In this paper we extend two theorems from [2] on p-adic subanalytic sets, where p is a fixed prime number, Qp is the field of p-adic numbers and Zp is the ring of p- adic integers. One of these theorems [2, 3.32] says that each subanalytic subset of Zp is semialgebraic. This is extended here as follows.

Mahler's measure of a polynomial can be written as a logarithmic integral over the torus. We propose a definition when the underlying group is an elliptic curve. Having reviewed some of the classical results in the toral case, we take some first steps towards realising elliptic analogues. In particular, we focus on elliptic analogues of Kronecker's theorem and Lehmer's problem. We wish to stress the fundamental role played by Jensen's formula in both the toral and elliptic formulations of these results.

It is proved that there is a certain degree of independence between stationary reflection phenomena at different cofinalities.

Let a=(a1, a2, a3, …) be an arbitrary infinite sequence in U=[0, 1). Let

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Van der Corput [5] conjectured that d(a, n) (n=1, 2, …) is unbounded, and this was proved in 1945 by van Aardenne-Ehrenfest [1]. Later she refined this [2], obtaining

formula here

for infinitely many n. Here and later c1, c2, … denote positive absolute constants.

In 1954, Roth [8] showed that the quantity

formula here

is closely related to the discrepancy of a suitable point set in U2.

It is well known that the multiplicity of a complex zero ρ=β+iγ of the zeta-function is O(log[mid ]γ[mid ]). This may be proved by means of Jensen's formula, as in Titchmarsh [7, Chapter 9]. It may also be seen from the formula for the number N(T) of zeros such that 0<γ<T,

formula here

due to Backlund [1], in which E(T) is a continuous function satisfying E(T)=O(1/T) and

formula here

We assume here that T is not the ordinate of a zero; with appropriate definitions of N(T) and S(T) the formula is valid for all T. We have S(T)=O(logT). On the Lindelöf Hypothesis S(T)=o(logT), (Cramér [2]), and on the Riemann Hypothesis

formula here

(Littlewood [5]). These results are over 70 years old.

formula here

Because the multiplicity problem is hard, it seems worthwhile to see what can be said about the number of distinct zeros in a short T-interval. We obtain the following result, which is independent of any unproved hypothesis.

Let Δ be a finite quiver, that is, a finite directed graph, k be a commutative ring, and kΔ be the semigroup ring of paths in Δ. In this paper we compute the Hochschild homology of the following classes of algebras:

(1) kΔ/[mfr ]n, where [mfr ] is the arrow ideal;

(2) kΔ/I where I is an ideal generated by quadratic monomials.

In Section 2 we establish the notation, and recall a projective resolution of kΔ0, the degree 0 part in kΔ, over a monomial algebra which is due to Anick and Green. This resolution is then used in Section 3 as the building block in the construction of a resolution of a truncated or quadratic monomial algebra A, over its enveloping algebra (in the Hochschild sense), Ae. The fact that a monomial algebra possesses a fine grading then enables us to compute the homology.

The results obtained extend results by Liu and Zhang [3] and Geller, Reid and Weibel [2].

If σ is an automorphism and δ is a σ-derivation of a ring R, then the subring of invariants is the set R(δ)={r∈R[mid ]δ(r)=0}. The main result of this paper is ‘let R be a semiprime ring with an algebraic σ-derivation δ such that R(δ) is central; then R is commutative’. This theorem generalizes results on the invariants of automorphisms and derivations and is proved by reducing down to the special cases of automorphisms and derivations.

It is proved that a graph K has an embedding as a regular map on some closed surface if and only if its automorphism group contains a subgroup G which acts transitively on the oriented edges of K such that the stabiliser Ge of every edge e is dihedral of order 4 and the stabiliser Gv of each vertex v is a dihedral group the cyclic subgroup of index 2 of which acts regularly on the edges incident with v. Such a regular embedding can be realised on an orientable surface if and only if the group G has a subgroup H of index 2 such that Hv is the cyclic subgroup of index 2 in Gv. An analogous result is proved for orientably-regular embeddings.

We study two aspects of generation of large exceptional groups of Lie type. First we show that any finite exceptional group of Lie rank at least four is (2,3)-generated, that is, a factor group of the modular group PSL2(ℤ). This completes the study of (2,3)-generation of groups of Lie type. Second, we complete the proof that groups of type E7 and E8 over fields of odd characteristic occur as Galois groups of geometric extensions of ℚab(t), where ℚab denotes the maximal Abelian extension field of ℚ. Finally, we show that all finite simple exceptional groups of Lie type have a pair of strongly orthogonal classes. The methods of proof in all three cases are very similar and require the Lusztig theory of characters of reductive groups over finite fields as well as the classification of finite simple groups.

We continue an investigation of complementation in closed ideals of commutative group algebras. The main result here is a sufficient condition for a closed ideal of a commutative group algebra to have a Banach space complement. This involves the comparison of uniformities on the hull of the ideal.

This paper is centred around a single question: can a minimal left ideal L in G[Lscr ][Uscr ][Cscr ], the largest semi-group compactification of a locally compact group G, be itself algebraically a group? Our answer is no (unless G is compact). In deriving this conclusion, we obtain for nearly all groups the stronger result that no maximal subgroup in L can be closed. A feature of our work is that completely different techniques are required for the connected and totally disconnected cases. For the former, we can rely on the extensive structure theory of connected, non-compact, locally compact groups to derive the solution from the commutative case, using some reduction lemmas. The latter directly involves topological dynamics; we construct a compact space and an action of G on it which has pathological properties. We obtain other results as tools towards our main goal or as consequences of our methods. Thus we find an extension to earlier work on the relationship between minimal left ideals in G[Lscr ][Uscr ][Cscr ] and H[Lscr ][Uscr ][Cscr ] when H is a closed subgroup of G with G/H compact. We show that the distal compactification of G is finite if and only if the almost periodic compactification of G is finite. Finally, we use our methods to show that there is no finite subset of G[Lscr ][Uscr ][Cscr ] invariant under the right action of G when G is an almost connected group or an IN-group.

Let F be a meromorphic function in the complex plane. We investigate the behaviour of the iterates of F in a Baker domain B. In particular, we describe the dynamics of the orbits with the help of conformal conjugacies; that is, we determine a function φ which is univalent in a large simply connected subdomain of B such that φ(F(z))=T(φ(z)) holds throughout B. Here T is either a parabolic or hyperbolic Möbius transformation mapping either a half plane or [Copf ] onto itself. This functional equation is always solvable in a Baker domain if F has only finitely many poles. Moreover, there is an example of a function with infinitely many poles where one cannot find an appropriate conformal conjugacy in an invariant Baker domain.

For an irrational rotation α of the circle group T=R/Z and a piecewise absolutely continuous function f@[ratio ]T→R, the unitary operator Vh(x)=e2πif(x)h(x+α) on L2(T) is studied. It is shown that if f has a single discontinuity with non-integer jump then V is κ-weakly mixing for some κ with 0<[mid ]κ[mid ]<1. In particular V has continuous singular spectrum. The property of κ-weak mixing (with possible change of the value of κ, 0<[mid ]κ[mid ]<1) holds for all irrational rotations and, given α, is stable under perturbations of f by functions with sufficiently small O(1/n)-norm. On the other hand, there exists a piecewise linear function f with two non-integer jumps such that the spectrum of V is continuous singular for one value of α and Lebesgue for another.

We investigate the location and nature of the spectrum of the fourth-order self-adjoint equation

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subject to certain asymptotic assumptions on the coefficients. The main tools are the theory of asymptotic integration and the Titchmarsh–Weyl M-matrix. Asymptotic integration yields asymptotic formulae for the solutions of the differential equation which are then used to derive properties of the M-matrix. The characterisation of spectral properties in terms of the boundary behaviour of M leads to the desired results.

Let f and g be two analytic function germs without common branches. We define the Jacobian quotients of (g, f), which are ‘first order invariants’ of the discriminant curve of (g, f), and we prove that they only depend on the topological type of (g, f). We compute them with the help of the topology of (g, f). If g is a linear form transverse to f, the Jacobian quotients are exactly the polar quotients of f and we affirm the results of D. T. Lê, F. Michel and C. Weber.

We consider the appearance of discrete spectrum in spectral gaps of magnetic Schrödinger operators with electric background field under strong, localised perturbations. We show that for compactly supported perturbations the asymptotics of the counting function of the occurring eigenvalues in the limit of a strong perturbation does not depend on the magnetic field nor on the background field.

We prove Tauberian theorems from Jp-summability methods of power series type to ordinary convergence, respectively Mp-summability methods of weighted means. Particular cases are the Abel and Cesàro, as well as logarithmic and harmonic summability. Besides numerical series, we also consider orthogonal series with coefficients from [lscr ]2. In the latter case, it turns out that one of our Tauberian conditions is satisfied almost everywhere on the underlying measure space, thereby proving the claim stated in the title.

In this paper we shall be dealing with best constants characterising the embedding of a Sobolev space of L2-type Hl=Wl2 into the space of bounded continuous functions when l>n/2. More specifically, we are interested in the value of the best constant cM(p, l) in the inequality

formula here

where M stands for Euclidean space Rn or the n-sphere Sn (in the latter case f is assumed to have zero average (f, 1)=0). Accordingly, Δ is either the classical Laplace operator or the Laplace–Beltrami operator acting on the surface of Sn[ratio ]Δf(s)= Δf(x/[mid ]x[mid ])[mid ]x=s, s∈Sn, n[ges ]2 (on S1, of course, Δf=f″).

Throughout ∥·∥ is the L2-norm, p and l are real numbers satisfying

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and θ=(2l−n)/(2(l−p)), 1−θ=(n−2p)/(2(l−p)).

Before describing the contents of the paper we recall the well-known references [3, 10, 11, 16] and the survey [18] where best constants and corresponding extremal functions of the Sobolev embeddings in Rn were dealt with.

We give necessary and sufficient conditions on the parameters s1, s2, …, sm, p1, p2, …, pm such that the Jacobian determinant extends to a bounded operator from [Hscr ]s1p1× [Hscr ]s2p2×…× [Hscr ]smpm into [Zscr ]′. Here all spaces are defined on ℝn and 2[les ]m[les ]n. In almost all cases the regularity of the Jacobian determinant is calculated exactly.

Several differential geometric aspects of the space of representations of amenable Banach algebras are studied: differential structure, natural connection, geodesics, reductive structure, and so on. As an application we get a characterisation of amenable groups in terms of the existence of a reductive structure in that space.