How the brain encodes information in sequences of voltage spikes is an open question. Past literature suggests the importance of bursts, high-frequency spike events, as a key step towards answering this question. In particular, it was recently shown that neurons could use bursts to communicate two streams of information simultaneously, resulting in higher information rates than seen with other neural code theories. However, it is unknown how a neuron’s spiking statistics might affect communication via this new code. To investigate the influence of spike statistics, we study a bursting neuron model with the goal of estimating its information rate as a function of its spike statistics. To this end we extend a recently proposed method for estimating information rate. We find the information rate in our burst-multiplexing model is robust to changes in spike-train statistics, providing evidence for the utility of a burst-multiplexing code to diverse brain networks.

Seismic hazard analysis aims to estimate human risk due to natural disasters such as earthquakes. To improve seismic hazard analysis, our group is focused on earthquake induced tsunamis and the use of statistical models to reconstruct historical earthquakes. Based on the estimated wave heights given in anecdotal historical descriptions, we created observational probability distributions to model the historically recorded observations and constructed a prior distribution on the relevant earthquake parameters based on known seismicity of a given region. Then we used the software package GeoClaw, and a Metropolis-Hastings sampler to obtain a posterior distribution of earthquake parameters that most closely matches the historically recorded tsunami. Our method was tested on two main events that occurred in 1820 and 1852 in central and eastern Indonesia respectively. The random walk Metropolis-Hastings sampler we employed appeared to recover the causal earthquake quite well, but the computational costs were prohibitive even though both scenarios we considered were relatively simple. To improve the sampling procedure, we have focused on advanced sampling techniques such as Hamiltonian Monte Carlo (HMC) where the gradient of the forward model (Geoclaw) is required. This is problematic however as this gradient is not available computationally. To mitigate this problem, we make use of a linearized adjoint solver for the shallow water equations, and exact gradient calculations for the Okada earthquake rupture model, yielding a surrogate gradient that leads to improved sampling.

Modern array sensing systems, such as distributed fiber optic sensing, are used in many applications which may record a mixture of responses to multiple physical quantities. In these applications, it may be helpful to be able to separate this mixture of responses into the signals resulting from the individual sources. This is similar to the cocktail party problem posed with Independent Component Analysis (ICA), in which we use gradient ascent and fixed point iteration optimization algorithms to achieve this separation. We then seek to apply the problem setup from ICA to mixed signals resulting from a sensor array with the goal of maintaining coherence throughout resulting spatial arrays. We propose a new post-processing technique after separation to pair up the signals from different types of physical quantities based on the Symmetric Reverse Cuthill-McKee (SRCM) and Symmetric Approximate Minimum Degree (SAMD) permutations of the coherence matrix. / Master of Science / Some modern sensing systems are able to collect data resulting from different types of sources, such as vibrations and electromagnetic waves, at the same time. This means we have signals resulting from a mixture of sources. An example of one such modern sensing system is distributed fiber optic sensors used in geoscience applications, such as seismology and subsurface imaging, which measures strain along the fiber optic cable. In many applications, it may be helpful to obtain the signals from each of these sources separately, instead of having a mixture of these sources. We propose the use of optimization algorithms, in particular two algorithms arising from Independent Component Analysis (ICA), which seek to maximize a function in order to separate these signals. We then explore changes required to the algorithms for scenarios in which we have multiple sensors spaced some distance away from each other which record signals from two different sources. We also present a method of determining which separated signals correspond to which sensors after performing signal separation.

We study minimizers of the Landau-de Gennes energy in the exterior region around a smooth 2-manifold in R3 with a constant external magnetic field present. Uniaxial boundary data and a strong tangential anchoring are imposed on the surface of the manifold and we consider the large particle limit in a regime where the magnetic field is relatively weak. Before studying the general manifold, we analyze a more simple case in which the manifold is spherical. After deriving a lower bound for the energy in this limiting regime, we prove that a director field on the boundary which maximizes its vertical component yields a minimal lower bound. We then construct a recovery sequence to show that this lower bound is in fact the optimal energy bound. These steps are later repeated in more generality for a larger class of smooth manifolds. / Thesis / Master of Science (MSc)

This thesis explores the application of hyperbolic geometry to deep variational autoencoders (VAEs) for learning low-dimensional latent representations of data. Hyperbolic geometry has gained increasing attention in machine learning due to its potential to embed hierarchical data structures in continuous, differentiable manifolds. We extend previous work investi- gating the Poincaré ball model of hyperbolic geometry and its integration into VAEs. By evaluating hyperbolic VAEs on the MNIST handwritten digit dataset and a single-cell RNA sequencing dataset of metastatic melanoma, we assess whether the inductive bias and math- ematical properties of hyperbolic spaces result in improved data representations compared to standard Euclidean VAEs, especially for single-cell RNA sequencing data. Our findings demonstrate the potential advantages of leveraging hyperbolic geometry for representation learning, while also highlighting some challenges. This work contributes to the growing field of geometric deep learning and provides insights for future research on non-Euclidean approaches to representation learning.

In this work, an attempt is made to lay out a framework in which models of tumor growth can be built, calibrated, validated, and differentiated in their level of goodness in such a manner that all the uncertainties associated with each step of the modeling process can be accounted for in the final model prediction. The study can be divided into four basic parts. The first involves the development of a general family of mathematical models of interacting species representing the various constituents of living tissue, which generalizes those previously available in the literature. In this theory, surface effects are introduced by incorporating in the Helmholtz free ` gradients of the volume fractions of the interacting species, thus providing a generalization of the Cahn-Hilliard theory of phase change in binary media and leading to fourth-order, coupled systems of nonlinear evolution equations. A subset of these governing equations is selected as the primary class of models of tumor growth considered in this work. The second component of this study focuses on the emerging and fundamentally important issue of predictive modeling, the study of model calibration, validation, and quantification of uncertainty in predictions of target outputs of models. The Bayesian framework suggested by Babuska, Nobile, and Tempone is employed to embed the calibration and validation processes within the framework of statistical inverse theory. Extensions of the theory are developed which are regarded as necessary for certain scenarios in these methods to models of tumor growth. The third part of the study focuses on the numerical approximation of the diffuse-interface models of tumor growth and on the numerical implementations of the statistical inverse methods at the core of the validation process. A class of mixed finite element models is developed for the considered mass-conservation models of tumor growth. A family of time marching schemes is developed and applied to representative problems of tumor evolution. Finally, in the fourth component of this investigation, a collection of synthetic examples, mostly in two-dimensions, is considered to provide a proof-of-concept of the theory and methods developed in this work. / text

This thesis investigated a method of barcode reconstruction to address the recovery of a blurred and convoluted one-dimensional barcode. There are a lot of types of barcodes used today, such as Code 39, Code 93, Code 128, etc. Our algorithm applies to the universal barcode, EAN 13. We extend the methodologies proposed by Iwen et al. (2013) in the journal article "A Symbol-Based Algorithm for Decoding barcodes." The algorithm proposed in the paper requires a signal measured by a laser scanner as an input. The observed signal is modeled as a true signal corrupted by a Gaussian convolution, additional noises, and an unknown multiplier. The known barcode dictionaries were incorporated into the forward map between the true barcode and the observed barcode. Unlike the one proposed by Iwen et al., we take dictionaries of different patterns into account, specifically for decoding book barcodes from images which are captured with smartphones. We also presented numerical experiments that examined the performance of the proposed algorithm and illustrated that the unique determination of barcode digits is possible even in the presence of noise.

Submitted by Renata Lopes (renatasil82@gmail.com) on 2015-12-10T10:32:17Z No. of bitstreams: 1 ricardoguimaraesdealmeida.pdf: 1238810 bytes, checksum: 7406c5368a23ac5eb633c275c1427013 (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2015-12-10T14:04:32Z (GMT) No. of bitstreams: 1 ricardoguimaraesdealmeida.pdf: 1238810 bytes, checksum: 7406c5368a23ac5eb633c275c1427013 (MD5) / Made available in DSpace on 2015-12-10T14:04:32Z (GMT). No. of bitstreams: 1 ricardoguimaraesdealmeida.pdf: 1238810 bytes, checksum: 7406c5368a23ac5eb633c275c1427013 (MD5) Previous issue date: 2015-08-15 / CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Este estudo tem como objetivo descrever os conceitos de razão e proporção através de situações que estão presentes no cotidiano popular, buscando estabelecer relações da vida prática do leitor com a matemática ensinada nas escolas por análise e resolução de problemas. Os modelos ora apresentados podem ser utilizados por professores e aplicados em salas de aula do ensino fundamental, sendo úteis àqueles que desejam compreender esses conceitos para aplicá-los em situações matemáticas que possam ser resolvidas através do uso dessas importantes ferramentas. Foi desenvolvido um estudo detalhado, aplicado e diversificado das razões, com a finalidade de garantir ao leitor fundamentos sólidos para a compreensão e resolução dos problemas de proporções, conhecidos por Regra de Três e Porcentagem, através de métodos que utilizam análises lógicas de fácil compreensão, inclusive para os não simpatizantes da matemática. / This study has as its objective to describe the concepts of ration and proportion through everyday situations, seeking to establish matches in the reader’s practical life with mathematics taught in schools by analysis and problem solving. The presented models can be used by teachers and applied in classes of elementary school, and also may be useful to those who wish to comprehend the concepts in order to apply them in mathematical situations which can be solved by the use of these important means. We aimed to develop a detailed study, applied and diversified, of the ratios with the objective of guaranteeing to the reader solid fundaments for its comprehension and to solve proportion problems, known as The Rule of Three and Percentage, through methods which use logical analysis of easy comprehension, even for those who are not familiar with mathematics.

Mathematical modeling provides a powerful framework for insight into current scientific theories as well as hypothesis generation for further research. Despite its undeniable potential to enrich scientific advancement, the application of mathematical modeling remains conspicuously scarce in the field of family science. The complexity inherent in family dynamics, coupled with the intricate interplay of emotions in the individual, underscores the necessity of a robust analytical approach. Addressing this critical gap in the literature, this thesis introduces a sophisticated mathematical model of family dynamics integrating essential elements from family systems theory, emotion dynamics, and appraisal theory. The model is implemented as a versatile computer program capable of simulating family interaction over time, and allows for customization to suit specific research needs. Noteworthy outcomes from the current model parameters include the emergence of both asymptotic and periodic emotion dynamics, the significant influence of family roles on behavior and rapport, and the ripple effect of a single individual's behavior on the system as a whole. Through the exploration of emergent behaviors, this model offers invaluable insights and paves the way for future mathematical modeling and advancements in family science research.

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