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

on the global convergence of a class of functional differential equations with applications in neural network theory abstract we study a system of retarded functional differential equations which generalise both the hopfield neural network model as well as hybrid network models of the cellular neural network type our main results give sufficient conditions for the global asymptotic stability of such systems and are milder than previously known conditions for the hybrid models when specialised to neural networks our models allow us to consider several different types of activation functions including piecewise linear sigmoids and unbounded activations as well as the usualc smooth sigmoids these issues are vital in the applications we also study neural network models with nonconstant delaysr t

on common diagonal lyapunov solutions abstract several recent results regarding common diagonal lyapunov solutions are further explored here the first one attributed to redheffer and revisited by shorten and narendra reduces the diagonal stability of a matrix to common diagonal lyapunov solutions on two matrices of order one less we present a shorter purely matrix theoretic proof of this result along with its extensions the second one concerns two different necessary and sufficient conditions due to oleng narendra and shorten for a pair of 2 2 matrices to share a common diagonal lyapunov solution we show that these two conditions are connected directly to each other

on discrete time diagonal and d stability abstract this paper introduces four discrete time analogs of different types of matrix stability diagonal simultaneous vertex and d stability the last three being defined in terms of a certain associated polytope of matrices the diagonal stability of any vertex of this polytope is shown to imply its simultaneous stability and hence d stability of the vertices it is shown by a counterexample that d stability as in the continuous time case is not equivalent to diagonal stability several important classes for which this equivalence is true are identified it is shown that simultaneous stability of the vertices is equivalent to the simultaneous stability of the whole polytope and it is conjectured that this equivalence holds without the requirement of simultaneity some other conjectures relating the four types of stability are made and it is shown that in the 2 x 2 case all four are equivalent

positivity of quasi local mass ii a spacetime is a four manifold with a pseudo metric of signature a hypersurface or a 2 surface in a spacetime is spacelike if the induced metric is pos itive definite a quasi local energy momentum vector is a vector in r3 1 associated to a spacelike 2 surface which depends on its first and second fundamental forms and the connection to its normal bundle in the spacetime the time component of the four vector is called quasi local energy mass similar to 10 8 we require the quasi local energy momentum vector to satisfy the following properties 1 it should be zero for the flat spacetime 2 the quasi local mass should be equivalent to the standard definition if the spacetime is spherically symmetric and the quasi local mass is evaluated on the spheres 7 we say that two masses mi and m2 are equivalent if there is a universal constant c 0 such that c lm area e 167r 6 the quasi local energy momentum vector should be nonspacelike and the quasi local mass should be nonnegative

solving quadratic distance problems an lmi based approach the computation of the minimum distance of a point to a surface in a finite dimensional space is a key issue in several system analysis and control problems the paper presents a general framework in which some classes of minimum distance problems are tackled via linear matrix inequality lmi techniques exploiting a suitable representation of homogeneous forms a lower bound to the solution of a canonical quadratic distance problem is obtained by solving a one parameter family of lmi optimization problems several properties of the proposed technique are discussed in particular tightness of the lower bound is investigated providing both a simple algorithmic procedure for a posteriori optimality testing and a structural condition on the related homogeneous form that ensures optimality a priori extensive numerical simulations are reported showing promising performances of the proposed method

random antagonistic matrices the ensemble of antagonistic matrices is introduced and studied in antagonistic matrices the entries and are real and have opposite signs or are both zero and the diagonal is zero this generalization of antisymmetric matrices is suggested by the linearized dynamics of competitive species in ecology

stability analysis of diagonally equipotent matrices diagonally equipotent matrices are diagonally dominant matrices for which dominance is never strict in any coordinate they appear e g as laplacian matrices of signed graphs we show in this paper that for this class of matrices it is possible to provide a complete characterization of the stability properties based only on the signs of the entries of the matrices

stability of acyclic matrices abstract necessary and sufficient conditions for stability of acyclic matrices whose principal minors are nonnegative are given also a characterization is given of additive d stable acyclic matrices

new conditions on global stability of cohen grossberg neural networks in this letter we discuss the dynamics of the cohen grossberg neural networks we provide a new and relaxed set of sufficient conditions for the cohen grossberg networks to be absolutely stable and exponentially stable globally we also provide an estimate of the rate of convergence

lmi characterization of structural and robust stability abstract this paper introduces several stability conditions for a given class of matrices expressed in terms of linear matrix inequalities lmi being thus simply and efficiently computable diagonal and simultaneous stability both characterized by polytopes of matrices are addressed using this approach a method particularly attractive to test a given matrix for d stability is proposed lyapunov parameter dependent functions are built in order to reduce conservativeness of the stability conditions the key idea is to relate hurwitz stability with a positive realness condition

fractional calculus a different approach to the analysis of viscoelastically damped structures fractional calculus is used to construct stress strain relationships for viscoelastic materials these relationships are used in the finite element analysis of viscoelastically damped structures and closed form solutions to the equations of motion are found the attractive feature of this approach is that very few empirical parameters are required to model the viscoelastic material and calculate the response of the structure for general loading conditions

p matrix properties injectivity and stability in chemical reaction systems in this paper we examine matrices which arise naturally as jacobians in chemical dynamics we are particularly interested in when these jacobians are p matrices up to a sign change ensuring certain bounds on their eigenvalues precluding certain behavior such as multiple equilibria and sometimes implying stability we first explore reaction systems and derive results which provide a deep connection between system structure and the p matrix property we then examine a class of systems consisting of reactions coupled to an external rate dependent negative feedback process and characterize conditions which ensure that the p matrix property survives the negative feedback the techniques presented are applied to examples published in the mathematical and biological literature

global stability of cellular neural networks with constant and variable delays this paper gives new conditions ensuring global asymptotic stability and global exponential stability for cellular neural networks with constant delay and variable delay respectively these conditions are derived by using the essence of piecewise linearity of the output function of cellular neural networks and by constructing lyapunov functions and functionals furthermore these conditions are significantly weaker than those given in existing literature

exponential stability and global stability of cellular neural networks in this paper we present two types of matrix stability complete stability and strong stability and some properties of them by using these properties we discuss stability of cellular neural networks and obtain some conditions ensuring uniqueness exponential stability and global asymptotic stability of the equilibrium point for cellular neural networks in particular in the two cell case the network is globally exponentially stable if and only if the feedback matrix minus the unit matrix is a complete stable matrix

deformations of the hemisphere that increase scalar curvature consider a compact riemannian manifold m of dimension n whose boundary m is totally geodesic and is isometric to the standard sphere s n 1 a natural conjecture of min oo asserts that if the scalar curvature of m is at least n n 1 then m is isometric to the hemisphere s n equipped with its standard metric this conjecture is inspired by the positive mass theorem in general relativity and has been verified in many special cases in this paper we construct counterexamples to min oo s conjecture in dimension n 3

stronger than lyapunov notions of matrix stability or how flowers help solve problems in mathematical ecology abstract persistent patterns of interactions in a multi component system e g intra and inter species relations in a community of n interacting species may imply a number of formalizations as special stronger than lyapunov notions of matrix stability like d stability qualitative stability volterra lyapunov stability and others a variety of these notions each having a certain motivation with regard to uncertainties inherent in model applications constitute a hierarchical topology sometimes very intricate and not yet well understood in a formal space of real n n matrices as visible forms of this hierarchy matrix flowers are suggested where petals correspond to subsets of particular stability kinds whose visible inclusion intersection represent logical implication junction the flowers are constructed under a few simple conventions and in the absence of ready characterizations to draw a petal often poses a challenging mathematical problem whose solution may reveal a new biological knowledge of a general nature particular petals concern the topics of strong and weak interactions in a community key species in its structure diffusion instability in its spatial dynamics where the flower plays a heuristic role in formulating a new problem or and stimulating a new application

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

boundary effect of ricci curvature on a compact riemannian manifold with boundary we study how ricci curvature of the interior affects the geometry of the boundary first we establish integral inequalities for functions defined solely on the boundary and apply them to obtain geometric inequalities involving the total mean curvature then we discuss related rigidity questions and prove ricci curvature rigidity results for manifolds with boundary

multiplicative perturbations of stable and convergent operators abstract some familiar classes of stable hilbert space operators are studied to determine how they overlap and where the unitary similarity classes of their members lie analogous but less familiar classes of convergent operators are examined with the same aim the classes considered are often sets of products m a where m is a given set of diagonal or hermitian matrices and a is a single matrix the a s for which m a is a set of stable or convergent operators are sometimes characterized

an lmi approach to constrained optimization with homogeneous forms abstract this paper considers the problem of determining the minimum euclidean distance of a point from a polynomial surface in r n it is well known that this problem is in general non convex the main purpose of the paper is to investigate to what extent linear matrix inequality lmi techniques can be exploited for solving this problem the first result of the paper shows that a lower bound to the global minimum can be achieved via the solution of a one parameter family of linear matrix inequalities lmis it is also pointed out that for some classes of problems the solution of a single lmi problem provides the lower bound the second result concerns the tightness of the bound it is shown that optimality of the lower bound amounts to solving a system of linear equations an application example is finally presented to show the features of the approach

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

stability of the positive mass theorem for rotationally symmetric riemannian manifolds we study the stability of the positive mass theorem using the intrinsic flat distance in particular we consider the class of complete asymptotically flat rotationally symmetric riemannian manifolds with nonnegative scalar curvature and no interior closed minimal surfaces whose boundaries are either outermost minimal hypersurfaces or are empty we prove that a sequence of these manifolds whose adm masses converge to zero must converge to euclidean space in the pointed intrinsic flat sense in fact we provide explicit bounds on the intrinsic flat distance between annular regions in the manifold and annular regions in euclidean space by constructing an explicit filling manifold and estimating its volume in addition we include a variety of propositions that can be used to estimate the intrinsic flat distance between riemannian manifolds without rotationally symmetry conjectures regarding the intrinsic flat stability of the positive mass theorem in the general case are proposed in the final section lee is partially supported by a psc cuny research grant and nsf dms 0903467 sormani is partially supported by a psc cuny research grant and nsf dms 1006059 1 ar x iv 1 10 4 26 57 v1 m at h d g 1 4 a pr 2 01 1 2 dan a lee and christina sormani

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

insufficient convergence of inverse mean curvature flow on asymptotically hyperbolic manifolds we construct a solution to inverse mean curvature flow on an asymptotically hyperbolic 3 manifold which does not have the convergence properties needed in order to prove a penrose type inequality this contrasts sharply with the asymptotically flat case the main idea consists in combining inverse mean curvature flow with work done by shi tam regarding boundary behavior of compact manifolds assuming the penrose inequality holds we also derive a nontrivial inequality for functions on s

absolute stability analysis in cellular neural networks with variable delays and unbounded delay abstract in this paper the existence and uniqueness of the equilibrium point and absolute stability of a class of cellular neural networks with globally lipschitz continue activation functions are investigated the neural networks contain both variable and unbounded delays the necessary and sufficient condition for the existence and uniqueness of the equilibrium point of the neural networks is obtained by constructing proper vector liapunov functions nonlinear integrodifferential inequalities involving both variable delays and unbounded delay which are corresponding to the neural networks are established by using m matrix theory and qualitative property of the integrodifferential inequalities the sufficient conditions for absolute stability global asymptotic stability are obtained

global exponential stability of high order hopfield type neural networks this paper studies the problem of global exponential stability for a class of high order hopfield type neural networks by utilizing lyapunov functions several sufficient conditions for the global exponential stability of equilibrium point are obtained the exponential convergence rate of the equilibrium point is also estimated as a special case several new criteria on global exponential stability for the corresponding first order hopfield neural networks are obtained those criteria generalize some of the recent results reported in the literature

notes on ω and τ matrices abstract results on and matrices are surveyed the question whether an matrix with positive leading principal minors is a matrix is answered negatively other open questions are discussed new conjectures and questions are introduced

absolute exponential stability of a class of continuous time recurrent neural networks this paper presents a new result on absolute exponential stability aest of a class of continuous time recurrent neural networks with locally lipschitz continuous and monotone nondecreasing activation functions the additively diagonally stable connection weight matrices are proven to be able to guarantee aest of the neural networks the aest result extends and improves the existing absolute stability and aest ones in the literature

an efficient algorithm for the evaluation of convolution integrals we propose an algorithm for the numerical evaluation of convolution integrals of the form 0 xk x y f y x dy for x 0 x our method is especially suitable in situations where the fundamental interval 0 x is very long and the kernel function k is expensive to calculate separate versions are provided where the forcing function f is known in advance and where it must be determined step by step along the solution path these methods are efficient with respect to both run time and memory requirements

on structural properties of the lyapunov matrix equation for optimal diagonal solutions we revisit the classical problem of finding a positive diagonal solution p to the lyapunov equation at p pa 0 that minimizes the lyapunov exponent the maximum eigenvalue of at p pa for a isin rnxn with the aim of identifying structural properties of the lyapunov matrix equation at the optimum using eigenvalue sensitivity notions together with optimization machinery we are able to obtain an explicit characterization of the minimum lyapunov exponent that provides such structural insight

positivity of principal minors sign symmetry and stability the relation between positivity of principal minors sign symmetry and stability of matrices is studied it is shown that for sign symmetric matrices having positive principal minors is equivalent to stability to d stability and to having a positive scaling into a stable matrix the relation between spectra of matrices some of whose powers have positive principal minors and matrices whose corresponding powers have positive sums of principal minors of each order is studied as well it is shown that for matrices of order less than 4 these two classes share the same spectra the relation of these classes and stability is studied in particular for sign symmetric matrices and for anti sign symmetric matrices

quasi local mass and the existence of horizons in this paper we obtain lower bounds for the brown york quasilocal mass and the bartnik quasilocal mass for compact three manifolds with smooth boundaries as a consequence we derive sufficient conditions for the existence of horizons for a certain class of compact manifolds with boundary and some asymptotically flat complete manifolds the method is based on analyzing hawking mass and inverse mean curvature flow

not all gkk τ matrices are stable abstract hermitian positive definite totally positive and nonsingular m matrices enjoy many common properties in particular a positivity of all principal minors b weak sign symmetry c eigenvalue monolonicity d positive stability the class of gkk matrices is defined by properties a and b whereas the class of nonsingular matrices by a and c it was conjectured that a b d d carlson j res nat bur standards sect b 78 1974 1 2 a c d g m engel and h schneider linear and multilinear algebra 4 1976 155 176 a b a property stronger than d r varga numerical methods in linear algebra 1978 pp 5 15 a b c d d hershkowitz linear algebra appl 171 1992 161 186 we describe a class of unstable gkk matrices thus disproving all four conjectures

on schur d stable matrices abstract it is shown that vertex stability implies schur d stability for real 2 2 matrices and real n n tridiagonal matrices additional results describing the class of n n complex schur d stable matrices are given

recent directions in matrix stability abstract matrix stability has been intensively investigated in the past two centuries we review work that has been done in this topic focusing on the great progress that has been achieved in the last decade or two we start with classical stability criteria of lyapunov routh and hurwitz and lienard and chipart we then study recently proven sufficient conditions for stability with particular emphasis on p matrices we investigate conditions for the existence of a stable scaling for a given matrix we review results on other types of matrix stability such as d stability additive d stability and lyapunov diagonal stability we discuss the weak principal submatrix rank property shared by lyapunov diagonally semistable matrices we also discuss the uniqueness of lyapunov scaling factors maximal lyapunov scaling factors cones of real positive semidefinite matrices and their applications to matrix stability and inertia preserving matrices

characterization of acyclic d stable matrices abstract necessary and sufficient conditions for d stability of acyclic matrices are given special cases of this characterization are recent results on tridiagonal matrices

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

localization of the spectra of p and p0 matrices abstract it is well known that if s 1 n is a spectrum of a p matrix then arg i n we show that if the number of elements of s in the right half plane or in the left half plane is given then the above bound can be improved namely there exists such that arg i n when s has exactly one element in the right half plane it is shown that barg i 2 3 independently of n

on two classes of matrices with positive diagonal solutions to the lyapunov equation abstract we characterize real indecomposable quasi jacobi matrices of class d i e those which satisfy the lyapunov equation pa a p q with p diagonal and both p and q positive definite the subclass d 2 of class d when also q is diagonal is also characterized in the case of general indecomposable real matrices

global exponential stability and global convergence in finite time of delayed neural networks with infinite gain this paper introduces a general class of neural networks with arbitrary constant delays in the neuron interconnections and neuron activations belonging to the set of discontinuous monotone increasing and possibly unbounded functions the discontinuities in the activations are an ideal model of the situation where the gain of the neuron amplifiers is very high and tends to infinity while the delay accounts for the finite switching speed of the neuron amplifiers or the finite signal propagation speed it is known that the delay in combination with high gain nonlinearities is a particularly harmful source of potential instability the goal of this paper is to single out a subclass of the considered discontinuous neural networks for which stability is instead insensitive to the presence of a delay more precisely conditions are given under which there is a unique equilibrium point of the neural network which is globally exponentially stable for the states with a known convergence rate the conditions are easily testable and independent of the delay moreover global convergence in finite time of the state and output is investigated in doing so new interesting dynamical phenomena are highlighted with respect to the case without delay which make the study of convergence in finite time significantly more difficult the obtained results extend previous work on global stability of delayed neural networks with lipschitz continuous neuron activations and neural networks with discontinuous neuron activations but without delays

yamabe flow and adm mass on asymptotically flat manifolds in this paper we investigate the behavior of adm mass and einstein hilbert functional under the yamabe flow through studying the yamabe flow by weighted spaces for parabolic operators we show that the asymptotically flat property is preserved under the yamabe flow we also obtain that adm mass is invariant under the yamabe flow and yamabe flow is the gradient flow of einstein hilbert functional on n dimensional n 3 asymptotically flat manifolds with order n 22 for n 3 4 or n 3 for n 4 moreover we show that adm mass and einstein hilbert functional are non increasing under the yamabe flow on n dimensional asymptotically flat manifolds if we only assume the order n 22 for n 4

some remarks on matrix stability with application to turing instability abstract an example of a 4 4 matrix is given that provides a counterexample to a result on turing diffusion driven instability and also answers negatively a conjecture on strong stability such instability is shown to arise from nonreal eigenvalues the relevance and the connection of our example to classes of matrix stability known in the literature are discussed

on sign symmetric circulant matrices abstract in the last four decades many researchers have studied and analyzed the study of sign symmetry and positivity of principal minors of matrices since these issues are related to stability in this work we extend the theory about sign symmetric basic p circulant permutation and sifted p circulant matrices we present and prove sufficient and necessary conditions for p matrices and necessary conditions for p2 matrices finally we present a class of matrices where the p2 matrices are stable

proof of the riemannian penrose inequality with charge for multiple black holes we present a proof of the riemannian penrose inequality with charge in the context of asymptotically flat initial data sets for the einstein maxwell equations having possibly multiple black holes with no charged matter outside the horizon and satisfying the relevant dominant energy condition the proof is based on a generalization of hubert bray s conformal flow of metrics adapted to this setting

beyond fractional derivatives local approximation of other convolution integrals dynamic systems involving convolution integrals with decaying kernels of which fractionally damped systems form a special case are non local in time and hence infinite dimensional straightforward numerical solution of such systems up to time t needs o t 2 computations owing to the repeated evaluation of integrals over intervals that grow like t finite dimensional and local approximations are thus desirable we present here an approximation method which first rewrites the evolution equation as a coupled in finite dimensional system with no convolution and then uses galerkin approximation with finite elements to obtain linear finite dimensional constant coefficient approximations for the convolution this paper is a broad generalization based on a new insight of our prior work with fractional order derivatives singh chatterjee 2006 nonlinear dyn 45 183 206 in particular the decaying kernels we can address are now generalized to the laplace transforms of known functions of these the power law kernel of fractional order differentiation is a special case the approximation can be refined easily the local nature of the approximation allows numerical solution up to time t with o t computations examples with several different kernels show excellent performance a key feature of our approach is that the dynamic system in which the convolution integral appears is itself approximated using another system as distinct from numerically approximating just the solution for the given initial values this allows non standard uses of the approximation e g in stability analyses

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

on the uniqueness of the lyapunov scaling factors sufficient conditions are given for a lyapunov diagonally semistable matrix a to have a unique up to scalar multiplication lyapunov scaling factor suppose d is a lyapunov scaling factor of a a necessary and sufficient condition is given for the uniqueness of d the results extend work of hershkowitz and schneider

on the stability of p matrices we establish two sufficient conditions for the stability of a p matrix first we show that a p matrix is positive stable if its skew symmetric component is sufficiently smaller in matrix norm than its symmetric component this result generalizes the fact that symmetric p matrices are positive stable and is analogous to a result by carlson which shows that sign symmetric p matrices are positive stable second we show that a p matrix is positive stable if it is strictly row column square diagonally dominant for every order of minors this result generalizes the fact that strictly row diagonally dominant p matrices are stable we compare our sufficient conditions with the sign symmetric condition and demonstrate that these conditions do not imply each other

a polyhedron comparison theorem for 3 manifolds with positive scalar curvature the study of comparison theorems in geometry has a rich history in this paper we establish a comparison theorem for polyhedra in 3 manifolds with nonnegative scalar curvature answering affirmatively a dihedral rigidity conjecture by gromov for a large collections of polyhedra with interior non negative scalar curvature and mean convex faces we prove the dihedral angles along its edges cannot be everywhere less or equal than those of the corresponding euclidean model unless it is isometric to a flat polyhedron

on global asymptotic stability of recurrent neural networks with time varying delays in this paper by constructing a new lyapunov functional and using m matrix and topological degree tool problem of the global asymptotic stability gas is discussed for a class of recurrent neural networks with time varying delays some simple and new sufficient conditions are obtained ensuring existence uniqueness of the equilibrium point and its gas of the neural networks some previous works are improved in addition this condition does not require the activation functions to be differentiable bounded and monotone nondecreasing and the weight connected matrices to be symmetric the neural network model considered in this paper include the delayed hopfield neural networks bidirectional associative memory networks and delayed cellular neural networks as its special cases

scalar curvature rigidity of geodesic balls in sn in this paper we prove a scalar curvature rigidity result for geodesic balls in s n this result contrasts sharply with the counterexamples to min oo s conjecture constructed in deformations of the hemisphere that increase scalar curvature invent math 185 2011 175 197

scaling a canonical design problem for networks motivated by decentralized control and numerical methods applications we explore the scaling problem that is we consider the design of a diagonal matrix k so as to i place the eigenvalues of the product kg in the olhp and ii optimize a scalar performance measure c kg of interest for a square topology matrix g we first illustrate our methodology for scaling design which combines optimization machinery with graph algebra notions for a class of topology matrices and a specific performance measure in particular for symmetric positive definite topologies and for a dominant eigenvalue ratio performance measure we find that the optimally scaled system kg has a special structure the right eigenvectors associated with the largest and smallest eigenvalues have corresponding entries of identical magnitude and hence differ only in sign pattern using this structure we develop both explicit and algorithmic methods for computing the optimal scaling and cost after illustrating our methodology with this

the gbc mass for asymptotically hyperbolic manifolds the paper consists of two parts in the first part by using the gauss bonnet curvature which is a natural generalization of the scalar curvature we introduce a higher order mass the gauss bonnet chern mass m mathbb h k for asymptotically hyperbolic manifolds and show that it is a geometric invariant moreover we prove a positive mass theorem for this new mass for asymptotically hyperbolic graphs and establish a relationship between the corresponding penrose type inequality for this mass and weighted alexandrov fenchel inequalities in the hyperbolic space mathbb h n in the second part we establish these weighted alexandrov fenchel inequalities in mathbb h n for any horospherical convex hypersurface sigma begin aligned int sigma v sigma 2k 1 d mu ge c n 1 2k 1 omega n 1 left left frac sigma omega n 1 right frac n k n 1 left frac sigma omega n 1 right frac n 2k k n 1 right k end aligned where sigma j is the j th mean curvature of sigma subset mathbb h n v cosh r and sigma is the area of sigma as an application we obtain an optimal penrose type inequality for the new mass defined in the first part begin aligned m k mathbb h ge frac 1 2 k left left frac sigma omega n 1 right frac n k n 1 left frac sigma omega n 1 right frac n 2k k n 1 right k end aligned for asymptotically hyperbolic graphs with a horizon type boundary sigma provided that a dominant energy type condition tilde l k ge 0 holds both inequalities are optimal

deformation of scalar curvature and volume the stationary points of the total scalar curvature functional on the space of unit volume metrics on a given closed manifold are known to be precisely the einstein metrics one may consider the modified problem of finding stationary points for the volume functional on the space of metrics whose scalar curvature is equal to a given constant in this paper we localize a condition satisfied by such stationary points to smooth bounded domains the condition involves a generalization of the static equations and we interpret solutions and their boundary values of this equation variationally on domains carrying a metric that does not satisfy the condition we establish a local deformation theorem that allows one to achieve simultaneously small prescribed changes of the scalar curvature and of the volume by a compactly supported variation of the metric we apply this result to obtain a localized gluing theorem for constant scalar curvature metrics in which the total volume is preserved finally we note that starting from a counterexample of min oo s conjecture such as that of brendle marques neves counterexamples of arbitrarily large volume and different topological types can be constructed

generalized lyapunov approach for convergence of neural networks with discontinuous or non lipschitz activations the paper considers a class of additive neural networks where the neuron activations are modeled by discontinuous functions or by continuous non lipschitz functions some tools are developed which enable us to apply a lyapunov like approach to differential equations with discontinuous right hand side modeling the neural network dynamics the tools include a chain rule for computing the time derivative along the neural network solutions of a nondifferentiable lyapunov function and a comparison principle for this time derivative which yields conditions for exponential convergence or convergence in finite time by means of the lyapunov like approach a general result is proved on global exponential convergence toward a unique equilibrium point of the neural network solutions moreover new results on global convergence in finite time are established which are applicable to neuron activations with jump discontinuities or neuron activations modeled by means of continuous non lipschitz holder functions

fractional order systems and pi sup spl lambda d sup spl mu controllers dynamic systems of an arbitrary real order fractional order systems are considered the concept of a fractional order pi sup spl lambda d sup spl mu controller involving fractional order integrator and fractional order differentiator is proposed the laplace transform formula for a new function of the mittag leffler type made it possible to obtain explicit analytical expressions for the unit step and unit impulse response of a linear fractional order system with fractional order controller for both open and closed loops an example demonstrating the use of the obtained formulas and the advantages of the proposed pi sup spl lambda d sup spl mu controllers is given

asymptotic stability of impulsive high order hopfield type neural networks in this paper we discuss impulsive high order hopfield type neural networks investigating their global asymptotic stability by using lyapunov function method sufficient conditions that guarantee global asymptotic stability of networks are given these criteria can be used to analyse the dynamics of biological neural systems or to design globally stable artificial neural networks two numerical examples are given to illustrate the effectiveness of the proposed method

a new characterization of simultaneous lyapunov diagonal stability via hadamard products abstract a well known characterization by kraaijevanger 14 for lyapunov diagonal stability states that a real square matrix a is lyapunov diagonally stable if and only if a s is a p matrix for any positive semidefinite s with nonzero diagonal entries this result is extended here to a new characterization involving similar hadamard multiplications for simultaneous lyapunov diagonal stability on a set of matrices among the main ingredients for this extension are a new concept called p sets and a recent result regarding simultaneous lyapunov diagonal stability by berman goldberg and shorten 2

on scalar curvature rigidity of vacuum static spaces in this paper we extend the local scalar curvature rigidity result in brendle and marques j differ geom 88 379 394 2011 to a small domain on general vacuum static spaces which confirms the interesting dichotomy of local surjectivity and local rigidity about the scalar curvature in general in the light of the paper corvino commun math phys 214 137 189 2000 we obtain the local scalar curvature rigidity of bounded domains in hyperbolic spaces we also obtain the global scalar curvature rigidity for conformal deformations of metrics in the domains where the lapse functions are positive on vacuum static spaces with positive scalar curvature and show such domains are maximal which generalizes the work in hang and wang commun anal geom 14 91 106 2006

analysis of complete stability for discrete time cellular neural networks with piecewise linear output functions this letter discusses the complete stability of discrete time cellular neural networks with piecewise linear output functions under the assumption of certain symmetry on the feedback matrix a sufficient condition of complete stability is derived by finite trajectory length because the symmetric conditions are not robust the complete stability of networks may be lost under sufficiently small perturbations the robust conditions of complete stability are also given for discrete time cellular neural networks with multiple equilibrium points and a unique equilibrium point these complete stability results are robust and available

asymptotic stability of systems with saturation constraints in the present paper the authors establish new sufficient conditions for the global asymptotic stability of linear systems operating on the unit hypercube in r sup n and of linear systems with partial state saturation constraints they also determine a set of necessary and sufficient conditions for the global asymptotic stability of second order linear systems with saturation constraints on both states using the above results systems of the type considered herein are widely used in several areas of applications including control systems signal processing and artificial neural networks

a comprehensive review of stability analysis of continuous time recurrent neural networks stability problems of continuous time recurrent neural networks have been extensively studied and many papers have been published in the literature the purpose of this paper is to provide a comprehensive review of the research on stability of continuous time recurrent neural networks including hopfield neural networks cohen grossberg neural networks and related models since time delay is inevitable in practice stability results of recurrent neural networks with different classes of time delays are reviewed in detail for the case of delay dependent stability the results on how to deal with the constant variable delay in recurrent neural networks are summarized the relationship among stability results in different forms such as algebraic inequality forms m matrix forms linear matrix inequality forms and lyapunov diagonal stability forms is discussed and compared some necessary and sufficient stability conditions for recurrent neural networks without time delays are also discussed concluding remarks and future directions of stability analysis of recurrent neural networks are given

maximal lyapunov scaling factors and their applications in the study of lyapunov diagonal semistability of block triangular matrices abstract for a lyapunov diagonally semistable matrix a we introduce the concept of maximal lyapunov scaling factor that is a lyapunov scaling factor d of a such that the range of ad da t is of maximal dimension we investigate properties of maximal lyapunov scaling factors and apply them in proving a characterization for block triangular lyapunov diagonally semistable matrices in terms of lyapunov diagonal semistability of the diagonal blocks the latter result is used to prove that a block triangular matrix is lyapunov diagonally semistable if and only if every 2 2 block principal submatrix of it is lyapunov diagonally semistable

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

curvature estimates for immersed hypersurfaces in riemannian manifolds we establish mean curvature estimate for immersed hypersurface with nonnegative extrinsic scalar curvature in riemannian manifold n n 1 bar g through regularity study of a degenerate fully nonlinear curvature equation in general riemannian manifold the estimate has a direct consequence for the weyl isometric embedding problem of mathbb s 2 g in 3 dimensional warped product space n 3 bar g we also discuss isometric embedding problem in spaces with horizon in general relativity like the anti de sitter schwarzschild manifolds and the reissner nordstrom manifolds

lyapunov diagonal semistability of real h matrices we characterize lyapunov diagonally stable real h matrices and those real h matrices which are lyapunov diagonally semistable but not lyapunov diagonally stable called lyapunov diagonally near stable the latter characterization is given in terms of the principal submatrix rank property defined here we apply our results to the numerical abscissas of real matrices one of our main tools is a slight strengthening of classical results of ostrowski which we derive from a fundamental theorem of wielandt

convex invertible sets and matrix sign function it is proved that in two dimensional unital real algebras a step by step construction of minimal convex invertible sets terminates in two steps an analogous construction of minimal convex conditionally invertible sets of the complex field also terminates in two steps and the complex field is the only finite dimensional complex algebra with this property connections are made with the matrix sign function the matrix sign function of a matrix belongs to the convex invertible set generated by the matrix an example is given to show that the matrix sign function cannot always be obtained in two steps by following the step by step construction

geometric inequalities for quasi local masses in this paper lower bounds are obtained for quasi local masses in terms of charge angular momentum and horizon area in particular we treat three quasi local masses based on a hamiltonian approach namely the brown york liu yau and wang yau masses the geometric inequalities are motivated by analogous results for the adm mass they may be interpreted as localized versions of these inequalities and are also closely tied to the conjectured bekenstein bounds for entropy of macroscopic bodies in addition we give a new proof of the positivity property for the wang yau mass which is used to remove the spin condition in higher dimensions furthermore we generalize a recent result of lu and miao to obtain a localized version of the penrose inequality for the static wang yau mass

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

consensus problems on networks with antagonistic interactions in a consensus protocol an agreement among agents is achieved thanks to the collaborative efforts of all agents expresses by a communication graph with nonnegative weights the question we ask in this paper is the following is it possible to achieve a form of agreement also in presence of antagonistic interactions modeled as negative weights on the communication graph the answer to this question is affirmative on signed networks all agents can converge to a consensus value which is the same for all agents except for the sign necessary and sufficient conditions are obtained to describe cases in which this is possible these conditions have strong analogies with the theory of monotone systems linear and nonlinear laplacian feedback designs are proposed