Systoles of Arithmetic Hyperbolic Surfaces and 3-manifolds

Our main result is that for all sufficiently large $x_0>0$, the set of commensurability classes of arithmetic hyperbolic 2- or 3-orbifolds with fixed invariant trace field $k$ and systole bounded below by $x_0$ has density one within the set of all commensurability classes of arithmetic hyperbolic 2- or 3-orbifolds with invariant trace field $k$. The proof relies upon bounds for the absolute logarithmic Weil height of algebraic integers due to Silverman, Brindza and Hajdu, as well as precise estimates for the number of rational quaternion algebras not admitting embeddings of any quadratic field having small discriminant. When the trace field is $\mathbf{Q}$, using work of Granville and Soundararajan, we establish a stronger result that allows our constant lower bound $x_0$ to grow with the area. As an application, we establish a systolic bound for arithmetic hyperbolic surfaces that is related to prior work of Buser-Sarnak and Katz-Schaps-Vishne. Finally, we establish an analogous density result for commensurability classes of arithmetic hyperbolic 3-orbifolds with small area totally geodesic $2$-orbifolds.Comment: v4: 17 pages. Revised according to referee report. Final version. To appear in Math. Res. Let

The Monastic Enclosure

The moral and physical enclosure of monks and nuns is central to the founding documents of Western monasticism. But even there it encountered the need for monasteries to interact with their societies, through recruits, hospitality, and the monastic economy. The increasing intensity of this tension is traced through key reforming texts, until later English visitations open up religious houses to closer scrutiny, ironically aided by inmates’ quandary over whether to conceal or reveal their secrets.Benjamin Thompson, ‘The Monastic Enclosure’, in Openness in Medieval Europe, ed. by Manuele Gragnolati and Almut Suerbaum, Cultural Inquiry, 23 (Berlin: ICI Berlin Press, 2022), pp. 249-69 <https://doi.org/10.37050/ci-23_13

The Reductive Couplings of Enones and Alkynes and Application to Heterocycle Synthesis.

A new method for the reductive coupling of alkynes to enones has been developed. This method provides an alternative to more traditional methods of conjugate addition and provides advances in terms of regioselectivity. Other methods rely on the use of preformed organometallic reagents that must be generated in situ stoichiometrically, while this approach utilizes simpler, more readily available starting materials. This method also offers an improved substrate scope over analogous methodologies. Pyrroles, furans and thiophenes make up a class of heterocycles that are present in a broad range of biologically active natural products and pharmaceutically relevant molecules. There is a wide variety of methods for their synthesis, and one of the most widely used strategies, the Stetter-Paal-Knorr sequence, relies on access to 1,4-diketones. The reductive coupling of enones and alkynes allows for access to products that, upon oxidative cleavage, can be used as precursors to five-membered heterocycles. This process allows access to a wide variety of diversely substituted and polycyclic products. The development of regiocontrol for a variety of coupling reactions was examined. Being able to selectively hydroborate or hydrosilylate π-systems such as dienes or alkynes provides access to useful synthetic intermediates. In an attempt to develop regiocontrol for the enone-alkyne reductive coupling, a new reaction was discovered, allowing for the stereoselective synthesis of 1,4-skipped dienes.Ph.D.ChemistryUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/91615/1/benthomp_1.pd

Hierarchically structured composites and porous materials

This thesis develops a hydrogel bead templating technique for the preparation of hierarchically structured composites and porous materials. This method involves using slurries of hydrogel beads with different size distributions as templates. Mixing hydrogel beads with a scaffolding material and then allowing the scaffold to harden, followed by drying of the composite leaves pores in the place of the hydrogel beads. These pores reflect the size and shape of the templates used and the porosity reflects the volume percentage of hydrogel bead slurry mixed with the scaffolding material. A viscous trapping technique has been developed which utilises the viscosity of methylcellulose to stop sedimentation of the scaffold particles during network formation. Both of these methods are attractive due to being cheap, non-toxic and they use food grade materials which allows their use in a multitude of applications.Porous and hierarchically porous gypsum composites have been prepared using both hydrogel bead templating and viscous trapping techniques, or a combination of the two. The level of control over the final microstructure of the dried composites offered by these techniques allowed for a systematic investigation of their thermal and mechanical properties as a function of the pore size, porosity and hierarchical microstructure. It has been shown that the thermal conductivity decreases linearly with increasing porosity, however it was not dependent on the pore sizes that were investigated here. The mechanical properties, however, were significantly different. The porous composites produced with either small hydrogel beads (100 μm) or methylcellulose solution had approximately twice the compressional strength and Young’s modulus compared to the ones produced with large hydrogel beads (600 μm).The sound insulating properties of porous and hierarchically porous gypsum composites have also been investigated. With increasing porosity, the sound transmission loss decreases, as expected. At constant porosity, it is shown that the composites with large pores perform significantly better than the ones with small pores in the frequency range of 75-2000 Hz. At higher frequencies (>2400 Hz) the composites with smaller pores begin to perform better. The material’s microstructure has been studied in an attempt to explain this effect.The hydrogel templating technique can be used to prepare composite materials if the drying step is not performed. Hydrogel beads have been incorporated into a soap matrix. The dissolution rate of these composites as a function of hydrogel bead size and volume percentage of hydrogel beads incorporated within the soap matrix has been investigated. It has been shown that the dissolution rate can be increased by increasing the volume percentage of hydrogel beads used during composite preparation but it is independent on their size distribution. Finally, three methods of controlling the release rate of encapsulated species from these soap-hydrogel bead composites have been shown. The first method involved varying the size distribution of the hydrogel beads incorporated within the soap matrix. The second involved changing the concentration of the gelling polymer and the final method required co-encapsulation of an oppositely charged polyelectrolyte.A binary hydrogel system has been developed and its rheological and thermal properties have been investigated. It consists of agar and methylcellulose and shows significantly improved rheological properties at high temperatures compared to agar alone. The storage modulus of the two component hydrogel shows a maximum at 55 °C which was explained by a sol-gel phase transition of methylcellulose, evidence of which was seen during differential scanning calorimetry measurements. After exposure of this binary hydrogel to high temperatures above the melting point of agar alone (> 120 °C), it maintains its structure. This suggests it could be used for high temperature templating or structuring of food products.The melt-resistant binary hydrogel was used for the preparation of pancake-hydrogel composites using hydrogel bead templating. Mixing slurry of hydrogel beads of this composition with pancake batter, followed by preparation at high temperatures produced pancakes with hydrogel beads incorporated within. Bomb calorimetry measurements showed that the caloric density could be reduced by a controlled amount by varying the volume percentage of hydrogel beads used during preparation of the composites. This method could be applied to other food products such as biscuits, waffles and breakfast bars. Furthermore, there is scope for development of this method by the encapsulation of flavour enhancing or nutritionally beneficial ingredients within the hydrogel beads

The meaning of charity in Locke's political thought

Copyright © 2009 by SAGE Publications. This is the authors final version, after peer-review. 12 month embargo by the publisher. Article will be released April 2010.The recent ‘religious turn’ within Locke scholarship has stressed the need to understand his theological commitments when approaching his political thought. One area of interpretation that has been completely transformed by this heightened sensitivity to the religious roots of Locke’s thought is his account of property ownership which, it is claimed, contains a ‘right to charity’—a subsistence entitlement that trumps established ownership rights. However, this increasingly accepted interpretive claim has been made without significant attention to the way in which charity is deployed throughout Locke’s writing. The aim of this article is to try and get to grips with Locke’s various usages of the term and determine whether the concept he deploys is a consistent one. After discussion of the uncertain role charity plays in his account of property, we examine how it is defined in the Essay Concerning Human Understanding, and then turn to the crucial position it occupies in his theological corpus. Though Locke’s understanding of charity seems fraught with ambiguities, the reason for these ambiguities relate to his configuration of charity as a disposition rather than a mere act, a configuration linked inextricably to his account of toleration