ADDITIVE FAMILIES OF INVARIANTS OF FORMS OF DEGREE d

Abstract

Let d be a positive integer, and F be a field of characteristic zero. Suppose that for each positive integer n, I _n, is a GL _n,(F)- invariant of forms of degree d in x₁, …, x _n, over F. We call {I _n} an additive family of invariants if I _p+q (f ⊥ g) = I _p(f).I _q(g) whenever f; g are forms of degree d over F in x _l, …, x _p; …, x _q respectively, and where (f ⊥ g)(x _l, …, x _p+q) = f(x ₁, …, x _p,) + g (x _p+1, …, x _p+q). It is well-known that the family of discriminants of the quadratic forms is additive. We prove that in odd degree d each invariant in an additive family must be a constant. We also give an example in each even degree d of a nontrivial family of invariants of the forms of degree d. The proofs depend on the symbolic method for representing invariants of a form, which we review.     

NOTES ON NILPOTENCY OF NIL SUBRINGS OF ENDOMORPHISM RINGS OF MODULES

Abstract

A family K of right R-modules is called a natural class if K is closed under submodules, direct sums, infective hulls, and isomorphic copies. The main result of this note is the following: Let K be a natural class on Mod-R and M ε K. If M satisfies a.c.c. (or d.c.c.) on the set of submodules {N ? M: M/N ε K}, then each nil subring of End(M_R ) is nilpotent.     

COMPARISON OF PRODUCTS OF THE VARIETIES OF LIE ALGEBRAS

Abstract

In Fortes (2001), we introduced a notion of order for associative pairs and we obtained a Goldie-like characterization of left orders in a semiprime pair coinciding with its socle. In this paper, we take up again that notion of order to establish a Faith-Utumi theorem, which studies left orders in a prime pair coinciding with its socle.     

函数集的完全性

1.:敲g(x)篇〔一二,二]上之非降的有界缝差两数,业具有性鬓(K)s‘二一0,一。(:);f--:.,。g。尹(:)!d:一郁匕,(‘一”,”;dg)篇在〔一二,司上定羲业且满足修件:,一{户,(柳dg(·)}青<一,>l的可测蝮值函数族{f(幻}.封龄一徊乙“(一二,侧d刃中之子族凌B(幻},若由f(劣)(乙,(一二,二:dg),夕>1生+上夕q=1,及f--:ha”“’“““’一0纷{B(x)}之任何B(哟成立必滇致f(幻在〔一二,司上规乎虚虚等焚零则释{B(x)}在乙“(一二,侧dg)中完全. 函数族的完全性是舆函数横造的一些简题很有阴保的.徙【l]我们知道{e‘”}豁。是在乙,(一二,州dg),,>1,中完全的,…     

A DESCRIPTION OF NORMAL SEMIGROUPS OF ENDOMORPHISMS OF PROPER INDEPENDENCE ALGEBRAS

Let A be a strong independence algebra of finite rank with at most one constant, and let G he the group of automorphisms of A. Let α be a singular endomorphism of α^G = 〈{α} U G〉. We describe the elements of α^G and give additional characterisations when A is a proper independence algebra and G is a periodic group.     

PROPERTIES OF FRACTIONAL κ-FACTORS OF GRAPHS

In this paper the properties of some maximum fractional [0, κ]-factors of graphs are presented. And consequently some results on fractional matchings and fractional 1-factors are generalized and a characterization of fractional k-factors is obtained.     

REPRESENTATIONS OF A CLASS OF DRINFELD'S DOUBLES

Let k be a field and A_n(ω) be the Taft's n²-dimensional Hopf algebra. When n is odd, the Drinfeld quantum double D(A_n(ω)) of A_n(ω) is a ribbon Hopf algebra. In the previous articles, we constructed an n₄-dimensional Hopf algebra H_n(p, q) which is isomorphic to D(A_n(ω)) if p ≠ 0 and q = ω⁻¹ , and studied the irreducible representations of H_n(1, q) and the finite dimensional representations of H₃(1, q). In this article, we examine the finite-dimensional representations of H_n(l q), equivalently, of D(A_n(ω)) for any n ≥ 2. We investigate the indecomposable left H_n(1, q)-module, and describe the structures and properties of all indecomposable modules and classify them when k is algebraically closed. We also give all almost split sequences in mod H_n(1, q), and the Auslander-Reiten-quiver of H_n(1 q).     

ADDITIVE FAMILIES OF INVARIANTS OF SKEWSYMMETRIC TENSORS

Abstract

Let d be a positive integer and F be a field of characteristic 0. Suppose that for each positive integer n, I _n is a polynomial invariant of the usual action of GL_n (F) on Λ^d(Fⁿ), such that for t ϵ Λ^d(F ^k) and s ϵ Λ^d(F ^l), I _{k + l} (t l s) = I _k(t)I _t (s), where t ⊥ s is defined in §1.4. Then we say that {I_n} is an additive family of invariants of the skewsymmetric tensors of degree d, or, briefly, an additive family of invariants. If not all the I_n are constant we say that the family is non-trivial. We show that in each even degree d there is a non-trivial additive family of invariants, but that this is not so for any odd d. These results are analogous to those in our paper [3] for symmetric tensors. Our proofs rely on the symbolic method for representing invariants of skewsymmetric tensors. To keep this paper self-contained we expound some of that theory, but for the proofs we refer to the book [2] of Grosshans, Rota and Stein.     

RESULTS ON SUBGROUPS OF THE GROUP OF HOMEOMORPHISMS OF THE RATIONAL NUMBERS

Abstract

If G is the group of homeomorphisms of the rationals with usual topology and H ⋚ G is such that |G:H| < 2^x0 then there exists a finite, non-empty subset Y of the rational numbers such that G(Y) ⋚ H ⋚ G^{Y} where G(Y) is the group o all homeomorphisms in G fixing a neighbourhood of Y and G^{Y} is the set-wise stabilizer of Y in G.     

EXISTENCE OF PERIODIC SOLUTIONS OF A CLASS OF SECOND-ORDER NON-AUTONOMOUS SYSTEMS

In this paper, we are concerned with the existence of periodic solutions of second-order non-autonomous systems. By applying the Schauder''s fixed point theorem and Miranda''s theorem, a new existence result of periodic solutions is established.     

A NOTION OF COMPLETENESS OF TOPOLOGICAL STRUCTURES

A connector U on a space S is a function from S to the power set of S such that each x in s belongs to its image. The image of x is denoted by xU. In other words, the relation {(x,y): y ϵ xU, x ϵ S) is a reflexive binary relation. A space with a certain set of connectors is a generalization of topological spaces as well as uniform spaces. In this paper, a notion of completeness of such a space is introduced. This completeness corresponds to completeness of uniform spaces if a set of cannectors meets the conditions of uniformity. Compactness of topological Spaces is a special case of the completeness.     

劳勃生的特殊星像函数和特殊凸像函数

<正> 设函数w在单位圆 E_z:|z|<1上是正则的.假如f(z)在 E_z上是单叶的,那末 D_f=f(E_z)是 w 平面上单叶的区域.记这种单叶函数f(z)的全体为 S_p,S_1=S.若 D_f 以原点 w=0 为星形中心,就是说若 w_0∈D_f则缐段■整个地落在区域 D_f 中,称这种函数 f(z)是 E_z 中的星像函数,其特徵是在 E_z     

ON THE INCOMPLETENESS OF A DESCENDING CHAIN OF EXTENSIONS OF IMPLICATIONAL S5

C5.ω is obtained by adding, schematically, to the strict-implicational fragment C5 of S5 the axiom ((p → q) → (q → p)) → (q → p). This paper presents a fully general proof that neither C5.ω nor any of a descending chain of its extensions is complete with respect to any class of frames, correcting the garbled details of a version skeched in an earlier paper (same Zeitschrift 31 (1985), 201-208).     

ON THE IRREDUCIBILITY OF THE DETERMINANT OF THE MATRIX OF MOVING PLANES

ABSTRACT

We show that, for generic bihomogeneous polynomials, the determinant of the matrix of moving planes is irreducible.     

PROPERTIES OF CHARACTERISTIC FUNCTIONS AND EXISTENCE OF LIMIT CYCLES OF LIENARD''S EQUATION

In this paper, the author considers Lienard''s equation, studies the properties of so-called characteristic functions and gives three theorems which ensure that the equation has at least one limit cycle. The theorems generlize Filippov''s Theorem which is a repesentative result, Dragilev''s Theorem, Theorem 1 2 of [6], Theorem 9 of [8] and Theorems 1, 2 of [9] respectively.     

BOUNDS FOR INDICES OF NILPOTENCY OF FINITELY GENERATED NIL RINGS OF FINITE INDEX

Abstract

An improved bound is given for the index of nil-potency of a finitely generated nil ring of index n in terms of the index of nilpotency of the ideal generated by T^m where m = [n/2] and T is a m-subset of the set of generators. If m = 3 it is proved that T¹⁰ is contained in an ideal generated by twenty-seven cubes and this is applied to get bounds for the index of nilpotency of a finitely generated nil ring of index 6 or 7, bounds which are less than one hundredth of the bounds we obtained in a previous paper.