In [4], we investigated the spaces of continuous functions on countable products of compact Hausdorff spaces. Our main object here is to extend the discussion to arbitrary products of compact Hausdorff spaces. We prove the following theorems in Section 3.

We are interested in the distribution of those zeros of the Riemann zeta-function which lie on the critical line ℜs = ½, and the maxima of the function between successive zeros. Our results are to be independent of any unproved hypothesis. Put

In this paper it is proved that, for x sufficiently large, every integer m with

can be written as m = Σ1≤n≤xεn/n, where εi, = 0 or 1.

This paper treats the reducibility of the quasiperiodic linear differential equations

where A is a constant matrix with multiple eigenvalues, Q(t) is a quasiperiodic matrix with respect to time t, and ε is a small perturbation parameter. Under some non-resonant conditions, rapidly convergent methods prove that, for most sufficiently small ε, the differential equations are reducible to a constant coefficient differential equation by means of a quasiperiodic change of variables with the same frequencies as Q(t).

In 1984 Heath-Brown [2] proved the conjecture of Erdős and Mirsky [1], and obtained the following result.

Theorem. There are infinitely many integers n for which d(n+ 1)= d(n). Indeed, for large x, the number of such n≤x is at least of order x ln-7 x.

Given a Banach space X and a norming subspace Z⊂X*, a geometrical method is introduced to characterize the existence of an equivalent σ(X, Z)-lsc LUR norm on X. A new simple proof of the Theorem of Troyanski: every rotund space with a Kadec norm is LUR renormable, and a generalization of the Moltó, Orihuela and Troyanski characterization of the LUR renormability, are provided without probability arguments. Among other applications, it is shown that a dual Banach space with a w*-Kadec norm admits a dual LUR norm.

We show that, if M is a connected binary matroid of cogirth at least five which does not have both an F7-minor and an F*7-minor, then M has a circuit C such that M − C is connected and r(M − C) = r(M).

In [1], recently published in Combinatorics, Probability and Computing, there was a typographical error. On p. 392, part (5) of Lemma 3.1, the formula should read as follows:

formula here

References

[1] Hadjicostas, P. (1998) The asymptotic proportion of subdivisions of a 2 × 2 table that result in Simpson's Paradox. Combinatorics, Probability and Computing 7 387–396.

We prove an Erdős–Ko–Rado-type theorem for intersecting k-chains of subspaces of a finite vector space. This is the q-generalization of earlier results of Erdős, Seress and Székely for intersecting k-chains of subsets of an underlying set. The proof hinges on the author's proper generalization of the shift technique from extremal set theory to finite vector spaces, which uses a linear map to define the generalized shift operation. The theorem is the following.

For c = 0, 1, consider k-chains of subspaces of an n-dimensional vector space over GF(q), such that the smallest subspace in any chain has dimension at least c, and the largest subspace in any chain has dimension at most n − c. The largest number of such k-chains under the condition that any two share at least one subspace as an element of the chain, is achieved by the following constructions:

(1) fix a subspace of dimension c and take all k-chains containing it,

(2) fix a subspace of dimension n − c and take all k-chains containing it.

Assemblies are decomposable combinatorial objects characterized by a sequence mi that counts the number of possible components of size i. Permutations on n elements, mappings from a set containing n elements into itself, 2-regular graphs on n vertices, and set partitions on a set of size n are all assemblies with natural decompositions. Logarithmic assemblies are characterized by constants θ > 0 and κ0 > 0 such that miκi0/(i−1)! → θ. Random mappings, permutations and 2-regular graphs are all logarithmic assemblies, but set partitions are not.

Given a logarithmic assembly, all representatives having total size n are chosen uniformly and a component counting process C(n) = (C1(n), C2(n), …, Cn(n)) is defined, where Ci(n) is the number of components of size i. Our results also apply to C(n) distributed as the Ewens sampling formula with parameter θ. Denote the component counting process up to size at most b by Cb(n) = (C1(n), C2(n), …, Cb(n)). It is natural to approximate Cb by Zb = (Z1, Z2, …, Zb), the b-dimensional process of independent Poisson variables Zi for which the ith variable has expectation []Zi = miκi0 exp((1−θ)i/n)/i!. We find asymptotics for the total variation distance between Cb(n) and Zb.

It has been conjectured that a connected matroid with largest circuit size c [ges ] 2 and largest cocircuit size c* [ges ] 2 has at most ½cc* elements. Pou-Lin Wu has shown that this conjecture holds for graphic matroids. We prove two special cases of the conjecture, not restricted to graphic matroids, thereby providing the first nontrivial evidence that the conjecture is true for non-graphic matroids. Specifically, we prove the special case of the conjecture in which c = 4 or c* = 4. We also prove the special case for binary matroids with c = 5 or c* = 5.

Each edge of the standard rooted binary tree is equipped with a random weight; weights are independent and identically distibuted. The value of a vertex is the sum of the weights on the path from the root to the vertex. We wish to search the tree to find a vertex of large weight. A very natural conjecture of Aldous states that, in the sense of stochastic domination, an obvious greedy algorithm is best possible. We show that this conjecture is false. We prove, however, that in a weaker sense there is no significantly better algorithm.

For a graph G on vertex set V = {1, …, n} let k = (k1, …, kn) be an integral vector such that 1 [les ] ki [les ] di for i ∈ V, where di is the degree of the vertex i in G. A k-dominating set is a set Dk ⊆ V such that every vertex i ∈ V[setmn ]Dk has at least ki neighbours in Dk. The k-domination number γk(G) of G is the cardinality of a smallest k-dominating set of G.

For k1 = · · · = kn = 1, k-domination corresponds to the usual concept of domination. Our approach yields an improvement of an upper bound for the domination number found by N. Alon and J. H. Spencer.

If ki = di for i = 1, …, n, then the notion of k-dominating set corresponds to the complement of an independent set. A function fk(p) is defined, and it will be proved that γk(G) = min fk(p), where the minimum is taken over the n-dimensional cube Cn = {p = (p1, …, pn) [mid ] pi ∈ ℝ, 0 [les ] pi [les ] 1, i = 1, …, n}. An [Oscr](Δ22Δn-algorithm is presented, where Δ is the maximum degree of G, with INPUT: p ∈ Cn and OUTPUT: a k-dominating set Dk of G with [mid ]Dk[mid ][les ]fk(p).

We prove an$n$-tuple analogue of the Putnam area inequality for the spectrum of asingle $p$-hyponormal operator.

Let $X$ be a Banach space and let $B(X)$ denote the space of bounded operators on $X$. Two elements $S,T\inB(X)$ are isometrically equivalent if there exists an invertible isometry $V$ such that $TV=VS$. If $X$ is a Hilbert space, then $V$ is a unitary operator and $S$ and $T$ are said to be unitarily equivalent.

By a quasi-permutation matrix we meana square matrix over the complex field [Copf] with non-negative integraltrace. Thus every permutation matrix over [Copf] is a quasi-permutationmatrix. For a given finite group G, let p(G) denote the minimal degreeof a faithful permutation representation of G (or a faithfulrepresentation of G by permutation matrices), let q(G) denote theminimal degree of a faithful representation of G by quasi-permutationmatrices over the rational field ℚ, and let c(G) be the minimaldegree of a faithful representation of G by complex quasi-permutationmatrices. See [1].

We denote by $\rm{Aut}_{sn}(G)$the set of all automorphisms that fix every subnormal subgroup of $G$setwise. In their paper [5], Franciosi and de Giovanni beganthe study of $\rm{Aut}_{sn}(G)$. Other authors have also considered thestructure of $\rm{Aut}_{sn}(G)$ under various restrictions on thestructure of $G$ (Robinson [11], Cossey [2], DalleMolle [4]). The inner automorphisms in $\rm{Aut}_{sn}(G)$ areprecisely the inner automorphisms induced by elements of $\omega(G)$,the Wielandt subgroup of $G$. Recall that the Wielandt subgroup of agroup $G$ is the set of all elements of $G$ that normalise eachsubnormal subgroup of $G$ and that $\zeta(G)$, the centre of $G$, iscontained in $\omega;(G)$. Thus $\rm{Aut}_{sn}(G)\cap;\rm{Inn}(G)$ isisomorphic to $\omega(G)/\zeta(G)$ and some of the results obtainedindicate that the structure of $\rm{Aut}_{sn}(G)$ is controlled by thestructure of $\omega(G)/\zeta(G)$; for example, Robinson [11,Corollary 3] shows that, for a finite group $G,\rm{Aut}_{sn}(G)$ isinsoluble if and only if $\omega(G)$ is insoluble. We shall prove aresult of a similar nature here. One of the main results (Theorem B) ofFranciosi and de Giovanni [5] is that, for a polycyclic group$G$, $\rm{Aut}_{sn}(G)$ is either finite or abelian. We shall show that$\rm{Aut}_{sn}(G)$ can indeed be infinite, but only if$\omega(G)/\zeta(G)$ is infinite.

An operator$T\in$[Lscr]$(H)$ is called a square root of a hyponormaloperator if $T^2$ is hyponormal. In this paper, we prove the followingresults: Let $S$ and $T$ be square roots of hyponormaloperators.

(1) If $\sigma(T)\cap[-\sigma(T)]=\phi$ or {0}, then$T$ is isoloid (i.e., every isolated point of $\sigma(T)$ is aneigenvalue of $T$).

(2) If $S$ and $T$ commute, then $ST$ is Weylif and only if $S$ and $T$ are both Weyl.

(3) If$\sigma(T)\cap[-\sigma(T)]=\phi$ or {0}, then Weyl's theorem holds for$T$.

(4) If $\sigma(T)\cap[-\sigma(T)]=\phi$, then $T$ issubscalar. As a corollary, we get that $T$ has a nontrivial invariantsubspace if $\sigma(T)$ has non-empty interior. (See[3].)

Letthe locally finite group G be the product of two locallynilpotent subgroups A and B, and assume thatH is a subgroup of G belonging to a group classℱ. The question is considered whether thereexists a subgroup X of G containing H whichbelongs to ℱ and satisfies $X=(X\capA)(X\capB)$.Under various assumptions on G and ℱ,necessary and sufficient conditions for the existence of such a subgroupX are obtained.