stability in the presence of degeneracy and error estimation given an approximation to a local minimizer to a nonlinear optimization problem and to associated multipliers we obtain a tight error estimate in terms of the violation of the first order conditions our results apply to degenerate optimization problems where independence of the active constraint gradients and strict complementarity can be violated

multi mode trimming of imperfect thin rings using masses at pre selected locations abstract this paper presents a practical method for trimming the natural frequencies of an initially imperfect ring to simultaneously eliminate certain of the frequency splits present compared with previous work the novel feature of this method is that the trimming masses are positioned at pre selected locations on the ring the basis for the proposed method is the concept of equivalent imperfection mass which allows any imperfect ring to be considered as a perfect ring with equivalent imperfection masses attached by considering this trimming problem it is deduced that it is possible to trim n pairs of modes simultaneously by removing a minimum of 2 n trimming masses at particular locations around the ring by positioning the trimming masses at pre selected locations it is shown that a simple set of trimming masses can be calculated easily and from this set an infinite number of solution sets can be found methods for generating these sets are outlined for the trimming of both a single and a dual pair of modes in practice it is likely that the trimming masses will be spaced regularly for this special case it is found that it is not possible to trim all single and dual mode pairs with any arrangement of masses validation of the derived simple solution set and the proposed procedure to generate further sets is achieved by studying a number of theoretical examples

free vibrations of elliptical rings with circumferentially variable thickness abstract this paper deals with the in plane free vibration of rings with a nominally elliptical centreline results are presented for rings of constant axial length that have a rectangular cross section the radial thickness of which is constant or has a simple analytically defined circumferential variation additionally and for the first time the effects of small variations in in plane profile such as those arising in practical rings due to manufacturing tolerances are considered the problem is tackled using an approach in which the true middle surface is determined numerically from the outer and inner surface profiles which can be defined either by exact analytical expressions or in a more general way using fourier series the rayleigh ritz method is used to obtain the natural frequencies and mode shapes results are presented for a range of cases including some that have previously been studied by other authors and some that have not the effects on frequency splitting due to profile variations and the aspect ratio of the ellipse are emphasized results obtained using the developed numerical approach show excellent agreement with finite element predictions

trace spaces of counterexamples to naimark s problem abstract a counterexample to naimark s problem is a c algebra that is not isomorphic to the algebra of compact operators on some hilbert space yet still has only one irreducible representation up to unitary equivalence it is well known that such algebras must be nonseparable and in 2004 akemann and weaver used the diamond principle a set theoretic axiom independent from zfc to give the first counterexamples for any such counterexample a the unitary group u a acts transitively on the pure states which are the extreme points of the state space s a it is conceivable that this implies as happens for finite dimensional simplexes that the action of u a on s a has at most one fixed point i e a has at most one trace we give a strong negative answer here assuming diamond in particular we adapt the akemann weaver construction to show that the trace space of a counterexample to naimark s problem can be affinely homeomorphic to any metrizable choquet simplex and can also be nonseparable

consistency of a counterexample to naimark s problem we construct a c algebra that has only one irreducible representation up to unitary equivalence but is not isomorphic to the algebra of compact operators on any hilbert space this answers an old question of naimark our construction uses a combinatorial statement called the diamond principle which is known to be consistent with but not provable from the standard axioms of set theory assuming that these axioms are consistent we prove that the statement there exists a counterexample to naimark s problem which is generated by elements is undecidable in standard set theory

multi mode trimming of imperfect rings abstract this paper proposes a method for trimming the natural frequencies of an imperfect ring to simultaneously eliminate certain of the frequency splits present initially the effect of the addition of a number of imperfection masses on a perfect ring is considered this is achieved by using a rayleigh ritz approach in which it is assumed that the mode shapes are identical to those of a perfect ring by considering the inverse the so called trimming problem it is deduced that it is possible to trim n pairs of modes simultaneously by removing a minimum of n trimming masses at particular locations around the ring to calculate the trimming mass locations it is necessary to solve n non linear algebraic equations once this has been achieved the magnitude of the trimming masses can be calculated easily for the special case of trimming a single pair of modes analytic solutions for the magnitude and position of the single required trimming mass are available to trim two pairs of modes it is shown that a simple analytic relationship exists between the angular positions of the two required trimming masses and that the magnitude of these masses can be obtained easily to trim more pairs of modes numerical techniques are required and for this purpose a numerical procedure is proposed validation of the derived analytic results and the proposed numerical procedure is achieved by studying a number of theoretical examples

an efficient algorithm for globally minimizing a quadratic function under convex quadratic constraints in this paper we investigate two approaches to minimizing a quadratic form subject to the intersection of finitely many ellipsoids the first approach is the d c difference of convex functions optimization algorithm abbr dca whose main tools are the proximal point algorithm and or the projection subgradient method in convex minimization the second is a branch and bound scheme using lagrangian duality for bounding and ellipsoidal bisection in branching the dca was first introduced by pham dinh in 1986 for a general d c program and later developed by our various work is a local method but from a good starting point it provides often a global solution this motivates us to combine the dca and our branch and bound algorithm in order to obtain a good initial point for the dca and to prove the globality of the dca in both approaches we attempt to use the ellipsoidal constrained quadratic programs as the main subproblems the idea is based upon the fact that these programs can be efficiently solved by some available polynomial and nonpolynomial time algorithms among them the dca with restarting procedure recently proposed by pham dinh and le thi has been shown to be the most robust and fast for large scale problems several numerical experiments with dimension up to 200 are given which show the effectiveness and the robustness of the dca and the combined dca branch and bound algorithm

a trust region strategy for minimization on arbitrary domains we present a trust region method for minimizing a general differentiable function restricted to an arbitrary closed set we prove a global convergence theorem the trust region method defines difficult subproblems that are solvable in some particular cases we analyze in detail the case where the domain is a euclidean ball for this case we present numerical experiments where we consider different hessian approximations

on the solution of a two ball trust region subproblem in this paper we investigate the structure of a two ball trust region subproblem arising frequently in nonlinear parameter identification problems and propose a method for its solution the method decomposes the subproblem and allows the application of efficient well studied methods for the solution of trust region subproblems arising in unconstrained optimization in the discussion of the structure we focus on the case where both constraints are active and on the treatment of the unconstrained problem

a new active set algorithm for box constrained optimization an active set algorithm asa for box constrained optimization is developed the algorithm consists of a nonmonotone gradient projection step an unconstrained optimization step and a set of rules for branching between the two steps global convergence to a stationary point is established for a nondegenerate stationary point the algorithm eventually reduces to unconstrained optimization without restarts similarly for a degenerate stationary point where the strong second order sufficient optimality condition holds the algorithm eventually reduces to unconstrained optimization without restarts a specific implementation of the asa is given which exploits the recently developed cyclic barzilai borwein cbb algorithm for the gradient projection step and the recently developed conjugate gradient algorithm cg descent for unconstrained optimization numerical experiments are presented using box constrained problems in the cuter and minpack 2 test problem libraries

simple nuclear c algebras not isomorphic to their opposites abstract we show that it is consistent with zermelo fraenkel set theory with the axiom of choice zfc that there is a simple nuclear nonseparable c c algebra which is not isomorphic to its opposite algebra we can furthermore guarantee that this example is an inductive limit of unital copies of the cuntz algebra o 2 o2 or of the canonical anticommutation relations car algebra

a class of methods for projection on the intersection of several ellipsoids the paper is concerned with the basic optimization problem of projecting a point onto the intersection of several ellipsoids we present a class of methods where the problem is reduced to a sequence of projections onto the intersection of several balls the subproblems are simpler and more tractable but the main advantage is that in so doing we can avoid solving linear systems completely and thus the methods are very suitable for large scale problems the methods have been shown to have nice convergence properties under slater s constraint qualification and to be very reliable and efficient in our testing

optimality conditions for the minimization of a quadratic with two quadratic constraints the trust region method has been proven to be very successful in both unconstrained and constrained optimization it requires the global minimum of a general quadratic function subject to ellipsoid constraints in this paper we generalize the trust region subproblem by allowing two general quadratic constraints conditions and properties of its solution are discussed

a simplicial branch and bound algorithm for solving quadratically constrained quadratic programs we propose a branch and bound algorithm for solving nonconvex quadratically constrained quadratic programs the algorithm is novel in that branching is done by partitioning the feasible region into the cartesian product of two dimensional triangles and rectangles explicit formulae for the convex and concave envelopes of bilinear functions over triangles and rectangles are derived and shown to be second order cone representable the usefulness of these new relaxations is demonstrated both theoretically and computationally

an ellipsoidal branch and bound algorithm for global optimization a branch and bound algorithm is developed for global optimization branching in the algorithm is accomplished by subdividing the feasible set using ellipses lower bounds are obtained by replacing the concave part of the objective function by an affine underestimate a ball approximation algorithm obtained by generalizing of a scheme of lin and han is used to solve the convex relaxation of the original problem the ball approximation algorithm is compared to sedumi as well as to gradient projection algorithms using randomly generated test problems with a quadratic objective and ellipsoidal constraints

natural frequencies and modes of rings that deviate from perfect axisymmetry given the experimental or theoretical natural frequencies and modes of an axisymmetric ring an analytical method is presented that allows one to obtain the natural frequencies and modes when this ring is non axisymmetric due to a mass or stiffness non uniformity of the ring a ring is the simplest model of an axisymmetric structure and treatment of its non uniformities paves the way for treatment of more complicated structures such as tires for example the receptance method is employed to determine the natural frequencies and mode shapes of a circular ring which is deviating from axisymmetry due to a local mass of stiffness non uniformity the receptance of the ring is derived by utilizing the modal expansion method

a simple separable exact c algebra not anti isomorphic to itself we give an example of an exact stably finite simple separable c algebra d which is not isomorphic to its opposite algebra moreover d has the following additional properties it is stably finite approximately divisible has real rank zero and stable rank one has a unique tracial state and the order on projections over d is determined by traces it also absorbs the jiang su algebra z and in fact absorbs the 3 uhf algebra we can also explicitly compute the k theory of d namely k 0 d cong mathbb z tfrac 1 3 with the standard order and k1 d 0 as well as the cuntz semigroup of d namely w d cong mathbb z tfrac 1 3 sqcup 0 infty

on the statistics of natural frequency splitting for rings with random mass imperfections abstract this paper considers the statistical distribution of natural frequency splits for an initially perfect ring with different types of random mass imperfection the analysis used to derive analytical expressions for the natural frequency splits is based on a rayleigh ritz approach in which it is assumed that the mode shapes of the imperfect rings are identical to those of a perfect ring the types of random mass imperfection investigated are i random harmonic variations in the mass per unit length around the circumference of the ring ii the attachment of random point masses at random locations on the ring and iii the attachment of random point masses at uniformly spaced positions on the ring for case i it is found that the frequency splits always have a rayleigh distribution for case ii an expression for the statistical distribution is deduced which tends to a rayleigh distribution as the number of attached masses increases for case iii it is found that the frequency split distribution is dependent upon the mode considered and the number of attached masses and that in some situations the frequency splits have a half gaussian i e non rayleigh distribution

the in plane vibration of thin rings with in plane profile variations part ii application to nominally circular rings abstract geometric profile variations always exist in nominally circular rings due to limitations in the manufacturing processes such profile variations are known to lead to frequency splitting between pairs of modes which are degenerate in a perfect ring in this paper the effects of circumferential profile variations on the in plane vibration characteristics of such rings are studied using a numerical method the inner and outer ring surfaces are described in a very general way by fourier series and the rayleigh ritz method is used to obtain the natural frequencies and mode shapes results are presented for a number of example cases which include single and multiple harmonic variations in profile the relationship between the patterns of frequency splitting and the harmonic content of the ring profile is investigated and the most important causes of frequency splitting are identified

outer automorphisms and reduced crossed products of simple c algebras every outer automorphism of a separable simplec algebra is shown to have a pure state which is mapped into an inequivalent state under this automorphism the reduced crossed product of a simplec algebra by a discrete group of outer automorphisms is shown to be simple

a note on quadratic forms we extend an interesting theorem of yuan 12 for two quadratic forms to three matrices let c 1 c 2 c 3 be three symmetric matrices in n n if max x t c 1 x x t c 2 x x t c 3 x 0 for all x n it is proved that there exist t i 0 i 1 2 3 such that i 1 3 t i 1 and i 1 3 t i c i has at most one negative eigenvalue

inexact variants of the proximal point algorithm without monotonicity this paper studies convergence properties of inexact variants of the proximal point algorithm when applied to a certain class of nonmonotone mappings the presented algorithms allow for constant relative errors in the line of the recently proposed hybrid proximal extragradient algorithm the main convergence result extends a recent work of the second author where exact solutions for the proximal subproblems were required we also show that the linear convergence property is preserved in the case when the inverse of the operator is locally lipschitz continuous near the origin as an application we give a convergence analysis for an inexact version of the proximal method of multipliers for a rather general family of problems which includes variational inequalities and constrained optimization problems

proximal methods for cohypomonotone operators conditions are given for the viability and the weak convergence of an inexact relaxed proximal point algorithm for finding a common zero of countably many cohypomonotone operators in a hilbert space in turn new convergence results are obtained for an extended version of the proximal method of multipliers in nonlinear programming

the in plane vibration of thin rings with in plane profile variations part i general background and theoretical formulation abstract this paper presents a methodology for the analysis of the free in plane vibration of thin rings with profile variations in the circumferential direction the methodology is suitable for any thin ring which is bounded by closed curves which are single valued functions of circumferential position the inner and outer profiles are expressed as fourier series thus allowing any profile to be approximated with any degree of accuracy an iterative numerical procedure for determining the true middle surface and the corresponding thickness at each cross section around the circumference is established a reduced plane stress form of novozhilov s thin shell theory is used to model the deformation mechanics of the ring the eigenvalue problem is then formulated using the rayleigh ritz method in conjunction with a harmonic series description of the displacements general expressions are presented for the corresponding mass and stiffness matrices a companion paper presents a comprehensive set of results which illustrates application of the theory

a simple theory for the analysis and correction of frequency splitting in slightly imperfect rings abstract the flexural radial vibration of circular rings with imperfections represented by small attached masses and springs is considered the concept of an equivalent imperfection mass is introduced this is used to quantify the frequency splitting which occurs in slightly imperfect axisymmetric bodies the concept is used to estimate the trimming mass which must be added to or removed from the ring and at which point in order to reduce or eliminate the frequency split in a given pair of modes the method can be applied to any axisymmetric body for which the mode shapes can be measured or computed

nonseparable uhf algebras i dixmier s problem abstract there are three natural ways to define uhf uniformly hyperfinite c algebras and all three definitions are equivalent for separable algebras in 1967 dixmier asked whether the three definitions remain equivalent for not necessarily separable algebras we give a complete answer to this question more precisely we show that in small cardinality two definitions remain equivalent and give counterexamples in other cases our results do not use any additional set theoretic axioms beyond the usual axioms namely zfc

an exact algorithm for graph partitioning an exact algorithm is presented for solving edge weighted graph partitioning problems the algorithm is based on a branch and bound method applied to a continuous quadratic programming formulation of the problem lower bounds are obtained by decomposing the objective function into convex and concave parts and replacing the concave part by an affine underestimate it is shown that the best affine underestimate can be expressed in terms of the center and the radius of the smallest sphere containing the feasible set the concave term is obtained either by a constant diagonal shift associated with the smallest eigenvalue of the objective function hessian or by a diagonal shift obtained by solving a semidefinite programming problem numerical results show that the proposed algorithm is competitive with state of the art graph partitioning codes

new results on quadratic minimization in this paper we present several new results on minimizing an indefinite quadratic function under quadratic linear constraints the emphasis is placed on the case in which the constraints are two quadratic inequalities this formulation is termed the extended trust region subproblem in this paper to distinguish it from the ordinary trust region subproblem in which the constraint is a single ellipsoid the computational complexity of the extended trust region subproblem in general is still unknown in this paper we consider several interesting cases related to this problem and show that for those cases the corresponding semidefinite programming relaxation admits no gap with the true optimal value and consequently we obtain polynomial time procedures for solving those special cases of quadratic optimization for the extended trust region subproblem itself we introduce a parameterized problem and prove the existence of a trajectory that will lead to an optimal solution combining this with a result obtained in the first part of the paper we propose a polynomial time solution procedure for the extended trust region subproblem arising from solving nonlinear programs with a single equality constraint

strong and weak convexity of sets and functions in this paper we study two classes of sets strongly and weakly convex sets for each class we derive a series of properties which involve either the concept of supporting ball an obvious extension of the concept of supporting hyperplane or the normal cone to the set we also study a class of functions denoted convex which satisfy for arbitrary points x1 and x2 and any value 0 1 the classical inequality of convex functions up to a term 1 x1 x2 2 depending on the sign of the constant the function is said to be strongly or weakly convex we provide characteristic properties of this class of sets and we relate it to strongly and weakly convex sets via the epigraph and the level sets finally we give three applications a separation theorem a sufficient condition for global optimum of a nonconvex programming problem and a sufficient geometrical condition for a set to be a manifold

frequency split and vibration localization in imperfect rings abstract the dynamics of linearly elastic imperfect rings vibrating in their own plane is considered in this paper imperfections are modeled as perturbations of the uniform linear mass density and bending stiffness of a perfect ring a perturbation expansion and a spectral representation are employed and a variational formulation of the vibration problem is obtained a linear theory is deduced by retaining only the leading order terms in the variational formulation the linear theory yields simple closed form expressions for the eigenfrequencies and the modal shapes which are accurate when the imperfections are sufficiently small an enhanced nonlinear theory is also derived which is accurate even when the ring imperfections are not small in this case an iterative solution procedure is developed the proposed theories are validated by considering some case study problems and using the ritz rayleigh solution as a benchmark finally the linear theory is applied to the frequency trimming problem of an imperfect ring a simple closed form expression for the trimming masses is presented valid for trimming any selected number of eigenmodes