properly embedded minimal planar domains in 1997 collin proved that any properly embedded minimal surface in r 3 with nite topology and more than one end has

explicit solution to the singular discrete time stationary linear filtering problem a closed form solution to the stationary discrete time linear filtering problem is obtained explicitly in terms of the system state space matrices in the limiting singular case where the measurement noise tends to zero simple expressions in closed form are obtained for the kalman gain matrix both for uniform and nonuniform rank systems and the explicit eigenstructure of the kalman filter closed loop matrix is derived the minimum error covariance matrices of the a priori and a posteriori filtered estimates are obtained using this special eigenstructure and a remarkably different behavior of the solution in the minimum and nonminimum phase cases is found

the role of the unitary interactor matrix in the explicit solution of the singular lq output feedback control problem abstract this paper is concerned with the unitary interactor matrix and its role in the explicit solution of the minimum variance and singular lq output feedback control problems the unitary interactor matrix an all pass factor that factors the infinite zeros while keeping the spectrum of the underlying system unchanged is also extended to include the weighted unitary interactor matrix with the weighted unitary interactor matrix the multivariate minimum variance control law as proposed by goodwin and sin 1984 is shown to be pairinginvariant and identical to the singular lq output feedback control law

on the negative squares of indefinite sturm liouville operators abstract the number of negative squares of all self adjoint extensions of a simple symmetric operator of defect one with finitely many negative squares in a krein space is characterized in terms of the behaviour of an abstract titchmarsh weyl function near 0 and these results are applied to a large class of symmetric and self adjoint indefinite sturm liouville operators with indefinite weight functions

analysis of the linearization around a critical point of an infinite dimensional hamiltonian system partial differential equations that conserve energy can often be written as infinite dimensional hamiltonian systems of the following general form du dt je u where j x x is a symplectic matrix i e jj 1 and e x r is a c 2 functional defined on some hilbert space x

loop transfer recovery techniques for discrete time optimal regulators using prediction estimators for discrete time lqg optimal regulators using current estimators it has been shown that perfect recoveries of the loop transfer matrices at the input or at the output are possible for square minimum phase systems however current estimators cannot be used when the processing time of the controller is significant in this note the loop transfer recoveries using prediction estimators for square minimum phase systems are considered it is shown that although the perfect recoveries are impossible the feedback properties obtained by the recovery techniques are those that can be recovered best in the presence of the delay in the controller

brief paper anti windup and bumpless transfer schemes this paper considers the problem of windup which occurs when a controller with unstable integrator modes is subjected to input saturation or multi mode operation an optimal strategy for the so called model based approach is proposed based on an h optimisation framework this has been found to give good results on numerical examples taken from the literature

one sided complete stable minimal surfaces we prove that there are no complete one sided stable minimal surfaces in the euclidean 3 space we classify least area surfaces in the quotient of mathbb r 3 by one or two linearly independent translations and we give sharp upper bounds of the genus of compact two sided index one minimal surfaces in non negatively curved ambient spaces finally we estimate from below the index of complete minimal surfaces in flat spaces in terms of the topology of the surface

a generalized inverse solution to the discrete time singular riccati equation a closed form solution to the discrete time singular riccati equation is presented which is based on the moore penrose matrix inverse applications to deadbeat control and stabilizability are discussed

brief limiting performance of optimal linear discrete filters the limiting variance of the estimation error is obtained for the optimal h 2 filter for discrete time systems when the intensity of the measurement noise tends to zero in particular an explicit expression is obtained for the lowest achievable mean square error when the measurement noise is slowly varying and has intensity which tends to zero this limitation can be characterized completely by the system order the dimension of the process noise the number of the system s transmission zeros and the location of the system s unstable transmission zeros

the rigidity of embedded constant mean curvature surfaces we study the rigidity of complete embedded constant mean curvature surfaces in r 3 among other things we prove that when such a surface has finite genus then intrinsic isometries of the surface extend to isometries of r 3 or its isometry group contains an index two subgroup of isometries that extend mathematics subject classification primary 53a10 secondary 49q05 53c42

spectral and modulational stability of periodic wavetrains for the nonlinear klein gordon equation abstract this paper is a detailed and self contained study of the stability properties of periodic traveling wave solutions of the nonlinear klein gordon equation u t t u x x v u 0 where u is a scalar valued function of x and t and the potential v u is of class c 2 and periodic stability is considered both from the point of view of spectral analysis of the linearized problem spectral stability analysis and from the point of view of wave modulation theory the strongly nonlinear theory due to whitham as well as the weakly nonlinear theory of wave packets the aim is to develop and present new spectral stability results for periodic traveling waves and to make a solid connection between these results and predictions of the formal modulation theory which has been developed by others but which we review for completeness

uniqueness of the riemann minimal examples we prove that a properly embedded minimal surface in r 3 of genus zero with infinite symmetry group is a plane a catenoid a helicoid or a riemann minimal example we introduce the language of hurwitz schemes to understand the underlying moduli space of surfaces in our setting

explicit solution to the singular lq regulation problem an explicit solution to the multivariable discrete linear quadratic lq regulation problem is obtained in the limiting singular case where the input weighting matrix tends to zero such a solution follows from a suitable spectral factorization of the input spectrum density matrix under the assumption that the system is stabilizable and detectable and that its transfer function matrix is of full rank the suitable spectral factor is shown to be the product of the system s minimum phase image and its unitary interactor matrix the unitary interactor matrix defined is a special case of the nilpotent interactor matrix defined by m w rogozinski et al 1987

the calabi yau conjectures for embedded surfaces in this talk i will discuss the proof of the calabi yau conjectures for embedded surfaces this is joint work with bill minicozzi cm9 the calabi yau conjectures about surfaces date back to the 1960s much work has been done on them over the past four decades in particular examples of jorge xavier from 1980 and nadirashvili from 1996 showed that the immersed versions were false we will show here that for embedded surfaces i e injective immersions they are in fact true their original form was given in 1965 in ca where e calabi made the following two conjectures about minimal surfaces see also s s chern page 212 of ch conjecture 1 prove that a complete minimal hypersurface in r must be unbounded calabi continued it is known that there are no compact minimal submanifolds of r or of any simply connected complete riemannian manifold with sectional curvature 0 a more ambitious conjecture is conjecture 2 a complete minimal hypersurface in r has an unbounded projection in every n 2 dimensional flat subspace

cheap optimal control of discrete single input single output systems this paper presents a solution to the linear discrete quadratic optimal control problem for a single input single output plant with a control weighting that tends to zero explicit expressions are derived for the optimal gain vector in terms of the first significant powers of the control weighting and the limiting values of these expressions are considered the optimal index of performance is obtained both for the minimum and non minimum phase plants the clear distinction between these two types of plants is discussed and the non vanishing value of the index of performance in the latter case is expressed in terms of the zero directions of the system extensions to the single input multi output case and the case where the index of performance is a function of dynamic operators are introduced

a closed form solution to the discrete time kalman filter and its applications abstract this paper presents a closed form solution to the discrete time kalman filter and its applications we first represent the kalman filter in terms of model parameters without using the riccati equation and requiring any artificial conditions such as invertibility of a system matrix and no system noises replacing the initial time with the fixed lag time to achieve the finite memory with respect to inputs and outputs and choosing the proper initial covariances on the recent finite horizon we easily obtain the minimum variance finite memory filter and then propose its iterative computation algorithm as another application a closed form solution to the difference riccati equation on the finite horizon is utilized to obtain a stabilizing gain matrix of a luenberger type filter as in ackermann s formula

sensitivity properties of a class of discrete time lqg controllers with computation delays sensitivity properties of the discrete time lqg controller which utilizes the optimal state predictor to compensate l step computation delays are discussed we obtain multiplicative decompositions of the sensitivity matrices at the input and output of a plant which clarify the effect of computation delays and imperfect observation furthermore these decompositions are used to show that duality between the sensitivity properties at the input and those at the output holds for l 1 and to identify the asymptotic sensitivity matrices obtained by loop transfer recovery procedures

the space of embedded minimal surfaces of fixed genus in a 3 manifold v fixed genus this paper is the fifth and final in a series on embedded minimal surfaces following our earlier papers on disks we prove here two main structure theorems for itnonsimply connected embedded minimal surfaces of any given fixed genus the first of these asserts that any such surface without small necks can be obtained by gluing together two oppositely oriented double spiral staircases the second gives a pair of pants decomposition of any such surface when there are small necks cutting the surface along a collection of short curves after the cutting we are left with graphical pieces that are defined over a disk with either one or two sub disks removed a topological disk with two sub disks removed is called a pair of pants both of these structures occur as different extremes in the two parameter family of minimal surfaces known as the riemann examples

stable standing waves of nonlinear klein gordon equations in this paper we give sufficient conditions for the stability of the standing waves of least energy for nonlinear klein gordon equations

a simple state space design of an interactor for a non square system via system matrix pencil approach abstract the construction of an interactor cancelling the infinite zeros of a non square proper transfer matrix is discussed in this paper along the line of approach of b r copeland m g safonov int j robust nonlinear control 2 1992 139 in this paper some new properties of the infinite eigenstructure of the system matrix pencil of the transfer matrix are presented a simple state space design of the interactor is proposed by using these properties then some new features about state space relations among the original system the interactor and the compensated system are presented also several conditions about the invariance of the stabilizability or controllability between the state space realizations of the original system and the compensated system are shown furthermore the proposed method of designing the interactor in this paper can be applied to a tall or fat transfer matrix with full normal rank

spectral stability analysis for special solutions of second order in time pdes the higher dimensional case abstract we develop a general theory to treat the linear stability of certain special solutions of second order in time evolutionary pde we apply these results to standing waves of the following problems the klein gordon equation for which we consider both ground states and certain excited states the klein gordon zakharov system and the beam equation we also discuss applications to excited states for the klein gordon model as well as multidimensional traveling waves not necessarily homoclinic to zero for general second order equations of this type in all cases our abstract results provide a complete characterization of the linear stability of such solutions

the classification of cmc foliations of bbb r 3 and bbb s 3 with countably many singularities in this paper we generalize our previous local removable singularity theorem for minimal laminations to the case of weak h laminations with h in bbb r constant in a punctured ball of a riemannian three manifold we also obtain a curvature estimate for any weak cmc foliation with possibly varying constant mean curvature from leaf to leaf of a compact riemannian three manifold n with boundary solely in terms of a bound of the absolute sectional curvature of n and of the distance to the boundary of n we then apply these results to classify weak cmc foliations of bbb r 3 and bbb s 3 with a closed countable set of singularities

singular filtering via spectral interactor matrix in this paper the notion of spectral interactor matrix sim for a given square transfer matrix is introduced and used in the singular filtering problem a sim is defined as an interactor matrix preserving the spectral properties of the underlying system after studying the class of all spectral interactor matrices associated with a given transfer matrix an algorithm for the computation of a sim is proposed by resorting to the sim it is possible to point out the role played by delays zeros at infinity and nonminimum phase zeros in the solution of the singular filtering problem

control under stochastic multiplicative uncertainties part ii optimal design for performance this paper studies the optimal control design problem for linear discrete time systems with stochastic multiplicative uncertainties these uncertainties are assumed to be present in the control inputs and modeled as independent and identically distributed i i d random processes the optimal performance under study is defined in the mean square sense referred to as the mean square optimal h 2 performance it is shown that the mean square optimal h 2 control problem via state feedback can be solved using a mean square stabilizing solution to a modified algebraic riccati equation mare a necessary and sufficient condition for the existence of this solution is presented which constitutes a generalization of the solution to the classic optimal h 2 state feedback design problem whereas the latter can be obtained by solving an algebraic riccati equation are it is also proven that the optimal control design problem can be cast as an eigenvalue problem evp for the output feedback case with possible input delays we show that the mean square optimal h 2 control problem also amounts to solving an mare when the plant has no nonminimum phase zeros from the inputs to the measurement outputs that is the global optimal solution is obtained by solving an mare incorporating the delays the implication is that in this case a separation principle still holds

an algorithm for the calculation of a nilpotent interactor matrix for linear multivariable systems a new algorithm for the calculation of the delay structure for a linear multivariable system is presented the algorithm evaluates a nilpotent interactor matrix from the matrix of coefficients of the numerator of a right matrix fraction description of the system the resulting nilpotent interactor is a product of first degree nilpotent polynomial matrices

prior information in the design of multivariable adaptive controllers this note discusses the design of a dynamic precompensator w c s for an m times m mimo plant w p s such that for almost all cases w p s w c s has a diagonal hermite form

the polynomial approach to the lq non gaussian regulator problem a new approach for the solution of the regulator problem for linear discrete time dynamical systems with non gaussian disturbances is proposed this approach generalizes a previous result concerning the definition of the quadratic optimal regulator it consists of the definition of the polynomial optimal algorithm of an order spl nu for the solution of the linear quadratic non gaussian stochastic regulator problem for systems with partial state information the validity of the separation principle has also been proved in this case numerical simulations show the high performance of the proposed method with respect to the classical linear regulation techniques

lieb thirring estimates for non self adjoint schrodinger operators for general nonsymmetric operators a we prove that the moment of order 1 of negative real parts of its eigenvalues is bounded by the moment of order of negative eigenvalues of its symmetric part h 12 a a as an application we obtain lieb thirring estimates for non self adjoint schrodinger operators in particular we recover recent results by frank et al lett math phys 77 309 2006 we also discuss moment of resonances of schrodinger self adjoint operators

instability of nonlinear bound states we establish a sharp instability theorem for the bound states of lowest energy of the nonlinear klein gordon equation utt u f u 0 and the nonlinear schrodinger equation iut u f u 0

on the focusing critical semi linear wave equation the wave equation partial derivative tt psi delta psi psi 5 0 in r 3 is known to exhibit finite time blowup for data of negative energy furthermore it admits the special static solutions phi x a 3a 1 4 1 a x 2 1 2 for all a 0 which are linearly unstable we view these functions as a curve in the energy space h over dot 1 x l 2 we prove the existence of a family of perturbations of this curve that lead to global solutions possessing a well defined long time asymptotic behavior as the sum of a bulk term plus a scattering term moreover this family forms a co dimension one manifold m of small diameter in a suitable topology loosely speaking m acts as a center stable manifold with the curve phi a as an attractor in m

the geometry of minimal surfaces of finite genus i curvature estimates and quasiperiodicity let mathcal m be the space of properly embedded minimal surfaces in 3 with genus zero and two limit ends normalized so that every surface m mathcal m has horizontal limit tangent plane at infinity and the vertical component of its flux equals one we prove that if a sequence m i i mathcal m has the horizontal part of the flux bounded from above then the gaussian curvature of the sequence is uniformly bounded this curvature estimate yields compactness results and the techniques in its proof lead to a number of consequences concerning the geometry of any properly embedded minimal surface in 3 with finite genus and the possible limits through a blowing up process on the scale of curvature of a sequence of properly embedded minimal surfaces with locally bounded genus in a homogeneously regular riemannian 3 manifold

discrete time lqg ltr technique for systems with time delays loop transfer recovery ltr is applied for lqg controllers with prediction estimators for a class of square minimum phase discrete time linear systems with time delays we consider the class of plants described by the triple a b c whose time delay structure is characterized by the following hypothesis on the plant impulse response cb cab cad 2b 0 det cad 1b 0 when the estimator gain is fixed by recovery technique it is shown that the loop transfer matrix at the plant input is equal to the transfer matrix obtained with the optimal digital regulator having d sample controller delays described in 6 the dual result corresponding to the situation where the feedback gain is fixed by recovery technique is also analysed

unstable ground state of nonlinear klein gordon equations on demontre l instabilite de l etat fondamental des equations de klein gordon non lineaires a n dimensions d espace n 3

the strong halfspace theorem for minimal surfaces d hoffman and w h meeks iii department of mathematics university of massachusetts amherst ma 01003 usa in 6 jorge and xavier constructed examples of complete nonplanar minimal surfaces contained between two parallel planes in 3 recently rosenberg and toubiana have found complete minimally immersed annuli that are proper in an open slab 1 12 on the basis of these and other results the question has arisen as to whether or not there could exist a properly immersed minimal surface in 3 that was contained between two parallel planes the answer is negative an immediate consequence of the following theorem

a numerical design of interactors for general descriptor systems in this technical note we develop a numerically reliable method to design an interactor for a general descriptor system relative to a stability region cg and an offending zero set our main result is based on a condensed form this condensed form is independent of the offending zeros and is computed using only orthogonal transformations and hence is numerically backward stable

optimal quadratic solution for the non gaussian finite horizon regulator problem in this paper the problem of the optimal quadratic regulator for non gaussian discrete time stochastic systems with a quadratic cost function is considered the main result here obtained is that such optimal control can be derived from the classical lqg solution by substituting the linear filtering part with a quadratic optimal filter numerical results show high performance of this method

counting eigenvalues via the krein signature in infinite dimensional hamiltonian systems spectra of nonlinear waves in infinite dimensional hamiltonian systems are investigated we establish a connection via the krein signature between the number of negative directions of the second variation of the energy and the number of potentially unstable eigenvalues of the linearization about a nonlinear wave we apply our results to determine the effect of symmetry breaking on the spectral stability of nonlinear waves in weakly coupled nonlinear schrodinger equations

stability theory of solitary waves in the presence of symmetry ii abstract consider an abstract hamiltonian system which is invariant under a group of operators we continue to study the effect of the group invariance on the stability of solitary waves applications are given to bound states and traveling wave solutions of nonlinear wave equations

the existence of ground states for fourth order wave equations abstract focusing on the fourth order wave equation u t t 2 u f u 0 we prove the existence of ground state solutions u u x c t for an optimal range of speeds c r n and a variety of nonlinearities f

nonradial solutions of a semilinear elliptic equation in two dimensions abstract we establish existence of an infinite family of exponentially decaying non radial c 2 solutions to the equation u u 0 on r 2 for a large class of nonlinearities these solutions have the form u r e im w r where r and are polar coordinates m is an integer and w 0 r is exponentially decreasing far from the origin we prove there is a solution with each prescribed number of nodes

triply periodic constant mean curvature surfaces abstract given a closed flat 3 torus n for each h 0 and each non negative integer g we obtain area estimates for closed surfaces with genus g and constant mean curvature h embedded in n this result contrasts with the theorem of traizet 31 who proved that every flat 3 torus admits for every positive integer g with g 2 connected closed embedded minimal surfaces of genus g with arbitrarily large area

homoclinic solutions for swift hohenberg and suspension bridge type equations we establish the existence of homoclinic solutions for a class of fourth order equations which includes the swift hohenberg model and the suspension bridge equation in the first case the nonlinearity has three zeros corresponding to a double well potential while in the second case the nonlinearity is asymptotically constant on one side the swift hohenberg model is a higher order extension of the classical fisher kolmogorov model its more complicated dynamics give rise to further possibilities of pattern formation the suspension bridge equation as studied by chen and mckenna j differential equations 136 1997 325 355 we give a positive answer to an open question raised by the authors c 2002 elsevier science usa

explicit solution to the unstable stationary filtering problem an expression in closed form for the constant kalman gain in stationary filtered estimation of linear unstable processes is obtained and bounds on the achievable accuracy of the estimate are derived this expression is used to find lower and upper bounds for the singular values of the solution of the algebraic matrix riccati equation and to derive the minimum phase image of linear systems that possess zeros in the right half plane

the space of embedded minimal surfaces of fixed genus in a 3 manifold iv locally simply connected this paper is the fourth in a series where we describe the space of all embedded minimal surfaces of fixed genus in a fixed but arbitrary closed 3manifold the key is to understand the structure of an embedded minimal disk in a ball in r3 this was undertaken in cm3 cm4 and the global version of it will be completed here see the discussion around figure 12 for the local case and cm15 for some more details our main results are theorem 0 1 the lamination theorem and theorem 0 2 the one sided curvature estimate the lamination theorem is stated in the global case where the lamination is in fact a foliation the first four papers of this series show that every embedded minimal disk is either a graph of a function or is a double spiral staircase where each staircase is a multivalued graph this is done by showing that if the curvature is large at some point and hence the surface is not a graph then it is a double spiral staircase like the helicoid to prove that such a disk is a double spiral staircase we showed in the first three papers of the series that it is built out of n valued graphs where n is a fixed number in this paper we will deal with how the multi valued graphs fit together and in particular prove regularity of the set of points of large curvature the axis of the double spiral staircase the first theorem is the global version of the statement that every embedded minimal disk is a double spiral staircase

the space of embedded minimal surfaces of fixed genus in a 3 manifold iii planar domains this paper is the third in a series where we describe the space of all embedded minimal surfaces of fixed genus in a fixed but arbitrary closed 3 manifold in cm3 cm5 we describe the case where the surfaces are topologically disks on any fixed small scale although the focus of this paper general planar domains is more in line with cm6 we will prove a result here namely corollary iii 3 5 below which is needed in cm5 even for the case of disks roughly speaking there are two main themes in this paper the first is that stability leads to improved curvature estimates this allows us to find large graphical regions these graphical regions lead to two possibilities

extended discrete time ltr synthesis of delayed control systems abstract this paper addresses compensation of dealys within a multi input multi output discrete time feedback loop for application to real time distributed control systems the delay compensation algorithm formulated in this paper is an extension of the standard loop transfer recovery ltr procedure from one step prediction to the general case of p step prediction p 1 it is shown that the steady state minimum variance filter gain is the h 2 minimization solution of the relative error between the target sensitivity matrix and the actual sensitivity matrix for p step prediction p 1 this concept forms the basis for synthesis of robust p step delay compensators p 1 the proposed control synthesis procedure for delay compensation is demonstrated via simulation of the flight control system of an advanced aircraft

discrete time loop transfer recovery for systems with nonminimum phase zeros and time delays abstract the purpose of this paper is to study what happens when the discrete time loop transfer recovery ltr procedure is applied to plants with nonminimum phase zeros and with uniform time delay in all channels explicit expressions are given for the asymptotic behavior of the resulting sensitivity function and loop transfer function these results yield a better understanding of the mechanism of the discrete time loop transfer recovery procedure and the design limitations due to nonminimum phase zeros and time delays

the space of embedded minimal surfaces of fixed genus in a 3 manifold ii multi valued graphs in disks this paper is the second in a series where we give a description of the space of all embedded minimal surfaces of fixed genus in a fixed but arbitrary closed 3 manifold the key for understanding such surfaces is to understand the local structure in a ball and in particular the structure of an embedded minimal disk in a ball in r 3 we show here that if the curvature of such a disk becomes large at some point then it contains an almost flat multi valued graph nearby that continues almost all the way to the boundary this will be proved by showing the existence of small multi valued graphs near points of large curvature and then using the extension result for multi valued graphs proved in the first paper in this series there are two local models for embedded minimal disks by an embedded disk we mean a smooth injective map from the closed unit ball in r 2 into r 3 one model is the plane or more generally a minimal graph the other is a piece of a helicoid in the first four papers of this series we will show that every embedded minimal disk is either a graph of a function or is a double spiral staircase like a helicoid recall that a double spiral staircase consists of two spiral staircases that spiral together around a common axis one inside the other this will be done by showing that if the curvature is large at some point and hence the surface is not a graph then it is a double spiral staircase to prove that it is a double spiral staircase we will first prove that it is built out of n valued graphs where n is a fixed number these n valued graphs are like a single spiral staircase connecting n floors the existence of the n valued graphs was initiated in the first paper and will be completed here the third and fourth papers of this series will deal with how the multi valued graphs fit together and in particular prove regularity of the set of points of large curvature the axis of the double spiral staircase

decoupling properties of singular lq regulation problem some properties of a limiting solution derived from a standard lq regulator problem are discussed in the case where the input weighting matrix tends to zero for a discrete time system the explicit limiting solution is given in some literature it is shown that the limiting solution is closely connected with a disturbance decoupling problem moreover it is pointed out that the limiting feedback gain with the help of feedforward gain constructs the decoupled system with the maximum number of blocks

optimal nearly singular estimation of continuous linear stationary uniform rank systems the problem of optimal estimation of linear continuous systems is considered for the case where all the noise signals are stationary white and the intensity of the measurement noise tends uniformly to zero explicit expressions for the estimation procedures and the corresponding minimum error covariance matrices of the estimate are obtained in terms of the measurement noise intensity and the zero properties of the system these expressions are derived for the case where the first non zero markov parameter of the system is of full rank for the infinite and the finite time filtering problems they provide a simple geometric interpretation to the structure of the error covariance matrices and they distinctly show the difference between the estimation of minimum and non minimum phase systems the theory is developed both for square and non square systems

multivariable adaptive control without a prior knowledge of the delay matrix this paper is concerned with the structure and role of the time delay matrix in discrete time multivariable processes interpretations and properties of this matrix as a multivariable generalization of the siso delay term are discussed it is formally shown that a finite time horizon or multi step cost function for the multivariable case can be minimized subject to a suitable choice of output and control horizons and without prior knowledge of the system delay matrix

the space of embedded minimal surfaces of fixed genus in a 3 manifold i estimates off the axis for disks this paper is the first in a series where we describe the space of all embedded minimal surfaces of fixed genus in a fixed but arbitrary closed riemannian 3 manifold the key for understanding such surfaces is to understand the local structure in a ball and in particular the structure of an embedded minimal disk in a ball in r3 with the flat metric this study is undertaken here and completed in cm6 these local results are then applied in cm7 where we describe the general structure of fixed genus surfaces in 3 manifolds there are two local models for embedded minimal disks by an embedded disk we mean a smooth injective map from the closed unit ball in r2 into r3 one model is the plane or more generally a minimal graph the other is a piece of a helicoid in the first four papers of this series we will show that every embedded minimal disk is either a graph of a function or is a double spiral staircase where each staircase is a multi valued graph this will be done by showing that if the curvature is large at some point and hence the surface is not a graph then it is a double spiral staircase to prove that such a disk

on the spectral problem mathcal l u lambda u and applications we develop a general instability index theory for an eigenvalue problem of the type mathcal l u lambda u for a class of self adjoint operators mathcal l on the line r1 more precisely we construct an evans like function to show a real eigenvalue instability in terms of a vakhitov kolokolov type condition on the wave if this condition fails we show by means of lyapunov schmidt reduction arguments and the kapitula kevrekidis sandstede index theory that spectral stability holds thus we have a complete spectral picture under fairly general assumptions on mathcal l we apply the theory to a wide variety of examples for the generalized bullough dodd tzitzeica type models we give instability results for travelling waves for the generalized short pulse ostrovsky vakhnenko model we construct almost explicit peakon solutions which are found to be unstable for all values of the parameters

an instability index theory for quadratic pencils and applications primarily motivated by the stability analysis of nonlinear waves in second order in time hamiltonian systems in this paper we develop an instability index theory for quadratic operator pencils acting on a hilbert space in an extension of the known theory for linear pencils explicit connections are made between the number of eigenvalues of a given quadratic operator pencil with positive real parts to spectral information about the individual operators comprising the coefficients of the spectral parameter in the pencil as an application we apply the general theory developed here to yield spectral and nonlinear stability instability results for abstract second order in time wave equations more specifically we consider the problem of the existence and stability of spatially periodic waves for the good boussinesq equation in the analysis our instability index theory provides an explicit and somewhat surprising connection between the stability of a given periodic traveling wave solution of the good boussinesq equation and the stability of the same periodic profile but with different wavespeed in the nonlinear dynamics of a related generalized korteweg de vries equation

linearized instability for nonlinear schrodinger and klein gordon equations in this paper i am introducing a new technique for proving instability of bound states for hamiltonian systems there are already two disparate types of instability results in the literature the approach developed by strauss shatah 20 gave an instability criterion coming from the variational structure of the problem on the other hand jones approach 11 produced a complementary criterion related to the difference between the number of negative eigenvalues of two selfadjoint operators using quite different techniques it turns out that with the methods developed in this paper these two criteria can be derived within a single framework that also leads to a generalization of the previous results finally in order to demonstrate how this method works i apply the instability criterion in some specific examples

local removable singularity theorems for minimal laminations in this paper we prove a local removable singularity theorem for certain minimal laminations with isolated singularities in a riemannian three manifold this removable singularity theorem is the key result used in our proof that a complete embedded minimal surface in r 3 with quadratic decay of curvature has nite total curvature