an assessment of the differential quadrature time integration scheme for nonlinear dynamic equations in 1996 xie an assessment of time integration schemes for non linear dynamic equations journal of sound and vibration 192 1 1996 321 331 presented an assessment on seven existing and commonly used time integration schemes for nonlinear dynamic equations in this work the differential quadrature dq time integration scheme proposed by fung in 2001 is assessed following the same procedures as xie s it is shown that accurate numerical results can be obtained by the dq method when using much larger time step over the commonly used time integration schemes based on the results reported herein some conclusions are drawn

exponential growth and an asymptotic formula for the ranks of homotopy groups of a finite 1 connected complex let x be an n dimensional finite simply connected cw complex and set alpha x lim sup i log rank pi i x i we prove that either rank pi i x 0 i 2n or else that 0 0 there is a k k epsilon such that e alpha x epsilon k k in particular this sum grows exponentially in k

a generalized approach for implementing general boundary conditions in the gdq free vibration analysis of plates abstract a new approach for implementing general boundary conditions in the gdq free vibration analysis of rectangular plates is presented in this paper the proposed approach directly couples the boundary conditions with the governing equations and is referred to as the cbcge approach it can be applied to any combination of simply supported clamped and free boundary conditions the generality and accuracy of the cbcge approach are demonstrated through its application to the vibration analysis of a rectangular plate with various combinations of free edges and corners in particular the effect of grid point distribution on the numerical results for plates with free boundary conditions is discussed and a new grid point distribution is suggested

a variable order approach to improve differential quadrature accuracy in dynamic analysis differential quadrature dq is a numerical technique which can produce highly accurate results by using a considerably small number of grid points when it is applied to dynamic equations however dq may exhibit dynamic numerical instability the present paper analyzed the sources of dynamic numerical instability through a simple example and the main finding is that dynamic stability is dominated by the grid points near and on boundaries based on this we propose a variable order approach which is characterized by applying different dq schemes to the grid points near boundaries and grid points far away from boundaries numerical examples of both linear and non linear dynamic equations show that the variable order approach presented in this paper may greatly improve dynamic stability producing convincing results

homological stability for topological chiral homology of completions abstract by proving that several new complexes of embedded disks are highly connected we obtain several new homological stability results our main result is homological stability for topological chiral homology on an open manifold with coefficients in certain partial framed e n algebras using this we prove a special case of a conjecture of vakil and wood on homological stability for complements of closures of particular strata in the symmetric powers of an open manifold and we prove that the bounded symmetric powers of closed manifolds satisfy homological stability rationally

higher order time step integration methods with complex time steps unconditionally stable higher order time step integration algorithms are presented the algorithms are based on the newmark method with complex time steps the numerical results at the complex sub step locations are combined linearly to give higher order accurate results at the end of the time step the ultimate spectral radius in the high frequency range is a controllable parameter for these algorithms among these algorithms the asymptotic annihilating algorithm and the non dissipative algorithm correspond to the first sub diagonal and diagonal pade approximations respectively the characteristics of the present algorithms with various numerical dissipations are found to be in between these two algorithms the algorithmic parameters for the third fifth and seventh order algorithms with various numerical dissipations are given explicitly the order of accuracy is increased by one if these algorithms are set to non dissipative the spectral radii algorithm damping ratios and relative period errors are compared favourably with other higher order algorithms

a function space model approach to the rational evaluation subgroups let f u x be a map from a connected nilpotent space u to a connected rational space x the evaluation subgroup g u x f which is a generalization of the gottlieb group of x is investigated the key device for the study is an explicit sullivan model for the connected component containing f of the function space of maps from u to x which is derived from the general theory of such a model due to brown and szczarba trans am math soc 349 4931 4951 1997 in particular we show that non gottlieb elements are detected by analyzing a sullivan model for the map f and by looking at non triviality of higher order whitehead products in the homotopy group of x the gottlieb triviality of a fibration in the sense of lupton and smith the evaluation subgroup of a fibre inclusion 2006 is also discussed from the function space model point of view moreover we proceed to consideration of the evaluation subgroup of the fundamental group of a nilpotent space in consequence the first gottlieb group of the total space of each s1 bundle over the n dimensional torus is determined explicitly in the non rational case

stability and accuracy of differential quadrature method in solving dynamic problems in this paper the differential quadrature method is used to solve dynamic problems governed by second order ordinary differential equations in time the legendre radau chebyshev chebyshev gauss lobatto and uniformly spaced sampling grid points are considered besides two approaches using the conventional and modified differential quadrature rules to impose the initial conditions are also investigated the stability and accuracy properties are studied by evaluating the spectral radii and truncation errors of the resultant numerical amplification matrices it is found that higher order accurate solutions can be obtained at the end of a time step if the gauss and radau sampling grid points are used however the conventional approach to impose the initial conditions in general only gives conditionally stable time step integration algorithms unconditionally stable algorithms can be obtained if the modified differential quadrature rule is used unfortunately the commonly used chebyshev gauss lobatto sampling grid points would not generate unconditionally stable algorithms

a new approach to the differential quadrature method for fourth order equations a generalized and more complete methodology for treating boundary conditions in the differential quadrature method dqm is presented this improved approach eliminates the deficiencies of the type grid arrangement which represents an approximation by applying the boundary conditions exactly two kinds of basis functions chebyshev and lagrange are used for concept demonstration it is found that the new approach cures most deficiencies of the current dqm 1997 by john wiley sons ltd

rational fibrations in differential homological algebra in this paper a result of 6 is generalized as follows given a fibration f e b of simply connected spaces in which either the fibre has finite dimensional rational cohomology or it has finite dimensional rational homotopy and induces a surjection in rational homotopy we construct an explicit isomorphism ext c b q q c b q 6aext c f q q c f q ext c e q q c e q this is deduced from its algebraic translation a more general result in the environment of graded differential homological algebra

on the rational homotopy type of function spaces the main result of this paper is the construction of a minimal model for the function space f x y of continuous functions from a finite type finite dimensional space x to a finite type nilpotent space y in terms of minimal models for x and y for the component containing the constant map 7r f x y 0 q 7r y 0 h x q in positive dimensions when x is formal there is a simple formula for the differential of the minimal model in terms of the differential of the minimal model for y and the coproduct of h x q we also give a version of the main result for the space of cross sections of a fibration

a conjecture based on somos 4 sequence and its extension abstract in two recent papers by barry 2010 29 and 2011 30 it is conjectured that somos 4 admits a solution expressed in terms of hankel determinant with its elements satisfying a convolution recursion relation in this paper barry s conjecture on somos 4 is firstly confirmed actually we present a more generalized result the proof is mainly based on new findings on properties for so called block hankel determinants the method can also be used to prove another conjecture proposed by michael somos which has been solved by guoce xin

weighting parameters for unconditionally stable higher order accurate time step integration algorithms part 2 second order equations in this paper unconditionally stable higher order accurate time step integration algorithms suitable for linear second order differential equations based on the weighted residual method are presented the second order equations are manipulated directly as in part 1 of this paper instead of specifying the weighting functions the weighting parameters are used to control the algorithm characteristics the algorithms are at least nth order accurate if the numerical solution for displacement is approximated by a polynomial of degree n 1 with n undetermined coefficients by choosing the weighting parameters carefully the order of accuracy can be improved the generalized pade approximations for the second order equations are considered the ultimate spectral radius is an algorithmic parameter by relating the approximate solutions to the equivalent formulations presented in part 1 of this paper the required weighting parameters are found explicitly any set of linearly independent functions can be used to construct the corresponding weighting functions from the weighting parameters the stabilizing weighting functions for the weighted residual method are found explicitly to ensure higher order accuracy in the general solution the accuracy of the particular solution due to excitation is also examined copyright 1999 john wiley sons ltd

buckling and post buckling analysis of extensible beam columns by using the differential quadrature method buckling and post buckling analysis of extensible beam columns is performed numerically in this paper it was experienced earlier that in some cases the numerical integration would not produce the convergent post buckling solutions especially under high loads therefore a new differential quadrature dq based iterative numerical integration method is proposed to solve post buckling differential equations of extensible beam columns six cases including five classical euler buckling cases are analyzed critical loads and convergent post buckling solutions under different applied loads are obtained the results are compared with the existing multiple scales solutions it is found that under high applied loads the small rotation assumption in obtaining multiple scales solutions is no longer valid the proposed iterative dq based numerical integration method can yield reliable and accurate post buckling solutions even at high applied loads for the cases investigated

static analysis of frame structures by the differential quadrature element method in this paper a new version of differential quadrature method dqm is proposed and then extended to analyse frame structures the new method called the differential quadrature element method dqem retains all advantages of the earlier version of the differential quadrature method and overcomes some critical shortcomings existing in the original dqm the proposed method is however different from the quadrature element method qem proposed earlier by striz et al the methodology is explained in details herein via a differential quadrature beam element and some numerical examples are given to show the efficiency of the present method 1997 by john wiley sons ltd

bilinear discrete painleve equations based on the results of singularity confinement we derive bilinear expressions for the discrete painleve equations in these cases where a bilinear expression is not sufficient we obtain trilinear or higher multilinear expressions we show that the bilinear approach provides a natural framework for the derivation of backlund and schlesinger transforms for the discrete painleve equations

a time integration algorithm for linear transient analysis based on the reproducing kernel method in this paper a new algorithm based on the reproducing kernel method is presented for linear transient analysis using very large time steps in the conventional time integration methods the required accuracy is always achieved by either decreasing the size of time steps or increasing the order of approximations the proposed method however does not require a restriction on the size of the time step the required approximating functions for integration of governing equations are determined using the concept of reproduction of the solution within each time step a recurrence relationship is obtained through solving the governing equation by the point collocation based reproducing kernel method within the time step in collocation points the values of external excitation can be obtained either through the approximation between the values at the beginning and the end of each time step or assigning the exact values of the external force hence one advantage of this new method is the independency of the sampling rates of external excitation from the size of the time step the proposed method is applied for solving first and second order differential equations and advantages of the method are illustrated through a number of numerical examples

hankel determinant solution for elliptic sequence abstract we show that the hankel determinants of a generalized catalan sequence satisfy the equations of the elliptic sequence as a consequence the coordinates of the multiples of an arbitrary point on the elliptic curve are expressed by the hankel determinants

on monomial algebras of finite global dimension let g be an associative monomial k algebra if g is assumed to be finitely presented then either g contains a free subalgebra on two monomials or else g has polynomial growth if instead g is assumed to have finite global dimension then either g contains a free subalgebra or else g has a finite presentation and polynomial growth also a graded hopf algebra with generators in degree one and relations in degree two contains a free hopf subalgebra if the number of relations is small enough

the norsett time integration methodology for finite element transient analysis this paper presents a set of methods for time integration of problems arising from finite element semidiscretizations the purpose is to obtain computationally efficient methods which possess higher order accuracy and controllable dissipation in the spurious high modes the methods are developed and analysed by a general collocation methodology which leads to the class of norsett approximants an algorithmic parameter is used to achieve an effective control over numerical dissipation moreover a simple and efficient implementation scheme is presented at each time step algorithms based on p order collocation polynomials require the solution of p sets of linear algebraic equations with the same coefficient matrix in this way a single factorization is needed and no transformations are required to recover the approximate solution at the end or within the time interval to demonstrate the performance of the proposed algorithms a wide experimental evaluation is carried out on typical test problems in finite element transient analysis

maurer cartan moduli and models for function spaces we set up a formalism of maurer cartan moduli sets for l algebras and associated twistings based on the closed model category structure on formal differential graded algebras a k a differential graded coalgebras among other things this formalism allows us to give a compact and manifestly homotopy invariant treatment of chevalley eilenberg and harrison cohomology we apply the developed technology to construct rational homotopy models for function spaces

andre quillen cohomology and rational homotopy of function spaces we develop a simple theory of andre quillen cohomology for commutative differential graded algebras over a field of characteristic zero we then relate it to the homotopy groups of function spaces and spaces of homotopy self equivalences of rational nilpotent cw complexes this puts certain results of sullivan in a more conceptual framework

algebraic models of non connected spaces and homotopy theory of l algebras we develop a homotopy theory of l algebras based on the lawrence sullivan construction a complete differential graded lie algebra which as we show satisfies the necessary properties to become the right cylinder in this category as a result we obtain a general procedure to algebraically model the rational homotopy type of non connected spaces 2012 elsevier ltd

on the evaluation map the evaluation map of a differential graded algebra or of a space is described under two different approaches this concept turns out to have geo metric implications i a 1 connected topological space with finite dimensional rational homotopy has finite dimensional rational cohomology if and only if it has nontrivial evaluation map ii let e b be a fibration of simply connected spaces if the rational cohomology of the fibre is finite dimensional and the evaluation map of the base is different from zero then the evaluation map of the total space is nonzero also if p is surjective in rational homotopy and the evaluation map of e is nontrivial then the evaluation map of the fibre is different from zero

two new approximate methods for analyzing free vibration of structural components two approximate methods which have not previously been used for structural dynamics problems are applied to the free vibration analysis of various structural components the first method is a new version of the complementary energy method it is shown to be considerably more accurate than the conventional rayleigh and rayleigh schmidt methods when applied to spatially one dimensional free vibration problems prismatic and tapered bars prismatic beams and axisymmetric motion of circular membranes the second method is the differential quadrature method introduced by bellman and his associates it is applied successfully here to all of the problems mentioned plus square membranes and circular and square plates

lie models for the components of sections of a nilpotent fibration we give an explicit lie model for any component of the space of free and pointed sections of a nilpotent fibration and in particular of the free and pointed mapping spaces among the applications presented we obtain a lie model of the exponential law and prove that in many cases the rank of the homotopy groups of the mapping space grows at the same rate as the rank of the homotopy groups of the target space

unconditionally stable collocation algorithms for second order initial value problems abstract in this paper unconditionally stable higher order accurate time step integration algorithms suitable for second order initial value problems in collocation form are presented the second order equations are manipulated directly if the approximate solution is expressed as a polynomial of degree n 1 there are n unknowns to be determined after taking into account the two given initial conditions it is well known that by suppressing the residuals of the governing equations at n distinct collocation points only the resultant algorithms are only conditionally stable in this paper linear combinations of the residuals at n 1 distinct collocation points are used to solve for the n unknowns the collocation points and the relative weights between the residuals are derived from the weighted residual method the weighting functions are arbitrary polynomials of degree not exceeding n 1 to control the accuracy and stability properties of the resultant algorithms the reduced integration technique is used to evaluate the integrals in the formulation once the reduced integration rules are decided the equivalent collocation form can be derived it is found that the resultant algorithms cast in the collocation form are easy to implement and can be used to tackle non linear problems directly numerical examples are given to illustrate the validity of the present formulation

solution of nonlinear initial value problems by the spline based differential quadrature method the spline based differential quadrature method sdqm is applied to the solution of nonlinear initial value problems explicit expressions of weighting coefficients for approximation of derivatives are presented dynamic systems with duffing type nonlinearity are solved to demonstrate the effectiveness of the method numerical results of three examples show that the spline based differential quadrature method is versatile and stable in the solution of nonlinear initial value problems it can be counted on to achieve satisfactory accuracy for long term integration

on the equivalence of the time domain differential quadrature method and the dissipative runge kutta collocation method numerical solutions for initial value problems can be evaluated accurately and efficiently by the differential quadrature method unconditionally stable higher order accurate time step integration algorithms can be constructed systematically from this framework it has been observed that highly accurate numerical results can also be obtained for non linear problems in this paper it is shown that the algorithms are in fact related to the well established implicit runge kutta methods through this relation new implicit runge kutta methods with controllable numerical dissipation are derived among them the non dissipative and asymptotically annihilating algorithms correspond to the gauss methods and the radau iia methods respectively other dissipative algorithms between these two extreme cases are shown to be b stable or algebraically stable as well and the order of accuracy is the same as the corresponding radau iia method through the equivalence it can be inferred that the differential quadrature method also enjoys the same stability and accuracy properties copyright 2001 john wiley sons ltd

imposition of boundary conditions by modifying the weighting coefficient matrices in the differential quadrature method one of the important issues in the implementation of the differential quadrature method is the imposition of the given boundary conditions there may be multiple boundary conditions involving higher order derivatives at the boundary points the boundary conditions can be imposed by modifying the weighting coefficient matrices directly however the existing method is not robust and is known to have many limitations in this paper a systematic procedure is proposed to construct the modified weighting coefficient matrices to overcome these limitations the given boundary conditions are imposed exactly furthermore it is found that the numerical results depend only on those sampling grid points where the differential quadrature analogous equations of the governing differential equations are established the other sampling grid points with no associated boundary conditions are not essential copyright 2002 john wiley sons ltd

rational homotopy of the space of sections of a nilpotent bundle we show that an algebraic construction proposed by sullivan is indeed a model for the rational homotopy type of the space of sections of a nilpotent bundle in his paper uhomologie des espaces fonctionnels r thom studied the homotopy type of the space f of continuous maps of x into f homotopic to a given map starting from a postnikov decomposition of f he built the functional space f step by step he also indicated how one could construct a differential graded algebra describing the rational homotopy type of f later on sullivan gave an algebraic model which mirrors this construction in terms of a dg algebra representing x and the minimal model of f the aim of this paper is to show following the method of thom that the model of sullivan is indeed a model for the functional space under suitable restrictions as in 3 we consider the slightly more general problem of the determination of the rational homotopy type of the space of sections ts of a nilpotent fiber space p y x homotopic to a given section s in 1 we explain thorn s geometric construction in 2 we describe an algebraic model for an abelian galois covering of a nilpotent space in 3 we show how the model of sullivan fits with the geometry i thank the referee for many improvements of this paper 1 a postnikov factorization of the space of sections 1 1 let g be a finitely generated abehan group and let x be a path connected space whose cohomology groups hk x g are finitely generated for each k to avoid difficulties with the topologies cf 4 we can work in the category of simplicial sets proposition thom 4 the space k g m x of continuous maps of x in the eilenberg mac lane complex k g m is homotopically equivalent to the product nr ra fthe eilenberg mac lane spaces ki k hm x g i more precisely let x g hm k g m g be the fundamental class of k g m if e k g m x xx k g m

rational homotopy of the homotopy fixed point sets of circle actions in this paper when g is the circle s 1 and m is a g space we study the rational homotopy type of the fixed point set m g the homotopy fixed point set m hg and the natural injection m g m hg c 2009 elsevier inc all rights reserved

on the autonomous limit of discrete painleve equations abstract we consider the autonomous limit of the known families of discrete painleve equations we show how these autonomous systems can be written as special cases of the qrt mapping for the latter we present in detail its integration in both the symmetric and asymmetric cases

application of differential quadrature to static analysis of structural components the numerical technique of differential quadrature for the solution of linear and non linear partial differential equations first introduced by bellman and his associates is applied to the equations governing the deflection and buckling behaviour of one and two dimensional structural components separate transformations are used for higher order derivatives as suggested by mingle thus extending the method to treat fourth order equations and to include multiple boundary conditions in the respective co ordinate directions results are obtained for various boundary and loading conditions and are compared with existing exact and numerical solutions by other methods the application of differential quadrature to this class of problems is seen to lead to accurate results with relatively small computational effort

combining differential quadrature method with simulation technique to solve non linear differential equations nowadays most of the ordinary differential equations odes can be solved by modelica based approaches such as matlab simulink dymola and labview which use simulation technique st however these kinds of approaches restrict the users in the enforcement of conditions at any instant of the time domain this limitation is one of the most important drawbacks of the st another method of solution differential quadrature method dqm leads to very accurate results using only a few grids on the domain on the other hand dqm is not flexible for the solution of non linear odes and it is not so easy to impose multiple conditions on the same location for these reasons the author aims to eliminate the mentioned disadvantages of the simulation technique st and dqm using favorable characteristics of each method in the other this work aims to show how the combining method cm works simply by solving some non linear problems and how the cm gives more accurate results compared with those of other methods copyright 2008 john wiley sons ltd

complex time step newmark methods with controllable numerical dissipation in this paper unconditionally stable higher order accurate time step integration algorithms with controllable numerical dissipation are presented the algorithms are based on the newmark method with complex time steps the ultimate spectral radius the sub step locations j and the weighting factors j are the algorithmic parameters for an algorithm that is 2n 1 th order accurate the sub step locations which may be complex are shown to be the roots of an nth degree polynomial the polynomial is given explicitly in terms of n and the weighting factors are then obtained by solving a system of n simultaneous equations it is further shown that the order of accuracy is increased by one for the non dissipative algorithms with 1 the stability properties of the present algorithms are studied it is shown that if the ultimate spectral radius is set between 1 and 1 the eigenvalues of the numerical amplification matrix are complex with magnitude less than or equal to unity the algorithms are therefore unconditionally c stable when the ultimate spectral radius is set to 0 or 1 the algorithms are found to be equivalent to the first sub diagonal and diagonal pade approximations respectively the present algorithms are more general as the numerical dissipation is controllable and are very suitable for parallel computers the accuracy of the excitation responses is found to be enhanced by the present complex time step procedure to maintain high order accuracy the excitation may need some modifications 1998 john wiley sons ltd

continuous cohomology and real homotopy type various aspects of homotopy theory in the category of simplicial spaces are developed topics covered include continuous cohomology continuous de rham cohomology the kan extension condition the homotopy relation fibrations the serre spectral sequence real homotopy type and its relation to graded commutative differential algebras over the reals

the de rham homotopy theory and differential graded category this paper is a generalization of moriya in j pure appl algebra 214 4 422 439 2010 we develop the de rham homotopy theory of not necessarily nilpotent spaces we use two algebraic objects closed dg categories and equivariant dg algebras we see these two objects correspond in a certain way proposition 3 3 4 theorem 3 4 5 we prove an equivalence between the homotopy category of schematic homotopy types toen in selecta math n s 12 1 39 135 2006 and a homotopy category of closed dg categories theorem 1 0 1 we give a description of homotopy invariants of spaces in terms of minimal models theorem 1 0 2 the minimal model in this context behaves much like the sullivan s minimal model we also provide some examples we prove an equivalence between fiberwise rationalizations bousfield and kan in lecture notes in mathematics vol 304 springer berlin 1972 and closed dg categories with subsidiary data theorem 1 0 4

rational l s category and its applications let s be a 1 connected cw complex of finite type and put cat0 s lusternik schnirelmann category of the localization sq this invariant is char acterized in terms of the minimal model of s it is shown that if is the fibre inclusion of a fibration it is also shown that if dim h s q oo then either dim it s q oo or the groups nk s q grow exponentially with k introduction in this paper topological space means a pointed path connected normal space s1 such that the inclusion of the basepoint is a cofibration and the rational singular homology is finite dimensional in each degree all vector spaces and algebras have q as ground field homology and cohomology of spaces is singular with rational coefficients we shall be dealing with category of spaces cat s in the sense of lusternik and schnirelmann 18 a recent survey is given in 15 but our definition differs by one from that of 15 so that for us spheres have category one not two our purpose is to provide a computationally useful characterization of the rational category cat0 5 of space 5 and to apply this in a variety of situations the definition of cat0 5 is sketched below and given precisely in 4 it satisfies cat0 s cauls when 5 is 1 connected cat0 5 coincides with the category of the localization sq which has been studied by toomer in 26 and 27 and lemaire and sigrist in 17 the characterization problem was posed by berstein 26 most of the applications depend on the following fundamental mapping theorem which we derive as a consequence of our characterization