Using naive algebraic geometric methods a new proof of the following celebrated theorem of Magnus is given: Let G be a group with a presentation having n generators and m relations. If G also has a presentation on n − m generators, then G is free of rank n − m.

The purpose of this paper is to present a short, self-contained proof of Euler’s relation. The ingredients of this proof are (i) the principle of inclusion and exclusion of combinatorics and (ii) the Euler characteristic; a development of the Euler characteristic is included.

Recently, F. H. Clarke and Y. Ledyaev established a multidirectional mean value theorem applicable to lower semi-continuous functions on Hilbert spaces, a result which turns out to be useful in many applications. We develop a variant of the result applicable to locally Lipschitz functions on certain Banach spaces, namely those that admit a C1-Lipschitz continuous bump function.

In this paper, we define the η-invariant for a cusped hyperbolic 3-manifold and discuss some of its applications. Such an invariant detects the chirality of a hyperbolic knot or link and can be used to distinguish many links with homeomorphic complements.

The primary purpose of this paper is to provide necessary and sufficient conditions for certain quadratic polynomials of negative discriminant (which we call Euler-Rabinowitsch type), to produce consecutive prime values for an initial range of input values less than a Minkowski bound. This not only generalizes the classical work of Frobenius, the later developments by Hendy, and the generalizations by others, but also concludes the line of reasoning by providing a complete list of all such primeproducing polynomials, under the assumption of the generalized Riemann hypothesis (GRH).We demonstrate how this prime-production phenomenon is related to the exponent of the class group of the underlying complex quadratic field. Numerous examples, and a remaining conjecture, are also given.

We show that the multiplier space , where X is BMOA, VMOA, B, B0 or disk algebra A. We give the multipliers from , we also give the multipliers from , C0, BMOA, and Hp.

For every field F of characteristic p ≥ 0, we construct an example of a finite dimensional nilpotent F-algebra R whose adjoint group A(R) is not centreby- metabelian, in spite of the fact that R is Lie centre-by-metabelian and satisfies the identities x2p = 0 when p > 2 and x8 > 0 when p = 2. The existence of such algebras answers a question raised by A. E. Zalesskii, and is in contrast to positive results obtained by Krasilnikov, Sharma and Srivastava for Lie metabelian rings and by Smirnov for the class Lie centre-by-metabelian nil-algebras of exponent 4 over a field of characteristic 2 of cardinality at least 4.

We consider rings as in the title and find the precise obstacle for them not to be Quasi-Frobenius, thus shedding new light on an old open question in Ring Theory. We also find several partial affirmative answers for that question.

In this paper, we consider Dirichlet series with Euler products of the form F(s) = Πp in > 1, and which are regular in ≥ 1 except for a pole of order m at s = 1. We establish criteria for such a Dirichlet series to be nonvanishing on the line of convergence. We also show that our results can be applied to yield non-vanishing results for a subclass of the Selberg class and the Sato-Tate conjecture.

Given a continuous map δ from the circle S to itself we want to find all self-maps σ: S → S for which δ o σ. If the degree r of δ is not zero, the transformations σ form a subgroup of the cyclic group Cr. If r = 0, all such invertible transformations form a group isomorphic either to a cyclic group Cn or to a dihedral group Dn depending on whether all such transformations are orientation preserving or not. Applied to the tangent image of planar closed curves, this generalizes a result of Bisztriczky and Rival [1]. The proof rests on the theorem: Let Δ: ℝ → ℝ be continuous, nowhere constant, and limx→−∞ Δ(x) = −∞, limx→−∞ Δ(xx) = +∞; then the only continuous map Σ: R → R such that Δ o Σ = Δ is the identity Σ = idℝ.

We classify the compact 3-manifolds whose boundary is a union of 2-spheres, and which embed in T ×I ×I, where T is a triod and I the unit interval. This class is described explicitly as the set of punctured handlebodies. We also show that any 3-manifold in T × I × I embeds in a punctured handlebody.

This paper is a study of summability methods that are based on Dirichlet convolution. If f(n) is a function on positive integers and x is a sequence such that then x is said to be Af-summable to L. The necessary and sufficient condition for the matrix Af to preserve bounded variation of sequences is established. Also, the matrix Af is investigated as ℓ − ℓ and G − G mappings. The strength of the Af-matrix is also discussed.

Statistics in the 20th century has been enlivened by a passionate, occasionally bitter, and still vibrant debate on the foundations of statistics and in particular on Bayesian vs. frequentist approaches to inference. In 1975 D. V. Lindley predicted a Bayesian 21st century for statistics. This prediction has often been discussed since, but there is still no consensus on the probability of its correctness. Recent developments in the asymptotic theory of statistics are, surprisingly, shedding new light on this debate, and may have the potential to provide a common middle ground.

This paper treats the quasilinear elliptic inequality

where N ≥ 2, m > 1, σ >m− 1, and p:ℝN → (0, ∞) is continuous. Sufficient conditions are given for this inequality to have no positive entire solutions. When p has radial symmetry, the existence of positive entire solutions can be characterized by our results and some known results.

Let H be the split, adjoint group of type E6 over a p-adic field. In this paper we study the restriction of the minimal representation of H to the closed subgroup PGL3 × G2.

We show that finite dimensional subfactors do not have subdiagonal algebras unless the Jones index is one.