Let f(x) denote a polynomial of degree d defined over a finite field k with q = pnelements. B. J. Birch and H. P. F. Swinnerton-Dyer [1] have estimated the number N(f) of distinct values of y in k for which at least one of the roots of

is in k. They prove, using A. Weil's deep results [12] (that is, results depending on the Riemann hypothesis for algebraic function fields over a finite field) on the number of points on a finite number of curves, that

where λ is a certain constant and the constant implied by the O-symbol depends only on d. In fact, if G(f) denotes the Galois group of the equation (1.1) over k(y) and G+(f) its Galois group over k+(y), where k+ is the algebraic closure of k, then it is shown that λ depends only on G(f), G+(f) and d. It is pointed out that “in general”

Applying Hopkins's Theorem asserting that each unitary right Artinian ring is right Noetherian, G. Köthe and K. Shoda proved the following theorem (cf. Köthe [7], p. 360, Theorem 1 and p. 363, Theorem 5): If R is a unitary right Artinian ring, then the following statements hold:

(i) Each nilpotent subring of R is contained in a maximal nilpotent subring of R.

(ii) The intersection of all maximal nilpotent subrings of R is the maximal nilpotent twosided ideal of R.

(iii) All maximal nilpotent subrings of R are conjugate.

Let S be a closed Riemann surface of genus

g > 1,

so that Ŝ, the universal covering surface of S, is hyperbolic. We can then uniformize S by a discrete, nonabelian group Γ1 of Möbius transformations of the upper half-plane ℋ. It follows that N1 = NΩ(Γ1) is discrete; here N1is the normalizer of Γ in Ω, the group of (conformal) automorphisms of ℋ. An automorphism of S can be lifted to a coset of Nl/Γl. Hence C(S), the group of automorphisms of S, is isomorphic to Nl/Γ1. The order of C = C(S) equals the index of Γ1 in N1, which in turn equals ⃒Γ1⃒ / ⃒Nl⃒, where ⃒Nl⃒ is the hyperbolic area of a fundamental region of Nl. Since Γ1 uniformizes a surface, we have ⃒Γ1⃒ = 4π(g – 1), while, by Siegel's results [7], ⃒N1 ⃒ ≧ π/21 and N1 can only be the triangle group (2, 3, 7). Hence in all cases the order of C(S) is at most 84(g–1), an old result of Hurwitz [1]. The surfaces that permit a maximal automorphism group (= automorphism group of maximum order) can therefore be obtained by studying the finite factor groups of (2, 3, 7). Such a treatment, purely algebraic in nature, has been promised by Macbeath [5].

If E is a subset of the real line of positive measure, then the associated Hilbert transform H = HE,

where the integral is a Cauchy principal value, is a bounded self-adjoint operator on L2(E) (cf. Muskhelishvili [4]). In case E = (-∞, ∞) the transformation is also unitary with a spectrum consisting of 1 and -1, each of infinite multiplicity (Titchmarsh [10]). If E is a inite interval the spectral representation of H has been given by Koppelman and Pincus [3]; see also Putnam [6]. In particular the spectrum of H is in this case the closed interval [-1, 1]. Moreover, according to Widom [11], the spectrum of H is [-1, 1] whenever E ≠ (-∞, ∞), that is, whenever

In [1], J. M. Howie considered the semigroup of transformations of sets and proved (Theorem 1) that every transformation of a finite set which is not a permutation can be written as a product of idempotents. In view of the analogy between the theories of transformations of finite sets and linear transformations of finite dimensional vector spaces, Howie's theorem suggests a corresponding result for matrices. The purpose of this note is to prove such a result.

According to Bourbaki [1, pp. 62–63, Exercise 11], a left (resp. right) A-module M is said to be pseudo-coherent if every finitely generated submodule of M is finitely presented, and is said to be coherent if it is both pseudo-coherent and finitely generated. This Bourbaki reference contains various results on pseudo-coherent and coherent modules. Then, in [1, p. 63, Exercise 12], a ring which as a left (resp. right) module over itself is coherent is said to be a left (resp. right) coherent ring, and various results on and examples of coherent rings are presented. The result stated in [1, p. 63, Exercise 12a] is a basic theorem of [2] and first appeared there. A variety of results on and examples of coherent rings and modules are presented in [3].

In 1926, I. J. Schur proved the following theorem on partitions [3].

The number of partitions of n into parts congruent to ±1 (mod 6) is equal to the number of partitions of n of the form 1 + …+bs = n, where bi–bi+1 ≧ 3 and, if 3 ∣ bi, then bi–bi+1 > 3.

Schur's proof was based on a lemma concerning recurrence relations for certain polynomials. In 1928, W. Gleissberg gave an arithmetic proof of a strengthened form of Schur's theorem [2]; however, the combinatorial reasoning in Gleissberg's paper becomes very intricate.

The asymptotic behavior of the solutions of ordinary nonlinear differential equations will be considered here. The growth of the solutions of a differential equation will be discussed by establishing criteria to determine when the differential equation does not possess a solution that is an element of the space Lp(0, ∞)(p ≧ 1).