The “k-dense” subgraphs of a connected graph G are connected and contain neighbours of all but at most k-1 points. We consider necessary and sufficient conditions that a point be in Γk, the union of the minimal k-dense subgraphs. It is shown that Γk contains all the [m, k]-isthmuses” and [m, k]-articulators“— minimal subgraphs which disconnect the graph into at least k + 1 disjoint graphs—and that an [m, k]-isthmus or [m, k]-articulator of Γk disconnects G. We define “central points,” “degree” of a point, and “chromatic number” and examine the relationship of these concepts to connectivity. Many theorems contain theorems previously proved (1) as special cases.

α-Recursion theory was invented simultaneously by Kripke [15] and Platek [22] and served to generalize the theories of Takeuti [34], Machover [20], Kreisel and Sacks [14] and others. Kripke (in [16]) derived machinery to construct an analogue to Kleene's T-predicate enabling him to assert that all of unrelativized ordinary recursion theory (as found in Kleene [13]) lifted to α-recursion theory. As a result, we were able to set down in [8] α-analogues to Blum's [1] well-studied axioms, thus, introducing the study of α-computational complexity theory.

A theorem of Eisenbud, Griffith, and Robson states that if R is hereditary and noetherian (on both the left and right) then every proper homomorphic image of R is a generalized unserial ring (see, for example, [3, p. 244]). Singh [11, p. 883] states a converse: If R is a right bounded, noetherian prime ring, all of whose proper homomorphic images are generalized uniserial rings, then (every divisible right R-module is injective, so) R is right hereditary. (Actually, Singh omitted the clearly necessary “bounded” condition.) Singh's theorem generalizes results of [9, Proposition 15], [2, Theorem 2.1], and [8], about commutative rings.

We will call a semi-prime ring R essentially right bounded if each essential right ideal contains a two-sided ideal which is essential as a right ideal. In case R is prime, “essentially right bounded” coincides with “right bounded”.

If f(z) is univalent (regular and one-to-one) in the open unit disk Δ, Δ = {z ∊ C:│z│ < 1}, and has a Maclaurin series expansion of the form

(1.1)

then, as de Branges has shown, │ak│ = k, for k = 2, 3, … and the Koebe function.

(1.1)

serves to show that these bounds are the best ones possible (see [3]). The functions defined above are generally said to constitute the class .

We investigate here the notion of a topological ring of quotients of a topological ring with respect to an arbitrary Gabriel (idempotent) filter of right ideals. We describe the topological ring of quotients first as a subring of the algebraic ring of quotients, and then show it is a topological bicommutator of a topological injective R-module. Unlike R. L. Johnson in [6] and F. Eckstein in [2] we do not always make the ring an open subring of its ring of quotients. This would exclude examples such as C(X), the ring of continuous real-valued functions on a compact space, and its ring of quotients as described in Fine, Gillman and Lambek [3].

On se propose d'étendre un résultat de Rorhlich [5] concernant les moyennes galoisiennes des valeurs en s = 1 des fonctions L associées à une forme modulaire de poids 2, tordue par certains caractères de Dirichlet. On considérera une représentation automorphe parabolique π de GL(n), n≦2, sur un corps de nombres K, et l'on s'intéressera aux valeurs des fonctions L(S,πχ) et de leurs dérivées L(m)(s, πχ), m ≧1, où χ parcourt certains caractères de Hecke de K, d'ordre fini, et où s appartient à la bande a < Re s < 1 — a, où a mesure la déviation de n par rapport à la conjecture de Petersson généralisée.

Equivariant holomorphic maps of Hermitian symmetric domains into Siegel upper half spaces can be used to construct families of abelian varieties parametrized by locally symmetric spaces, which can be regarded as complex torus bundles over the parameter spaces. We extend the construction of such torus bundles using 2-cocycles of discrete subgroups of the semisimple Lie groups associated to the given symmetric domains and investigate some of their properties. In particular, we determine their cohomology along the fibers.

We shall call a linear function on the elements of a ring R skew-symmetric if it vanishes whenever at least two of the variables are equal. Here we shall study rings R of characteristic not 2 which satisfy the following two identities:

1

(2)is skew-symmetric.

Both of these identities hold in alternative rings. The fact that F(w, x, y, z) is skew-symmetric in alternative rings is an important tool in the study of such rings. It is also obvious that both identities hold in commutative rings. But unlike other recent generalizations of commutative and alternative rings it turns out that there exist simple, finite dimensional algebras of degree two which are neither alternative nor commutative and satisfy (1) and (2).

We verify a generalization of (3.3) from [Lê73] proving that the homotopy type of the Milnor fiber of a reduced hypersurface singularity depends only on the embedded topological type of the singularity. In particular, using [Zariski68, Lipman83, Oh93, Gau88] for irreducible quasi-ordinary germs, it depends only on the normalized distinguished pairs of the singularity. The main result of the paper provides an explicit formula for the Euler-characteristic of the Milnor fiber in the surface case.

Nous allons montrer que tout système faible de Tchebycheff est voisin d'un système de Tchebycheff. Ce fait permet de prolonger aux systèmes faibles de Tchebycheff plusieurs résultats valides pour les systèmes de Tchebycheff, tels le théorème d'alternance et le théorème de de la Vallée- Poussin qui sont reliés au problème de la meilleure approximation uniforme par une fonction d'un système de Tchebycheff. Il arrive que certaines classes de fonctions splines forment un système faible de Tchebycheff. Pour ces classes, on pourra donc caractériser les meilleures approximations par des propriétés d'alternance de l'écart à la fonction à approcher.

N. Wiener [12] introduced the idea of the class Vp. A 2 π-periodic function ƒ is said to have bounded p-variation Vp(f)(1 ≦ p < ∞), or to belong to the class Vp, if

The density u(x, t) of an ideal gas flowing through a homogeneous porous media satisfies the equation Here m > 1 is a physical constant and u also satisfies the initial condition If the initial data is not strictly positive it is necessary to work with generalized solutions of the Cauchy problem (1), (2) (see [1]). By a weak solution we shall mean a function u(x, t) such that for (in the sense of distributions) and

In the Euclidean plane a curve C has a one-parameter family of parallel involutes and a unique evolute C* which coincides with the locus of the centres of the osculating circles of C. If is parallel to C, C* is also the evolute of . We will study parallel curves in n-dimensional Euclidean space and obtain generalizations of the properties given above.

We will study parallel curves in n-dimensional Euclidean space and obtain generalizations of the properties given above.

In [10], Witt laid the foundation for the study of quadratic forms over fields. Suppose Q is a quadratic form defined on a finite dimensional vector space V over a field of characteristic not equal to 2. Witt showed that non-zero vectors x and y in V satisfying Q(x) = Q(y) can be mapped into each other via an isometry of the vector space V. More generally, if τ : W ⟶ W’ is an isometry between subspaces of V, then τ extends to an isometry ϕ of V.

In this article, we study complete surfaces $\sum $, isometrically immersed in the product spaces ${{\mathbb{H}}^{2}}\,\times \,\mathbb{R}$ or ${{\mathbb{S}}^{2\,}}\times \,\mathbb{R}$ having positive extrinsic curvature ${{K}_{e}}$. Let ${{K}_{i}}$ denote the intrinsic curvature of $\sum $. Assume that the equation $a{{K}_{i\,}}\,+\,b{{K}_{e\,}}\,=\,c$ holds for some real constants $a\,\ne \,0$, $b\,>\,0$, and $c$. The main result of this article states that when such a surface is a topological sphere, it is rotational.

We construct a scheme $B(r; {\mathbb {A}}^n)$ such that a map $X \to B(r; {\mathbb {A}}^n)$ corresponds to a degree-n étale algebra on X equipped with r generating global sections. We then show that when $n=2$ , i.e., in the quadratic étale case, the singular cohomology of $B(r; {\mathbb {A}}^n)({\mathbb {R}})$ can be used to reconstruct a famous example of S. Chase and to extend its application to showing that there is a smooth affine $r-1$ -dimensional ${\mathbb {R}}$ -variety on which there are étale algebras ${\mathcal {A}}_n$ of arbitrary degrees n that cannot be generated by fewer than r elements. This shows that in the étale algebra case, a bound established by U. First and Z. Reichstein in [6] is sharp.

Hyperbolic differential operators with constant coefficients introduced and studied systematically by Gårding (4), were characterized by the existence of the fundamental solution with some cone condition, according to Hörmander (6). Recently Ehrenpreis, extending the notion of hyperbolicity due to Gårding, has defined hyperbolic operators for distributions with compact support in the convolution sense. Under the hypothesis that the operator is invertible as a distribution, he has established a theorem analogous to the theorem of Hörmander mentioned above (3).

Let X be a normal projective variety of dimension n and G an abelian group of automorphisms such that all elements of $G\setminus \{\operatorname {id}\}$ are of positive entropy. Dinh and Sibony showed that G is actually free abelian of rank $\le n - 1$ . The maximal rank case has been well understood by De-Qi Zhang. We aim to characterize the pair $(X, G)$ such that $\operatorname {rank} G = n - 2$ .

The Bernoulli polynomials of order k, where k is a positive integer, are defined by

Bm(k)(x) is a polynomial of degree m with rational coefficients, and the constant term of Bm(k)(x) is the mth Bernoulli number of order k, Bm(k). In a previous paper (3) we obtained some conditions, in terms of k and m, which imply that Bm(k)(x) is irreducible (all references to irreducibility will be with respect to the field of rational numbers). In particular, we obtained the following two results.

The present work stems from the following classical result, due to G. H. Hardy, J. E. Littlewood, G. Pólya [7], and R. Rado [10].

THEOREM 1. Concerning a pair of n-tuples x, y ϵ Rn, the following four statementsare equivalent:

(a) for every continuous, convex function f : R → R