Applied mathematics
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A discovery of neural network architectures for contextdependent computations
All human and animal behavior from seeing, hearing, running, and falling in love, is the result of complex dynamics in a web of intricate networks in the brain. The human brain, in particular, contains close to 100 billion ... 
Lanczosbased methods for matrix functions
We study Lanczosbased methods for tasks involving matrix functions. We begin by resurfacing a range of ideas regarding matrixfree quadrature which, to the best of our knowledge, have not been treated simultaneously. This ... 
Branching process models for cancer evolution: overview and an application to colorectal cancer initiation
We study a multitype branching process model associated with a transitional network between types. In particular, we are interested in determining the waiting time to each type in the network, employing an approximation ... 
A Complete Asymptotic Analysis of the Spectral Instabilities of SmallAmplitude Periodic Water Waves
Euler's equations govern the behavior of gravity waves on the surface of an incompressible, inviscid, and irrotational fluid (water, in this case). We consider the smallamplitude, periodic travelingwave solutions of ... 
Lanczos Algorithm and Conjugate Gradient
We review results from the literature on the conjugate gradient algorithm for solving symmetric positive definite linear systems and the related Lanczos algorithm. We derive the conjugate gradient algorithm from the more ... 
Dimensionality Reduction for Supervised and Unsupervised Learning: New Algorithms, Analysis and Application
Dimensionality reduction is an essential topic in data science, particularly when data are highdimensional or have more features than samples. The process of reducing the data dimension usually involves solving an eigenvalue ... 
Dynamical Modeling and Numerical Methods for CART Cell Therapy and Viral Tweets
The development of CART cell immunotherapies has been one of the most exciting advancements in the field of cancer research over the last decade. Many mathematical models have been proposed to better understand the nonlinear ... 
The Unified Transform Method and its semidiscrete analogue for numerically solving IBVPs
Finitedifference schemes are a popular and intuitive approach to numerically solve nonlinear initialboundary value problems (IBVPs). Often, this leads to the introduction of ghost points, where the numerical method depends ... 
Life Together: Modeling the Collective Behavior of Cellular Communities
Our lives as eukaryotic organisms are defined by the collective behaviors of cellular systems. Together groups of cells form intricate structures and accomplish complex tasks. Where an individual cell cycles through growth, ... 
Applications of Optimization and Machine Learning to Healthcare
As the field of healthcare becomes increasingly datadriven, optimization and machine learning methods provide the scientific community and practitioners with powerful tools to extract insights from the data with potential ... 
Multiscale Financial Signal Processing and Machine Learning
Financial time series such as market indices and asset prices are shown to be driven by multiscale factors, ranging from longterm market regimes to rapid fluctuations. Multiscale analysis and signal processing not only ... 
Thermodynamic Principles of Stochastic Dynamics: Time Symmetries and Data Infinitum
Dynamical systems theory has played a central role in applied mathematics for nearly a century. Besides providing a geometric understanding for difference and differential equations, it is also the main framework for ... 
Representations in Biological and Artificial Neural Networks
Remarkably, artificial neural networks (ANNs) have shown astounding success in almost all aspects of artificial intelligence. Meanwhile, large scale experiments have gathered an unprecedented amount of data about the ... 
Understanding Variational Autoencoders and Disentanglement Metrics
In this thesis, we conduct a thorough study of "Variational Autoencoders". We explain the limitations of ``supervised learning" and emphasize the need for ``generative models" to solve complex problems. Variational Autoencoder ... 
Neural Networks for Nonlinear Dynamical Systems
Nonlinear dynamical systems are ubiquitous in many fields of sciences and engineering. Throughout the history, differential equations were widely accepted as effective tools for describing the evolution of such systems in ... 
DataDriven Methods for Time Series Forecasting, Classification, and Uncertainty Quantification
The increased availability of time series data has led to a burgeoning interest in datadriven modeling and time series analysis. The ability to model temporal data can not only enable us to ... 
Affine Structures and Stochastic Thermodynamics on the Space of Measures
Kolmogorov’s theory of probability emphasizes a given state space Ω and a given probabilitymeasure P, then constructs the entire calculus of measurable functions X : Ω → R. From this perspective, the properties and dynamics ... 
A Stochastic RecordValue Approach to Global Simulation Optimization
Blackbox optimization is ubiquitous in machine learning, operations research and engineering simulation. Blackbox optimization algorithms typically do not assume structural information about the objective function and ... 
Invasions with FatTailed and LongDistance Dispersal
Ecologists have recognized fattailed and longdistance dispersal (LDD) as critical to our understanding of population spread and invasions; heavy and fattailed kernels fit empirical dispersal data better than classical ... 
From worms to wars, modeling and controlling networked dynamical systems
Networks in nature regularly exhibit dynamics that are difficult to characterize due to their nonlinear nature and use of obscure control signals. These systems are often marked by lowdimensional dynamics, multiple stable ...