Let f(x) be an irreducible polynomial of degree n with coefficients in a field L and r be an integer prime to the characteristic of L. The object of this paper is to describe the galois group g of f(xr) over L when the galois group G of f(x) itself over L is either the full symmetric group Snor the alternating group An. We shall call f standard if G = Sn or An with |G|>2.

We present some results on locally (soluble-by-finite) groups with various restrictions on subgroups of infinite rank.

A homology theory based on measures, first mentioned by Thurston, is naturally defined here as a functor into the category of locally convex topological vector spaces. It is proved that the first homology space is Hausdorff.

We investigate questions on the existence of nontrivial solution for a class of the critical Kirchhoff-type problems in Hyperbolic space. By the use of the stereographic projection the problem becomes a singular problem on the boundary of the open ball $B_1(0)\subset \mathbb{R}^n$ Combining a version of the Hardy inequality, due to Brezis–Marcus, with the mountain pass theorem due to Ambrosetti–Rabinowitz are used to obtain the nontrivial solution. One of the difficulties is to find a range where the Palais Smale converges, because our equation involves a nonlocal term coming from the Kirchhoff term.

For p ≥ 1, one can define a generalisation of the unknotting number tup called the pth untwisting number, which counts the number of null-homologous twists on at most 2p strands required to convert the knot to the unknot. We show that for any p ≥ 2 the difference between the consecutive untwisting numbers tup–1 and tup can be arbitrarily large. We also show that torus knots exhibit arbitrarily large gaps between tu1 and tu2.

In this article, we study localizations of hearts of cotorsion pairs ($\mathcal{U}, \mathcal{V}$) where $\mathcal{U}$ is rigid on an extriangulated category $\mathcal{B}$ . The hearts of such cotorsion pairs are equivalent to the functor categories over the stable category of $\mathcal{U}$ ( $\bmod \underline{\mathcal{U}}$ ). Inspired by Marsh and Palu (Nagoya Math. J.225(2017), 64–99), we consider the mutation (in the sense of Iyama and Yoshino, Invent. Math.172(1) (2008), 117–168) of $\mathcal{U}$ that induces a cotorsion pair ( $\mathcal{U}^{\prime}, \mathcal{V}^{\prime}$ ). Generally speaking, the hearts of ( $\mathcal{U}, \mathcal{V}$ ) and ( $\mathcal{U}^{\prime}, \mathcal{V}^{\prime}$ ) are not equivalent to each other, but we will give a generalized pseudo-Morita equivalence between certain localizations of their hearts.

In the paper we describe Kahler QCH surfaces. We prove that any Calabi type and orthotoric Kahler surfaces are QCH Kahler surfaces. We also classify locally homogeneous QCH surfaces.

We show that the theorem stated in the title is a corollary to a result of K. A. Zaretskii [5] and a theorem of G. Birkhoff [1]. The construction we use further shows that all groups with cardinal less than or equal to the cardinal of the given group are simultaneously realised as maximal subgroups of the same semigroup of binary relations ℬx. For finite or countable groups, when Xmay be taken to be finite or countable, respectively, and for an entirely different method of proof, the paper of J. S. Montague and R. J. Plemmons [3] should be consulted. For two further proofs of the theorem of the title to this note, this time for any X, see also R. J. Plemmons and B. M. Schein [4] and A. H. Clifford [2].

In his unpublished manuscripts (referred to by Birch [1] as Fragment V, pp. 247–249), Ramanujan [3] gave a whole list of assertions about various (transforms of) modular forms possessing naturally associated Euler products, in more or less the spirit of his extremely beautiful paper entitled “On certain arithmetical functions” (in Trans. Camb. Phil. Soc. 22 (1916)). It is simply amazing how Ramanujan could write down (with an ostensibly profound insight) a basis of eigenfunctions (of Hecke operators) whose associated Dirichlet series have Euler products, anticipating by two decades the famous work of Hecke and Petersson. That he had further realized, in the event of a modular form f not corresponding to an Euler product, the possibility of restoring the Euler product property to a suitable linear combination of modular forms of the same type as f, is evidently fantastic.

In this paper we study some questions proposed by B. Schein [8] regarding the semigroup of binary relations Bx for a finite set X: what is the ideal structure of Bx, what are the congruences on Bx, what are the endomorphisms of Bx? For |X| = nit is convenient to regard Bx as the semigroup Bn of n×n (0, l)-matrices under Boolean matrix multiplication.

A well-known fact concerning a prime right Goldie ring R, proved in [4, Section 5], is that an essential right ideal is generated by the regular elements which it contains. There is a modification of that proof which shows that each element of R is the sum of at most two regular elements. This suggested that the recent results of Chatters and Ginn [1] concerning rings generated by their regular elements might possibly be refined a little, since their arguments actually show that elements of R are sums of at most three regular elements.

We begin this paper by considering a Boolean algebra as a lattice which is relatively pseudo-complemented (i.e., residuated with respect to intersection) and give, in this case, certain properties of the equivalences of types A, B and F(as introduced by Molinaro [1]). We then show how these results carry over to the case of Boolean matrices, which form a Boolean algebra residuated also with respect to matrix multiplication. Other properties of matrix residuals are established and we conclude with three algebraic characterisations of invertible Boolean matrices.

In § 2 a number of integrals in which the integrand contains a product of a hypergeometric function and an E-function will be evaluated. The following formulae will be employed in the proofs.

If ρ +σ = α + β + γ + 1, and if α, β or γ is zero or a negative integer,

this is Sallschütz's theorem [1].

If R(γ - ½α - ½β)< - ½,

This theorem was given by Wastson [2] for negative integral values of α and later by Whipple [3] for general values α.

R(γ) > 0,

This formula was given by Whipple [3]

l is a positive integer,

An inverse semigroup R is said to be reduced (or proper) if ℛ∩σ= i (where σ is the minimum group congruence on R). McAlister has shown ([3], [4]) that every reduced inverse semigroup is isomorphic with a “P-semigroup” P(G, , ), for some semilattice , poset containing as an ideal, and group G acting on by order-automorphisms; (and, conversely, every P-semigroup is reduced). In [4], he also found the morphisms between P-semigroups, in terms of morphisms between the respective groups, and between the respective posets.

In [1], the natural representation module of the symmetric groups, hereafter called the first natural representation module of the symmetric groups, was analysed. It is the purpose of this paper to analyse the second natural representation module of the symmetric groups.

The formula to be proved is

where R(z)>0.

Let G be a finite abelian group, and Y be a closed surface. The problems of classifying and enumerating the free and effective G-actions on Y modulo selfhomeomorphisms of Y and X = Y/G can be transferred into ones of classifying regular G-coverings on X. P. A. Smith [7], proved that for any prime number p there are pr(r–1)/2 equivalence classes of free (ℤp)r actions on Y provided that rℤgenus of X. This paper is devoted to the classification and the enumeration of regular G-covering surfaces, when G is any finite abelian group. Recently, A. Edmonds [2] classified the G-actions on closed surfaces by their G-bordism classes in the set (G) of free oriented G-cobordism classes of free oriented G-surfaces.

Let ℂn,n denote the space of n × n matrices with complex entries and let ℋn denote the set of n × n hermitian matrices. Given any matrix A∊ℂn,n, the Lyapunov transformation corresponding to A is defined by ℐA(H) = AH+HA*, where H∊ℋn. Let PSD(n) be the set of all n × n hermitian positive semidefinite matrices. Taussky [8, 9] raised the problems of determining

and

Both of these problems seem to be difficult.

§ 1. Introductory. The formula to be established is

where m is a positive integer,

and the constants are such that the integral converges.

Let σ1(n) denote the sum of the tth powers of the divisors of n, σ(n) = σ1(n). Also place

where γ is Euler's constant, ζ(s) is the Riemann ζ-function and x ≧ 2. The function Δ(x) is the remainder term arising in the divisor problem for σ((m, n)). Cesàro proved originally [1], [6, p. 328] that Δ(x) = o(x2 log x). More recently in I [2, (3.14)] it was shown by elementary methods that . This estimate was later improved to in II [3, (3.7)]. In the present paper (§ 3) we obtain a much more substantial reduction in the order of Δ(x), by showing that Δ(x) can be expressed in terms of the remainder term in the classical Dirichlet divisor problem. On the basis of well known results for this problem, it follows easily that . The precise statement of the result for σ((m, n)) is contained in (3.2).