Let Fn, Mn and Un denote the free group of rank n, the free metabelian group of rank n, and the free abelian-by-(nilpotent of class 2) group of rank n, respectively. Thus Mn≅ Fn/F″″n and Un≅Fn/ [γ3(Fn), γ3(Fn)], where γ3(Fn) =[F′n, Fn], the third term of the lower central series of Fn.

Consider an arbitrary epimorphism θ[ratio ]Fn[Rarr ]Fk, where k<n. A routine argument (see [10, Theorem 3.3]) using the Nielsen reduction process shows that there exists a free basis {y1, y2, …, yn} of Fn such that kerθ is the normal closure in Fn of {y1, y2, …, yn−k}. A similar result has been obtained for free metabelian groups by C. K. Gupta, N. D. Gupta and G. A. Noskov [5]. Indeed, it follows immediately from [5, Theorem 3.1] that if k<n and θ[ratio ]Mn[Rarr ]Mk is an epimorphism, then there exists a free basis {m1, m2, …, mn} of Mn such that kerθ is the normal closure in Mn of {m1, m2, …, mn−k}.

A Lefschetz hyperplane theorem for the determinantal loci of a morphism between two holomorphic vector bundles $E$ and $F$ over a complex manifold is proved, under the condition that $E^* \otimes F$ is Griffiths $k$-positive. This result is applied to find some homotopy groups of the Brill–Noether loci for a generic curve.

Consider the parabolic equation \[ \begin{array}{@{}l@{\qquad}l@{}} u_t=\Delta u+|u|^p& \text{in}\ R^d\times(0,T), u(x,0)=\phi(x)& \text{in}\ R^d, \end{array} \] up to a maximal time $T=T_\infty$, where $p>1$. Let $p^{\ast}=1+2/d$. It is a classical result that if $p\le p^{\ast}$, then there exist no non-negative, global solutions to the above equation for any choice of $\phi\geq0$; that is, necessarily $T_\infty<\infty$ and the solution blows up in some sense. On the other hand, if $p>p^{\ast}$, then there do exist non-negative, global solutions for appropriate choices of $\phi$. A recent paper by Zhang seems to be the first to consider the existence of global solutions for the above equation with sign-changing initial data. He proved that if $p\le p^{\ast}$ and $\int_{R^d}\phi(x)dx>0$, then the solution to the above equation is not global. In this paper it is shown that this result continues to hold if $\int_{R^d}\phi(x)dx=0$, and then it is shown that for each $p{>}1$ and each $l\in(-\infty,0)$, there exists a $\phi$ satisfying $\int_{R^d}\phi(x)dx=l$ and such that the corresponding solution to the above equation is global, and there exists a $\phi$ satisfying $\int_{R^d}\phi(x)dx=l$ and such that the corresponding solution is non-global.

The infimal Heegaard gradient of a 3-manifold was defined and studied by Marc Lackenby in an approach towards proving the well-known virtually Haken conjecture. As instructive examples, Seifert fibered 3-manifolds are considered in this paper. The author shows that a compact orientable Seifert fibered 3-manifold has zero infimal Heegaard gradient if and only if it virtually fibers over either the circle or a surface other than the 2-sphere or, equivalently, if it has infinite fundamental group.

This paper provides a new, simpler proof of the fact that the smallest volume hyperbolic 3-manifold has betti number at least 3. In the process, the best known lower bound on the volume of such a manifold is improved.

Eric Charles Milner was born on 17 May 1928 and brought up in London. His father was an engineer, but times were hard and work was often difficult to obtain. So his mother had to help out by working as a seamstress, and Eric was often looked after by his grandmother. At the age of 11, he won a scholarship to the Haberdashers' Aske's Boys' School, but never attended it in its permanent London buildings because the outbreak of the Second World War caused all London schools and their pupils to be evacuated to safer parts of the country. As a result, Eric, an only child and knowing none of his new schoolfellows, was billeted at a home near Reading where he was extremely unhappy. In despair, he ran away and returned to London, where, after unsuccessful attempts to find him another billet, he roamed the streets and missed school. After some time, he was eventually found another billet where he received kindness and was much happier. Despite these disruptions and the other inevitable shortcomings of a war-time education, Eric's intelligence more than sufficed to surmount such hurdles, and in later life he could speak and write better than most of us.

From 1946 to 1951, Eric attended King's College, London. He graduated with First Class Honours in 1949, when he was awarded the Drew Gold Medal as the most distinguished Mathematics student in that year, and a Research Studentship. He then studied for an MSc degree, taking ‘Modern algebra’ and ‘Quantum mechanics (Wave mechanics)’ as his selected subjects, his supervisors being Richard Rado (then a Reader at King's College) and Professor Charles Coulson. He received the MSc degree, with distinction, in 1950. This was followed by a year's research in quantum mechanics under the supervision of Professor Coulson.

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 G be the group given by the following presentation:

formula here

The subgroup generated by ab is infinite-cyclic and normal, with quotient the dihedral group of order 6, so G is cyclic-by-finite. The subgroups H = 〈a, c〉 and K = 〈b, c〉 are both dihedral of order 6, and G is isomorphic to the free product of H and K amalgamating L = H∩K. We study K0(kG), the Grothendieck group of isomorphism classes of finitely generated projective kG-modules, and, in particular, the dependence of K0(kG) on the choice of field k. As usual, let ℚ, ℝ and [Copf ] stand for the rationals, reals and complex numbers, respectively. We prove the following.

The purpose of this paper is to give sufficient conditions for all nontrivial solutions of the nonlinear differential equation x″ +a(t)g(x) = 0 to be nonoscillatory. Here, g(x) satisfies the sign condition xg(x) > 0 if x ≠ 0, but is not assumed to be monotone increasing. This differential equation includes the generalized Emden–Fowler equation as a special case. Our main result extends some nonoscillation theorems for the generalized Emden–Fowler equation. Proof is given by means of some Liapunov functions and phase-plane analysis.

Let G be a finite group, let V be an [ ]G-module of finite dimension d, and denote by β(V, G) the minimal number m such that the invariant ring S(V)G is generated by finitely many elements of degree at most m. A classical result of E. Noether says that β(V, G)[les ][mid ]G[mid ] provided that char [ ] is coprime to [mid ]G[mid ]!. If char [ ] divides [mid ]G[mid ], then no bounds for β(V, G) are known except for very special choices of G. In this paper we present a constructive proof of the following. If H[les ]G with [G[ratio ]H]∈[ ]*, and if the restriction V[mid ]H is a permutation module (for example, if V is a projective [ ]G-module and H∈Sylp(G)), then β(V, G)[les ]max{[mid ]G[mid ], d([mid ]G[mid ]−1)} regardless of char [ ].

A sufficient condition is provided for a compact set to be removable for analytic functions in the Zygmund spaces in terms of a porosity. As a consequence, it is shown that the classes of removable sets for the Zygmund space and the little Zygmund space are different.

We construct a c0-valued random variable X such that (Sn/√n)n∈ℕ has a convergent subsequence, but X does not satisfy the central limit theorem.

For a wide class of set ideals (including, for example, all uniform ideals), a criterion of completeness of their symmetry groups is provided in terms of ideal quotients (polars). We apply it to partition ideals, and derive the extended Sierpiński–Erdös duality principle. We demonstrate that if the measure and category ideals I0 and I1 on the real line ℝ are partition (or, equivalently, if just I0 is partition), then not only are they isomorphic via an involution, but they also have complete (and distinct) symmetry groups coinciding, respectively, with the symmetry groups of the polars I⊥0 and I⊥1. The measure and category ideals on ℝ (and in more general spaces) are partition (Oxtoby) ideals assuming Martin's Axiom. In this case their polars are, respectively, the ideals generated by c-Sierpiński and c-Lusin sets. It is well known that the isomorphism of the measure and category ideals is not provable in ZFC. We show that the isomorphism of their symmetry groups is likewise unprovable.

Let $\mathfrak{g}=\mathfrak{g}_0\oplus\mathfrak{g}_1$ be a $\mathbb Z_2$-graded simple Lie algebra. Fix a Borel subalgebra $\mathfrak b_0\subset\mathfrak{g}_0$. Let $\mathfrak a\subset\mathfrak{g}_1$ be a $\mathfrak b_0$-stable subalgebra. Then $\mathfrak a$ is automatically commutative. It is known that if $\overline{G{\cdot}\mathfrak a}$ is a spherical $G$-variety, then $G_0{\cdot}\mathfrak a$ is a spherical $G_0$-variety. We describe all $\mathbb Z_2$-gradings having the property that $\overline{G{\cdot}\mathfrak a}$ is a spherical $G$-variety for any $\mathfrak a$.

We prove uniqueness for the nonharmonic trigonometric series [sum ]∞k=0akeiλkx under the weaker condition (*) where [sum ]∞k=0 [mid ]ak[mid ]/exp[mid ]λk[mid ]γ < ∞, for some 0 < γ < 1. In other words, if {λk}∞0 satisfies the above condition (*), and if [sum ]∞k=0akeiλkx = 0, then ak = 0 for all k = 0, 1,…. Finally, we state an improvement of Zygmund's uniqueness result as a corollary.

It is shown that if a C*-algebra A contains a semi-scattered C*-algebra B such that the pure states of B and the zero functional extend uniquely to A, and the canonical mapping Bˆ → Â is injective, then there exists a (unique) projection of norm one R : A → B. In certain circumstances, the conditional expectation R can be effected by a unitary averaging process using unitary elements in the centre of the multiplier algebra M(B).