The relative rank $\mbox{\rm rank}(S:U)$ of a subsemigroup $U$ of a semigroup $S$ is the minimum size of a set $V\subseteq S$ such that $U$ together with $V$ generates the whole of $S$. As a consequence of a result of Sierpiński, it follows that for $U\leq \mathcal{T}_{X}$, the monoid of all self-maps of an infinite set $X$, $\mbox{\rm rank}(\mathcal{T}_{X}:U)$ is either 0, 1 or 2, or uncountable. In this paper, the relative ranks $\mbox{\rm rank}(\mathcal{T}_{X}:\mathcal{O}_{X})$ are considered, where $X$ is a countably infinite partially ordered set and $\mathcal{O}_{X}$ is the endomorphism monoid of $X$. We show that $\mbox{\rm rank}(\mathcal{T}_{X}:\mathcal{O}_{X})\leq 2$ if and only if either: there exists at least one element in $X$ which is greater than, or less than, an infinite number of elements of $X$; or $X$ has $|X|$ connected components. Four examples are given of posets where the minimum number of members of $\mathcal{T}_{X}$ that need to be adjoined to $\mathcal{O}_{X}$ to form a generating set is, respectively, 0, 1, 2 and uncountable.

Power series $\sum_{n=0}^\infty f(n)x^n$ with non-zero convergence radius $R(f)$ are considered, and the arithmetical nature (that is, irrationality, or even transcendence) of the corresponding multivariate series $\smash{\sum_{i_1,\ldots,i_m\!=0}^\infty f(i_1+\ldots +i_m)x_1^{i_1}\cdot\ldots \cdot x_m^{i_m}}$ is studied if $x_1,\ldots,x_m$ and the sequence $(f(n))$ satisfy appropriate arithmetical conditions. It follows that such arithmetical results can be written down easily if linear independence results on the function $F(x)$, defined in $|x|\,{<}\,R(f)$ by the original one-dimensional power series, and possibly its derivatives at the points $x_\mu$ are known. Some typical applications are explicitly stated.

We prove tight closure analogues of results of Watanabe about chains and families of integrally closed ideals.

In a recent paper Ahlgren, Ono and Penniston described the L-series of K3 surfaces from a certain one-parameter family in terms of those of a particular family of elliptic curves. The Tate conjecture predicts the existence of a correspondence between these K3 surfaces and certain Kummer surfaces related to these elliptic curves. A geometric construction of this correspondence is given here, using results of D. Morrison on Nikulin involutions.

A base of an action of a group G on a set $\Omega$ is a subset $B \subseteq \Omega$ such that the pointwise stabiliser of $B$ in $G$ is the identity. We prove that if $\Omega$ is the set of partitions of $[ 1, kl ]$ into $l$ subsets of size $k$, then the action of $S_{kl}$ on $\Omega$ has a base of size two if and only if $k \geq 3$ and $l \geq \max \{ k + 3, 8 \}$. This result completes a classification of the primitive base 2 actions of the symmetric groups. During the proof we show that there exists a $k$-regular bipartite graph $\mathcal{G}$ on $2l$ vertices with no non-trivial automorphisms fixing the bipartite blocks if and only if $k \geq 3$ and $l \geq \max \{ k + 3, 8 \}$.

I exhibit a pair of non-symmetric operads that, while not themselves isomorphic, induce isomorphic monads. The existence of such a pair implies that if ‘algebraic theory’ is understood as meaning ‘monad’ then operads cannot be regarded as algebraic theories of a special kind.

A result is proved which implies the following conjecture of Osgood and Yang from 1976: if $f$ and $g$ are non-constant entire functions, such that $ T(r, f) = O(T(r, g))$ as $ r \to \infty $ and such that $g(z) \in {\mathbb Z}$ implies that $f(z) \in {\mathbb Z}$, then there exists a polynomial $G$ with coefficients in ${\mathbb Q}$, such that $G({\mathbb Z}) \subseteq {\mathbb Z}$ and $f = G \circ g$.

A proof is given to show that a positive function on the unit disk admits a harmonic majorant if and only if it has a certain explicit upper envelope that admits a superharmonic majorant. The (logarithmic) Lipschitz regularity of this superharmonic majorant is discussed.

Let $ S_n(x)=\sum_{k=1}^{n}({\sin(kx)})/{k} $ be Fejér's sine polynomial. We prove the following statements.

1. The inequality $ (S_n(x+y))^{\alpha} (x+y)^{\beta} \leq (S_n(x))^{\alpha}x^{\beta}+ (S_n(y))^{\alpha}y^{\beta} {(n\in \mathbb{N}; \; \alpha, \beta \in \mathbb{R})}$ holds for all $x,y \in (0,\pi)$ with $x+y<\pi$ if and only if $\alpha\geq 0$ and $\alpha+\beta \leq 1$.

2. The converse of the above inequality is valid for all $x,y \in (0,\pi)$ with $x+y<\pi$ if and only if $\alpha\leq 0$ and $\alpha+\beta \geq 1$.

3. For all $n\in\mathbb{N}$ and $x,y \in [0,\pi]$ we have $ 0\leq S_n(x)+S_n(y)- S_n(x+y)\leq \frac{3}{2}\sqrt{3}.$ Both bounds are best possible.

There are many references showing that a classical solution to the Black–Scholes equation is a stochastic solution. However, it is the converse of this theorem that is most relevant in applications, and the converse is also more mathematically interesting. In this paper we establish such a converse. We find a Feynman–Kac-type theorem showing that the stochastic representation yields a classical solution to the corresponding Black–Scholes equation with appropriate boundary conditions under very general conditions on the coefficients. We also obtain additional regularity results in the one-dimensional case.

This paper treats the existence of multiple positive solutions of the semi-positone Sturm–Liouville boundary value problem. \begin{eqnarray*} &\lambda (p(t)y'(t))'+g(t)f(t,y(t))=0 \qquad\mbox{almost everywhere on } [R_{0},R_{1}],\\ &\alpha z(R_{0})-\beta p(R_{0})z'(R_{0})=0,\\ &\gamma z(R_{1})+\delta p(R_{1})z'(R_{1})=0, \end{eqnarray*} where $g\in L_{+}^{\infty}[R_{0}, R_{1}]$ and $f$ is allowed to take negative values (that is, $f$ is semi-positone). When $\lambda=1$, new results on the existence of one or two nonzero positive solutions are obtained. These results generalize previous results for positone cases (that is, $f\ge 0$) to the semi-positone cases. We illustrate our results with an explicit example which has two nonzero positive solutions. These results are used to deduce results on intervals of eigenvalues for which there exist one or two nonzero positive eigenfunctions. Applications of these eigenvalue results are provided.

We give a simple direct proof of the interpolation inequality $ \|\nabla f\|^2_{L^{2p}} \le C \|f\|_{\rm BMO} \|f\|_{W^{2,p}}$, where $ 1<p<\infty$. For $p=2$ this inequality was obtained by Meyer and Rivière via a different method, and it was applied to prove a regularity theorem for a class of Yang–Mills fields. We also extend the result to higher derivatives, sharpening all those cases of classical Gagliardo–Nirenberg inequalities where the norm of the function is taken in $L^\infty$ and other norms are in $L^q$ for appropriate $q>1$.

In this paper we discuss the asymptotic distribution of the approximation numbers of the finite sections for a Toeplitz operator $T(a) \in \mathcal{L}(\ell^{p,\mu}_N)$, $1 < p < \infty$ and $\mu \in \mathbb{R}$, with a continuous matrix-valued generating function $a$. We prove that the approximation numbers of the finite sections $T_n(a) = P_n T(a) P_n$ have the $k$-splitting property, provided $T(a)$ is a Fredholm operator on $\ell^{p,\mu}_N$.

In this paper we use the Hausdorff metric to prove that two compact metric spaces are homeomorphic if and only if their canonical complements are uniformly homeomorphic. We take one of the two steps needed to prove that the difference between the homotopical and topological classifications of compact connected ANRs depends only on the difference between continuity and uniform continuity of homeomorphisms in their canonical complements, which are totally bounded metric spaces. The more important step was provided by the Chapman complement and the Curtis–Schori–West theorems. We also improve the multivalued description of shape theory given by J. M. R. Sanjurjo but only in the class of locally connected compacta.

Real flag manifolds are the isotropy orbits of noncompact symmetric spaces $G/K$. Any such manifold $M$ is acted on transitively by the (noncompact) Lie group $G$, and it is embedded in euclidean space as a taut submanifold. The aim of this paper is to show that the gradient flow of any height function is a one-parameter subgroup of $G$, where the gradient is defined with respect to a suitable homogeneous metric $s$ on $M$; this generalizes the Kähler metric on adjoint orbits (the so-called complex flag manifolds).

Ali Fröhlich was one of the truly great mathematicians of his generation. In addition to making fundamental contributions, he is one of the very few of whom it can be said, “he founded a new subject”: namely that area now known as Galois module theory. In addition to establishing a theoretical basis for the subject, he also formulated some very deep and wonderful conjectures that would thrust him onto the world's stage. He also built up a school of researchers that played a central part in establishing British algebraic number theory as a world force. The outstanding nature of his contributions to mathematics was recognised in 1992 when the London Mathematical Society awarded him the De Morgan Medal: their highest and most distinguished prize, which is awarded triennially.