On the Witt ring and some arithmetical invariants of quadratic forms

The Witt ring of a field serves as an effective medium to study certain arithmetical invariants of quadratic forms, such as: s = the Stufe (the least number of summands to represent ?1 as a sum of squares), q = the number of square classes, u = the maximal anisotropic dimension of a quadratic form over the given field, and h = the height (the minimal 2-power that kills the torsion subgroup of the Witt group). These invariants may also be defined over commutative rings. This paper discusses these invariants and extend the investigations to some commutative rings, e.g. valuation rings, connected semilocal rings, Prüfer rings.     

Generating Sets of Finite Transformation Semigroups PK(n,r) and K (n,r)

Let P _n and T _n be the partial transformation and the full transformation semigroups on the set {1,…, n}, respectively. In this paper we find necessary and sufficient conditions for any set of partial transformations of height r in the subsemigroup PK(n, r) = {α ∈P _n : |im (α)| ≤r} of P _n to be a (minimal) generating set of PK(n, r); and similarly, for any set of full transformations of height r in the subsemigroup K(n, r) = {α ∈T _n : |im (α)| ≤r} of T _n to be a (minimal) generating set of K(n, r) for 2 ≤ r ≤ n ? 1.     

函数芽的有限R*r(S; n)-决定性

光滑映射芽的有限决定性是奇点理论中一个重要专题．对函数芽的有限决定性问题，主要是在右等价群及其一些子群作用下来讨论的．本文在[1]和[4]的基础上讨论函数芽在右等价群的正规子群Rr^*(S；n)作用下的有限决定性，并组出函数芽有限Rr^*(S；n)-决定的一个充分必要条件．     

An algorithm for computingSU(n) invariants

An algorithm and its computer implementation for calculating the values of three types of invariants in any irreducible representation ofSU(n) are described. It is not necessary to calculate the structure constants or the matrix elements of generators. Computing times forn10 are quite reasonable.     

(r n−1 ,r) Hjelmslev-Ebenen des Typsn

Ohne ZusammenfassungHerrn Günter Pickert zum 60. Geburtstag gewidmetDer Autor dankt der Stiftung Volkswagenwerk für die Unterstützung während der Abfassung dieser Arbeit     

Second-order differential invariants of the rotation group O(n) and of its extensions: E(n), P(1, n), G(1, n)

Functional bases of second-order differential invariants of the Euclid, Poincaré, Galilei, conformal, and projective algebras are constructed. The results obtained allow us to describe new classes of nonlinear many-dimensional invariant equations.     

半群PO_(n,r)的秩

设POn为Xn上的保序部分变换半群.对任意的2≤r≤n一1,考虑半群PO_(n,r)={α∈PO_n:Im(α)■[r]}([r]={1,2,…,r}),证明了PO_(n,r)的秩为Σn-1k=r(nk)((k-1)(r-1))+r-1.     

Explicit norm one elements for ring actions of finite abelian groups

It is known that the norm map N _G for the action of a finite groupG on a ringR is surjective if and only if for every elementary abelian subgroupU ofG the norm map N _U is surjective. Equivalently, there exists an elementx _G ∈R satisfying N _G (x _G )=1 if and only if for every elementary abelian subgroupU there exists an elementx _U ∈R such that N _U (x _U )=1. When the ringR is noncommutative, it is an open problem to find an explicit formula forx _G in terms of the elementsx _U . We solve this problem when the groupG is abelian. The main part of the proof, which was inspired by cohomological considerations, deals with the case whenG is a cyclicp-group. Supported by TMR-Grant ERB FMRX-CT97-0100 of the European Union.     

半群K(n,r)中的幂等生成元

设Singn是由一个n元集上的所有奇异变换所构成的奇异变换半群，I是由Singn中一些亏数为1的幂等元组成的集合，Howie利用有向图证明了：I是Singn的一个生成集当且仅当与其相应的有向图D（I）是强连通的完全图，本文利用多重有向图将这一结果推广到Singn的每个理想K（,r)上。     

Blocking Sets in PG(r, q n )

Let $\mathcal S$ be a Desarguesian (n – 1)-spread of a hyperplane Σ of PG(rn, q). Let Ω and ${\bar B}$ be, respectively, an (n – 2)-dimensional subspace of an element of $\mathcal S $ and a minimal blocking set of an ((r – 1)n + 1)-dimensional subspace of PG(rn, q) skew to Ω. Denote by K the cone with vertex Ω and base ${\bar B}$ , and consider the point set B defined by $$B=\left(K\setminus\Sigma\right)\cup \{X\in \mathcal S\, : \, X\cap K\neq \emptyset\}$$ in the Barlotti–Cofman representation of PG(r, q ⁿ ) in PG(rn, q) associated to the (n – 1)-spread $\mathcal S$ . Generalizing the constructions of Mazzocca and Polverino (J Algebraic Combin, 24(1):61–81, 2006), under suitable assumptions on ${\bar B}$ , we prove that B is a minimal blocking set in PG(r, q ⁿ ). In this way, we achieve new classes of minimal blocking sets and we find new sizes of minimal blocking sets in finite projective spaces of non-prime order. In particular, for q a power of 3, we exhibit examples of r-dimensional minimal blocking sets of size q ⁿ⁺² + 1 in PG(r, q ⁿ ), 3 ≤ r ≤ 6 and n ≥ 3, and of size q ⁴ + 1 in PG(r, q ²), 4 ≤ r ≤ 6; actually, in the second case, these blocking sets turn out to be the union of q ³ Baer sublines through a point. Moreover, for q an even power of 3, we construct examples of minimal blocking sets of PG(4, q) of size at least q ² + 2. From these constructions, we also get maximal partial ovoids of the hermitian variety H(4, q ²) of size q ⁴ + 1, for any q a power of 3.     

Explicit Plancherel formula for the p-adic group GL(n)

We provide an explicit Plancherel formula for the p-adic group GL(n). We determine explicitly the Bernstein decomposition of Plancherel measure, including all numerical constants. We also prove a transfer-of-measure formula for GL(n). To cite this article: A.-M. Aubert, R. Plymen, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Consecutive-(r,f,k)out-of-n:F和Consecutive(f,g)-out-of-(r,n):F线性系统的可靠性

Consecutive-(r,f,k)-out-of-n:F系统由n个单元顺序连结而成,仅当在连续的r个单元中,至少有f个失效或者至少连续k个失效,整个系统才失效;而Consecutive-(f,g)-out-of-(r,n):F系统由n个单元顺序连结而成,仅当在整个系统中至少有f个失效或者在连续的r个单元中,至少有g个失效,整个系统才失效。本文运用马氏链嵌入方法,在单元之间相互独立以及单元之间马氏相关这两种情况下,给出线性系统的可靠性。     

Ternary forms as invariants of Markoff forms and other SL2 (Z)-bundles

Consider the free group Γ = {A,B} generated by matrices A, B in SL₂(Z). We can construct a ternary form Φ(x,y,z) whose GL₃(Z) equivalence class is invariant, as it depends on Γ and not the choice of generators. If Γ is the commutator of SL₂(Z), then the generating matrices have fixed points corresponding to different fields and inequivalent Markoff forms, but they are all biuniquely determined by Φ = -z²+ y(2x+y+z) to within equivalence. When referred to transformations A, B of the upper half plane, this phenomenon is interpreted in terms of inequivalent homotopy elements which are primitive for the perforated torus.