The three families of classical groups of linear transformations (complex, orthogonal, symplectic) give rise to the three great branches of differential geometry (complex analytic, Riemannian and symplectic). Complex analytic geometry derives most of its interest from complex algebraic geometry, while symplectic geometry provides the general framework for Hamiltonian mechanics.

These three classical groups “intersect” in the unitary group and the three branches of differential geometry correspondingly “intersect” in Kähler geometry, which includes the study of algebraic varieties in projective space. This is the basic reason why Hodge was successful in applying Riemannian methods to algebraic geometry in his theory of harmonic forms.

It is well known that the mean value theorem (MVT) does not, in general, hold for analytic functions. The most familiar example to this effect is f(z) = ez since e2πi−e0≠2πiez0 for any z0∈ℂ. On the other hand, it is easy to show that the MVT holds in ℂ if f(z) is a polynomial of degree at most 2. Thus it is natural to ask what conditions on a function f(z) analytic in a domain D are necessary and sufficient for f(z) to satisfy the MVT in D. This is one of the questions answered in this paper.

The object of this paper is to complete and continue some matters in [1].

In [1], Section 2, the torsion and torsion-free functors, whose operation on the category of abelian groups are well known, were extended to the category of all groups as follows. For a group A, put t0(A)= 0 and t1(A) = the subgroup of A generated by the torsion elements of A. Inductively define tn+1(A)/tn(A)=t1(A)/tn(A)), for every positive integer n. Then T(A)=∪ntn(A) is the smallest subgroup H of A such that A/H is torsion-free, [1], Th. 2.2. A group A satisfying T(A) = A was called a pre-torsion group. In [1], 2.12 an example was constructed of a group A satisfying t1(A)≠t2(A)=A. The question was posed whether for every positive integer n there exist groups A, satisfying tn–1(A)≠tn(A)=A. Here we give an affirmative answer. In fact, such groups will be constructed, as well as pre-torsion groups A with tk(A)≠A for every positive integer k, see Section 1.

In [3] we initiated our study of the automorphism groups of a certain class of near-rings. Specifically, let P be any complex polynomial and let P denote the near-ring of all continuous selfmaps of the complex plane where addition of functions is pointwise and the product fg of two functions f and g in P is defined by fg=f∘P∘g. The near-ring P is referred to as a laminated near-ring with laminating element P. In [3], we characterised those polynomials P(z)=anzn + an−1zn−1 +…+a0 for which Aut P is a finite group. We are able to show that Aut P is finite if and only if Deg P≧3 and ai ≠ 0 for some i ≠ 0, n. In addition, we were able to completely determine those infinite groups which occur as automorphism groups of the near-rings P. There are exactly three of them. One is GL(2) the full linear group of all real 2×2 nonsingular matrices and the other two are subgroups of GL(2). In this paper, we begin our study of the finite automorphism groups of the near-rings P. We get a result which, in contrast to the situation for the infinite automorphism groups, shows that infinitely many finite groups occur as automorphism groups of the near-rings under consideration. In addition to this and other results, we completely determine Aut P when the coefficients of P are real and Deg P = 3 or 4.

Let S(X) denote the semigroup of all continuous selfmaps of the topological space X. Let ℒ(S(X)) and ℛ(S(X)) denote the partially ordered families of all ℒ-classes and ℛ-classes, respectively, of S(X) where the partial orders are the usual ones [3, p. 29]. In [6] we made the following

Conjecture. The following statements are equivalent about any two compact 0-dimensional metric spaces X and Y:

(1) ℒ(S(X)) and ℒ(S(Y)) are order isomorphic.

(2) ℛ(S(X)) and (S{Y)) are order isomorphic.

(3) The semigroupsS(X) and S(Y) are isomorphic.

(4) The spaces X and Y are homeomorphic.

This account of our Society is based to some extent on my Presidential address, which was given on 19 October 1977 and was devoted to the first fifty years.

In the latter half of the nineteenth century there was an upsurge of interest in mathematics that resulted in the foundation of a number of mathematical societies in different countries. The London Mathematical Society (1865), the Moscow Mathematical Society (1867), the Société Mathématique de France (1873), the Edinburgh Mathematical Society (1883) and the New York (later American) Mathematical Society (1888) were all founded in this period. There had, of course, been earlier more local societies, such as the Spittalfields Mathematical Society, which flourished over a long period before becoming defunct, as well as one or two much older bodies, for example the Mathematische Gesellschaft in Hamburg (1690), which still survive.

1. It is well known [1] that there is a one-to-one relation between solutions of the Darboux equation and the wave equation. The purpose of this paper is to show that some recent results in the fractional calculus can be used to obtain a similar connection between solutions of Darboux's equation and second order linear hyperbolic differential equations with constant coefficients.

Let R be a ring, a linear sequential machine over R is a quintuple ℳ = (Q, A, B, F, G) where

Q, A, B are R-modules

F:Q × A →Q and G:Q × A→B are R-homomorphisms.

After preliminary results and definitions in Section 1, we show in Section 2 that any finite regular semigroup is saturated, in the sense of Howie and Isbell [8] (that is, the dominion of a finite regular semigroup U in a strictly containing semigroup S is never S). This is equivalent of course to showing that in the category of semigroups any epi from a finite regular semigroup is in fact onto. Note for inverse semigroups the stronger result, that any inverse semigroup is absolutely closed [11, Theorem VII. 2.14] or [8, Theorem 2.3]. Further, any inverse semigroup is in fact an amalgamation base in the class of semigroups [10], in the sense of [5]. These stronger results are known to be false for finite regular semigroups [8, Theorem 2.9] and [5, Theorem 25]. Whether or not every regular semigroup is saturated is an open problem.

Let 2* denote the dual of the mod two Steenrod algebra. In [5] an algebraic filtration B*(n) of H*(BO; ℤ2) was constructed such that each B*(n) is a bipolynomial sub Hopf algebra and sub 2*-comodule of H*(BO; ℤ2). In Lemma 3.1 we prove that the Thom isomorphism determines a corresponding filtration of H*(MO;ℤ2) by polynomial subalgebras and sub 2*-comodules M*(n). Let (n) denote the subalgebra of 2 generated by Sq2k, 0 ≦ k < n, and let *(n) be its dual, a quotient Hopf algebra of 2*. In Section 3 we construct a polynomial algebra and *(n)-comodule R(n) such that M*(n)≃2*□*(n)R(n) as algebras and 2*-comodules. Here □ denotes the cotensor product defined in [9, §2]. Dually it will follow that M*(n) has a sub (n)-module and subcoalgebra T(n) such that M*(n)≃2⊗n)T(n) as coalgebras and 2-modules. We also show that M*(n) can not be realised as the homology of a spectrum for n≧4. Of course M*(0)=H*(MO;ℤ2), M*(1)=H*(MSO;ℤ2), M*(2)=H* (MSpin;ℤ2) and M*(3)=H*(MO<8>;ℤ2). Moreover, it follows from [4; Thm. 2.10, Cor. 2.11] that M*(n)=Images[H*(MO<ϕ(n)>;ℤ2)→H*(MO;ℤ2)] and M*(n) ≃ Image [H*(MO;ℤ2)→ H*(MO<ϕ(n)>;ℤ2)]. Here MO<k> id the Thom spectrum of BO<k>, the (k−1)-connected covering of BO, and ϕ(n)=8s + 2t where n = 4s + t, 0≦t≦3. In Section 4 we sketch the odd primary analogue—a filtration pM*(n) of H*(MUp, 0;ℤp) for p an odd prime. MUp, 0 is the Thom spectrum of the (2p-3)-connected factor of the Adams splitting [2] of BU(p).

Following Lotz, Peck and Porta [9], a continuous linear operator from one Banach space into another is called a semi-embedding if it is one-to-one and maps the closed unit ball of the domain onto a closed (hence complete) set. (Below we shall allow the codomain to be an F-space, i.e., a complete metrisable topological vector space.) One of the main results established in [9] is that if X is a compact scattered space, then every semi-embedding of C(X) into another Banach space is an isomorphism ([9], Main Theorem, (a)⇒(b)).

A subgroup Q of a group G is called quasinormal in G if Q permutes with every subgroup of G. Of course a quasinormal subgroup Q of a group G may be very far from normal. In fact, examples of Iwasawa show (for a convenient reference see [8]) that we may have Q core-free and the normal closure QG of Q in G equal to G so that Q is not even subnormal in G. We note also that the core of Q in G, QG, is of infinite index in QG in this example. If G is finitely generated then any quasinormal subgroup Q is subnormal in G [8] and although Q is not necessarily normal in G we have that |QG:Q| is finite and |QG:Q| is a nilpotent group of finite exponent [5].

The classical von Neumann–Oxtoby–Ulam Theorem states the following:

Given non-atomic Borel probability measures μ, λ on In such that

there exists a homeomorphism h of In onto itself fixing the boundary pointwise such that for any λ-measurable set S

It is known that the above theorem remains valid if In is replaced by any compact finite dimensional manifold [2], [4] or with I∞, the Hilbert cube, [8].

A liquid crystal is a transversely isotropic liquid consisting of large, relatively rigid, elongated molecules which align more or less parallel to their neighbours. Three distinct types of liquid crystal occur, namely nematic, cholesteric and smectic. In the absence of any external influences, nematics tend to orientate with their anisotropic axis uniformly aligned, whereas cholesterics prefer a characteristic helical configuration and smectics are more highly organised in layered structures. However, it is possible to influence the orientation of the anisotropic axis by a variety of external means. In particular, solid surfaces affect the alignment through the action of surface torques, while electromagnetic fields exert body torques which tend to align the anisotropic axis either parallel or perpendicular to the applied field. Detailed descriptions of the physical properties of liquid crystals may be found in the books by de Gennes [1] and Chandrasekhar [2] and the review by Stephen and Straley [3].

A theorem due to Milutin [12] (see also [13]) asserts that for any two uncountable compact metric spaces Ω1 and Ω2 the spaces of continuous real-valued functions C(Ω1) and C(Ω2) are linearly isomorphic. It immediately follows from consideration of tensor products that if X is any Banach space then C(Ω1;X) and C(Ω2;X) are isomorphic.

In this note we consider a singular perturbation problem for the equation

where K(y) = sgn y and. Ε is a small (positive) parameter. This equation for ε≠O is elliptic for y<0 and hyperbolic for y>0. Many of the results carry over to more difficult and interesting problems for equations of mixed type. The particularly simple model treated here permits the elimination of some complications in the analysis involving singular integral equations while preserving the main qualitative features of more general cases. For a special Tricomi-like problem for (1.1) we construct asymptotic expansions in ε, including boundary layer corrections, of the solution. A proof of uniform asymptotic validity of the lowest order terms is given.

In a previous paper [1] we constructed a free resolution for a class of groups which include Fuchsian groups with compact orbit spaces [2, 3], infinite polyhedral groups, plane crystallographic groups p2, p3, p4 and p6 and Dyck's groups [4], and used this resolution for computation of the integral homology and cohomology of these groups. Lyndon [5] determined the cohomology of groups with a single defining relation. The plane crystallographic groups p1 and pg and Artin's braid group B3 are among these groups. In this paper we have constructed free resolutions for certain classes of groups–resolutions which are particularly suitable for direct computation of the homology and the cohomology of these groups for any coefficient module. These classes of groups include the plane crystallographic groups pm, cm and pgg. We have computed the integral homology and cohomology from each of the free resolutions obtained.

The purpose of this article is to illustrate how the theorem of Lévy about conformal invariance of Brownian motion can be used to obtain information about boundary behaviour and removable singularity sets of analytic functions. In particular, we prove a Frostman–Nevanlinna–Tsuji type result about the size of the set of asymptotic values of an analytic function at a subset of the boundary of its domain of definition (Theorem 1). Then this is used to prove the following extension of the classical Radó theorem: If φ is analytic in B\K, where B is the unit ball of ℂ;n and K is a relatively closed subset of B, and the cluster set of φ at K has zero harmonic measure w.r.t. φ(B\K)\≠Ø, then φ extends to a meromorphic function in B (Theorem 2).

Let E be a real Hausdorff locally convex space with topological dual E′, topologised by the strong topology. Let (x, x′) denote the bilinear mapping defining the duality between E and E′ (x∈E, x′∈E′). By a unitary representation of E′ we mean an operator valued function U(x′) = Ux′. defined on E′, whose values are unitary operators in a separable Hilbert space H such that

Consider the Klein-Gordon equation

where q, Aj are real valued functions on Rn, m and e positive constants. Equation (1) describes the motion of a relativistic particle of mass m and charge e in an external field described by the electrostatic potential q and the electromagnetic potential A = (Aj); units are chosen so that the speed of light is one.