Understanding crystal nucleation through pressure-driven phase transformations

This thesis investigates the nucleation and the discovery of new phases using solubility and high-pressure measurements of chiral and racemic solids of (RS)/(S)-2-(2-oxopyrrolidin-1-yl)butanamide. The cocrystallisation of this compound with a range of different coformers has been investigated using solubility measurements to alter the eutectic composition to aid the chiral resolution. Only one successful cocrystal was isolated and was observed to move the eutectic composition. However, this moved towards the chiral phase rather than the racemic composition hence reducing the phase space available for the chiral resolution. Whilst this result is less ideal, the ability to move the eutectic composition is demonstrated and the role of the cocrystal solubility in this process identified. To follow this study, the response of Levetiracetam and Etiracetam to high pressure was investigated with a view to identifying phase transitions that may be used in future measurements of nucleation using high pressure. In this work, both materials undergo phase transformations to new high-pressure phases. The changes to both were very subtle and difficult to analyse using Raman spectroscopy, which would have been the best method for phase identification during nucleation due to the speed of collection. The difference in the compression are analysed through void analysis as well as energy calculations to confirm the phase transition had taken place. Lastly, hydrochlorothiazide was identified as a potential compound to use in the nucleation measurements. The phase transition to a new phase occurs at ~0.5 GPa and the Raman spectrum indicated a distinct change over the phase transformation which was one of the key concepts for analysis. Unfortunately, the phase transformation is not reversible within a short time scale. The high-pressure form is stable as a solid for many months without transformation. This observation led the study on a different path to explore the formation and recovery of this high-pressure phase. The high-pressure parameters for the successful recovery of the phase were explored as well as the indexing of the new phase. A potential indexing of this new phase is identified, however, the solution of the new phase has eluded characterisation.This thesis investigates the nucleation and the discovery of new phases using solubility and high-pressure measurements of chiral and racemic solids of (RS)/(S)-2-(2-oxopyrrolidin-1-yl)butanamide. The cocrystallisation of this compound with a range of different coformers has been investigated using solubility measurements to alter the eutectic composition to aid the chiral resolution. Only one successful cocrystal was isolated and was observed to move the eutectic composition. However, this moved towards the chiral phase rather than the racemic composition hence reducing the phase space available for the chiral resolution. Whilst this result is less ideal, the ability to move the eutectic composition is demonstrated and the role of the cocrystal solubility in this process identified. To follow this study, the response of Levetiracetam and Etiracetam to high pressure was investigated with a view to identifying phase transitions that may be used in future measurements of nucleation using high pressure. In this work, both materials undergo phase transformations to new high-pressure phases. The changes to both were very subtle and difficult to analyse using Raman spectroscopy, which would have been the best method for phase identification during nucleation due to the speed of collection. The difference in the compression are analysed through void analysis as well as energy calculations to confirm the phase transition had taken place. Lastly, hydrochlorothiazide was identified as a potential compound to use in the nucleation measurements. The phase transition to a new phase occurs at ~0.5 GPa and the Raman spectrum indicated a distinct change over the phase transformation which was one of the key concepts for analysis. Unfortunately, the phase transformation is not reversible within a short time scale. The high-pressure form is stable as a solid for many months without transformation. This observation led the study on a different path to explore the formation and recovery of this high-pressure phase. The high-pressure parameters for the successful recovery of the phase were explored as well as the indexing of the new phase. A potential indexing of this new phase is identified, however, the solution of the new phase has eluded characterisation

A fiedler-type theorem for the determinant of J-positive matrices

https://thekeep.eiu.edu/commencement_spring2014/1030/thumbnail.jp

Numerical ranges of Toeplitz operators with matrix symbols

For Toeplitz operators acting on the vector Hardy space H-2 with definite or indefinite metric, the closure of the respective numerical range is completely described. In the definite case, some observations regarding its boundary are also made. (C) 2011 Elsevier Inc. All rights reserved

Reciprocal matrices: properties and approximation by a transitive matrix

N. Bebiano: partially supported by project UID/MAT/00324/2019. R. Fernandes: partially supported by project UID/MAT/00297/2019. S. Furtado: partially supported by project UID/MAT/04721/2019.Reciprocal matrices and, in particular, transitive matrices, appear in several applied areas. Among other applications, they have an important role in decision theory in the context of the analytical hierarchical process, introduced by Saaty. In this paper, we study the possible ranks of a reciprocal matrix and give a procedure to construct a reciprocal matrix with the rank and the off-diagonal entries of an arbitrary row (column) prescribed. We apply some techniques from graph theory to the study of transitive matrices, namely to determine the maximum number of equal entries, and distinct from ± 1 , in a transitive matrix. We then focus on the n-by-n reciprocal matrix, denoted by C(n, x), with all entries above the main diagonal equal to x> 0. We show that there is a Toeplitz transitive matrix and a transitive matrix preserving the maximum possible number of entries of C(n, x), whose distances to C(n, x), measured in the Frobenius norm, are smaller than the one of the transitive matrix proposed by Saaty, constructed from the right Perron eigenvector of C(n, x). We illustrate our results with some numerical examples.authorsversionpublishe