The cubic Fermat equation and complex multiplication on the Deuring normal form

The arithmetic on elliptic curves in Deuring normal form is shown to be related to solutions of the Fermat equation 27X ³+27Y ³=X ³ Y ³. This arithmetic is used to give conditions for the existence of multipliers μ on supersingular elliptic curves in characteristic p for which μ ²=?3p. Together with an explicit factorization of a certain class equation, these conditions imply that the number of irreducible binomial quadratic factors (mod p) of the Legendre polynomial P _(p?e)/3(x) of degree (p?e)/3 is a simple linear function of the class number of the quadratic field \(\mathbb{Q}(\sqrt{-3p})\).     

Supersingular quartic surfaces

In this paper, we prove that, over an algebraically closed field of odd characteristic p ($p\ne 5$), every supersingular K3 surface is isomorphic to a quartic surface in the three dimensional projective space.     

Supersingular quotients of Fermat curves

We give formulas for the genera of all the possible quotient of a Fermat curve by a group of automorphisms in characteristic zero and for many classes of quotient curves also in positive characteristic. We use those results for giving evidence to the conjecture that, in any fixed positive characteristic, there should exist supersingular curves for any genus.     

Generalized theta functions and the Deuring polynomial

In this paper we present a calculation to study a relationship between the generalized theta function and the Deuring polynomial.     

Aspects of the planetary Birkhoff normal form

The discovery of the Birkhoff normal form around circular, co-planar motions for the planetary system opened new insights and hopes for the comprehension of the dynamics of this problem. Remarkably, it allowed to give a direct proof (after the proof in [18]) of the celebrated Arnold’s Theorem [5] on the stability of planetary motions. In this paper, after reviewing the story of the proof of this theorem, we focus on technical aspects of this normal form. We develop an asymptotic formula for it that may turn to be useful in applications. Then we provide two simple applications to the three-body problem: we prove that the “density” of the Kolmogorov set of the spatial three-body problem does not depend on eccentricities and the mutual inclination but depends only on the planets’ masses and the separation among semi-axes (going in the direction of an assertion by V. I.Arnold [5]) and, using Nehoro?ev Theory [33], we prove, in the planar case, stability of all planetary actions over exponentiallylong times, provided mean-motion resonances are excluded. We also briefly discuss difficulties for full generalization of the results in the paper.     

The multiplicativity of the smith normal form

The problem of multiplicativity of the Smith normal form of nonsingular matrices over a principal ideal rings is investigated.     

Multi-output dependent observability normal form

This article gives a sufficient condition which guarantees the existence of a reference frame in which a multi-output nonlinear system is linearizable with a linear part depending on its outputs. Our method is based on the design of a reference frame associated with nonlinear observable systems. Moreover, we give a generalization of the result obtained in [D. Boutat, G. Zheng, J.P. Barbot, Observer error linearization multi-output depending, in: Proc. of IEEE CDC 06, 2006] and [G. Zheng, D. Boutat, J.P. Barbot, Single output-dependent observability normal form, SIAM Journal on Control and Optimization 46 (6) (2007)]. And some examples are given in order to illuminate our thinking.     

Symbolic computation of normal form for Hopf bifurcation in a retarded functional differential equation with unknown parameters

Based on the normal form theory for retarded functional differential equations by Faria and Magalhães, a symbolic computation scheme together with the Maple program implementation is developed to compute the normal form of a Hopf bifurcation for retarded functional differential equations with unknown parameters. Not operating as the usual way of computing the center manifold first and normal form later, the scheme features computing them simultaneously. Great efforts are made to package this task into one Maple program with an input interface provided for defining different systems. The applicability of the Maple program is demonstrated via three kinds of delayed dynamic systems such as a delayed Liénard equation, a simplified drilling model and a delayed three-neuron model. The effectiveness of Maple program is also validated through the numerical simulations of those three systems.     

A constructive generalised Goursat normal form

We provide necessary and sufficient conditions on the derived type of a vector field distribution V in order that it be locally equivalent to a partial prolongation of the contact distribution , on the 1st order jet bundle of maps from R to Rq, q?1. This result fully generalises the Goursat normal form from the theory of exterior differential systems. Our proof is constructive: it is proven that if V is locally equivalent to a partial prolongation of then the explicit construction of contact coordinates algorithmically depends upon the integration of a sequence of geometrically defined and algorithmically determined integrable Pfaffian systems on M. Though the tools required are rather different, our main theorem may be regarded as a generalisation of the work of R.B. Gardner and W.F. Shadwick on the feedback linearisation of autonomous nonlinear control systems.     

Smith normal form of some distance matrices

We determine the Smith normal form of the distance matrices of unicyclic graphs and of the wheel graph with trees attached to each vertex.     

Estimating the parameters of a truncated normal distribution

Let ø(x) be a truncated normal pdf over the interval [a,b], that is, assume ø(x)=exp[-(x–μ)²/2σ²]/∝^b_aexp[-(x–μ)²/2σ²]dx for - ∞<a?x?b?< + ∞ and zero elsewhere. Suppose that X₁,X₂,…,X_n is a random sample of size n from this truncated distribution. Using known properties of exponential families of distributions and the system of Legendre polynomials over the interval [-1,1], we examine the maximum likelihood estimation of the parameters μ and σ².     

A new normal form of Boolean functions

A new normal form of Boolean functions based on the sum (mod 2), product and negation is presented. Let n = {1, 2,…, n}, let A_s be the family of s-element subsets of a set A and let π_a?φx_a = 1. Then every Boolean function ?(x₁,x₂,…,x_n) has a normal form

$\text{?(x}{}_1\text{,x}{}_2\text{,…,x}{}_{\mathrm{n}}\text{=}\text{∑}\text{s=0}\text{n}\text{⊕}\text{Π}\text{A∈n}{}_{\mathrm{s}}\text{1⊕d}{}_{\mathrm{A}}\text{Π}\text{a∈A}\text{x}{}_{\mathrm{a}}$

with unique coefficients d_A? {0, 1}. A transformation of Galois normal form into the present normal form is also shown.     

On the computation of the Smith normal form of compound matrices

Compound matrices are encountered in many fields such as Matrix Theory, Systems Theory, Control Theory, etc. In the present paper we develop an efficient algorithm computing the Smith normal form of compound matrices. This algorithm is based on a new theorem establishing an equivalence relation between the Smith normal form of the compounds of a given matrix and the compounds of the Smith normal form of the given matrix. This revised version was published online in June 2006 with corrections to the Cover Date.