In a series of papers [6], [7], [8], [10], Munn has considered the problem of constructing all irreducible representations of a semigroup by matrices over a field. In [10], he showed how to construct all the irreducible representations of an arbitrary inverse semigroup from those of associated Brandt semigroups. In this paper, we generalize the method of [10] to give a construction for the irreducible representations of an arbitrary semigroup from those of certain associated semigroups

We write

and

so that p(n) is the number of unrestricted partitions of n. Ramanujan [1] conjectured in 1919 that if q = 5, 7, or 11, and 24m ≡ 1 (mod qn), then p(m) ≡ 0 (mod qn). He proved his conecture for n = 1 and 2†, but it was not until 1938 that Watson [4] proved the conjecture for q = 5 and all n, and a suitably modified form for q = 7 and all n. (Chowla [5] had previously observed that the conjecture failed for q = 7 and n = 3.) Watson's method of modular equations, while theoretically available for the case q = 11, does not seem to be so in practice even with the help of present-day computers. Lehner [6, 7] has developed an essentially different method, which, while not as powerful as Watson's in the cases where Γ0(q) has genus zero, is applicable in principle to all primes q without prohibitive calculation. In particular he proved the conjecture for q = 11 and n = 3 in [7]. Here I shall prove the conjecture for q = 11 and all n, following Lehner's approach rather than Watson's. I also prove the analogous and essentially simpler result for c(m), the Fourier coefficient‡ of Klein's modular invariant j (τ) as

Many q-identities have been proved combinatorially. For example

Let B(X) denote the Banach algebra of all bounded linear operators on a Banach space X. Let t be an element of B(X), and let edenote the identity operator on X. Since the earliest days of the theory of Banach algebras, ithas been understood that the natural setting within which to study spectral properties of t is the Banach algebra B(X), or perhaps a closed subalgebra of B(X) containing t and e. The effective application of this method to a given class of operators depends upon first translating the data into terms involving only the Banach algebra structure of B(X) without reference to the underlying space X. In particular, the appropriate topology is the norm topology in B(X) given by the usual operator norm. Theorem 1 carries out this translation for the class of compact operators t. It is proved that if t is compact, then multiplication by t is a compact linear operator on the closed subalgebra of B(X) consisting of operators that commute with t.

In recent years, a number of special integral equations of the first kind was discussed by several authors (see [l]–[4], [6], [7], [9]–[18]). The kernels of these integral equations are special functions of the hypergeometric family, and it was necessary to restrict the parameters appearing in these functions to secure convergence of the integrals. If these restrictions are removed, the integral fails to converge but it may possess a finite part (in Hadamard's sense), and the question arises whether the methods used in the restricted case will alsoapply in the new situation. Indeed, one could pose the moregeneral problem of Volterra integral equations involving finite parts of divergent integrals [19]

In the terminology of Clifford and Preston [2], a band B is a semigroup in which every element is idempotent. On such a semigroup there is a natural (partial) order relation defined by the rule

If the order relation ≧ is compatible with the multiplication in B, in the sense that e ≧ f and g ≧ h together imply that eg ≧ fh, we shall say that B is a naturally ordered band. The object of this note is to describe the structure of naturally ordered bands.

A Hadamard matrix H is an orthogonal square matrix of order m all the entries of which are either + 1 or - 1; i. e.

where H′ denotes the transpose of H and Im is the identity matrix of order m. For such a matrix to exist it is necessary [1] that

It has been conjectured, but not yet proved, that this condition is also sufficient. However, many values of m have been found for which a Hadamard matrix of order m can be constructed. The following is a list of such m (p denotes an odd prime).

Let R denote the space of real numbers and let D(R) denote the family of all functions mapping R into R that are (finitely) differentiable at each point of R. Since the composition f o g of two differentiable functions is also differentiable and since the composition operation is associative, it follows that D(R) is a semigroup with this operation. Such semigroups have been studied previously. Nadler, in [4], has shown that the semigroup of al differentiable functions mapping the closed unit interval into itself has no idempotent elements other than the identity function and the constant functions. The proof of that result carries over easily to the semigroup D(R).