Degree 2n Research Articles

On the construction of normal wildly ramified extensions over Q 2

It is well known that a finite totally ramified extension of a local field can be generated by a uniformising element the minimal polynomial of which is also Eisenstein. The quadratic and the quartic normal totally ramified extensions of Q2 are well known and well characterized. In this note we characterize the Eisenstein polynomials of degree 4 with coefficients in Z2 that define normal totally ramified extensions of Q2. Furthermore we give some necessary conditions for the cyclic case of degree 2n. Also examples are given.

Failure detection and isolation in robotic manipulators using joint torque sensors

SUMMARYReliability of any model-based failure detection and isolation (FDI) method depends on the amount of uncertainty in a system model. Recently, it has been shown that the use of joint torque sensing results in a simplified manipulator model that excludes hardly identifiable link dynamics and other nonlinearities such as friction, backlash, and flexibilities. In this paper, we show that the application of the simplified model in a fault detection algorithm increases reliability of fault monitoring system against modeling uncertainty. The proposed FDI filter is based on a smooth velocity observer of degree 2n where n stands for the number of manipulator joints. No velocity measurement and assumptions on smoothness of faults are used in the fault detection process. The paper focuses on actuator faults and investigates the effect of torque sensor noise on threshold selection. The FDI filter is further improved to become robust against an unknown bias in torque sensor reading. The effect of position sensor noise together with position sensor faults are also investigated. Simulation example on a 6-degrees of freedom manipulator is carried out to illustrate the performance of the proposed FDI method.

Maass relations in higher genus

For an arbitrary even genus 2n we show that the subspace of Siegel cusp forms of degree 2n generated by Ikeda lifts of elliptic cusp forms can be characterized by certain linear relations among Fourier coefficients. This generalizes the classical Maass relations in degree two to higher degrees.

New cubature formulae and hyperinterpolation in three variables

A new algebraic cubature formula of degree 2n+1 for the product Chebyshev measure in the d-cube with ≈nd/2d−1 nodes is established. The new formula is then applied to polynomial hyperinterpolation of degree n in three variables, in which coefficients of the product Chebyshev orthonormal basis are computed by a fast algorithm based on the 3-dimensional FFT. Moreover, integration of the hyperinterpolant provides a new Clenshaw-Curtis type cubature formula in the 3-cube.

Discriminant of Symplectic Involutions

We define an invariant of torsors under adjoint linear algebraic groups of type C_n-equivalently, central simple algebras of degree 2n with symplectic involution-for n divisible by 4 that takes values in H^3(F, mu_2). The invariant is distinct from the few known examples of cohomological invariants of torsors under adjoint groups. We also prove that the invariant detects whether a central simple algebra of degree 8 with symplectic involution can be decomposed as a tensor product of quaternion algebras with involution.

Explicit orthogonal polynomials for reciprocal polynomial weights on $(-\infty ,\infty )$

Let S be a polynomial of degree 2n + 2, that is, positive on the real axis, and let w = 1/S on (―∞, ∞). We present an explicit formula for the nth orthogonal polynomial and related quantities for the weight w. This is an analogue for the real line of the classical Bernstein-Szegő formula for (―1,1).

FINITE TYPE INVARIANTS AND MILNOR INVARIANTS FOR BRUNNIAN LINKS

A link L in the 3-sphere is called Brunnian if every proper sublink of L is trivial. In a previous paper, Habiro proved that the restriction to Brunnian links of any Goussarov–Vassiliev finite type invariant of (n + 1)-component links of degree < 2n is trivial. The purpose of this paper is to study the first nontrivial case. We show that the restriction of an invariant of degree 2n to (n + 1)-component Brunnian links can be expressed as a quadratic form on the Milnor link-homotopy invariants of length n + 1.

The zeros of Dedekind zeta functions and class numbers of CM-fields

Let F'/F be a finite normal extension of number fields with Galois group Gal(F'/F). Let Χ be an irreducible character of Gal(F'/F) of degree greater than one and L(s, x) the associated Artin L-function. Assuming the truth of Artin's conjecture, we have explicitly determined a zero-free region about 1 for L(s,x). As an application we show that, for a CM-field K of degree 2n with solvable normal closure over Q, if n > 370 as well as n ∉ {384,400,416,448,512}, then the relative class number of K is greater than one.

Euler-Rodrigues and Cayley Formulae for Rotation of Elasticity Tensors

It is well known that rotation in three dimensions can be expressed as a quadratic in a skew symmetric matrix via the Euler-Rodrigues formula. A generalized Euler-Rodrigues polynomial of degree 2n in a skew symmetric generating matrix is derived for the rotation matrix of tensors of order n. The Euler-Rodrigues formula for rigid body rotation is recovered by n = 1. A Cayley form of the nth-order rotation tensor is also derived. The representations simplify if there exists some underlying symmetry, as is the case for elasticity tensors such as strain and the fourth-order tensor of elastic moduli. A new formula is presented for the transformation of elastic moduli under rotation: as a 21-vector with a rotation matrix given by a polynomial of degree 8. Explicit spectral representations are constructed from three vectors: the axis of rotation and two orthogonal bivectors. The tensor rotation formulae are related to Cartan decomposition of elastic moduli and projection onto hexagonal symmetry.

The range of multiplicative functions on ℂ[x ], ℝ[x ] and ℤ[x

Mahler's measure is generalized to create the class of {\it multiplicative distance functions}. These functions measure the complexity of polynomials based on the location of their zeros in the complex plane. Following work of S.-J. Chern and J. Vaaler in \cite{chern-vaaler}, we associate to each multiplicative distance function two families of analytic functions which encode information about its range on \C[x] and \R[x]. These {\it moment functions} are Mellin transforms of distribution functions associated to the multiplicative distance function and demonstrate a great deal of arithmetic structure. For instance, we show that the moment function associated to Mahler's measure restricted to real reciprocal polynomials of degree 2N has an analytic continuation to rational functions with rational coefficients, simple poles at integers between -N and N, and a zero of multiplicity 2N at the origin. This discovery leads to asymptotic estimates for the number of reciprocal integer polynomials of fixed degree with Mahler measure less than T as $T \to \infty$. To explain the structure of this moment functions we show that the real moment functions of a multiplicative distance function can be written as Pfaffians of antisymmetric matrices formed from a skew-symmetric bilinear form associated to the multiplicative distance function.

More limit cycles than expected in Liénard equations

The paper deals with classical polynomial Lienard equations, i.e. planar vector fields associated to scalar second order differential equations x+ f(x)x' + x = 0 where f is a polynomial. We prove that for a well-chosen polynomial f of degree 6, the equation exhibits 4 limit cycles. It induces that for n ≥ 3 there exist polynomials f of degree 2n such that the related equations exhibit more than n limit cycles. This contradicts the conjecture of Lins, de Melo and Pugh stating that for Lienard equations as above, with f of degree 2n, the maximum number of limit cycles is n. The limit cycles that we found are relaxation oscillations which appear in slow-fast systems at the boundary of classical polynomial Lienard equations. More precisely we find our example inside a family of second order differential equations ex + f μ (x)x' + x = 0. 0. Here, f μ is a well-chosen family of polynomials of degree 6 with parameter μ ∈ R 4 and e is a small positive parameter tending to 0. We use bifurcations from canard cycles which occur when two extrema of the critical curve of the layer equation are crossing (the layer equation corresponds to e = 0). As was proved by Dumortier and Roussarie (2005) these bifurcations are controlled by a rational integral computed along the critical curve of the layer equation, called the slow divergence integral. Our result is deduced from the study of this integral.

Embeddability of quadratic extensions in cyclic extensions

For an algebraic number field K we study the quadratic extensions of K which can be embedded in a cyclic extension of K of degree 2n for all natural numbers n, as well as the quadratic extensions which can be embedded in an infinite normal extension with the additive group of 2-adic integers as Galois group.

Strong Asymptotics for Multiple Laguerre Polynomials

We consider multiple Laguerre polynomials l~ of degree 2n orthogonal on (0;1) with respect to the weights x e 1x and x e 2x , where 1 < , 0 < 1 < 2, and we study their behavior in the large n limit. The analysis diers among three dieren t cases which correspond to the ratio 2= 1 being larger, smaller, or equal to some specic critical value . In this paper, the rst two cases are investigated and strong uniform asymptotics for the scaled polynomials l~(nz) are obtained in the entire complex plane by using the Deift{Zhou steepest descent method for a 3 3{matrix Riemann{Hilbert problem. Asymptotique forte pour les polyn^ omes de Laguerre multiples R esum e

On the Kontsevich integral of Brunnian links

The purpose of the paper is twofold. First, we give a short proof using the Kontsevich integral for the fact that the restriction of an invariant of degree 2n to (n+1)-component Brunnian links can be expressed as a quadratic form on the Milnor mu-bar link-homotopy invariants of length n+1. Second, we describe the structure of the Brunnian part of the degree 2n-graded quotient of the Goussarov--Vassiliev filtration for (n+1)-component links.

On the semicontinuity of ramification invariants in dimension 2

We consider a cyclic extension L/K of degree 2n of the field K = k[[T,U]] of characteristic 2. It is shown that for all sufficiently large N, the jets of order N of all curves which are not components of the ramification locus and for which the corresponding valuation of the function field has a unique extension, the valuations of the coefficients of the Inaba equation are positive, and the ramification jumps are maximal, is an open set. In the case of a general (not cyclic) extension, it is shown that the set of jets with a fixed value of the kth jump is the intersection of an open and a closed set. Bibliography: 4 titles.

Elliptic Elastic Contact Between High Order Symmetrical Surfaces

This paper proves that a generalized Hertz pressure (the product of Hertz square root and an even polynomial of degree 2n with respect to coordinates) applied over elastic half-space boundary generates a polynomial normal displacement of degree 2n+2. Polynomial surface coefficients are combinations of elliptical integrals. The equation of rigid punch surface generating this pressure is derived, as well as the conditions in which an elliptical contact occurs. For second order surfaces, n=0, these results yield all Hertz formulas, whereas new formulas are derived for contact parameters between fourth, sixth, and eight order surfaces.

A Moment Matrix Approach to Multivariable Cubature

We develop an approach to multivariable cubature based on positivity, extension, and completion properties of moment matrices. We obtain a matrix-based lower bound on the size of a cubature rule of degree 2n + 1; for a planar measure μ, the bound is based on estimating \( \rho (C): = \inf \{ {\text{rank}}(T - C):T{\text{ Toeplitz and }}T \geq C\} , \) where C:=C# [μ ] is a positive matrix naturally associated with the moments of μ. We use this estimate to construct various minimal or near-minimal cubature rules for planar measures. In the case when C = diag(c1,...,cn) (including the case when μ is planar measure on the unit disk), ρ(C) is at least as large as the number of gaps ck >ck+1.

An efficient numerical method for the solution of sliding contact problems

In this paper, an efficient numerical method to solve sliding contact problems is proposed. Explicit formulae for the Gauss–Jacobi numerical integration scheme appropriate for the singular integral equations of the second kind with Cauchy kernels are derived. The resulting quadrature formulae for the integrals are valid at nodal points determined from the zeroes of a Jacobi polynomial. Gaussian quadratures obtained in this manner involve fixed nodal points and are exact for polynomials of degree 2n − 1, where n is the number of nodes. From this Gauss–Jacobi quadrature, the existing Gauss–Chebyshev quadrature formulas can be easily derived. Another apparent advantage of this method is its ability to capture correctly the singular or regular behaviour of the tractions at the edge of the region of contact. Also, this analysis shows that once if the total normal load and the friction coefficient are given, the external moment M and contact eccentricity e (for incomplete contact) in fully sliding contact are uniquely determined. Finally, numerical solutions are computed for two typical contact cases, including sliding Hertzian contact and a sliding contact between a flat punch with rounded corners pressed against the flat surface of a semi-infinite elastic solid. These results provide a demonstration of the validity of the proposed method. Copyright © 2005 John Wiley & Sons, Ltd.

Cantor necklaces and structurally unstable Sierpinski curve Julia sets for rational maps

In this paper we consider families of rational maps of degree 2n on the Riemann sphereF λ:\(F_\lambda :\bar {\mathbb{C}} \to \bar {\mathbb{C}}\) given by $$F_\lambda (z) = z^n + \frac{\lambda }{{z^n }}$$ where\(\lambda \in \mathbb{C} - \{ 0\} \) andn>-2. One of our goals in this paper is to describe a type of structure that we call aCantor necklace that occurs in both the dynamical and the parameter plance forF λ. Roughly speaking, such a set is homeomorphic to a set constructed as follows. Start with the Cantor middle thirds set embedded on thex-axis in the plane. Then replace each removed open interval with an open circular disk whose diameter is the same as the length of the removed interval. A Cantor necklace is a set that is homeomorphic to the resulting union of the Cantor set and the adjoined open disks.

CM–fields and skew–symmetric matrices

Cohen and Odoni prove that every CM–field can be generated by an eigenvalue of some skew–symmetric matrix with rational coefficients. It is natural to ask for the minimal dimension of such a matrix. They show that every CM–field of degree 2n is generated by an eigenvalue of a skew–symmetric matrix over Q of dimension at most 4n+2. The aim of the present paper is to improve this bound.