We provide an existence theorem for stable sets which is equivalent to Zorn's lemma and study the connections between the unique stable set for majorization and the stable sets for the dominance relation.

We give a new type of geometric construction that allows for the construction of families of quintic irrationalities, and is quite rich in algebraic properties. This construction may be considered as our first attempt at characterizing points constructible with compass and twice-notched ruler, a problem that seems to have been known in some form for more than two millennia.

The cancellation law is a necessary condition for a semigroup to be embedded in a group. In general, this condition is not sufficient; necessary and sufficient conditions are rather complicated (see [1]). It is, therefore, of interest to find large classes of semigroups for which the cancellation law is sufficient to ensure embeddability in a group.

In this paper we shall be interested in the Weierstrass transform defined by (1.1.) converging (conditionally) for x in some interval, where (1.2) .

We show that the Kato conjecture is true for m-accretive operators with highly singular coefficients. For operators of the form A = *F, where formally corresponds to d/dx + zδ on L2 (R), we prove that Dom (A1/2) = Dom() = e-zHH1(R) where H is the Heavysied function. By adapting recent methods of Auscher and Tchamitchian, we characterize Dom (A) in terms of an unconditional wavelet basis for L2(R).

We prove that if $G_{P}$ is a finitely constrained group of binary rooted tree automorphisms (a group binary tree subshift of finite type) defined by an essential pattern group $P$ of pattern size $d$, $d\geq 2$, and if $G_{P}$ has maximal Hausdorff dimension (equal to $1-1/2^{d-1}$), then $G_{P}$ is not topologically finitely generated. We describe precisely all essential pattern groups $P$ that yield finitely constrained groups with maximal Hausdorff dimension. For a given size $d$, $d\geq 2$, there are exactly $2^{d-1}$ such pattern groups and they are all maximal in the group of automorphisms of the finite rooted regular tree of depth $d$.

Recursively presented topological spaces are topological spaces with a recursive system of basic neighbourhoods. A recursively enumerable (r.e.) open set is a r.e. union of basic neighbourhoods. A set is everywhere r.e. open if its intersection with each basic neighbourhood is r.e. Similarly we define everywhere creative, everywhere simple, everywhere r.e. non-recursive sets and show that there exist sets both with and without these everywhere properties.

Let X be a (real or complex) rearrangement-invariant function space on Ω (where Ω = [0, 1] or Ω ⊆ N) whose norm is not proportional to the L2-norm. Let H be a separable Hilbert space. We characterize surjective isometries of X (H). We prove that if T is such an isometry then there exist Borel maps a: Ω → + K and σ: Ω → Ω and a strongly measurable operator map S of Ω into B (H) so that for almost all ω, S(ω) is a surjective isometry of H, and for any f ∈ X(H), T f(ω) = a(ω)S(ω)(f(σ(ω))) a.e. As a consequence we obtain a new proof of the characterization of surjective isometries in complex rearrangement-invariant function spaces.

Ritt introduced the concepts of prime and composite polynomials and proved three fundamental theorems on factorizations (in the sense of compositions) of polynomials in 1922. In this paper, we shall give a density estimate on the set of composite polynomials.

It is known that the minimal 3-spheres of CR type with constant sectional curvature have been classified explicitly, and also that the weakly Lagrangian case has been studied. In this paper, we provide some examples of minimal 3-spheres with constant curvature in the complex projective space, which are neither of CR type nor weakly Lagrangian, and give the adapted frame of a minimal 3-sphere of CR type with constant sectional curvature.

All inverse semigroups with idempotents dually well-ordered may be constructed inductively. The techniques involved are the constructions of ordinal sums, direct limits and Bruck-Reilly extensions.

The paper discusses modules over free nilpotent groups and demonstrates that faithful modules are more restricted than might appear at first glance. Some discussion is also made of applying the techniques more generally.

Let (P,Y ) be a bundle gerbe over a fibre bundle Y →M. We show that if M is simply connected and the fibres of Y →M are connected and finite-dimensional, then the Dixmier–Douady class of (P,Y ) is torsion. This corrects and extends an earlier result of the first author.

We provide an upper bound for the order of a nilpotent injector of a finite solvable group with Fitting subgroup of order n. We also show that the same bound is an upper bound for the number of conjugacy classes, provided that the k(G V)-conjecture holds for solvable G all primes dividing n.

For every $k \geq 2$ and $n \geq 2$, we construct n pairwise homotopically inequivalent simply connected, closed $4k$-dimensional manifolds, all of which are stably diffeomorphic to one another. Each of these manifolds has hyperbolic intersection form and is stably parallelisable. In dimension four, we exhibit an analogous phenomenon for spin$^{c}$ structures on $S^2 \times S^2$. For $m\geq 1$, we also provide similar $(4m-1)$-connected $8m$-dimensional examples, where the number of homotopy types in a stable diffeomorphism class is related to the order of the image of the stable J-homomorphism $\pi _{4m-1}(SO) \to \pi ^s_{4m-1}$.

Let p be a prime. In this paper, we use techniques from Iwasawa theory to study questions about rank jump of elliptic curves in cyclic extensions of degree p. We also study growth of the p-primary Selmer group and the Shafarevich–Tate group in cyclic degree-p extensions and improve upon previously known results in this direction.

This paper gives an explicit version of Selberg’s mean-value estimate for the prime number theorem in intervals, assuming the Riemann hypothesis [25]. Two applications are given to short-interval results for primes and for Goldbach numbers. Under the Riemann hypothesis, we show there exists a prime in $(y,y+32\,277\log ^2 y]$ for at least half the $y\in [x,2x]$ for all $x\geq 2$, and at least one Goldbach number in $(x,x+9696 \log ^2 x]$ for all $x\geq 2$.

In this note the L2-angle between two concentric rings and between the ring and the exterior of the disc in the complex plane are calculated. In the second part we prove that the L2-angles between domains A and B and between A × C and B × C are equal. We give also some examples of discontinuity of the L2-angle between domains.

Let $G$ be a finite group and $p$ a prime. We say that a $p$-regular element $g$ of $G$ is $p$-nonvanishing if no irreducible $p$-Brauer character of $G$ takes the value $0$ on $g$. The main result of this paper shows that if $G$ is solvable and $g\in G$ is a $p$-regular element which is $p$-nonvanishing, then $g$ lies in a normal subgroup of $G$ whose $p$-length and $p^{\prime }$-length are both at most 2 (with possible exceptions for $p\leq 7$), the bound being best possible. This result is obtained through the analysis of one particular orbit condition in linear actions of solvable groups on finite vector spaces, and it generalizes (for $p>7$) some results in Dolfi and Pacifici [‘Zeros of Brauer characters and linear actions of finite groups’, J. Algebra 340 (2011), 104–113].

Let k ≥ 0 be an integer, T = (T(t))t ∈R a C0-group of bounded operators and A the infinitesimal generator of T. We prove that if, and , and if the spectrum of A is equal to {λ}, then A is bounded and (A – λ)k+1 = 0. Examples are given to show that these conditions are, essentially, the best possible.