This paper solves a question posed by P. Erdös:

If A1, A2, …, A are distinct subsets of n elements and if |Ai ∩ Ai| ≡ 0 mod 2 (i ≠ j), then

and for each n, there exists a collection of subsets which achieves this bound with equality.

The permanent of an n-square matrix A = (aij) is defined by

where the summation extends over all permutations σ of the symmetric group Sn. A matrix is said to be a (0, 1)-matrix if each of its entries is either 0 or 1. A (0, 1)-matrix of n-1 the form , where θj = 0 or 1, j = 1,…, n, and Pn is the n-square permutation matrix with ones in the (1, 2), (2, 3),…, (n-1, n), (n, 1) positions, is called a (0, 1)-circulant. Denote the (0, 1)-circulant . It has been conjectured that

1

A path of maximum length in a graph G is referred to as a detour path of G and the length of such a path is called the detour number of G. It is not surprising that the study of detour paths is closely associated with the problem of investigating hamiltonian paths in graphs. Evidently few results have been obtained in this area, although Ore [3] has shown that any two detour paths intersect. It is the purpose of this article to further investigate these concepts. In particular, we obtain bounds for several graph theoretic parameters in terms of the detour number and also present formulae for the detour numbers of several important classes of graphs.

A well known result of Levitzki [2, Lemma 1.1] is the following:

Theorem. Let R be a ring and U a non-zero one-sided ideal of R. Suppose that given a∈U, an = 0 for a fixed integer n ≥ 1; then R has a non-zero nilpotent ideal.

The purpose of this note is to observe some additional results which are related to the above.

Theorem 1. Let R be a ring with no non-zero nil ideals and U an ideal of R. Suppose that a∊R is such that for every x∊U, axn(x) = 0 where n(x) ≥ 1 depends on x; then aU= Ua = 0.

In connection with a statistical problem concerning the Galtontest Cśaki and Vincze [1] gave for an equivalent Bernoullian symmetric random walk the joint distribution of g and k, denoting respectively the number of positive steps and the number of times the particle crosses the origin, given that it returns there on the last step.

The purpose of this paper is to present a short, self-contained proof of Euler’s relation. The ingredients of this proof are (i) the principle of inclusion and exclusion of combinatorics and (ii) the Euler characteristic; a development of the Euler characteristic is included.

In this paper, we study the warped structures of Finsler metrics. We obtain the differential equation that characterizes Finsler warped product metrics with vanishing Douglas curvature. By solving this equation, we obtain all Finsler warped product Douglas metrics. Some new Douglas Finsler metrics of this type are produced by using known spherically symmetric Douglas metrics.

Let 5 be a monoid. A right S-system A is called strongly flat if the functor A ⊗ — (from the category of left S-systems into the category of sets) preserves pullbacksand equalizers. (This concept arises in B. Stenström, Math. Nachr. 48(1971), 315-334 under the name weak flatness). The main result of the present paper is a proof that for A to be strongly flat it is in fact sufficient that A ⊗ — preserve only pullbacks. The approach taken is to develop an "interpolation" condition for pullback-preservation, and then to show its equivalence to Stenström's conditions for strong flatness.

We prove in this paper that fifteen classes of rings coincide with the class of rings named in the title. One of them is the class of rings R such that X2 = X for each R-ideal X: we shall refer to rings with this property (and thus to the rings of the title) as fully idempotent rings. The simple rings and the (von Neumann) regular rings are fully idempotent. Indeed, every finitely generated right or left ideal of a regular ring is generated by an idempotent [l, p. 42], so that X2 = X holds for every one-sided ideal X.

Let C(X) be the ring of continuous real-valued functions on a (completely regular) topological space X. The structure of the prime ideals and the prime z-ideals of C(X) has been the subject of much investigation (see e-g- [1], [3], [5]). One of the surprising facts about C(X) is that the sum of two prime ideals is again prime.

It is shown that a two-sided continuous ring with ascending chain condition on left annihilators is quasi-Frobenius.

Let σ*(N) denote the sum of the unitary divisors of N, that is,

It is easily seen that σ*(N) is multiplicative. In fact σ*(l) = 1 if N > 1 has the prime decomposition . Let us define a positive integer to be unitary perfect whenever σ*(N) = 2N. The first four such numbers are 6, 60, 90 and 87, 360. In a recent abstract [ l ] published by one of us, the last of these numbers was overlooked. No other unitary perfect numbers are known to the authors.

Let $b\,\in \,\text{BMO}\left( {{\mathbb{R}}^{n}} \right)$ and ${{T}_{\Omega }}$ be the singular integral operator with kernel $\Omega \left( x \right)/{{\left| x \right|}^{n}}$, where $\Omega$ is homogeneous of degree zero, integrable, and has mean value zero on the unit sphere ${{S}^{n-1}}$. In this paper, using Fourier transform estimates and approximation to the operator ${{T}_{\Omega }}$ by integral operators with smooth kernels, it is proved that if $b\,\in \,\text{CMO}\left( {{\mathbb{R}}^{n}} \right)$ and $\Omega$ satisfies certain minimal size condition, then the commutator generated by $b$ and ${{T}_{\Omega }}$ is a compact operator on ${{L}^{p}}\left( {{\mathbb{R}}^{n}} \right)$ for appropriate index $p$. The associated maximal operator is also considered.

In this paper, we prove a global rigidity theorem for negatively curved Finsler metrics on a compact manifold of dimension $n\,\ge \,3$. We show that for such a Finsler manifold, if the flag curvature is a scalar function on the tangent bundle, then the Finsler metric is of Randers type. We also study the case when the Finsler metric is locally projectively flat.

We present a construction of singular rearrangement invariant functionals on Marcinkiewicz function/operator spaces. The functionals constructed differ from all previous examples in the literature in that they fail to be symmetric. In other words, the functional $\phi$ fails the condition that if $x\prec \prec \,Y$ (Hardy-Littlewood-Polya submajorization) and $0\,\le \,x,\,y$, then $0\,\le \,\phi \left( x \right)\,\le \,\phi \left( y \right)$. We apply our results to singular traces on symmetric operator spaces (in particular on symmetrically-normed ideals of compact operators), answering questions raised by Guido and Isola.

Given a finite group $G$, we examine the classification of all frame representations of $G$ and the classification of all $G$-frames, i.e., frames induced by group representations of $G$. We show that the exact number of equivalence classes of $G$-frames and the exact number of frame representations can be explicitly calculated. We also discuss how to calculate the largest number $L$ such that there exists an $L$-tuple of strongly disjoint $G$-frames.

Let R be a commutative ring with identity and let X be an indeterminate. In [5] and [16] it was shown that if f, g∈R[X], then for some integer n≥l, c(f)n+1c(g) = c(g)nc(fg), where c(h) denotes the additive subgroup of JR generated by the coefficients of h ∈ R[X]. Actually the statements in [5J and [16] are not so general as this; however, the proofs are. Specifically, in [16] Mertens considers only the case that R is a polynomial ring over the integers, but this gives the result for any ring by specializing the coefficients of f and g. In [5] Dedekind considers only the case that JR is a ring of algebraic integers, but his proof is completely general. Further, the above formula then holds if one lets c(h) denote the S-submodule of R generated by the coefficients of h ∈ R[X], S a subring of R, and it is in this form that it usually appears, especially the case S = R. Dedekind′s very elegant proof is reproduced in [15, p. 9, Lemma 6.1].

An intrinsic characterization is given of those von Neumann algebras which are injective objects in the category of C*-algebras with completely positive maps. For countably generated von Neumann algebras several such characterizations have been given, so it is in fact enough to observe that an injective von Neumann algebra is generated by an upward directed collection of injective countably generated sub von Neumann algebras. The present work also shows that three of the intrinsic characterizations known in the countably generated case hold in general.

Given a group automorphism $\phi :\,\Gamma \,\to \,\Gamma $, one has an action of $\Gamma $ on itself by $\phi $-twisted conjugacy, namely, $g.x\,=\,gx\phi ({{g}^{-1}})$. The orbits of this action are called $\phi $-twisted conjugacy classes. One says that $\Gamma $ has the ${{R}_{\infty }}$-property if there are infinitely many $\phi $-twisted conjugacy classes for every automorphism $\phi $ of $\Gamma $. In this paper we show that $\text{SL(}n\text{,}\mathbb{Z}\text{)}$ and its congruence subgroups have the ${{R}_{\infty }}$-property. Further we show that any (countable) abelian extension of $\Gamma $ has the ${{R}_{\infty }}$-property where $\Gamma $ is a torsion free non-elementary hyperbolic group, or $\text{SL(}n\text{,}\mathbb{Z}\text{)},\text{Sp(2}n\text{,}\mathbb{Z}\text{)}$ or a principal congruence subgroup of $\text{SL(}n\text{,}\mathbb{Z}\text{)}$ or the fundamental group of a complete Riemannian manifold of constant negative curvature.

The Hardy space Hp=Hp(U), where 1≤p≤∞ and U is the unit disc |z|<1, is shown to be an algebra under the product

Moreover for each p, 1≤p≤∞, there exists a constant Cp such that ‖f*g‖p,≤ Cp‖f‖p ‖g‖p.