chaotic functions with zero topological entropy on caracterise la classe m c i i des applications chaotiques en ce sens on montre que m contient certaines appliquations qui ont a la fois une entropie topologique nulle et des attracteurs infinis on montre que le complement de m consiste en applications qui ont seulement des trajectoires approchables par des cycles

three dimensional vibrations of thick circular and annular plates abstract the ritz method is applied in a three dimensional 3 d analysis to obtain accurate frequencies for thick circular and annular plates the method is formulated in a manner which allows one to have any combination of free or fixed plate boundaries admissible functions for the three displacement components are chosen as trigonometric functions in the circumferential co ordinate and algebraic polynomials in the radial and axial co ordinates upper bound convergence of the non dimensional frequencies to at least four significant figures is demonstrated comparisons of results are made with ones obtained by others using 2 d mindlin thick plate theory and with other 3 d solutions extensive and accurate four significant figure frequencies are presented for completely free circular plates having thickness to diameter ratios of 0 2 0 3 0 4 and 0 5 for poisson s ratios 0 0 3 and 0 499 frequencies are also given for thick annular plates having a thickness to outer diameter of 0 2 inside to outside diameter ratios of 0 1 0 5 and 0 9 and 0 3 all 3 d modes are included in the analyses e g flexural thickness shear in plane stretching and torsional the circular and annular plate frequency data given is exact to at least four digits thus being benchmark data against which results from 2 d thick plate theories or other approximate methods e g finite elements may be compared

three dimensional vibration analysis of circular and annular plates via the chebyshev ritz method abstract a three dimensional free vibration analysis of circular and annular plates is presented via the chebyshev ritz method the solution procedure is based on the linear small strain three dimensional elasticity theory selecting chebyshev polynomials which can be expressed in terms of cosine functions as the admissible functions a convenient governing eigenvalue equation can be derived through the ritz method according to the geometric properties of circular and annular plates the vibration is divided into three distinct categories axisymmetric vibration torsional vibration and circumferential vibration each vibration category can be further subdivided into the antisymmetric and symmetric ones in the thickness direction convergence and comparison study demonstrated the high accuracy and efficiency of the present method the present approach shows a distinct advantage over some other ritz solutions in that stable numerical operation can be guaranteed even when a large number of admissible functions is employed therefore not only lower order but also higher order eigenfrequencies can be obtained by using sufficient terms of the chebyshev polynomials finally some valuable results for annular plates with one or both edges clamped are given and discussed in detail

three dimensional vibration analysis of cantilevered and completely free isosceles triangular plates the free vibration of cantilevered and completely free isosceles triangular plates based on exact three dimensional elasticity theory is investigated the actual plate domain is first mapped onto a basic cubic domain then the ritz method is applied to derive the eigenfrequency equation from the strain energy and the kinetic energy of the plate a set of chebyshev polynomial series multiplied by a boundary function chosen to satisfy the essential geometry boundary conditions of the plate is developed as the admissible functions of each displacement component the convergence and comparison studies show that rather accurate results can be obtained by using this approach the effects of thickness to width ratio and apex angle on eigenfrequencies of the plates are studied in detail sets of valuable results are presented which may serve as the benchmark values for future numerical techniques in thick plate vibration analysis data for the completely free isosceles triangular plates are presented for the first time

an example of algebraic cycles with nontrivial abel jacobi images abstract in this paper we will study a family of quintic hypersurfaces and its abel jacobi images we will construct codimension 2 algebraic cycles on the blowing up of certain dimension 3 quintic hypersurfaces defined over number fields and show that they are homologically trivial but have nontrivial images under the 3 adic abel jacobi map

transverse vibration of simply supported elliptical and circular plates using boundary characteristic orthogonal polynomials in two variables abstract the first four frequencies and mode shapes have been computed for elliptical and circular plates of uniform thickness with simply supported edges by employing orthogonal polynomials in the rayleigh ritz method in general this is found to give better results than traditional methods successive approximations have been worked out to ensure convergence comparisons have been made with existing results in some cases calculations have been done for several values of b a i e the ratio of the semi major and semi minor axes mode shapes have been used to obtain three dimensional plots of plates in displaced configurations

interval ellipsoidal and super ellipsoidal calculi for experimental and theoretical treatment of uncertainty which one ought to be preferred abstract in this study we compare three calculi listed in the title for analysis of structures involving uncertainty the main idea is based on the consideration that the maximum structural response predicted by the preferred theory ought to be minimal and the minimum structural response predicted by the preferred theory ought to be maximal to constitute a lower overestimation we present analytic results that allow one to calculate the structural response via the interval ellipsoidal or super ellipsoidal calculus we provide several examples of truss structures and illustrate that in different situations depending on the available data one of these calculi ought to be preferred conclusion is made on the preferable approach to be the super ellipsoidal calculus

use of characteristic orthogonal polynomials in two dimensions for transverse vibration of elliptic and circular plates with variable thickness abstract free flexural vibrations of elliptic and circular plates with variable thickness have been studied by using characteristic orthogonal polynomials satisfying the essential boundary conditions and the rayleigh ritz method two types of variable thickness have been considered in the first case it varies linearly or quadratically parallel to the major axis in the second case it is taken to be the same along concentric ellipses but varies linearly or quadratically as we move from one ellipse to another the results for a circular plate with variable thickness follow as a special case computations have been carried out for clamped simply supported and free boundary comparison has been made with known results in special cases

forced vertical vibrations of an elastic elliptic plate on an elastic half space a direct approach using orthogonal polynomials the forced vibration of an elastic half space produced by a rigid elliptic indenter oscillating about an axis perpendicular to the plane face of the half space is considered the boundary conditions lead to a two dimensional dual integral equation in terms of the unknown normal stress by appropriate substitution the dual integral equation is first reduced to a two dimensional fredholm integral equation this is transformed to an infinite set of equations using abelian transformations next the abel transformed variable of the unknown normal stress is expanded in terms of orthogonal jacobi polynomials and by solving the system of linear equations orthogonal polynomial solutions are obtained the method used to obtain the orthogonal polynomial solutions of this problem is new and the major advantage of this expansion technique is that it is valid for all frequencies detailed numerical work has been given for the total load on the disc for different values of frequencies

free vibration analysis of orthotropic elliptical plates resting on arbitrarily distributed point supports the free vibration of point supported elliptical plates is studied analytically with consideration of rectangular orthotropy apparently this problem has been previously unsolved in the literature an analytical method used is an accurate variational approach the ritz lagrange multiplier method which constitutes an extension of the classical ritz method in numerical examples elliptical plates on some symmetric point supports are considered and it is shown that the method is straightforwardly applicable to determination of frequencies for elliptical plates resting on various support configurations

free vibration of super elliptical plates with constant and variable thickness by ritz method in this study free vibration of simply supported and clamped super elliptical plates is investigated this class of plates includes a wide range of external boundaries varying from an ellipse to a rectangle although studies on the upper and lower bounds of these plate geometries namely circle and rectangle are quite extensive contributions on the mid shapes especially for simply supported boundary edges are very limited the kirchhoff plate model with isotropic and homogeneous material is studied the super elliptical powers are chosen from 1 to 10 the ritz method is employed for the solution of the energy equations of the plates the effects of poisson s ratio which should not be neglected for simply supported plates with curved boundaries and the aspect ratio of the plate are both examined in detail the effect of thickness variation is also considered in this study in order to avoid long computational run times physically pertinent trial functions are utilized the frequency parameters obtained are presented and compared with published results for plate shapes that match the current cases

three dimensional vibration analysis of thick rectangular plates using chebyshev polynomial and ritz method this paper describes a method for free vibration analysis of rectangular plates with any thicknesses which range from thin moderately thick to very thick plates it utilises admissible functions comprising the chebyshev polynomials multiplied by a boundary function the analysis is based on a linear small strain three dimensional elasticity theory the proposed technique yields very accurate natural frequencies and mode shapes of rectangular plates with arbitrary boundary conditions a very simple and general programme has been compiled for the purpose for a plate with geometric symmetry the vibration modes can be classified into symmetric and antisymmetric ones in that direction in such a case the computational cost can be greatly reduced while maintaining the same level of accuracy convergence studies and comparison have been carried out taking square plates with four simply supported edges as examples it is shown that the present method enables rapid convergence stable numerical operation and very high computational accuracy parametric investigations on the vibration behaviour of rectangular plates with four clamped edges have also been performed in detail with respect to different thickness side ratios aspect ratios and poissons ratios these results may serve as benchmark solutions for validating approximate two dimensional theories and new computational techniques in future 2002 elsevier science ltd all rights reserved

vibration of symmetrically laminated thick super elliptical plates this paper reports a free vibration analysis of thick plates with rounded corners subject to a free simply supported or clamped boundary condition the plate perimeter is defined by a super elliptic function with a power defining the shape ranging from an ellipse to a rectangle to incorporate transverse shear deformation the reddy third order plate theory is employed the energy integrals incorporating shear deformation and rotary inertia are formulated and the p ritz procedures are used to derive the governing eigenvalue equation numerical examples for plates with different shapes and boundary conditions are solved and their frequency parameters where possible are compared with known results parametric studies are carried out to show the sensitivities of frequency parameters by varying the geometry fibre stacking sequence and boundary condition c 1999 academic press

complex multiplication cycles and a conjecture of beilinson and bloch a generalization of the conjecture of birch and swinnerton dyer is investigated using complex multiplication cycles on a particular kuga fiber variety a weak finiteness result consistent with the conjecture is proved the image of complex multiplication cycles under the abel jacobi map is computed explicitly the results provide numerical evidence supporting the conjecture they also give evidence for a relationship between complex multiplication cycles and a modular form of weight 5 2 and raise questions for further investigation

comments on free vibration of super elliptical plates with constant and variable thickness by ritz method where the maximum dimensions of the plate are 2a and 2b in the x and y directions respectively the coefficient n is the power of the super ellipse and the limiting cases of super elliptical shapes are an ellipse when n 1 and for larger values of n the curve gets gradually more rectangular shapes until for n 1 the curve takes up a rectangular shape in their study c eribas i and altay have calculated the perimeters and areas of different super ellipses and they are presented in their table 2 1 they wrote in their work any deviation from the area of the super ellipse causes extra error in the results calculating the area and perimeter of the region may give an idea about the convergence of the integration the discussers have found that some differences appeared in those basic results when they recalculated the perimeters and areas of the same geometries the perimeters calculated by c eribas i and altay have slight differences with the recalculated values but the differences are more important for the recalculated areas of the super elliptical plates to be sure about the good precision of their values the discussers also obtained the areas and perimeters using a finite element

on the birch and swinnerton dyer conjecture let f be a finite extension of the rational field q if e is an elliptic curve defined over f then the mordell weil group e f of points on e with coordinates in f is a finitely generated abelian group let l e f s be the hasseweil zeta function of e over f which at least in the case where e has complex multiplication is known to be an analytic function on the entire complex plane birch and swinnerton dyer have conjectured that l e f s has a zero at s l of order precisely equal to the rank of e f over z the strongest result known in support of this conjecture is the following theorem of coates and wiles 4

3 d vibration analysis of generalized super elliptical plates using chebyshev ritz method the three dimensional free vibration of generalized super elliptical plates is analysed based on the exact small strain and linear elasticity theory the ritz method is applied to derive the frequency equation the triplicate chebyshev polynomial series form the backbones of the admissible functions as modified by a characteristic boundary function to ensure the satisfaction of geometric boundary conditions of the plate utilizing the symmetry of the plate under consideration eight distinct vibration modes can be classified and individually solved while maintaining the same level of accuracy the accuracy of the present method has been examined by the convergence and comparison studies the effect of geometric parameters on vibration behaviour of the generalized super elliptical plates with free and fixed perimeters have been studied for different powers thickness ratios and aspect ratios

on the three dimensional vibrations of the cantilevered rectangular parallelepiped a solution is presented for the three dimensional problem of determining the free vibration frequencies and mode shapes for a rectangular parallelepiped which is completely fixed on one face and free on the other five faces this problem apparently is previously unsolved in the published literature the ritz method is used with displacements assumed in the form of algebraic polynomials convergence is studied numerical results are given for the first five frequencies of each of the four symmetry classes of vibration for five thick parallelepiped configurations including the cube contour plots are exhibited for the modal displacements of the cube the effects of varying poisson s ratio are also observed

elasticity solutions for free vibrations of annular plates from three dimensional analysis abstract this article examines the vibrational characteristics of annular plates by using the three dimensional elasticity theory it aims to raise the quality of the investigation beyond that provided by the two dimensional plate theories by resorting to a full three dimensional analysis a polynomials ritz model based on sets of orthogonally generated polynomial functions to approximate the spatial displacements of the plates in cylindrical polar coordinates is presented the model is then used to extract the full vibration spectrum of natural frequencies and mode shapes the vibration responses due to the variations of boundary conditions and thickness are investigated frequency parameters and three dimensional deformed mode shapes are presented in vivid graphical forms the accuracy of the method is validated through appropriate convergence and comparison studies

modeling three dimensional vibration of elliptic bars this paper presents a three dimensional vibration analysis of elliptic bars with various end constraints the theoretical derivation is established based on the linear small strain three dimensional elasticity principle with the expansion of the displacements in a three dimensional spatial coordinate the integral expressions for strain and kinetic energies are developed using the orthogonal polynomials to represent the three dimensional displacement field a governing eigenvalue equation is obtained by minimizing the energy functional according to the ritz principle to demonstrate the applicability of this method several numerical examples are solved the accuracy of the frequency solutions is established by checking with the existing refined beam theories first known results in terms of frequency parameters and displacement mode shape plots of the elliptic bar with different end support conditions are presented the new results may serve as a useful reference to future research into the refined b

a continuum three dimensional vibration analysis of thick rectangular plates abstract this paper presents a continuum three dimensional ritz formulation for the vibration analysis of homogeneous thick rectangular plates with arbitrary combinations of boundary constraints this model is formulated on the basis of the linear three dimensional small deformation elasticity theory to predict the vibratory responses of these thick rectangular plates the displacement fields in the transverse and in plane directions are expressed by sets of orthogonally generated polynomial functions these shape functions are intrinsically a product of a class of orthogonal polynomial functions and a basic function which are chosen to satisfy the essential geometric boundary conditions at the outset sets of frequency data for plates with various aspect ratios and thickness ratios have been presented these data are used to examine the merits and limitations of the classical plate theory and mindlin plate theory by direct comparisons finally using the three dimensional continuum approach sets of first known deformed mode shapes have been generated thus helping to understand the vibratory motion furthermore these results may also serve as the benchmark to further research into the refined plate theories

p adic l functions and the coniveau filtration on chow groups we give evidence for the refined version of the bloch beilinson conjecture involving coniveau filtrations by studying several infinite families of cm motives indexed by the integers r 1 that are irreducible of hodge type 2r 1 0 0 2r 1 and whose l functions vanish at the center in each case we construct a corresponding algebraic cycle that is homologically trivial but nontrivial in the top graded piece for the coniveau filtration consequently we obtain new explicit examples of cycles that are nontorsion in the griffiths group the proof of the main result relies crucially on p adic hodge theoretic methods and in particular on the relation between images of homologically trivial cycles under the p adic abel jacobi map and the values of p adic l functions

three dimensional elasticity solutions to vibration of cantilevered skewed trapezoids an accurate three dimensional elasticity model for determining the free vibration response of skewed trapezoidal plates is presented the method is formulated on the basis of a linear small strain three dimensional elasticity theory with the ritz minimization procedure the solution method uses sets of uniquely defined one and twodimensional polynomial functions to approximate the trial displacements in the thickness and surface directions respectively these functions are orthogonally generated and have been proven to be highly efficient and numerically stable the proposed technique yields natural frequencies and mode shapes for a wide range of cantilevered skewed trapezoidal plates of arbitrary planform the accuracy of these results whenever possible is validated through comparison with the analytical and experimental data parametric investigations on the vibration behavior of cantilevered trapezoidal plates with respect to different thickness ratios skew angles and chord ratios have been carried out for the first time the presence of surface parallel symmetric thickness modes for these plates is demonstrated in vivid three dimensional mode shape plots

a characterization of nonchaotic continuous maps of the interval stable under small perturbations recent results of the second author show that every continuous map of the interval to itself either has every trajectory approximable by cycles some times this is possible even in the case when the trajectory is not asymptotically periodic or is e chaotic for some e 0 in certain cases the first property is stable under small perturbations this means that a perturbed map can be chaotic but the chaos must be small whenever the perturbation is small in other words there are nonchaotic maps without chaos explosions in the paper we give a characterization of these maps along with some consequences using the main result it is possible to prove that generically the nonchaotic maps are stable asymptotic properties of trajectories have been intensively considered since the middle of the sixties when excellent papers by a n sarkovskii appeared the main reason is applicability of these results in mathematical modelling important are maps with trajectories having regular asymptotical behavior like asymptotical periodicity however this property can be easily destroyed by ar bitrarily small perturbations of the maps and the corresponding unstable models in connection with this there appeared some results guaranteeing the stability in 15 it is shown that every map having no periodic orbits of period dif ferent from 1 2 2 for some n with per nowhere dense is stable this result was generalized 16 to the maps with per nowhere dense which have all trajectories asymptotically periodic these maps can have cycles of period 2 for every n but no other cycles and are characterized e g by the property that per is closed 12 cf also 10

transverse vibrations of an elliptic plate with variable thickness abstract galerkin s method has been used to obtain the frequencies of symmetric transverse vibrations of an elliptic plate with clamped edges and variable thickness the procedure can be used to generate a sequence of approximations which may be truncated when the required number of frequencies have converged to the desired accuracy the numerical values of the first four frequencies are reported correct to five significant digits for various values of the taper parameter and the ratio of the semi axes mode shapes and the nodal ellipses have also been worked out for the second and third modes the results for an elliptic plate with uniform thickness circular plate with parabolically varying thickness and a circular plate with uniform thickness have been obtained as special cases

vibration and buckling of super elliptical plates this paper is concerned with the vibration and buckling of a new class of plates the periphery shape of which is defined by a super elliptical function such a plate shape has practical applications as the advantageous curved corners help to diffuse stress concentrations the loading considered for the buckling problem is that of in plane uniform pressure along the periphery accurate frequency and buckling factors are tabulated for such plates with either simply supported or clamped edges the solutions are obtained using the pb 2 rayleigh ritz method

free vibration of symmetrically laminated thick perforated plates abstract this paper investigates the free vibration of symmetrically laminated thick doubly connected plates of arbitray plate perimeter for the outer boundary and a hole defined by a super elliptical equation which is able to describe a rectangle ellipse or quasi rectangle the laminated perforated plates can be subject to free simply supported or clamped edge conditions convergence and comparisons with established work have been studied to ensure accuracy of results efforts are made to interpret the frequency results to provide physical insight to the problem