Quantum traces for SLn(C)\mathrm{SL}_n(\mathbb{C}): the case n=3n=3

We generalize Bonahon and Wong's $\mathrm{SL}_2(\mathbb{C})$-quantum trace map to the setting of $\mathrm{SL}_3(\mathbb{C})$. More precisely, for each non-zero complex number $q$, we associate to every isotopy class of framed oriented links $K$ in a thickened punctured surface $\mathfrak{S} \times (0, 1)$ a Laurent polynomial $\mathrm{Tr}_\lambda^q(K) = \mathrm{Tr}_\lambda^q(K)(X_i^q)$ in $q$-deformations $X_i^q$ of the Fock-Goncharov coordinates $X_i$ for a higher Teichm\"{u}ller space, depending on the choice of an ideal triangulation $\lambda$ of the surface $\mathfrak{S}$. Along the way, we propose a definition for a $\mathrm{SL}_n(\mathbb{C})$-version of this invariant.Comment: 34 pages, 23 figures + 1 appendix; 48 pages in tota

Tropical Fock-Goncharov coordinates for SL3\mathrm{SL}_3-webs on surfaces II: naturality

In a companion paper (arXiv 2011.01768) we constructed non-negative integer coordinates $\Phi_\mathcal{T}$ for a distinguished collection $\mathcal{W}_{3, \widehat{S}}$ of $\mathrm{SL}_3$-webs on a finite-type punctured surface $\widehat{S}$, depending on an ideal triangulation $\mathcal{T}$ of $\widehat{S}$. We prove that these coordinates are natural with respect to the choice of triangulation, in the sense that if a different triangulation $\mathcal{T}^\prime$ is chosen then the coordinate change map relating $\Phi_\mathcal{T}$ and $\Phi_{\mathcal{T}^\prime}$ is a prescribed tropical cluster transformation. Moreover, when $\widehat{S}=\Box$ is an ideal square, we provide a topological geometric description of the Hilbert basis (in the sense of linear programming) of the non-negative integer cone $\Phi_\mathcal{T}(\mathcal{W}_{3, \Box}) \subset \mathbb{Z}_{\geq 0}^{12}$, and we prove that this cone canonically decomposes into 42 sectors corresponding topologically to 42 families of $\mathrm{SL}_3$-webs in the square.Comment: 39 pages, 27 figure

The Five Planets in the Kepler-296 Binary System All Orbit the Primary: A Statistical and Analytical Analysis

Kepler-296 is a binary star system with two M-dwarf components separated by 0.2 arcsec. Five transiting planets have been confirmed to be associated with the Kepler-296 system; given the evidence to date, however, the planets could in principle orbit either star. This ambiguity has made it difficult to constrain both the orbital and physical properties of the planets. Using both statistical and analytical arguments, this paper shows that all five planets are highly likely to orbit the primary star in this system. We performed a Markov-Chain Monte Carlo simulation using a five transiting planet model, leaving the stellar density and dilution with uniform priors. Using importance sampling, we compared the model probabilities under the priors of the planets orbiting either the brighter or the fainter component of the binary. A model where the planets orbit the brighter component, Kepler-296A, is strongly preferred by the data. Combined with our assertion that all five planets orbit the same star, the two outer planets in the system, Kepler-296 Ae and Kepler-296 Af, have radii of 1.53 +/- 0.26 and 1.80 +/- 0.31 R_earth, respectively, and receive incident stellar fluxes of 1.40 +/- 0.23 and 0.62 +/- 0.10 times the incident flux the Earth receives from the Sun. This level of irradiation places both planets within or close to the circumstellar habitable zone of their parent star.Comment: Accepted for publication in Ap

Do Electrostatic Interactions with Positively Charged Active Site Groups Tighten the Transition State for Enzymatic Phosphoryl Transfer?

The effect of electrostatic interactions on the transition-state character for enzymatic phosphoryl transfer has been a subject of much debate. In this work, we investigate the transition state for alkaline phosphatase (AP) using linear free-energy relationships (LFERs). We determined kcat/KM for a series of aryl sulfate ester monoanions to obtain the Brønsted coefficient, βlg, and compared the value to that obtained previously for a series of aryl phosphorothioate ester dianion substrates. Despite the difference in substrate charge, the observed Brønsted coefficients for AP-catalyzed aryl sulfate and aryl phosphorothioate hydrolysis (−0.76 ± 0.14 and −0.77 ± 0.10, respectively) are strikingly similar, with steric effects being responsible for the uncertainties in these values. Aryl sulfates and aryl phosphates react via similar loose transition states in solution. These observations suggest an apparent equivalency of the transition states for phosphorothioate and sulfate hydrolysis reactions at the AP active site and, thus, negligible effects of active site electrostatic interactions on charge distribution in the transition state

Electric utilities, fuel use, and responsiveness to fuel prices

This research tests the impact of changes in fuel price to explain fuel use by electric utilities. We employ a three-stage least squares model that explains changes in fuel use as a function of changes in three fuel prices. This model is repeated across sub-samples of data aggregated at the plant level and operating holding company level. We expect that plants and holding companies reduce fuel use when fuel prices rise. Several fuel substitution effects within and across plants and holding companies are demonstrated, as well as several frictions. At the plant level, higher prices of natural gas lead to less natural gas consumption, less coal consumption, and more fuel oil consumption. At the operating holding company level, results demonstrate the inelasticity of coal use and the increases of natural gas in response to higher coal prices. Subsamples demonstrate heterogeneity of results across different plants. Results emphasize that technological, market, and regulatory frictions may hinder the performance of energy policies