The relationships between various notions of completeness of eigenvectors and root vectors of the eigenvalue problem $Af=\lambda Bf$ are investigated. Here $A$ and $B$ are self-adjoint operators in Hilbert space with $B$ bounded and positive semidefinite, and with $A$ having compact resolvent.

AMS 2000 Mathematics subject classification: Primary 47A75. Secondary 34B24; 35P10

Montel introduced the concept of quasi-normal families $f:\varOmega\to\mathbb{C}$ in 1922: $\mathcal{F}$ is quasi-normal of order $N$ if every sequence $\{f_n\}$ from $\mathcal{F}$ has a subsequence which converges uniformly on compact subsets of $\varOmega\setminus Z^\dagger$, where $Z^\dagger\subset\varOmega$ contains at most $N\in\mathbb{N}$ elements. ($\mathcal{F}$ is of order $N:=\infty$ if every such exceptional set $Z^\dagger$ is finite.) The problem is that $Z^\dagger$ normally depends on the subsequence. So even if every sequence has a subsequence which converges to a given function $f$ in $\varOmega$ except at $N$ points, the sequence itself may not converge in any domain $D\subseteq\varOmega$.

In this paper we introduce the concept of general convergence. Indeed, $\{f_n\}$ above converges generally to $f$. We also introduce a related concept, restrained sequences, and study some of their properties. The definitions extend earlier concepts introduced for sequences of linear fractional transformations.

AMS 2000 Mathematics subject classification: Primary 20H10; 30B70; 30D45; 30F35; 58F08

It is well known that the square matrix, of rank n−k + 1,

which we shall denote by B where any element to the left of, or below the nonzero diagonal b1, k, b2, k + 1, . …, bn−k + 1, n is zero, can be resolved into factors Z−1DZ; where D is a square matrix of order n having the elements d1, k, d2, k + 1, . …, dn−k + 1, n all unity and all the other elements zero, and where Z is a non-singular matrix. In this paper we shall show in a particular case that this is so, and in the case in question we shall exhibit the matrix Z explicitly. Application of this is made to find the classical canonical form of a rational integral function of a square matrix A.

The equation

need not have a solution z in the complex plane, even when ƒ is entire. For example, let ƒ(z) = ez, z1 = z0+2kπi. Thus the classical mean value theorem does not extend to the complex plane. McLeod has shown (2) that if ƒ is analytic on the segment joining z1 and z0, then there are points w1 and w2 on the segment such that where

The purpose of this article is to give a local mean value theorem in the complex plane. We show that there is at least one point z satisfying (1), which we will call a mean value point, near z1 and z0 but not necessarily on the segment joining them, provided z1 and z0 are sufficiently close. The proof uses Rouché's Theorem (1).

In this note we intend to discuss the method of A. Córdoba and R. Fefferman of using covering lemmas to control maximal functions, and make some simplifications which allow us to obtain alternative proofs of some of their results.

Perhaps the simplest elementary proof of the prime number theorem, see Erdös (2) and Selberg (5), is Wright's modification (8), (3, p. 362) of Selberg's original proof (5). Another variant is due to V. Nevanlinna (4). Wright's proof uses Selberg's idea of smoothing the weighting process which occurs in the Selberg inequality, (1.2) below, by iterating this inequality. Here it will be shown that the proof requires less ingenuity if use is made of a further smoothing operation, namely first integrating the Selberg inequality itself. Integration has been used on a related inequality by Breusch (1 to obtain a remainder term. This method also makes proof by contradiction unnecessary.

If a locally compact group G acts on a C*-algebra B, we have both full and reduced crossed products and each has a coaction of G. We investigate ‘exotic’ coactions in between the two, which are determined by certain ideals E of the Fourier–Stieltjes algebra B(G); an approach that is inspired by recent work of Brown and Guentner on new C*-group algebra completions. We actually carry out the bulk of our investigation in the general context of coactions on a C*-algebra A. Buss and Echterhoff have shown that not every coaction comes from one of these ideals, but nevertheless the ideals do generate a wide array of exotic coactions. Coactions determined by these ideals E satisfy a certain ‘E-crossed product duality’, intermediate between full and reduced duality. We give partial results concerning exotic coactions with the ultimate goal being a classification of which coactions are determined by ideals of B(G).

We introduce full diffeomorphism-invariant Colombeau algebras with added ε-dependence in the basic space. This unites the full and special settings of the theory into one single framework. Using locality conditions we find the appropriate definition of point values in full Colombeau algebras and show that special generalized points suffice to characterize elements of full Colombeau algebras. Moreover, we specify sufficient conditions for the sheaf property to hold and give a definition of the sharp topology in this framework.

For complex connected, reductive, affine, algebraic groups G, we give a Lie-theoretic characterization of the semistability of principal G-co-Higgs bundles on the complex projective line ℙ1 in terms of the simple roots of a Borel subgroup of G. We describe a stratification of the moduli space in terms of the Harder–Narasimhan type of the underlying bundle.

The Lusternik–Schnirelmann category of the quaternionic Grassmannian

is known to be k(n − k). In this paper we show that this result can be deduced from Morse theory.

Let $X\subset\mathbb{R}^2$ be the graph of a Pfaffian function $f$ in the sense of Khovanskii. Suppose that $X$ is non-algebraic. This note gives an estimate for the number of rational points on $X$ of height less than or equal to $H$; the estimate is uniform in the order and degree of $f$.

It is known that a polynomial local diffeomorphism $(f,\, g): {\mathbb {R}}^{2} \to {\mathbb {R}}^{2}$ is a global diffeomorphism provided the higher homogeneous terms of $f f_x+g g_x$ and $f f_y+g g_y$ do not have real linear factors in common. Here, we give a weight-homogeneous framework of this result. Our approach uses qualitative theory of differential equations. In our reasoning, we obtain a result on polynomial Hamiltonian vector fields in the plane, generalization of a known fact.

There are various methods in existence for the practical solution of a set of simultaneous equations

Some of these methods are appropriate to special systems, as for example to the axisymmetric “normal equations” of Least Squares. In many applications, however, as in problems of statistical correlation of many variables, it may be desired not merely to solve a given set of equations but to obtain as much knowledge as possible about the system or matrix of coefficients; perhaps to evaluate its determinant and various minors, such as principal minors, possibly also to determine the elements of the adjugate matrix, or the reciprocal matrix. The examination of the sign of successive principal minors of an axisymmetric determinant, in order to find the signature of the corresponding quadratic form, is a case in point; and there are many such applications.

In this paper we give a method for the solution of the dual integral equations

where Jv and Yv are Bessel functions of the first and second kind, −½≦α≦½, f1(ρ) and f2(ρ) are known functions and ψ(ξ) is to be determined. Such equations arise in the discussion of boundary value problems for half-spaces containing a cylindrical cavity. For example, let us take the problem of finding a potential function φ(ρ, θ, z) which satisfies Laplace's equation for

subject to the usual regularity conditions and the following boundary conditions:

For a fixed integer q≧2, every positive integer k = Σr≧0ar(q, k)qr where each ar(q, k)∈{0,1,2,…, q−1}. The sum of digits function α(q, k) Σr≧0ar(q, k) behaves rather erratically but on averaging has a uniform behaviour. In particular if , where n>1, then it is well known that A(q, n)∼½((q − 1)/log q)n logn as n → ∞. For odd values of q, a lower bound is now obtained for the difference 2S(q, n) = A(q, n)−½(q − 1))[log n/log q, where [log n/log q] denotes the greatest integer ≦log n /log q. This complements an upper bound already found.

Normal right submodules and right ideals need not coincide in an arbitrary near-ring. Berman and Silverman (1) have shown that in a near-ring (N, +, ·) with a two-sided zero (i.e. x · 0 = 0 · x = 0, for all x ∈ N) a right ideal is also a right submodule. If (N, +, ·) is in fact a distributively generated near-ring, then all normal right submodules are also right ideals. (See (5).)

The concept of a laminated near-ring was introduced in [2]. We recall briefly what it is. Let N be a near-ring and let a∈N. Define a new multiplication on N by x * y = xay for all x,y∈N. With this new multiplication and the same addition as before we have another near-ring which we denote by Na. The near-ring Na is referred to as a laminated near-ring, the original near-ring N is the base near-ring and a is the laminator or laminating element.

Let d(<0) denote a squarefree integer. The ideal class group of the imaginary quadratic field has a cyclic 2-Sylow subgroup of order ≦8 in precisely the following cases (see for example [5] and [6]):

where p and q denote primes and g, h, u and v are positive integers. The class number of is denoted by h(d) and in the above cases h(d) = 0(mod 8). For cases (i), (ii) and (iii) the authors [6] have given necessary and sufficient conditions for h(d) to be divisible by 16. In this paper we do the same for case (iv) extending the results of Brown [4].

Zemanian [17] obtained abelian theorems for the Hankel and K-transforms of functions and then extended his results to the corresponding transforms of distributions in the sense of Schwartz [11]. Jones [6] has discussed at length asymptotic behaviours of transforms generalized in his sense. Following the technique of Zemanian many authors have obtained abelian theorems for more general transforms of functions and distributions in the sense of Schwartz. Mention may be made of the works of Joshi and Saxena [7], Lavoine and Misra [8] and Pathak [10]. However, these authors were confined to the transforms of real variables only.