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.

A finite rewriting system is presented that does not satisfy the homotopical finiteness condition FDT, although it satisfies the homological finiteness condition FHT. This system is obtained from a group $G$ and a finitely generated subgroup $H$ of $G$ through a monoid extension that is almost an HNN extension. The FHT property of the extension is closely related to the ${\rm FP}_2$ property for the subgroup $H$, while the FDT property of the extension is related to the finite presentability of $H$. The example system separating the FDT property from the FHT property is then obtained by applying this construction to an example group.

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]).

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.

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.

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.

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.

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].

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 $p>n$.

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.

An asymptotically perturbed cylinder is a manifold which, to the exterior of a compact set, is of the form R×M, where M is a compact manifold (with or without boundary), and for which the metric approaches the product metric as the axial variable tends to infinity. The properties of eigenfunctions of the Laplacian on asymptotically perturbed cylinders, with either Dirichlet or Neumann boundary conditions if the boundary is non-empty, are studied. For a large class of asymptotic perturbations, the eigenfunctions are proved to decay faster than the reciprocal of any polynomial as the axial variable tends to infinity. The decay estimates are then used to prove that the eigenvalues are of finite multiplicity, and can accumulate at the thresholds only from below. The results are shown to apply to a large class of acoustic and quantum waveguides.

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

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].

We define a sequence of real functions which coincide with Li's coefficients at one and which allow us to extend Li's criterion for the Riemann Hypothesis to yield a necessary and sufficient condition for the existence of zero-free strips inside the critical strip $0<\Re(z)<1$. We study some of the properties of these functions, including their oscillatory behaviour.

Let M1 and M2 be two simply connected closed manifolds of the same dimension. It is proved that

(1) if k is a coefficient field such that neither M1 nor M2 has the same cohomology as a sphere, then the sequence (bk)k[ges ]1 of Betti numbers of the free loop space on M1 #M2 is unbounded;

(2) if, moreover, the cohomology H*(M1;k) is not generated as algebra by only one element, then the sequence (bk)k[ges ]1 has an exponential growth.

Thanks to theorems of Gromoll and Meyer and of Gromov, this implies, in case 1, that there exist infinitely many closed geodesics on M1#M2 for each Riemannian metric, and, in case 2, that for a generic metric, the number of closed geodesics of length [les ]t grows exponentially with t.

An approach of Swinnerton-Dyer is extended to obtain new upper bounds for the number of lattice points close to a smooth curve. One consequence of these bounds is a new asymptotic result for the distribution of squarefull numbers in short intervals.

The main result provides mild conditions under which a closed, orientable, PL 4-manifold $N = N_1\,\#\,N_2$ with $\pi_1(N_e)$ residually finite ($e=1,2$) is a codimension-5 PL fibrator. The paper also presents a rich variety of conditions on a closed 4-manifold $N^4$ under which every PL map between manifolds, where the domain is orientable and all point inverses are copies of $N^4$, must be an approximate fibration.

The paper proves two univalence criteria involving harmonic mappings in multiply connected domains. These generalise a classical result of Radó, Kneser and Choquet and a recent result of Bshouty and Hengartner.