The exponent three class group problem for some real cyclic cubic number fields

We determine all the simplest cubic fields whose ideal class groups have exponent dividing , thus generalizing the determination by G. Lettl of all the simplest cubic fields with class number and the determination by D. Byeon of all all the simplest cubic fields with class number . We prove that there are simplest cubic fields with ideal class groups of exponent (and simplest cubic fields with ideal class groups of exponent , i.e. with class number one).

     

LCM-splitting sets in some ring extensions

Let be a saturated multiplicative set of an integral domain . Call an lcm splitting set if and are principal ideals for every and . We show that if is an -stable overring of (that is, if whenever and is principal, it follows that and if is an lcm splitting set of , then the saturation of in is an lcm splitting set in . Consequently, if is Noetherian and is a (nonzero) prime element, then is also a prime element of the integral closure of . Also, if is Noetherian, is generated by prime elements of and if the integral closure of is a UFD, then so is the integral closure of .

     

An automatic adjoint theorem and its applications

In this paper, we prove the following automatic adjoint theorem: For any sequence spaces and , if has the signed-weak gliding hump property and is an infinite matrix which transforms into , then the transpose matrix of transforms into , and for any and , . That is, the adjoint operator of automatically exists and is just the transpose matrix of . From the theorem we obtain a class of infinite matrix topological algebras , and prove also a -multiplier convergence theorem of Orlicz-Pettis type. The theorem improves substantially the famous Stiles' Orlicz-Pettis theorem.

     

Relative Brauer groups and -torsion

Let be a field and its Brauer group. If is a field extension, then the relative Brauer group is the kernel of the restriction map . A subgroup of is called an algebraic relative Brauer group if it is of the form for some algebraic extension . In this paper, we consider the -torsion subgroup consisting of the elements of killed by , where is a positive integer, and ask whether it is an algebraic relative Brauer group. The case is already interesting: the answer is yes for squarefree, and we do not know the answer for arbitrary. A counterexample is given with a two-dimensional local field and .

     

Parallel tangent hyperplanes

Let be a smooth strictly convex closed hypersurface in and let be any oriented smooth connected manifold immersed in Suppose that is a continuous function from to Then there is at least one point such that the hyperplane tangent to at is parallel to the hyperplane tangent to the immersed manifold at the point corresponding to If there did not exist at least two such points, would have to be compact and the Hurewicz homomorphism of into would have to be surjective. If in addition our immersion was an embedding, the Euler characteristic of would have to be equal to For any and any immersed we could always get maps for which the number of points satisfying the conditions of our theorem exactly equaled two. An example can be given in which both and are the unit sphere about the origin in and there is only one such point .

     

Metacompact subspaces of products of ordinals

Let be a subspace of the product of finitely many ordinals. is countably metacompact, and is metacompact iff has no closed subset homeomorphic to a stationary subset of a regular uncountable cardinal. A theorem generalizing these two results is: is -metacompact iff has no closed subset homeomorphic to a -stationary set where .

     

The extension of positive definite operator-valued functions defined on a symmetric interval of an ordered group

Let be an ordered abelian group and . Let be an abelian group and an operator-valued positive definite function on . We prove that admits a positive definite extension to , generalizing in this way existing results for the case when and is continuous.

     

A uniqueness result for harmonic functions

Let , , and suppose is harmonic in and on the closure of . If the gradient of vanishes continuously on a subset of of positive -dimensional Lebesgue measure and satisfies certain regularity conditions, then must be identically constant.

     

Proof of the prime power conjecture for projective planes of order

Let be an abelian collineation group of order of a projective plane of order . We show that must be a prime power, and that the -rank of is at least if for an odd prime .

     

A global pinching theorem for surfaces with constant mean curvature in

Let be a compact immersed surface in the unit sphere with constant mean curvature . Denote by the linear map from into , , where is the linear map associated to the second fundamental form and is the identity map. Let denote the square of the length of . We prove that if , then is either totally umbilical or an -torus, where is a constant depending only on the mean curvature .