Phase transition in random tensors with multiple spikes

University of Minnesota Ph.D. dissertation. 2019. Major: Mathematics. Advisors: Gilad Lerman, Wei-Kuo Chen. 1 computer file (PDF); 102 pages.This thesis is concerned with the problem of detecting and recovering a low-rank tensor in noise. A spiked random tensor is composed of a symmetric Gaussian p-tensor and a fixed number of spikes. Each spike is a rank one p-tensor formed by a vector whose entries are drawn i.i.d. from a probability measure on the real line with bounded support. Each spike is weighted by a signal-to-noise ratio (SNR). For a random tensor with a single spike, it is possible to detect the presence of the spike when the SNR exceeds a critical threshold, and impossible when the SNR is below this threshold. For a random tensor with multiple spikes, detection of the low-rank structure is possible when the SNR of at least one spike exceeds its critical threshold. Additionally, recovery of the spikes by the minimum mean square error estimator has the same phase transition. When at least one SNR is above its critical threshold, the minimum mean square error estimator performs better than a random guess. It is shown that the spike detection problem is equivalent to distinguishing between the high- and low-temperature regimes of certain mean field spin glass models. The set of SNRs for which detection is impossible is equal to the high-temperature regime of a certain $p$-spin model. Thus the main tools to investigate the detection problem come from the study of spin glasses

Splines in geometry and topology

This survey paper describes the role of splines in geometry and topology, emphasizing both similarities and differences from the classical treatment of splines. The exposition is non-technical and contains many examples, with references to more thorough treatments of the subject.Comment: 18 page

Switching dynamics of surface stabilized ferroelectric liquid crystal cells: effects of anchoring energy asymmetry

We study both theoretically and experimentally switching dynamics in surface stabilized ferroelectric liquid crystal cells with asymmetric boundary conditions. In these cells the bounding surfaces are treated differently to produce asymmetry in their anchoring properties. Our electro-optic measurements of the switching voltage thresholds that are determined by the peaks of the reversal polarization current reveal the frequency dependent shift of the hysteresis loop. We examine the predictions of the uniform dynamical model with the anchoring energy taken into account. It is found that the asymmetry effects are dominated by the polar contribution to the anchoring energy. Frequency dependence of the voltage thresholds is studied by analyzing the properties of time-periodic solutions to the dynamical equation (cycles). For this purpose, we apply the method that uses the parameterized half-period mappings for the approximate model and relate the cycles to the fixed points of the composition of two half-period mappings. The cycles are found to be unstable and can only be formed when the driving frequency is lower than its critical value. The polar anchoring parameter is estimated by making a comparison between the results of modelling and the experimental data for the shift vs frequency curve. For a double-well potential considered as a deformation of the Rapini-Papoular potential, the branch of stable cycles emerges in the low frequency region separated by the gap from the high frequency interval for unstable cycles.Comment: 35 pages, 15 figure

Optoelectronic Chip for the Implementation of Back Error Propagation

We present a CMOS silicon chip that optically implements the back error propagation (BEP) algorithm [1] of a two layer neural network. The chip has eight units (or "neurons") on a area of approximately 2x2 mm. Each unit consists of a phototransistor as the detector, a modulator pad for light modulation, sample-and-hold circuits, and additional circuits necessary to perform the BEP algorithm

A searchable database and mass spectral comparison tool for the Aerosol Mass Spectrometer (AMS) and the Aerosol Chemical Speciation Monitor (ACSM)

The Aerodyne Aerosol Mass Spectrometer (AMS) and Aerosol Chemical Speciation Monitor (ACSM) are the most widely applied tools for in situ chemical analysis of the non-refractory bulk composition of fine atmospheric particles. The mass spectra (MS) of many AMS and ACSM observations from field and laboratory studies have been reported in peer-reviewed literature and many of these MS have been submitted to an open-access website. With the increased reporting of such datasets, the database interface requires revisions to meet new demands and applications. One major limitation of the web-based database is the inability to automatically search the database and compare previous MS with the researcher's own data. In this study, a searchable database tool for the AMS and ACSM mass spectral dataset was built to improve the efficiency of data analysis using Igor Pro, consistent with existing AMS and ACSM software. The database tool incorporates the published MS and sample information uploaded on the website. This tool allows the comparison of a target mass spectrum with the reference MS in the database, calculating cosine similarity, and provides a range of MS comparison plots, reweighting, and mass spectrum filtering options. The aim of this work is to help AMS and ACSM users efficiently analyze their own data for possible source or atmospheric processing features by comparison to previous studies, enhancing information gained from past and current global research on atmospheric aerosol.</p

Front propagation into unstable and metastable states in Smectic C* liquid crystals: linear and nonlinear marginal stability analysis

We discuss the front propagation in ferroelectric chiral smectics (SmC*) subjected to electric and magnetic fields applied parallel to smectic layers. The reversal of the electric field induces the motion of domain walls or fronts that propagate into either an unstable or a metastable state. In both regimes, the front velocity is calculated exactly. Depending on the field, the speed of a front propagating into the unstable state is given either by the so-called linear marginal stability velocity or by the nonlinear marginal stability expression. The cross-over between these two regimes can be tuned by a magnetic field. The influence of initial conditions on the velocity selection problem can also be studied in such experiments. SmC$^*$ therefore offers a unique opportunity to study different aspects of front propagation in an experimental system

Combined X-Ray and Fully Leaky Guided Mode Studies of the Smectic Layer and Optic Tensor Configuration in a Ferroelectric Liquid-Crystal Cell

B. Hodder, J. Roy Sambles, S. Jenkins, and R. M. Richardson, Physical Review Letters, Vol. 85, pp. 3181 - 3184 (2000). "Copyright © 2000 by the American Physical Society."X-ray scattering together with optical characterization using fully leaky guided modes have been used for the first time to study the same ferroelectric liquid-crystal cell. This enables direct calculation of an accurate cone and chevron description of the liquid-crystal director profile since the layer structure and optic tensor configuration are both well known