Free analysis and planar algebras

We study 2-cabled analogs of Voiculescu's trace and free Gibbs states on Jones planar algebras. These states are traces on a tower of graded algebras associated to a Jones planar algebra. Among our results is that, with a suitable definition, finiteness of free Fisher information for planar algebra traces implies that the associated tower of von Neumann algebras consists of factors, and that the standard invariant of the associated inclusion is exactly the original planar algebra. We also give conditions that imply that the associated von Neumann algebras are non-$\Gamma$ non-$L^2$ rigid factors

Non-commutative connections of the second kind

A connection-like objects, termed {\em hom-connections} are defined in the realm of non-commutative geometry. The definition is based on the use of homomorphisms rather than tensor products. It is shown that hom-connections arise naturally from (strong) connections in non-commutative principal bundles. The induction procedure of hom-connections via a map of differential graded algebras or a differentiable bimodule is described. The curvature for a hom-connection is defined, and it is shown that flat hom-connections give rise to a chain complex.Comment: 13 pages, LaTe

Absence of room temperature ferromagnetism in bulk Mn-doped ZnO

Structural and magnetic properties have been studied for polycrystalline Zn_1-xMn_xO (x=0.02, 0.03, 0.05). Low-temperature (~500 oC) synthesis leaves unreacted starting ZnO and manganese oxides. Contrary to a recent report, no bulk ferromagnetism was observed for single-phase materials synthesized in air at temperatures above 900 oC. Single-phase samples show paramagnetic Curie-Weiss behavior.Comment: Accepted for publication in J. Appl. Phys., RevTeX, 3 pages, 4 figure

Structural and physical properties of SrMn1−xRuxO3SrMn_{1-x}Ru_xO_3 perovskites

We combine the results of magnetic and transport measurements with neutron diffraction data to construct the structural and magnetic phase diagram of the entire family of SrMn$_{1-x}$Ru$_{x}$O$_3$ ($0 \leqslant x \leqslant 1$) perovskites. We have found antiferromagnetic ordering of the C type for lightly Ru-substituted materials ($0.06 \leqslant x \leqslant 0.5$) in a similar manner to $R_{y}$Sr$_{1-y}$MnO$_3$ ($R$=La, Pr), due to the generation of Mn$^{3+}$ in both families of manganite perovskites by either $B$-site substitution of Ru$^{5+}$ for Mn$^{4+}$ or $A$-site substitution of $R^{3+}$ for Sr$^{2+}$. This similarity is driven by the same ratio of $d^4$ / $d^3$ ions in both classes of materials for equivalent substitution level. In both cases, a tetragonal lattice distortion is observed, which for some compositions ($0.06 \leqslant x \leqslant 0.2$) is coupled to a C-type AF transition and results in a first order magnetic and resistive transition. Heavily substituted SrMn$_{1-x}$Ru$_{x}$O$_3$ materials are ferromagnetic due to dominating exchange interactions between the Ru$^{4+}$ ions. Intermediate substitution ($0.6 \leqslant x \leqslant 0.7$) leads to a spin-glass behavior instead of a quantum critical point reported previously in single crystals, due to enhanced disorder.Comment: 9 pages, 10 figures, accepted for publication in Physical Review

Radium isotopes and radon-222 as tracers of sediment-water interaction in Arctic coastal and lacustrine environments

Submitted in partial fulfillment of the requirements for the degree of Master of Science in Chemical Oceanography at the Massachusetts Institute of Technology and the Woods Hole Oceanographic Institution September 2020.Arctic marine and lacustrine systems are experiencing rapid warming due to climate change. These changes are especially important at the interface between sediments and surface waters because they are hotspots for biogeochemical transformations such as redox reactions, nutrient consumption and regeneration, organic matter leaching and degradation, and mineral weathering. Radium isotopes (223Ra, 224Ra, 226Ra, 228Ra) and radon-222, naturally occurring radioactive isotopes produced in sediments, are well-suited as tracers of nutrients, trace metals, and organic matter cycling processes at the sediment-water interface. In this thesis, I have applied radon-222 and the quartet of radium isotopes to study fundamental processes in subarctic lakes and on the Arctic continental shelf. First, radon-222 is used to quantify groundwater discharge into a shallow, tundra lake on the Yukon-Kuskokwim Delta in Alaska in summer of 2017. Radon-derived groundwater fluxes were then paired with methane (CH4) measurements to determine delivery rates of methane into the lake via groundwater. Groundwater CH4 fluxes significantly exceeded diffusive air-water fluxes from the lake to the atmosphere, suggesting that groundwater is an important source of CH4 to Arctic lakes and may drive observed CH4 emissions. Higher CH4 emissions were observed compared to those reported previously in high latitude lakes, like due to higher CH4 concentrations in groundwater. These findings indicate that deltaic lakes across warmer permafrost regions may act as important hotspots for methane release across Arctic landscapes. Then, the quartet of radium isotopes is used to study the impacts of storms and sea ice formation as drivers of sediment-water interaction on the Alaskan Beaufort shelf. The timeseries presented in this study is among the first to document the combined physical and chemical signals of winter water formation in the Beaufort Sea, made possible by repeat occupations of the central Beaufort shelf. Radium measurements are combined with inorganic nitrogen and hydrographic measurements to elucidate the episodic behavior of winter water formation and its ability to drive exchange with bottom sediments during freeze-up.Financial support for Chapter 2 was funded by National Science Foundation awards OCE-1458305 to M.A.C., 1561437 to S.M.N, J.D.S., and R.M.H and 1624927 to S.M.N., P.J.M. and R.M.H. The work completed for Chapter 3 was funded by the Montrym Fund at the Massachusetts Institute of Technology, the Academic Programs Office at Woods Hole Oceanographic Institution, and the NSF Arctic GEOTRACES (OCE-1458305), Pacific GEOTRACES (OCE-1736277), and Arctic Observing Network programs (OPP-1733564)

Star product formula of theta functions

As a noncommutative generalization of the addition formula of theta functions, we construct a class of theta functions which are closed with respect to the Moyal star product of a fixed noncommutative parameter. These theta functions can be regarded as bases of the space of holomorphic homomorphisms between holomorphic line bundles over noncommutative complex tori.Comment: 12 page