The title is somewhat misleading, since the classical modular group Г= PSL(2, ℤ) is certainly not a subgroup of GL(2, ℤ). What is meant of course are the faithful representations of F as a subgroup of GL(2, ℤ), where Г is to be thought of as the free product of a cyclic group of order 2 and a cyclic group of order 3. No such representation is possible as a subgroup of SL(2, ℤ); it is necessary to have matrices of determinant −1 as well.

We are interested in two parameter eigenvalue problems of the form

subject to Dirichlet boundary conditions

The weight function 5 and the potential q will both be assumed to lie in L2[0,1]. The problem (1.1), (1.2) generates eigencurves

in the sense that for any fixed λ, ν(λ) is the nth eigenvalue ν, (according to oscillation indexing) of (1.1), (1.2). These curves are in fact analytic functions of λ and have been the object of considerable study in recent years. The survey paper [1] provides background in this area and itemises properties of eigencurves.

The purpose of this paper is to show the equivalence of convergence, an associated projective system of measures and a martingale decomposition for a uniformly integrable stochastic process. Emphasis is placed on a direct juxtaposition of these concepts and on displaying underlying mechanisms.

The impact of the martingale convergence theorem on contemporary probability theory has been immense. Therein lies the reason for numerous generalizations of both the basic martingale convergence theorem and the martingale concept itself.

Let E[τ] be a locally convex Hausdorff topological vector space. An extended decomposition of E[τ] is a family {Ea}α∈A of closed subspaces of E such that, for each x in E and each α in A, there exists a unique point xα in Eα, with Here convergence will have the following meaning. Let Ф denote the set of all finite subsets of A. The sum is said to be convergent to x if for each neighbourhood U of 0 in E, there is an element φ0 of Ф such that , for all φ in Ф containing φ0. It follows that is Cauchy if and only if, for each neighbourhood U of 0 in E, there is an element φ0 of Ф such that , for all φ in Ф disjoint from φ0.

Free monoids play a central role in the theory of formal languages. Their endomorphisms appear naturally in the context of deterministic OL-schemes which trace their origin to biology. Closely related to such a scheme is a DOL-system which consists of a triple (X, φ, w) where X is a finite set, φ is an endomorphism of the free monoid X* and w ∈ X. The associated language is defined as the set {w, φw, φ2w,…} called a DOL-language. For a full discussion of this subject, we recommend the book [2] by Herman and Rozenberg.

Let f: (ℝn, 0)→ (ℝ,0) be a germ of a real analytic function. Let L and F(f) denote the link of f and the Milnor fibre of fc respectively, i. e., L = {x ∈ Sn−1 | f(x) = 0}, , where 0 ≤ ξ ≪ r ≪ 1, . In [2] Szafraniec introduced the notion of an -germ as a generalization of a germ defined by a weighted homogeneous polynomial satisfying some condition concerning the relation between its degree and weights (definition 1). He also proved that if f is an -germ (presumably with nonisolated singularity) then the number χ(F(f)/d mod 2 is a topological invariant of f, where χ(F(f)) is the Euler characterististic of F(f), and gave the formula for χ(L)/2 mod 2 (it is a well-known fact that F(L) is an even number). As a simple consequence he got the fact that χ(F(f)mod 2 is a topological invariant for any f, which is a generalization of Wall's result [3] (he considered only germs with an isolated singularity).

En algèbre non-commutative, on dit qu'un anneau noethérien A est local si:

(i) le radical de Jacobson M de A est un idéal maximal,

(ii) ∩ Mn = (0),

(iii) A/M est artinien simple.

Dans [9], Walker definit un anneau local régulier comme un anneau local A dont le radical de Jacobson M est engendré par une A-suite centralisante x1; x2, …, xt, [4], et demontre alors que:

t = cldim A = Kdim A = rgldim A = pdAA/M.

A Banach space X is said to have property (PROXBID) if the canonical image of X in its bidual X** is proximal. In other words, if J: X → X** is the canonical embedding, then it is required that every element of X** have at least one best approximation (i.e., nearest point) from the closed subspace J(X). We show below that, if X is the space of (real or complex) continuous functions on a compact set, or the space of (real or complex) continuous functions that vanish at infinity on a locally compact set, then X has property (PROXBID). At this point we should mention the existence of a variety of examples [2, 8] of Banach spaces which lack property (PROXBID).

Let C be the Cayley algebra denned over the real field. If, for given elements α, β, and γ of a quaternion subalgebra of C, α = βγ, it follows, by associativity, that for any nonzero element δ of the same quaternion subalgebra, α = (βδ)(δ-1γ). For Cayley numbers ζ ξ, and η with ζ = ξη, the relation ζ = (ξδ)(δ-1η) in general only holds when δ is a nonzero real number. Because of the existence of factorization results [1, 2] in the orders of C, the question naturally arises of whether it is possible to choose one-to-one mappings, θ and φ, of C onto itself such that ζ = θξ. φη whenever ζ = ξη. To discuss this question, we make the following definition.

Let Mn be an n-dimensional manifold immersed in an (n+p)-dimensional unit sphere Sn+p, with mean curvature H and second fundamental form B. We put φ(X, Y) = B(X, Y)–(X, Y)H where X and Y are tangent vector fields on Mn. Assume that the mean curvature is parallel in the normal bundle of Mn in Sn+p. Following Alencar and do Carmo [1] we denote by BH the square of the positive root of

In the theory of self-adjoint operators in Hilbert space and of formally self-adjoint linear differential equations there are many situations involving analytic functions on the complex plane whose singularities are confined to the real axis and where the growth of the function at such singular points is strictly limited.

Let K be a quadratic number field with 2-class group of type (2,2). Thus if Sk is the Sylow 2-subgroup of the ideal class group of K, then Sk = ℤ/2ℤ × ℤ/2ℤ Let

K ⊂ K1 ⊂ K2 ⊂ K3 ⊂…

the 2-class field tower of K. Thus K1 is the maximal abelian unramified extension of K of degree a power of 2; K2 is the maximal abelian unramified extension of K of degree a power of 2; etc. By class field theory the Galois group Ga1 (K1/K) ≅ Sk ≅ ℤ/2ℤ × ℤ/2ℤ, and in this case it is known that Ga(K2/Kl) is a cyclic group (cf. [3] and [10]). Then by class field theory the class number of K2 is odd, and hence K2 = K3 = K4 = …. We say that the 2-class field tower of K terminates at K1 if the class number of K1 is odd (and hence K1 = K2 = K3 = … ); otherwise we say that the 2-class field tower of K terminates at K2. Our goal in this paper is to determine how likely it is for the 2-class field tower of K to terminate at K1 and how likely it is for the 2-class field tower of K to terminate at K2. We shall consider separately the imaginary quadratic fields and the real quadratic fields.

In § 2 a product of two modified Bessel Functions of the Second Kind is expressed as an integral with a function of the same type as a factor of the integrand. In § 3 an integral involving a product of these functions, regarded as functions of their orders, is evaluated in terms of another function of this kind. These results were suggested by a study of Mellin's inversion formula.

The cone length Cl(f) of a map f: X → Y is defined to be the least number of attaching maps possible in a conic (or iterated mapping cone) structure for f. Cone length is a homotopy invariant in the sense that if φ: X → X and ρ: Y → Y are homotopy equivalences then Cl (ρ°f°φ) = Cl(f). Furthermore Cl(f) depends only on the homotopy class of f. It was shown by Ganea [8] that the cone length of the map * → X coincides with the strong Lusternik-Schnirelmann category of X as a space (see Proposition 1.6 below). Recent work of Cornea ([3]–[6]) is much concerned with cone length and its role in critical point theory. For example, let f be a smooth real valued function on a manifold triad (M; V0, V1) with V0 ≠ θ. Under certain conditions, if f has only “reasonable” critical points then it must have at least Cl(V0↪M) of them (see [6]).

The development of the theory of absolute integrals derives from certain key facts. Among them are:

(I) An integral is a positive linear functional on a vector lattice, which is continuous in a certain sense.

(II) A function equal almost everywhere to a summable function is itself summable.

(III) Every measurable function is the pointwise limit of a sequence of elementary step functions.

A device that often plays an important role in measure theory, but which has not beenfully exploited in the theory of abstract integrals is that of

(IV) the smallest class containing a given class and having a certain property

(such as being a σ-ring of sets). It is our purpose in this paper to examine the theory of abstract real-valued absolute integrals axiomatically, in such a way as to isolate and clarify the roles of (I) through (IV).

If A is a semiprime Banach algebra, soc A, rad A the socle and radical of A, then Soc A ∩ rad A = (0). This elementary result enables us to prove some results concerning algebraic ideal and algebraic elements modulo the socle of A. We also deduce several conditions for A equivalent to the condition dim A <+∞.

Let k be a field and G an Abelian group of finite torsion-free rank. Brewer, Costa and Lady [1, Theorem A] showed that if k has characteristic 0 then each Localization of the group algebra kG at a prime ideal is a regular local ring. They also showed (in the same theorem) that if k has characteristic p>0, then kG is locally Joetherian (i.e. each localization of kG at a prime ideal is a Noetherian ring) if and only if G is an extension of a finitely generated group by a torsion p′-group. The purpose of this note is to examine this theorem in a more general setting.

Let n = be the factorization of an integer n(>1) into prime powers, and set Φ(n):= . In particular, for squarefree n, Φ(n) = phi;(n). Consider the set

.

It is known (from [5]) that A consists precisely of those integers n for which there is no non-abelian group of order n. It is also known (from [7]) that the set

consists solely of integers n with the property that every group of order n is cyclic. We set C′ = A – C.