The third homology of the special linear group of a field

We prove that for any infinite field homology stability for the third integral homology of the special linear groups $SL(n,F)$ begins at $n=3$. When $n=2$ the cokernel of the map from the third homology of $SL(2,F)$ to the third homology of $SL(3,F)$ is naturally isomorphic to the square of Milnor $K_3$. We discuss applications to the indecomposable $K_3$ of the field and to Milnor-Witt K-theory.Comment: PDFLatex, 21 page

Large gaps between consecutive prime numbers

Let $G(X)$ denote the size of the largest gap between consecutive primes below $X$. Answering a question of Erdos, we show that $$G(X) \geq f(X) \frac{\log X \log \log X \log \log \log \log X}{(\log \log \log X)^2},$$ where $f(X)$ is a function tending to infinity with $X$. Our proof combines existing arguments with a random construction covering a set of primes by arithmetic progressions. As such, we rely on recent work on the existence and distribution of long arithmetic progressions consisting entirely of primes.Comment: v2. very minor corrections. To appear in Ann. Mat

Learning, adaptive expectations, and technology shocks

This study explores the macroeconomic implications of adaptive expectations in a standard real business cycle model. When rational expectations are replaced by adaptive expectations, we show that the self-confirming equilibrium is the same as the steady-state rational expectations equilibrium for all admissible parameters but that dynamics around the steady state are substantially different between the two equilibria. The differences are driven mainly by the dampened wealth effect and the strengthened intertemporal substitution effect, not by the escapes emphasized by Williams (2003). As a result, adaptive expectations can be an important source of frictions that amplify and propagate technology shocks and seem promising for generating plausible labor market dynamics.Equilibrium (Economics)

Learning, adaptive expectations, and technology shocks

This study explores the macroeconomic implications of adaptive expectations in a standard real business cycle model. When rational expectations are replaced by adaptive expectations, we show that the self-confirming equilibrium is the same as the steady state rational expectations equilibrium for all admissible parameters, but that dynamics around the steady state are substantially different between the two equilibria. The differences are driven mainly by the dampened wealth effect and the strengthened intertemporal substitution effect, not by the escapes emphasized by Williams (2003). As a result, adaptive expectations can be an important source of frictions that amplify and propagate technology shocks and seem promising for generating plausible labor market dynamics.Macroeconomics

Long gaps in sieved sets

For each prime $p$, let $I_p \subset \mathbb{Z}/p\mathbb{Z}$ denote a collection of residue classes modulo $p$ such that the cardinalities $|I_p|$ are bounded and about $1$ on average. We show that for sufficiently large $x$, the sifted set $\{ n \in \mathbb{Z}: n \pmod{p} \not \in I_p \hbox{ for all }p \leq x\}$ contains gaps of size at least $x (\log x)^{\delta}$ where $\delta>0$ depends only on the density of primes for which $I_p\ne \emptyset$. This improves on the ``trivial'' bound of $\gg x$. As a consequence, for any non-constant polynomial $f:\mathbb{Z}\to \mathbb{Z}$ with positive leading coefficient, the set $\{ n \leq X: f(n) \hbox{ composite}\}$ contains an interval of consecutive integers of length $\ge (\log X) (\log\log X)^{\delta}$ for sufficiently large $X$, where $\delta>0$ depends only on the degree of $f$.Comment: Major revision. We replaced the PNT-type assumption with (a) a Mertens estimate; (b) that the density $\rho$ of nonempty $I_p$ exists. Our main theorem now gives an exponent which is a function of $\rho$, and is completely explicit. In particular, the exponent $e^{-1-4/\rho}$ is admissible. Various notational simplifications. Many remarks added to help the reade

Identifying (BN)2-pyrene as a new class of singlet fission chromophores: significance of azaborine substitution

Singlet fission converts one photoexcited singlet state to two triplet excited states and raises photoelectric conversion efficiency in photovoltaic devices. However, only a handful of chromophores have been known to undergo this process, which greatly limits the application of singlet fission in photovoltaics. We hereby identify a recently synthesized diazadiborine-pyrene ((BN)2-pyrene) as a singlet fission chromophore. Theoretical calculations indicate that it satisfies the thermodynamics criteria for singlet fission. More importantly, the calculations provide a physical chemistry insight into how the BN substitution makes this happen. Both calculation and transient absorption spectroscopy experiment indicate that the chromophore has a better absorption than pentacene. The convenient synthesis pathway of the (BN)2-pyrene suggests an in situ chromophore generation in photovoltaic devices. Two more (BN)2-pyrene isomers are proposed as singlet fission chromophores. This study sets a step forward in the cross-link of singlet fission and azaborine chemistry

The viscosity of silica fibres

The viscosity of an optical fibre over 1000 to 1150 {\deg}C is studied by inscribing an optical fibre Bragg grating that can withstand temperatures up to 1200 {\deg}C and monitoring fibre elongation under load through the Bragg wavelength shift. This optical interrogation offers high accuracy and reliability compared to direct measurements of elongation, particularly at lower temperatures, thus avoiding significant experimental error. An excellent Arrhenius fit is obtained from which an activation energy for viscous flow of Ea = 450 kJ/mol is extracted; addition of an additional temperature dependent pre-exponential does not change this value. This value is less than that idealised by some literature but consistent with other literature. The log plot of viscosity is overall found to be consistent with that reported in the literature for silica measurements on rod and beams, but substantially higher to past work reported for optical fibres. The discrepancy from an idealised activation energy Ea ~ 700 kJ/mol may be explained by noting the higher fictive temperature of the fibre. On the other hand, past optical fibre results obtained by beam bending with much lower values leave questions regarding the method of viscosity measurement and the time taken for structural equilibration. We note that because regenerated gratings already involve post-annealing to stabilise their operation at higher temperature, the structures are much more relaxed compared to normal fibres. This work highlights the need to stabilize components for operation in harsh environments before their application, despite some mechanical compromise. Given the increasing expectation of all-optical waveguide technologies operating above 1000 {\deg}C, the need to study the behaviour of glass over the long term brings added significance to the basic understanding of glass in this regime.Comment: Submitted to Acta Material