The number of independent sets in a graph with small maximum degree

Let ${\rm ind}(G)$ be the number of independent sets in a graph $G$. We show that if $G$ has maximum degree at most $5$ then $${\rm ind}(G) \leq 2^{{\rm iso}(G)} \prod_{uv \in E(G)} {\rm ind}(K_{d(u),d(v)})^{\frac{1}{d(u)d(v)}}$$ (where $d(\cdot)$ is vertex degree, ${\rm iso}(G)$ is the number of isolated vertices in $G$ and $K_{a,b}$ is the complete bipartite graph with $a$ vertices in one partition class and $b$ in the other), with equality if and only if each connected component of $G$ is either a complete bipartite graph or a single vertex. This bound (for all $G$) was conjectured by Kahn. A corollary of our result is that if $G$ is $d$-regular with $1 \leq d \leq 5$ then $${\rm ind}(G) \leq \left(2^{d+1}-1\right)^\frac{|V(G)|}{2d},$$ with equality if and only if $G$ is a disjoint union of $V(G)/2d$ copies of $K_{d,d}$. This bound (for all $d$) was conjectured by Alon and Kahn and recently proved for all $d$ by the second author, without the characterization of the extreme cases. Our proof involves a reduction to a finite search. For graphs with maximum degree at most $3$ the search could be done by hand, but for the case of maximum degree $4$ or $5$, a computer is needed.Comment: Article will appear in {\em Graphs and Combinatorics

Magnetic phase diagram of a frustrated ferrimagnetic ladder: Relation to the one-dimensional boson Hubbard model

We study the magnetic phase diagram of two coupled mixed-spin $(1,{1/2})$ Heisenberg chains as a function of the frustration parameter related to diagonal exchange couplings. The analysis is performed by using spin-wave series and exact numerical diagonalization techniques. The obtained phase diagram--containing the Luttinger liquid phase, the plateau phase with a magnetization per rung $M=1/2$, and the fully polarized phase--is closely related to the generic $(J/U,\mu/U)$ phase diagram of the one-dimensional boson Hubbard model.Comment: 4 pages, 2 figure

Glycosylation of hyperthermostable designer cellulosome components yields enhanced stability and cellulose hydrolysis

Biomass deconstruction remains integral for enabling second‐generation biofuel production at scale. However, several steps necessary to achieve significant solubilization of biomass, notably harsh pretreatment conditions, impose economic barriers to commercialization. By employing hyperthermostable cellulase machinery, biomass deconstruction can be made more efficient, leading to milder pretreatment conditions and ultimately lower production costs. The hyperthermophilic bacterium Caldicellulosiruptor bescii produces extremely active hyperthermostable cellulases, including the hyperactive multifunctional cellulase CbCel9A/Cel48A. Recombinant CbCel9A/Cel48A components have been previously produced in Escherichia coli and integrated into synthetic hyperthermophilic designer cellulosome complexes. Since then, glycosylation has been shown to be vital for the high activity and stability of CbCel9A/Cel48A. Here, we studied the impact of glycosylation on a hyperthermostable designer cellulosome system in which two of the cellulosomal components, the scaffoldin and the GH9 domain of CbCel9A/Cel48A, were glycosylated as a consequence of employing Ca. bescii as an expression host. Inclusion of the glycosylated components yielded an active cellulosome system that exhibited long‐term stability at 75 °C. The resulting glycosylated designer cellulosomes showed significantly greater synergistic activity compared to the enzymatic components alone, as well as higher thermostability than the analogous nonglycosylated designer cellulosomes. These results indicate that glycosylation can be used as an essential engineering tool to improve the properties of designer cellulosomes. Additionally, Ca. bescii was shown to be an attractive candidate for production of glycosylated designer cellulosome components, which may further promote the viability of this bacterium both as a cellulase expression host and as a potential consolidated bioprocessing platform organism

Spin and exchange coupling for Ti embedded in a surface dipolar network

We have studied the spin and exchange coupling of Ti atoms on a Cu$_2$N/Cu(100) surface using density functional theory. We find that individual Ti have a spin of 1.0 (i.e., 2 Bohr Magneton) on the Cu$_2$N/Cu(100) surface instead of spin-1/2 as found by Scanning Tunneling Microscope. We suggest an explanation for this difference, a two-stage Kondo effect, which can be verified by experiments. By calculating the exchange coupling for Ti dimers on the Cu$_2$N/Cu(100) surface, we find that the exchange coupling across a `void' of 3.6\AA\ is antiferromagnetic, whereas indirect (superexchange) coupling through a N atom is ferromagnetic. We confirm the existence of superexchange interactions by varying the Ti-N angle in a model trimer calculation. For a square lattice of Ti on Cu$_2$N/Cu(100), we find a novel spin striped phase

Multi-Period Trading via Convex Optimization

We consider a basic model of multi-period trading, which can be used to evaluate the performance of a trading strategy. We describe a framework for single-period optimization, where the trades in each period are found by solving a convex optimization problem that trades off expected return, risk, transaction cost and holding cost such as the borrowing cost for shorting assets. We then describe a multi-period version of the trading method, where optimization is used to plan a sequence of trades, with only the first one executed, using estimates of future quantities that are unknown when the trades are chosen. The single-period method traces back to Markowitz; the multi-period methods trace back to model predictive control. Our contribution is to describe the single-period and multi-period methods in one simple framework, giving a clear description of the development and the approximations made. In this paper we do not address a critical component in a trading algorithm, the predictions or forecasts of future quantities. The methods we describe in this paper can be thought of as good ways to exploit predictions, no matter how they are made. We have also developed a companion open-source software library that implements many of the ideas and methods described in the paper

An invitation to grieve: reconsidering critical incident responses by support teams in the school setting

This paper proposes that consideration could be given to an invitational intervention rather than an expectational intervention when support personnel respond to a critical incident in schools. Intuitively many practitioners know that it is necessary for guidance/counselling personnel to intervene in schools in and following times of trauma. Most educational authorities in Australia have mandated the formulation of a critical incident intervention plan. This paper defines the term critical incident and then outlines current intervention processes, discussing the efficacy of debriefing interventions. Recent literature suggests that even though it is accepted that a planned intervention is necessary, there is scant evidence as to the effectiveness of debriefing interventions in stemming later symptoms of post traumatic stress disorder. The authors of this paper advocate for an expressive therapy intervention that is invitational rather than expectational, arguing that not all people respond to trauma in the same way and to expect that they will need to recall and retell what has happened is most likely a dangerous assumption. A model of invitation using Howard Gardner’s (1983) multiple intelligences is proposed so that students are invited to grieve and understand emotionally what is happening to them following a critical incident

Interobserver Reliability in Describing Radiographic Lung Changes After Stereotactic Body Radiation Therapy

Purpose Radiographic lung changes after stereotactic body radiation therapy (SBRT) vary widely between patients. Standardized descriptions of acute (≤6 months after treatment) and late (\u3e6 months after treatment) benign lung changes have been proposed but the reliable application of these classification systems has not been demonstrated. Herein, we examine the interobserver reliability of classifying acute and late lung changes after SBRT. Methods and materials A total of 280 follow-up computed tomography scans at 3, 6, and 12 months post-treatment were analyzed in 100 patients undergoing thoracic SBRT. Standardized descriptions of acute lung changes (3- and 6-month scans) include diffuse consolidation, patchy consolidation and ground glass opacity (GGO), diffuse GGO, patchy GGO, and no change. Late lung change classifications (12-month scans) include modified conventional pattern, mass-like pattern, scar-like pattern, and no change. Five physicians scored the images independently in a blinded fashion. Fleiss\u27 kappa scores quantified the interobserver agreement. Results The Kappa scores were 0.30 at 3 months, 0.20 at 6 months, and 0.25 at 12 months. The proportion of patients in each category at 3 and 6 months was as follows: Diffuse consolidation 11% and 21%; patchy consolidation and GGO 15% and 28%; diffuse GGO 10% and 11%; patchy GGO 15% and 15%; and no change 49% and 25%, respectively. The percentage of patients in each category at 12 months was as follows: Modified conventional 46%; mass-like 16%; scar-like 26%; and no change 12%. Uniform scoring between the observers occurred in 26, 8, and 14 cases at 3, 6, and 12 months, respectively. Conclusions Interobserver reliability scores indicate a fair agreement to classify radiographic lung changes after SBRT. Qualitative descriptions are insufficient to categorize these findings because most patient scans do not fit clearly into a single classification. Categorization at 6 months may be the most difficult because late and acute lung changes can arise at that time

The Chandra Iron-L X-Ray Line Spectrum of Capella

An analysis of the iron L-shell emission in the publicly available spectrum of the Capella binary system, as obtained by the High Energy Transmission Grating Spectrometer on board the Chandra X-ray Observatory, is presented. The atomic-state model, based on the HULLAC code, is shown to be especially adequate for analyzing high-resolution x-ray spectra of this sort. Almost all of the spectral lines in the 10 - 18 Angstrom wavelength range are identified. It is shown that, for the most part, these lines can be attributed to emission from L-shell iron ions in the Capella coronae. Possibilities for electron temperature diagnostics using line ratios of Fe16+ are demonstrated. It is shown that the observed iron-L spectrum can be reproduced almost entirely by assuming a single electron temperature of kTe= 600 eV. This temperature is consistent with both the measured fractional ion abundances of iron and with the temperature derived from ratios of Fe16+ lines. A volume emission measure of 1053 cm-3 is calculated for the iron L-shell emitting regions of the Capella coronae indicating a rather small volume of 1029 cm3 for the emitting plasma if an electron density of 1012 cm-3 is assumed.Comment: Accepted to Ap

High-Resolution X-Ray Spectroscopy of the Accretion Disk Corona Source 4U 1822-37

We present a preliminary analysis of the X-ray spectrum of the accretion disk corona source, 4U 1822-37, obtained with the High Energy Transmission Grating Spectrometer onboard the Chandra X-ray Observatory. We detect discrete emission lines from photoionized iron, silicon, magnesium, neon, and oxygen, as well as a bright iron fluorescence line. Phase-resolved spectroscopy suggests that the recombination emission comes from an X-ray illuminated bulge located at the predicted point of impact between the disk and the accretion stream. The fluorescence emission originates in an extended region on the disk that is illuminated by light scattered from the corona.Comment: 12 pages, 6 figures; Accepted for publication in ApJ Letter