optimal rigidity estimates for nearly umbilical surfaces let r 3 be a smooth compact connected surface without boundary and denote by a its second fundamental form we prove the existence of a universal constant c such that

embedded tori with prescribed mean curvature abstract we construct a sequence of compact oriented embedded two dimensional surfaces of genus one into euclidean 3 space with prescribed almost constant mean curvature of the form h x 1 a x for x large when a 0 and 0 2 such surfaces are close to sections of unduloids with small neck size folded along circumferences centered at the origin and with larger and larger radii the construction involves a deep study of the corresponding jacobi operators an application of the lyapunov schmidt reduction method and some variational argument

algebraic and differential operator equations abstract explicit expressions for solutions of boundary value problems and cauchy problems related to the operator differential equation x n a n 1 x n 1 a 0 x 0 are given in terms of solutions of the algebraic operator equation x n a n 1 x n 1 a 0 0 a method for solving this algebraic equation is studied

contraction of information in systems far from equilibrium the relationship between a physical model and a time series model or an innovation model is discussed from the viewpoint of contraction of information due to the way of observation in this framework measurable correlation functions are expressed in terms of both models by using the riccati type equation and the singular value decomposition of the hankel matrix a data oriented model is derived from measurable correlation functions there occurs the contraction in a noise source space or random forces from the mechanism of contraction random forces of systems can be identified by using two innovation models in different states

a mass transportation approach to quantitative isoperimetric inequalities a sharp quantitative version of the anisotropic isoperimetric inequality is established corresponding to a stability estimate for the wulff shape of a given surface tension energy this is achieved by exploiting mass transportation theory especially gromov s proof of the isoperimetric inequality and the brenier mccann theorem a sharp quantitative version of the brunn minkowski inequality for convex sets is proved as a corollary

some sphere theorems in linear potential theory in this paper we analyze the capacitary potential due to a charged body in order to deduce sharp analytic and geometric inequalities whose equality cases are saturated by domains with spherical sy

embedded constant mean curvature surfaces in euclidean three space in this paper we refine the construction and related estimates for complete constant mean curvature surfaces in euclidean three space developed in kapouleas ann math 131 239 330 1990 by adopting the more precise and powerful version of the methodology which was developed in kapouleas invent math 119 3 443 518 1995 as a consequence we remove the severe restrictions in establishing embeddedness for complete constant mean curvature surfaces in kapouleas ann math 131 239 330 1990 and we produce a very large class of new embedded examples of finite topology

the sharp quantitative isoperimetric inequality a quantitative sharp form of the classical isoperimetric inequality is proved thus giving a positive answer to a conjecture by hall

a simple characterization of optimal arma predictors abstract the coefficients of the optimal steady state k step ahead predictor for an arma process in general depend on k it is shown that a simple formula completely characterizes all these coefficients

on the linear quadratic guassian problem with correlated noise and its relation to minimum variance control the linear quadratic gaussian log stochastic control problem with correlated dynamic and observation noise and no information delay is studied an explicit feedback solution is given for finite as well as infinite time problems these results are then applied to minimum variance control of single input single output armax systems the lqg controller and the minimum variance controller obtained using input output methods are shown to be identical for any system delay extending a result of 5

type ii ancient compact solutions to the yamabe flow nsf 0604657 1266172 0905749 1056387 fondo basal cmm millenium nucleus capde nc130017 fondecyt 1150066

a realization approach to stochastic model reduction the problem of discrete time stochastic model reduction approximation is considered using the canonical correlation analysis approach of akaike 1975 a new order reduction algorithm is developed furthermore it is shown that the inverse of the reduced order realization is asymptotically stable next an explicit relationship between canonical variables and the linear least squares estimate of the state vector is established using this a more direct approach for order reduction is presented and also a new design for reduced order kalman filters is developed finally the uniqueness and symmetry properties for the new realization the balanced stochastic realization along with a simulation result are presented

soliton surfaces in the mechanical equilibrium of closed membranes for a closed material membrane in equilibrium in a force field we investigate whether the external observables membrane geometry and force field determine the internal membrane response the stress tensor t when the mean stress open image in new window tr t is known for membranes with boundary the indeterminacy of the response is classical for closed membranes the geometry decides the question we show uniqueness for all but a class of soliton surfaces the globally isothermic surfaces 1 2 the physical phenomenon exhibited by any closed globally isothermic membrane in equilibrium is that with all observables static there is in total a 1 parameter family of responses with the same mean stress all canonically determined by membrane geometry theorem 1 there exist closed embedded globally isothermic surfaces of every genus the recognition of the role of these soliton surfaces settles the old classification problem for the space of static shears in any closed membrane explicitly identifying all static shears where before only a genus dependent dimension bound was known theorem 2

state space design of multivariable self tuning regulators abstract a state space design for a multivariable self tuning regulator is presented which is based upon eigenvalue assignment the parameters of a mimo stochastic process are estimated at each time interval by recursive least squares and are used to construct an equivalent state space representation eigenvalue assignment of the closed loop system is accomplished by state feedback where the state estimate is obtained from the output of a full order luenberger observer the algorithm can be used to regulate non minimum phase and open loop unstable systems three examples are given

solutions without any symmetry for semilinear elliptic problems abstract we prove the existence of infinitely many solitary waves for the nonlinear klein gordon or schrodinger equation u u u 3 0 in r 2 which have finite energy and whose maximal group of symmetry reduces to the identity

the global structure of constant mean curvature surfaces in a recent ground breaking paper kapouleas constructed this century s first examples of complete finite topology properly embedded surfaces with non zero constant mean curvature ka l geometrically these surfaces are quite easy to visualize as we explain below the purpose of the present paper is to characterize the extent to which kapouleas examples are typical of all embedded constant mean curvature surfaces first and throughout this paper we scale space so that the constant mean curvature the trace of the second fundamental form of our surface l is identically one and we call such a surface an mc1 surface the only complete embedded mc1 surfaces of finite topology known before kapouleas work arise from the 1parameter family of axially symmetric and periodic examples found by delaunay in 1841 using the maximum axial radius r of such a surface as a parameter this family begins with the cylinder r 1 and evolves through embedded periodic surfaces until the family degenerates to a linear chain of touching spheres r 2 the delaunay family continues for all r 2 as periodic immersions kapouleas construction of embedded surfaces begins with a piecewise linear graph in r 3 having the property that edges are either finite segments with length a multiple of 4 or half infinite rays and with the further property that at each vertex some linear combination of their direction vectors with positive coefficients adds to zero see fig 1 he constructs an approximate solution to the problem by centering radius r 2 spheres at all vertices and connecting these smoothly by delaunay segments having r 2 but near 2 with axes along the graph edges since the periods of the delaunay surfaces are slightly longer than 4 he must assume also a flexibility condition on the graph to guarantee that such a gluing

solving second order matrix differential equations with regular singular points avoiding the increase of the problem dimension in this paper we propose a method for solving second order matrix differential systems of the form t 2x ta t x b t x 0 where a and b are analytic matrix functions in a neighborhood of t 0 series solutions of the problem are constructed without increasing the problem dimension by introducing the concept of a fundamental set of solutions of the above equation an analogous result to the frobenius theorem to solve the corresponding scalar equation is presented

analytic solutions of coupled second order delay differential equations abstract in this paper initial value problems for systems of second order delay differential equations with variable coefficients are treated by means of a combined use of the method of steps and the method of frobenius the analytic solution of the problem is constructed

equivalent random force and time series model in systems far from equilibrium under the condition that observed time series data is given a stochastic markovian equation for a physical system can be transformed into an observable non markovian equation used in the time series analysis the physical random force satisfying the fluctuation dissipation theorem is also transformed into a stochastically equivalent random force in the derivation of the time series model of observable variables statistical quantities i e correlation and power spectral density functions for observable variables can be expressed not only by the physical random force but also by the equivalent random force a relation between the variance of physical random force and that of equivalent random force is also found

self tuning regulators a state space approach abstract this paper employs a state space system description to provide a pole placement scheme via state feedback it is shown that when a recursive least squares estimation scheme is used the feedback employed can be expressed simply in terms of the estimated system parameters to complement the state feedback approach a method employing both state feedback and linear output feedback is discussed both methods arc then compared with the previous output polynomial type feedback schemes

a noncommutative version of the bispectral problem we consider a matrix valued version of the bispectral problem involving a block tridiagonal doubly infinite matrix and a first order differential operator with matrix coefficients we give a set of necessary conditions that the coefficients need to satisfy and solve these equations under a variety of conditions the situations discussed here should make it plain that while the corresponding problem in the scalar case is relatively trivial and devoid of any interest the noncommutative version of the problem is much richer and subtle the results here should be useful for instance in the study of a noncommutative version of the nonlinear toda lattice

on the area of constant mean curvature discs and annuli with circular boundaries it is still an open question whether a constant mean curvature cmc disc which is bounded by a circle is necessarily a spherical cap or a flat disc the authors together with lopez 1 recently showed that the only stable cmc discs which are bounded by a circle are spherical caps in this paper we derive lower bounds for the area of constant mean curvature discs and annuli with circular boundaries in 3 dimensional space forms

on the stability of the serrin problem abstract we investigate stability issues concerning the radial symmetry of solutions to serrin s overdetermined problems in particular we show that if u is a solution to u n in a smooth domain r n u 0 on and d u is close to 1 on then is close to the union of a certain number of disjoint unitary balls

notes on poles of autoregressive type model part ii robust singular pole abstract the non robust singular pole of a scalar t ar model which is defined by truncating the taylor expansion of physical ar ma model has been introduced in the previous paper this journal a two dimensional t ar model is obtained when two state variables are observed from a linear gauss markovian system described with three degrees of freedom the vector t ar model has a robust singular pole which has no direct correspondence to poles and or zeros of the original vector ar ma model the comparison of pole locations between t ar model and ar model is studied in a simple example

minimum variance control of discrete time multivariable armax systems we consider multivariable armax stochastic systems these systems can incorporate the following complicating features general delay structures nonminimum phase transfer functions different dimensions for input and output vectors we obtain the control laws which minimize the variance of the output process while maintaining system stability

optimal observers for arma models a polynomial based arma model when posed in a state space framework can be regarded in many different ways in this paper two particular state space forms of the arma model are considered and although both are canonical in structure they differ in respect of the mode in which disturbances are fed into the state and output equations for both forms a solution is found to the optimal discrete time observer problem and algebraic connections between the two optimal observers are shown the purpose of the paper is to highlight the fact that the optimal observer obtained from the first state space form commonly known as the innovations form is not that employed in an optimal controller in the minimum output variance sense whereas the optimal observer obtained from the second form is hence the second form is a much more appropriate state space description to use for controller design particularly when employed in self tuning control schemes

hybrid state space fuzzy model based controller with dual rate sampling for digital control of chaotic systems we develop a hybrid state space fuzzy model based controller with dual rate sampling for digital control of chaotic systems a takagi sugeno ts fuzzy model is used to model the chaotic dynamic system and the extended parallel distributed compensation technique is proposed and formulated for designing the fuzzy model based controller under stability conditions the optimal regional pole assignment technique is also adopted in the design of the local feedback controllers for the multiple ts linear state space models the proposed design procedure is as follows an equivalent fast rate discrete time state space model of the continuous time system is first constructed by using fuzzy inference systems to obtain the continuous time optimal state feedback gains the constructed discrete time fuzzy system is then converted into a continuous time system the developed optimal continuous time control law is finally converted into an equivalent slow rate digital control law using the proposed intelligent digital redesign method the main contribution of the paper is the development of a systematic and effective framework for fuzzy model based controller design with dual rate sampling for digital control of complex such as chaotic systems the effectiveness and the feasibility of the proposed controller design method is demonstrated through numerical simulations on the chaotic chua circuit

gluing eguchi hanson metrics and a question of page in 1978 gibbons pope and page proposed a physical picture for the ricci flat kahler metrics on the k3 surface based on a gluing construction in this construction one starts from a flat torus with 16 orbifold points and resolves the orbifold singularities by gluing in 16 small eguchi hanson manifolds that all have the same orientation this construction was carried out rigorously by topiwala lebrun singer and donaldson in 1981 page asked whether the above construction can be modified by reversing the orientations of some of the eguchi hanson manifolds this is a subtle question if successful this construction would produce einstein metrics that are neither kahler nor self dual in this paper we focus on a configuration of maximal symmetry involving eight small eguchi hanson manifolds of each orientation that are arranged according to a chessboard pattern by analyzing the interactions between eguchi hanson manifolds with opposite orientation we identify a nonvanishing obstruction to the gluing problem thereby destroying any hope of producing a metric of zero ricci curvature in this way using this obstruction we are able to understand the dynamics of such metrics under ricci flow as long as the eguchi hanson manifolds remain small in particular for the configuration described above we obtain an ancient solution to the ricci flow with the property that the maximum of the riemann curvature tensor blows up at a rate of t 1 2 while the maximum of the ricci curvature converges to 0 2016 wiley periodicals inc

explicit solutions for second order operator differential equations with two boundary value conditions ii abstract boundary value problems for second order operator differential equations with two boundary value conditions are studied explicit expressions of the solutions in terms of data problems are given by means of the application of algebraic techniques analogous expressions to the ones known for the scalar case are obtained

constant mean curvature surfaces with boundary on a sphere in this article we study the shape of a compact surface of constant mean curvature of euclidean space whose boundary is contained in a round sphere we consider the case that the boundary is prescribed or that the surface meets the sphere with a constant angle we study under what geometric conditions the surface must be spherical our results apply in many scenarios in physics where in absence of gravity a liquid drop is deposited on a round solid ball and the air liquid interface is a critical point for area under all variations that preserve the enclosed volume

detectability and stabilizability of time varying discrete time linear systems the concepts of detectability and stabilizability are explored for time varying systems we study duality invariance under feedback an extended version of the lemma of lyapunov existence of stabilizing feedback laws linear quadratic filtering and control and the existence of approximate canonical forms