one parameter families of elliptic curves over ℚ with maximal galois representations let e be an elliptic curve over q and let q e n be its nth division field in 1972 serre showed that if e is without complex multiplication then the galois group of q e n q is as large as possible that is gl2 z nz for all integers n coprime to a constant integer m e q depending at most on e q serre also showed that the best one can hope for is to have gl2 z nz gal q e n q 2 for all positive integers n we study the frequency of this optimal situation in a one parameter family of elliptic curves over q and show that in essence for almost all one parameter families almost all elliptic curves have this optimal behavior

moduli interpretations for noncongruence modular curves we consider the moduli of elliptic curves with g structures where g is a finite 2 generated group when g is abelian a g structure is the same as a classical congruence level structure there is a natural action of text sl 2 mathbb z on these level structures if gamma is a stabilizer of this action then the quotient of the upper half plane by gamma parametrizes isomorphism classes of elliptic curves equipped with g structures when g is sufficiently nonabelian the stabilizers gamma are noncongruence using this we obtain arithmetic models of noncongruence modular curves as moduli spaces of elliptic curves equipped with nonabelian g structures as applications we describe a link to the inverse galois problem and show how our moduli interpretations explains the bad primes for the unbounded denominators conjecture and allows us to translate the conjecture into the language of geometry and galois theory

shimura curves of genus at most two we enumerate all shimura curves x d 0 r of genus at most two there are exactly 858 such curves up to equivalence

a general solution procedure for the forced vibrations of a continuous system with cubic nonlinearities primary resonance case abstract nonlinear vibrations of a general model of continuous system is considered the model consists of arbitrary linear and cubic operators the equation of motion is solved by the method of multiple scales a perturbation method the primary resonances of external excitation is analysed the amplitude and phase modulation equations are presented approximate analytical solution is derived steady state solutions and their stability are discussed finally the solution algorithm is applied to two different engineering problems one of the application is the transverse vibration of an axially moving euler bernoulli beam and the other is a viscoelastic beam

constrained degree reduction of polynomials in bernstein bezier form over simplex domain in this paper we show that the orthogonal complement of a subspace in the polynomial space of degree n over d dimensional simplex domain with respect to the l 2 inner product and the weighted euclidean inner product of bb bezier bernstein coefficients are equal using it we also prove that the best constrained degree reduction of polynomials over the simplex domain in bb form equals the best approximation of weighted euclidean norm of coefficients of given polynomial in bb form from the coefficients of polynomials of lower degree in bb form

eine arithmetische eigenschaft automorpher formen zu gewissen nicht arithmetischen gruppen let sl2 open image in new window denote a fuchsian triangle group anda an elliptic fixed point of for any weightk and any multiplier systemv whose values are roots of unity there is a basis of automorphic forms with expansions f left z right left z bar a right k sum limits n 0 infty r n b n left frac z a z a right n

torsion free genus zero congruence subgroups of psl2 ℝ we study and classify all the conjugacy classes of the genus zero congruence subgroups of psl2 r with no elliptic elements we show that it suffices to classify those inside the modular group and determine them completely we also discuss an application to modular curves

wohlfahrt s theorem for the hecke group g5 abstract let k be a subgroup of the inhomogeneous hecke group g 5 of geometric level r then k is congruence if and only if k contains the principal congruence subgroup g 2 r in the case r 0 mod 4 k is congruence if and only if k contains the principal congruence subgroup g r

normalisers of subgroups of the modular group abstract let g be a subgroup of finite index of the modular group and let n g be the normaliser of g in psl2 r in this article we give an algorithm that determines n g

arithmetic and geometry of the hecke groups abstract we study the arithmetic and geometry properties of the hecke group g q in particular we prove that g q has a subgroup x of index d genus g with v cusps and 2 resp v r i conjugacy classes of elliptic elements that are conjugates of s resp r q r i if and only if i 2 g 2 2 2 i 1 k v r i 1 1 r i v d 1 2 1 q and ii m 0 4 g 4 2 2 v i 1 k v r i 2 q r i 0 is a multiple of q 2 note that if q is odd resp prime then m 0 q 2 z resp n 0 is a consequence of i

werte hypergeometrischer funktionen a partial result on a problem of siegel is given hypergeometric functions with rational parameters have transcendental values in almost all algebraic points up to some natural exceptions these exceptions are the well known algebraic functions and an unexpected second class of examples related to certain shimura curves

coupled longitudinal transverse dynamics of an axially accelerating beam abstract the coupled longitudinal transverse nonlinear dynamics of an axially accelerating beam is numerically investigated this problem is classified as a parametrically excited gyroscopic system the axial speed is assumed to be comprised of a constant mean value along with harmonic fluctuations hamilton s principle is employed to derive the equations of motion of the system which are in the form of two coupled partial differential equations the equations are discretized using the galerkin method which yields a set of coupled second order nonlinear ordinary differential equations with time dependent coefficients the sub critical dynamics of the system is examined via the pseudo arclength continuation technique while the global dynamics is investigated using direct time integration the mean axial speed and the amplitude of the speed variations are varied so as to construct the bifurcation diagrams of poincare maps the vibration specifications of the system are investigated more detailed via plotting time histories phase plane portraits and fast fourier transforms ffts

nonlinear free transverse vibrations of in plane moving plates without and with internal resonances in this paper nonlinear free transverse vibrations of in plane moving plates subjected to plane stresses are investigated the hamilton principle is applied to derive the governing equation and the associated boundary conditions the method of multiple scales is employed to analyze the nonlinear partial differential equation the solvability conditions are established in the cases without internal resonance and with 3 1 or 1 1 internal resonances some numerical examples are presented to demonstrate the effects of in plane moving speeds on the frequencies the nonlinear frequencies of the in plane moving plate without internal resonances are numerically calculated the relationship between the nonlinear frequencies and the initial amplitudes is showed at different in plane moving speeds and the nonlinear coefficients respectively it is feasible to investigate resonances without the modes not involved in the resonances the effects of the related parameters are demonstrated for the case of 3 1 and 1 1 internal resonances respectively the differential quadrature scheme is developed to solve numerically the governing equation and confirm results via the method of multiple scales

stability in parametric resonance of axially moving viscoelastic beams with time dependent speed stability in transverse parametric vibration of axially accelerating viscoelastic beams is investigated the governing equation is derived from newton s second law the kelvin constitution relation and the geometrical relation when the axial speed is a constant mean speed with small harmonic variations the governing equation can be regarded as a continuous gyroscopic system under small periodically parametric excitations and a damping term the method of multiple scales is applied directly to the governing equation without discretization the stability conditions are obtained for combination and principal parametric resonance numerical examples are presented for beams with simple supports and fixed supports respectively to demonstrate the effect of viscoelasticity on the stability boundaries in both cases

steady state response of axially moving viscoelastic beams with pulsating speed comparison of two nonlinear models abstract principal parametric resonance in transverse vibration is investigated for viscoelastic beams moving with axial pulsating speed a nonlinear partial differential equation governing the transverse vibration is derived from the dynamical constitutive and geometrical relations under certain assumption the partial differential reduces to an integro partial differential equation for transverse vibration of axially accelerating viscoelastic nonlinear beams the method of multiple scales is applied to two equations to calculate the steady state response closed form solutions for the amplitude of the vibration are derived from the solvability condition of eliminating secular terms the stability of straight equilibrium and nontrivial steady state response are analyzed by use of the lyapunov linearized stability theory numerical examples are presented to highlight the effects of speed pulsation viscoelascity and nonlinearity and to compare results obtained from two equations

defining equations of modular curves we obtain defining equations of modular curves x0 n x1 n and x n by explicitly constructing modular functions using generalized dedekind eta functions as applications we describe a method of obtaining a basis for the space of cusp forms of weight 2 on a congruence subgroup we also use our model of x0 37 to find explicit modular parameterization of rational elliptic curves of conductor 37

level and index in the modular group it is shown that the index of a congruence subgroup of the modular group cannot be less than the level of the subgroup this allows a number of existence theorems about non congruence subgroups the level of a subgroup of the modular group can be defined in terms of the action on q we define a similar action to get information on congruence subgroups in fact we get a more powerful result but this appears to be the most useful version

best polynomial degree reduction on q lattices with applications to q orthogonal polynomials we study the best polynomial degree reduction with respect to the q l2 norm we study a finite analogue with respect to finite q lattices we present applications to q orthogonal polynomials we show that a weighted least squares approximation of q bezier coefficients provides the best polynomial degree reduction in the q l2 norm we also provide a finite analogue of this result with respect to finite q lattices and we present applications of these results to q orthogonal polynomials

the signature of γ 0 n abstract the signature of 0 en2 where e is square free is completely determined if e 3 or n is odd or e where is the set of all square free integers e n such that i if is odd then admits no divisors of the form 8 3 ii if is even then admits no divisors of the form 8 7 in particular v2 number of elliptic classes of period 2 of 0 n is expressed as a linear combination of multiplicative functions

on the cuspidal spectrum of the arithmetic hecke groups let rf vh m 1 2 3 be the three arithmetic hecke groups generated by the translation z z v i and the inversion z i r 1 is the modular group psl 2 z the purpose of this article is to present several theoretical results on x v hi the weight zero cuspidal spectrum of if v m which are of interest from a computational perspective since they have application to the numerical study both of the spectra themselves and the fourier coefficients of the associated maass wave forms the first of these is the theorem that x 1 c x v hi for m 2 and m 3 additional results explicate the action of the hecke operators for the three groups in question upon maass wave forms in particular upon their fourier coefficients these results are motivated in part by work of stark and more recent work of hejhal and hejhal and arno which demonstrates the importance of the hecke operators for the numerical study of the fourier coefficients of maass wave forms on 1

three to one internal resonances in a general cubic non linear continuous system a general continuous system with an arbitrary cubic non linearity is considered the non linearity is expressed in terms of an arbitrary cubic operator three to one internal resonance case is considered a general approximate solution is presented for the system amplitude and phase modulation equations are derived steady state solutions and their stability are discussed in the general sense the sufficiency condition for such resonances to occur is derived finally the algorithm is applied to a beam resting on a non linear elastic foundation

approximating tensor product bezier surfaces with tangent plane continuity we present a simple method for degree reduction of tensor product bezier surfaces with tangent plane continuity in l 2 norm continuity constraints at the four corners of surfaces are considered so that the boundary curves preserve endpoints continuity of any order a we obtain matrix representations for the control points of the degree reduced surfaces by the least squares method a simple optimization scheme that minimizes the perturbations of some related control points is proposed and the surface patches after adjustment are c continuous in the interior and g 1 continuous at the common boundaries we show that this scheme is applicable to surface patches defined on chessboard like domains

a general solution procedure for vibrations of systems with cubic nonlinearities and nonlinear time dependent internal boundary conditions abstract the subject of this paper is the development of a general solution procedure for the vibrations primary resonance and nonlinear natural frequency of systems with cubic nonlinearities subjected to nonlinear and time dependent internal boundary conditions this is a commonly occurring situation in the vibration analysis of continuous systems with intermediate elements the equations of motion form a set of nonlinear partial differential equations with nonlinear time dependent and coupled internal boundary conditions the method of multiple timescales an approximate analytical method is applied directly to each partial differential equation of motion as well as coupled boundary conditions i e on each sub domain and the corresponding internal boundary conditions for a continuous system with intermediate elements which ultimately leads to approximate analytical expressions for the frequency response relation and nonlinear natural frequencies of the system these closed form solutions provide direct insight into the relationship between the system parameters and vibration characteristics of the system moreover the suggested solution procedure is applied to a sample problem which is discussed in detail

direct treatment and discretizations of non linear spatially continuous systems abstract approximate analytical methods for the study of non linear vibrations of spatially continuous systems with general quadratic and cubic non linearities are discussed the cases of an external primary resonance of a non internally resonant mode and of a sub harmonically excited two to one internal resonance are investigated it is shown in a general fashion that application of the method of multiple scales to the original partial differential equations and boundary conditions produces the same approximate dynamics as those obtained by applying the reduction method to the full basis galerkin discretized system using the complete set of eigenfunctions of the associated linear system or to convenient low order rectified galerkin models as a corollary it is shown that due to the effects of the quadratic non linearities all of the modes from the relevant eigenspectrum in principle contribute to the non linear motions hence classical low order galerkin models may be inadequate to describe quantitatively and qualitatively the dynamics of the original continuous system although the direct asymptotic and rectified galerkin procedures seem to be more appealing from a computational standpoint the full basis galerkin discretization procedure furnishes a remarkably interesting spectral representation of the non linear motions

a general solution procedure for the forced vibrations of a system with cubic nonlinearities three to one internal resonances with external excitation a general vibrational model of a continuous system with arbitrary linear and cubic operators is considered approximate analytical solutions are found using the method of multiple scales the primary resonances of the external excitation and three to one internal resonances between two arbitrary natural frequencies are treated the amplitude and phase modulation equations are derived the steady state solutions and their stability are discussed the solution algorithm is applied to two specific problems 1 axially moving euler bernoulli beam and 2 axially moving viscoelastic beam

degree reduction of bezier curves by l1 approximation with endpoint interpolation abstract we consider the one degree reduction problem with endpoint interpolation in the l1 norm we obtain the best one degree reduction of bezier curve of the degree n 5 with endpoint interpolation by using perfect splines for the general degree n we propose a good one degree reduction by use of an appropriate transform of the tchebycheff polynomials un x of the second kind of degree n by use of the good one degree reduction subdivision algorithm is given to get one degree reduced bezier curve within a given tolerance some numerical experiments are also given

principal parametric resonances of a general continuous system with cubic nonlinearities abstract a generalized vibrational model of cubic nonlinear continuous system with arbitrary parametric excitation is considered the method of multiple scales a perturbation method is used to find an approximate analytical solution the primary parametric resonance of the parametric excitation is considered the amplitude and phase modulation equations are derived steady state solutions and their stability are discussed the solution algorithm is applied to the problem of nonlinear vibrations of viscoelastic pipes conveying fluids natural frequencies of viscoelastic pipes are found frequency response curves and bifurcation points are drawn stable and unstable regions of trivial and nontrivial solutions are obtained

application of chebyshev ii bernstein basis transformations to degree reduction of bezier curves a polynomial curve on 0 1 can be expressed in terms of bernstein polynomials and chebyshev polynomials of the second kind we derive the transformation matrices that map the bernstein and chebyshev coefficients into each other and examine the stability of this linear map in the p 1 and norms the condition number of the chebyshev bernstein transformation matrix grows at a significantly slower rate with n than in the power bernstein case and the rate is very close somewhat faster to the legendre bernstein case using the transformation matrices we present a method for the best multi degree reduction with respect to the t t 2 weighted square norm for the unconstrained case which is further developed to provide a good approximation to the best multi degree reduction with constraints of endpoints continuity of orders r s r s 0 this method has a quadratic complexity and may be ill conditioned when it is applied to the curves of high degree we estimate the posterior l 1 error bounds for degree reduction

symbolic dynamics for the modular surface and beyond in this expository article we describe the two main methods of representing geodesics on surfaces of constant negative curvature by symbolic sequences and their development a geometric method stems from a 1898 work of j hadamard and was developed by m morse in the 1920s it consists of recording the successive sides of a given fundamental region cut by the geodesic and may be applied to all finitely generated fuchsian groups another method of arithmetic nature uses continued fraction expansions of the end points of the geodesic at infinity and is even older it comes from the gauss reduction theory introduced to dynamics by e artin in a 1924 paper this method was used to exhibit dense geodesics on the modular surface for 80 years these classical works have provided inspiration for mathematicians and a testing ground for new methods in dynamics geometry and combinatorial group theory we present some of the ideas results old and recent and interpretations that illustrate the multiple facets of the subject

on rademacher s conjecture congruence subgroups of genus zero of the modular group we list all genus zero congruence subgroups of psl2 z there are altogether 132 of them up to conjugation in psl2 z geometrical invariants genus v2 v3 number of cusps index in psl2 z fundamental polygons farey symbol and independent generators of all such groups are determined generators of the function fields associated to such groups are also determined http www math nus edu sg matlml

vibrations of continuous systems with a general operator notation suitable for perturbative calculations the operator notation previously developed to analyze vibrations of continuous systems has been further generalized to model a system with an arbitrary number of coupled differential equations linear parts of the equations are expressed with an arbitrary linear differential and or integral operators and non linear parts are expressed with arbitrary quadratic and cubic operators equations of motion are solved in their general form using the method of multiple scales a perturbation technique the case of primary resonances of the external excitation and one to one internal resonances between the natural frequencies of the equations is considered the algorithm developed is applied to a non linear cable vibration problem having small sag to span ratios

a comparison of different versions of the method of multiple scales for partial differential equations abstract applications of the methods of multiple scales a perturbation method to partial differential systems arising in non linear vibrations of continuous systems are considered two different versions of the method of multiple scales are applied to two general non linear models in one of the models the small parameter multiplies an arbitrary non linear cubic operator whereas in the other model arbitrary quadratic and cubic non linearities exist the linear parts of both models are represented by arbitrary operators general solutions are found by applying different versions of the method of multiple scales results of the first version reconstitution method and the second version proposed by rahman and burton 8 are compared for both models from the comparisons of both methods it is found that the second version yields better results applications of the general models to specific problems are also presented a final recommendation is to use the second version of the method of multiple scales combined with the direct perturbation method in finding steady state solutions of partial differential equations

a unified approach for degree reduction of polynomials in the bernstein basis part i real polynomials the polynomials determined in the bernstein bezier basis enjoy considerable popularity and the algorithms for reducing their degree are of practical importance in computer aided design applications on the other hand the conversion between the bernstein and the power basis is ill conditioned thus only the degree reduction algorithms which may be carried out without using this conversion are of practical value our unified approach enables us to describe all the algorithms of this kind known in the literature to construct a number of new ones which are better conditioned and cheaper than some of the currently used ones and to study the errors of all of them in a simple homogeneous way all these algorithms can be applied componentwise to a vector valued polynomial representing a bezier curve consider the values of the derivatives whose orders vary successively from 1 to a given number i or k at the start and end point respectively of this curve the current algorithms allow us to preserve these points and values for i equal to k the new ones do that also without the latter constraint