Using uniformization of Riemann surfaces by Fuchsian groups and the equisymmetric stratification of the branch locus of the moduli space of surfaces of genus 4, we prove its connectedness. As a consequence, one can deform a surface of genus 4 with automorphisms, i.e. symmetric, to any other symmetric genus 4 surface through a path consisting entirely of symmetric surfaces.

By applying the Sears non-terminating transformations, we establish four general transformation theorems for double basic hypergeometric series of type Φ2:1;λ2:0;μ. Moreover, several transformation, reduction and summation formulae on the double basic hypergeometric series Φ2:1;22:0;1, Φ2:1;32:0;2 and Φ2:1;42:0;3 are also derived through parameter specialisation.

We prove that for each Riemann surface of finite analytic type (p, n) with p ≥ 2, there exist uncountably many Teichmüller disks Δ in the Teichmüller space T(S), where S = - {a point a}, with these properties: (1) the natural projection j: T(S) → T() defined by forgetting a induces an isometric embedding of each Δ into T(); and (2) the stabilizer of each Teichmüller disk Δ in the a-pointed mapping class group of S is trivial.

Let a, b and n be non-negative integers such that 1 ≤ a ≤ b, and let G be a graph of order p with and f be an integer-valued function defined on V(G) such that a ≤ f(x) ≤ b for all x ∈ V(G). Let h: E(G) → [0, 1] be a function. If ∑e∋xh(e) = f(x) holds for any x ∈ V(G), then we call G[Fh] a fractional f-factor of G with indicator function h, where Fh = {e ∈ E(G): h(e) > 0}. A graph G is called a fractional (f, n)-critical graph if after deleting any n vertices of G the remaining graph of G has a fractional f-factor. In this paper, it is proved that G is a fractional (f, n)-critical graph if for every non-empty independent subset X of V(G), and . Furthermore, it is shown that the result in this paper is best possible in some sense.

Let R be a commutative ring with identity. We will say that an R-module M has Nakayama property, if IM = M, where I is an ideal of R, implies that there exists a ∈ R such that aM = 0 and a − 1 ∈ I. Nakayama's Lemma is a well-known result, which states that every finitely generated R-module has Nakayama property. In this paper, we will study Nakayama property for modules. It is proved that R is a perfect ring if and only if every R-module has Nakayama property (Theorem 4.9).

Type I domains are the domains of the self-adjoint operators determined by the weak formulation of formally self-adjoint differential expressions ℓ. This class of operators is defined by the requirement that the sesquilinear form q(u, v) obtained from ℓ by integration by parts agrees with the inner product 〈ℓu, v〉. A complete characterisation of the boundary conditions assumed by functions in these domains for second-order differential expressions is given in this paper. In the singular case, the boundary conditions are stated in terms of certain ‘boundary condition’ functions and in the regular case they are given in terms of classical function values.

In this paper, we investigate linear Weingarten hypersurfaces with two distinct principal curvatures in a real space form Mn+1(c), we obtain two rigidity results and give some characterization of the Riemannian product Sk(a) × Sn−k(), 1 ≤ k ≤ n − 1 in Mn+1(c)(c = 1), the Riemannian product Rk × Sn−k(a), 1 ≤ k ≤ n −1 in Mn+1(c)(c = 0) and the Riemannian product Hk(tanh2 ρ−1) × Sn−k(coth2 ρ−1), 1 ≤ k ≤ n −1 in Mn+1(c)(c = −1).

Let H be a Hopf algebra, A/B be an H-Galois extension. Let D(A) and D(B) be the derived categories of right A-modules and of right B-modules, respectively. An object M⋅ ∈ D(A) may be regarded as an object in D(B) via the restriction functor. We discuss the relations of the derived endomorphism rings EA(M⋅) = ⊕i∈ℤHomD(A)(M⋅, M⋅[i]) and EB(M⋅) = ⊕i∈ℤHomD(B)(M⋅, M⋅[i]). If H is a finite-dimensional semi-simple Hopf algebra, then EA(M⋅) is a graded sub-algebra of EB(M⋅). In particular, if M is a usual A-module, a necessary and sufficient condition for EB(M) to be an H*-Galois graded extension of EA(M) is obtained. As an application of the results, we show that the Koszul property is preserved under Hopf Galois graded extensions.

We present two new proofs of the q-commuting property holding among certain pairs of quantum minors of a q-generic matrix. The first uses elementary quasi-determinantal arithmetic; the second involves paths in a directed graph. Together, they indicate a means to build the multi-homogeneous coordinate rings of flag varieties in other non-commutative settings.

We develop a general theory of Hopf image of a Hopf algebra representation, with the associated concept of inner faithful representation, modelled on the notion of faithful representation of a discrete group. We study several examples, including group algebras, enveloping algebras of Lie algebras, pointed Hopf algebras, function algebras, twistings and cotwistings, and we present a Tannaka duality formulation of the notion of Hopf image.

The ‘polaroid’ property transfers from Banach algebra elements to their tensor product, and hence also to their induced multiplications on ‘ultraprime’ Banach bimodules.

Walter Wilson Stothers was born in Glasgow on 8 November 1946. A third (youngest) son, he had the identical name to his father. From childhood, however, he had always been known by his middle name ‘Wilson’, so that his father, a Glasgow GP, would never be referred to as ‘Old Walter’. His mother, as Jean Young Kyle, had herself graduated in Mathematics in 1927, a rare achievement for a woman at that time. After attending the local primary school 1952–1956, Wilson completed his primary education in the preparatory classes in Allan Glen's School, then a distinguished Glasgow boys school with a scientific emphasis, progressing to the secondary school in 1958 and ending by becoming Dux in 1964. He also played in the school rugby first XV. From 1964–1968 he was a student in the Science Faculty of Glasgow University. His original intention was to take Honours in Chemistry and, indeed, he won the Chemistry prize in his first year. But he excelled in all subjects, winning the Faraday medal in the Intermediate Honours (second year) class in Natural Philosophy (Physics). After that he concentrated on Mathematics and became the top student, gaining a First Class Honours degree, as well as the Cunninghame Medal and a Jack Scholarship to Peterhouse College, Cambridge (1968). Before commencing postgraduate studies he married Andrea Watson in September 1968. At Cambridge from 1968–1971 he studied for a Ph.D. in number theory under the supervision of Peter Swinnerton-Dyer and graduated in 1972 with the thesis Some Discrete Triangle Groups. By then he was becoming aware of the strange realm inhabited by mathematicians that he seemed to be entering. So when his Cambridge room-mate Bob Odoni, at a research meeting they were attending as postgrads, asked, ‘Wilson, do you realise that we are the only normal people here’, Wilson felt compelled to respond, ‘What makes you think that we are normal?’