Let $k$ be an arbitrary positive integer and let $\unicode[STIX]{x1D6FE}(n)$ stand for the product of the distinct prime factors of $n$. For each integer $n\geqslant 2$, let $a_{n}$ and $b_{n}$ stand respectively for the maximum and the minimum of the $k$ integers $\unicode[STIX]{x1D6FE}(n+1),\unicode[STIX]{x1D6FE}(n+2),\ldots ,\unicode[STIX]{x1D6FE}(n+k)$. We show that $\liminf _{n\rightarrow \infty }a_{n}/b_{n}=1$. We also prove that the same result holds in the case of the Euler function and the sum of the divisors function, as well as the functions $\unicode[STIX]{x1D714}(n)$ and $\unicode[STIX]{x1D6FA}(n)$, which stand respectively for the number of distinct prime factors of $n$ and the total number of prime factors of $n$ counting their multiplicity.

In this paper we study the zero sets of harmonic functions on open sets in ${{\mathbb{R}}^{N}}$ and holomorphic functions on open sets in ${{\mathbb{C}}^{N}}$. We show that the non-extendability of such zero sets is a generic phenomenon.

In this article we will show that there are infinitely many symmetric, integral 3 × 3 matrices, with zeros on the diagonal, whose eigenvalues are all integral. We will do this by proving that the rational points on a certain non-Kummer, singular $\text{K3}$ surface are dense. We will also compute the entire Néron–Severi group of this surface and find all low degree curves on it.

Let $F$ be a local non-archimedean field of characteristic zero. We prove that a representation of $GL\left( n,\,F \right)$ obtained from irreducible parabolic induction of supercuspidal representations is distinguished by an orthogonal group only if the inducing data is distinguished by appropriate orthogonal groups. As a corollary, we get that an irreducible representation induced from supercuspidals that is distinguished by an orthogonal group is metic.

Suppose throughout that {sn} is a sequence of real numbers, λ > - 1, a > 0, and β is real. Let N be any non-negative integer such that αN + β > l.

We are concerned primarily with the logarithmic summability method L and the Borel-type method (B, α, β). Some known results involve the Abel-type summability method Aγ.

We establish variants of stability estimates in norms somewhat stronger than the ${{H}^{1}}$-norm under Arnold's stability hypotheses on steady solutions to the Euler equations for fluid flow on planar domains.

It is shown that a Lie algebra having a nilpotent radical has a fundamental set of invariants consisting of Casimir operators. A different proof is given in the well known special case of an abelian radical. A result relating the number of invariants to the dimension of the Cartan subalgebra is also established.

The aim of this article is to obtain an upper bound for the exponential sums $\sum{e(f(x)\,/\,q)}$, where the summation runs from $x=1$ to $x=q$ with $(x,q)=1$ and $e(\alpha )$ denotes $\exp (2\pi i\alpha )$.

We shall show that the upper bound depends only on the values of $q$ and $s$, where $s$ is the number of terms in the polynomial $f(x)$.

Let $A$ be a stable, separable, real rank zero ${{C}^{*}}$-algebra, and suppose that $A$ has an AF-skeleton with only finitely many extreme traces. Then the corona algebra $\mathcal{M}\left( A \right)/A$ is purely infinite in the sense of Kirchberg and Rørdam, which implies that $A$ has the corona factorization property.

A subset of a topological space is called discrete iff every point in the space has a neighborhood which meets the set in at most one point. Discrete sets are useful for decomposing the images of certain maps and for generalizing closed maps. All discrete sets are closed iff the space is T1. As a result of characterizing discrete and countably discrete maps, theorems due to Vaĭnšteĭn and Engelking are extended to these maps.

In this note we do not intend to establish new results but only to suggest a very simple proof of Lusin's theorem, direct for σ-finite regular measures, a proof that bypasses the usual procedure of first establishing this theorem for sets of finite measure only. The proposed proof utilizes the notion of subuniform convergence, a method which seems not yet to have been used, despite its simplicity and adaptability. Simultaneously, a useful supplement to Egoroff's theorem will be obtained.

Using a classical generalization of Jacobi’s derivative formula, we give an explicit expression for Gunning’s prime form in genus 2 in terms of theta functions and their derivatives.

In this Bulletin Ligh [2] generalized to finite near rings with identity a theorem Zassenhaus [5] used to prove every finite near field has abelian addition. B. H. Neumann [4] extended Zassenhaus’ result, using similar techniques and showed that all near fields are abelian. It has been an open question whether Ligh’s generalization could be carried out to infinite near rings with identity. The purpose of this paper is to show that Ligh’s theorem cannot be so extended. In particular, it cannot be extended even to distributively generated near rings, a type of near ring which has been useful in studying endomorphism rings of non-abelian groups [1,3].

The nonsolvable R-groups as defined by Walter [3] are groups of orders (q3+l)q3(q — 1), q = 32n+1, n ≥ 0. These are the groups of Ree type discussed by Ward [4] together with the Ree group R(3) of order 28.27.2. The R-group with parameter q has a doubly transitive representation of degree q3+1 but in this note we will prove that it cannot contain a sharply doubly transitive subset.

In this paper, we are concerned with the functional inequality

1

where 0 < Pi < l, 0 < qi < l, fi(p)≠0, for 0 < P < 1, (i = 1, 2,..., n) and n is a fixed positive integer, n ≥ 2.

Inequality (1) was studied by Rényi and Fischer, (see [1], [3]) in the special case

2

and this provided a characterization of Rényi's entropy.

If I is a right ideal of a ring R, I is said to be an annihilator right ideal provided that there is a subset S in R such that

I is said to be injective if it is injective as a submodule of the right regular R-module RR. The purpose of this note is to prove that a prime ring R (not necessarily with 1) which satisfies the ascending chain condition on annihilator right ideals is a simple ring with descending chain condition on one sided ideals if R contains a nonzero right ideal which is injective.

A. M. Macbeath, in November 1965, communicated the following theorem to me which he proved with the aid of the Lefschetz fixed point formula.

Theorem. If Γ is a Fuchsian group and N a torsion free normal subgroup, then the rank of N/[Γ, N] is twice the genus of the orbit space D/Γ where D denotes the hyperbolic plane which Γ acts.

This theorem will follow from a consideration of the exact sequence

*

Anderson and Ohm have introduced valuations of monoid rings k[Γ] where k is a field and Γ a cancellative torsion-free commutative monoid. We study the residue class fields in question and solve a problem concerning the pure transcendence of the residue fields.

We obtain an asymptotic formula for the number of (k, r)-free integers that do not exceed x. By definition, a (k, r)-free integer is one in whose canonical representation no prime power is in the interval [r, k−1] where 1 < r < k are fixed integers. These include as special cases the r-free integers, the semi r-free integers and the k-full integers. We obtain an asymptotic formula for the number of representations of an integer as the sum of a prime and a (k, r)-free integer, and use the result to prove that every sufficiently large integer can be represented as the sum of a prime and m = abk where a and b are both square free, (a, b) = 1, b > 1 and k is any fixed integer, k ≥ 3.

We construct a positive linear contraction T of all LP(X, μ)- spaces, 1 ≦ p ≦ ∞, μ(X) = 1 such that T1 = 1, T* 1 = 1 and also Tf > 0 a.e. for all f ≧ 0 a.e., f ≢ 0 but for which there is an f ∊ L∞ such that (Tnf — ∫ fdμ) does not converge in L1-norm. We also show that if T is a contraction of a Hilbert space H, there exists an isometry Q and a contraction R such that ∥Tnx - QnRx∥ —> 0 as n —» ∞ for all x in H