Let Fq be a finite field of characteristic r and order q=ra, V=Fnq the vector space of finite dimension n[ges ]1 over Fq, and Mat (n, q) the matrix algebra on V. We denote by Hn(q) (or Hn if q is clear from the context) one of the following classical groups: Sp (n, q) (with n even), the symplectic group on V; O(n, q) (with q odd), one of the full orthogonal groups on V in odd characteristic; U(n, q) (with q a square), the full unitary group on V.

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

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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

formula here

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.

For any closed oriented manifold $M$, the top degree multi-vector fields transverse to the zero section of $\wedge^{{\rm top}}TM$ are classified, up to orientation preserving diffeomorphism, in terms of the topology of the arrangement of their zero locus and a finite number of numerical invariants. The group governing the infinitesimal deformations of such multi-vector fields is computed, and an explicit set of generators exhibited. For the sphere $S^n$, a correspondence between certain isotopy classes of multi-vector fields and classes of weighted signed trees is established.

Let $\phi$ be a bounded linear functional on $A$, where $A$ is a commutative Banach algebra, then the bilinear functional $\skew5\check{\phi}$ is defined as $\skew5\check{\phi} (a,b)\,{=}\,\phi (ab)-\phi (a) \phi (b)$ for each $a$ and $b$ in $A$. If the norm of $\skew5\check{\phi}$ is small then $\phi$ is approximately multiplicative, and it is of interest whether or not $\|\skew5\check{\phi}\|$ being small implies that $\phi$ is near to a multiplicative functional. If this property holds for a commutative Banach algebra then $A$ is an AMNM algebra (approximately multiplicative functionals are near multiplicative functionals). The main result of the paper shows that ${\text{C}}^N {[0,1]}^M$ (the complex valued functions defined on ${[0,1]}^M$ with all $N$th order partial derivatives continuous) is AMNM. It is also shown that a similar proof can be applied to certain Lipschitz algebras.

Ramanujan claimed in his first letter to Hardy (16 January 1913) that $$\frac{1}{2}e^k-\sum_{\nu=0}^{k-1}\frac{k^{\nu}}{\nu !}=\frac{k^k}{k!}\bigg(\frac{1}{3}+\frac{4}{135(k+\theta(k))}\bigg) \qquad{(k=1,2,{\ldots})},$$ where $\theta(k)$ lies between $2/21$ and $8/45$. This conjecture was proved in 1995 by Flajolet et al. The paper establishes the following refinement. $$\frac{1}{2}e^k-\sum_{\nu=0}^{k-1}\frac{k^{\nu}}{\nu !}=\frac{k^k}{k!} \bigg(\frac{1}{3}+\frac{4}{135 k}-\frac{8}{2835(k+\theta^*(k))^2}\bigg) \qquad{(k=1,2,{\ldots})},$$ where $$-\frac{1}{3}<\theta^*(k)\leq -1+\frac{4}{\sqrt{21(368-135e)}}=-0.140\,74{\ldots}\,.$$ Both bounds for $\theta^*(k)$ are sharp.

Given a smooth family ${\hbox{\ac L}}$ of real or complex variable taking values within the class of Fredholm operators of index zero in a Banach space, there are some available definitions in the literature of the concept of algebraic multiplicity of the family ${\hbox{\ac L}}$ at a point $x_0$ of the parameter at which the operator ${\hbox{\ac L}}(x_0)$ becomes non-invertible. The purpose of the paper is to show that the algebraic multiplicity is uniquely determined by a few of its properties, independently of its construction. The main technical tools to obtain this uniqueness result are a Lyapunov–Schmidt reduction, the local Smith form and a new factorization result for general families at non-algebraic eigenvalues obtained in the paper.

A conjecture is formulated which relates the equivariant local epsilon constant of a Galois extension of $p$-adic fields to a natural algebraic invariant coming from étale cohomology. Some evidence for the conjecture is provided and its relation to a conjecture for the equivariant global epsilon constant of an extension of number fields formulated by Bley and Burns is established.

The paper defines a new type of product of association schemes (and of the related objects, permutation groups and orthogonal block structures), which generalizes the direct and wreath products (which are referred to as ‘crossing’ and ‘nesting’ in the statistical literature). Given two association schemes $\mathcal{Q}_r$ for $r\,{=}\,1,2$, each having an inherent partition $F_r$ (that is, a partition whose equivalence relation is a union of adjacency relations in the association scheme), a product of the two schemes is defined, which reduces to the direct product if $F_1=U_1$ or $F_2=E_2$, and to the wreath product if $F_1 = E_1$ and $F_2 = U_2$, where $E_r$ and $U_r$ are the relation of equality and the universal relation on $\mathcal{Q}_r$. The character table of the crested product is calculated, and it is shown that, if the two schemes $\mathcal{Q}_1$ and $\mathcal{Q}_2$ have formal duals, then so does their crested product (and a simple description of this dual is given). An analogous definition for permutation groups with intransitive normal subgroups is created, and it is shown that the constructions for association schemes and permutation groups are related in a natural way.

The definition can be generalized to association schemes with families of inherent partitions, or permutation groups with families of intransitive normal subgroups. This time the correspondence is not so straightforward, and it works as expected only if the inherent partitions (or orbit partitions) form a distributive lattice.

The paper concludes with some open problems.

We consider a class of reductive linear groups defined in terms of weighted oriented graphs of a special sort that we call signed quivers. Each of these yields a symmetric quiver, that is, a quiver endowed with an involutive anti-automorphism together with signs for the vertices and arrows fixed by the involution. The orbits of the groups can be described in terms of the indecomposable symmetric representations of symmetric quivers. We provide a general description of the indecomposable symmetric representations and prove that their dimensions in the finite and tame cases constitute root systems corresponding to certain symmetrizable generalized Cartan matrices.

The paper gives an explicit classification of the finite groups which are homogeneous in the sense of Fraïssé.

Let $\Omega=\{1,2,\dotsc,n\}$ where $n \ge 2$. The shape of an ordered set partition $P=(P_1,\dotsc, P_k)$ of $\Omega$ is the integer partition $\lambda=(\lambda_1,\dotsc,\lambda_k)$ defined by $\lambda_i = |P_i|$. Let G be a group of permutations acting on $\Omega$. For a fixed partition $\lambda$ of n, we say that G is $\lambda$-transitive if G has only one orbit when acting on partitions P of shape $\lambda$. A corresponding definition can also be given when G is just a set. For example, if $\lambda=(n-t,1,\dotsc,1)$, then a $\lambda$-transitive group is the same as a t-transitive permutation group, and if $\lambda=(n-t,t)$, then we recover the t-homogeneous permutation groups.

We use the character theory of the symmetric group Sn to establish some structural results regarding $\lambda$-transitive groups and sets. In particular, we are able to generalize a celebrated result of Livingstone and Wagner [Math. Z. 90 (1965) 393–403] about t-homogeneous groups. We survey the relevant examples coming from groups. While it is known that a finite group of permutations can be at most 5-transitive unless it contains the alternating group, we show that it is possible to construct a nontrivial t-transitive set of permutations for each positive integer t. We also show how these ideas lead to a combinatorial basis for the Bose–Mesner algebra of the association scheme of the symmetric group and a design system attached to this association scheme.

Convergent sums $T = {\sum}_{n=1}^{\infty} P(n)/Q(n)$ and $U = {\sum}_{n=1}^{\infty}(-1)^n P(n)/Q(n)$, where $P(X), Q(X) \in {\mathbb Q}[X]$, and $Q(X)$ has only simple rational roots, are investigated. Adhikari, Shorey and the authors have shown that $T$ and $U$ are either rational or transcendental. Simple necessary and sufficient conditions are formulated for the transcendence of $T$ and $U$ if the degree of $Q$ is 3 and 2, respectively.

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.

A matrix $A$ with minimum polynomial $m_A$ and characteristic polynomial $c_A$ is said to be cyclic if $m_A = c_A$ , semisimple if $m_A$ has no repeated factors, and separable if it is both cyclic and semisimple. For any set $T$ of matrices, we write $C_T$ for the proportion of cyclic matrices in $T, SS_T$ for the proportion of semisimple matrices, and $S_T$ for the proportion of separable matrices. We will write $C_{{\rm GL}(\infty,q)}$ for $\lim_{d\rightarrow\infty}C_{{\rm GL}(d,q)}$ , and so on.

We show that for every natural number m a finitely generated metabelian group G embeds in a quotient of a metabelian group of type $\textit{FP}_m$. Furthermore, if $m \leq 4$, the group G can be embedded in a metabelian group of type $\textit{FP}_m$. For L a finitely generated metabelian Lie algebra over a field K and a natural number m we show that, provided the characteristic p of K is 0 or $p > m$, then L can be embedded in a metabelian Lie algebra of type $\textit{FP}_m$. This result is the best possible as for $0 < p\leq m$ every metabelian Lie algebra over K of type $\textit{FP}_m$ is finite dimensional as a vector space.

The paper gives the best quantitative forms of Kronecker's theorem.

The bordism class of a self-transverse immersion f:Mn−k ℝn corresponds to an element α of the homotopy group πnΩ∞Σ∞MO(k). It is explained how the ℤ/2 Hurewicz image h(α)∈Hn(Ω∞Σ∞MO(k);ℤ/2) may be used to determine the characteristic numbers of the self-intersection manifolds Δr(f) of the immersion f.

Let $\omega \subseteq {\bb R}^d$ be an open domain. The sequentially complete DF-spaces $E$ are characterized such that for each (some) discrete sequence $(z_n) \subseteq \omega$ , a sequence of natural numbers $(k_n)$ and any family $(x_{n, \alpha})_{n \in {\bb N}, \vert \alpha\vert \leqslant k_n} \subseteq E$ the infinite system of equations \[ \left(\frac{\partial^{\vert \alpha\vert } f}{\partial z^{\alpha}}\right)(z_n) = x_{n,\alpha} \quad \hbox{for } n \in {\bb N}, \alpha \in {\bb N}^{d}, \vert \alpha\vert \leqslant k_n, \] has an $E$ -valued real analytic solution $f$ .

We investigate the finer fractal structure of the set of points escaping to infinity under iteration of an arbitrary exponential map. Providing exact formulas, we show how sensitively the Hausdorff dimension depends on the rate of growth of canonical Devaney–Krych codes.

The trace-norm convergence of the Trotter product formula is proved for nonself-adjoint Gibbs semigroups. For any m-sectorial generators A and B such that e−tReA is in the trace-class for t > 0, the Trotter product formula converges in the trace-norm. With smallness conditions on B with respect to A, we give error bound estimates of the convergence rate in this topology.