negative curvature on a compact surface without boundary an inequality between the marked length spectra implies the same inequality between the areas with equality implying isometry