We prove a general result on Bailey pairs and show that two Bailey pairs of Bringmann and Kane are special cases. We also show how to use a change of base formula to pass from the pairs of Bringmann and Kane to pairs used by Andrews in his study of Ramanujan's seventh order mock theta functions. We derive several more Bailey pairs of a similar type and use these to construct a number of new q-hypergeometric double sums which are mock theta functions. Finally, we prove identities between some of these mock theta double sums and classical mock theta functions.

Over a field of prime characteristic p the group algebra of a finite group has a non-trivial radical if and only if the order of the group is divisible by the prime p. In two earlier papers [7,8] we have imposed certain restrictions on the radical, namely that the radical be contained in the centre of the group algebra and that the radical be of square zero, and we have considered what influence these conditions have on the structure of the group itself. These conditions are, at first sight, of different types and our present paper is an attempt to generalise them by merely assuming that the radical is commutative.

In [4] an identity is given which relates the product of two Fourier coefficients of a Hecke eigenform g of half-integral weight and level 4N with N odd and squarefree to the integral of a Hecke eigenform f of even integral weight associated to g under the Shimura correspondence along a geodesic period on the modular curve X0(N) This formula contains as a special case a refinement of a result of Waldspurger [6] about special values of L-series attached to f at the central point.

We define “star reducible” Coxeter groups to be those Coxeter groups for which every fully commutative element (in the sense of Stembridge) is equivalent to a product of commuting generators by a sequence of length-decreasing star operations (in the sense of Lusztig). We show that the Kazhdan–Lusztig bases of these groups have a nice projection property to the Temperley–Lieb type quotient, and furthermore that the images of the basis elements $C'_w$ (for fully commutative $w$) in the quotient have structure constants in ${\mathbb Z}^{\geq 0}[v, v^{-1}]$. We also classify the star reducible Coxeter groups and show that they form nine infinite families with two exceptional cases.

Let us consider the Cauchy problem

in QT= ℝ3 × [0, T], where f(x, t), ρ0(x) and v0(x) are given, while the density ρ(x, t), the velocity vector v(x, t)= (υ1(x, t), υ2(x, t), υ3(x, t)) and the pressure p(x, t) are unknowns. The viscosity coefficient μ is assumed to be nonnegative. In these equations, the pressure p is automatically determined (up to a function of t) by ρ and v, namely, by solving the equation

Thus we mention (ρ, v) when we talk about the solution of (1.1:μ).

A module M is called a CS-module or an extending module if every submodule is essential in a direct summand of M. A ring R is called a right CS-ring or a right extending ring if R_R is a CS-module. For several types of right CS-rings it is known that either all right ideals or some large class of right ideals inherit the CS property. For example, by a result of Dung-Smith or Vanaja-Purav, a ringR is (right and left) Artinian, serial, and J(R)^2 = 0 if and only if every R-module is CS. In particular, if R is a QF-ring and J(R)^2 = 0 (hence R is serial), then everyR -module is CS. However we exhibit a finite, serial, strongly bounded QF group algebra R with J(R)^3 = 0 for which there is a principal right ideal which is a right essential extension of a CS-module and essential inR_R but not CS itself.

We show that if F is a free Lie algebra of rank at least 2 and if G is a non-trivial finite group of automorphisms of F then the fixed point subalgebra F^G is not finitely generated. Some similar results are proved for relatively free Lie algebras.

A right [left] unitary S-system is a set M with right [left] operators in a semigroup S with 1, where x1 = x [1x = x] for all x ∈ M. We define a semigroup S with 1 to be completely right [left] injective provided that every right [left] unitary S-system is injective. The main purpose of this paper is to determine a structure for completely right [left] injective semigroups whose idempotents are in the centre.

Many of the classical inequalities of analysis can be written in the form P(x) ≥ 0 for x ∈ I or P(x) > 0 for x ∈ I′, where P(x) is a polynomial and I′ ⊂ I are certain intervals on the real line. This gives rise to the question of where the zeros of P(x) are located. For example, if f is a polynomial with real zeros, then an inequality of Laguerre [8, p. 171 f.] asserts that

for all x. A detailed study of the zeros of this particular P(x) has been made [5].

We give the necessary and sufficient conditions for Lagrangian submanifolds in Kähler manifolds to be biharmonic. We classify biharmonic PNMC Lagrangian H-umbilical submanifolds in the complex space forms. Furthermore, we classify biharmonic PNMC Lagrangian surfaces in the two-dimensional complex space forms.

A free boundary problem for a parabolic system arising from the mathematical theory of combustion will be considered in the one dimensional case. The existence and uniqueness of the classical solution locally in time will be obtained by the use of a fixed point theorem. Also the existence of the classical solution globally in time and a convergence result with respect to a parameter $\lambda$ will be proved under some reasonable assumptions.

Let S be a prime Noetherian ring and G a finite group acting on 5 such that Gis x-outer on S. We give sufficient conditions for the skew group ring S * Gto be a prime maximal order. If we impose the further hypothesis that the order of Gbe a unit of S, then these conditions are also necessary. Moreover, if S is a commutative Noetherian domain, then there are necessary and sufficient conditions for S*Gto be a prime maximal order, without requiring that the order of G be a unit in S.

Based on the ideas of Bessa, Jorge and Montenegro (Comm. Anal. Geom., vol. 15, no. 4, 2007, pp. 725–732) we show that a complete submanifold M with tamed second fundamental form in a complete Riemannian manifold N with sectional curvature KN ≤ κ ≤ 0 is proper (compact if N is compact). In addition, if N is Hadamard, then M has finite topology. We also show that the fundamental tone is an obstruction for a Riemannian manifold to be realised as submanifold with tamed second fundamental form of a Hadamard manifold with sectional curvature bounded below.

This note draws together and extends two recent results on Diophantine approximation and Hausdorff dimension. The first, by Hinokuma and Shiga [12], considers the oscillating error function | sinq|/qτ rather than the strictly decreasing function qτ of Jarnik's theorem. The second is Rynne's extension [17] to systems of linear forms of Borosh and Fraenkel's paper [3] on restricted Diophantine approximation with real numbers. Rynne's result will be extended to a class of general error functions and applied to obtain a more general form of [12] in which the error function is any positive function.

Let Г(1) denote the homogeneous modular group of 2 × 2 matrices with integral entries and determinant 1. Let (1) be the inhomogeneous modular group of 2 × 2 integral matrices of determinant 1 in which a matrix is identified with its negative. (N), the principal congruence subgroup of level N, is the subgroup of (1) consisting of all T ∈ (1) for which T ≡ ± I (mod N), where N is a positive integer and I is the identity matrix. A subgroup of (1) is said to be a congruence group of level N if contains (N) and N is the least such integer. Similarly, we denote by Г(N) the principal congruence subgroup of level N of Г(1), consisting of those T∈(1) for which T ≡ I (mod N), and we say that a sub group of Г(1) is a congruence group of level N if contains Г (N) and N is minimal with respect to this property. In a recent paper [9] Rankin considered lattice subgroups of a free congruence subgroup of rank n of (1). By a lattice subgroup of we mean a subgroup of which contains the commutator group . In particular, he showed that, if is a congruence group of level N and if is a lattice congruence subgroup of of level qr, where r is the largest divisor of qr prime to N, then N divides q and r divides 12. He then posed the problem of finding an upper bound for the factor q. It is the purpose of this paper to find such an upper bound for q. We also consider bounds for the factor r.

We consider the isomorphism problem for partial group rings $R_{\hbox{\scriptsize\it par}}G$ and show that, in the modular case, if $\textit{char}(R)\,{=}\,p$ and $R_{\hbox{\scriptsize\it par}}G_1\,{\cong}\, R_{\hbox{\scriptsize\it par}}G_2$ then the corresponding group rings of the Sylow $p$-subgroups are isomorphic. We use this to prove that finite abelian groups having isomorphic modular partial group algebras are isomorphic. Moreover, in the integral case, we show that the isomorphism of partial group rings of finite groups $G_1$ and $G_2$ implies $\Z G_1\,{\cong}\, \Z G_2$.

Motivated by the recent work due to Warnaar (2005), two new and elementary proofs are presented for a very useful q-difference equation on eight shifted factorials of infinite order. As the common source of theta function identities, this q-difference equation is systematically explored to review old and establish new identities on Ramanujan's partition functions. Most of the identities obtained can be interpreted in terms of theorems on classical partitions.

We develop a theory of group actions and coverings on Brauer graphs that parallels the theory of group actions and coverings of algebras. In particular, we show that any Brauer graph can be covered by a tower of coverings of Brauer graphs such that the topmost covering has multiplicity function identically one, no loops, and no multiple edges. Furthermore, we classify the coverings of Brauer graph algebras that are again Brauer graph algebras.

The object of this paper is to derive, using a version of the large sieve for function fields due to J. Johnsen [6], explicit lower boundsfor the average number of distinct values taken by a polynomial over a finite field.

The notion of a completely hyperexpansive operator has been introduced in [1] by Athavale. In this paper hyperexpansive operator valued unilateral weighted shifts are investigated. A unique semispectral measure is associated with a bounded completely hyperexpansive operator valued unilateral weighted shift with invertible weights. Examples of bounded and unbounded hyperexpansive operator valued unilateral weighted shifts with invariant domains are given.