Let M = SL(2, Z) be the classical modular matrix group. One form of the Poincaré series on M is

here z ∈ H={z = x + iy: y >0}, q ≧ 2 and m ≧ 1 are integers, and the summation is over a complete system of matrices (ab: cd) in M with different lower row. The problem of the identical vanishing of the Poincaré series for different values of m and q goes back to Poincaré.

Recently several papers on varieties of topological groups have appeared. In this note we investigate the question: if Ω is a class of topological groups, what topological groups are in the variety V(Ω) generated by Ω that is, what topological groups can be “manufactured” from Ω using repeatedly the operations of taking subgroups, quotient groups and arbitrary cartesian products? We seeka general theorem which will be useful for investigating V(Ω) for well-known classesΩ.

We establish two separation theorems in which the classic interior is replaced by the quasi-relative interior.

W. H. Cornish (2) has investigated congruences on pseudo-complemented distributive lattices and has identified those ideals (resp. filters) that are congruence kernels (resp. cokernels). In this paper we show that many of the principal results concerning congruence kernels and cokernels hold in a semilattice and therefore do not depend on distributivity, nor on the existence of unions.

Let Fn be the free group on {ai: i ∈ Zn}, where the set of congruence classes mod n is used as an index set for the generators. Let φ be the permutation (1, 2, 3, …, n) of Zn and denote by θ the automorphism of Fn induced by φ, namely

This paper answers two questions posed by P. M. Neumann and K. Hickin on embeddings of infinite permutation groups. We first give some definitions that would make their questions understood.

We study certain convexity-type properties of homogeneous functions on topological vector lattices, focusing on a concept of 0+-convexity, and using some probabilistic inequalities.

The general linear matrix equation has been investigated by Sylvester in a series of interesting papers published in the Comptes Rendus. His method, however, does not show that the solution can be exhibited as an analytical function of the coefficients. The object of this note is to supply such a solution.

The following pages continue a line of enquiry begun in a work On Differentiating a Matrix, (Proceedings of the Edinburgh Mathematical Society (2) 1 (1927), 111-128), which arose out of the Cayley operator , where xij is the ijth element of a square matrix [xij] of order n, and all n2 elements are taken as independent variables. The present work follows up the implications of Theorem III in the original, which stated that

where s (Xr) is the sum of the principal diagonal elements in the matrix Xr. This is now written ΩsXr = rXr – 1 and Ωs is taken as a fundamental operator analogous to ordinary differentiation, but applicable to matrices of any finite order n.

In (7), Wright gives an enumerative proof of an identity algebraically equivalent to that of Jacobi, namely

Here, and in the sequel, products run from 1 to oo and sums from - oo to oo unless otherwise indicated. We give here a simplified version of his argument by working directly with (1), the substitution leading to equation (3) of his paper being omitted. We then supply an alternative proof of (1) by means of a generalisation of the Durfee square concept utilising the rectangle of dimensions v by v + r for fixed r and maximal v contained in the Ferrers graph of a partition.

A linear algebra of order n, in general non-commutative and non-associative, may be regarded as being determined by the “cubic matrix ”consisting of its n3 constants of multiplication, and conversely. This requires that the n basis elements (units) of the algebra should be specified, and should be given in a definite order. Then the various “transpositions” of the cubic matrix induce corresponding “transpositions” of the algebra, for which a notation is given in §2.

Let Γ be a torsion-free geometrically finite Kleinian group. In this paper, we investigate which systems of loxodromic conjugacy classes of Γ can be simultaneously made parabolic in a group on the boundary of the quasi-conformal deformation space of Γ. We shall prove that for this, it is sufficient that the classes of the system are represented by disjoint primitive simple closed curves on the ideal boundary of H3/Γ.

A tournamentTn consists of a finite set of nodes1, 2, …, n such that each pair of distinct nodes i and j is joined by exactly one of the arcsij or ji. If the arc ij is in Tn we say that i beats j or j loses to i and write i→j. If each node of a subtournament A beats each node of a subtournament B we write A→B and let A + B denote the tournament determined by the nodes of A and B.

We give various realizations of the adjoint orbits of a semi-simple Lie group and describe their symplectic geometry. We then use these realizations to identify a family of Lagrangian submanifolds of the orbits.

A pair of coupled nonlinear differential equations is studied and asymptotic properties of its non-oscillatory solutions are obtained. In particular, we provide classification schemes for these solutions which are justified by existence criteria.

For a Banach space $\mathfrak{X}$, let $\mathcal{B}(\mathfrak{X})$ denote the Banach algebra of all continuous linear operators on $\mathfrak{X}$. First, we study the lattice of closed ideals in $\mathcal{B}(\mathfrak{J}_p)$, where $1 \lt p \t \infty$ and $\mathfrak{J}_p$ is the $p$th James space. Our main result is that the ideal of weakly compact operators is the unique maximal ideal in $\mathcal{B}(\mathfrak{J}_p)$. Applications of this result include the following.

(i) The Brown–McCoy radical of $\mathcal{B}(\mathfrak{X})$, which by definition is the intersection of all maximal ideals in $\mathcal{B}(\mathfrak{X})$, cannot be turned into an operator ideal. This implies that there is no ‘Brown–McCoy’ analogue of Pietsch’s construction of the operator ideal of inessential operators from the Jacobson radical of $\mathcal{B}(\mathfrak{X})/\mathcal{A}(\mathfrak{X})$.

(ii) For each natural number $n$ and each $n$-tuple $(m_1,\dots,m_n)$ in $\{k^2\mid k\in\mathbb{N}\}\cup\{\infty\}$, there is a Banach space $\mathfrak{X}$ such that $\mathcal{B}(\mathfrak{X})$ has exactly $n$ maximal ideals, and these maximal ideals have codimensions $m_1,\dots,m_n$ in $\mathcal{B}(\mathfrak{X})$, respectively; the Banach space $\mathfrak{X}$ is a finite direct sum of James spaces and $\ell_p$-spaces.

Second, building on the work of Gowers and Maurey, we obtain further examples of Banach spaces $\mathfrak{X}$ such that all the maximal ideals in $\mathcal{B}(\mathfrak{X})$ can be classified. We show that the ideal of strictly singular operators is the unique maximal ideal in $\mathcal{B}(\mathfrak{X})$ for each hereditarily indecomposable Banach space $\mathfrak{X}$, and we prove that there are $2^{2^{\aleph_0}}$ distinct maximal ideals in $\mathcal{B}(\mathfrak{G})$, where $\mathfrak{G}$ is the Banach space constructed by Gowers to solve Banach’s hyperplane problem.

AMS 2000 Mathematics subject classification: Primary 47D30; 47D50; 46H10; 16D30

This paper deals with the class of Schrödinger–Kirchhoff-type biharmonic problems

where Δ2 denotes the biharmonic operator, and f ∈ C(ℝN × ℝ, ℝ) satisfies the Ambrosetti–Rabinowitz-type conditions. Under appropriate assumptions on V and f, the existence of infinitely many solutions is proved by using the symmetric mountain pass theorem.

We study the eigenproblem

where

and Tm, Vmn are self-adjoint operators on separable Hilbert spaces Hm. We assume the Tm to be bounded below with compact resolvents, and the Vmn to be bounded and to satisfy an “ellipticity” condition. If k = 1 then ellipticity is automatic, and if each Tm is positive definite then the problem is “left definite”.

We study totally positive definite quadratic forms over the ring of integers $\mathcal {O}_K$ of a totally real biquadratic field $K=\mathbb {Q}(\sqrt {m}, \sqrt {s})$. We restrict our attention to classic forms (i.e. those with all non-diagonal coefficients in $2\mathcal {O}_K$) and prove that no such forms in three variables are universal (i.e. represent all totally positive elements of $\mathcal {O}_K$). Moreover, we show the same result for totally real number fields containing at least one non-square totally positive unit and satisfying some other mild conditions. These results provide further evidence towards Kitaoka's conjecture that there are only finitely many number fields over which such forms exist. One of our main tools are additively indecomposable elements of $\mathcal {O}_K$; we prove several new results about their properties.

We study affine invariants of plane curves from the view point of the singularity theory of smooth functions