25,149 research outputs found
Convolution, Separation and Concurrency
A notion of convolution is presented in the context of formal power series
together with lifting constructions characterising algebras of such series,
which usually are quantales. A number of examples underpin the universality of
these constructions, the most prominent ones being separation logics, where
convolution is separating conjunction in an assertion quantale; interval
logics, where convolution is the chop operation; and stream interval functions,
where convolution is used for analysing the trajectories of dynamical or
real-time systems. A Hoare logic is constructed in a generic fashion on the
power series quantale, which applies to each of these examples. In many cases,
commutative notions of convolution have natural interpretations as concurrency
operations.Comment: 39 page
Niceness theorems
Many things in mathematics seem lamost unreasonably nice. This includes
objects, counterexamples, proofs. In this preprint I discuss many examples of
this phenomenon with emphasis on the ring of polynomials in a countably
infinite number of variables in its many incarnations such as the representing
object of the Witt vectors, the direct sum of the rings of representations of
the symmetric groups, the free lambda ring on one generator, the homology and
cohomology of the classifying space BU, ... . In addition attention is paid to
the phenomenon that solutions to universal problems (adjoint functors) tend to
pick up extra structure.Comment: 52 page
Rationality of Hilbert series in noncommutative invariant theory
It is a fundamental result in commutative algebra and invariant theory that a
finitely generated graded module over a commutative finitely generated graded
algebra has rational Hilbert series, and consequently the Hilbert series of the
algebra of polynomial invariants of a group of linear transformations is
rational, whenever this algebra is finitely generated. This basic principle is
applied here to prove rationality of Hilbert series of algebras of invariants
that are neither commutative nor finitely generated. Our main focus is on
linear groups acting on certain factor algebras of the tensor algebra that
arise naturally in the theory of polynomial identities.Comment: Examples both from commutative and noncommutative invariant theory
are included, a problem is formulated and references are added. Comments for
v3: references added, minor revisio
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