We strengthen a characterization of the $p$-adic binomial series and a special case of a formal analogue of Hilbert's Theorem 90 for $p$-adic power series.

A basis $\{x_n\}$ for a Hilbert space H is called a Riesz basis if it has the property that $\sum a_nx_n$ converges in H if and only if $\sum|a_n|^2<\infty$, and hence if and only if $\{x_n\}$ is the isomorphic image of some orthonormal basis for H. A consequence of a classical result of Bary [1] is that any basis for H that is quadratically near an orthonormal basis must be a Riesz basis. Motivated by this result, we study in this paper the class of normalized bases in a Hilbert space that are quadratically near some orthonormal basis, bases we call almost-orthonormal bases. In particular, we prove that any such basis must be quadratically near its Gram-Schmidt orthonormalization, and derive an internal characterization of these bases that indicates how restrictive the property of being almost-orthonormal is.

The aim of this paper is to prove that the Ricci curvature ${\rm Ric}_M$ of a complete hypersurface $M^n$, $n\,{\ge}\,3$, of the Euclidean sphere $\mathbb{S}^{n+1}$, with two distinct principal curvatures of multiplicity 1 and $n-1$, satisfies $\sup {\rm Ric}_M\,{\ge}\,\inf\, f(H)$, for a function\, $f$ depending only on $n$ and the mean curvature $H$. Supposing in addition that $M^n$ is compact, we will show that the equality occurs if and only if $H$ is constant and $M^n$ is isometric to a Clifford torus $S^{n-1}(r) \times S^1(\sqrt{1-r^2})$.

The conformal geometry of the Schwarzian Davey-Stewartson II hierarchy and its discrete analogue is investigated. Connections with discrete and continuous isothermic surfaces and generalised Clifford configurations are recorded. An interpretation of the Schwarzian Davey-Stewartson II flows as integrable deformations of conformally immersed surfaces is given.

Let $\mu$ be a finite positive Borel measure defined on a $\sigma$-algebra of subsets of a set $\X$. Using operator techniques we provide several criteria for finitely generated algebras to be dense in the space $L^2(\mu)$.

We present some basic properties of two distinguished discretizations of elliptic operators: the self-adjoint 5-point and 7-point schemes on a two dimensional lattice. We first show that they allow us to solve Dirichlet boundary value problems; then we present their Moutard transformations (distinguished examples of transformation of Darboux type in two dimensions). Finally we construct their Lelieuvre formulae and we show that, at the level of the normal vector and in full analogy with their continuous counterparts, the self-adjoint 5-point scheme characterizes a two dimensional quadrilateral lattice (a lattice whose elementary quadrilaterals are planar), while the self-adjoint 7-point scheme characterizes a generic 2D lattice.

A generalization due to Gessel [3] of Miki's identity between Bernoulli numbers is shown to be a direct consequence of a functional equation for the generating function.

In this note, we show that results of U. Baumgartner and G. A. Willis concerning contraction groups of automorphisms of metrizable totally disconnected, locally compact groups remain valid also in the non-metrizable case, if one restricts attention to automorphisms for which small tidy subgroups exist.

Associated with $T=U|T|$ (polar decomposition) in ${\cal{L}}({\bf H})$ is a related operator $\skew3\tilde{T} = |T|^{\frac{1}{2}}U|T|^{\frac{1}{2}}$, called the Aluthge transform of $T$. In this paper we study some connections between $T$ and $\skew3\tilde{T}$, including the following relations; the single valued extension property, an analogue of the single valued extension property on $W^{m}(D, {\bf H})$, Dunford's property $(C)$ and the property $(\beta)$.

We consider a special class of solutions of the BKP hierarchy which we call $\tau$-functions of hypergeometric type. These are series in Schur $Q$-functions over partitions, with coefficients parameterised by a function of one variable $\xi$, where the quantities $\xi(k)$, $k\in\mathbb{Z^+}$, are integrals of motion of the BKP hierarchy. We show that this solution is, at the same time, a infinite soliton solution of a dual BKP hierarchy, where the variables $\xi(k)$ are now related to BKP higher times. In particular, rational solutions of the BKP hierarchy are related to (finite) multi-soliton solution of the dual BKP hierarchy. The momenta of the solitons are given by the parts of partitions in the Schur $Q$-function expansion of the $\tau$-function of hypergeometric type. We also show that the KdV and the NLS soliton $\tau$-functions coinside the BKP $\tau$-functions of hypergeometric type, evaluated at special point of BKP higher time; the variables $\xi$ (which are BKP integrals of motions) being related to KdV and NLS higher times.

A global existence result for solutions $u(t)$ of the differential equation $x^{\prime \prime }+f(t,x)=p(t)$, $t\geq t_0\geq 1$, that can be written as $u(t)=P(t)+o(1)$ for all large $t$, where $P^{\prime \prime}(t)=p(t)$, is established by means of the Schauder-Tikhonov theorem. It generalizes the recent work of Lipovan [On the asymptotic behaviour of the solutions to a class of second order nonlinear differential equations, Glasgow Math. J.45 (2003), 179–87] and allows for a unifying treatment of the existence problems concerning asymptotically linear and oscillatory solutions of second order nonlinear differential equations.

A $q$-discrete analog of the Toda molecule equation and its $N$-soliton solution are constructed by using the bilinear method. The solution is expressed in the Casorati determinant form whose elements are given in terms of the $q$-orthogonal polynomials.

Let $R$ be a semi-prime Noetherian ring of injective dimension 1. Let $P$ be a minimal prime ideal of $R$. In this paper it is shown that $R/P$ need not have injective dimension 1. Necessary and sufficient conditions are given for $R/P$ to have injective dimension 1.

We give characterisations of the linear relations that are injective and open (respectively, almost open with dense range) in terms of the stability of the nullity (respectively, of the deficiency). Results of Mbekhta about bounded operators in Banach spaces (J. Operator Theory35 (1996), 191–201) are covered.

We extend a solution method used for the one-dimensional Toda lattice in [1], [2] to the two-dimensional Toda lattice. The idea is

1. to study the lattice not with values in $\mathbb{C}$ but in the Banach algebra ${\cal L}$ of bounded operators and

2. to derive solutions of the original lattice ($\mathbb{C}$-solutions) by applying a functional $\tau$ to the ${\cal L}$-solutions constructed in 1.

In the second part we focus on applications. We start by rederiving line-solitons and briefly discuss discrete resonance phenomena. Moreover, we are able to find conditions under which it is possible to superpose even countably many line-solitons.

We show that the Fourier expansion in spherical $h$-harmonics (from Dunkl's theory) of a function $f$ on the sphere converges uniformly to $f$ if this function is sufficiently differentiable.

Given a sequence $(\pi_n)$ of irreducible representations of a liminal $C^*$-algebra $A$, and a sequence $(b_n)$ of trace class operators with $b_n\in \pi_n(A)$, we investigate necessary conditions and sufficient conditions for the existence of a simultaneous lifting $a\in A$ such that, for each $n$, the trace of $\sigma(a)$ is bounded for irreducible representations $\sigma$ in a neighbourhood of $\pi_n$.

A closer look at Laguerre and Meixner polynomials shows that they interact, with a more active role played by the latter. We intend here to expound on this development of the story up to some level of abstraction.

We introduce the notion of valuation of a dense near polygon. The valuations of a dense near polygon $F$ describe the possible relations between a point of a dense near polygon $\cS$ and any geodetically closed sub near polygon of $\cS$ isomorphic to $F$. Several nice properties of valuations are given and several classes of these objects are defined. Valuations are an important tool for classifying dense near polygons.

A number theoretical aspect of the fundamental cycle of a periodic box-ball system is investigated. Using the formulae for the fundamental cycle of a class of initial states, we point out that the asymptotic behaviour of the fundamental cycle is closely related to the celebrated Riemann hypothesis.