The paper is devoted to the study of groups whose word problem can be solved by a Turing machine which operates in real time. A recent result of the first author for word hyperbolic groups is extended to prove that under certain conditions the generalised Dehn algorithms of Cannon, Goodman and Shapiro, which clearly run in linear time, can be programmed on real-time Turing machines. It follows that word-hyperbolic groups, finitely generated nilpotent groups and geometrically finite hyperbolic groups all have real-time word problems.

The non-degeneracy of the canonical $p$-adic height pairing defined by Perrin-Riou and Schneider on an elliptic curve over a number field with good, ordinary reduction is still unknown.

Following the work done for the real-valued pairing, the behaviour of the $p$-adic height is analysed as a point varies on a section of a family of elliptic curves, and so new information is obtained about this pairing. In particular, the variation is $p$-adically continuous and the non-degeneracy of a set of sections can be checked simultaneously for almost all elements of the family. The paper contains some conjectures about the valuation of the $p$-adic regulator and its consequences for the growth of the Mordell–Weil group in cyclotomic $\mathbb{Z}_p$-extensions.

We develop a theory of harmonic analysis and duality for finite commutative hypergroups by considering somewhat more general objects called signed hypergroups. A notion of entropy is defined, and a Second Law of Thermodynamics is established. Applications to group theory and to the fusion rule algebras of conformal field theory are given.

JB*-triples occur in the study of bounded symmetric domains in several complex variables and in the study of contractive projections on C*-algebras. These spaces are equipped with a ternary product {·,·,·}, the Jordan triple product, and are essentially geometric objects in that the linear isometries between them are exactly the linear bijections preserving the Jordan triple product (cf. [23]).

A noninjective bounded-to-one factor map from an irreducible shift of finite type onto a sofic system can be factored as a composition of other such maps in only finitely many ways (up to isomorphism). This generalizes to factor maps from systems with canonical coordinates to finitely presented dynamical systems.

We generalise group algebras to other algebraic objects with bounded Hilbert space representation theory; the generalised group algebras are called ‘host’ algebras. The main property of a host algebra is that its representation theory should be isomorphic (in the sense of the Gelfand–Raikov theorem) to a specified subset of representations of the algebraic object. Here we obtain both existence and uniqueness theorems for host algebras as well as general structure theorems for host algebras. Abstractly, this solves the question of when a set of Hilbert space representations is isomorphic to the representation theory of a C*-algebra. To make contact with harmonic analysis, we consider general convolution algebras associated to representation sets, and consider conditions for a convolution algebra to be a host algebra.

The determination of the modular character tables of the sporadic O'Nan group, its automorphism group and its covering group is completed by the calculation of the 7-modular decomposition numbers. The results are obtained with the assistance of the systems GAP, MOC, and MeatAxe, and by applying new condensation methods.

A simple geometrical argument shows that every pair of projections on a finite-dimensional complex vector space has a common invariant subspace of dimension 1 or 2. The idea extends to certain pairs of projections on an infinite-dimensional Hilbert space H. In particular every projection on H has a reducing subspace, although a finite-dimensional one need not exist. In a final section, the results are extended to the existence of hyperinvariant subspaces for pairs of projections.

Let $H$ be a finite dimensional non-semisimple Hopf algebra over an algebraically closed field $k$ of characteristic 0. If $H$ has no nontrivial skew-primitive elements, some bounds are found for the dimension of $H_1$, the second term in the coradical filtration of $H$. Using these results, it is shown that every Hopf algebra of dimension 14 is semisimple and thus isomorphic to a group algebra or the dual of a group algebra. Also a Hopf algebra of dimension $pq$ where $p$ and $q$ are odd primes with $p < q \leq 1 + 3p$ and $q \leq 13$ is semisimple and thus a group algebra or the dual of a group algebra. Some partial results in the classification problem for dimension 16 are also given.

Let $\Bbb H^n$ denote the (2n+1)-dimensional Heisenberg group. Given an operator-valued function $M$, define the operator $T_{M}$ by $(T_{M}f)\hat{\vphantom{f}}=\skew4\hat{f} M$ with ‘$\hat{}$’ denoting the Fourier transform. Hörmander-type sufficient conditions are determined on $M$ for the $H^p$-boundedness, $p\le 1$, of the operator $T_{M}$ on $\Bbb H^n$.

We describe complemented copies of $\ell_2$ both in $C(K_1)\hat{\otimes}_{\pi} C(K_2)$ when at least one of the compact spaces $K_i$ is not scattered and in $L_1(\mu_1)\hat{\otimes}_{\epsilon} L_1(\mu_2)$ when at least one of the measures is not atomic. The corresponding local construction gives uniformly complemented copies of the $\ell_2^n$ in $c_0\hat{\otimes}_{\pi} c_0$. We continue the study of $c_0\hat{\otimes}_{\pi} c_0$ showing that it contains a complemented copy of Stegall's space $c_0(\ell_2^n)$ and proving that $(c_0\ppi c_0)''$ is isomorphic to $\ell_\infty(\ell_\infty^n\hat{\otimes}_{\pi} \ell_\infty^n)$, together with other results. In the last section we use Hardy spaces to find an isomorphic copy of $L_p$ in the space of compact operators from $L_q$ to $L_r$, where $1<p,q,r<\infty$ and $1/r=1/p+1/q$.

Given a sequence of integers aj, $j\ge 1$, a multiset is a combinatorial object composed of unordered components, such that there are exactly aj one-component multisets of size j. When $a_j\asymp j^{r-1} y^j$ for some r > 0, $y\geq 1$, then the multiset is called expansive. Let cn be the number of multisets of total size n. Using a probabilistic approach, we prove for expansive multisets that $c_n/c_{n+1}\to 1$ and that cn/cn+1 < 1 for large enough n. This allows us to prove monadic second-order limit laws for expansive multisets. The above results are extended to a class of expansive multisets with oscillation.

Moreover, under the condition $a_j=Kj^{r-1}y^j + O(y^{\nu j})$, where K > 0, r > 0, y > 1, $\nu\in (0,1)$, we find an explicit asymptotic formula for cn. In a similar way we study the asymptotic behavior of selections, which are defined as combinatorial objects composed of unordered components of distinct sizes.

Let T be an ergodic and free ℤdrotation on the d-dimensional torus [ ]d given by

formula here

where (m1, …, md) ∈ ℤd, (z1, …, zd) ∈ [ ]d and [αjk]j,k=1 …, d ∈ Md(ℝ). For a continuous circle cocycle ϕ[ratio ]ℤd × [ ]d → [ ](ϕm+n(z) = ϕm(Tnz)ϕn(z) for any m, n ∈ ℤd), the winding matrix W(ϕ) of a cocycle ϕ, which is a generalization of the topological degree, is defined. Spectral properties of extensions given by

formula here

are studied. It is shown that if ϕ is smooth (for example ϕ is of class C1) and det W(ϕ) ≠ 0, then Tϕ is mixing on the orthocomplement of the eigenfunctions of T. For d = 2 it is shown that if ϕ is smooth (for example ϕ is of class C4), det W(ϕ) ≠ 0 and T is a ℤ2-rotation of finite type, then Tϕ has countable Lebesgue spectrum on the orthocomplement of the eigenfunctions of T. If rank W(ϕ) = 1, then Tϕ has singular spectrum.

In this paper, we are mainly concerned with $n$ -dimensional simplices in hyperbolic space ${\bb H}^n$ . We will also consider simplices with ideal vertices, and we suggest that the reader keeps the Poincaré unit ball model of hyperbolic space in mind, in which the sphere at infinity ${\bb H}^n(\infty)$ corresponds to the bounding sphere of radius 1. It is known that all hyperbolic simplices (even the ideal ones) have finite volume. However, explicit calculation of their volume is generally a very difficult problem (see, for example, [1] or [16]). Our first theorem states that, amongst all simplices in a closed geodesic ball, the simplex of maximal volume is regular. We call a simplex regular if every permutation of its vertices can be realized by an isometry of ${\bb H}^n$ . A corresponding result for simplices in the sphere has been proved by Böröczky [4].

This paper is mainly devoted to the study of the index of a map at a zero, and the index of a polynomial map over ℝn. For semi-quasi-homogeneous maps we prove that the index at a zero coincides with the index at this zero of its quasi-homogeneous part. For a class of polynomial maps with finite zero set we provide a method which makes easier the computation of its index over ℝn. Finally we relate the index and the multiplicity.

given an $n$-dimensional compact closed oriented manifold $m$ and a field $\mathbb k$, f. cohen and l. taylor have constructed a spectral sequence, ${\mathcal e}(m,n,\mathbb k)$, converging to the cohomology of the space of ordered configurations of $n$ points in $m$. the symmetric group $\sigma_n$ acts on this spectral sequence giving a spectral sequence of $\sigma_n$-differential graded commutative algebras. here, an explicit description is provided of the invariants algebra $(e_1,d_1)^{\sigma_n}$ of the first term of $\mathcal e(m,n,\mathbb q)$. this determination is applied in two directions.

(a) in the case of a complex projective manifold or of an odd-dimensional manifold $m$, the cohomology algebra $h^*(c_n(m);\mathbb q)$ of the space of unordered configurations of $n$ points in $m$ is obtained (the concrete example of $p^2(\mathbb c)$ is detailed).

(b) the degeneration of the spectral sequence formed of the $\sigma_n$-invariants $\mathcal e(m,n,\mathbb q)^{\sigma_n}$ at level 2 is proved for any manifold $m$.

these results use a transfer map and are also true with coefficients in a finite field $\mathbb f_p$ with $p>n$.

Let E2(T) denote the error term in the asymptotic formula for ∫T0[mid ]ζ(½+it)[mid ]4dt. It is proved that there exist constants A>0, B>1 such that for T[ges ]T0>0 every interval [T, BT] contains points T1, T2 for which

formula here

and that ∫T0[mid ]E2(t) [mid ]adt[Gt ]T1+(a/2) for any fixed a[ges ]1. These results complement earlier results of Motohashi and Ivić that ∫T0E2(t) dt[Lt ]T3/2 and that ∫T0E22(t) dt[Lt ]T2logCT for some C>0. Omega-results analogous to the above ones are obtained also for the error term in the asymptotic formula for the Laplace transform of [mid ]ζ(½+it)[mid ]4.

This paper is devoted to the study of a family of oscillatory multipliers of the Laplace–Beltrami operator on symmetric spaces of the noncompact type. This family is related to the regularity (in space, for fixed time) of the solutions to the wave equation. Our method relies on estimates for the Poisson semigroup in complex time, which were obtained in our previous paper [6].

A $5$-local characterization of the Lyons sporadic simple group is presented.

In this paper we study the value distribution of the least primitive root to a prime modulus, as the modulus varies. For each odd prime number p, we shall denote by g(p) and G(p) the least primitive root and the least prime primitive root (mod p), respectively. Numerical examples show that, in most cases, g(p) and G(p) are very small (cf. §4). We can support this observation by a probabilistic argument [14, §1]. In fact, on the assumption of a good distribution of the primitive residue classes modulo p, we can surmise that

formula here

where π(x) denotes the number of primes not exceeding x.