On twisted Fourier analysis and convergence of Fourier series on discrete groups

We study norm convergence and summability of Fourier series in the setting of reduced twisted group $C^*$-algebras of discrete groups. For amenable groups, F{\o}lner nets give the key to Fej\'er summation. We show that Abel-Poisson summation holds for a large class of groups, including e.g. all Coxeter groups and all Gromov hyperbolic groups. As a tool in our presentation, we introduce notions of polynomial and subexponential H-growth for countable groups w.r.t. proper scale functions, usually chosen as length functions. These coincide with the classical notions of growth in the case of amenable groups.Comment: 35 pages; abridged, revised and update

Evolutionary tradeoffs in cellular composition across diverse bacteria

One of the most important classic and contemporary interests in biology is the connection between cellular composition and physiological function. Decades of research have allowed us to understand the detailed relationship between various cellular components and processes for individual species, and have uncovered common functionality across diverse species. However, there still remains the need for frameworks that can mechanistically predict the tradeoffs between cellular functions and elucidate and interpret average trends across species. Here we provide a comprehensive analysis of how cellular composition changes across the diversity of bacteria as connected with physiological function and metabolism, spanning five orders of magnitude in body size. We present an analysis of the trends with cell volume that covers shifts in genomic, protein, cellular envelope, RNA and ribosomal content. We show that trends in protein content are more complex than a simple proportionality with the overall genome size, and that the number of ribosomes is simply explained by cross-species shifts in biosynthesis requirements. Furthermore, we show that the largest and smallest bacteria are limited by physical space requirements. At the lower end of size, cell volume is dominated by DNA and protein content—the requirement for which predicts a lower limit on cell size that is in good agreement with the smallest observed bacteria. At the upper end of bacterial size, we have identified a point at which the number of ribosomes required for biosynthesis exceeds available cell volume. Between these limits we are able to discuss systematic and dramatic shifts in cellular composition. Much of our analysis is connected with the basic energetics of cells where we show that the scaling of metabolic rate is surprisingly superlinear with all cellular components

Retroelement-guided protein diversification abounds in vast lineages of Bacteria and Archaea

Major radiations of enigmatic Bacteria and Archaea with large inventories of uncharacterized proteins are a striking feature of the Tree of Life1-5. The processes that led to functional diversity in these lineages, which may contribute to a host-dependent lifestyle, are poorly understood. Here, we show that diversity-generating retroelements (DGRs), which guide site-specific protein hypervariability6-8, are prominent features of genomically reduced organisms from the bacterial candidate phyla radiation (CPR) and as yet uncultivated phyla belonging to the DPANN (Diapherotrites, Parvarchaeota, Aenigmarchaeota, Nanoarchaeota and Nanohaloarchaea) archaeal superphylum. From reconstructed genomes we have defined monophyletic bacterial and archaeal DGR lineages that expand the known DGR range by 120% and reveal a history of horizontal retroelement transfer. Retroelement-guided diversification is further shown to be active in current CPR and DPANN populations, with an assortment of protein targets potentially involved in attachment, defence and regulation. Based on observations of DGR abundance, function and evolutionary history, we find that targeted protein diversification is a pronounced trait of CPR and DPANN phyla compared to other bacterial and archaeal phyla. This diversification mechanism may provide CPR and DPANN organisms with a versatile tool that could be used for adaptation to a dynamic, host-dependent existence

Cu-Ni-Sn: A key system for lead-free soldering

Being the most complex constituent of the quaternary system Ag-Cu-Ni-Sn, the ternary system Cu-Ni-Sn is the key system for the investigation of the interactions of Ag-Cu-Sn solder alloys with Ni as a contact material. Although this system has been thoroughly studied in the literature, there are still many uncertainties left. In the present work, a study of the phase equilibria in four isothermal sections at 220, 400, 500, and 700C of the Cu-Ni-Sn system was carried out following a comprehensive literature study. The methods employed were x-ray diffraction (XRD), metallography, and scanning electron microscopy including electron probe microanalysis. The ternary solubilities of the Ni3Sn2-Cu6Sn5 and Ni3Sn-Cu3Sn fields were characterized in detail. So far no continuous solubility between the respective phases has been found. At 25 at.% Sn the existence of two ternary compounds formed from the BiF3-type (Cu,Ni)3Sn phase and reported in literature could be confirmed. On the other hand, our results differ significantly from the very recent literature related to lead-free soldering

Cu-Ni-Sn: A key system for lead-free soldering

Being the most complex constituent of the quaternary system Ag-Cu-Ni-Sn, the ternary system Cu-Ni-Sn is the key system for the investigation of the interactions of Ag-Cu-Sn solder alloys with Ni as a contact material. Although this system has been thoroughly studied in the literature, there are still many uncertainties left. In the present work, a study of the phase equilibria in four isothermal sections at 220, 400, 500, and 700C of the Cu-Ni-Sn system was carried out following a comprehensive literature study. The methods employed were x-ray diffraction (XRD), metallography, and scanning electron microscopy including electron probe microanalysis. The ternary solubilities of the Ni3Sn2-Cu6Sn5 and Ni3Sn-Cu3Sn fields were characterized in detail. So far no continuous solubility between the respective phases has been found. At 25 at.% Sn the existence of two ternary compounds formed from the BiF3-type (Cu,Ni)3Sn phase and reported in literature could be confirmed. On the other hand, our results differ significantly from the very recent literature related to lead-free soldering

Enthalpies of mixing of metallic systems relevant for lead-free soldering : Ag-Pd and Ag-Pd-Sn

The partial and integral enthalpies of mixing of binary liquid Ag–Pd alloys (with compositions up to about 55 at.% Pd) were determined at 1400 °C. A Calvet type microcalorimeter was used for the measurements employing a drop calorimetric technique. Additionally, eight sections in the Sn-rich corner of the ternary Ag–Pd–Sn system, which is relevant for lead-free soldering, were investigated at 900 °C. The ternary interaction parameters were fitted using the Redlich–Kister–Muggiano model for substitutional solutions. The isoenthalpy curves for Ag–Pd–Sn at 900 °C were constructed for the integral molar enthalpy of mixing

Th In-Pd-Sn phase diagram (xPd

The ternary In¿Pd¿Sn phase diagram was investigated in the composition range up to approximately 60 at.% Pd which includes the phase relations most relevant for lead-free soldering. The investigation included X-ray powder diffraction (XRD) and scanning electron microscopy (SEM). The ternary equilibrium phase relations were studied for three isothermal sections at 200, 500, and 700 ¿C, respectively, and the ternary solubilities of several binary compounds were determined for 500 and 700 ¿C using quantitative SEM. No ternary compound was found to exist in the investigated part of the system, but some of the binary phases show a considerable solid solubility in the ternary system, including most probably a continuous solid solution between ¿Pd2 and Pd2Sn