Set Theory with Urelements

This dissertation aims to provide a comprehensive account of set theory with urelements. In Chapter 1, I present mathematical and philosophical motivations for studying urelement set theory and lay out the necessary technical preliminaries. Chapter 2 is devoted to the axiomatization of urelement set theory, where I introduce a hierarchy of axioms and discuss how ZFC with urelements should be axiomatized. The breakdown of this hierarchy of axioms in the absence of the Axiom of Choice is also explored. In Chapter 3, I investigate forcing with urelements and develop a new approach that addresses a drawback of the existing machinery. I demonstrate that forcing can preserve, destroy, and recover the axioms isolated in Chapter 2 and discuss how Boolean ultrapowers can be applied in urelement set theory. Chapter 4 delves into class theory with urelements. I first discuss the issue of axiomatizing urelement class theory and then explore the second-order reflection principle with urelements. In particular, assuming large cardinals, I construct a model of second-order reflection where the principle of limitation of size fails.Comment: arXiv admin note: text overlap with arXiv:2212.13627. Definition 15 in the previous versions is flawed, which is fixed in this versio

An asymptotic preserving scheme for kinetic models with singular limit

We propose a new class of asymptotic preserving schemes to solve kinetic equations with mono-kinetic singular limit. The main idea to deal with the singularity is to transform the equations by appropriate scalings in velocity. In particular, we study two biologically related kinetic systems. We derive the scaling factors and prove that the rescaled solution does not have a singular limit, under appropriate spatial non-oscillatory assumptions, which can be verified numerically by a newly developed asymptotic preserving scheme. We set up a few numerical experiments to demonstrate the accuracy, stability, efficiency and asymptotic preserving property of the schemes.Comment: 24 pages, 6 figure

Broad Learning for Healthcare

A broad spectrum of data from different modalities are generated in the healthcare domain every day, including scalar data (e.g., clinical measures collected at hospitals), tensor data (e.g., neuroimages analyzed by research institutes), graph data (e.g., brain connectivity networks), and sequence data (e.g., digital footprints recorded on smart sensors). Capability for modeling information from these heterogeneous data sources is potentially transformative for investigating disease mechanisms and for informing therapeutic interventions. Our works in this thesis attempt to facilitate healthcare applications in the setting of broad learning which focuses on fusing heterogeneous data sources for a variety of synergistic knowledge discovery and machine learning tasks. We are generally interested in computer-aided diagnosis, precision medicine, and mobile health by creating accurate user profiles which include important biomarkers, brain connectivity patterns, and latent representations. In particular, our works involve four different data mining problems with application to the healthcare domain: multi-view feature selection, subgraph pattern mining, brain network embedding, and multi-view sequence prediction.Comment: PhD Thesis, University of Illinois at Chicago, March 201

Forcing with Urelements

ZFCU$_{\rm R}$ is ZFC (with the Replacement Scheme) modified to allow a class of urelements. I first isolate a hierarchy of axioms based on ZFCU$_{\rm R}$ and argue that the Collection Principle should be included as an axiom in order to obtain a more robust set theory with urelements. I then turn to forcing over countable transitive models of ZFCU$_{\rm R}$. A new definition of $\mathbb{P}$-names is given. The resulting forcing relation is full just in case the Collection Principle holds in the ground model. While forcing preserves ZFCU$_{\rm R}$ and many axioms in the hierarchy, it can also destroy the DC$_{\omega_1}$-scheme and recover the Collection Principle. The ground model definability fails when the ground model contains a proper class of urelements

Human-Style Text Parsing System

The present disclosure relates to a text parsing system and related method for accurately parsing the content of text in messages and providing an output that can be used by various systems including systems used to detect spam and advertising content. The text parsing system can include a computing system that can parse the content of text (e.g., using a computing system including a machine-learned model or a rules based text parsing system) and provide an output including a list of potential parsed words along with associated word types, language, and confidence of word matching. Furthermore, the text parsing system can further determine the content of text through use of a knowledge base that includes a structured data repository represented as a graph. The knowledge base can be used to generate further output associated with the content of the text including related information drawn from the knowledge base

HitFraud: A Broad Learning Approach for Collective Fraud Detection in Heterogeneous Information Networks

On electronic game platforms, different payment transactions have different levels of risk. Risk is generally higher for digital goods in e-commerce. However, it differs based on product and its popularity, the offer type (packaged game, virtual currency to a game or subscription service), storefront and geography. Existing fraud policies and models make decisions independently for each transaction based on transaction attributes, payment velocities, user characteristics, and other relevant information. However, suspicious transactions may still evade detection and hence we propose a broad learning approach leveraging a graph based perspective to uncover relationships among suspicious transactions, i.e., inter-transaction dependency. Our focus is to detect suspicious transactions by capturing common fraudulent behaviors that would not be considered suspicious when being considered in isolation. In this paper, we present HitFraud that leverages heterogeneous information networks for collective fraud detection by exploring correlated and fast evolving fraudulent behaviors. First, a heterogeneous information network is designed to link entities of interest in the transaction database via different semantics. Then, graph based features are efficiently discovered from the network exploiting the concept of meta-paths, and decisions on frauds are made collectively on test instances. Experiments on real-world payment transaction data from Electronic Arts demonstrate that the prediction performance is effectively boosted by HitFraud with fast convergence where the computation of meta-path based features is largely optimized. Notably, recall can be improved up to 7.93% and F-score 4.62% compared to baselines.Comment: ICDM 201