Indices of Kummer extensions of prime degree

Let p be a prime and let ${\zeta }_p$ be a primitive p-th root of unity. For a finite extension k of $\mathbb{Q}$ containing ${\zeta }_p$, we consider a Kummer extension $L/k$ of degree p. In this paper, we show that if $k=\mathbb{Q}({\zeta }_p)$ and the class number of k is one, the index of $L/k$ is one. We also show that if $L/k$ is tamely ramified with a normal integral basis, the index is at most a power of p. In the last section, we show that there exist infinitely many cubic Kummer extensions of $\mathbb{Q}({\zeta }_3)$ for both wildly and tamely ramified cases, whose integer rings do not have a power integral basis over that of $\mathbb{Q}({\zeta }_3)$.     

Unramified Kummer extensions of prime power degree

We determine, for which xR^* the Kummer extension K(Pⁿx) is unramified over K, where K is a local field of char. 0 and res.char. p, _pnK, and R is the valuation ring of K. This builds on techniques of Hasse. Actually, a much more general result concerning cyclic Galois extensions of commutative rings is proved.     

Topological degree and the Sperner lemma

In this paper, we give a new proof of Sperner's lemma, in its superstrong from, using the topological degree. Thus, we point out a relation between several methods for fixed-point theorems using either the topological degree, or the KKM lemma, or the Sperner lemma.The author would like to thank Dr. G. Leitmann for his remarks and suggestions.     

The shadowing chain lemma for singular Hamiltonian systems involving strong forces

We consider a planar autonomous Hamiltonian system :q+?V(q) = 0, where the potential V: ?² \{??}?? ? has a single well of infinite depth at some point ?? and a strict global maximum 0at two distinct points a and b. Under a strong force condition around the singularity ?? we will prove a lemma on the existence and multiplicity of heteroclinic and homoclinic orbits ?? the shadowing chain lemma ?? via minimization of action integrals and using simple geometrical arguments.     

The denormalized 3×3 lemma

The classical 3×3 lemma holds in any regular protomodular category with a zero object. It is investigated here whether there is a “denormalized” version when the category no longer has a zero object, as, for instance, any slice category of the category , or any slice category of an abelian category. The answer is actually positive in the weaker context of regular Mal'cev categories.     

Isotropic subspaces in symmetric composition algebras and Kummer subspaces in central simple algebras of degree 3

The maximal isotropic subspaces in split Cayley algebras were classified by van der Blij and Springer in Nieuw Archief voor Wiskunde VIII(3):158–169, 1960. Here we translate this classification to arbitrary composition algebras. We study intersection properties of such spaces in a symmetric composition algebra, and prove two triality results: one for two-D isotropic spaces, and another for isotropic vectors and maximal isotropic spaces. We bound the distance between isotropic spaces of various dimensions, and study the strong orthogonality relation on isotropic vectors, with its own bound on the distance. The results are used to classify maximal p-central subspaces in central simple algebras of degree p = 3. We prove various linkage properties of maximal p-central spaces and p-central elements. Analogous results are obtained for symmetric p-central elements with respect to an involution of the second kind inverting a third root of unity.     

Tensor products of cyclic algebras of degree 4 and their Kummer subspaces

We prove that the maximal dimension of a Kummer space in the generic tensor product of n cyclic algebras of degree 4 is 4n + 1.     

The Schwarz-Pick lemma and Julia lemma for real planar harmonic mappings

The classical Schwarz-Pick lemma and Julia lemma for holomorphic mappings on the unit disk D are generalized to real harmonic mappings of the unit disk, and the results are precise. It is proved that for a harmonic mapping U of D into the open interval I = (?1, 1), $$\frac{{\Lambda _U (z)}} {{\cos \tfrac{{U(z)\pi }} {2}}} \leqslant \frac{4} {\pi }\frac{1} {{1 - \left| z \right|^2 }}$$ holds for z ∈ D, where Λ _U (z) is the maximum dilation of U at z. The inequality is sharp for any z ∈ D and any value of U(z), and the equality occurs for some point in D if and only if $U(z) = \tfrac{4} {\pi }\operatorname{Re} \{ \arctan \phi (z)\}$ , z ∈ D, with a Möbius transformation φ of D onto itself.     

The averaging lemma

Averaging lemmas deduce smoothness of velocity averages, such as

from properties of . A canonical example is that is in the Sobolev space whenever and are in . The present paper shows how techniques from Harmonic Analysis such as maximal functions, wavelet decompositions, and interpolation can be used to prove versions of the averaging lemma. For example, it is shown that implies that is in the Besov space , . Examples are constructed using wavelet decompositions to show that these averaging lemmas are sharp. A deeper analysis of the averaging lemma is made near the endpoint .

     

The asymptotically optimal meshsize function for bi-p degree interpolation over rectangular elements

A method is presented to recover near optimal interpolation on finite element meshes based on information in the approximation error on an initial mesh. Only a certain class of admissable meshes with rectangular elements in the computational domains are allowed. The method attempts to reach the optimal mesh in one step from the initial mesh, and is based on the notion of meshsize function components or mesh density functions. Asymptotical results showing the optimality of the recovered meshes are given, and extensive computational verification of the method in the special case of Lagrange polynomial interpolation is provided.     

An existence lemma for groups generated by 3-transpositions

Oblatum 29-VI-1991 & 28-I-1992     

The Schwarz-Pick lemma for planar harmonic mappings

The classical Schwarz-Pick lemma for holomorphic mappings is generalized to planar harmonic mappings of the unit disk D completely. (I) For any 0 < r < 1 and 0 ρ < 1, the author constructs a closed convex domain Er,ρ such that F((z,r))  eiαEr,ρ = {eiαz : z ∈ Er,ρ} holds for every z ∈ D, w = ρeiα and harmonic mapping F with F(D)D and F(z) = w, where △(z,r) is the pseudo-disk of center z and pseudo-radius r; conversely, for every z ∈ D, w = ρeiα and w ∈ eiαEr,ρ, there exists a harmonic mapping F such that ...     

Réseaux de Kummer et surfaces K3

Sans résumé