A remark on nonlocal algebras

The construction of examples of nonlocal uniform Banach algebras (similar to the well-known example given by E. Kallin) required always considerable efforts and it could create the impression that nonlocal algebras are infrequent. It turns out that one can construct such algebras by starting practically with any uniform algebras. One gives a simple and general method for constructing nonlocal algebras, based on the use of nonlocal point derivations. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 65, pp. 172–177, 1976.     

A remark on simplicity of vertex algebras and Lie conformal algebras

I give a short proof of the following algebraic statement: in a simple vertex algebra, the underlying Lie conformal algebra is either abelian, or it is an irreducible central extension of a simple Lie conformal algebra. This provides many examples of non-finite simple Lie conformal algebras, and should prove useful for classification purposes.     

A remark on tilting theory and DG algebras

Techniques of homological algebra over DG algebras [2, VI, 10] provide an alternative approach to ‘Morita theory for derived categories’ [5]. Supported by a grant of the Swiss National Foundation     

A remark on Burnside's theorem on matrix algebras

We present a simple proof of Burnside's theorem using techniques from elementary linear algebra.     

A remark on the C 2-cofiniteness condition on vertex algebras

We show that a finitely strongly generated, non-negatively graded vertex algebra is C ₂-cofinite if and only if it is lisse in the sense of Beilinson et al. (preprint). This shows that the C ₂-cofiniteness is indeed a natural finiteness condition.     

A remark on rank varieties for a class of local algebras

We answer a question raised in [9] by showing the equality of two different definitions of rank variety for finitely generated modules over truncated polynomial algebras. We do this by establishing an isomorphism of algebras used in the two definitions of rank variety. Received: 23 April 2007     

A remark on equational classes generated by very small free algebras

Archiv der Mathematik -     

A remark on the liftable derivation of moduli algebras of isolated hypersurface singularities

An example is given to show that not every derivation in the nilradical of the Lie algebra of derivations of moduli algebras can be liftable and the dimension of the nilradical of the Lie algebra of derivations of moduli algebras is not a topological invariant for an isolated hypersurface singularity.

     

A remark on Mackey-functors

In the following note we characterize the category of Mackey functors from a categoryC, satisfying a few assumptions, to a categoryD as the category of functors from Sp(C), the category of spans inC, toD which preserve finite products. This caracterization permits to apply all results on categories of functors preserving a given class of limits to the case of Mackey-functors.     

A remark on accessibility

This note obtains some characteristics of accessibility and Kato’s chaos. Applying these results, an accessible dynamical system whose product system is not accessible is constructed, giving a negative answer to a question in [Li R, Wang H, Zhao Y. Kato’s chaos in duopoly games. Chaos Solit Fract 2016;84:69–72]. Besides, it is proved that every transitive interval self-map is accessible.     

A remark on quasi-isometries

We show that if is an quasi-isometry, with , defined on the unit ball of , then there is an affine isometry with where is a universal constant. This result is sharp.

     

A remark on banks theorem

ABSTRACT

For discrete autonomous dynamical systems, it was found that in the three conditions defining Devaney chaos, topological transitivity and dense periodic points together imply sensitive dependence on initial condition (Banks, Brooks, Cairns, Davis and Stacey, On Devaney's definition of chaos, The Amer. Math. Monthly 99 (1992), pp. 332–334). In this paper, we give the definition of finitely generated non-autonomous dynamical systems (NADS) and generalize the Banks et al. theorem to the finitely generated NADS.     

A remark on symplectic spreads

We first note that each element of a symplectic spread of PG(2n − 1, 2 ^r ) either intersects a suitable nonsingular quadric in a subspace of dimension n − 2 or is contained in it, then we prove that this property characterises symplectic spreads of PG(2n − 1, 2 ^r ). As an application, we show that a translation plane of order q ⁿ , q even, with kernel containing GF(q), is defined by a symplectic spread if and only if it contains a maximal arc of the type constructed by Thas (Europ J Combin 1:189–192, 1980).