Finite free convolutions via Weingarten calculus

We consider the three finite free convolutions for polynomials studied in a recent paper by Marcus, Spielman, and Srivastava. Each can be described either by direct explicit formulae or in terms of operations on randomly rotated matrices. We present an alternate approach to the equivalence between these descriptions, based on combinatorial Weingarten methods for integration over the unitary and orthogonal groups. A key aspect of our approach is to identify a certain \emph{quadrature property}, which is satisfied by some important series of subgroups of the unitary groups (including the groups of unitary, orthogonal, and signed permutation matrices), and which yields the desired convolution formulae.Comment: Major revision: includes unitary and hyperoctahedral versions of the convolution formulae, via a "quadrature property" which yields the relevant quadrature result

Quantum Catalysis of Magnetic Phase Transitions in a Quantum Simulator

We control quantum fluctuations to create the ground state magnetic phases of a classical Ising model with a tunable longitudinal magnetic field using a system of 6 to 10 atomic ion spins. Due to the long-range Ising interactions, the various ground state spin configurations are separated by multiple first-order phase transitions, which in our zero temperature system cannot be driven by thermal fluctuations. We instead use a transverse magnetic field as a quantum catalyst to observe the first steps of the complete fractal devil's staircase, which emerges in the thermodynamic limit and can be mapped to a large number of many-body and energy-optimization problems.Comment: New data in Fig. 3, and much of the paper rewritte

Finitary approximations of free probability, involving combinatorial representation theory

This thesis contributes to two theories which approximate free probability by finitary combinatorial structures. The first is finite free probability, which is concerned with expected characteristic polynomials of various random matrices and was initiated by Marcus, Spielman, and Srivastava in 2015. An alternate approach to some of their results for sums and products of randomly rotated matrices is presented, using techniques from combinatorial representation theory. Those techniques are then applied to the commutators of such matrices, uncovering the non-trivial but tractable combinatorics of immanants and Schur polynomials. The second is the connection between symmetric groups and random matrices, specifically the asymptotics of star-transpositions in the infinite symmetric group and the gaussian unitary ensemble (GUE). For a continuous family of factor representations of $S_{\infty}$, a central limit theorem for the star-transpositions $(1,n)$ is derived from the insight of Gohm-K\"{o}stler that they form an exchangeable sequence of noncommutative random variables. Then, the central limit law is described by a random matrix model which continuously deforms the well-known traceless GUE by taking its gaussian entries from noncommutative operator algebras with canonical commutation relations (CCR). This random matrix model generalizes results of K\"{o}stler and Nica from 2021, which in turn generalized a result of Biane from 1995

Constraints on Mars Aphelion Cloud Belt Phase Function and Ice Crystal Geometries

This study constrains the lower bound of the scattering phase function of Martian water ice clouds (WICs) through the implementation of a new observation aboard the Mars Science Laboratory (MSL). The Phase Function Sky Survey (PFSS) was a multiple pointing all-sky observation taken with the navigation cameras (Navcam) aboard MSL. The PFSS was executed 35 times during the Aphelion Cloud Belt (ACB) season of Mars Year 34 over a solar longitude range of L_s=61.4{\deg}-156.5{\deg}. Twenty observations occurred in the morning hours between 06:00 and 09:30 LTST, and 15 runs occurred in the evening hours between 14:30 and 18:00 LTST, with an operationally required 2.5 hour gap on either side of local noon due the sun being located near zenith. The resultant WIC phase function was derived over an observed scattering angle range of 18.3{\deg} to 152.61{\deg}, normalized, and compared with 9 modeled phase functions: seven ice crystal habits and two Martian WIC phase functions currently being implemented in models. Through statistical chi-squared probability tests, the five most probable ice crystal geometries observed in the ACB WICs were aggregates, hexagonal solid columns, hollow columns, plates, and bullet rosettes with p-values greater than or equal to 0.60, 0.57,0.56,0.56, and 0.55, respectively. Droxtals and spheres had p-values of 0.35, and 0.2, making them less probable components of Martian WICs, but still statistically possible ones. Having a better understanding of the ice crystal habit and phase function of Martian water ice clouds directly benefits Martian climate models which currently assume spherical and cylindrical particles.Comment: Accepted Manuscript by Planetary and Space Scienc

Demographic characteristics of exploited tropical lutjanids: a comparative analysis

Demographic parameters from seven exploited coral reef lutjanid species were compared as a case study of the implications of intrafamily variation in life histories for multispecies harvest management. Modal lengths varied by 4 cm among four species (Lutjanus fulviflamma, L. vitta, L. carponotatus, L. adetii), which were at least 6 cm smaller than the modal lengths of the largest species (L. gibbus, Symphorus nematophorus, Aprion virescens). Modal ages, indicating ages of full selection to fishing gear, were 10 years or less for all species, but maximum ages ranged from 12 (L. gibbus) to 36 years (S. nematophorus). Each species had a unique growth pattern, with differences in length-at-age and mean asymptotic fork length (L∞), but smaller species generally grew fast during the first 1–2 years of life and larger species grew more slowly over a longer period. Total mortality rates varied among species; L. gibbus had the highest mortality and L. fulviflamma, the lowest mortality. The variability in life history strategies of these tropical lutjanids makes generalizations about lutjanid life histories difficult, but the fact that all seven had characteristics that would make them particularly vulnerable to fishing indicates that harvest of tropical lutjanids should be managed with caution

A mechanochemical approach for organocatalysis and improved synthesis of gemini surfactants

1 online resource (ix, 63 pages) : illustrationsIncludes abstract.Includes bibliographical references (pages 59-63).Mechanochemistry and organocatalysis provide green synthetic methods for chemical reactions. Reported herein are Morita-Baylis-Hilman reactions co-catalyzed by 1,4-diazabicyclo[2.2.2]octane, DABCO, and 1-(4-(3-(3,5-bis(trifluoromethyl)phenyl)thioureido)-benzyl)-1-methylpyrrolidin-1-ium hexafluorophosphate using a mechanochemical approach known as Liquid Assisted Grinding, LAG. These room temperature reactions used methanol as a liquid additive to achieve an 85.7 % conversion and 14.8 % isolated yields, which is a slight improvement over conventional methods. Gemini cationic surfactants have a lower critical micelle concentration (CMC) when compared to conventional analogues. This led to many improvements in materials such as soaps, detergents, wetting agents and foaming agents. There is a particular interest from a physical and material chemistry standpoint on how gemini cationic surfactants will behave with other surfactants in mixed micellular systems. Previously reported syntheses of quaternary amine gemini surfactants in the literature noted long reaction times, poor yields and large solvent use, making large scale production troublesome. Reported herein is an improved synthesis of mcm type cationic gemini surfactants using microwave irradiation

Development of a Digital Noise Exposure System for Research on Noise Induced Hearing Loss

Over thirty million Americans suffer from noise induced hearing loss (NIHL). Previous research demonstrated that different types of noises, even with equal sound energies, could produce different amounts of hearing loss. In this project, a novel digital noise exposure system has been developed for generating various noise signals (e.g. pure-tone, Gaussian, impulsive, and complex noise). The developed system can be used to study NIHL in animal models. The system could produce impulse noise with peak sound pressure level (SPL) up to 160 dB, which effectively mimics the noise generated by a military weapon (e.g., M-16 rifle). The preliminary results of in an animal study showed significant permanent threshold shift (PTS) produced by 90 shocks impulse noise with peak SPL = 155 dB generated by the system. In summary, the digital noise exposure system replicates environmental noise allowing researchers to study impulse noise induced hearing loss in a controlled situation