multi stability in monotone input output systems this paper studies the emergence of multi stability and hysteresis in those systems that arise under positive feedback starting from monotone systems with well defined steady state responses such feedback configurations appear routinely in several fields of application and especially in biology characterizations of global stability behavior are stated in terms of easily checkable graphical conditions

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

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

a dual to lyapunov s stability theorem lyapunov s second theorem is a standard tool for stability analysis of ordinary differential equations here we introduce a theorem which can be viewed as a dual to lyapunov s result from existence of a scalar function satisfying certain inequalities it follows that almost all trajectories of the system tend to zero the scalar function has a physical interpretation as the stationary density of a substance that is generated in all points of the state space and flows along the system trajectories if the stationary density is bounded everywhere except at a singularity in the origin then almost all trajectories tend towards the origin the weaker notion of stability allows for applications also in situations where lyapunov s theorem cannot be used moreover the new criterion has a striking convexity property related to control synthesis

what is wrong with the hausdorff measure in finsler spaces we construct a class of finsler metrics in three dimensional space such that all their geodesics are lines but not all planes are extremal for their hausdorff area functionals this shows that if the hausdorff measure is used as notion of volume on finsler spaces then totally geodesic submanifolds are not necessarily minimal filling results such as those of ivanov on two dimensional minimal fillings st petersburg math j 13 2002 17 25 do not hold and integral geometric formulas do not exist on the other hand using the holmes thompson definition of volume we prove a general crofton formula for finster spaces and give an easy proof that their totally geodesic hypersurfaces are minimal c 2005 elsevier inc all rights reserved

synchronization and transient stability in power networks and nonuniform kuramoto oscillators motivated by recent interest for multiagent systems and smart grid architectures we discuss the synchronization problem for the network reduced model of a power system with nontrivial transfer conductances our key insight is to exploit the relationship between the power network model and a first order model of coupled oscillators assuming overdamped generators possibly due to local excitation controllers a singular perturbation analysis shows the equivalence between the classic swing equations and a nonuniform kuramoto model here nonuniform kuramoto oscillators are characterized by multiple time constants nonhomogeneous coupling and nonuniform phase shifts extending methods from transient stability synchronization theory and consensus protocols we establish sufficient conditions for synchronization of nonuniform kuramoto oscillators these conditions reduce to necessary and sufficient tests for the standard kuramoto model combining our singular perturbation and kuramoto analyses we derive

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

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

singular numbers of contractions in spaces with an indefinite metric and yamamoto s theorem abstract a theorem of yamamoto on singular numbers of matrices is extended to g bounded matrices

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

global bounded synchronization of general dynamical networks with nonidentical nodes this note addresses the problem of synchronization for general dynamical networks with nonidentical nodes the coupling strength outer coupling configuration and inner connection in such networks are all time varying neither an equilibrium for each node nor a synchronization manifold is assumed to exist an estimate of the convergence domain for a general class of time varying nonlinear systems is given by introducing the average dynamics of all nodes and based on this estimate a criterion of global synchronization in the sense of boundedness of the maximum state deviation between nodes is developed an explicit bound of the maximum state deviation between nodes is obtained by the maximum difference between each node dynamics and the average dynamics the proposed criterion is an extension of several related synchronization criteria for the case of identical nodes to the case of nonidentical nodes

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

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

bistable biological systems a characterization through local compact input to state stability many biological systems have the capacity to operate in two distinct modes in a stable manner typically the system can switch from one stable mode to the other in response to a specific external input mathematically these bistable systems are usually described by models that exhibit at least two distinct stable steady states on the other hand to capture biological variability it seems more natural to associate to each stable mode of operation an appropriate invariant set in the state space rather than a single fixed point a general formulation is proposed in this paper which allows freedom in the form of kinetic interactions and is suitable for establishing conditions on the existence of one or more disjoint forward invariant sets for the given system stability with respect to each set is studied in terms of a local notion of input to state stability with respect to compact sets two well known systems that exhibit bistability are analyzed in this framework the lac operon and an apoptosis network for the first example the question of designing an input that drives the system to switch between modes is also considered

pencils of complex and real symmetric and skew matrices abstract this expository paper establishes the canonical forms under congruence for pairs of complex or real symmetric or skew matrices the treatment is in the spirit of the well known book of gantmacher on matrix theory and may be regarded as a supplement to gantmacher s chapters on pencils of matrices

forward completeness unboundedness observability and their lyapunov characterizations a finite dimensional continuous time system is forward complete if solutions exist globally for positive time this paper shows that forward completeness can be characterized in a necessary and sufficient manner by means of smooth scalar growth inequalities moreover a version of this fact is also proved for systems with inputs and a generalization is also provided for systems with outputs and a notion unboundedness observability of relative completeness we apply these results to obtain a bound on reachable states in terms of energy like estimates of inputs

examples of unoriented area minimizing surfaces a comprehensive study is made of constructions of area minimizing flat chains modulo two many have singularities for instance any bounded area mini mizing submanifold of rn occurs as the singular set of some area minimizing flat chain modulo two in some rn introduction given a k l dimensional boundary in r one can seek to minimize area in some class of surfaces we want to take a large class that includes nonorientable as well as orientable surfaces without restrictions on the singularities or topological type such a class appears in geometric measure theory as the flat chains modulo two the existence and regularity of area minimizing flat chains modulo two are already well established cf 7 4 2 26 9 unfortunately examples have been rare this paper presents several methods for constructing area minimizing flat chains modulo two the first five theorems include the following five examples 1 a mobius strip 2 the graph over a convex region of any function satisfying the minimal surface equation

a minimax characterization for eigenvalues of hermitian pencils ii abstract we establish a minimax characterization for extreme real eigenvalues of a general hermitian matrix pencil the results extend the previous generalizations for real diagonable hermitian pencils and the classical courant fischer theorem

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

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

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

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

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

boundary and lens rigidity of lorentzian surfaces let g be a lorentzian metric on the plane 2 that agrees with the standard metric g0 dx2 dy2 outside a compact set and so that there are no conjugate points along any time like geodesic of 2 g then 2 g and 2 g0 are isometric further if m g and m p are two dimensional compact time oriented lorentzian manifolds with space like boundaries and so that all time like geodesies of m g maximize the distances between their points and m g and m g are boundary isometric then there is a conformal diffeomorphism between m g and m g and they have the same areas similar results hold in higher dimensions under an extra assumption on the volumes of the manifolds

robust synchronization for multistable systems in this note we study a robust synchronization problem for multistable systems evolving on manifolds within an input to state stability iss framework based on a recent generalization of the classical iss theory to multistable systems a robust synchronization protocol is designed with respect to a compact invariant set of the unperturbed system the invariant set is assumed to admit a decomposition without cycles that is with neither homoclinic nor heteroclinic orbits numerical simulation examples illustrate our theoretical results

determination of the spacetime from local time measurements we consider an inverse problem for a lorentzian spacetime m g and show that time measurements that is the knowledge of the lorentzian time separation function on a submanifold sigma subset m determine the c infty jet of the metric in the fermi coordinates associated to sigma we use this result to study the global determination of the spacetime m g when it has a real analytic structure or is stationary and satisfies the einstein scalar field equations in addition to this we require that m g is geodesically complete modulo scalar curvature singularities the results are lorentzian counterparts of extensively studied inverse problems in riemannian geometry the determination of the jet of the metric and the boundary rigidity problem we give also counterexamples in cases when the assumptions are not valid and discuss inverse problems in general relativity

inverse problem for the wave equation with a white noise source we consider a smooth riemannian metric tensor g on mathbb r n and study the stochastic wave equation for the laplace beltrami operator partial t 2 u delta g u f here f f t x is a random source that has white noise distribution supported on the boundary of some smooth compact domain m subset mathbb r n we study the following formally posed inverse problem with only one measurement suppose that g is known only outside of a compact subset of mint and that a solution u t x omega 0 is produced by a single realization of the source f t x omega 0 we ask what information regarding g can be recovered by measuring u t x omega 0 on mathbb r times partial m we prove that such measurement together with the realization of the source determine the scattering relation of the riemannian manifold m g with probability one that is for all geodesics passing through m the travel times together with the entering and exit points and directions are determined in particular if m g is a simple riemannian manifold and g is conformally euclidian in m the measurement determines the metric g in m

characterizations of integral input to state stability for systems with multiple invariant sets we generalize the theory of input to state stability iss and of its characterizations by means of lyapunov dissipation inequalities to the study of systems admitting invariant sets which are not necessarily stable in the sense of lyapunov but admit a suitable hierarchical decomposition it is the latter which allows to greatly extend the class of systems to which iss theory can be applied allowing in a unified treatement to deal with oscillators in euclidean coordinates almost globally asymptotically stable systems on manifolds systems with multiple equilibria in bbr n just to name a few

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

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

trace minimization principles for positive semi definite pencils abstract this paper is concerned with x h ax subject to x h bx j for a hermitian matrix pencil a b where j is diagonal and j 2 i the identity matrix of apt size the same problem was investigated earlier by kovac striko and veselic linear algebra appl 216 1995 139 158 for the case in which b is assumed nonsingular but in this paper b is no longer assumed nonsingular and in fact a b is even allowed to be a singular pencil it is proved among others that the infimum is finite if and only if a b is a positive semi definite pencil in the sense that there is a real number 0 such that a 0 b is positive semi definite the infimum when finite can be expressed in terms of the finite eigenvalues of a b sufficient and necessary conditions for the attainability of the infimum are also obtained

global lyapunov analysis of multistable nonlinear systems a new type of stability is introduced and its equivalent lyapunov characterization is presented the problem of global stability for the compact set composed of all invariant solutions of a nonlinear system several equilibriums for instance is studied such a problem statement allows us to analyze global stability properties for multistable systems it is shown that several well known multistable systems satisfy this new stability property

an almost global notion of input to state stability a new definition of almost global input to state stability for systems on differentiable manifolds is discussed some of its properties are investigated and specific examples are shown which can be treated by means of dual lyapunov techniques

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

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

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

analysis of interconnected oscillators by dissipativity theory this paper employs dissipativity theory for the global analysis of limit cycles in particular dynamical systems of possibly high dimension oscillators are regarded as open systems that satisfy a particular dissipation inequality it is shown that this characterization has implications for the global stability analysis of limit cycle oscillations i in isolated oscillators ii in interconnections of oscillators and iii for the global synchrony analysis in interconnections of identical oscillators