partial compact quantum groups abstract compact quantum groups of face type as introduced by hayashi form a class of quantum groupoids with a classical finite set of objects using the notions of weak multiplier bialgebras and weak multiplier hopf algebras resp due to bohm gomez torrecillas lopez centella and van daele wang we generalize hayashi s definition to allow for an infinite set of objects and call the resulting objects partial compact quantum groups we prove a tannaka kre n woronowicz reconstruction result for such partial compact quantum groups using the notion of partial fusion c categories as examples we consider the dynamical quantum su 2 groups from the point of view of partial compact quantum groups

finite element solution of two dimensional inverse elastic problems using spatial smoothing a finite element method fem is presented for calculating the surface tractions on a body from internal measurements of displacement at discrete sensor locations the solution algorithm employs a sensitivity analysis which minimizes the difference between the calculated and measured displacements at each sensor location spatial regularization is one technique employed to stabilize the minimization process by imposing various degrees of smoothness on the solution it also allows the problem to be solved with fewer sensors than traction boundary nodes as an alternative to spatial regularization a method based on keynodes is introduced which assumes that each component of the boundary traction distribution can be described by a polynomial of specified order the methods are applied to several two dimensional examples including a rolling contact problem the effects of parameters such as the number of sensors the location of the sensors and the error in the sensor displacements are discussed

an iterative method for the cauchy problem in linear elasticity with fading regularization effect in this paper an iterative method for solving the cauchy problem in linear elasticity is introduced this problem consists in recovering missing data displacements and forces on some parts of a domain boundary from the knowledge of overspecified data displacements and forces on the remaining parts the algorithm reads as a least square fitting of the given data with a regularization term whose effect fades as the iterations go on so the algorithm converges to the solution of the cauchy problem numerical simulations using the finite element method highlight the algorithm s efficiency accuracy robustness to noisy data as well as its ability to deblur noisy data

the minimal error method for the cauchy problem in linear elasticity numerical implementation for two dimensional homogeneous isotropic linear elasticity abstract in this paper yet another iterative procedure namely the minimal error method mem for solving stably the cauchy problem in linear elasticity is introduced and investigated furthermore this method is compared with another two iterative algorithms i e the conjugate gradient cgm and landweber fridman methods lfm previously proposed by marin et al marin l hao d n lesnic d 2002b conjugate gradient boundary element method for the cauchy problem in elasticity quarterly journal of mechanics and applied mathematics 55 227 247 and marin and lesnic marin l lesnic d 2005 boundary element landweber method for the cauchy problem in linear elasticity ima journal of applied mathematics 18 817 825 respectively in the case of two dimensional homogeneous isotropic linear elasticity the inverse problem analysed in this paper is regularized by providing an efficient stopping criterion that ceases the iterative process in order to retrieve stable numerical solutions the numerical implementation of the aforementioned iterative algorithms is realized by employing the boundary element method bem for two dimensional homogeneous isotropic linear elastic materials

a meshless method for solving the cauchy problem in three dimensional elastostatics the application of the method of fundamental solutions to the cauchy problem in three dimensional isotropic linear elasticity is investigated the resulting system of linear algebraic equations is ill conditioned and therefore its solution is regularized by employing the first order tikhonov functional while the choice of the regularization parameter is based on the l curve method numerical results are presented for both under and equally determined cauchy problems in a piece wise smooth geometry the convergence accuracy and stability of the method with respect to increasing the number of source points and the distance between the source points and the boundary of the solution domain and decreasing the amount of noise added into the input data respectively are analysed

transverse vibrations of the elastically connected rectangular double membrane compound system abstract the free and forced vibrations of a system of two rectangular membranes attached together by a winkler elastic layer are studied analytically the motion of the system is described by two non homogeneous partial differential equations the solutions of the free vibrations are obtained by the bernoulli fourier method solving the boundary value problem the natural frequencies and the mode shape functions are found the initial value problem is also solved the free vibrations of the double membrane system are realised by synchronous and asynchronous deflections the forced vibrations of membranes subjected to arbitrarily distributed continuous loads are determined by using the classical method of the expansion in a series of the normal modes of vibrations discussing the vibrations caused by the harmonic exciting forces it is shown that the dynamic absorption phenomenon appears therefore the double membrane system can be used as a dynamic vibration absorber as a numerical example the vibrations of the system consisting of two identical membranes subjected to harmonic uniform distributed load are treated in detail

dynamic response of a double euler bernoulli beam due to a moving constant load in this paper the dynamic response of a double beam system traversed by a constant moving load is studied the system consists of two elastic homogeneous isotropic euler bernoulli beams the two simply supported prismatic beams are identical parallel one upon the other and connected continuously by a viscoelastic layer the dynamic deflections of both beams are given in analytical closed forms the effects of the moving speed of the load and the damping and the elasticity of the viscoelastic layer on the dynamic responses of the beams are investigated in detail also several plots of the deflections of the beams are given and discussed for different values of speed parameter damping ratio and stiffness parameter furthermore the force transmitted between the beams is determined and the effects of the system parameters on this force are studied plots of the maximum responses of the beams are presented

multiplier bialgebras in braided monoidal categories abstract multiplier bimonoids or bialgebras in arbitrary braided monoidal categories are defined they are shown to possess monoidal categories of comodules and modules these facts are explained by the structures carried by their induced functors

on duality of algebraic quantum groupoids abstract like quantum groups quantum groupoids frequently appear in pairs of mutually dual objects we develop a general pontrjagin duality theory for quantum groupoids in the algebraic setting that extends van daele s duality theory for multiplier hopf algebras and overcomes the finiteness restrictions of the approach of kadison szlachanyi bohm and schauenburg our construction is based on the integration theory for multiplier hopf algebroids and yields as a corollary a duality theory for weak multiplier hopf algebras with integrals we compute the duals in several examples and introduce morphisms of multiplier hopf algebroids to succinctly describe their structure moreover we show that such morphisms preserve the antipode

an invariant method of fundamental solutions for two dimensional isotropic linear elasticity abstract in this paper we propose an invariant method of fundamental solutions for solving boundary value problems in two dimensional isotropic linear elasticity the invariant method of fundamental solutions keeps a very basic natural property the so called invariance property under trivial coordinate changes in the problem description e g dilations and or contractions the problems are solved by the tikhonov regularization method in conjunction with morozov s discrepancy principle finally five examples of boundary value problems are investigated to show the efficiency of this method the numerical convergence accuracy and stability of the proposed method with respect to the number of source points the distance between the sources and the boundary and the amount of noise added into the input data respectively are also analyzed

weak hopf algebras i integral theory and c structure we give an introduction to the theory of weak hopf algebras proposed as a coassociative alternative of weak quasi hopf algebras we follow an axiomatic approach keeping as close as possible to the classical theory of hopf algebras the emphasis is put on the new structure related to the presence of canonical subalgebras al and ar in any weak hopf algebra a that play the role of non commutative numbers in many respects a theory of integrals is developed in which we show how the algebraic properties of a such as the frobenius property or semisimplicity or innerness of the square of the antipode are related to the existence of non degenerate normalized or haar integrals in case of c weak hopf algebras we prove the existence of a unique haar measure h a and of a canonical grouplike element g a implementing the square of the antipode and factorizing into left and right elements g glg 1r gl al gr ar further discussion of the c case will be presented in part ii

transverse vibrations of elastically connected double string complex system part i free vibrations this paper analyzes the forced transverse vibrations of an elastically connected double string complex continuous system the general solutions of forced vibrations of strings subjected to arbitrarily distributed continuous loads are found by using the method of expansion in a series of the mode shape functions different cases of exciting loadings are analyzed the action of stationary harmonic loads and moving concentrated forces is considered vibrations caused by the harmonic exciting forces are discussed and conditions of resonance and dynamic vibration absorption are formulated thus the string type dynamic absorber can be used to suppress the excessive vibrations of corresponding string systems the dynamic string type damper is a new concept of a continuous dynamic vibration absorber cdva it is shown that a corresponding two degree of freedom discrete system is an analogue of an elastically connected double string complex system theoretical analysis is illustrated by a numerical example

free transverse vibrations of elastically connected simply supported double beam complex system abstract in this paper the free vibration analysis of two parallel simply supported beams continuously joined by a winkler elastic layer is presented the motion of the system is described by a homogeneous set of two partial differential equations which is solved by using the classical bernoulli fourier method the natural frequencies of the system are determined the initial value problem is considered to find the final form of the free vibrations the free vibrations of an elastically connected double beam complex system are realized by synchronous and asynchronous deflections the presented theoretical analysis is illustrated by a numerical example in which the effect of physical parameters characterizing the vibrating system on the natural frequencies is investigated

free transverse vibrations of an elastically connected rectangular simply supported double plate complex system abstract in this paper the free transverse vibrations of a system of two rectangular simply supported thin plates connected by a homogeneous winkler elastic layer are investigated analytically the small vibrations of the system are described by a set of two partial differential equations based on the kirchhoff love plate theory next the homogeneous equations of motion are solved by using the classical navier method the natural frequencies of the system in the form of two infinite sequences are determined and the corresponding mode shapes of vibration are shown as a consequence an elastically connected double plate complex system executes two kinds of the free vibrations synchronous and asynchronous the initial value problem is then considered to find the final form of the free vibrations the theoretical analysis presented is illustrated by a numerical example in which the free vibrations of a system of two identical plates are discussed in detail

free transverse vibrations of an elastically connected complex beam string system abstract the present work is devoted to theoretical vibration analysis of a complex mixed continuous system undamped free transverse vibrations of an elastically connected beam string system are considered solutions of the problem are formulated by using the modal expansion method two infinite sequences of the natural frequencies corresponding to two possible kinds of vibration motions synchronous and asynchronous are determined in a numerical example illustrating the theory presented the effect of string tension force on the natural frequencies of the system is investigated in detail

the method of fundamental solutions for the cauchy problem in two dimensional linear elasticity in this paper the application of the method of fundamental solutions to the cauchy problem in two dimensional isotropic linear elasticity is investigated the resulting system of linear algebraic equations is ill conditioned and therefore its solution is regularised by employing the first order tikhonov functional while the choice of the regularisation parameter is based on the l curve method numerical results are presented for both smooth and piecewise smooth geometries as well as for constant and linear stress states the convergence and the stability of the method with respect to increasing the number of source points and the distance between the source points and the boundary of the solution domain and decreasing the amount of noise added into the input data respectively are analysed

free vibrations of elastically connected stretched beams abstract a general theory for the determination of natural frequencies and mode shapes for a set of elastically connected axially loaded euler bernoulli beams is developed a normal mode solution is applied to a set of non dimensional coupled partial differential equations the natural frequencies are the eigenvalues of a matrix of differential operators the matrix operator is shown to be self adjoint leading to an orthogonality condition for the mode shapes in the special case of identical beams it is shown that the natural frequencies are organized into sets of intramodal frequencies in which each mode shape is a product of a spatial mode and a discrete mode an exact solution is available for the general case however the natural frequencies and mode shapes are then determined using a complicated numerical method a rayleigh ritz method using mode shapes of the corresponding unstretched beams is developed as an alternative

an energy error based method for the resolution of the cauchy problem in 3d linear elasticity a new method is described for the problem of expanding known displacement fields at the boundary of a solid together with the surface tractions on it towards the solid interior up to the inaccessible part of the boundary the solid is supposedly linearly elastic with known elastic moduli but not necessarily homogeneous nor isotropic the problem is the cauchy problem for the lame operator a new form of this cauchy problem suited for applications associated with surface tangential fields measurements is also stated and studied the method is based on the splitting of the elastic fields into two separate solutions of wellposed problems the gap between these fields is subsequently minimised with respect to the unknown boundary data in order to produce the desired expanded elastic fields the gap used here is an energy error associated with the elastic energy of the system various 3d applications are given including non linear boundary conditions on the unreachable boundary

fading regularization mfs algorithm for inverse boundary value problems in two dimensional linear elasticity abstract we investigate the numerical reconstruction of the missing displacements dirichlet data and tractions neumann data on an inaccessible part of the boundary in the case of two dimensional linear isotropic elastic materials from the knowledge of over prescribed noisy measurements taken on the remaining accessible boundary part this inverse problem is solved using the fading regularization method originally proposed by cimetiere et al 2000 2001 for the laplace equation in conjunction with a meshless method namely the method of fundamental solutions mfs the stabilisation of the numerical method proposed herein is achieved by stopping the iterative procedure according to morozov s discrepancy principle morozov 1966

new modified multi level residue harmonic balance method for solving nonlinearly vibrating double beam problem abstract in this study a new modified multi level residue harmonic balance method is presented and adopted to investigate the forced nonlinear vibrations of axially loaded double beams although numerous nonlinear beam or linear double beam problems have been tackled and solved there have been few studies of this nonlinear double beam problem the geometric nonlinear formulations for a double beam model are developed the main advantage of the proposed method is that a set of decoupled nonlinear algebraic equations is generated at each solution level this heavily reduces the computational effort compared with solving the coupled nonlinear algebraic equations generated in the classical harmonic balance method the proposed method can generate the higher level nonlinear solutions that are neglected by the previous modified harmonic balance method the results from the proposed method agree reasonably well with those from the classical harmonic balance method the effects of damping axial force and excitation magnitude on the nonlinear vibrational behaviour are examined

computational vibration and buckling analysis of microtubule bundles based on nonlocal strain gradient theory abstract a nonlocal strain gradient model is proposed to study buckling and vibrational responses of microtubules in axons with attention to different size effect parameters based on finite element method supporting effects of surrounding cytoplasm and map tau proteins are considered microtubules are modeled as elastically connected improved timoshenko nano beams resting on a two parameter pasternak foundation differential equations are discretized using galerkin method finally two eigenvalue problems are solved to achieve critical buckling loads and frequencies of single and doubled microtubule systems the nonlocal strain gradient model is employed in order to show both hardening and softening effects of structural stiffness based on relative magnitudes of nanoscale parameters influence of size effects including nonlocal nanoscale parameter gradient coefficient and surface effects are examined for various boundary conditions and some benchmark results are reported it is observed that these effects are more prominent at higher modes based on presented numerical results map tau proteins strengthen doubled microtubule systems to bear 11 7 more buckling load than a single microtubule furthermore vibration frequencies of microtubules depend on their physical surrounding condition in cell such as cell matrix and membrane and since microtubules could be employed as biosensors this property may be used in order to detect malignant tumors based on vibrational damping

damping of beam vibrations by means of a thin constrained viscoelastic layer evaluation of a new theory abstract in this paper a new one dimensional theory of beams damped by means of a thin viscoelastic film is presented based upon a prescribed piecewise linear longitudinal displacement field and a shear stress field satisfying continuity conditions at the interfaces the beam formulation of the proposed theory is derived in order to demonstrate the accuracy of this beam model numerical tests were carried out by using the finite element method the present results are compared with well known analytical finite element with the msc nastran computer program and experimental solutions under various material temperature and boundary conditions it is found that the proposed theory is very simple and efficient in predicting the dynamic response of beams that are damped by means of a thin viscoelastic film

a relaxation method of an alternating iterative algorithm for the cauchy problem in linear isotropic elasticity we propose two algorithms involving the relaxation of either the given dirichlet data boundary displacements or the prescribed neumann data boundary tractions on the over specified boundary in the case of the alternating iterative algorithm of kozlov et al 16 applied to cauchy problems in linear elasticity a convergence proof of these relaxation methods is given along with a stopping criterion the numerical results obtained using these procedures in conjunction with the boundary element method bem show the numerical stability convergence consistency and computational efficiency of the proposed method

relaxation procedures for an iterative mfs algorithm for the stable reconstruction of elastic fields from cauchy data in two dimensional isotropic linear elasticity we investigate two numerical procedures for the cauchy problem in linear elasticity involving the relaxation of either the given boundary displacements dirichlet data or the prescribed boundary tractions neumann data on the over specified boundary in the alternating iterative algorithm of kozlov et al 1991 the two mixed direct well posed problems associated with each iteration are solved using the method of fundamental solutions mfs in conjunction with the tikhonov regularization method while the optimal value of the regularization parameter is chosen via the generalized cross validation gcv criterion an efficient regularizing stopping criterion which ceases the iterative procedure at the point where the accumulation of noise becomes dominant and the errors in predicting the exact solutions increase is also presented the mfs based iterative algorithms with relaxation are tested for cauchy problems for isotropic linear elastic materials in various geometries to confirm the numerical convergence stability accuracy and computational efficiency of the proposed method

free vibration analysis of elastically connected multiple beams by using the adomian modified decomposition method abstract the adomian modified decomposition method amdm is employed in this paper to investigate the free vibrations of n elastically connected parallel euler bernoulli beams which are continuously joined by a winkler type elastic layer the proposed amdm method can be used to analyze the vibration of beam system consisting of an arbitrary number of beams by using boundary conditions the natural frequencies and corresponding mode shapes can be easily obtained simultaneously the numerical results for different boundary conditions beam numbers and the stiffness of the winkler type elastic layer are presented it is shown that the amdm offers an accurate and effective method of free vibration analysis of multiple connected beams with arbitrary boundary conditions

vibrations of double string complex system under moving forces closed solutions in this paper the dynamic response of a double string system traversed by a constant or a harmonically oscillating moving force is considered the force is moving with a constant velocity on the top string the strings are identical parallel one upon the other and continuously coupled by a linear winkler elastic element the classical solution of the response of a double string system subjected to a force moving with a constant velocity has a form of an infinite series the main goal of this paper is to show that in the considered case a part of the solution can be presented in a closed analytical form instead of an infinite series the presented method of finding the solution in a closed analytical form is based on the observation that the solution of the system of partial differential equations in the form of an infinite series is also a solution of an appropriate system of ordinary differential equations

boundary element solution for the cauchy problem in linear elasticity using singular value decomposition abstract in this paper the singular value decomposition svd truncated at an optimal number is analysed for obtaining approximate solutions to ill conditioned linear algebraic systems of equations which arise from the boundary element method bem discretisation of an ill posed boundary value problem in linear elasticity the regularisation parameter namely the optimal truncation number is chosen according to the discrepancy principle the numerical results obtained confirm that the svd bem produces a convergent and stable numerical solution with respect to decreasing the mesh size discretisation and the amount of noise added into the input data

the generalized finite difference method for the inverse cauchy problem in two dimensional isotropic linear elasticity abstract in this study the generalized finite difference method gfdm was used to stably and accurately solve two dimensional 2d inverse cauchy problems in linear elasticity by using the navier equations in cauchy problems overdetermined boundary conditions are imposed on parts of the boundary whereas there are missing boundary conditions on some parts of the boundary in cauchy problems conventional numerical methods generally generate highly ill conditioned matrices and thus provide unstable numerical solutions moreover even if a slight noise is added in the boundary conditions numerical errors are evidently magnified the gfdm one of the most promising meshless methods and an extension of the classical finite difference method can avoid time consuming tasks of mesh generation and numerical quadrature the gfdm was applied in this study to stably solve the 2d cauchy problems in linear elasticity and four numerical examples are provided to illustrate the consistency and accuracy of the presented meshless numerical scheme moreover the stability of the presented scheme for inverse cauchy problems was proved by adding noise into the boundary conditions

fading regularization mfs algorithm for the cauchy problem associated with the two dimensional helmholtz equation abstract in this paper we combine the fading regularization method with the method of fundamental solutions mfs and investigate its application to the cauchy problem for the two dimensional helmholtz equation we present a numerical reconstruction of the missing data on an inaccessible part of the boundary from the knowledge of overprescribed noisy data taken on the remaining accessible boundary part for both smooth and piecewise smooth two dimensional geometries the accuracy convergence stability and efficiency of the proposed numerical algorithm as well as its capability to deblur the noisy data are validated by three numerical examples

application of fractional operators to the analysis of damped vibrations of viscoelastic single mass systems free damped vibrations of an oscillator whose viscoelastic properties are described in terms of the fractional calculus kelvin voight model maxwell model and standard linear solid model are determined the problem is solved by the laplace transform method when passing from image to pre image one is led to find the roots of an algebraic equation with fractional exponents the method for solving such equations is proposed which allows one to investigate the roots behaviour in a wide range of single mass system parameters a comparison between the results obtained on the basis of the three models has been carried out it has been shown that for all models the characteristic equations do not possess real roots but have one pair of complex conjugates i e the test single mass systems subjected to the impulse excitation do not pass into an aperiodic regime in none of magnitudes of the relaxation and creep times main characteristics of vibratory motions of the single mass system as functions of the relaxation time or creep time which are equivalent to the temperature dependencies are constructed and analyzed for all three models