Wandering intervals and absolutely continuous invariant probability measures of interval maps

For piecewise $C^1$ interval maps possibly containing critical points and discontinuities with negative Schwarzian derivative, under two summability conditions on the growth of the derivative and recurrence along critical orbits, we prove the nonexistence of wandering intervals, the existence of absolutely continuous invariant measures, and the bounded backward contraction property. The proofs are based on the method of proving the existence of absolutely continuous invariant measures of unimodal map, developed by Nowicki and van Strien.Comment: 16 pages, 2 figure

Computational prediction of organic crystal structures

Crystal Structure Prediction (CSP) is used by the pharmaceutical industry to assess whether other polymorphs of active pharmaceutical ingredients (API) might cause problems during manufacturing processes. In the 7th Blind Test of CSP, organized by the Cambridge Crystallographic Data Centre (CCDC), one of the targets (XXX) was to predict the possible stoichiometries of two co−crystals of cannabinol (CBN) and tetramethylpyrazine (TMP). This thesis describes the methodology used for the submission of predicted structures of these co−crystals, concluding that the likely stoichiometries were 1CBN:1TMP and 1CBN:2TMP, as these were more stable than the component structures and had plausible crystal packings. Following submission, this thesis analysed the crystal structures of TMP and have proposed starting points for the crystal structure refinement of a structure (MPYRAZ03) on the Cambridge Structural Database (CSD) that has no atomic coordinates. The CBN search failed to find the Z’=2 experimental crystal structure (CANNOL) that is on the CSD, which has a high energy molecular conformation. This failure was found to be due to the limits on the structure generation program (Sobol sequence and density setting) and was exacerbated by the point charge model failing to model the CANNOL hydrogen bonding adequately. Alternative strategies to find the experimental structure were proposed, but they were deemed too expensive to run a full search. As this thesis was being completed, experimental co−crystal structures were provided by CCDC. After comparing with experimental structures, there was no experimental co−crystal structure in co−crystal CSP searches used in this thesis. This problem was caused by the folded pentane tail instead of the hydroxyl group

A New Strategy For Collection Of High-temperature Broad-band Absorption Spectra For Gas-phase Molecules In The Mid-infrared

\begin{wrapfigure}{r}{0pt} \includegraphics[scale=0.28]{Ethylene_nu_7_band_995K_2.3atm.eps} \end{wrapfigure} To address the notable lack of knowledge on high-temperature absorption cross sections of important molecular species in combustion and exoplanets, a new strategy is proposed and deployed to collect broad-band absorption spectra in shock-heated gases. The methodology utilizes a broad-scan, rapid-tuning external-cavity quantum cascade laser in conjunction with a shock tube and is capable of providing quantitative spectroscopic information across full vibrational bands spanning over 200 \wn within 6 ms ($>$ 30,000 \wn/s), with a spectral resolution between 0.3 – 0.6 \wn. This experimental approach is demonstrated with absorption spectra measurements on the \nub{7} vibrational band of ethylene (\chem{C_2H_4}) from 8.4 $\mu$m to 11.7 $\mu$m at temperature/pressure conditions between 800 – 1600 K, 1 – 5 atm. The measured spectra are compared against spectral simulations using existing spectroscopic databases, showing better agreement with the line list of Rey et al.\footnote{M. Rey, T. Delahaye, A. V. Nikitin, and V. G. Tyuterev, “First theoretical global line lists of ethylene ($^{12}$\chem{C_2H_4}) spectra for the temperature range 50-700 K in the far-infrared for quantification of absorption and emission in planetary atmospheres,” Astron. Astrophys., vol. 594, pp. 1–16, 2016.} than of HITRAN 2016\footnote{I. E. Gordon et al., “The HITRAN2016 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transf., vol. 203, pp. 3–69, 2017.}. With the current set of instruments available, this methodology could be applied to numerous gas-phase molecules that have attractive absorption features in the spectral range of 3.6 – 11.7 $\mu$m and opens an efficient pathway towards improving knowledge on radiation absorption in the mid-infrared at high temperatures