In this paper we study several kinds of maximal almost disjoint families. In the main result of this paper we show that for successor cardinals $\kappa$, there is an unexpected connection between invariants $\frac{a}_{e}(\kappa), \frac{b} (\kappa)$ and a certain cardinal invariant $\frac{m}_{d}(\kappa^{+})$ on $\kappa^{+}$. As a corollary we get for example the following result. For a successor cardinal $\kappa$, even assuming that $\kappa^{<\kappa}=\kappa$ and $2^{\kappa}=\kappa^{+}$, the following is not provable in Zermelo–Fraenkel set theory. There is a $\kappa^{+}$-cc poset which does not collapse $\kappa$ and which forces $\frac{a} (\kappa)=\kappa^{+}<\frac{a}_{e}(\kappa)=\kappa^{++}=2^{\kappa}$. We also apply the ideas from the proofs of these results to study $\frac{a} =\frac{a} (\omega)$ and $\hbox{\rm non}(\cal{M})$.

The purpose of this note is to give a proof of a theorem of Serre, which states that if G is a p-group which is not elementary abelian, then there exist an integer m and non-zero elements x1, … xm∈H1 (G, Z/p) such that

formula here

with β the Bockstein homomorphism. Denote by mG the smallest integer m satisfying the above property. The theorem was originally proved by Serre [5], without any bound on mG. Later, in [2], Kroll showed that mG[les ]pk−1, with k=dimZ/pH1 (G, Z/p). Serre, in [6], also showed that mG[les ](pk−1)/ (p−1). In [3], using the Evens norm map, Okuyama and Sasaki gave a proof with a slight improvement on Serre's bound; it follows from their proof (see, for example, [1, Theorem 4.7.3]) that mG[les ](p+1) pk−2. However, mG can be sharpened further, as we see below.

For convenience, write H*ast;(G, Z/p)=H*(G). For every xi∈H1(G), set

formula here

It is proved that the number of ${\bb F}_q$-rational points of an irreducible projective smooth 3-dimensional geometrically unirational variety defined over the finite field ${\bb F}_q$ with $q$ elements is congruent to 1 modulo $q$. Some Fermat 3-folds, some classes of rationally connected 3-folds and some weighted projective $d$-folds also having this property are given.

This paper provides a proof that an n-dimensional complete open Riemannian manifold M with sectional curvature KM [ges ] −1 is diffeomorphic to a Euclidean n-space Rn if the volume growth of geodesic balls in M is close to that of the balls in an n-dimensional hyperbolic space Hn(−1) of sectional curvature −1.

The aim of this paper is to prove that for certain generalized Weyl algebras A (including the first Weyl algebra A1 over a field of characteristic zero) and for every simple left (right) A-module M, there are infinitely many non-isomorphic simple left (right) A-modules {Ni} such that Ext1A (M, Ni) ≠ 0 (respectively Ext1A(Ni, M) ≠ 0.

Let F be a field of characteristic other than 2. Let F(2) denote the compositum over F of all quadratic extensions of F, let F(3) denote the compositum over F(2) of all quadratic extensions of F(2) that are Galois over F, and let F{3} denote the compositum over F(2) of all quadratic extensions of F(2). This paper shows that F(3) = F{3} if and only if F is a rigid field, and that F(3) = K(3) for some extension K of F if and only if F is Pythagorean and K = F(√−1). The proofs depend mainly on the behavior of quadratic forms over quadratic extensions, and the corresponding norm maps.

Keedwell has shown that none of the groups of order less than 5 has a weak uniquely completable set. We prove that a weak uniquely completable set exists in a latin square based on a finite group if and only if the group is of order greater than 5.

Regular homotopy classes of immersions of homotopy $n$-spheres into $(n+q)$-space form an Abelian group under connected summation. The subgroup of immersions of those homotopy spheres which bound parallelizable manifolds is computed for $n=4k-1$ and arbitrary codimension $q$. A consequence of this computation is that in codimensions $q<2k+1$, the group of immersed homotopy $(4k-1)$-spheres is not the direct sum of the group of immersed standard spheres and the group of homotopy spheres. Also, our results shed light on Brieskorn's famous equations: for any exotic sphere (bounding a parallelizable manifold), a countable number of distinct equations give rise to codimension-two embeddings of this exotic sphere. It follows from our results that these embeddings lie in pairwise distinct regular homotopy classes, and that any regular homotopy class containing embeddings is representable by a map arising from such an equation.

a proof is given to show that for an inner form of $\gl_n$ over a global field of zero characteristic, there exist only a finite number of automorphic representations with fixed local factor (up to equivalence) at almost every place. what is new in comparison to earlier work (see a. i. badulescu and p. broussous, ‘un théorème de finitude’, compositio math. 132 (2002) 177–190) is the case when the local factors are not fixed at the infinite places, as well as the statement of the result for the automorphic spectrum, rather than the cuspidal one.

It is shown that a hypercentral group that has all subgroups subnormal and every non-nilpotent subgroup of bounded defect is nilpotent. As a consequence, a hypercentral group of length at most ω in which every subgroup is subnormal is nilpotent.

It is a well-known problem of Von Neumann to discover whether the countable chain condition and weak distributivity of a complete Boolean algebra imply that it carries a strictly positive probability measure. It was shown recently by Balcar, Jech and Pazák, and by Veličković, that it is consistent with ZFC, modulo the consistency of a supercompact cardinal, that every ccc weakly distributive complete Boolean algebra carries a contiuous strictly positive submeasure – that is, it is a Maharam algebra. We use some ideas of Gitik and Shelah and implications from the inner model theory to show that some large cardinal assumptions are necessary for this result.

It is known that if a is an algebraic element of a Banach algebra A, then its spectrum σ(a) is finite, and there exists γ > 0 such that the Hausdorff distance to spectra of nearby elements satisfies

(formula here)

We prove that the converse is also true, provided that A is semisimple.

It is shown that if Ω is an infinite, metrizable, compact space such that some finite order iterated derived set of Ω is empty, then $\dg C(\om)=\db C(\om)=2$.

In this paper, a geometric interpretation is provided of a new rational Landen transformation. The convergence of its iterates is also established.

This paper provides a method of generating infinitely many integer solutions of the simultaneous equations $x_1^r+x_2^r+x_3 ^r=y_1^r+y_2^r+y_3^r$ where $r$ is equal to $2$, $3$ or $4$. Three numerical solutions of this system of equations have also been given in this paper.

In the paper, three lower bounds are given for the Morse index of a constant mean curvature torus in Euclidean $3$ -space, in terms of its spectral genus $g$ . The first two lower bounds grow linearly in $g$ and are stronger for smaller values of $g$ , while the third grows quadratically in $g$ but is weaker for smaller values of $g$ .

Let G be a residually-finite virtually soluble minimax group. We give upper and lower linear bounds for the degree of subgroup growth of G in terms of the Hirsch length of G.