Minimal transformations with no common factor need not be disjoint

A countable family of minimal transformations (X, Z) is described for which no pair have a non-trivial common factor, and so that no pair is disjoint. This answers in the negative a question of H. Furstenberg.     

Finely Harmonic Functions need not be Quasi-Analytic

If a function f is harmonic on a connected open set R^d andis constant in a neighbourhood of one point in then it is identicallyconstant. We give an example of a non-constant finely harmonicfunction defined on E={zC||z|<2}\p which is identically zeroon |z|1. The exceptional set P is of capacity zero so E is finelyopen and finely connected. This example therefore shows thatfinely harmonic functions are not globally determined by theirlocal behaviour.     

Subpullback flat S-posets need not be subequalizer flat

If S is a monoid, a right S-act A _S is a set A, equipped with a “right S-action” A×S→A sending the pair (a,s)∈ A×S to as, that satisfies the conditions (i) a(st)=(as)t and (ii) a1=a for all a∈A and s,t∈S. If, in addition, S is equipped with a compatible partial order and A is a poset, such that the action is monotone (when A×S is equipped with the product order), then A _S is called a right S-poset. Left S-acts and S-posets are defined analogously. For a given S-act (resp. S-poset) a tensor product functor A _S ?? from left S-acts to sets (resp. left S-posets to posets) exists, and A _S is called pullback flat or equalizer flat (resp. subpullback flat or subequalizer flat) if this functor preserves pullbacks or equalizers (resp. subpullbacks or subequalizers). By analogy with the Lazard-Govorov Theorem for R-modules, B. Stenström proved in 1971 that an S-act is isomorphic to a directed colimit of finitely generated free S -acts if and only if it is both pullback flat and equalizer flat. Some 20 years later, the present author showed that, in fact, pullback flatness by itself is sufficient. (A new, more direct proof of that result is contained in the present article.) In 2005, Valdis Laan and the present author obtained a version of the Lazard-Govorov Theorem for S-posets, in which subpullbacks and subequalizers now assume the role previously played by pullbacks and equalizers. The question of whether subpullback flatness implies subequalizer flatness remained unsolved. The present paper provides a negative answer to this question.     

Universal deformation rings need not be complete intersections

We answer a question of M. Flach by showing that there is a linear representation of a profinite group whose (unrestricted) universal deformation ring is not a complete intersection. We show that such examples arise in arithmetic in the following way. There are infinitely many real quadratic fields F for which there is a mod 2 representation of the Galois group of the maximal unramified extension of F whose universal deformation ring is not a complete intersection. Finally, we discuss bounds on the singularities of universal deformation rings of representations of finite groups in terms of the nilpotency of the associated defect groups. The first author was supported in part by NSF Grant DMS01-39737 and NSA Grant H98230-06-1-0021. The second author was supported in part by NSF Grants DMS00-70433 and DMS05-00106.     

Universal deformation rings need not be complete intersections

We answer a question of M. Flach by showing that there is a linear representation of a profinite group whose universal deformation ring is not a complete intersection. We show that such examples arise in arithmetic in the following way. There are infinitely many real quadratic fields F for which there is a mod 2 representation of the Galois group of the maximal unramified extension of F whose universal deformation ring is not a complete intersection. To cite this article: F.M. Bleher, T. Chinburg, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Multiplicity-free Hamiltonian actions need not be Kähler

Multiplicity-free actions are symplectic manifolds with a very high degree of symmetry. Delzant [2] showed that all compact multiplicity-free torus actions admit compatible K?hler structures, and are therefore toric varieties. In this note we show that Delzant's result does not generalize to the non-abelian case. Our examples are constructed by applying U(2)-equivariant symplectic surgery to the flag variety U(3)/T ³. We then show that these actions fail a criterion which Tolman [9] shows is necessary for the existence of a compatible K?hler structure. Oblatum IX-1995 & 21-IV-1997     

Polynomial extensions of SCN rings need not be SCN

In this article, we show that there exists an SCN ring R such that the polynomial ring R[x] is not SCN. This answers a question posed by T. K. Kwak et al. in [2 Kwak, T. K., Lee, M. J., Lee, Y. (2014). On sums of coe?cients of products of polynomials. Comm. Algebra 42(9):4033–4046.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]].     

Simultaneous Lagrange interpolating approximation need not always be convergent

Recently, several publications have been devoted to investigation of simultaneous Lagrange interpolating approximation. In this paper we carefully construct a counterexample with a system of nodes showing that the simultaneous Lagrange interpolating approximation need not always be convergent. It is especially interesting to note that the system of nodes behaving “badly” in this case is exactly the “near optimal choice” in the ordinary Lagrange interpolating case, the zeros of the Chebyshev polynomials.     

2-affine complete algebras need not be affine complete

For each we exhibit a finite algebra R _k such that R _k is k-affine complete, but not (k+1)-affine complete; this means that every k-ary congruence preserving function on R _k lies in , but there is a (k +1)-ary congruence preserving function of R _k that does not lie in . Received September 27, 2001; accepted in final form February 9, 2002.     

Ideal determined varieties need not be congruence 3-permutable

Presented by W. Taylor.     

Uncountable products of determined groups need not be determined

If H is a dense subgroup of G, we say that H determines G if their groups of characters are topologically isomorphic when equipped with the compact open topology. If every dense subgroup of G determines G, then we say that G is determined. The importance of this property is justified by the recent generalizations of Pontryagin-van Kampen duality to wider classes of topological Abelian groups. Among other results, we show (a) _⊕i∈IR determines the product _∏i∈IR if and only if I is countable, (b) a compact group is determined if and only if its weight is countable. These answer questions of Comfort, Raczkowski and the third listed author. Generalizations of the above results are also given.     

Quartically hyponormal weighted shifts need not be 3-hyponormal

We give the first example of a quartically hyponormal unilateral weighted shift which is not 3-hyponormal.     

Subgroups of free idempotent generated semigroups need not be free

We study the maximal subgroups of free idempotent generated semigroups on a biordered set by topological methods. These subgroups are realized as the fundamental groups of a number of 2-complexes naturally associated to the biorder structure of the set of idempotents. We use this to construct the first example of a free idempotent generated semigroup containing a non-free subgroup.