An introduction to second degree forms

We are dealing with orthogonal sequences with respect to forms verifying a second degreee equation, i.e. that its formal Stieltjes functionS(u)(z) satisfies a quadratic equation of the formB(z)S ²(u)(z)+C(z)S(u)(z)+D(z)=0, whereB, C, D are polynomials. Various algebraic properties are given, especially those concerning the quasi-orthogonality of associated sequences. A classification is outlined. Some examples are quoted. In particular, we give the representation of Tchebychev co-recursive forms for any complex value of the parameter.     

An introduction to second degree forms

We are dealing with orthogonal sequences with respect to forms verifying a second degreee equation, i.e. that its formal Stieltjes functionS(u)(z) satisfies a quadratic equation of the formB(z)S ²(u)(z)+C(z)S(u)(z)+D(z)=0, whereB, C, D are polynomials. Various algebraic properties are given, especially those concerning the quasi-orthogonality of associated sequences. A classification is outlined. Some examples are quoted. In particular, we give the representation of Tchebychev co-recursive forms for any complex value of the parameter.     

A family of second degree semi-classical forms of class <Emphasis Type="Italic">s</Emphasis>=1

Let u(β ₀) be the regular form fulfilling (u(β ₀))_2n+1=β ₀(u(β ₀))_2n ,n≥0 where β ₀ is an arbitrary complex number in such a way that for β ₀=0 one has the symmetric forms. Recently, the second degree symmetric semi-classical forms (when β ₀=0) of class s≤1 were determined. In this paper, we determine all the semi-classical forms u(β ₀) of class s=1, which are also of second degree when β ₀≠0.     

Groups with a character of large degree

Let be a finite group of order and a simple -module of dimension . For some nonnegative number , we have . If is small, then the character of has unusually large degree. We fix and attempt to classify such groups. For we give a complete classification. For any other fixed we show that there are only finitely many examples.

     

A family of centered forms for a polynomial

The general centered form for multi-variate polynomials is investigated and a computing procedure is proposed that results in a certain superset. Based on this procedure the optimal centered forms for monomials and for some special cases of polynomials are investigated.     

A class of finite groups with complete character degree graphs

In this paper, we construct a new class of finite groups whose common divisor graphs are complete graphs, while there is no prime dividing all the nontrivial degrees.     

A sphere in a second degree polynomial creeping flow parallel to a wall

Comprehensive results are provided for the creeping flow arounda spherical particle in a viscous fluid close to a plane wall,when the external velocity is parallel to the wall and variesas a second degree polynomial in the coordinates. By linearityof Stokes equations, the solution is a sum of flows for typicalunperturbed flows: a pure shear flow, a ‘modulated shearflow’, for which the rate of shear varies linearly inthe direction normal to the wall, and a quadratic flow. Solutionsconsidered here use the bipolar coordinates technique. Theycomplement the accurate results of Chaoui and Feuillebois (2003)for the pure shear flow. The solution of Goren and O'Neill (1971)for the quadratic flow is reconsidered and a new analyticalsolution is derived for the ambient modulated shear flow. Theperturbed flow fields for these two cases are presented in detailand discussed. Results for the force and torque friction factorsare provided with a 5 x 10^–17 accuracy as a reference.For the quadratic flow, there is a force and a torque on a fixedsphere. A minimum value of the torque is found for a gap ofabout 0·18a, where a is the sphere radius. This minimumis interpreted in term of the corresponding flow structure.For the modulated shear flow, there is only a torque. The freemotion of a sphere in an ambient quadratic flow is also determined.     

Indecomposable forms of higher degree

All nondegenerate indecomposable forms of higher degree over a perfect field k can be realized as traces of nondegenerate absolutely indecomposable forms of higher degree over a suitable algebraic field extension of k. With the help of trace forms of certain nonassociative algebras we construct classes of indecomposable forms of degree d≥3.     

A characterization of second order efficiency in a curved exponential family

Summary The optimality of estimation method is investigated in a curved exponential family. A risk function, which is an extension of a residual sum of squares in regression analysis, is introduced. It is shown that second order efficiency of an estimation method is equivalent to attain the minimum among limiting risks of all estimation methods.     

Some characterizations of quasi-symmetric designs with a spread

The designPG ₂ (4,q) of the points and planes ofPG (4,q) forms a quasi-symmetric 2-design with block intersection numbersx=1 andy=q+1. We give some characterizations of quasi-symmetric designs withx=1 which have a spread through a fixed point. For instance, it is proved that if such a designD is also smooth, thenDPG ₂ (4,q).     

Estimates for a family of multi-linear forms

We consider a special class of the multi-linear forms studied by Brascamp and Lieb (1976) [7]. For these forms, we give a characterization of the Lp spaces for which the form is bounded. This characterization appears in earlier work of Barthe (1998) [1]. We use this characterization to study a non-linear map that arises in scattering theory.     

Projective character degree patterns of 2-groups

Abstract

We discuss the prospects for finding a “core class,” i.e., a well-behaved class of non-free abelian groups of cardinality ?₁ such that every non-free abelian group of cardinality ?₁ has a subgroup in the core class.