Fractional glassy relaxation and convolution modules of distributions

Solving fractional relaxation equations requires precisely characterized domains of definition for applications of fractional differential and integral operators. Determining these domains has been a longstanding problem. Applications in physics and engineering typically require extension from domains of functions to domains of distributions. In this work convolution modules are constructed for given sets of distributions that generate distributional convolution algebras. Convolutional inversion of fractional equations leads to a broad class of multinomial Mittag-Leffler type distributions. A comprehensive asymptotic analysis of these is carried out. Combined with the module construction the asymptotic analysis yields domains of distributions, that guarantee existence and uniqueness of solutions to fractional differential equations. The mathematical results are applied to anomalous dielectric relaxation in glasses. An analytic expression for the frequency dependent dielectric susceptibility is applied to broadband spectra of glycerol. This application reveals a temperature independent and universal dynamical scaling exponent.

Spectral Invariance of $$*$$-Representations of Twisted Convolution Algebras with Applications in Gabor Analysis

We show spectral invariance for faithful $$*$$ ∗ -representations for a class of twisted convolution algebras. More precisely, if G is a locally compact group with a continuous 2-cocycle c for which the corresponding Mackey group $$G_c$$ G c is $$C^*$$ C ∗ -unique and symmetric, then the twisted convolution algebra $$L^1 (G,c)$$ L 1 ( G , c ) is spectrally invariant in $${\mathbb {B}}({\mathcal {H}})$$ B ( H ) for any faithful $$*$$ ∗ -representation of $$L^1 (G,c)$$ L 1 ( G , c ) as bounded operators on a Hilbert space $${\mathcal {H}}$$ H . As an application of this result we give a proof of the statement that if $$\Delta $$ Δ is a closed cocompact subgroup of the phase space of a locally compact abelian group $$G'$$ G ′ , and if g is some function in the Feichtinger algebra $$S_0 (G')$$ S 0 ( G ′ ) that generates a Gabor frame for $$L^2 (G')$$ L 2 ( G ′ ) over $$\Delta $$ Δ , then both the canonical dual atom and the canonical tight atom associated to g are also in $$S_0 (G')$$ S 0 ( G ′ ) . We do this without the use of periodization techniques from Gabor analysis.

Topological tensor product of bimodules, complete Hopf algebroids and convolution algebras

Given a finitely generated and projective Lie–Rinehart algebra, we show that there is a continuous homomorphism of complete commutative Hopf algebroids between the completion of the finite dual of its universal enveloping Hopf algebroid and the associated convolution algebra. The topological Hopf algebroid structure of this convolution algebra is here clarified, by providing an explicit description of its topological antipode as well as of its other structure maps. Conditions under which that homomorphism becomes an homeomorphism are also discussed. These results, in particular, apply to the smooth global sections of any Lie algebroid over a smooth (connected) manifold and they lead a new formal groupoid scheme to enter into the picture. In the appendices we develop the necessary machinery behind complete Hopf algebroid constructions, which involves also the topological tensor product of filtered bimodules over filtered rings.