In this paper we study the algebraic structure of the hyperelliptic mapping class group of Klein surfaces, which is closely related to the mapping class group of punctured discs. This group plays an important role in the study of the moduli space of hyperelliptic real algebraic curves. Our main result provides a presentation by generators and relations for the hyperelliptic mapping class group of surfaces of prescribed topological type.

AMS 2000 Mathematics subject classification: Primary 14H10; 20H10; 30F50

The necessary and sufficient conditions that guarantee the boundedness and compactness of integral operators with positive kernels from $L^p(a,b)$ to $L^q_{\nu}(a,b)$, where $p,q\in(1,\infty)$ or $0lt q\leq1lt plt\infty$, for a non-negative Borel measure $\nu$ on $(a,b)$ are found.

AMS 2000 Mathematics subject classification: Primary 46B50; 47B34; 47B38

We follow the notations and basic equations of Chen (2). Let M be a surface immersed in an m-dimensional space form Rm(c) of curvature c = 1, 0 or −1. We choose a local field of orthonormal frames e1, …, em in Rm(c) such that, restricted to M, the vectors e1, e2 are tangent to M. Let ω1, …, ωm be the field of dual frames. Then the structure equations of Rm(c) are given by

The following notes are intended to introduce a simple method of treating elementary geometrical conics, and at the same time to supply a missing link in the chain of continuity between Euclidean geometry and the modern methods of treating the conics, which at present are treated more as different subjects than as a continuous whole.

We recall the following definition (see [1]):

A finite group G is said to be a Frobenius–Wielandt group provided that there exists a proper subgroup H of G and a proper normal subgroup N of H such that H∩Hg≦N if g∈G–H. Then H/N is said to be the complement of (G, H, N) (see [1] for more details and notation).

It is well known that the elliptic integral of the second kind may be represented by the arc of an ellipse, and mathematicians have sought with various success to represent similarly by the arc of an algebraic curve the elliptic integral of the first kind. The general solution of the problem has not been obtained, but Serret and Cayley have given solutions of a very general character.

The substance of this paper is contained in Chrystal, chap, xxv., §§13, and 15 to 20, with some applications thereof occurring in chap. xxvi. But it is treated here in a fresh manner which would seem simpler on several points. This mode of presentation was, in the start, suggested by Peano's method given by Prof. Gibson in his “Note on the Fundamental Inequality Theorems Connected with ex and xm,” in Vol. XVIII. of the Proceedings.

A real matrix is called non-negative (positive) if all its entries are non-negative (positive). Two matrices A and B are said to be cogredient if there exists a permutation matrix Q such that QAQT = B. A square non-negative matrix is called reducible if it is cogredient to a matrix of the form

where the blocks X and Y are square. Otherwise it is called irreducible.

Let v = (a1 …, an) be a real n-tuple and be the numbers a1 …, an arranged in decreasing order. Let denote the sum of m greatest components of v and the sum of m smallest components of v, i.e.,

We prove that, for every extension of Banach algebras 0 → B →A → D → 0 such that B has a left or right bounded approximate identity, the existence of an associated long exact sequence of Banach simplicial or cyclic cohomology groups is equivalent to the existence of one for homology groups. It follows from the continuous version of a result of Wodzicki that associated long exact sequences exist. In particular, they exist for every extension of C*-algebras.

The main theorem of this paper is a little involved (though the proof is straightforward using a well-known idea) but the immediate corollaries are interesting. For example, take a complex normed vector space A which is also a normed algebra with identity under each of two multiplications * and ∘. Then these multiplications coincide if and only if there exists α such that ‖a ∘ b ‖ ≦ α ‖ a * b ‖ for a, b in A. This is a condition for the two Arens multiplications on the second dual of a Banach algebra to be identical. By taking * to be the multiplication of a Banach algebra and ∘ to be its opposite, we obtain the condition for commutativity given in (3). Other applications are concerned with conditions under which a bilinear mapping between two algebras is a homomorphism, when an element lies in the centre of an algebra, and a one-dimensional subspace of an algebra is a right ideal. An example shows that the theorem is false for algebras over the real field, but Theorem 2 gives the parallel result in this case.

Multiresolution is investigated on the basis of shift-invariant spaces. Given a finitely generated shift-invariant subspace S of L2(ℝd), let Sk be the 2k-dilate of S (k∈ℤ). A necessary and sufficient condition is given for the sequence {Sk}k∈ℤ to fom a multiresolution of L2(ℝd). A general construction of orthogonal wavelets is given, but such wavelets might not have certain desirable properties. With the aid of the general theory of vector fields on spheres, it is demonstrated that the intrinsic properties of the scaling function must be used in constructing orthogonal wavelets with a certain decay rate. When the scaling function is skew-symmetric about some point, orthogonal wavelets and prewavelets are constructed in such a way that they possess certain attractive properties. Several examples are provided to illustrate the general theory.