The (classical, small quantum, equivariant) cohomology ring of the grassmannian $G\left( k,\,n \right)$ is generated by certain derivations operating on an exterior algebra of a free module of rank $n$ (Schubert calculus on a Grassmann algebra). Our main result gives, in a unified way, a presentation of all such cohomology rings in terms of generators and relations. Using results of Laksov and Thorup, it also provides a presentation of the universal factorization algebra of a monic polynomial of degree $n$ into the product of two monic polynomials, one of degree $k$.

The enumeration method for finite groups, the so-called Todd-Coxeter process, has been described in [2], [3]. Leech [4] and Trotter [5] carried out the process of coset enumeration for groups on a computer. However Mendelsohn [1] was the first to present a formal proof of the fact that this process ends after a finite number of steps and that it actually enumerates cosets in a group. Dietze and Schaps [7] used Todd-Coxeter′s method to find all subgroups of a given finite index in a finitely presented group. B. H. Neumann [8] modified Todd-Coxeter′s method to enumerate cosets in a semigroup, giving however no proofs of the effectiveness of this method nor that it actually enumerates cosets in a semigroup.

For each positive integer n, the radix representation of the complex numbers in the base —n + i gives rise to a tiling of the plane. Each tile consists of all the complex numbers representable in the base -n + i with a fixed integer part. We show that the fractal dimension of the boundary of each tile is 2 log λn/log(n2 + 1), where λn is the positive root of λ3 - (2n - 1) λ2 - (n - 1) 2λ - (n2 + 1).

We consider semilinear periodic problems with the right-hand side nonlinearity satisfying a double resonance condition between two successive eigenvalues. Using a combination of variational and degree theoretic methods, we prove the existence of at least two nontrivial solutions.

The relations between the adjoint group and the additive group of a radical ring and its nilpotency are investigated. It is shown that certain finiteness conditions carry over from the adjoint group to the additive group and that the converse holds for the class of minimax groups.

Abstract. A subset $W$ of the vertex set of a graph $G$ is called a resolving set of $G$ if for every pair of distinct vertices $u,\,v$, of $G$, there is $w\,\in \,W$ such that the distance of $w$ and $u$ is different from the distance of $w$ and $v$. The cardinality of a smallest resolving set is called the metric dimension of $G$, denoted by $\dim\left( G \right)$. The circulant graph ${{C}_{n}}\left( 1,\,2,\,.\,.\,.\,,\,t \right)$ consists of the vertices ${{v}_{0}},\,{{v}_{1\,}},\,.\,.\,.\,,{{v}_{n\,-\,1}}$ and the edges ${{v}_{i}}{{v}_{i\,+\,j}}$, where $0\,\le \,i\,\le \,n\,-\,1,1\,\le \,j\,\le \,t\,\left( 2\,\le \,t\,\le \,\left\lfloor \frac{n}{2} \right\rfloor \right)$, the indices are taken modulo $n$. Grigorious, Manuel, Miller, Rajan, and Stephen proved that $\dim\left( {{C}_{n}}\left( 1,\,2,\,.\,.\,.\,,\,t \right) \right)\,\ge \,t\,+\,1$ for $t\,<\,\left\lfloor \frac{n}{2} \right\rfloor ,\,n\,\ge \,3$, and they presented a conjecture saying that $\dim\left( {{C}_{n}}\left( 1,\,2,\,.\,.\,.\,,\,t \right) \right)\,=\,t\,+\,p\,-\,1$ for $n\,=\,2tk\,+\,t\,+\,p$, where $3\,\le \,p\,\le \,t\,+\,1$. We disprove both statements. We show that if $t\,\ge \,4$ is even, there exists an infinite set of values of $n$ such that $\dim\left( {{C}_{n}}\left( 1,\,2,\,.\,.\,.\,,t \right) \right)\,=\,t$. We also prove that $\dim\left( {{C}_{n}}\left( 1,\,2,\,.\,.\,.\,,\,t \right) \right)\,\le \,t\,+\,\frac{p}{2}$ for $n\,=\,2tk\,+\,t\,+\,p$, where $t$ and $p$ are even, $t\,\ge \,4,\,2\,\le \,p\,\le \,t$, and $k\,\ge \,1$.

In a group, a nonidentity element is called a generalized torsion element if some product of its conjugates equals the identity. We show that for many classical knots one can ûnd generalized torsion in the fundamental group of its complement, commonly called the knot group. It follows that such a group is not bi-orderable. Examples include all torus knots, the (hyperbolic) knot ${{5}_{2}}$ , and algebraic knots in the sense of Milnor.

Let $P\left( z \right)$ be a polynomial of degree not exceeding $n$ and let $W\left( z \right)\,=\,\prod\nolimits_{j=1}^{n}{\left( z\,-\,{{a}_{j}} \right)}$ where $\left| {{a}_{j}} \right|\,>\,1$, $j\,=\,1,\,2,\,.\,.\,.\,,\,n$. If the rational function $r\left( z \right)\,=\,{P\left( z \right)}/{W\left( z \right)}\;$ does not vanish in $\left| z \right|\,<\,k$, then for $k\,=\,1$ it is known that

$$\left| {{r}^{'}}\left( z \right) \right|\le \frac{1}{2}\left| {{B}^{'}}(z) \right|_{\left| z \right|=1}^{\text{Sup}}\left| r(z) \right|$$

where $B\left( Z \right)\,=\,{{{W}^{*}}\left( z \right)}/{W\left( z \right)}\;$ and ${{W}^{*}}\left( z \right)\,=\,{{z}^{n}}\overline{W\left( {1}/{\overline{z}}\; \right)}$. In the paper we consider the case when $k\,>\,1$ and obtain a sharp result. We also show that

$$\underset{\left| z \right|=1}{\mathop{\text{Sup}}}\,\left\{ \left| \frac{{{r}^{\prime }}\left( z \right)}{{{B}^{\prime }}\left( z \right)} \right|+\left| \frac{{{\left( {{r}^{*}}\left( z \right) \right)}^{\prime }}}{{{B}^{\prime }}\left( z \right)} \right| \right\}=\underset{\left| z \right|=1}{\mathop{\text{Sup}}}\,\left| r\left( z \right) \right|$$

where ${{r}^{*}}\left( z \right)\,=\,B\left( z \right)\overline{r\left( {1}/{\overline{z}}\; \right)}$, and as a consquence of this result, we present a generalization of a theorem of O’Hara and Rodriguez for self-inversive polynomials. Finally, we establish a similar result when supremum is replaced by infimum for a rational function which has all its zeros in the unit circle.

The purpose of this paper is to extend results of Ball [1] concerning maximal subgroups of the group S(X) of all permutations of the infinite set X. The basic idea is to consider S(X) as a group of operators on objects more complicated than X. The objects we consider here are subspaces of the Stone-Čech compactification of the discrete space X and the Boolean algebra of “big setoids” of X.

The paper considers the relationship between positive polynomials, sums of squares and the multi-dimensional moment problem in the general context of basic closed semi-algebraic sets in real $n$-space. The emphasis is on the non-compact case and on quadratic module representations as opposed to quadratic preordering presentations. The paper clarifies the relationship between known results on the algebraic side and on the functional-analytic side and extends these results in a variety of ways.

This article consists of a list of problems connected with the theory of convex sets in the Euclidean space En, although in some cases the connection is indirect and/or inessential. The assumptions may often be relaxed or otherwise changed, thus giving rise to a wide variety of questions.

Several authors have recently considered the problem of obtaining sufficient conditions for the oscillation of the quasilinear matrix differential equation

(1)

and the associated inequality VTLV ≤ 0 (as a form). Here A, B, and V are m x m matrix functions, A(x) is symmetric, positive semidefinite and continuous in an interval [a, ∞) and B(x,V, V') is symmetric and continuous in a≤ x < ∞ for all V and V'.

Let G ⊂ Aut (C) be a (finite) group of automorphisms of a curve C defined over a field K and, for each subgroup H ≤ G, let gH denote the genus of the quotient curve CH = C/H (briefly: quotient genus of H).

In this paper we show that certain idempotent relations in the rational group ring [G] imply relations between the quotient genera {gH}H=G this generalizes two theorems of Accola. Moreover, we show that in the case of char (K) = p ≠ 0, a similar statement holds for the Hasse-Witt invariants σH of the curves CH

After the publication of the important paper by Rota [9] on Möbius functions a large number of papers have appeared in which the ideas are applied or generalized in various directions, the papers by Crapo [3], Smith [10] and Tainiter [11] are some of them. The theory of Möbius functions is now recognized as a valuable tool in combinatorial and arithmetical research.

Classically, there has been, for obvious reasons, an intimate relation between the concepts "rings of quotients" and "divisible modules". Recently, however, their generalizations have appeared to diverge.

For example, Hattori ([9]) and Levy ([15]) have generalized the concept of "divisibility" as follows: Hattori (respectively Levy) defines a left R-module M over a ring R to be divisible if and only if Ext1R(R/I, M)=0 for every principal left ideal I ⊂ R (respectively, every principal left ideal I ⊂ R which is generated by a regular element of R).

Let h(m) denote the class number of the quadratic field Q(√m). In this paper necessary and sufficient conditions for h (m) to be divisible by 16 are determined when m = −p, where p is a prime congruent to 1 modulo 8, and when m = −2p, where p is a prime congruent to ±1 modulo 8.

The motivation for this paper lies in the following remarkable property of certain probability distributions. The distribution law of the r. v. (random variable) X is exactly the same as that of 1/ X, and in the case of a r. v. with p. d. f. (probability density function) f(x; a, b) where a, b are parameters, the p. d. f. of 1/X is f(x; b, a). In the latter case the p. d. f. of the reciprocal is obtained from the p. d. f. of X by merely switching the parameters. The existence of random variables with this property is perhaps familiar to statisticians, as is evidenced by the use of the classical 'F' distribution. The Cauchy law is yet another example which illustrates this property. It seems, therefore, reasonable to characterize this class of random variables by means of this rather interesting property.

Let T be a linear transformation acting on the space of n x n complex matrices. Let G(k) be the set of all hermitian matrices with k positive and n — k negative eigenvalues. Let T map some indefinite inertia class G(k) onto itself. We classify all such T. The possibilities are congruence, congruence followed by transposition, and, if n = 2k, it is possible that — T can be a congruence or a congruence followed by transposing. In other words, negation is an admissible transformation when n = 2k.

Let E be a finite set containing n elements, n ≡ 1, 3 (mod 6), S = S(E) a Steiner triple system on E, i.e. each unordered pair of elements of E is a subset of exactly one triple in S. Let T be a subset of E such that none of the triples of elements of T is a member of S. Erdös has asked (in a recent letter to the authors) for the maximal size of such a set T. Denote max |T| for fixed n and S by f(n, S). We prove in this note the following result:

1. (i)

1. (ii) for every n ≡ 1, 3 (mod 6) there exists a Steiner triple system S0 such that equality holds in i.

If Y is a subspace of a Banach space X, either Y contains an isomorphic copy of ℓ1 or each weak Cauchy sequence in X/Y has a subsequence that is the image under the quotient mapping of a weak Cauchy sequence in X. If X is weakly sequentially complete and Y is reflexive, X/Y is weakly sequentially complete. Related structural results are given.