line search acceleration of iterative methods abstract this paper describes and analyzes a simple technique that accelerates the convergence of iterative methods for solving large linear systems the proposed technique can be used with any convergent method however it is especially useful when solving unstructured systems since in this case traditional methods as sor or chebyshev semiiteration have difficulty in obtaining optimal parameters special attention is given to column relaxation and row relaxation it is demonstrated experimentally that an accelerated gauss seidel scheme runs faster than an optimally tuned sor scheme

on modified factorizations for large scale linearly constrained optimization we consider the algebraic issues concerning the solution of general large scale linearly constrained nonlinear optimization problems particular attention is given to suitable methods for solving the linear systems that occur at each iteration of such methods the main issue addressed is how to ensure that a quadratic model of the objective function is positive definite in the null space of the constraints while neither adversely affecting the convergence of newton s method nor incurring a significant computational overhead numerical evidence to support the theoretical developments is provided

the computational complexity of knot genus and spanning area we investigate the computational complexity of some problems in three dimensional topology and geometry we show that the problem of determining a bound on the genus of a knot in a 3 manifold is np complete using similar ideas we show that deciding whether a curve in a metrized pl 3 manifold bounds a surface of area less than a given constant c is np hard

an augmented lagrangian interior point method using directions of negative curvature we describe an efficient implementation of an interior point algorithm for non convex problems that uses directions of negative curvature these directions should ensure convergence to second order kkt points and improve the computational efficiency of the procedure some relevant aspects of the implementation are the strategy to combine a direction of negative curvature and a modified newton direction and the conditions to ensure feasibility of the iterates with respect to the simple bounds the use of multivariate barrier and penalty parameters is also discussed as well as the update rules for these parameters we analyze the convergence of the procedure both the linesearch and the update rule for the barrier parameter behave appropriately as the main goal of the paper is the practical usage of negative curvature a set of numerical results on small test problems is presented based on these results the relevance of using directions of negative curvature is discussed

inertia controlling methods for general quadratic programming active set quadratic programming qp methods use a working set to define the search direction and multiplier estimates in the method proposed by fletcher in 1971 and in several subsequent mathematically equivalent methods the working set is chosen to control the inertia of the reduced hessian which is never permitted to have more than one nonpositive eigenvalue such methods will be called inertia controlling this paper presents an overview of a generic inertia controlling qp method including the equations satisfied by the search direction when the reduced hessian is positive definite singular and indefinite recurrence relations are derived that define the search direction and lagrange multiplier vector through equations related to the karush kuhn tucker system discussion is included of connections with inertia controlling methods that maintain an explicit factorization of the reduced hessian matrix

on the rhs description of resonances and virtual states we discuss the role of the m ller wave operators in a rigged hilbert space rhs model for resonances we also construct a new rhs of analytic functions valid for the description of resonances and virtual states these become generalized state vectors in a general physically realizable representation

superlinear convergence of primal dual interior point algorithms for nonlinear programming the local convergence properties of a class of primal dual interior point methods are analyzed these methods are designed to minimize a nonlinear nonconvex objective function subject to linear equality constraints and general inequalities they involve an inner iteration in which the log barrier merit function is approximately minimized subject to satisfying the linear equality constraints and an outer iteration that specifies both the decrease in the barrier parameter and the level of accuracy for the inner minimization under nondegeneracy assumptions it is shown that asymptotically for each value of the barrier parameter solving a single primal dual linear system is enough to produce an iterate that already matches the barrier subproblem accuracy requirements the asymptotic rate of convergence of the resulting algorithm is q superlinear and may be chosen arbitrarily close to quadratic furthermore this rate applies componentwise these results hold in particular for the method described in a r conn n i m gould d orban and p l toint math program ser b 87 2000 pp 215 249 and indicate that the details of its inner minimization are irrelevant in the asymptotics except for its accuracy requirements

on practical conditions for the existence and uniqueness of solutions to the general equality quadratic programming problem we present practical conditions under which the existence and uniqueness of a finite solution to a given equality quadratic program may be examined different sets of conditions allow this examination to take place when null space range space or lagrangian methods are used to find stationary points for the quadratic program

on the convegence of a sequential penalty function method for constrained minimization the convergence behaviour of a class of iterative methods for solving the constrained minimization problem is analysed the methods are based on the sequential minimization of a simple differentiable penalty function they are sufficiently general to ensure global convergence of the iterates to the solution of the problem at an asymptotic two step q superlinear rate

natural preconditioning and iterative methods for saddle point systems the solution of quadratic or locally quadratic extremum problems subject to linear ized constraints gives rise to linear systems in saddle point form this is true whether in the continuous or the discrete setting so saddle point systems arising from the discretization of partial differential equation problems such as those describing electromagnetic problems or incompressible flow lead to equations with this structure as do for example interior point methods and the sequential quadratic programming approach to nonlinear optimization this survey concerns iterative solution methods for these problems and in particular shows how the problem formulation leads to natural preconditioners which guarantee a fast rate of convergence of the relevant iterative methods these preconditioners are related to the original extremum problem and their effectiveness in terms of rapidity of convergence is established here via a proof of general bounds on the eigenvalues of the preconditioned saddle point matri

pl equivariant surgery and invariant decompositions of 3 manifolds using normal surface theory h j2 we introduce the notion of least weight normal surfaces the weight of a normal surface is a nonnegative integer invariant of the normal isotopy class of the surface if we focus on a particular class of normal surfaces and choose representatives which minimize the weight over the class then we have least weight normal surfaces it is remarkable how these least weight normal surfaces exhibit many of the same useful properties as least area minimal surfaces they provide a piecewise linear pl environment to obtain the recent topological results coming from the analysis and geometry of least area surfaces in an impressive series of papers meeks and yau m y i m y z m y 3 m y m y meeks simon and yau m s y scott s meeks and scott m s and freedman hass and scott f h s introduced the analysis and geometry of least area surfaces into the study of topological questions about 3 manifolds the consequences have led to the resolution of many outstanding problems in the topology of 3 manifolds since most of the problems being solved with these new techniques are topological in nature it has been felt that there should be a more topological approach to the proofs of these results besides the bulk of the work in the theory of

formulas for calculating the extremum ranks and inertias of a four term quadratic matrix valued function and their applications abstract this paper studies the quadratic matrix valued function x dxax d dxb b x d c through some expansion formulas for ranks and inertias of hermitian matrices where a b c and d are given complex matrices with a and c hermitian x is a variable matrix and denotes the conjugate transpose of a complex matrix we first introduce an algebraic linearization method for studying this matrix valued function and establish a group of explicit formulas for calculating the global maximum and minimum ranks and inertias of this matrix valued function with respect to the variable matrix x we then use these rank and inertia formulas to derive i necessary and sufficient conditions for the matrix equation x 0 to have a solution as well as the four matrix inequalities x 0 in the lowner partial ordering to be feasible respectively ii necessary and sufficient conditions for the four matrix inequalities x 0 in the lowner partial ordering to hold for all matrices x respectively iii the two matrices x and x such that the inequalities x x and x x hold for all matrices x in the lowner partial ordering respectively an application of the quadratic matrix valued function in control theory is also presented

successive refinement of large multicell models this paper presents and analyzes a new relaxation method for minimizing a function of many variables the minimized function is associated with a model that is composed of a large number of cells in some applications the terms elements mesh points or facilities are more appropriate such problems often arise in models that are based on finite difference or finite element methods the new method which we call the successive refinement method utilizes the special structure of the model to save on both storage and computations although the s r method has many features in common with the nonlinear s o r method there is a basic difference between the two methods the difference is that the steps of the s r method are not using relaxation parameters instead each of its steps is aimed to reduce the objective function value as much as possible this gives us a robust algorithm that is able to solve problems that the s o r method cannot handle such problems are for example minimization of a

domain decomposition methods for advection dominated linear quadratic elliptic optimal control problems abstract we present an optimization level domain decomposition dd preconditioner for the solution of advection dominated elliptic linear quadratic optimal control problems which arise in many science and engineering applications the dd preconditioner is based on a decomposition of the optimality conditions for the elliptic linear quadratic optimal control problem into smaller subdomain optimality conditions with dirichlet boundary conditions for the states and the adjoints on the subdomain interfaces these subdomain optimality conditions are coupled through robin transmission conditions for the states and the adjoints the parameters in the robin transmission condition depend on the advection this decomposition leads to a schur complement system in which the unknowns are the state and adjoint variables on the subdomain interfaces the schur complement operator is the sum of subdomain schur complement operators the application of which is shown to correspond to the solution of subdomain optimal control problems which are essentially smaller copies of the original optimal control problem we show that under suitable conditions the application of the inverse of the subdomain schur complement operators requires the solution of a subdomain elliptic linear quadratic optimal control problem with robin boundary conditions for the state numerical tests for problems with distributed and with boundary control show that the dependence of the preconditioners on mesh size and subdomain size is comparable to its counterpart applied to a single advection dominated equation these tests also show that the preconditioners are insensitive to the size of the control regularization parameter

computing the nearest low rank correlation matrix by a simplified sqp algorithm in this paper we propose a numerical method for computing the nearest low rank correlation matrix lrcm motivated by the fact that the nearest lrcm problem can be reformulated as a standard nonlinear equality constrained optimization problem with matrix variables via the gramian representation we propose a new algorithm based on the sequential quadratic programming sqp method on each iteration we do not solve the quadratic program qp corresponding to the exact hessian but a modified qp with a simpler hessian this qp subproblem can be solved efficiently by equivalently transforming it to a sparse linear system global convergence is established and preliminary numerical results are presented to demonstrate the proposed method is potentially useful

the rigged hilbert space approach to the lippmann schwinger equation ii the analytic continuation of the lippmann schwinger bras and kets the analytic continuation of the lippmann schwinger bras and kets is obtained and characterized it is shown that the natural mathematical setting for the analytic continuation of the solutions of the lippmann schwinger equation is the rigged hilbert space rather than just the hilbert space it is also argued that this analytic continuation entails the imposition of a time asymmetric boundary condition upon the group time evolution resulting in a semigroup time evolution physically the semigroup time evolution is simply a retarded or advanced propagator

a topological proof of the equivariant dehn lemma an elementary topological proof is given for a completely general version of the equivariant dehn lemma in the spirit of the original proof of the nonequivariant version due to c d papakyriakopolous in 1957

embedded minimal surfaces exotic spheres and manifolds with positive ricci curvature let n be a three dimensional riemannian manifold let e be a closed embedded surface in n then it is a question of basic interest to see whether one can deform in its isotopy class to some canonical embedded surface from the point of view of geometry a natural canonical surface will be the extremal surface of some functional defined on the space of embedded surfaces the simplest functional is the area functional the extremal surface of the area functional is called the minimal surface such minimal surfaces were used extensively by meeks yau my21 in studying group actions on three dimensional manifolds in my2 the theory of minimal surfaces was used to simplify and strengthen the classical dehn s lemma loop theorem and the sphere theorem in the setting there one minimizes area among all immersed surfaces and proves that the extremal object is embedded in this paper we minimize area among all embedded surfaces isotopic to a fixed embedded surface in the category of these surfaces we prove a general existence theorem theorem 1 a particular consequence of this theorem is that for irreducible manifolds an embedded incompressible surface is isotopic to an embedded incompressible surface with minimal area we also prove that there exists an embedded sphere of least area enclosing a fake cell provided the complementary volume is not a standard ball and provided there exists no embedded one sided rp2 by making use of the last result and a cutting and pasting argument we are able to settle a well known problem in the theory of three dimensional manifolds we prove that the covering space of any irreducible orientable three dimensional manifold is irreducible it is possible to exploit our existence theorem to study

neumann neumann domain decomposition preconditioners for linear quadratic elliptic optimal control problems we present a class of domain decomposition dd preconditioners for the solution of elliptic linear quadratic optimal control problems our dd preconditioners are extensions of neumann neumann dd preconditioners which have been successfully applied to the solution of single pdes the dd preconditioners are based on a decomposition of the optimality conditions for the elliptic linear quadratic optimal control problem into smaller subdomain optimality conditions with dirichlet boundary conditions for the states and the adjoints on the subdomain interfaces these subdomain optimality conditions are coupled through neumann interface conditions for the states and the adjoints this decomposition leads to a schur complement system in which the unknowns are the state and adjoint variables on the subdomain interfaces the schur complement operator is the sum of subdomain schur complement operators the application of which is shown to correspond to the solution of subdomain elliptic linear quadratic optimal control problems which are essentially smaller copies of the original optimal control problem we show that under suitable conditions the application of the inverse of the subdomain schur complement operators requires the solution of a subdomain elliptic linear quadratic optimal control problem with neumann interface conditions for the state the subdomain schur complement operators are analyzed in the variational setting of the problem as well as the algebraic setting obtained after a finite element discretization of the problem definiteness properties of the algebraic form of the subdomain schur complement operator s are studied numerical tests show that the dependence of these preconditioners on mesh size and subdomain size is comparable to its counterpart applied to elliptic equations only these tests also show that the preconditioners are insensitive to the size of the control regularization parameter

modifying the inertia of matrices arising in optimization applications in constrained optimization and other areas produce symmetric matrices with a natural block 2 2 structure an optimality condition leads to the problem of perturbing the 1 1 block of the matrix to achieve a specific inertia we derive a perturbation of minimal norm for any unitarily invariant norm that increases the number of nonnegative eigenvalues by a given amount and we show how it can be computed efficiently given a factorization of the original matrix we also consider an alternative way to satisfy the optimality condition based on a projection approach theoretical tools developed here include an extension of ostrowski s theorem on congruences and some lemmas on inertias of block 2 2 symmetric matrices

reply to comment on on the inconsistency of the bohm gadella theory with quantum mechanics in this reply we show that when we apply standard distribution theory to the lippmann schwinger equation the resulting spaces of test functions would comply with the hardy axiom only if classic results of paley and wiener of gelfand and shilov and of the theory of ultradistributions were wrong also we point out several differences between the standard method of constructing rigged hilbert spaces in quantum mechanics and the method used in time asymmetric quantum theory

resonance poles and gamow vectors in the rigged hilbert space formulation of quantum mechanics after a summary of the rigged hilbert space formulation of quantum mechanics and a brief statement of its advantages over von neumann s formulation a mathematically correct definition of gamow s exponentially decaying vectors as generalized energy eigenvectors is suggested it is shown that exponentially decaying vectors are obtained from the s matrix poles in the lower half of the second sheet and exponentially growing vectors from the s matrix poles in the upper half of the second sheet decaying state vectors are defined as functionals over half of the space of physical states and growing state vectors are defined as functionals over the other half on functionals over these subspaces the dynamical group of time development splits into two semigroups one for t 0 and the other for t 0 the generalized basis system connected with the spectrum of the hamiltonian is transformed into a new basis system in which the exponentially decaying component of the density matrix is separated

convergence theory for nonconvex stochastic programming with an application to mixed logit monte carlo methods have extensively been used and studied in the area of stochastic programming their convergence properties typically consider global minimizers or first order critical points of the sample average approximation saa problems and minimizers of the true problem and show that the former converge to the latter for increasing sample size however the assumption of global minimization essentially restricts the scope of these results to convex problems we review and extend these results in two directions we allow for local saa minimizers of possibly nonconvex problems and prove under suitable conditions almost sure convergence of local second order solutions of the saa problem to second order critical points of the true problem we also apply this new theory to the estimation of mixed logit models for discrete choice analysis new useful convergence properties are derived in this context both for the constrained and unconstrained cases and associated estimates of the simulation bias and variance are proposed

on solving block structured indefinite linear systems we consider 2 2 block indefinite linear systems whose 2 2 block is zero such systems arise in many applications we discuss two techniques that are based on modifying the 1 1 block in a way that makes the system easier to solve the main part of the paper focuses on an augmented lagrangian approach a technique that modifies the 1 1 block without changing the system size the choice of the parameter involved the spectrum of the linear system and its condition number are discussed and some analytical observations are provided a technique of deflating the 1 1 block is then introduced finally numerical experiments that validate the analysis are presented

finite group actions on 3 manifolds if g is a finite group acting smoothly on a closed surface f it is well known that g leaves invariant some riemannian metric of constant curvature on f thus any action of g on the 2 sphere s 2 is conjugate in diff s 2 to an orthogonal action if g acts on the torus sx s 1 there is a g invariant flat metric on s a s 1 and if g acts on a surface f with negative euler number then f admits a g invariant hyperbolic metric recently thurston th 1 th 2 th 3 has described the eight 3 dimensional geometries which provide geometric structures for closed 3 manifolds in the same way that the 2 sphere s 2 the euclidean plane e 2 and the hyperbolic plane h 2 provide geometric structures for surfaces see also the survey article by scott sc4 thurston also conjectured that if m is a closed 3 manifold with a geometric structure modelled on one of these eight geometries say x then any smooth action of a finite group g on m should leave invariant some metric on m inducing the geometry x we will say that g preserves the geometric structure on m in this case it should be noted that the restriction to smooth actions of g on m is essential for bing bi showed that there are involutions of s 3 whose fixed set is a wild 2 sphere however in dimension two it was proved by eilenberg ei that any action of a finite group on a surface is conjugate to a smooth action in this paper our main result asserts that thurston s conjecture holds for five of the eight geometries the result is the following

an inexact newton method combined with hestenes multipliers scheme for the solution of karush kuhn tucker systems in this work a newton interior point method for the solution of karush kuhn tucker systems is presented a crucial feature of this iterative method is the solution at each iteration of the inner subproblem this subproblem is a linear quadratic programming problem that can solved approximately by an inner iterative method such as the hestenes multipliers method a deep analysis on the choices of the parameters of the method perturbation and damping parameters has been done the global convergence of the newton interior point method is proved when it is viewed as an inexact newton method for the solution of nonlinear systems with restriction on the sign of some variables the newton interior point method is numerically evaluated on large scale test problems arising from elliptic optimal control problems which show the effectiveness of the approach

on the inconsistency of the bohm gadella theory with quantum mechanics the bohm gadella theory sometimes referred to as the time asymmetric quantum theory of scattering and decay is based on the hardy axiom the hardy axiom asserts that the solutions of the lippmann schwinger equation are functionals over spaces of hardy functions the preparation registration arrow of time provides the physical justification for the hardy axiom in this paper it is shown that the hardy axiom is incorrect because the solutions of the lippmann schwinger equation do not act on spaces of hardy functions it is also shown that the derivation of the preparation registration arrow of time is flawed thus hardy functions neither appear when we solve the lippmann schwinger equation nor should they appear it is also shown that the bohm gadella theory does not rest on the same physical principles as quantum mechanics and that it does not solve any problem that quantum mechanics cannot solve the bohm gadella theory must therefore be abandoned

crosscap numbers and the jones polynomial abstract we give sharp two sided linear bounds of the crosscap number non orientable genus of alternating links in terms of their jones polynomial our estimates are often exact and we use them to calculate the crosscap numbers for several infinite families of alternating links and for several alternating knots with up to twelve crossings we also discuss generalizations of our results for classes of non alternating links

the existence of embedded minimal surfaces and the problem of uniqueness it was in the early nineteen thirties that douglas and radd solved the plateau problem for a rectifiable jordan curve in euclidean space the minimal surfaces that they obtained are realized by conformal harmonic maps from the unit disk in 1948 morrey generalized the theorem of douglas and radd to minimal surfaces in a homogeneously regular riemannian manifold in both cases the solutions are defective as they may possess branch points it was not until 1968 that osserman 24 was able to prove that interior branch points do not exist on the douglas radd morrey solution osserman s theorem in this full generality was proved by alt 2 3 and gulliver 8 if the jordan curve is real analytic gulliver and lesley 9 showed that the solution surface is free of branch points even on its boundary hence at least in this case we know that the douglas radd morrey solution is an immersion and represents a classical surface however the douglas radd morrey solution may in general have selfintersections it was then a classical problem to find conditions on the jordan curve to guarantee that the solution is an embedded disk in 1932 radd was the first to produce a general condition of this type he proved that if the curve can be projected onto a convex curve in the plane then it bounds a unique embedded minimal disk which is a graph over this plane it was not until the seventies that there was further progress on the problem of embeddedness osserman had conjectured that if a jordan curve is extreme that is lies on the boundary of its convex hull then the douglas rad6 solution is embedded gulliver and spruck 10 proved this conjecture under the additional assumption that the absolute total curvature of the jordan curve is less than 4 z right after this work tomi and tromba 32 showed that an extreme curve must bound an embedded minimal disk which need not be stable in 1976 almgren and simon 1 established the existence for such a curve of an embedded stable minimal disk which need not be a douglas radd solution finally in the same year the authors 18 proved that any douglas rad6 solution must be embedded

0 efficient triangulations of 3 manifolds efficient triangulations of 3 manifolds are defined and studied it is shown that any triangulation of a closed orientable irreducible 3 manifold m can be modified to a 0 efficient triangulation orm can be shown to be one of the manifolds s 3 rp 3 or l 3 1 similarly any triangulation of a com pact orientable irreducible irreducible 3 manifold can be modified to a 0 efficient triangulation the notion of a 0 efficient ideal tr is de fined it is shown if m is a compact orientable irreducible irreducible 3 manifold having no essential annuli and distinct from the 3 cell then m admits an ideal triangulation furthermore it is shown that any ideal trian gulation of such a 3 manifold can be modified to a 0 efficient ideal triangula tion a 0 efficient triangulation of a closed manifold has only one vertex or the manifold is s 3 and the triangulation has precisely two vertices 0 efficient tri angulations of 3 manifolds with boundary and distinct from the 3 cell have all their vertices in the boundary and then just one vertex in each bound ary component as tools we introduce the concepts of barrier surface and shrinking as well as the notion of crushing a triangulation along a normal sur face a number of applications are given including an algorithm to construct an irreducible decomposition of a closed orientable 3 manifold an algorithm to construct a maximal collection of pairwise disjoint normal 2 spheres in a closed 3 manifold an alternate algorithm for the 3 sphere recognition prob lem results on edges of low valence in minimal triangulations of 3 manifolds and a construction of irreducible knots in closed 3 manifolds

the number of reidemeister moves needed for unknotting there is a positive constant c1 such that for any diagram d representing the unknot there is a sequence of at most 2 c1n reidemeister moves that will convert it to a trivial knot diagram where n is the number of crossings in d a similar result holds for elementary moves on a polygonal knot k embedded in the 1 skeleton of the interior of a compact orientable triangulated p l 3 manifold m there is a positive constant c2 such that for each t 1 if m consists of t tetrahedra and k is unknotted then there is a sequence of at most 2 c 2t elementary moves in m which transforms k to a triangle contained inside one tetrahedron of m we obtain explicit values for c1 and c2

a description of virtual scattering states in the rigged hilbert space formulation of quantum mechanics we define a pair of gel fand triplets i hi i i 1 2 where the linear spaces i are formed by suitably chosen entire analytic functions on the riemann surface associated to the transformation z2 virtual scattering states are well defined as pairs of functionals respectively in 1 and 2

on the location of directions of infinite descent for nonlinear programming algorithms there is much current interest in general equality constrained quadratic programming problems both for their own sake and for their applicability to active set methods for nonlinear programming in the former case typically the issues are existence of solutions and their determination in the latter instance nonexistence of solutions gives rise to directions of infinite descent such directions may subsequently be used to determine a more desirable active set the generalised cholesky decomposition of relevant matrices enables us to answer the question of existence and to determine directions of infinite descent when applicable in an efficient and stable manner the methods examined are related to implementations that are suitable for null space range space and lagrangian methods

solving optimization problems on ranks and inertias of some constrained nonlinear matrix functions via an algebraic linearization method abstract we establish in this paper a group of closed form formulas for calculating the global maximum and minimum ranks and inertias of the quadratic hermitian matrix function x q x p x with respect to the variable matrix x by using a linearization method and some known formulas for extremum ranks and inertias of linear hermitian matrix functions where both p and q are complex hermitian matrices and x is the conjugate transpose of x we then derive the global maximum and minimum ranks and inertias of the two quadratic hermitian matrix functions 1 x q 1 x p 1 x and 2 x q 2 x p 2 x subject to a consistent matrix equation a x b respectively by using some pure algebraic operations of matrices and their generalized inverses as consequences we establish necessary and sufficient conditions for the solutions of the matrix equation a x b to satisfy the quadratic hermitian matrix equalities x p 1 x q 1 and x p 2 x q 2 respectively and for the quadratic matrix inequalities x p 1 x q 1 and x p 2 x q 2 in the lowner partial ordering to hold respectively in addition we give complete solutions to four lowner partial ordering optimization problems on the matrix functions 1 x and 2 x subject to a x b examples are also presented to illustrative applications of the equality constrained quadratic optimizations in some matrix completion problems

partial pivoting strategies for symmetric gaussian elimination in an earlier paper 6 it is shown how the use of symmetric additions of rows and columns enables a stableldlt factorization of symmetric indefinite matrices in this paper we describe partial pivoting strategies these strategies are faster than the complete pivoting strategies that were introduced in the first paper numerical experiments are included the results show that some of the new strategies share the stable behaviour of complete pivoting while running almost as fast as the well known cholesky decomposition

a note on the augmented hessian when the reduced hessian is semidefinite certain matrix relationships play an important role in optimality conditions and algorithms for nonlinear and semidefinite programming let h be an n n symmetric matrix a an m n matrix and z a basis for the null space of a in a typical optimization context h is the hessian of a smooth function and a is the jacobian of a set of constraints when the reduced hessian zthz is positive definite augmented lagrangian methods rely on the known existence of a finite bar rho ge 0 such that for all rho bar rho the augmented hessian h rho ata is positive definite in this note we analyze the case when zthz is positive semidefinite i e singularity is allowed and show that the situation is more complicated in particular we give a simple necessary and sufficient condition for the existence of a finite bar rho so that h rho ata is positive semidefinite for rho ge bar rho a corollary of our result is that if h is nonsingular and indefinite while zthz is positive semidefinite and singular no such bar rho exists

a rigged hilbert space of hardy class functions applications to resonances the explicit construction of a dense subspace of square integrable functions on the positive half of the real line is given this space has the properties that 1 it is endowed with a metrizable nuclear topology 2 it is stable under multiplication by x and 3 the functions in have suitable analytical continuation to a half plane the space of functions which are conjugate to elements of is also considered then the triplets l2 0 and l2 0 are used to give a description of resonances

comment on on the inconsistency of the bohm gadella theory with quantum mechanics rigged hilbert spaces of hardy functions lead to a consistent theory of resonance scattering and decay contrary to the claims of a recent article 8 the theory holds for a wide range of potentials and rigorously describes the asymmetric time evolution of resonances

one fixed point actions and homology 3 spheres one of the best known three dimensional manifolds is the poincare homology three sphere 1 2 3 5 this closed 3 manifold has the same homology groups as the standard sphere s3 but is not simply connected see pnc p 106 or br2 section 1 8 numerous investigations during the past century have shown that this manifold has many remarkable properties several equivalent descriptions of 1 2 3 5 are summarized in kis from the viewpoint of transformation groups one noteworthy property is that 1 2 3 5 is the only nonsimply connected homology sphere admitting a transitive action of a compact lie group brl the manifold 1 2 3 5 also figures importantly in regularity questions for group actions on spherelike manifolds linear or orthogonal actions of finite groups on spheres are generally viewed as the simplest and most regular examples of such actions and one of the themes of transformation groups is to determine the extent to which arbitrary actions on spherelike manifolds resemble linear actions one distinguishing feature of linear actions is that their fixed point sets if nonempty are also spheres the well known results of p a smith cf br2 show that continuous p group actions on homology spheres all resemble linear actions in this respect i e the fixed point sets are always zp homology cohomology spheres on the other hand outside the family of p groups there are many exotic group actions perhaps the most elementary example of this type involves 1 2 3 5 if a is the subgroup of the rotation group so3 given by the symmetries of a regular icosahedron so that a is isomorphic to the alternating group a5 then the quotient space s03 a is the poincare homology 3 sphere and the induced action of a5 a has exactly one fixed point this example suggests that general actions of finite groups on homology spheres can be quite different from linear actions and much of the work on exotic finite group actions during the past three decades was at least partially motivated by the existence of this group action

the computational complexity of knot and link problems we consider the problem of deciding whether a polygonal knot in 3 dimensional euclidean space is unknotted ie capable of being continuously deformed without self intersection so that it lies in a plane we show that this problem unknotting problem is in np we also consider the problem splitting problem of determining whether two or more such polygons can be split or continuously deformed without self intersection so that they occupy both sides of a plane without intersecting it we show that it also is in np finally we show that the problem of determining the genus of a polygonal knot a generalization of the problem of determining whether it is unknotted is in pspace we also give exponential worst case running time bounds for deterministic algorithms to solve each of these problems these algorithms are based on the use of normal surfaces and decision procedures due to w haken with recent extensions by w jaco and j l tollefson

partitioned variable metric updates for large structured optimization problems this paper presents a minimization method based on the idea of partitioned updating of the hessian matrix in the case where the objective function can be decomposed in a sum of convex element functions this situation occurs in a large class of practical problems including nonlinear finite elements calculations some theoretical and algorithmic properties of the update are discussed and encouraging numerical results are presented

huber approximation for the non linear l1 problem abstract the smooth huber approximation to the non linear l 1 problem was proposed by tishler and zang 1982 and further developed in yang 1995 in the present paper we use the ideas of gould 1989 to give a new algorithm with rate of convergence results for the smooth huber approximation results of computational tests are reported

least area incompressible surfaces in 3 manifolds let m be a riemannian manifold and let f be a closed surface a map f f m is called least area if the area of f is less than the area of any homotopic map from f to m note that least area maps are always minimal surfaces but that in general minimal surfaces are not least area as they represent only local stationary points for the area function the existence of least area immersions in a homotopy class of maps has been established when the homotopy class satisfies certain injectivity conditions on the fundamental group 18 17 in this paper we shall consider the possible singularities of such immersions our results show that the general philosophy is that least area surfaces intersect least meaning that the intersections and self intersections of least area immersions are as small as their homotopy classes allow when measured correctly one should note that evidence supporting this view had been found by meeks yau in their embedding theorems for minimal disks and 2 spheres 13 143 our main result asserts that if a least area immersion is homotopic to an embedding then it has no self intersections which clearly exemplifies the above philosophy the precise result is the following

constraint deletion strategy in the inertia controlling quadratic programming method abstract the inertia controlling strategy in active set methods consists of choosing the working set so that the reduced hessian never has more than one non positive eigenvalue usually this strategy has been implemented by permitting to delete constraints only at stationary points this paper concerns the general inertia controlling quadratic programming method in which constraints may be deleted at non stationary points we consider the determination of the search direction when the reduced hessian is positive definite positive semidefinite and singular and indefinite or negative definite recurrence formulas are presented to update the search direction and multiplier estimates when the working set changes

convergence analysis of a regularized interior point algorithm for the barrier problems with singular solutions abstract wachter and biegler 2006 25 presented and implemented an interior point filter line search method called ipopt for large scale nonlinear programming to make ipopt efficient and robust they add some safeguards in their implementation one crucial safeguard is to apply the inertia correction technique to modify the kkt matrix we focus on the degenerate problems in which the kkt matrix might be singular to complement theoretical foundation of this technique in the degenerate case we propose and analyze a regularized interior point method for the barrier problems with singular solutions which largely relies on ipopt based on the inertia correction technique the matrix arising in the kkt equation is corrected to be a nonsingular one so that the newton step can be calculated at each iteration even in the degenerate case we emphasize that the special inertia correction for the kkt matrix does not impose any negative impact on the fast convergence of our algorithm under some local error bound condition we prove that the rate of quadratic convergence is achieved when the second order condition or some constraint qualification condition fails to hold or both

the rigged hilbert space approach to the lippmann schwinger equation part i we exemplify the way the rigged hilbert space deals with the lippmann schwinger equation by way of the spherical shell potential we explicitly construct the lippmann schwinger bras and kets along with their energy representation their time evolution and the rigged hilbert spaces to which they belong it will be concluded that the natural setting for the solutions of the lippmann schwinger equation and therefore for scattering theory is the rigged hilbert space rather than just the hilbert space