finite time asynchronous h control for markov jump repeated scalar non linear systems with input constraints finite time control issue for a quite comprehensive class of system model that is markov jump repeated scalar nonlinear systems mjrsnss has been addressed an asynchronous controller is first proposed in mjrsnss which has less conservatism than mode dependent when the partly available mode information is encountered the input constraints associated with physical limitations in some practical applications have been fully considered which make our model more realistic this paper is concerned with the problem of finite time h control for markov jump repeated scalar nonlinear systems mjrsnss with input constraints the jumps between mjrsnss and desired mode dependent controllers are considered to be nonsynchronous the mode transition of the controllers is governed by a nonstationary markov chain which can model the different degrees of asynchronous jumps between mjrsnss and controllers with the aid of lyapunov krasovskii methodology some h performance criteria for the mjrsnss are proposed based on these criteria sufficient conditions on the existence of the desired controllers are established and then the gains of the controllers can be obtained by solving a convex optimization problem finally an example is presented to illustrate the effectiveness of our method

gradient systems in view of information geometry abstract dualistic properties of a gradient flow on a manifold m associated with a dualistic structure g is studied from an information geometrical viewpoint some useful applications are also investigated

contact geometric descriptions of vector fields on dually flat spaces and their applications in electric circuit models and nonequilibrium statistical mechanics contact geometry has been applied to various mathematical sciences and it has been proposed that a contact manifold and a strictly convex function induce a dually flat space that is used in information geometry here such a dually flat space is related to a legendre submanifold in a contact manifold in this paper contact geometric descriptions of vector fields on dually flat spaces are proposed on the basis of the theory of contact hamiltonian vector fields based on these descriptions two ways of lifting vector fields on legendre submanifolds to contact manifolds are given for some classes of these lifted vector fields invariant measures in contact manifolds and stability analysis around legendre submanifolds are explicitly given throughout this paper legendre duality is explicitly stated in addition to show how to apply these general methodologies to applied mathematical disciplines electric circuit models and some examples taken from nonequilibrium statistical mechanics are analyzed

extrema of sums of heterogeneous quadratic forms abstract we analyze the following problem arising in various situations in multivariate statistical analysis we are given k symmetric positive definite n n matrices a1 a2 ak k n and we would like to maximize the function ki 1 xtiaixi under the constraint that x1 x2 xk rn form an orthonormal system some theoretical results as well as an algorithm are presented

rigid body constrained noisy point pattern matching noisy pattern matching problems arise in many areas e g computational vision robotics guidance and control stereophotogrammetry astronomy genetics and high energy physics least squares pattern matching over the euclidean space e sup n for unordered sets of cardinalities p and q is commonly formulated as a combinatorial optimization problem having complexity p p 1 spl middot spl middot spl middot p q 1 q spl les p since p and q may be 10 sup 3 or larger in typical applications less than satisfactory suboptimal methods are usually employed a hybrid approach is described for solving the pattern matching problem under rigid motion constraints which often apply the method reduces the complexity to l sub 21 spl middot n sup 4 l sub 12 spl middot p sup 3 where l sub 12 and l sub 21 are the number of iterations required by steepest ascent and singular value decomposition svd based procedures respectively

neural learning by geometric integration of reduced rigid body equations in previous contributions the second author of this paper presented a new class of algorithms for orthonormal learning of linear neural networks with p inputs and m outputs based on the equations describing the dynamics of a massive rigid frame on the stiefel manifold these algorithms exhibit good numerical stability strongly binding to the sub manifold of constraints and good controllability of the learning dynamics but are not completely satisfactory from a computational complexity point of view in the recent literature efficient methods of integration on the stiefel manifold have been proposed by various authors see for example phys d 156 2001 219 numer algorithms 32 2003 163 j numer anal 21 2001 463 numer math 83 1999 599 inspired by these approaches in this paper we propose a new and efficient representation of the mentioned learning equations and a new way to integrate them the numerical experiments show how the new formulation leads to significant computational savings especially when p m the effectiveness of the algorithms is substantiated by numerical experiments concerning principal subspace analysis and independent component analysis these experiments were carried out with both synthetic and real world data

limiting weak type behavior for multilinear fractional integrals abstract in this paper we establish the limiting weak type behavior for multilinear fractional integrals i a and i a which were introduced by kenig and stein see the definition in section 1 although kenig and stein proved that these two kinds of multilinear fractional operators have the same weak and strong type estimates we provide that their limiting behaviors are totally different when p 1 p 2 p k 1 specifically motivated by janakiraman s result we establish when f i 0 1 i k there holds lim 0 x r n i k f 1 f k x k n n k k n v n k n n i 1 k f i l 1 r n which extend the first author s result about linear fractional integral to multilinear case but for i a there holds lim 0 x r n i a f 1 f k x k n n 0 limiting behaviors for the other indexes are also presented and at last we set up limiting weak type behaviors for multilinear fractional integrals when

vector fields global flows and the calculus of pulses the point of this paper is to show a remarkable property of functional expansions in the form of iterated integrals in capturing the dynamics of digital computation when realized in the form of pulse spike driven controlled smooth dynamical systems the key features of the structure thus captured by the formalism are a the combinatorics of spike trains in various input channels and their effect on the discrete time dynamics of the realized automaton and b the relative timing precedence of the spike trains in various input channels we do this by providing globally valid functional expansions for a certain kind of hybrid dynamical system when subject to inputs which resemble pulse trains we give these functional expansions in the form of a product of a nonlinear holonomy factor and a local fliess functional expansion we also obtain a computational scheme to determine the response of finite automata to input sequences which is meant as a substitute for the naive iteration of the next state maps we obtain this by embedding finite automata in suitable controlled dynamical systems on the special orthogonal group incidentally we show that our realization of automata can also be modified to realize hybrid machines in which a finite automaton is interacting with an external system some of whose analog variables are converted into interspike time intervals which in turn are simultaneously tokenized and acted upon by the modified realization of the finite automaton the latter is achieved intrinsically by exploiting the dynamics of the realization

a learning result for continuous time recurrent neural networks abstract the following learning problem is considered for continuous time recurrent neural networks having sigmoidal activation functions given a black box representing an unknown system measurements of output derivatives are collected for a set of randomly generated inputs and a network is used to approximate the observed behavior it is shown that the number of inputs needed for reliable generalization the sample complexity of the learning problem is upper bounded by an expression that grows polynomially with the dimension of the network and logarithmically with the number of output derivatives being matched

recurrent neural network training with feedforward complexity this paper presents a training method that is of no more than feedforward complexity for fully recurrent networks the method is not approximate but rather depends on an exact transformation that reveals an embedded feedforward structure in every recurrent network it turns out that given any unambiguous training data set such as samples of the state variables and their derivatives we need only to train this embedded feedforward structure the necessary recurrent network parameters are then obtained by an inverse transformation that consists only of linear operators as an example of modeling a representative nonlinear dynamical system the method is applied to learn bessel s differential equation thereby generating bessel functions within as well as outside the training set

a lie bracket decomposition and its application to flows on symmetric matrices abstract the paper studies the problem of finding a canonical form for differential equations on symmetric matrices that applies in particular to isospectral flows and spectrum increasing flows such a canonical form is obtained using the result for any lie algebra u and elements w z u such that z is ad semisimple there exist unique a b u with w z a b such that a z c for some c u and z b 0 taking u gl n r the ad semisimplicity condition holds for symmetric matrices z and differential equations then have a canonical form z t z t a t b t where a t z c t and z b t 0 a flow on symmetric matrices is isospectral if and only if b t 0 and spectrum increasing if and only if b t is positive definite applying this canonical form twice to the finite nonperiodic toda flow we obtain a double bracket form for this differential equation that differs from one recently found by bloch

on realizability of neural networks based input output models in the classical state space form this paper proves that the typical neural network based input output model does not have a state space realization and suggests the additive nonlinear auto regressive with exogenous input anarx structure as an excellent choice for neural network based input output models the advantage of the anarx model is that the time steps in the argument are pair wise decomposed which allows the anarx model to be realized in state space and to be linearized via dynamic output feedback moreover accessibility of the state space realization has been proved

maps on statistical manifolds exactly reduced from the perron frobenius equations for solvable chaotic maps maps on a parameter space for expressing distribution functions are exactly derived from the perron frobenius equations for a generalized boole transform family here the generalized boole transform family is a one parameter family of maps where it is defined on a subset of the real line and its probability distribution function is the cauchy distribution with some parameters with this reduction some relations between the statistical picture and the orbital one are shown from the viewpoint of information geometry the parameter space can be identified with a statistical manifold and then it is shown that the derived maps can be characterized also with an induced symplectic structure from a statistical structure symplectic and information geometric aspects of the derived maps are discussed

dynamical methods for polar decomposition and inversion of matrices we show how one may obtain polar decomposition as well as inversion of fixed and time varying matrices using a class of nonlinear continuous time dynamical systems first we construct a dynamic system that causes an initial approximation of the inverse of a time varying matrix to flow exponentially toward the true time varying inverse using a time parameterized homotopy from the identity matrix to a fixed matrix with unknown inverse and applying our result on the inversion of time varying matrices we show how any positive definite fixed matrix may be dynamically inverted by a prescribed time without an initial guess at the inverse we then construct a dynamical system that solves for the polar decomposition factors of a time varying matrix given an initial approximation for the inverse of the positive definite symmetric part of the polar decomposition as a byproduct this method gives another method of inverting time varying matrices finally using homotopy again we show how dynamic polar decomposition may be applied to fixed matrices with the added benefit that this allows us to dynamically invert any fixed matrix by a prescribed time

dual connections and affine geometry on etudie des immersions affines de varietes statistiques dans l espace affine plus particulierement on etablit deux resultats sur le rapport entre l existence de telles immersions et le fait que les varietes statistiques connexes en question ont des courbures constantes

information geometry of boltzmann machines a boltzmann machine is a network of stochastic neurons the set of all the boltzmann machines with a fixed topology forms a geometric manifold of high dimension where modifiable synaptic weights of connections play the role of a coordinate system to specify networks a learning trajectory for example is a curve in this manifold it is important to study the geometry of the neural manifold rather than the behavior of a single network in order to know the capabilities and limitations of neural networks of a fixed topology using the new theory of information geometry a natural invariant riemannian metric and a dual pair of affine connections on the boltzmann neural network manifold are established the meaning of geometrical structures is elucidated from the stochastic and the statistical point of view this leads to a natural modification of the boltzmann machine learning rule

lur e systems with multilayer perceptron and recurrent neural networks absolute stability and dissipativity sufficient conditions for absolute stability and dissipativity of continuous time recurrent neural networks with two hidden layers are presented in the autonomous case this is related to a lur e system with multilayer perceptron nonlinearity such models are obtained after parametrizing general nonlinear models and controllers by a multilayer perceptron with one hidden layer and representing the control scheme in standard plant form the conditions are expressed as matrix inequalities and can be employed for nonlinear h sub spl infin control and imposing closed loop stability in dynamic backpropagation

an information geometric approach to a theory of pragmatic structuring 1 1 the motivation of the approach in the field of neural networks so called infomax principles like the principle of maximum information preservation by linsker 20 are formulated to derive learning rules that improve the information processing properties of neural systems see 12 these principles which are based on information theoretic measures are intended to describe the mechanism of learning in the brain there the starting point is a low dimensional and biophysiologically motivated parametrization of the neural system which need not necessarily be compatible with the given optimization principle in contrast to this we establish theoretical results about the low complexity of optimal solutions for the optimization problem of frequently used measures like the mutual information in an unconstrained and more theoretical setting in the present paper we do not comment on applications to modeling neural networks this is intended to be done in a further step where the results can be used for the characterization of good parameter sets that on the one hand are compatible with the underlying optimization and on the other hand are biologically motivated

complete controllability of continuous time recurrent neural networks this paper studies controllability for the class of control systems commonly called continuous time recurrent neural networks it is shown that under a generic condition on the input matrix the system is controllable for every possible state matrix the result holds when the activation function is the hyperbolic tangent

dynamical systems that sort lists diagonalize matrices and solve linear programming problems abstract we establish a number of properties associated with the dynamical system h h h n where h and n are symmetric n by n matrices and a b ab ba the most important of these come from the fact that this equation is equivalent to a certain gradient flow on the space of orthogonal matrices we are especially interested in the role of this equation as an analog computer for example we show how to map the data associated with a linear programming problem into h 0 and n in such a way as to have h h h n evolve to a solution of the linear programming problem this result can be applied to find systems which solve a variety of genetic combinatorial optimization problems and it even provides an algorithm for diagonalizing symmetric matrices

recurrent neural networks coupled with linear systems observability in continuous and discrete time we give necessary and sufficient conditions for observability of a class of recurrent neural networks having a subsystem where the activation function is the identity an algorithm for computing all pairs of indistinguishable states is also given

asymptotic theory of sequential estimation differential geometrical approach sequential estimation continues observations until the observed sample satisfies a prescribed criterion its properties are superior on the average to those of nonsequential estimation in which the number of observations is fixed a priori a higher order asymptotic theory of sequential estimation is given in the framework of geometry of multidimensional curved exponential families this gives a design principle of the second order efficient sequential estimation procedure it is also shown that a sequential estimation can be designed to have a covariance stabilizing effect at the same time

bounds of the induced norm and model reduction errors for systems with repeated scalar nonlinearities the class of nonlinear systems described by a discrete time state equation containing a repeated scalar nonlinearity as in recurrent neural networks is considered sufficient conditions are derived for the stability and induced norm of such systems using positive definite diagonally dominant lyapunov functions or storage functions satisfying appropriate linear matrix inequalities results are also presented for model reduction errors for such systems

martingale transforms and related singular integrals byrodrigo banuelosabstract the operators obtained by taking conditional expectation of continuoustime martingale transforms are studied both on the circle t and on r using aburkholder gundy inequality for vector valued martingales it is shown that thevector formed by any number of these operators is bounded on lp r 1 p oo with constants that depend only on p and the norms of the matrices involved as acorollary we obtain a recent result of stein on the boundedness of the riesztransforms on lp k 1 p oo with constants independent of n

model indexing and object recognition using 3d viewpoint invariance object recognition and location of 3d objects from range data are described based on invariant representations of characteristic views of single objects depth data is processed to remove outliers then smoothed and segmented into surface patches fitted by polynomial equations mathematical invariants are derived from groups of these surface patches and used for indexing a model database stored in the form of a kd tree the pose of the known object may then be determined from the euclidean centre and orientation of the higher order surfaces formed from the feature groups the approach is complete scalable and extendable the robustness of the method is evaluated in terms of the stability and discriminatory power of the representation and the accuracy of the determined pose

the nonlinear geometry of linear programming iii projective legendre transform coordinates and hilbert geometry this paper studies projective scaling trajectories which are the tra jectories obtained by following the infinitesimal version of karmarkar s linear programming algorithm a nonlinear change of variables projective legendre transform coordinates is introduced to study these trajectories the projective legendre transform mapping has a coordinate free geometric interpretation in terms of the notion of centering by a projective transformation let h be a set of linear programming constraints aj x bj 1 s j s m on rn such that its polytope of feasible solutions ph is bounded and contains 0 in its interior the projective legendre transform mapping i h is given by m a where ph x l b j 1 aj x i x ph x h m ph x x here ph x is the legendre transform coordinate mapping introduced in part ii l h x is a one to one and onto mapping of the interior of the feasible solu tion polytope int ph to the interior of its polar polytope int p the set of projective scaling trajectories with objective function c x co are mappcd under l h to the set of straight line segments in int p passing through the boundary point clco of p as a consequence the projective scaling tra jectories for all objective functions can be interpreted as the complete set of geodesics actually distinguished chords of a projectively invariant metric ge ometry on int ph which is isometric to hilbert geometry on the interior of the polar polytope p

geometrical theory of higher order asymptotics of test interval estimator and conditional inference higher order asymptotic powers of one sided and two sided tests both conditional and unconditional are evaluated for a one parameter curved exponential family of distributions by using differential geometrical notions higher order asymptotic powers and the expected size of one sided and two sided interval estimators both conditional and unconditional are also obtained the tests and interval estimators which are third order most powerful at any arbitrary specified one point are explicitly designed with the help of the approximate ancillary statistic the characteristics of the conditional inference are elucidated by proving the equivalence up to the third order of the conditional inference and likelihood ratio inference geometrical notions such as curvatures and angles play a fundamental role in the present theory

uniqueness of circuits and systems containing one nonlinearity we study systems containing one memoryless nonlinearity we show that two such systems have the same i o operator only when they are related by simple scaling delay and loop transformations the theory is applied to one port networks containing one nonlinear element

recurrent neural nets as dynamical boolean systems with application to associative memory discrete time discrete state recurrent neural networks are analyzed from a dynamical boolean systems point of view in order to devise new analytic and design methods for the class of both single and multilayer recurrent artificial neural networks with the proposed dynamical boolean systems analysis we are able to formulate necessary and sufficient conditions for network stability which are more general than the well known but restrictive conditions for the class of single layer networks 1 symmetric weight matrix with 2 positive diagonal and 3 asynchronous update in terms of design we use a dynamical boolean systems analysis to construct a high performance associative memory with this boolean memory we can guarantee that all fundamental memories are stored and also guarantee the size of the basin of attraction for each fundamental memory

model construction of boolean network via observed data in this paper a set of data is assumed to be obtained from an experiment that satisfies a boolean dynamic process for instance the dataset can be obtained from the diagnosis of describing the diffusion process of cancer cells with the observed datasets several methods to construct the dynamic models for such boolean networks are proposed instead of building the logical dynamics of a boolean network directly its algebraic form is constructed first and then is converted back to the logical form firstly a general construction technique is proposed to reduce the size of required data the model with the known network graph is considered motivated by this the least in degree model is constructed that can reduce the size of required data set tremendously next the uniform network is investigated the number of required data points for identification of such networks is independent of the size of the network finally some principles are proposed for dealing with data with errors

vector valued estimates on limiting weak type behaviors of singular integrals and maximal operators abstract this paper is devoted to studying the vector valued estimates for the limiting weak type behaviors of singular and fractional integrals as well as maximal operators with homogeneous kernels under the assumptions of that the kernels satisfy certain dini type conditions the limiting weak type behaviors for the corresponding vector valued operators are given our results imply that the classical vector valued weak type endpoint estimates for singular and fractional integrals or maximal operators are not sharp

estimation in the presence of infinitely many nuisance parameters geometry of estimating functions when there exist nuisance parameters whose number increases in proportion to that of independent observations it is in general difficult to get a consistent or efficient estimator of a common structural parameter the present paper proposes a new theory based on a vector bundle consisting of certain random variables over the statistical model structures and properties of estimating functions are elucidated in the class of consistent estimators a necessary and sufficient condition is obtained for the existence of a consistent estimator given by an estimating function a necessary and sufficient condition is then given for the existence of the optimal estimator in this class which is further obtained when it exists in their derivations the concept of dual connections and parallel transports plays an essential role the results are applied to a special type of exponential family and the optimal estimators are explicitly obtained in some examples this explains the reason why the conditional score plays an important role

least squares matching problems abstract a large class of matching problems can be thought of in the following way one is given a group g which acts on a riemannian manifold and one is given two points x 1 and x 2 in the manifold the problem is to find g g such that d g x 1 x 2 is as small as possible if the group has finite cardinality this is a combinatorial optimization problem whereas if g is a lie group one can use the calculus to obtain necessary conditions our main results establish circumstances under which one can relax the combinatorial problem to an appropriate lie group problem and then solve it by the method of steepest descent an interesting class of applications involves embedding the permutation group on n letters in the orthogonal group o n with the latter acting on r n n 1 2 via a symmetricized tensor product the convergence properties of the descent equation which arises in this way are analyzed in some detail

stabilization and performance synthesis for systems with repeated scalar nonlinearities the class of nonlinear systems described by a discrete time state equation containing a repeated scalar nonlinearity as in recurrent neural networks is considered given a plant of this form sufficient conditions are derived for 1 the parametrization of all controllers of the same form such that the closed loop is stable in the sense of a diagonally dominant lyapunov function and 2 the synthesis of a controller of the same form so that the induced norm of the closed loop is under a prescribed level several of these conditions can be written into linear matrix inequalities

differential geometry of curved exponential families curvatures and information loss the differential geometrical framework is given for analyzing statistical problems related to multi parameter families of distributions the dualistic structures of the exponential families and curved exponential families are elucidated from the geometrical viewpoint the duality connected by the legendre transformation is thus extended to include two kinds of affine connections and two kinds of curvatures the second order information loss is calculated for fisher efficient estimators and is decomposed into the sum of two non negative terms one is related to the exponential curvature of the statistical model and the other is related to the mixture curvature of the estimator only the latter term depends on the estimator and vanishes for the maximum likelihood estimator a set of statistics which recover the second order information loss are given the second order efficiency also is obtained the differential geometry of the function space of distributions is discussed

escort evolutionary game theory abstract a family of replicator like dynamics called the escort replicator dynamic is constructed using information geometric concepts and generalized information divergences from information geometry and statistical thermodynamics a single formula lyapunov function is given that covers the entire class of dynamics which includes the replicator dynamic and the projection dynamic as well as several new dynamics a further class is discussed that allows for more variation such as variable intensities of selection

induced l sub 2 and generalized h sub 2 filtering for systems with repeated scalar nonlinearities this paper provides complete results on the filtering problem for a class of nonlinear systems described by a discrete time state equation containing a repeated scalar nonlinearity as in recurrent neural networks both induced l sub 2 and generalized h sub 2 indexes are introduced to evaluate the filtering performance for a given stable discrete time systems with repeated scalar nonlinearities our purpose is to design a stable full order or reduced order filter with the same repeated scalar nonlinearities such that the filtering error system is asymptotically stable and has a guaranteed induced l sub 2 or generalized h sub 2 performance sufficient conditions are obtained for the existence of admissible filters since these conditions involve matrix equalities the cone complementarity linearization procedure is employed to cast the nonconvex feasibility problem into a sequential minimization problem subject to linear matrix inequalities which can be readily solved by using standard numerical software if these conditions are feasible a desired filter can be easily constructed these filtering results are further extended to discrete time systems with both state delay and repeated scalar nonlinearities the techniques used in this paper are very different from those used for previous controller synthesis problems which enable us to circumvent the difficulty of dilating a positive diagonally dominant matrix a numerical example is provided to show the applicability of the proposed theories

global stability and stabilization of switched boolean network with impulsive effects the present paper investigates the global stability and stabilization of switched boolean network with impulsive effects by using the semi tensor product the algebraic form of the switched boolean network with impulsive effects is obtained then sufficient and necessary condition is presented for the global stability of the switched boolean network with impulsive effects furthermore sufficient and necessary conditions are provided for the stabilization of the switched boolean network with impulsive effects under two types of controls finally examples are presented to illustrate the main results

complete affine hypersurfaces part i the completeness of affine metrics on etudie les proprietes des hypersurfaces invariantes sous le groupe des transformations affines unimodulaires

a variation on karmarkar s algorithm for solving linear programming problems the algorithm described here is a variation on karmarkar s algorithm for linear programming it has several advantages over karmarkar s original algorithm in the first place it applies to the standard form of a linear programming problem and produces a monotone decreasing sequence of values of the objective function the minimum value of the objective function does not have to be known in advance secondly in the absence of degeneracy the algorithm converges to an optimal basic feasible solution with the nonbasic variables converging monotonically to zero this makes it possible to identify an optimal basis before the algorithm converges

hessian information geometric formulation of hamiltonian systems and generalized toda s dual transform in this paper a class of classical hamiltonian systems is geometrically formulated this class is such that a hamiltonian can be written as the sum of a kinetic energy function and a potential energy function in addition these energy functions are assumed strictly convex for this class of hamiltonian systems hessian and information geometric formulation is given with this formulation a generalized toda s dual transform is proposed where his original transform was used in deriving his integrable lattice system then a relation between the generalized toda s dual transform and the legendre transform of a class of potential energy functions is shown as an extension of this formulation dissipation less electric circuit models are also discussed in the geometric viewpoint above

on explicitly solvable gradient systems of moser karmarkar type abstract a generalization and a classification of explicitly solvable gradient systems which appear in linear programming and neurodynamics are studied it is shown that nonlinear dynamical systems of moser karmarkar type generically take both a lax representation and a gradient form a related optimization problem on a simplex is discussed an integration procedure and a complete description of the invariant manifolds and the phase portrait of the dynamical systems of moser karmarkar type are also investigated

further results on controllability of recurrent neural networks this paper studies controllability properties of recurrent neural networks the new contributions are 1 an extension of a previous result to a slightly different model 2 a formulation and proof of a necessary and sufficient condition and 3 an analysis of a low dimensional case for which the hypotheses made in previous work do not apply

recognition by linear combinations of models an approach to visual object recognition in which a 3d object is represented by the linear combination of 2d images of the object is proposed it is shown that for objects with sharp edges as well as with smooth bounding contours the set of possible images of a given object is embedded in a linear space spanned by a small number of views for objects with sharp edges the linear combination representation is exact for objects with smooth boundaries it is an approximation that often holds over a wide range of viewing angles rigid transformations with or without scaling can be distinguished from more general linear transformations of the object by testing certain constraints placed on the coefficients of the linear combinations three alternative methods of determining the transformation that matches a model to a given image are proposed

the continuous time rayleigh quotient flow on the sphere a continuous time differential equation inspired by the rayleigh quotient iteration for a symmetric matrix a is studied the set of all continuous solutions termed the rayleigh quotient flow is shown to be a time scaled version of the newton flow for rayleigh quotient cost functional the scaling factor ensures that the rate of variation of the rayleigh quotient is constant and positive along solutions this interpretation leads to a precise phase portrait for the rayleigh quotient flow it is shown that a complete non degenerate solution of the rayleigh quotient flow visits each of the eigenvectors of a in ascending order

limiting weak type behavior for singular integral and maximal operators the following limit result holds for the weak type 1 1 constant of dilation commuting singular integral operator t in r n for f l 1 r n f 0 lim m x r n 0 tf x 1 n s n 1 x d x f 2 for the maximal operator m the corresponding result is lim

bayesian segmentation via asymptotic partition functions asymptotic approximations to the partition function of gaussian random fields are derived textures are characterized via gaussian random fields induced by stochastic difference equations determined by finitely supported stationary linear difference operators adjusted to be nonstationary at the boundaries it is shown that as the scale of the underlying shape increases the log normalizer converges to the integral of the log spectrum of the operator inducing the random field fitting the covariance of the fields amounts to fitting the parameters of the spectrum of the differential operator induced random field model matrix analysis techniques are proposed for handling textures with variable orientation examples of texture parameters estimated from training data via asymptotic maximum likelihood are shown isotropic models involving powers of the laplacian and directional models involving partial derivative mixtures are explored parameters are estimated for mitochondria and actin myocin complexes in electron micrographs and clutter in forward looking infrared images deformable template models are used to infer the shape of mitochondria in electron micrographs with the asymptotic approximation allowing easy recomputation of the partition function as inference proceeds

inverse eigenvalue problems a collection of inverse eigenvalue problems are identified and classified according to their characteristics current developments in both the theoretic and the algorithmic aspects are summarized and reviewed in this paper this exposition also reveals many open questions that deserve further study an extensive bibliography of pertinent literature is attached

observability of automata networks fixed and switching cases automata networks are a class of fully discrete dynamical systems which have received considerable interest in various different areas this brief addresses the observability of automata networks and switched automata networks in a unified framework and proposes simple necessary and sufficient conditions for observability the results are achieved by employing methods from symbolic computation and are suited for implementation using computer algebra systems several examples are presented to demonstrate the application of the results

positive and negative circuits in discrete neural networks we study the relationships between the positive and negative circuits of the connection graph and the fixed points of discrete neural networks dnns as main results we give necessary conditions and sufficient conditions for the existence of fixed points in a dnn moreover we exhibit an upper bound for the number of fixed points in terms of the structure and number of positive circuits in the connection graph this allows the determination of the maximum capacity for storing vectors in dnns as fixed points depending on the architecture of the network

computational capabilities of random automata networks for reservoir computing this paper underscores the conjecture that intrinsic computation is maximal in systems at the edge of chaos we study the relationship between dynamics and computational capability in random boolean networks rbn for reservoir computing rc rc is a computational paradigm in which a trained readout layer interprets the dynamics of an excitable component called the reservoir that is perturbed by external input the reservoir is often implemented as a homogeneous recurrent neural network but there has been little investigation into the properties of reservoirs that are discrete and heterogeneous random boolean networks are generic and heterogeneous dynamical systems and here we use them as the reservoir a rbn is typically a closed system to use it as a reservoir we extend it with an input layer as a consequence of perturbation the rbn does not necessarily fall into an attractor computational capability in rc arises from a tradeoff between separability and fading memory of inputs we find the balance of these properties predictive of classification power and optimal at critical connectivity these results are relevant to the construction of devices which exploit the intrinsic dynamics of complex heterogeneous systems such as biomolecular substrates

diffusion equations from master equations a discrete geometric approach in this paper continuous time master equations with finite states employed in nonequilibrium statistical mechanics are formulated in the language of discrete geometry in this formulation chains in algebraic topology are used and master equations are described on graphs that consist of vertices representing states and of directed edges representing transition matrices it is then shown that master equations under the detailed balance conditions are equivalent to discrete diffusion equations where the laplacians are defined as self adjoint operators with respect to introduced inner products an isospectral property of these laplacians is shown for non zero eigenvalues and its applications are given the convergence to the equilibrium state is shown by analyzing this class of diffusion equations in addition a systematic way to derive closed dynamical systems for expectation values is given for the case that the detailed balance conditions are not imposed master equations are expressed as a form of a continuity equation