on the solution of s ω x 0 by a newtonian procedure the method 1 developed for the solution of e a x 0 where e and a are real symmetric matrices and a is positive definite is extended to deal with s x 0 where s is a real symmetric structural dynamic stiffness matrix whose elements are trancendental functions of the radian natural frequency the method enables all natural frequencies latent roots 2 and mode shapes latent vectors x across a prescribed frequency range to be determined infallibly and in ascending frequency order at a speed approaching twice that associated with bisection and sign counting methods 3 it is ideally suited for use on micro and minicomputers where single precision working is the norm and on the commodore 4032 computer can cope with matrices s of order n and bandwidth p provided that np 4200 much larger problems of course may be dealt with on mainframe machines

a newtonian procedure for the solutions of ex λax abstract newtonian iteration on a scalar function which is obtained by condensation of the pencil e a is used in conjunction with sturm sequence properties of the pencil to provide an infallible quadratically convergent algorithm for all eignevalues of a 1 e within any stated range the algorithm has been programmed for use on a desk top machine and large sparse matrix eigenvalue problems have been solved a brief outline of the essential steps in the program is given along with some particular solutions

an exact model reduction procedure for mechanical systems abstract this paper gives details of an exact model reduction procedure for the dynamic analysis of general mechanical systems in this procedure the system is first transformed into an equivalent one dimensional id lumped mass spring damping system by coordinate transformations the 1d lumped system is further reduced into a lower order model with arbitrarily selected degrees of freedom three numerical examples are given for a distributed lumped mass spring system a truss frame structure and a cantilever plate

formulation and solution of the non linear damped eigenvalue problem for skeletal systems this paper presents and discusses an arnoldi based eigensolution technique for evaluating the complex natural frequencies and mode shapes from frequency dependent quadratic eigenproblems associated with vibration analysis of damped structures the new solution technique is used in conjunction with a mixed finite element modelling procedure which utilizes both the polynomial and frequency dependent displacement fields in formulating the system matrices this modelling provides the ability to represent a frequency dependent damping matrix in vibration analysis of skeletal systems the eigensolution methodology presented here is based upon the ability to evaluate a specific set of parametrized curves for the non linear eigenvalue problem at given values of the parameter numerical examples illustrate that this method used in conjunction with a secant interpolation accurately evaluates the complex natural frequencies and modes of the quadratic non linear eigenproblem and verifies that the new eigensolution technique coupled with the mixed finite element modelling procedure is more accurate than the conventional finite element models

coupled bending torsional dynamic stiffness matrix of an axially loaded timoshenko beam element abstract analytical expressions for the coupled bending torsional dynamic stiffness matrix terms of an axially loaded uniform timoshenko beam element are derived in an exact sense by solving the governing differential equations of motion of the element the symbolic computing package reduce has been used to generate an analytical expression for each of the dynamic stiffness terms in a concise form for check purposes numerical values of the dynamic stiffness matrix terms were obtained using the derived explicit expressions as well as by an alternative nonanalytical method based on matrix inversions and matrix multiplications stiffnesses obtained from both methods agreed with each other to machine accuracy application of the developed theory is discussed with particular reference to an established algorithm the influence of axial force shear deformation and rotatory inertia on the natural frequencies of a bending torsion coupled beam with cantilever end conditions is demonstrated by numerical results such results are not generally available in the literature therefore results obtained by partially restricting the present theory are compared with the existing literature wherever possible the results indicate that the method is accurate and efficient

geometry of entangled states bloch spheres and hopf fibrations we discuss a generalization of the standard bloch sphere representation for a single qubit to two qubits in the framework of hopf fibrations of highdimensional spheres by lower dimensional spheres the single qubit hilbert space is the three dimensional sphere s 3 t hes 2 base space of a suitably oriented s 3 hopf fibration is nothing but the bloch sphere while the circular fibres represent the overall qubit phase degree of freedom for the two qubits case the hilbert space is a seven dimensional sphere s 7 which also allows for a hopf fibration with s 3 fibres and a s 4 base the most striking result is that suitably oriented s 7 hopf fibrations are entanglement sensitive the relation with the standard schmidt decomposition is also discussed

extrapolation locking and its sanitization in koiter s asymptotic analysis abstract this paper shows that the fem implementation of koiter s asymptotic method w t koiter on the stability of elastic equilibrium 1970 ph d thesis delft 1945 english transl nasa tt f10 883 1967 affdl tr70 25 outlined by casciaro et al finite element asymptotic analysis of slender elastic structures a simple approach int j num meth eng 35 1992 1397 1426 provides accurate and reliable results in the critical and post critical analysis of non linear elastic structures care however does have to be taken in implementing apparently minor details to avoid locking effects which adversely affect accuracy and which can destroy the method reliability as the effects related to the finite element interpolation have been discussed before this paper focuses on the non linear locking due to the use implicit in the method of finite distance extrapolations within this scope it is shown that perturbation algorithms based on compatible formulations can imply a strong critical and post critical locking when analysing structures characterized by high stiffness ratios in the presence of moderate pre critical rotations on the contrary perturbation algorithms based on independent extrapolations of displacements and stresses furnish reliable results in excellent agreement with those provided by step by step analysis at a small fraction of its computational cost

the member stiffness determinant and its uses for the transcendental eigenproblems of structural engineering and other disciplines transcendental stiffness matrices are well established in vibration and buckling analysis having been derived from exact analytical solutions of the differential equations for many structural members without recourse to finite element discretization their assembly into the overall structural stiffness matrix gives a transcendental eigenproblem solvable with certainty by the wittrick williams ww algorithm instead of the usual linear algebraic eigenproblem this paper establishes a normalized member stiffness determinant being the value of the stiffness matrix determinant if the member were modelled by infinitely many finite elements and its ends were clamped it is derived for beams with uncoupled axial and bernoulli euler flexural behaviour by methods applicable to any member possessing transcendental stiffnesses multiplying the product of a structure s member stiffness determinants by the determinant of its transcendental overall stiffness matrix gives a determinant which unlike the determinant of the transcendental overall stiffness matrix itself has no poles when plotted against the eigenparameter leading to a more secure eigenvalue location numerical results demonstrate the resulting advantages in addition the member stiffness determinant can give exact determinants for members with various standard end conditions and also provides limits for finite element solutions as the number of elements approaches infinity analogous advantages occur in all other disciplines where the ww algorithm is used e g fluid vibrating in pipes heat and mass diffusion and sturm liouville problems

substructure mode synthesis with overlapping connecting elements abstract an approach is presented to synthesize the modal properties of two substructures to obtain the modal properties of the combined structure the substructures are assumed to be fixed at the interfacing boundary and have overlapping elements the eigenproperties of the overlapping elements are also used in the synthesis process the approach can be used with any finite element discretization scheme the approach avoids the solution of a second eigenvalue problem by conventional means instead the eigenvalues are obtained as the solution of characteristic equations and eigenvectors are defined in closed form if the complete set of eigenproperties are used the approach provides the exact eigenproperties

a lanczos based technique for exact vibration analysis of skeletal structures this paper presents and discusses a lanczos based eigensolution technique for evaluating the natural frequencies and modes from frequency dependent eigenproblems in structural dynamics the new solution technique is used in conjunction with a mixed finite element modelling procedure which utilizes both the polynomial and frequency dependent displacement fields in formulating the system matrices the method is well suited to the solution of large scale problems the new solution methodology presented here is based on the ability to evaluate a specific set of parameterized non linear eigenvalue curves at given values of the parameter through an implicitly restarted lanczos technique numerical examples illustrate that the implicitly restarted lanczos method with secant interpolation accurately evaluates the exact natural frequencies and modes of the non linear eigenproblem and verifies that the new eigensolution technique coupled with the mixed finite element modelling procedure is more accurate than the conventional finite element models in addition the eigenvalue technique presented here is shown to be far more computationally efficient on large scale problems than the determinant search techniques traditionally employed for solving exact vibration problems

the geometry of concurrence as a measure of entanglement it is widely recognized that concurrence can be regarded as a measure of two qubit entanglement this paper studies the geometry of concurrence for a two qubit system to show that the concurrence serves as a coordinate of the factor space where is the space of normalized two qubit states and where g u 1 su 2 su 2 any monotonically increasing function of the concurrence can serve as a measure for entanglement from the viewpoint of riemannian geometry a state with concurrence r is shown to be distant from the separable states by where r ranges over 0 r 1 in addition measures of entanglement for n qubits are discussed on the basis of a bipartite decomposition with m n they are invariant under the local unitary transformation group u 2 u 2m

development of a block lanczos algorithm for free vibration analysis of spinning structures this paper is concerned with the development of an efficient eigenproblem solution algorithm and an associated computer program for the economical solution of the free vibration problem of complex practical spinning structural systems thus a detailed description of a newly developed block lanczos procedure is presented in this paper that employs only real numbers in all relevant computations and also fully exploits sparsity of associated matrices the procedure is capable of computing multiple roots and proves to be most efficient compared to other existing similar techniques

safeguarded use of the implicit restarted lanczos technique for solving non linear structural eigensystems this paper presents a new algorithm for evaluating the eigenvalues and their corresponding eigenvectors for large scale non linear eigensystems in structural dynamics the algorithm is based on solving a sequence of algebraic eigenproblems and updating the parameter the implicitly restarted lanczos method had been determined to be well suited for solving the linear eigenproblems that arise in this context a zero finder approach that uses rational interpolation to approximate the generalized eigenvalues has been developed to update the methodology of the new algorithm developed here is designed to evaluate a subset of the parametrized non linear eigencurves at specific values of numerical experiments show that the new eigensolution technique is superior to existing approaches for large scale problems and competitive for small size problems the main emphasis of this contribution is the derivation and analysis of this scheme for non linear structural eigensystems

geometry of the three qubit state entanglement and division algebras we present a generalization to three qubits of the standard bloch sphere representation for a single qubit and of the seven dimensional sphere representation for two qubits presented in mosseri et al mosseri r and dandoloff r 2001 j phys a math gen 34 10243 the hilbert space of the three qubit system is the 15 dimensional sphere s15 which allows for a natural last hopf fibration with s8 as base and s7 as fibre a striking feature is as in the case of one and two qubits that the map is entanglement sensitive and the two distinct ways of un entangling three qubits are naturally related to the hopf map we define a quantity that measures the degree of entanglement of the three qubit state conjectures on the possibility of generalizing the construction for higher qubit states are also discussed

coupled bending torsional dynamic stiffness matrix for beam elements explicit expressions for the coupled bending torsional dynamic stiffness matrix of a uniform beam element are derived in an exact sense by solving the governing differential equation of the beam implementation of the derived dynamic stiffness matrix in a space frame computer program is discussed with particular reference to an established algorithm to enable vibration analysis of coupled systems to be made the application of the theory is demonstrated by an illustrative example wherein the results for a cantilever beam with a substantial amount of coupling between bending and torsion are highlighted the correctness of the theory is confirmed to a high degree of accuracy by computed results and numerical checks

on roots of functional lambda matrices the problem of determining the roots or eigenvalues of functional lambda matrices and their sensitivities are considered an accurate and efficient computational technique is proposed it is shown that sensitivities are computed virtually for free as a byproduct of the algorithm examples demonstrate applications

a solution of the eigenproblem for undamped gyroscopic systems with the lanczos algorithm this paper presents an efficient numerical solution of the quadratic eigenproblem arising in the analysis of gyroscopic systems such problems are known to reduce to a generalized linear eigenproblem defined by two real non singular matrices one symmetric and one skew symmetric for this class of problems the general lanczos algorithm for unsymmetric matrices is shown to simplify considerably and yields an efficient solution of the problem full advantage can be taken of the sparsity of the matrices and of the specific nature of gyroscopic systems numerical examples are presented which demonstrate the efficiency and accuracy of the solution procedure

reliable use of determinants to solve non linear structural eigenvalue problems efficiently a multiple determinant parabolic interpolation method is described and evaluated principally by using a plane frame test bed program it is intended primarily for solving the transcendental eigenvalue problems arising when the exact member equations obtained by solving the governing differential equations of members are used to find the eigenvalues i e critical buckling loads or undamped natural frequencies of structures the method has five stages which together ensure successful convergence on the required eigenvalues in all circumstances thus whenever checks indicate its suitability parabolic interpolation is used to obtain eigenvalues more rapidly than would the popular bisection alternative the checks also ensure a wise choice of the determinant used by the interpolation the determinants available are all usually zero at eigenvalues and comprise the determinant of the overall stiffness matrix kn and the determinants which result with negligible extra computation from effectively considering all except the last m m 1 2 n 1 freedoms to which kn corresponds as internal substructure freedoms tests showed time savings compared to bisection of 31 per cent when finding non coincident eigenvalues to relative accuracy 10 4 increasing to 62 per cent when 10 8 the tests also showed time savings of about 10 per cent compared with an earlier newtonian approach the method requires no derivatives and its use in the widely available space frame program bunvis rg has demonstrated how easily it can replace bisection which was used in the earlier program bunvis

thomas rotation and the mixed state geometric phase it is shown that uhlmann s parallel transport of purifications along a path of mixed states represented by 2 2 density matrices is just the path ordered product of thomas rotations these rotations are invariant under hyperbolic translations inside the bloch sphere that can be regarded as the poincare ball model of hyperbolic geometry a general expression for the mixed state geometric phase for an arbitrary geodesic triangle in terms of the bures fidelities is derived the formula gives back the solid angle result well known from studies of the pure state geometric phase it is also shown that this mixed state anholonomy can be reinterpreted as the pure state non abelian anholonomy of entangled states living in a suitable restriction of the quaternionic hopf bundle in this picture uhlmann s parallel transport is just the pancharatnam transport of quaternionic spinors

local symmetry properties of pure three qubit states entanglement types of pure states of three spin particles are classified by means of their stabilizers in the group of local unitary transformations it is shown that the stabilizer is generically discrete and that a larger stabilizer indicates a stationary value for some local invariant we describe all the exceptional states with enlarged stabilizers

exact dynamic stiffness matrix of a three dimensional shear beam with doubly asymmetric cross section abstract the dynamic member stiffness matrix of a three dimensional shear beam with doubly asymmetric cross section is derived exactly from the governing sixth order differential equation of motion such a formulation accounts for the uniform distribution of mass in the member and necessitates the solution of a transcendental eigenvalue problem this is achieved using the wittrick williams algorithm where the necessary parameters are developed using a generalised procedure an example is given to clarify the theory together with a small parametric study that indicates when lateral torsional coupling may safely be ignored the work also holds considerable potential in its application to the approximate analysis of asymmetric multi storey three dimensional frame structures

on development of a finite dynamic element and solution of associated eigenproblem by a block lanczos procedure the paper first presents the details of the development of a new six noded plane triangular finite dynamic element a block lanczos algorithm is developed next for the accurate and efficient solution of the quadratic matrix eigenvalue problem associated with the finite dynamic element formulation the resulting computer program fully exploits matrix sparsity inherent in such a discretization and proves to be most efficient for the extraction of the usually required first few roots and vectors including repeated ones most importantly the present eigenproblem solution effort is shown to be comparable to that of the corresponding finite element analysis thereby rendering the associated dynamic element method rather attractive owing to superior convergence characteristics of such elements presented herein

solving polynomial eigenvalue problems by means of the ehrlich aberth method abstract given the n n matrix polynomial p x i 0 k p i x i we consider the associated polynomial eigenvalue problem this problem viewed in terms of computing the roots of the scalar polynomial det p x is treated in polynomial form rather than in matrix form by means of the ehrlich aberth iteration the main computational issues are discussed namely the choice of the starting approximations needed to start the ehrlich aberth iteration the computation of the newton correction the halting criterion and the treatment of eigenvalues at infinity we arrive at an effective implementation which provides more accurate approximations to the eigenvalues with respect to the methods based on the qz algorithm the case of polynomials having special structures like palindromic hamiltonian symplectic etc where the eigenvalues have special symmetries in the complex plane is considered a general way to adapt the ehrlich aberth iteration to structured matrix polynomials is introduced numerical experiments which confirm the effectiveness of this approach are reported

stochastic vibration of axially loaded monosymmetric timoshenko thin walled beam abstract an analytical method is presented to perform the flexure torsion coupled stochastic response analysis of monosymmetric axially loaded timoshenko thin walled beam subjected to various kinds of concentrated and distributed stochastic excitations with stationary and ergodic properties the effects of warping stiffness axial force shear deformation and rotary inertia are included in the present formulations first the damped general governing differential equations of motion of axially loaded timoshenko thin walled beam are developed and the free vibration analysis is performed once the natural frequencies and mode shapes are obtained mode superposition method in conjunction with receptance method is used to compute the mean square displacement response of the axially loaded thin walled beam finally the method is illustrated by its application to two test examples to investigate the effects of warping stiffness axial force shear deformation and rotary inertia on the stochastic response of the thin walled beams

on rankin triple product l functions over function fields central critical values the aim of this article is to study the special values of rankin triple product l functions associated to drinfeld type newforms of equal square free levels the functional equation of these l functions is deduced from a garrett type integral representation and the functional equation of eisenstein series on the group of similitudes of a symplectic vector space of dimension 6 when the associated root number is positive we present a function field analogue of gross kudla formula for the central critical value this formula is then applied to the non vanishing of l functions coming from elliptic curves over function fields

beam bending torsion dynamic stiffness method for calculation of exact vibration modes abstract the exact dynamic stiffness matrix is derived for a straight and uniform beam element whose elastic and inertial axes are not coincident elementary bending torsion beam theory is used and bending translation is restricted to one direction the element matrix can be used in the dynamic stiffness method for calculation of exact natural frequencies mode shapes and generalized masses for planar assemblages of connected bending torsion beams the dynamic stiffness method is outlined and details pertinent to the bending torsion beam element are given an example of a three beam assemblage representing an airplane wing is posed and exact numerical solutions are tabulated for natural frequencies and generalized masses solutions calculated by the finite element method are also tabulated for comparison

a successive bounding method to find the exact eigenvalues of transcendental stiffness matrix formulations an alternative algorithm for finding exact natural frequencies or buckling loads from the transcendental e g dynamic stiffness matrix method is presented in this paper and evaluated by using the plate assembly testbed program viconopt the method is based on the bounding properties of the eigenvalues provided by either linear or quadratic matrix pencils on the exact solutions of the transcendental eigenvalue problem the procedure presented has five stages including two accuracy checking stages which prevent unnecessary calculations numerical tests on buckling of general anisotropic plate assemblies show that significant time savings can be achieved compared with an earlier multiple determinant parabolic interpolation method

natural vibration and buckling of general periodic lattice structures a method is presented for vibration and buckling analysis of arbitrary lattice structures having repetitive geometry in any combination of coordinate directions the approach is based on exact member theory representing the stiffness of an individual member subject to axial load and in the case of vibration undergoing harmonic oscillation the method is an extension of previous work that was limited to specific geometries the resulting eigenvalue problem is of the size associated with the repeating element of the structure a computer program has been developed incorporating the theory and results are given for vibration of rectangular platforms and a large antenna structure having rotational symmetry buckling and vibration results for cablestiffened rings are also given

power series expansions of dynamic stiffness matrices for tapered bars and shafts summary stiffness and consistent mass matrices for tapered bars and shafts are derived with the aid of static displacement functions moreover the corresponding dynamic stiffness matrices are developed in the laplace transform domain from the exact solutions of axial torsional governing equations power series expansions of the bessel functions comprising the dynamic stiffness influence coefficients show that the stiffness and consistent mass matrices can be mathematically derived from the dynamic stiffness matrices a discussion on the convergence of the power series expansions is also presented the developments provide further insight into the approximations present in conventional consistent mass formulations of frameworks with tapered members

dynamic stiffness for structures with distributed deterministic or random loads abstract the dynamic stiffness method applies mainly to excitations of harmonic nodal forces for distributed loads modal analysis is generally required in the case of a clamped clamped beam the modal decomposition of a uniformly distributed load by the eigenbeam functions inherits slow convergence because the finite loads at the beam ends cannot be represented efficiently by the zero deflection and zero slope of the clamped clamped beam functions the computed reactions at the supports do not converge at all the problem is eliminated in this paper by using the finite element interpolation functions for the distributed load if the distributed load is adequately represented explicit exact solutions are found otherwise the residual load is expanded in the modal space as the residual modal force is much smaller and agrees well with the clamped clamped conditions fast convergence is achieved by means of the principle of superposition a structure with members having distributed loads can be analyzed by two systems one is associated with the individual members having distributed loads and the other is associated with resulting equivalent nodal forces the required frequency functions are given for all possible cases the results presented are exact if the load is interpolated adequately by finite element shape functions both deterministic and random loads are considered closed form solutions are obtained for the first time

beam element matrices derived from vlasov s theory of open thin walled elastic beams a uniform beam element of open thin walled cross section is studied under stationary harmonic end excitation an exact dynamic transcendentally frequency dependent 14 14 element stiffness matrix is derived from vlasov s coupled differential equations special attention is paid to the computational problems arising when coefficients vanish in these equations because of symmetric cross section zero warping stiffness etc the dynamic element stiffness matrix is established via a generalized linear eigenvalue problem and a system of linear algebraic equations with complex matrices a static stiffness matrix is also derived and the associated consistent mass and geometric stiffness matrices are given modal masses are evaluated a fortran program and a numerical example are included

une base explicite de symboles modulaires sur les corps de fonctions modular symbols for the congruence subgroup gamma 0 mathfrak n of gl 2 mathbf f q t have been defined by teitelbaum they have a presentation given by a finite number of generators and relations in a formalism similar to manin s for classical modular symbols we completely solve the relations and get an explicit basis of generators when mathfrak n is a prime ideal of odd degree as an application we give a non vanishing statement for l functions of certain automorphic cusp forms for mathbf f q t the main statement also provides a key step for a result towards the uniform boundedness conjecture for drinfeld modules of rank 2

on the geometry of a class of n qubit entanglement monotones a family of n qubit entanglement monotones invariant under stochastic local operations and classical communication slocc is defined this class of entanglement monotones includes the well known examples of the concurrence the 3 tangle and some of the four five and n qubit slocc invariants introduced recently the construction of these invariants is based on bipartite partitions of the hilbert space in the form with l 2n n l 2n such partitions can be given a nice geometrical interpretation in terms of grassmannians gr l l of l planes in cl that can be realized as the zero locus of quadratic polynomials in the complex projective space of suitable dimension via the plucker embedding the invariants are neatly expressed in terms of the plucker coordinates of the grassmannians

exact natural frequencies of a three dimensional shear torsion beam with doubly asymmetric cross section using a two dimensional approach abstract this paper presents an exact analytical approach to the calculation of the natural frequencies of structures comprising contiguous three dimensional shear torsion beams with doubly asymmetric cross section such component members have the unusual theoretical property that they allow for coupled torsional and shearing deformation but not bending deformation initially exact dynamic member stiffness matrices exact finite elements are developed for planar shear and torsional motion these matrices can be combined in the usual way to model stepped and continuous beams whose uncoupled frequencies can then be determined exactly it is then shown how the corresponding coupled frequencies can be established easily from the uncoupled values through an exact relationship this enables coupled three dimensional vibration problems to be solved very efficiently using a two dimensional approach the paper is concluded with an example that clarifies the theory together with a parametric study that enables guidance to be given as to when lateral torsional coupling may safely be ignored

two step component mode synthesis for the eigensolution of large systems component mode synthesis is a dynamic substructuring technique for approximate eigensolutions of large degree of freedom systems divisible into two or more components system synthesis using component modes results in approximate eigenparameters the approximation is improved by using more component modes in synthesis typically all component modes up to x times the frequency range of interest are required to ensure reasonable accuracy where x varies from 2 to 10 a two step component mode synthesis approach is presented the first step involves system synthesis using the minimum number of component modes required to obtain approximate cigenvalues up to a preselected frequency and the second step uses additional component modes in a convergence scheme operating on the system eigenparameters calculated in the first step this results in an infinite series eigenvalue problem that is solved by a modified shooting method a perturbation approach is adopted to derive the convergence scheme in which separate iterations are performed for each eigenvalue and eigenvector in each step selection of initial values for the convergence scheme is also presented in a few instances an eigenvalue nearly reached the convergence tolerance but then wandered away and converged to an adjacent value the algorithm was modified to automatically detect this problem and abort iteration with a small penalty in accuracy this two step approach provides the analyst with some insight into the convergence and accuracy of the eigenparameters without resolving an increasingly larger system over and over with more component modes numerical examples are included to demonstrate the applicability of the method

geometry of a two qubit state and intertwining quaternionic conformal mapping under local unitary transformations in this paper the geometry of two qubit systems under a local unitary group so 2 su 2 is discussed it is shown that the quaternionic conformal map intertwines between this local unitary subgroup of sp 2 and the quaternionic mobius transformation which is rather a generalization of the results of lee et al 2002 quantum inf process 1 129