multiple solutions for 2mth order sturm liouville boundary value problems growth conditions are imposed on f such that the following boundary value problem 1 my 2m f t y i 1y 2i 0 i 1y 2i 1 0 i 1y 2i 1 i 1y 2i 1 1 0 0 i m 1 has an arbitrary number of positive solutions

comparison of eigenvalues for sturm liouville boundary value problems on a measure chain under consideration is a class of even ordered linear differential equations 1 m 2m t i 0m 1pi t 2i t with sturm liouville boundary conditions i 2i 0 i 2i 0 yi 1 2i 1 i 1 2i 1 0 for 0 i m 1 the derivative in this dynamic equation is the generalized delta derivative defined on a measure chain for a pair of eigenvalue problems for this dynamic equation we first verify the existence of smallest positive eigenvalues and then establish a comparison between the smallest eigenvalues of each eigenvalue problem

triple positive solutions and dependence on higher order derivatives abstract in this paper we consider the lidstone boundary value problem y 2 m t f y t y 2 j t y 2 m 1 t 0 t 1 y 2 i 0 0 y 2 i 1 0 i m 1 where 1 m f 0 growth conditions are imposed on f and inequalities involving an associated green s function are employed which enable us to apply the leggett williams fixed point theorem to cones in ordered banach spaces this in turn yields the existence of at least three positive symmetric concave solutions the emphasis here is that f depends on higher order derivatives

h2 optimal model order reduction by two sided technique on grassmann manifold via the cross gramian of bilinear systems abstractin this paper we discuss the optimal h2 model order reduction mor problem for bilinear systems the h2 optimal mor problem of bilinear systems is considered as the minimisation problem on grassmann manifold which is stored as a quotient space of the noncompact stifiel manifold grassmann manifold whose tangent space is endowed with a riemannian metric is a riemannian manifold in its tangent space equipped with the riemannian metric we derive the negative gradients of the cost function i e the steepest descent direction of the cost function after that the formulas of geodesic on grassmann manifold are given then we perform linear searches along geodesics to obtain the optimal solutions thereby a two sided mor iterative algorithm is proposed to construct an order reduced bilinear system which is used to simulate the output and input responses of the original bilinear system numerical examples demonstrate the effectiveness of our algorithm

lyapunov equations energy functionals and model order reduction of bilinear and stochastic systems we discuss the relation of a certain type of generalized lyapunov equations to gramians of stochastic and bilinear systems together with the corresponding energy functionals while gramians and energy functionals of stochastic linear systems show a strong correspondence to the analogous objects for deterministic linear systems the relation of gramians and energy functionals for bilinear systems is less obvious we discuss results from the literature for the latter problem and provide new characterizations of input and output energies of bilinear systems in terms of algebraic gramians satisfying generalized lyapunov equations in any of the considered cases the definition of algebraic gramians allows us to compute balancing transformations and implies model reduction methods analogous to balanced truncation for linear deterministic systems we illustrate the performance of these model reduction methods by showing numerical experiments for different bilinear systems

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

a note on the numerical approximate solutions for generalized sylvester matrix equations with applications abstract in the present paper we propose a krylov subspace method for solving large and sparse generalized sylvester matrix equations the proposed method is an iterative projection method onto matrix krylov subspaces as a particular case we show how to adapt the ilu and the ssor preconditioners for solving large sylvester matrix equations numerical examples and applications to some pde s will be given

triple positive solutions of conjugate boundary value problems ii we consider the following boundary value problems 1 n py n a t h y n 2 t 0 1 y i 0 0 0 i p 1 y i 1 0 0 i n p 1 and 1 n p ny f k y y n 1y n 2 0 k m iy 0 0 i p 1 iy m p 1 0 0 i n p 1 where 1 p n 1 is fixed by employing fixed point theorems for operators on a cone existence criteria are developed for multiple at least three positive solutions of the boundary value problems as an application we also establish the existence of radial solutions of certain partial difference equations several examples are included to dwell upon the importance of the results obtained

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

on the existence of positive solutions for semilinear elliptic equations in the annulus abstract we study the existence of positive radial solutions of u g x u 0 in annuli with dirichlet dirichlet neumann boundary conditions we prove that the problems have positive radial solutions on any annulus if is sublinear at 0 and

frequency interval model reduction of bilinear systems in this technical note a new method for model reduction of bilinear systems is presented the method is developed in particular for many applications in which one is interested to approximate a system in a given frequency interval to this end new generalized frequency interval gramians are introduced for bilinear systems it is shown that these gramians are the solutions to the so called frequency interval generalized lyapunov equations the conditions for these equations to be solvable are derived and an algorithm is proposed to solve such equations iteratively the method is further illustrated with the help of an illustrative example the numerical results show that the method is more accurate than its previous counterpart which is based on the ordinary gramians

existence of three positive solutions to a second order boundary value problem on a measure chain growth conditions are imposed on f such that the boundary value problem x t f x t t t1 t2 x t1 x t1 0 and x t2 x t2 0 where t1 t2 from a measure chain t has at least three positive solutions by way of the five functionals fixed point theorem

green s function for a third order generalized right focal problem abstract we determine the green s function for the third order three point generalized right focal boundary value problem x t 0 t 1 t t 3 x t 1 x t 2 0 x t 3 x t 3 0 where 0 and determine conditions on the coefficients and boundary points to ensure its positivity then we use the krasnoselskii and leggett and williams fixed point theorems to prove the existence of solutions to the nonlinear problem x t f t x t t 1 t t 3 x t 1 x t 2 0 x t 3 x t 3 0 for certain constraints placed on nonnegative f

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

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

higher order three point boundary value problem on time scales in this paper we consider a higher order three point boundary value problem on time scales we study the existence of solutions of a non eigenvalue problem and of at least one positive solution of an eigenvalue problem later we establish the criteria for the existence of at least two positive solutions of a non eigenvalue problem examples are also included to illustrate our results

twin solutions of boundary value problems for differential equations on measure chains a new twin fixed point theorem is applied to obtain the existence of at least two positive solutions for the right focal boundary value problem on a measure chain t y t f y t 0 t 0 1 t y 0 y 1 0

numerical calculation of integrals involving oscillatory and singular kernels and some applications of quadratures numerical methods for strongly oscillatory and singular functions are given in this paper beside a summary of standard methods and product integration rules we consider a class of complex integration methods several applications of quadrature processes in problems in telecom munications and physics are also presented 1998 elsevier science ltd all rights reserved integration oscillatory kernel singular kernel orthogonal polynomials product rules gaussian quaclratures error function bessel functions legendre functions

positive solutions for p n p conjugate boundary value problems a new existence result is presented for the p n p conjugate boundary value problem 1 n py n h t y 0 t 1 withy i 0 0 0 i p 1 andy i 1 0 0 i n p 1 our nonlinearityhmay be singular aty 0 t 0 and ort 1

constant sign solutions for a system of integral equations on time scales we consider the following system of integral equations on a time scale t u i t email protected a s b g i t s f i s u 1 s u 2 s u n s ds email protected a s l b t email protected email protected n using a nonlinear alternative of leray schauder type and krasnosel skii s fixed point theorem we establish the existence of one or more constant sign solutions of the system the generality of the results obtained is illustrated through application to boundary value problems

contracting convex hypersurfaces in riemannian manifolds by their mean curvature which shrink towards the center of the initial sphere in finite time it was shown in 3 that this behaviour is very typical if the initial hypersurface m o o r 1 is uniformly convex then the surfaces m t contract smoothly to a single point in finite time and the shape of the surfaces becomes spherical at the end of the contraction if the ambient space n is a general riemannian manifold the curvature of n will interfere with the mot ion of the surfaces m r we want to show here that the contract ion first to a small sphere and then to a single point is still

local convexity and nonnegative curvature gromov s proof of the sphere theorem an immersed hypersurface s in a riemannian manifold m will be called e convex for some e 0 if all principal curvatures have the same sign and absolute value at least e can one characterize all compact e convex immersed hypersurfaces of a complete manifold m if m is euclidean n space n 3 this problem is solved by a theorem of hadamard 14 generalized by hopf 15 s is embedded and bounds a convex n disk see 6 6 6 3 if m is a flat space form s is no longer necessarily embedded but it still bounds an immersed convex n disk the aim of this paper is to show the same fact for a manifold m of curvature k 0

the lipschitz continuity of the distance function to the cut locus let n be a closed submanifold of a complete smooth riemannian manifold m and uv the total space of the unit normal bundle of n for each v c uv let p v denote the distance from n to the cut point of n on the geodesic y with the velocity vector tyv 0 v the continuity of the function p on uv is well known in this paper we prove that p is locally lipschitz on which p is bounded in particular if m and n are compact then p is globally lipschitz on uy therefore the canonical interior metric 6 may be introduced on each connected component of the cut locus of n and this metric space becomes a locally compact and complete length space let n be an immersed submanifold of a complete cc riemannian manifold m and 7 uv n the unit normal bundle of n for each positive integer k and vector v c uv let a number k v denote the parameter value of fy where 7yv denotes the geodesic for which the velocity vector is v at t 0 such that yv ak v is the k th focal point conjugate point for the case in which n is a point of n along y counted with focal or conjugate multiplicities a unit speed geodesic segment 0 a m emanating from n is called an n segment if t d n y t on 0 a by the first variation formula an n segment is orthogonal to n a point 7y to on an n segment 7y v e uv is called a cut point of n if there is no n segment properly containing y o to for each v c uv let p v denote the distance from n to the cut point on y of n whitehead 27 investigated the structure of the conjugate locus and the cut locus of a point on a real analytic finsler manifold he determined the structure of the conjugate locus around a conjugate point for which the conjugate multiplicity is locally constant on its neighborhood cf also 25 and proved the continuity of the function p in compact symmetric spaces t sakai 19 and m takeuchi 23 determined the detailed structure of the cut locus of a point the detailed structure of the cut locus of a point in a 2 dimensional riemannian manifold has been investigated by poincare myers and others 7 11 13 hartman first tried to show the absolute continuity of the function p when m is 2 dimensional he proved in 8 that if p is of bounded variation then p is absolutely continuous recently hebda 11 and the first named author 13 independently proved ambrose s problem by showing that p is absolutely continuous on a closed arc on which p is bounded when n is a point in a 2 dimensional riemannian manifold therefore the cut locus of a point in a compact 2 dimensional received by the editors october 14 1998 and in revised form april 13 1999 2000 mathematics subject classification primary 53c22 secondary 28a78 supported in part by a grant in aid for scientific research from the ministry of education science sports and culture japan 2000 american mathematical society

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

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

double solutions of boundary value problems for 2mth order differential equations and difference equations abstract a double fixed point theorem is applied to obtain the existence of at least two positive solutions for the boundary value problem 1 m y 2 m t f y t t 0 1 y 2 i 0 y 2 i 1 1 0 0 i m 1 it is later applied to obtain the existence of at least two positive solutions for the analogous discrete boundary value problem 1 m 2 m u k g u k k 0 n 2 i u 0 2 i 1 u n 1 0 0 m 1

a three point boundary value problem for nonlinear fourth order differential equations abstract in this paper the authors consider the nonlinear fourth order ordinary differential equation e u t g t f u 0 with the boundary conditions b u 0 u 1 u 0 u p u 1 0 some results on the existence and nonexistence of positive solutions to problem e b are obtained results on the existence of infinitely many positive solutions are also presented examples are included to demonstrate that the results are sharp

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

even order nonlinear eigenvalue problems on a measure chain we consider the even order nonlinear eigenvalue problem 1 mu 2m t f t u t u 2i 0 u 2i 1 0 0 i m 1 on a measure chain t results on existence and nonexistence of positive solutions are obtained for evaluated in different intervals under certain assumptions the complete scenario for all 0 is established our work develops and improves many known results in the literature even for the case that t is the real number line we also interpret our general results on measure chains to the discrete case which yields a new set of conditions for the existence and nonexistence of positive solutions of eigenvalue problems for difference equations

control configuration selection for bilinear systems via generalised hankel interaction index array decentralised and partially decentralised control strategies are very popular in practice to come up with a suitable decentralised or partially decentralised control structure it is important to select the appropriate input and output pairs for control design this procedure is called control configuration selection it is well known that a suitable control configuration selection is an important prerequisite for a successful industrial control in this paper the problem of control configuration selection for multiple input and multiple output mimo bilinear processes is addressed first the concept of the cross gramian is developed for bilinear systems the conditions for the existence of generalised cross gramian are derived it is shown that if the cross gramian exists it is the solution to the generalised sylvester equation to obtain the cross gramian in a more computationally efficient way an iterative method for solving the generalised sylvester equation is proposed the generalised cross gram

triple solutions of focal boundary value problems on time scale we consider the following differential equation on a time scale t y d d t p t y s t 0 t a b t together with focal boundary conditions y a 0 y d s b 0 where a b e t and a s b by using two different fixed point theorems criteria are established for the existence of triple positive solutions of the boundary value problem we also include some examples to illustrate the results obtained

higher order self adjoint boundary value problems on time scales in this study higher order self adjoint differential expressions on time scales and their associated self adjoint boundary conditions are discussed the symmetry property of the corresponding green s functions is shown together with specific formulas of green s functions for select time scales

solutions of two point boundary value problems for even order differential equations abstract the existence of solutions of the two point boundary value problems consisting of the even order differential equations x 2 n t f t x t x t x 2 n 2 t r t 0 t 1 and the boundary value conditions i x 2 i 0 i x 2 i 1 0 0 i x 2 i 1 i x 2 i 1 1 0 i 0 1 n 1 is studied sufficient conditions for the existence of at least one solution of above bvps are established it is interesting that the nonlinearity f in the equation depends on all lower derivatives especially odd order derivatives and the growth conditions imposed on f are allowed to be super linear the degrees of phases variables are allowed to be greater than 1 if it is a polynomial the results are different from known ones since we do not apply the green s functions of the corresponding problem and the method to obtain a priori bounds of solutions is different from known ones examples that cannot be solved by known results are given to illustrate our theorems

invariant distributions beurling transforms and tensor tomography in higher dimensions in the recent articles paternain et al j differ geom 98 147 181 2014 invent math 193 229 247 2013 a number of tensor tomography results were proved on two dimensional manifolds the purpose of this paper is to extend some of these methods to manifolds of any dimension a central concept is the surjectivity of the adjoint of the geodesic ray transform or equivalently the existence of certain distributions that are invariant under the geodesic flow we prove that on any anosov manifold one can find invariant distributions with controlled first fourier coefficients the proof is based on subelliptic type estimates and a pestov identity we present an alternative construction valid on manifolds with nonpositive curvature based on the fact that a natural beurling transform on such manifolds turns out to be essentially a contraction finally we obtain uniqueness results in tensor tomography both on simple and anosov manifolds that improve earlier results by assuming a condition on the terminator value for a modified jacobi equation