The inner kernel theorem for a certain Segal algebra

The Segal algebra ${\textbf{S}}_{0}(G)$ is well defined for arbitrary locally compact Abelian Hausdorff (LCA) groups $G$. Despite the fact that it is a Banach space it is possible to derive a kernel theorem similar to the Schwartz kernel theorem, of course without making use of the Schwartz kernel theorem. First we characterize the bounded linear operators from ${\textbf{S}}_{0}(G_1)$ to ${\textbf{S}}_{0}'(G_2)$ by distributions in ${\textbf{S}}_{0}'(G_1 \times G_2)$. We call this the "outer kernel theorem". The "inner kernel theorem" is concerned with the characterization of those linear operators which have kernels in the subspace ${\textbf{S}}_{0}(G_1 \times G_2)$, the main subject of this manuscript. We provide a description of such operators as regularizing operators in our context, mapping ${\textbf{S}}_{0}'(G_1)$ into test functions in ${\textbf{S}}_{0}(G_2)$, in a $w^{*}$-to norm continuous manner. The presentation provides a detailed functional analytic treatment of the situation and applies to the case of general LCA groups, without recurrence to the use of so-called Wilson bases, which have been used for the case of elementary LCA groups. The approach is then used in order to describe natural laws of composition which imitate the composition of linear mappings via matrix multiplications, now in a continuous setting. We use here that in a suitable (weak) form these operators approximate general operators. We also provide an explanation and mathematical justification used by engineers explaining in which sense pure frequencies "integrate" to a Dirac delta distribution

Gabor Duality Theory for Morita Equivalent C∗C^*-algebras

The duality principle for Gabor frames is one of the pillars of Gabor analysis. We establish a far-reaching generalization to Morita equivalent $C^*$-algebras where the equivalence bimodule is a finitely generated projective Hilbert $C^*$-module. These Hilbert $C^*$-modules are equipped with some extra structure and are called Gabor bimodules. We formulate a duality principle for standard module frames for Gabor bimodules which reduces to the well-known Gabor duality principle for twisted group $C^*$-algebras of a lattice in phase space. We lift all these results to the matrix algebra level and in the description of the module frames associated to a matrix Gabor bimodule we introduce $(n,d)$-matrix frames, which generalize superframes and multi-window frames. Density theorems for $(n,d)$-matrix frames are established, which extend the ones for multi-window and super Gabor frames. Our approach is based on the localization of a Hilbert $C^*$-module with respect to a trace.Comment: 36 page

Time-frequency analysis on the adeles over the rationals

We show that the construction of Gabor frames in $L^{2}(\mathbb{R})$ with generators in $\mathbf{S}_{0}(\mathbb{R})$ and with respect to time-frequency shifts from a rectangular lattice $\alpha\mathbb{Z}\times\beta\mathbb{Z}$ is equivalent to the construction of certain Gabor frames for $L^{2}$ over the adeles over the rationals and the group $\mathbb{R}\times\mathbb{Q}_{p}$. Furthermore, we detail the connection between the construction of Gabor frames on the adeles and on $\mathbb{R}\times\mathbb{Q}_{p}$ with the construction of certain Heisenberg modules.Comment: minor revisions, added more references, added a Balian-Low type result in the form of Proposition 4.

The endoribonucleolytic N-terminal half of Escherichia coli RNase E is evolutionarily conserved in Synechocystis sp. and other bacteria but not the C-terminal half, which is sufficient for degradosome assembly

Escherichia coli RNase E, an essential single-stranded specific endoribonuclease, is required for both ribosomal RNA processing and the rapid degradation of mRNA. The availability of the complete sequences of a number of bacterial genomes prompted us to assess the evolutionarily conservation of bacterial RNase E. We show here that the sequence of the N-terminal endoribonucleolytic domain of RNase E is evolutionarily conserved in Synechocystis sp. and other bacteria. Furthermore, we demonstrate that the Synechocystis sp. homologue binds RNase E substrates and cleaves them at the same position as the E. coli enzyme. Taken together these results suggest that RNase E-mediated mechanisms of RNA decay are not confined to E. coli and its close relatives. We also show that the C-terminal half of E. coli RNase E is both sufficient and necessary for its physical interaction with the 3'-5' exoribonuclease polynucleotide phosphorylase, the RhlB helicase, and the glycolytic enzyme enolase, which are components of a "degradosome" complex. Interestingly, however, the sequence of the C-terminal half of E. coli RNase E is not highly conserved evolutionarily, suggesting diversity of RNase E interactions with other RNA decay components in different organisms. This notion is supported by our finding that the Synechocystis sp. RNase E homologue does not function as a platform for assembly of E. coli degradosome components

Investigations of lubricant rheology as applied to elastohydrodynamic lubrication

The pressure viscometer was modified to permit the measurement of viscosity at elevated pressures and shear stresses up to 5 x 10 to the 6th power N/sq m (720 psi). This shear stress is within a factor of three of the shear stress occurring in a sliding ehd point contact such as occurs in the ehd simulator. Viscosity data were taken on five lubricant samples, and it was found that viscous heating effects on the viscosity were predominant and not non-Newtonian behavior at the high shear stresses. The development of the infrared temperature measuring technique for the ehd simulator was completed, and temperature data for a set of operating conditions and one lubricant are reported. The numerical analysis of the behavior of nonlinear lubricants in the lubrication of rollers is reported

Frequency Dependent Specific Heat from Thermal Effusion in Spherical Geometry

We present a novel method of measuring the frequency dependent specific heat at the glass transition applied to 5-polyphenyl-4-ether. The method employs thermal waves effusing radially out from the surface of a spherical thermistor that acts as both a heat generator and thermometer. It is a merit of the method compared to planar effusion methods that the influence of the mechanical boundary conditions are analytically known. This implies that it is the longitudinal rather than the isobaric specific heat that is measured. As another merit the thermal conductivity and specific heat can be found independently. The method has highest sensitivity at a frequency where the thermal diffusion length is comparable to the radius of the heat generator. This limits in practise the frequency range to 2-3 decades. An account of the 3omega-technique used including higher order terms in the temperature dependency of the thermistor and in the power generated is furthermore given.Comment: 17 pages, 15 figures, Substantially revised versio