If we seek solutions of the hyperbolic differential equation

which depend only on the variables i and , we see that these solutions must be even in r and satisfy the differential equation

The object of this paper is to show that some recent results in the fractional calculus can be used to prove the following theorem.

The study of modular forms has been deeply influenced by famous conjectures and hypotheses concerning

where T(n) denotes Ramanujan's function. The fundamental discriminant Δ is a cusp form of weight 12 with respect to the modular group. Its associated Dirichlet series

defines an entire function of s and satisfies the functional equation

The most penetrating statements that have been made on T(n) and LΔ(s)are:

Of these four problems only A1 has been established so far. This was done by Deligne [1] using methods from algebraic geometry and number theory. While B1 trivially holds with ε > 1/2, it was established in [2] for every ε>1/3. Serre [12] proved A2 for a positive proportion of the integers and Hafner [5] showed that LΔ has a positive proportion of its non-trivial zeros on the line σ=6. The proofs of the last three results are largely analytic in nature.

In this note we describe a class of functors on the category of associative rings with unity (hereafter “rings”) and of ring homomorphisms which, loosely speaking, ‘preserve the properties’ of two-sided ideals, but can be chosen to be arbitrarily ‘bad’ for one-sided properties of rings.

In ([6]; pages 36–41), Lambek constructs the maximal ring of quotients Q(R) of a commutative ring R by denning a multiplication on Homr(D, R) where D ranges over all the dense ideals of R, and this generalizes the classical construction of ring of quotients, (cf. [6] for all the references on the subject.)

This programme is carried over, in the first section of this article, to the categoryof commutative reductive semigroups. Examples show that the maximal semigroup of quotients of a commutative monoid can be different from the classical one.

Let I denote the interval [0,1] in its usual topology and let S = S(I) be the semigroup of continuous mappings of I into itself with function composition as the semigroup operation. In a survey talk given at Oberwolfach in 1989 (see [5]), K. D. Magill Jr. pointed out that some elementary algebraic properties of S are still unknown. We shall answer one of the questions he raised (which appears as Problem 4.6 in [5] and which he had asked earlier, as long ago as 1975 [3]) by showing that S has infinitely many distinct two-sided ideals. In fact, we shall produce an infinite descending sequence of distinct ideals. As Magill points out, this also solves his Problem 4.5: S has infinitely many distinct congruences. We believe that S must have c distinct ideals, but we have been unable to prove this.

The purpose of this paper is to point out a flaw in H. B. Griffiths' covering space approach to residual properties of groups [3]. One is led to this paper from Lyndon and Schupp's book [4, pp. 114, 141] where it is cited for covering space methods and a proof that F-groups are residually finite. However the main result of [3] is false.

In [3], McAlister introduced a class of semigroups, called covering semigroups, which were shown to play an important role in the theory of E-unitary covers of semigroups. Strangely, this class of semigroups appears to have received little attention subsequently. It is the aim of this paper to rehabilitate them and to study their properties in more detail. As a first step, we have chosen to rename them almost factorisable semigroups, since they can be regarded as the semigroup analogues of factorisable inverse monoids. Before discussing the contents of this paper in more detail we recall some standard terminology.

Let be a group-theoretic property. We say a group has a finite covering by -subgroups if it is the set-theoretic union of finitely many -subgroups. The topic of this paper is the investigation of groups having a finite covering by nilpotent subgroups, n-abelian subgroups or 2-central subgroups.

R. Baer [12; 4.16] characterized central-by-finite groups as those groups having a finite covering by abelian subgroups. In [6] it was shown that [G: ZC (G)] finite implies the existence of a finite covering by subgroups of nilpotency class c, i.e. ℜc-groups. However, an example of a group is given there which has a finite covering by ℜ2-groups, but Z2(G) does not have finite index in the group. These results raise two questions, on which we will focus our investigations.

Much work has been carried out on the classification of finite simple groups in terms of the structures of centralisers of involutions. However, it is sometimes the case that these classification results cannot be applied to particular problems even although information is available about one conjugacy class of involutions. The trouble is that information about the other classes can be almost non-existent. In this paper we deal with a situation where character theory can be employed to give a strong connection between the orders of centralisers of different classes of involutions, enabling information about one class to be used to give information about other classes. We prove the following result.

In this paper all graphs will be ordinary graphs, i.e. finite, undirected, and without loops or multiple edges. For points x and y of a graph G, we shall indicate that x is adjacent to y by writing x ⊥ y, and if x is not adjacent to y we shall write xy. We shall denote the degree of a point x by δ(x) and the minimal degree of G by δ(G).

By the line graph of a graph G we shall mean the graph L(G) whose points are the edges of G, with two points of L(G) adjacent whenever they are adjacent in G. A graph G is said to be a line graph if there exists a graph H such that G = L(H).

The object of the present paper is to establish the equivalence of the well-known theorem of the double-six of lines in projective space of three dimensions and a certain theorem in Euclidean plane geometry. The latter theorem is of considerable interest in itself for two reasons. In the first place, it is a natural extension of Euler's classical theorem connecting the radii of the circumscribed and the inscribed (or the escribed) circles of a triangle with the distance between their centres. Secondly, it gives in a geometrical form the invariant relation between the circle circumscribed to a triangle and a conic inscribed in the triangle. For a statement of the theorem, see § 13 (4).

We denote by F the field R of real numbers, the field C of complex numbers or the skew-field H of real quaternions, and by Fn an n-dimensional left vector space over F. If A is a matrix with elements in F, we denote by A* its conjugate transpose. In all three cases of F, an n × n matrix A is said to be hermitian (unitary resp.) if A = A* (AA*= identity matrix resp.). An n ×x n hermitian matrix A is said to be definite (semidefinite resp.) if uAu*vAv* ≥ 0 (uAu*vAv* ≧ 0 resp.) for all nonzero u and v in Fn. If A and B are n × n hermitian matrices, then we say that A and B can be diagonalized simultaneously into blocks of size less than or equal to m (abbreviated to d. s. ≧ m) if there exists a nonsingular matrix U with elements in F such that UAU* = diag{A1,…, Ak} and UBU* = diag{B1…, Bk}, where, for each i = 1, …, k, Ai and Bk are of the same size and the size is ≧ m. In particular, if m = 1, then we say A and B can be diagonalized simultaneously (abbreviated to d. s.).

Normal-convex embeddings are introduced for inverse semigroups, generalizing the group-theoretic concept, due to Papakyriakopoulos [4]. It is shown that every E-unitary inverse semigroup admits a normal-convex embedding into a semidirect product of a semilattice by a group, a stronger version of a result by O'Carroll [3]. A general embedding result for inverse semigroups is also obtained.

In [1] the following question was posed by R. W. Irving (see also Conjecture 4.10 in [4]): Is there an edge 2-colouring of the complete bipartite graph K13.17 with no monochromatic K3.3? We give a negative answer in this note (Theorem 2). Furthermore we prove Conjecture 4.11 (i) of [4] (Theorem 1), that is, any edge 2-coloured K2n+i,4n-3 contains a monochromatic K2,n with the 2 and n vertices a subset of the 2n +1 and 4n–3 vertices, respectively. Theorem 1 is a consequence of Satz 4 in [3], however, we give a direct proof here.

It has been shown ([8], [2], [1], [3], [9]) that a collection of physical operations or experiments can be represented by a nonempty set of nonempty sets satisfying certain technical conditions. Such a set is called a manual. The operations in are looked at as having no “before” and no “after”, i.e., they are isolated in time. If we wish to look at connected sequences of operations—in particular, if we wish to condition by events in —we must look at the compound manual c whose elements represent compound operations built up from the operations in the base manual .

We let X be an arbitrary infinite set. A semigroup S of total or partial transformations of X is called -normal if hSh-1 = S, for all h in , the symmetric group on X. For example, the full transformation semigroup , the semigroup of all partial transformations , the semigroup of all 1–1 partial transformations and all ideals of and are -normal.

Let G be a compact, simply-connected Lie group. The cohomology of the loop space ΏG has been described by Bott, both in terms of a cell decomposition [1] and certain homogeneous spaces called generating varieties [2]. It is possible to view ΏG as an infinite dimensional “Grassmannian” associated to an appropriate infinite dimensional group, cf. [3], [7]. From this point of view the above cell-decomposition of Bott arises from a Bruhat decomposition of the associated group. We choose a generator H ∈ H2(ΏG, ℤ) and call it the hyperplane class. For a finite-dimensional Grassmannian the highest power of H carries geometric information about the variety, namely, its degree. An analogous question for ΏG is: What is the largest integer Nk = Nk(G) which divides Hk ∈ H2k(ΏG, ℤ)?

Of course, if G = SU(2) = S3, one knows Nk = h!. In general, the deviation of Nk from k! measures the failure of H to generate a divided polynomial algebra in H*(ΏG, ℤ).

The structure theorem for completely 0-simple semigroups established by Rees [5] in 1940 has proved a very powerful tool in the investigation of such semigroups. In this paper the theorem is applied to an investigation of the subsemigroup of a completely 0-simple semigroup generated by its idempotents. Previous work on this problem has been carried out by Kim [4], but the present note offers a more direct approach.

Let X be a complex Banach space and H be a hermitian operator on X. Then in [7] Sinclair proved that r(H) = ¶H¶, where r(H) and ¶H¶ are the spectral radius and the operator norm of H, respectively.

For a commuting n-tuple T = (T1,…, Tn) of operators on X, we denote the (Taylor) joint spectrum of T by σ(T) (see [9]) and define the joint operator norm ¶T¶ and the joint spectral radius r(T) by

and

respectively.