Central to the Fourier development of automorphic forms on GL(n, ℝ) are the class 1 principal series Whittaker functions Wn,a(z), which were first studied systematically by Jacquet [13]. (See Section 2 below for the definition of Wn,a(z).)

Of particular interest are the Mellin transforms Mn,a(s) (s ∈ [Copf ]n−1) of Wn,a(z). (See equation (2.11) below.) For example, such transforms, and analogous Mellin transforms of products of Whittaker functions, arise as archimedean Euler factors for certain automorphic L-functions (see [5, 6, 8, 21, 22] for discussions and examples.) Moreover, Mn,a(s) has relevance to the problem of special values of Whittaker functions; cf. [7] in the case n = 3.

Friedberg and Goldfeld [11] have shown Mn,a(s), for general n, to have analytic continuation and to satisfy certain recurrence relations. However, explicit formulae for these transforms have, until the present work, been deduced only for n [les ] 4. In particular, both M2,a(s) (cf. [4, 25]) and M3,a(s) (cf. [7, 9, 19]) are expressible as (some powers of 2 and π times) ratios of gamma functions. On the other hand, M4,a(s) (cf. [22]) may be realized essentially as a hypergeometric series of type 7F6(1), or equivalently as a sum of two series of type 4F3(1). (See Section 2 below for a brief general discussion of hypergeometric series.)

In an attempt to prove all the identities given in Ramanujan's first two letters to G. H. Hardy, G. N. Watson devoted a paper to the evaluation of G1353. At the end of his paper, Watson remarked that his proof was not rigorous as he had assumed certain identities (see (1.1) and (1.2)) which he found empirically. In this paper, we use class field theory, Galois theory and Kronecker's limit formula to justify Watson's assumptions. We shall then use our results to compute some new values of Gn.

An explicit geometrical study of the curves

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is presented. These are non-singular curves of genus 3, defined over ℚ(a). By exploiting their symmetries, it is possible to determine most of their geometric invariants, such as their bitangent lines and their period lattice. An explicit description is given of the bijection induced by the Abel–Jacobi map between their bitangent lines and odd 2-torsion points on their jacobian. Finally, three elliptic quotients of these curves are constructed that provide a splitting of their jacobians. In the case of the curve [Cscr ]1±√2, which is isomorphic to the Fermat curve of degree 4, the computations yield a finer splitting of its jacobian than the classical one.

The commonly known properties of continuous measures such as the symmetry of the A∞ condition, its equivalence to the reverse Hölder inequality, the left-openness of the Ap condition, etc., are no longer necessarily true when the underlying measure is allowed to have atoms. The measures that preserve these properties are called good measures. The class of good measures is investigated and various criteria for a measure to belong to this class are presented.

Let $p$ be a prime and let $q = p^a$, where $a$ is a positive integer. Let $G = G({\mathbb{F}}_q)$ be a Chevalley group over $\mathbb{ F}_q$, with associated system of roots $\Phi$ and Weyl group $W$. Steinberg showed in 1957 that $G$ has an irreducible complex representation whose degree equals the $p$-part of $|G|$ [11]. This representation, now known as the Steinberg representation, has remarkable properties, which reflect the structure of $G$, and there have been many research papers devoted to its study. The module constructed in [11] is in fact a right ideal in the integral group ring $\mathbb{ Z}G$ of $G$, and is thus a $\mathbb{ Z}G$-lattice, which we propose to call the Steinberg lattice of $G$. It should be noted that lattices not integrally isomorphic to the Steinberg lattice may also afford the Steinberg representation, and such lattices may differ considerably in their properties compared with the Steinberg lattice.

Let k be an algebraically closed field of characteristic p > 0, and let G be a connected, reductive algebraic group over k. In [8] and [11], conditions on the dimension of rational G modules were seen to imply semisimplicity of these modules. In [8], certain of these conditions were extended to cover the finite groups of Lie type. In this paper, we extend some of the results of [11] to cover these finite Lie type groups. The main such extension is the following result.

The paper describes some qualitative properties of minimizers on a manifold $\mathcal M$ endowed with a discontinuous metric. The discontinuity occurs on a hypersurface $\Sigma$ disconnecting $\mathcal M$. Denote by $\Omega_1$ and $\Omega_2$ the open subsets of $\mathcal M$ such that $\mathcal M\setminus\Sigma=\Omega_1\cup\Omega_2$. Assume that $\overline\Omega_1$ and $\overline\Omega_2$ are endowed with metrics $\left\langle\cdot,\cdot\right\rangle_{\left(1\right)}$ and $\left\langle\cdot,\cdot\right\rangle_{\left(2\right)}$, respectively, such that $\overline\Omega_i (i=1, 2)$ is convex or concave. The existence of a minimizer of the length functional on curves joining two given points of $\mathcal M$ is proved. The qualitative properties obtained allows the refraction law in a very general situation to be described.

Consider the Sobolev embedding operator from the space of functions in $W^{1,p}(I)$ with average zero into $L^p$, where $I$ is a finite interval and $p\,{>}\,1$. This operator plays an important role in recent work. The operator norm and its approximation numbers in closed form are calculated. The closed form of the norm and approximation numbers of several similar Sobolev embedding operators on a finite interval have recently been found. It is proved in the paper that most of these operator norms and approximation numbers on a finite interval are the same.

Sun and Wilson defined the notion of infinite determinacy of a smooth function germ singular along a line, and related this notion to some good geometric properties of derived objects related to the given function germ. The paper extends their results to a wider class of smooth function with prescribed non-isolated singularities. For this purpose, it was necessary to study the behaviour of the function germ along a transverse direction of the given singular set, and to relate these properties to geometric properties of the function and derived objects, expressed in terms of relative Łojasiewicz conditions.

The complex Stiefel manifolds admit a stable decomposition as Thom spaces of certain bundles over Grassmannians. The purpose of the paper is to identify the splitting in any complex oriented cohomology theory.

The paper characterizes the reproducing kernel Hilbert spaces with orthonormal bases of the form {(an,0+an,1z+…+an,JzJ)zn, n [ges ] 0}. The primary focus is on the tridiagonal case where J = 1, and on how it compares with the diagonal case where J = 0. The question of when multiplication by z is a bounded operator is investigated, and aspects of this operator are discussed. In the diagonal case, Mz is a weighted unilateral shift. It is shown that in the tridiagonal case, this need not be so, and an example is given in which the commutant of Mz on a tridiagonal space is strikingly different from that on any diagonal space.

In this paper we develop a structure theory for transitive permutation groups definable in o-minimal structures. We fix an o-minimal structure [Mscr ], a group G definable in [Mscr ], and a set Ω and a faithful transitive action of G on Ω definable in [Mscr ], and talk of the permutation group (G, Ω). Often, we are concerned with definably primitive permutation groups (G, Ω); this means that there is no proper non-trivial definable G-invariant equivalence relation on Ω, so definable primitivity is equivalent to a point stabiliser Gα being a maximal definable subgroup of G. Of course, since any group definable in an o-minimal structure has the descending chain condition on definable subgroups [23] we expect many questions on definable transitive permutation groups to reduce to questions on definably primitive ones.

Recall that a group G definable in an o-minimal structure is said to be connected if there is no proper definable subgroup of finite index. In some places, if G is a group definable in [Mscr ] we must distinguish between definability in the full ambient structure [Mscr ] and G-definability, which means definability in the pure group G:= (G, .); for example, G is G-definably connected means that G does not contain proper subgroups of finite index which are definable in the group structure. By definable, we always mean definability in [Mscr ]. In some situations, when there is a field R definable in [Mscr ], we say a set is R-semialgebraic, meaning that it is definable in (R, +, .). We call a permutation group (G, Ω) R-semialgebraic if G, Ω and the action of G on Ω can all be defined in the pure field structure of a real closed field R. If R is clear from the context, we also just write ‘semialgebraic’.

The paper is an addendum to D. Andrica and L. Funar, ‘On smooth maps with finitely many critical points’, J. London Math. Soc. (2) 69 (2004) 783–800.

The paper establishes estimates for ideal measures of operators interpolated by the real method in terms of the measures of their restrictions to a sequence of spaces modelled on the intersection. It also shows that the estimates are optimal.

Certain results which have been proved or conjectured for the subclass consisting of those in L2(ℝ) are generalized to entire functions of exponential type bounded on the real axis.

It is shown that a weakly closed subspace ${\cal S}$ of a nest algebra ${\cal A}$ is closed under conjugation by invertible elements in ${\cal A}$ , that is, $a^{-1}{\cal S}\,a={\cal S}$ if and only if ${\cal S}$ is a Lie ideal. A similar result holds for not-necessarily-closed subspaces of algebras of infinite multiplicity. An explicit characterisation of weakly closed Lie ideals in a nest algebra is given.

It is proved that if $F$ is an infinite field with characteristic different from $2$, whose theory is supersimple, and $C$ is an elliptic or hyperelliptic curve over $F$ with generic ‘modulus’, then $C$ has a generic $F$-rational point. The notion of generity here is in the sense of the supersimple field $F$.

Maximal accretive realizations and bound-preserving self-adjoint extensions are two fundamental problems in applications of semi-bounded operator theory to differential equations. On the basis of using differential operator theory in direct sum spaces and Phillips theory for maximal accretive extensions of accretive operators, a complete characterization of the set of maximal accretive boundary conditions for Sturm–Liouville differential operators is presented. As an application, all possible forms of bound-preserving self-adjoint extensions of regular Sturm–Liouville operators are also characterized via various explicit boundary conditions. The methodology can also be applied to dealing with general classes of semi-bounded symmetric differential operators.

A method is given to solve any Thue–Mahler equation when the coefficients and variables come from the ring of integers of a number field. The method involves using an algorithm for solving an S-unit equation and a method of reducing the number of exponential variables. The method is used to solve the equation

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the interplay between the behaviour of approximately convex (and approximately affine) functions on the unit ball of a banach space and the geometry of banach $k$-spaces is studied.