We determine the automorphism group for a large class of affine quadrics over a field, viewed as affine algebraic varieties. The proof uses a fundamental theorem of Karpenko's in the theory of quadratic forms [13], along with some useful arguments of birational geometry. In particular, we find that the automorphism group of the n-sphere {x02+···+xn2=1} over the real numbers is just the orthogonal group O(n+1) whenever n is a power of 2. It is not known whether the same is true for arbitrary n. This result is reminiscent of Wood's theorem that when n is a power of 2, every real polynomial mapping from the n-sphere to a lower-dimensional sphere is constant [22].

In this paper we completely solve a family of Thue inequalities defined over the field of functions , namely deg (X4−4cX3Y+(6c+2)X2Y2+4cXY3+Y4) ≤ deg c, where the solutions x,y come from the ring and the parameter is some non-constant polynomial.

Let G be a locally compact group and be its largest semigroup compactification. Then sx≠x in whenever s is an element in G other than e. This result was proved by Ellis in 1960 for the case G discrete (and so is the Stone–Čech compactification βG of G), and by Veech in 1977 for any locally compact group. We study this property in the WAP – compactification of G; and in , we look at the situation when xs≠x. The points are separated by some weakly almost periodic functions which we are able to construct on a class of locally compact groups, which includes the so-called E-groups introduced by C. Chou and which is much larger than the class of SIN groups. The other consequences deduced with these functions are: a generalization of some theorems on the regularity of due to Ruppert and Bouziad, an analogue in of the “local structure theorem” proved by J. Pym in , and an improvement of some earlier results proved by the author and J. Baker on .

We first study the minimum total curvature of a knot when it is embedded on the cubic lattice. Let be a knot or link with a lattice embedding of minimum total curvature among all possible lattice embeddings of . We show that there exist positive constants c1 and c2 such that for any knot type . Furthermore we show that the powers of in the above inequalities are sharp hence cannot be improved in general. Our results and observations show that lattice embeddings with minimum total curvature are quite different from those with minimum or near minimum lattice embedding length. In addition, we discuss the relationship between minimal total curvature and minimal ropelength for a given knot type. At the end of the paper, we study the total curvatures of smooth thick knots and show that there are some essential differences between the total curvatures of smooth thick knots and lattice knots.

The reversible elements of a group are those elements that are conjugate to their own inverse. A reversible element is said to be reversible by an involution if it is conjugate to its own inverse by an involution. In this paper, we classify the reversible elements and the elements reversible by involutions in the group of quaternionic Möbius transformations.

We show that any closed hyperbolic surface admitting a conformal automorphism with “many” fixed points is uniformly quasiconformally homogeneous, with constant uniformly bounded away from 1. In particular, there is a uniform lower bound on the quasiconformal homogeneity constant for all hyperelliptic surfaces. In addition, we introduce more restrictive notions of quasiconformal homogeneity and bound the associated quasiconformal homogeneity constants uniformly away from 1 for all hyperbolic surfaces.

Suppose that f(z) is a transcendental entire function and that the Fatou set F(f)≠∅. Setandwhere the supremum supU is taken over all components of F(f). If B1(f)<∞ or B2(f)<∞, then we say F(f) is strongly uniformly bounded or uniformly bounded respectively. We show that, under some conditions, F(f) is (strongly) uniformly bounded.

We prove that spectral synthesis holds on a discrete Abelian group G if and only if the torsion free rank of G is finite.

Pre-symmetric complex Banach spaces have been proposed as models for state spaces of physical systems. A neutral GL-projection on a pre-symmetric space represents an operation on the corresponding system, and has as its range a further pre-symmetric space which represents the state space of the resulting system. Two neutral GL-projections S and T on the pre-symmetric space A* are said to be L-orthogonal if for all elements x in SA* and y in TA*,By studying the algebraic properties of the dual space A of A*, which is a JBW*-triple, it is shown that, provided that the orthogonal neutral GL-projections S and T satisfy a certain geometrical condition, there exists a smallest neutral GL-projection S∨T majorizing both S and T, and that S, T and S∨T form a compatible family.

We consider random functions formed as sums of pulseswhere Xn are independent random vectors, 0<α<1, and G is an elementary “pulse” or “bump”. Typically such functions have fractal graphs and we find the Hausdorff dimension of these graphs using a novel variant on the potential theoretic method.

For a polyhedral domain , and a Hilbert metric nonexpansive map T:Σ→Σ which does not have a fixed point in Σ, we prove that the omega limit set ω(x;T) of any point x ∈ Σ is contained in a convex subset of the boundary ∂Σ. We also identify a class of order-preserving homogeneous of degree one maps on the interior of the standard cone which demonstrate that there are Hilbert metric nonexpansive maps on an open simplex with omega limit sets that can contain any convex subset of the boundary.

A coupled cell system is a collection of dynamical systems, or ‘cells’, that are coupled together. The associated coupled cell network is a labelled directed graph that indicates how the cells are coupled, and which cells are equivalent. Golubitsky, Stewart, Pivato and Török have presented a framework for coupled cell systems that permits a classification of robust synchrony in terms of the concept of a ‘balanced equivalence relation’, which depends solely on the network architecture. In their approach the network is assumed to be finite. We prove that the set of all balanced equivalence relations on a network forms a lattice, in the sense of a partially ordered set in which any two elements have a meet and a join. The partial order is defined by refinement. Some aspects of the theory make use of infinite networks, so we work in the category of networks of ‘finite type’, a class that includes all locally finite networks. This context requires some modifications to the standard framework. As partial compensation, the lattice of balanced equivalence relations can then be proved complete. However, the intersection of two balanced equivalence relations need not be balanced, as we show by a simple example, so this lattice is not a sublattice of the lattice of all equivalence relations with its usual operations of meet and join. We discuss the structure of this lattice and computational issues associated with it. In particular, we describe how to determine whether the lattice contains more than the equality relation. As an example, we derive the form of the lattice for a linear chain of identical cells with feedback.

We study the spectral gaps of the Schrödinger operatorwhere κ∈(0,2π) and are parameters. Let τ=2π−κ. Suppose that the ratio κ0:=τ/κ is irrational. We denote the jth gap of the spectrum of H by Gj, its length by |Gj|. We obtain a relationship between the asymptotic behaviour of |Gj| as j→∞ and the Diophantine properties of κ0. In particular, we show that if β1+β2=0, thenwhere M(κ0) stands for the Markov constant of κ0.

Stochastic generators of completely positive and contractive quantum stochastic convolution cocycles on a C*-hyperbialgebra are characterised. The characterisation is used to obtain dilations and stochastic forms of Stinespring decomposition for completely positive convolution cocycles on a C*-bialgebra.

A new approach for studying initial-boundary value problems for linear partial differential equations (PDEs) with variable coefficients was introduced recently by the second author, and was applied to PDEs involving second order derivatives. Here, we extend this approach further to solve an initial-boundary value problem for a third-order evolution PDE with a space-dependent coefficient. The analysis is presented in such a way that it can be applied to PDEs with higher derivatives, and thus provides a method for solving initial-boundary value problems for a certain class of linear evolution equations with variable coefficients of arbitrary order.

The Stokes operator E2 governs the irrotational axisymmetric Stokes flow and its square governs the corresponding rotational flow. In spheroidal coordinates the elements of the solution space ker E2 enjoy a spectral decomposition into separable eignefunction, while the elements of the ker E4 accept a spectral decomposition in terms of semiseparable eigensolutions involving 3D-by-3D eigenfunctions of the Gegenbauer operator. These spectral characteristics are utilized to construct the fundamental solutions for both the E2 and the E4 operators in spheroidal geometry. The fundamental solution for E2 is expressed in terms of the elements of the irrotational space ker E2, while the fundamental solution for E4 is expressed in terms of the corresponding generalized eigenfunctions alone.

We fill a gap in the construction of the extended Johnson homomorphism, stated in Section 2 of the paper The abelianization of the congruence IA-automorphism group of a free group.