A formula for the grössencharacter of a parametrized elliptic curve

A formula for the grössencharacter of an elliptic curve with complex multiplication, in a family parametrized by modified Weierstrass functions or classical theta-functions, is given. The method is based on Shimura's Reciprocity Law for modular functions, and applies to Legendre, Jacobi, and Hesse curves. As an application, the conductors of the CM curves in these families are determined.     

Oblique derivative problems for second order nonlinear equations of mixed type with degenerate hyperbolic curve

The present paper deals with oblique derivative problems for second order nonlinear equations of mixed type with degenerate hyperbolic curve, which include the Tricomi problem as a special case. Firstly the formulation of the problems for the equations is given, next the representation and estimates of solutions for the above problems are obtained, finally the existence of solutions for the problems is proved by the successive iteration of solutions of the equations and the fixed-point principle. In this paper, we use the complex analytic method, namely the new partial derivative notations, elliptic complex functions in the elliptic domain and hyperbolic complex functions in the hyperbolic domain are introduced, such that the second order equations of mixed type with degenerate curve are reduced to the first order mixed complex equations with singular coefficients, and then the advantage of complex analytic method can be applied.     

Lifting the j-invariant: Questions of Mazur and Tate

In this paper we analyze the j-invariant of the canonical lifting of an elliptic curve as a Witt vector. We show that its coordinates are rational functions on the j-invariant of the elliptic curve in characteristic p. In particular, we prove that the second coordinate is always regular at j=0 and j=1728, even when those correspond to supersingular values. A proof is given which yields a new proof for some results of Kaneko and Zagier about the modular polynomial.     

Constructions of almost complex 2-tori of type (III) in the nearly Kähler 6-sphere

In this paper, we give a machinery method of constructing almost complex curve of type (III) in the nearly Kähler 6-sphere. As application, a first non-trivial example of almost complex 2-tori of type (III) will be described in terms of the Jacobi elliptic functions. In the final section, a general solution of such tori will be described in terms of the Prym-theta functions using the known results from the integrable system theory.     

On the reduction (mod 1) of completely monotone functions (0, ∞)

Summary The Euler-Maclaurin summation formula and its harmonic analysis (Poisson) are applied to the case of functions which are completely monotone on an open half-line. What thus results is a curious class of Fourier series, which can be determined explicitly and which represent completely monotone functions on the first half of the period. A by-product is the complete monotony (on the first half-period) of the Bernoulli functions, whether the index is integral or fractional.     

Construction of an optimal envelope for a cone of nonnegative functions with monotonicity properties

The paper is devoted to problems at the intersection of formal group theory, the theory of Hirzebruch genera, and the theory of elliptic functions. In the focus of our interest are Tate formal groups corresponding to the general five-parametric model of the elliptic curve as well as formal groups corresponding to the general four-parametric Krichever genus. We describe coefficient rings of formal groups whose exponentials are determined by elliptic functions of levels 2 and 3.     

Differential geometry and the red blood cell

The usual mathematical expressions for the shape of the red blood cell, whether in rectangular or polar coordinates, are very complicated and difficult to use analytically. We have found that by using parametric equations and Jacobean elliptic functions to describe the cell things are much easier. The usual differential geometric concepts can be adapted to find interesting new formulas for ideas like Fick's Law, etc. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

长水波近似方程组的新精确解

依据齐次平衡法的思想 ,首先提出了求非线性发展方程精确解的新思路 ,这种方法通过改变待定函数的次序 ,优势是使求解的复杂计算得到简化 .应用本文的思路 ,可得到某些非线性偏微分方程的新解 .其次我们给出了长水波近似方程组的一些新精确解 ,其中包括椭圆周期解 ,我们推广了有关长波近似方程的已有结果 .     

The Trefoil

Dixon’s elliptic functions parameterize the real sextic trefoil curve by arc length and the complex curve as an embedded Platonic surface with 18 (or 108) faces.     

Convergence of the arithmetic-geometric mean procedure for the complex variables and the calculation of the complete elliptic integrals with complex modulus

The convergence of the arithmetic-geometric mean procedure is checked for complex variables. The procedure is shown to be useful for the evaluation of the complete elliptic integrals of the first and second kinds with complex modulus. It is suggested that the procedure will be useful also for the numerical calculation of the elliptic integrals and the Jacobian elliptic functions with complex modulus in general.     

A singular multi-dimensional piston problem in compressible flow

This paper concerns the multi-dimensional piston problem, which is a special initial boundary value problem of multi-dimensional unsteady potential flow equation. The problem is defined in a domain bounded by two conical surfaces, one of them is shock, whose location is also to be determined. By introducing self-similar coordinates, the problem can be reduced to a free boundary value problem of an elliptic equation. The existence of the problem is proved by using partial hodograph transformation and nonlinear alternating iteration. The result also shows the stability of the structure of shock front in symmetric case under small perturbation.     

The electrically induced stress in the dielectric of a transmission coaxial line

The electrically induced stress in the dielectric material of a rectangular coaxial line with a centered inner cross is investigated by use of a conformal mapping method in complex plane. The electric field is analytically expressed through elliptic functions. The components of the stress parallel and perpendicular to the electric field are computed. The location where the maximum stress occurs is determined. The centered strip line and square coaxial line are discussed as sample problems. The stress distribution in dielectric is improved through electrostriction coefficient when the dependence of permittivity on strain is considered. This methodology can be applicable to other transmission lines.     

New Jacobi elliptic functions solutions for the variable-coefficient mKdV equation

In this work, a new generalized Jacobi elliptic functions expansion method based upon four new Jacobi elliptic functions is described and abundant new Jacobi-like elliptic functions solutions for the variable-coefficient mKdV equation are obtained by using this method, some of these solutions are degenerated to solitary-like solutions and triangular-like functions solutions in the limit cases when the modulus of the Jacobi elliptic functions m→1 or 0, which shows that the new method can be also used to solve other nonlinear partial differential equations in mathematical physics.     

Certain values of Hecke L-functions and generalized hypergeometric functions

We compare two calculations due to Bloch and the author of the regulator of an elliptic curve with complex multiplication which is a quotient of a Fermat curve, and express the special value of its L-function at s=0 in terms of special values of generalized hypergeometric functions.     

Modeling the angular capability of the ball joints in a complex mechanism with two degrees of mobility

The paper puts forward a complex linkage mechanism with two degrees of mobility and three kinematic loops, which is used for the guiding (suspension and steering) system of the vehicles. The geometric parameters and the coordinates frames that define the mechanical system are presented, as well as the specific kinematic functions. For this complex mechanical system, the angular capability of the ball (spherical) joints is defined by two angles. The equations for these angles have been determined by matrix algebra tools, considering the transformation matrices between the bodies reference frames. The diagrams of the angular capability of the ball joints, which are represented in angular coordinates, describe the form, orientation and size of the sockets from the spherical casings. Wears, shocks, functional locks or the compromising of the joint strength can occur if scarce sockets are implemented. The risk points, in which the angular parameters have maximum values, have been determined, the simulation being performed for a real system (vehicle).     

A new look at Jarvis' distribution formula

Starting from the well-known transformation law for the Klein functions, we give a proof of a fairly general multiplicative distribution formula for the Siegel functions associated to isogenous complex lattices. This formula has as an immediate consequence the remarkable distribution formula proved by Jarvis in 2000 on the occasion of Rolshausen's thesis on the second K-group of an elliptic curve.     

Algebraic-geometric solutions of the Krichever-Novikov equation

A zero-curvature representation with constant poles on an elliptic curve is obtained for the Krichever-Novikov equation. Algebraic-geometric solutions of this equation are constructed. The consideration is based on reducing the theta function of a two-sheet covering of an elliptic curve to the Prym theta functions of codimension one. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 121, No. 3, pp. 367–373, December, 1999.     

An addition formula for the Jacobian theta function and its applications

In this paper, we prove an addition formula for the Jacobian theta function using the theory of elliptic functions. It turns out to be a fundamental identity in the theory of theta functions and elliptic function, and unifies many important results about theta functions and elliptic functions. From this identity we can derive the Ramanujan cubic theta function identity, Winquist's identity, a theta function identities with five parameters, and many other interesting theta function identities; and all of which are as striking as Winquist's identity. This identity allows us to give a new proof of the addition formula for the Weierstrass sigma function. A new identity about the Ramanujan cubic elliptic function is given. The proofs are self contained and elementary.     

Spectral methods using generalized Laguerre functions for second and fourth order problems

Spectral methods using generalized Laguerre functions are proposed for second-order equations under polar (resp. spherical) coordinates in ?² (resp. ?³) and fourth-order equations on the half line. Some Fourier-like Sobolev orthogonal basis functions are constructed for our Laguerre spectral methods for elliptic problems. Optimal error estimates of the Laguerre spectral methods are obtained for both second-order and fourth-order elliptic equations. Numerical experiments demonstrate the effectiveness and the spectral accuracy.     

A note on the elliptic equation method

The elliptic equation method is improved for constructing exact travelling wave solutions of nonlinear partial differential equations (PDEs). The rational forms of Jacobi elliptic functions are presented. By using new Jacobi elliptic function solutions of the elliptic equation, new doubly periodic solutions are obtained for some important PDEs. This method can be applied to many other nonlinear PDEs.