Triangles with Three Rational Medians

We present a characterization of all rational sided triangles with three rational medians. It turns out that they each correspond to a point on a one-parameter family of elliptic curves. It is possible to show that the rank of this family is at least two and in fact some reasonably high rank curves appear among them.     

The Igusa local zeta functions of elliptic curves

We determine the explicit form of the Igusa local zeta function associated to an elliptic curve. The denominator is known to be trivial. Here we determine the possible numerators and classify them according to the Kodaira-Néron classification of the special fibers of elliptic curves as determined by Tate's algorithm.

     

The complexity of numerical methods for elliptic partial differential equations

We consider three Ritz-Galerkin procedures with Hermite bicubic, bicubic spline and linear triangular elements for approximating the solution of self-adjoint elliptic partial differential equations and a collocation with Hermite bicubics method for general linear elliptic equations defined on general two dimensional domains with mixed boundary conditions. We systematically evaluate these methods by applying them to a sample set of problems while measuring various performance criteria. The test data suggest that collocation is the most efficient method for general use.     

Some families of Mordell curves associated to cubic fields

We introduce some Mordell curves of two different natures both of which are associated to cubic fields. One set of them consists of those elliptic curves whose rational points over the rational number field are described by or closely related to cubic fields. The other is a one-parameter family of Mordell curves which gives all (cyclic) cubic twists and all quadratic twists of the Fermat curve X³+Y³+Z³=0.     

Logarithmic transformations of rigid analytic elliptic surfaces

We give new examples of algebraic elliptic surfaces and non-algebraic rigid analytic elliptic surfaces by means of logarithmic transformations. In the complex analytic case, it is known that all multiple fibers of elliptic surfaces are obtained by logarithmic transformations. Using rigid analytic geometry, we construct similar transformations of elliptic surfaces over complete non-Archimedean valuation base fields. These operations yield rigid analytic elliptic fibrations with multiple fibers. When the resulting surface admits an ample line bundle, we may algebraize the surface. In the positive characteristic case, we obtain new types of algebraic elliptic surfaces. We also obtain a non-algebraic rigid analytic surface the combination of whose invariants appears neither in the algebraic case nor in the complex analytic case.     

Polynomial Poisson Algebras with Regular Structure of Symplectic Leaves

We study polynomial Poisson algebras with some regularity conditions. Linear (Lie–Berezin–Kirillov) structures on dual spaces of semisimple Lie algebras, quadratic Sklyanin elliptic algebras, and the polynomial algebras recently described by Bondal, Dubrovin, and Ugaglia belong to this class. We establish some simple determinant relations between the brackets and Casimir functions of these algebras. In particular, these relations imply that the sum of degrees of the Casimir functions coincides with the dimension of the algebra in the Sklyanin elliptic algebras. We present some interesting examples of these algebras and show that some of them arise naturally in the Hamiltonian integrable systems. A new class of two-body integrable systems admitting an elliptic dependence on both coordinates and momenta is among these examples.     

Generating More MNT Elliptic Curves

In their seminal paper, Miyaji et al. [13] describe a simple method for the creation of elliptic curves of prime order with embedding degree 3, 4, or 6. Such curves are important for the realisation of pairing-based cryptosystems on ordinary (non-supersingular) elliptic curves. We provide an alternative derivation of their results, and extend them to allow for the generation of many more suitable curves. Research supported by Enterprise Ireland grant IF/2002/0312/N.     

Average frobenius distributions for elliptic curves with 3-torsion

In this paper, we examine the Lang-Trotter conjecture for elliptic curves which possess rational 3-torsion points. We prove that if one averages over all such elliptic curves then one obtains an asymptotic similar to the one predicted by Lang and Trotter.     

Elliptic solutions of nonlinear integrable equations and related topics

The theory of elliptic solitons for the Kadomtsev-Petviashvili (KP) equation and the dynamics of the corresponding Calogero-Moser system is integrated. It is found that all the elliptic solutions for the KP equation manifest themselves in terms of Riemann theta functions which are associated with algebraic curves admitting a realization in the form of a covering of the initial elliptic curve with some special properties. These curves are given in the paper by explicit formulae. We further give applications of the elliptic Baker-Akhiezer function to generalized elliptic genera of manifolds and to algebraic 2-valued formal groups.Dedicated to the memory of J.-L. Verdier     

Natural equivariant transversally elliptic Dirac operators

We introduce a new class of natural, explicitly defined, transversally elliptic differential operators over manifolds with compact group actions. Under certain assumptions, the symbols of these operators generate all the possible values of the equivariant index. We also show that the components of the representation-valued equivariant index coincide with those of an elliptic operator constructed from the original data.     

Elliptic General Analytic Solutions

To find analytically the traveling waves of partially integrable autonomous nonlinear partial differential equations, many methods have been proposed over the ages: "projective Riccati method,""tanh-method,""exponential method,""Jacobi expansion method,""new … ," etc. The common default to all these "truncation methods" is that they provide only some solutions, not all of them. By implementing three classical results of Briot, Bouquet, and Poincaré, we present an algorithm able to provide in closed form  all  those traveling waves that are elliptic or degenerate elliptic, i.e., rational in one exponential or rational. Our examples here include the Kuramoto–Sivashinsky equation and the cubic and quintic complex Ginzburg–Landau equations.     

Rank of <Emphasis Type="Italic">K</Emphasis><Subscript>2</Subscript> of elliptic curves

We prove that (i) rank(K2(E)) 1 for all elliptic curves E defined over Q with a rational torsion point of exact order N 4; (ii) rank(K2(E)) 1 for all but at most one R-isomorphism class of elliptic curves E defined over Q with a rational torsion point of exact order 3. We give some sufficient conditions for rank(K2(EZ)) 1.     

Characterization of sub‐Gaussian heat kernel estimates on strongly recurrent graphs

Sub‐Gaussian estimates for random walks are typical of fractal graphs. We characterize them in the strongly recurrent case, in terms of resistance estimates only, without assuming elliptic Harnack inequalities. © 2005 Wiley Periodicals, Inc.     

Correspondences between modular Calabi–Yau fiber products

We describe two ways to construct finite rational morphisms between fiber products of rational elliptic surfaces with section and some Calabi–Yau varieties. We use them to construct correspondences between such fiber products that admit at most five singular fibers and rigid Calabi–Yau threefolds.     

基于第二类椭圆积分的椭圆弧长公式变换与应用

以第二类椭圆积分为理论基础,通过推导,将椭圆弧长公式变换为以椭圆离心角、极角等常用角度参数为自变量的第二类椭圆积分的标准形式,建立起椭圆弧长公式与第二类椭圆积分标准形式之间的关系,并分析了椭圆上的弧微分变化规律及椭圆周长与离心率的变化关系.公式反映了椭圆弧长的本质问题即为第二类椭圆积分问题.因此,各类涉及椭圆弧长计算的应用问题,均可化为第二类椭圆的计算问题,应用时直接调用各类编程软件的函数库中的第二类椭圆积分函数,无需复杂编程即可实现椭圆弧长的高精度计算.文章以GPS采用的WGS-84椭球子午线弧长为例进行计算分析,验证了给出的公式及相关分析的正确性及应用价值.     

On differential modular forms and some analytic relations between Eisenstein series

In the present article we define the algebra of differential modular forms and we prove that it is generated by Eisenstein series of weight 2, 4 and 6. We define Hecke operators on them, find some analytic relations between these Eisenstein series and obtain them in a natural way as coefficients of a family of elliptic curves. The fact that a complex manifold over the moduli of polarized Hodge structures in the case h ¹⁰=h ⁰¹=1 has an algebraic structure with an action of an algebraic group plays a basic role in all of the proofs.      

二阶非线性椭圆型微分方程解的Sturm比较定理

本文通过对一类含阻尼项的椭圆型微分方程建立几个微分不等式,得到几个新的Sturm型比较定理,将经典的Sturm比较定理推广到含阻尼项椭圆型微分方程,并给出一些具体应用的例子.     

Orthonormal polynomial wavelets on the interval

We use special functions and orthonormal wavelet bases on the real line to construct wavelet-like bases. With these wavelets we can construct polynomial bases on the interval; moreover, we can use them for the numerical resolution of degenerate elliptic operators.

     

On the Shapiro–Lopatinkii condition for elliptic problems

This paper considers elliptic problems with high-order derivatives multiplied by a small parameter. We found the algebraic conditions for an operator and the boundary conditions that guarantee the Fredholm property. An a priori estimate for the solution with a constant independent of the small parameter is proved. These results are known for elliptic boundary-value problems with small parameter in the half-space R _n ⁺. We extend them to the case of bounded domains with smooth boundary. The small-parameter coercive conditions are formulated, and a two-sided estimate is proved.     

A class of motions of a top in the Goryachev-Chaplygin case

A solution of the Euler-Poisson equations is studied for the Goryachev-Chaplygin case /1/ under the condition when the ultraelliptic integrals degenerate to elliptic /2/. A solution is constructed for a class of motions in which both quantities, u and v, brought in by Chaplygin, vary with time, but one of them tends asymptotically to a constant when the time increases without limit. The dependence of the Euler-Poisson variables on time is expressed in terms of elliptic functions and an elliptic integral of the third kind. Fairly simple approximate formulas are given for determining all six variables sought.