The multiplicative structure of the algebra of stable operations for ρ-local complex K-theory is studied, and the units and zero divisors are identified.

Let M be the group defined by the presentation 〈 x, y, z | xy = xm, yz = yn, zx = zr〉,m,n,r ∊ Z. M is one of the few 3-generator finite groups of deficiency zero. These groups have been considered by Mennicke [3], Macdonald, Wamsley [10], Johnson and Robertson [7], and Albar. Properties like the order of M, the nilpotency and solvability were studied. In this paper we give a better upper bound for M than the one given by Johnson and Robertson [7]. We also describe the structure of some cases of M.

Let {Xa, a ∊ A} be a family of completely regular Hausdorff spaces, {βXa} the corresponding family of their Stone-Čech compactifications and ΠaXa the usual topological product. The following theorem was proved by Glicksberg [2] and subsequently by Frolík [1].

Let G be a group, let RG be the group ring of the group G over the unital commutative ring R and let U(RG) be its group of units. Conditions which imply that U(RG) contains no free noncyclic group are studied, when R is a field of characteristic p ≠ 0, not algebraic over its prime field, and G is a solvable-by-finite group without p-elements. We also consider the case R = ℤp, the ring of p-adic integers and G torsionby- nilpotent torsion free group. Finally, the residual nilpotence of U(ℤpG) is investigated.

J. Ch. Boland suggested, and Mrs. Sheehan named, the idea of the slimming number of a graph G, i.e. the minimum number, s(G), of edges, e1e2,…, es, which must be removed from G in order that G—∪ ei be planar.

For the complete graph, Kn (n≥3), it may be seen by Euler's formula that a planar subgraph contains at most 3n—6 edges; moreover one may construct such a subgraph inductively, starting from K3, and adding points successively, joining them to the three vertices of the region in which they lie, so

1

All spaces in this note are regular, Hausdorff topological spaces.

At the topology conference in Pullman, Washington, in March 1970, E. Michael posed the following question: if X is Lindelôf Y is separable and metrizable and I × Y is paracompact, must X × Y be Lindelöf?

Let $({\mathcal{X}},d,\unicode[STIX]{x1D707})$ be a metric measure space satisfying the geometrically doubling condition and the upper doubling condition. In this paper, the authors establish the John-Nirenberg inequality for the regularized BLO space $\widetilde{\operatorname{RBLO}}(\unicode[STIX]{x1D707})$.

We show that a class of semilinear Laplace–Beltrami equations on the unit sphere in ${{\mathbb{R}}^{n}}$ has infinitely many rotationally symmetric solutions. The solutions to these equations are the solutions to a two point boundary value problem for a singular ordinary differential equation. We prove the existence of such solutions using energy and phase plane analysis. We derive a Pohozaev-type identity in order to prove that the energy to an associated initial value problem tends to infinity as the energy at the singularity tends to infinity. The nonlinearity is allowed to grow as fast as ${{\left| s \right|}^{p-1}}s$ for $\left| s \right|$ large with $1\,<\,p\,<\,\left( n+5 \right)/\left( n-3 \right)$.

A question which Chalk and L. Moser asked me several years ago led me to the following problem: Let 0 < x ≤ y. Estimate the smallest f(x) so that there should exist integers u and v satisfying

1

I am going to prove that for every ∊ > 0 there exist arbitrarily large values of x satisfying

2

but that for a certain c > 0 and all x

3

A sharp estimation of f(x) seems to be a difficult problem. It is clear that f(p) = 2 for all primes p. I can prove that f(x)→ ∞ and f(x)/loglog x → 0 if we neglect a sequence of integers of density 0, but I will not give the proof here.

In this paper, we investigate the common approximate fixed point sequences of nonexpansive semigroups of nonlinear mappings ${{\left\{ {{T}_{t}} \right\}}_{t\ge 0}}$, i.e., a family such that ${{T}_{0}}\left( x \right)\,=\,x,\,{{T}_{s+t}}\,=\,{{T}_{s}}\left( {{T}_{t}}\left( x \right) \right)$, where the domain is a metric space $\left( M,\,d \right)$. In particular, we prove that under suitable conditions the common approximate fixed point sequences set is the same as the common approximate fixed point sequences set of two mappings from the family. Then we use the Ishikawa iteration to construct a common approximate fixed point sequence of nonexpansive semigroups of nonlinear mappings.

Let R be a commutative domain, I an ideal of R and write E1 for the infective envelope of R / I. In this note the following theorem will be proved:

Theorem. For a prime ideal P of a commutative domain R the following are equivalent:

1. (i) Every factor module of Ep is an indecomposable infective module;

1. (ii) Every non-zero prime ideal P′ ⊆ P is contained in only one maximal ideal M of R, and RM is an almost maximal valuation ring.

We prove that a Banach space $X$ has the Schur property if and only if every $X$-valued weakly differentiable function is Fréchet differentiable. We give a general result on the Fréchet differentiability of $f\,\circ \,T$, where $f$ is a Lipschitz function and $T$ is a compact linear operator. Finally we study, using in particular a smooth variational principle, the differentiability of the semi norm ${{\left\| {} \right\|}_{\text{lip}}}$ on various spaces of Lipschitz functions.

Using Komlos' Theorem, a sequence decomposition result due to Gaposhkin, and two results due to Artstein, we prove a result concerning the properties of Komlos limits. We then show that a stronger version of Fatou's Lemma in several dimensions can be deduced from Artstein's version of the Lemma. The version of Fatou's Lemma proved here subsumes the most recent version of the Lemma in several dimensions given by Balder.

We consider a concrete family of $2$ -towers $(\mathbb {Q}(x_n))_n$ of totally real algebraic numbers for which we prove that, for each $n$ , $\mathbb {Z}[x_n]$ is the ring of integers of $\mathbb {Q}(x_n)$ if and only if the constant term of the minimal polynomial of $x_n$ is square-free. We apply our characterization to produce new examples of monogenic number fields, which can be of arbitrary large degree under the ABC-Conjecture.

We show that any exceptional non-trivial Dehn surgery on a twist knot, except the trefoil, yields a 3-manifold whose fundamental group is left-orderable. This is a generalization of a result of Clay, Lidman, and Watson, and also gives a new supporting evidence for a conjecture of Boyer, Gordon, andWatson.

It is known that the $K$-groups that appear in the calculation of the $K$-theory of a large class of groups can be computed from the $K$-theory of their virtually infinite cyclic subgroups. On the other hand, Nil-groups appear to be the obstacle in calculations involving the $K$-theory of the latter. The main difficulty in the calculation of Nil-groups is that they are infinitely generated when they do not vanish. We develop methods for computing the exponent of $\text{N}{{\text{K}}_{0}}$-groups that appear in the calculation of the ${{K}_{0}}$-groups of virtually infinite cyclic groups.

The generalized Beurling–Ahlfors operator $S$ on ${{L}^{p}}({{\mathbb{R}}^{n}};\,\Lambda )$, where $\Lambda \,:=\,\Lambda ({{\mathbb{R}}^{n}})$ is the exterior algebra with its natural Hilbert space norm, satisfies the estimate

$$||S||L\left( {{L}^{p}}\left( {{\mathbb{R}}^{n}};\Lambda \right) \right)\le \left( n/2+1 \right)\left( {{p}^{*}}-1 \right),\,\,\,{{p}^{*}}:=\,\max \{p,\,{{p}^{'}}\}.$$

This improves on earlier results in all dimensions $n\,\ge \,3$. The proof is based on the heat extension and relies at the bottom on Burkholder's sharp inequality for martingale transforms.

We prove that if $f\,:\,{{\mathbb{R}}^{N}}\,\to \,\overline{\mathbb{R}}$ is quasiconvex and $U\,\subset \,{{\mathbb{R}}^{N}}$ is open in the density topology, then $\underset{U}{\mathop{\sup }}\,f=\text{ess}\,\underset{U}{\mathop{\sup }}\,f$, while ${{\inf }_{U}}\,f\,=\,\text{ess}\,{{\inf }_{U}}\,f$ if and only if the equality holds when $U\,\subset \,{{\mathbb{R}}^{N}}$. The first (second) property is typical of $\text{lsc}\,\text{(usc)}$ functions, and, even when $U$ is an ordinary open subset, there seems to be no record that they both hold for all quasiconvex functions.

This property ensures that the pointwise extrema of $f$ on any nonempty density open subset can be arbitrarily closely approximated by values of $f$ achieved on “large” subsets, which may be of relevance in a variety of situations. To support this claim, we use it to characterize the common points of continuity, or approximate continuity, of two quasiconvex functions that coincide away from a set of measure zero.

We identify when a tubular group (the fundamental group of a finite graph of groups with ${{\mathbb{Z}}^{2}}$ vertex and $\mathbb{Z}$ edge groups) is free by cyclic and show, using Wise’s equitable sets criterion, that every tubular free by cyclic group acts freely on a $\text{CAT}\left( 0 \right)$ cube complex.

We study infinitesimal deformations of holomorphic maps of compact, complex, Kähler manifolds. In particular, we describe a generalization of Bloch's semiregularity map that annihilates obstructions to deform holomorphic maps with fixed codomain.