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It is well known that for a p -group, the invariant field is purely transcendental (T. Miyata, Invariants of certain groups I, Nagoya Math. J. 41 (1971), 69–73). In this note, we show that a minimal generating set of this field can be chosen as homogeneous invariants from the invariant ring. As...
We estimate the density of integers which have more than one divisor in an interval ( y , z ) with z ≈ y + y /(log y ) log 4 − 1 . As a consequence, we determine the precise range of z such that most integers which have at least one divisor in ( y , z ) have exactly one such divisor.
Two results are proved about the Banach space X = 𝒞( K ), where K is compact and Hausdorff. The first concerns smooth approximation: let m be a positive integer or ∞; we show that if there exists on X a non-zero function of class 𝒞 m with bounded support, then all continuous real-valued...
We show that for a finite reductive group tensoring with a quotient of the complex inducing the Alvis–Curtis character duality induces a self-derived equivalence on a quotient of the group algebra. In the case of general linear groups, this equivalence realizes an equivalence between derived...
It is proved that if a Lie algebra L has a nilpotent ideal of nilpotency class c and of finite codimension r , then L has also a nilpotent ideal of class ≤ c and of finite codimension bounded in terms of r and c that is invariant under all automorphisms of L . In a similar result for groups,...
We prove that the weighted differences of ergodic averages, induced by a Cesàro bounded, strongly continuous, one-parameter group of positive, invertible, linear operators on L p , 1 < p < ∞, converge almost every where and in the L p -norm. We obtain first the boundedness of the ergodic...
For a finite abelian group G , let s ( G ) denote the smallest integer l such that every sequence S over G of length | S | ≥ l has a zero-sum subsequence of length exp ( G ). We derive new upper and lower bounds for s ( G ), and all our bounds are sharp for special types of groups. The results...
We prove a result about the possible dimensions of modules over the completed 𝔽 group algebra of a Heisenberg pro- p group that are not torsion qua modules over the centre. We explain why this result is analogous to a result of Bernstein for modules over Weyl algebras in characteristic 0.
We study unitary Banach algebras, as defined by M. L. Hansen and R. V. Kadison in 1996, as well as some related concepts like maximal or uniquely maximal Banach algebras. We show that a norm-unital Banach algebra is uniquely maximal if and only if it is unitary and has minimality of the...
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