Noether’s problem for GL(2, 3)

We prove that Noether’s problem has an affirmative answer for the group GL(2, 3), over every field K. In particular, the group admits a generic polynomial over . As a consequence, so does the group . Research was partially supported by MCYT grant BFM2003-01898.     

A Description of an algebraic automorphism in terms of the radical of its minimal polynomial

It is shown in [5, Theorem 3] that if s is an algebraic automorphism of a k-vector space V with minimalpolynomial μ(T) ∈ K[T], then the extension σ of s to a k-automorphism of the field of quotients of the symmetric k-algebra k(V) of V is completely determined by μ(T) (and dim_kV). In Theorem 4 of this article, we show that σ is almost completely determined by the radical μ₀(T) of μ(T) and we see in particular that if μ(T) is separable then the rationality of (the fixed field of) σ depends only on μ₀(T). In Theorem 5, the rationality of σ is established under certain assumptions on the Galois group of μ₀(T).     

A remark on a theorem of Bavula on Lüroth extensions

Using a theorem of Roquette-Ohm [P. Roquette, Isomorphisms of generic splitting fields of simple algebras, J. Reine Angew. Math. 214-215 (1964) 207-226 and J. Ohm, On subfields of rational function fields, Arch. Math. 42 (1984) 136-138], we indicate a short proof that a rational extension of a Lüroth extension of a field k is a Lüroth extension of k. This assertion was proved by Bavula [V. Bavula, Lüroth field extensions, J. Pure Appl. Algebra 199 (2005) 1-10] with the added hypothesis that .     

Field Theoretic Cogalois Theory via Abstract Cogalois Theory

Abstract Cogalois Theory for arbitrary profinite groups was initiated by T. Albu and ?.A. Basarab [An Abstract Cogalois Theory for profinite groups, J. Pure. Appl. Algebra 200 (2005) 227-250]. The aim of this paper is twofold: firstly, to present the abstract group theoretic versions of various types of Kummer field extensions, and secondly, to show how some basic results of the (Field Theoretic) Cogalois Theory can be very easily deduced from their abstract versions.     

Parametric tail copula estimation and model testing

Parametric models for tail copulas are being used for modeling tail dependence and maximum likelihood estimation is employed to estimate unknown parameters. However, two important questions seem unanswered in the literature: (1) What is the asymptotic distribution of the MLE and (2) how does one test the parametric model? In this paper, we answer these two questions in the case of a single parameter for ease of illustration. A simulation study is provided to investigate the finite sample performance of the proposed estimator and test.     

Writing certain commutators as products of cubes in free groups

Let Cu(γ) be the minimal number of cubes required to express an element γ of a free group F. We establish a method for showing that certain equations do not have solutions in free groups. Using it, we find Cu(γ) for certain elements of the derived subgroup of F. If is the wreath product of F by the infinite cyclic group, we also show that every element of W′ is a product of at most one commutator and three cubes in W.     

Connected components of attractors and other stable sets

Summary We study the action of a continuous mapf on a locally connected metric space, with an emphasis on the action induced byf on the space of connected components of an attractorA or a stable setY off. The main result for attractors is that, ifA is an attractor in the sense of C. Conley, thenf permutes the components ofA and that the permutation is cyclic if and only if the attractor is indecomposable. This is related to similar results obtained recently by Buescu and Stewart. In the case of an indecomposable stable invariant setY we show that either the action off on the components ofY is as described above for the action on the attractor, or elseY has infinitely many components, the action off on them is a generalized adding machine, so that there are no periodic points inY. Results along these lines are due to Buescu and Stewart and to Melbourne, Dellnitz and Golubitsky.     

Mixed boundary value problems for concentric cylinders—an algebraic operator formulation

Unitary operators are introduced which act on the Fourier transforms of the boundary values on concentric cylinders of a harmonic function in E ₃. The boundary values on the concentric cylinders are determined from given mixed boundary conditions on the same cylinders, and the solution for them is expressed explicitly and simply through the use of these unitary operators plus certain other self adjoint positive operators. The boundary values on the cylinders then determine the harmonic function everywhere in E ₃.     

Generic polynomials are descent-generic

Let be a generic polynomial for a group G in the sense that every Galois extension N/L of infinite fields with group G and K≤L is given by a specialization of g(X). We prove that then also every Galois extension whose group is a subgroup of G is given in this way. Received: 15 January 2001     

Remark on infinite unramified extensions of number fields with class number one

We modify an idea of Maire to construct biquadratic number fields with small root discriminants, class number one, and having an infinite, necessarily non-solvable, strictly unramified Galois extension.     

Sur les extensions purement inséparables

If K/k is a finite purely inseparable extension of fields, we are interested in the factorizations of K as a tensor product over k of intermediates fields of K/k. We introduce the notion of e-factorization that generalizes the notion of modular factorization. Contrary to modular factorization, K/k has always an e-factorization and its factors, when their number is maximum, are quasi-invariants.Received: 4 April 2002     

Estimation in mixed effects model with errors in variables

In this paper, we consider a linear mixed-effects model with measurement errors in both fixed and random effects and find the moment of estimators for the parameters of interest. The strong consistency and asymptotic normality of the estimators are obtained under regularity conditions. Moreover, we obtain the strong consistent estimators of the asymptotic covariance matrices involved in the limiting theory. Simulations are reported for illustration.     

On a vector space analogue of Kneser’s theorem

In a previous paper [X. Hou, K.H. Leung, Q. Xiang, A generalization of an addition theorem of Kneser, J. Number Theory 97 (2002) 1-9], the following result was established: let E⊂K be fields such that the algebraic closure of E in K is separable over E. Let A,B be E-subspaces of K such that 0<dim_EA<∞ and 0<dim_EB<∞. Then dim_EAB?dim_EA+dim_EB-dim_EH(AB), where AB is the E-space generated by {ab:a∈A,b∈B} and H(AB)={x∈K:xAB⊂AB}. The separability assumption was essential in the proof of this result. However, even without the separability assumption, no counterexample is known. The present paper shows that no counterexample can be found if dim_EA?5.     

MOMENT ESTIMATION FOR MULTIVARIATE EXTREME VALUE DISTRIBUTION

Moment estimation for multivariate extreme value distribution is described in this paper. Asymptotic covariance matrix of the estimators is given. The relative efficiencies of moment estimators as compared with the maximum likelihood and the stepwise estimators are computed. We show that when there is strong dependence between the variates, the generalized variance of moment estimators is much lower than the stepwise estimators. It becomes more obvious when the dimension increases.     

Asymptotic properties of conditional maximum likelihood estimator in a certain exponential model

The conditional maximum likelihood estimator is suggested as an alternative to the maximum likelihood estimator and is favorable for an estimator of a dispersion parameter in the normal distribution, the inverse-Gaussian distribution, and so on. However, it is not clear whether the conditional maximum likelihood estimator is asymptotically efficient in general. Consider the case where it is asymptotically efficient and its asymptotic covariance depends only on an objective parameter in an exponential model. This remand implies that the exponential model possesses a certain parallel foliation. In this situation, this paper investigates asymptotic properties of the conditional maximum likelihood estimator and compares the conditional maximum likelihood estimator with the maximum likelihood estimator. We see that the bias of the former is more robust than that of the latter and that two estimators are very close, especially in the sense of bias-corrected version. The mean Pythagorean relation is also discussed.