Blow-Ups of Three-Dimensional Terminal Singularities: The cA Case

Divisors with minimal discrepancy over points are classified.     

The Maximal and Minimal Ranks of Some Expressions of Generalized Inverses of Matrices

In this paper, we first determine the maximal and minimal ranks of A — BXC with respect to X. Using those results, we then find the maximal and minimal ranks of the expressions AA^– — A^– A — BB^– A — AC^– C and B^– BA — ACC^– with respect to the choice of generalized inverses A^–, B^– and C^–. In particular, we consider the commutativity of A and A^–, A^k and A^–.The research of the author was supported in part by the Natural Sciences and Engineering Research Council of Canada.     

Algebras in the Positive Cone of po-Groups

We systemize a number of algebras that are especially known in the field of quantum structures and that in particular arise from the positive cones of partially ordered groups. Generalized effect algebras, generalized difference posets, cone algebras, commutative BCK-algebras with the relative cancellation property, and positive minimal clans are included in the text.All these structures are conveniently characterizable as special cases of generalized pseudoeffect algebras, which we introduced in a previous paper. We establish the exact relations between all mentioned structures, thereby adding new structures whenever necessary to make the scheme of order complete.Generalized pseudoeffect algebras were under certain conditions proved to be representable by means of a po-group. From this fact, we will easily establish representation theorems for all of the structures included in discussion.     

Willmore Surfaces in S n

A surface x> : M S ⁿ is called a Willmore surface if it is a critical surface of the Willmore functional. It is well known that any minimal surface is a Willmore surface and that many nonminimal Willmore surfaces exists. In this paper, we establish an integral inequality for compact Willmore surfaces in S ⁿ and obtain a new characterization of the Veronese surface in S ⁴ as a Willmore surface. Our result reduces to a well-known result in the case of minimal surfaces.     

Isomorphism of commutative group algebras of closed -local algebraically compact groups

Let be an abelian group and let be a field of 0$">. It is shown via a universal algorithm that if the modified Direct-Factor Problem holds, then the -isomorphism for some group yields provided is a closed -group or a -local algebraically compact group. In particular, this is the case when is closed -primary of arbitrary power, or is -local algebraically compact with cardinality at most and is in cardinality not exceeding . The last claim completely settles a question raised by W. May in Proc. Amer. Math. Soc. (1979) and partially extends our results published in Rend. Sem. Mat. Univ. Padova (1999) and Southeast Asian Bull. Math. (2001).

     

On locally finite -groups satisfying an Engel condition

For a given positive integer and a given prime number , let be the integer satisfying . We show that every locally finite -group, satisfying the -Engel identity, is (nilpotent of -bounded class)-by-(finite exponent) where the best upper bound for the exponent is either or if is odd. When the best upper bound is or . In the second part of the paper we focus our attention on -Engel groups. With the aid of the results of the first part we show that every -Engel -group is soluble and the derived length is bounded by some constant.

     

“Minimal Surfaces in ℍ<Superscript>2</Superscript> × ℝ”

In ² × ℝ one has catenoids, helicoids and Scherk-type surfaces. A Jenkins-Serrin type theorem holds here. Moreover there exist complete minimal graphs in ² with arbitrary continuous asymptotic values. Finally, a graph on a domain of ² cannot have an isolated singularity. Received: 20 June 2002     

Semilinear Perturbations of Harmonic Spaces,Liouville Property and a Boundary Value Problem

Let (X,) be a P-harmonic Bauer space and let be a Borel measurable function on X×R satisfying conditions (A) through (D) of Section 2 (e.g., (x,t)=t|t|^–1 where >1). For every Kato family M of potential kernels on X let ^M U(X) denote the set of all real continuous functions on X such that u+K ^M _D (,u)(D) for every open relatively compact subset D of X. We study the existence of a non-trivial function in ^M U(X) which is dominated by a given positive harmonic function on X. If X is a domain of R ^d , is a positive Kato measure on X and L is a second-order differential operator in R ^d , we apply our study to derive a characterization of finite positive measures on the minimal Martin boundary ^M ₁ X for which the boundary value problem Lu=(,u) in X and u= on ^M ₁ X is solvable.     

On Non-Congruence Subgroups of the Analogue of the Modular Group in Characteristic p

Let k[t] be the polynomial ring over a finite field k. The group SL ₂(k[t]) is often referred to as the analogue, in characteristic p, of the classical modular group SL ₂( ), where is the ring of rational integers. It is well-known that the smallest index of a non-congruence subgroup of SL ₂( ) is 7. Here we compute this index for SL ₂(k[t]). (In all but 6 cases it turns out to be 1 + q, where q is the order of k.)     

An n-Dimensional Hahn-Banach Extension Theorem and Minimal Projections

Let T~=_i=1 ⁿ _irv_i:V V=[v₁,. . . .,v_n] X, where _i V* and X is a Banach space. Let T= _i=1 ⁿu_iv_i: X V be an extension of T~ to all of X (i.e., u_i X*) such that T has minimal (operator) norm. (E.g., if T~=I, T is a minimal projection from X onto V.) Then it is necessary and sufficient that u:=u_1,. . . ,u_n is given by (v:=v₁,. . . ,v_n)ext_v(u) Vⁿ,where the notion of a v-extremal (ext_v) of u is properly defined.The condition above leads in many important cases to a simple geometric interpretation of minimal projections. Furthermore, by applying this formula to the case X=L^p, we obtain a linear n-dimensional analog of the Hölder equality condition (M is given by ext_v(u)=Mv)1/p u · Mv = 1/q u · Mv,wherever v is differentiable.We point out several applications, including the determination of the absolute projection constant of _n ^p