Applied Mathematics Commons^™

Towards Practical Homomorphic Encryption And Efficient Implementation, Gyana R. Sahu

Dissertations

Cloud computing has gained significant traction over the past few years and its application continues to soar as evident from its rapid adoption in various industries. One of the major challenges involved in cloud computing services is the security of sensitive information as cloud servers have been often found to be vulnerable to snooping by malicious adversaries. Such data privacy concerns can be addressed to a greater extent by enforcing cryptographic measures. Fully homomorphic encryption (FHE), a special form of public key encryption has emerged as a primary tool in deploying such cryptographic security assurances without sacrificing many of the …

Binary Neutron Star Mergers: Testing Ejecta Models For High Mass-Ratios, Allen Murray

The Journal of Purdue Undergraduate Research

Neutron stars are extremely dense stellar corpses which sometimes exist in orbiting pairs known as binary neutron star (BNS) systems. The mass ratio (q) of a BNS system is defined as the mass of the heavier neutron star divided by the mass of the lighter neutron star. Over time the neutron stars will inspiral toward one another and produce a merger event. Although rare, these events can be rich sources of observational data due to their many electromagnetic emissions as well as the gravitational waves they produce. The ability to extract physical information from such observations relies heavily on numerical …

An Effective Method For Synthesizing The Abbreviated Disjunctive Normal Form Of A Boolean Function, Erkin Urunbaev

Scientific Journal of Samarkand University

In discrete mathematics, minimizing Boolean functions in the class of disjunctive normal forms is one of the necessary tasks. This paper presents an effective method for synthesizing the reduced disjunctive normal form of a Boolean function.

Diagrams Of ⋆-Trisections, José Román Aranda, Jesse Moeller

Department of Mathematics: Faculty Publications

In this note we provide a generalization for the definition of a trisection of a 4-manifold with boundary. We demonstrate the utility of this more general definition by finding a trisection diagram for the Cacime Surface, and also by finding a trisection-theoretic way to perform logarithmic surgery. In addition, we describe how to perform 1-surgery on closed trisections. The insight gained from this description leads us to the classification of an infinite family of genus three trisections. We include an appendix where we extend two classic results for relative trisections for the case when the trisection surface is closed.

Studies Of Two-Phase Flow With Soluble Surfactant, Ryan Peter Atwater

Dissertations

Numerical methods are developed for accurate solution of two-phase flow in the zero Reynolds number limit of Stokes flow, when surfactant is present on a drop interface and in its bulk phase interior. The methods are designed to achieve high accuracy when the bulk Péclet number is large, or equivalently when the bulk phase surfactant has small diffusivity

In the limit of infinite bulk Péclet number the advection-diffusion equation that governs evolution of surfactant concentration in the bulk is singularly perturbed, indicating a separation of spatial scales. A hybrid numerical method based on a leading order asymptotic reduction in this …

Analysis Of Surface Temperature Trends Of Global Lakes Using Satellite Remote Sensing And In Situ Observations, Christal Jean Soverall, Zahida Yasmin, Mahoutin Godnou, Wen Yong Huang, Ryan Chen, Abdou Bah, Hamidreza Norouzi, Reginald Blake

Publications and Research

Even though lakes make up a small percentage of the water bodies on the global land surface, lakes provide critically important ecosystem services. Unfortunately, however, several lake surface areas around the globe have been changing with many of them drastically decreasing due to climate variability and local mismanagement at the basin-scale level. Lake Surface Water Temperature (LSWT) is recognized as a critical indicator of climate change in lakes. The changes in water and the surrounding land temperatures may be an indicator of climate variability if there is consistency between changes in both temperatures. This project focuses on the application of …

Modeling Braids With Space-Varying And Time-Varying Stranded Cellular Automata, Brian Chan

Mathematical Sciences Technical Reports (MSTR)

Braids in a traditional sense and braids in a mathematical sense are wildly different outlooks on the same concept. Using cellular automata to represent and analyze braids is a way to bridge the gap between them. Joshua and Lana Holden and Hao Yang have previously worked on developing and expanding upon a Stranded Cellular Automata (SCA) model capable of representing many different braids and weaves. Continuing their work, we were able to devise a more user-friendly method for interacting with the model such that even those without a mathematical background can construct and analyze braids of their own. This paper …

Cell Assembly Detection In Low Firing-Rate Spike Train Data, Phan Minh Duc Truong

Mathematics Theses and Dissertations

Cell assemblies, defined as groups of neurons forming temporal spike coordination, are thought to be fundamental units supporting major cognitive functions. However, detecting cell assemblies is challenging since they can occur at a range of time scales and with a range of precisions, from synchronous spikes to co-variations in firing rate. In this dissertation, we use a recently published cell assembly detection (CAD) algorithm that is capable of detecting assemblies at a range of time scales and precisions. We first showed that the CAD method can be applied to sparser spike train data than what have previously been reported. This …

Advection-Reaction-Diffusion Model Of Drug Concentration In A Lymph Node, Ting Yan

Mathematics Theses and Dissertations

It is recognized that there exist reservoirs of HIV located outside the bloodstream, and that these reservoirs hinder the efficacy of antiretroviral medication regimens in combating the virus. The prevailing theories regarding these reservoirs point to the lymphatic system. In this work, we discuss a novel computational model of viral dynamics in the lymph node, to allow numerical studies of viral “reservoirs” causing reinfection. Our model consists of a system of advection-reaction-diffusion partial differential equations (PDEs), where the diffusion coefficients vary between species (virus, drugs, lymphocytes) and include discontinuous jumps to capture differing properties of internal lymph node structures. We …

Why Cutting Trajectories Into Small Pieces Helps To Learn Dynamical Systems Better: A Seemingly Counterintuitive Empirical Result Explained, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In general, the more information we use in machine learning, the more accurate predictions we get. However, recently, it was observed that for prediction of the behavior of dynamical systems, the opposite effect happens: when we replace the original trajectories with shorter pieces -- thus ignoring the information about the system's long-term behavior -- the accuracy of machine learning predictions actually increases. In this paper, we provide an explanation for this seemingly counterintuitive result.

Variable Compact Multi-Point Upscaling Schemes For Anisotropic Diffusion Problems In Three-Dimensions, James Quinlan

Dissertations

Simulation is a useful tool to mitigate risk and uncertainty in subsurface flow models that contain geometrically complex features and in which the permeability field is highly heterogeneous. However, due to the level of detail in the underlying geocellular description, an upscaling procedure is needed to generate a coarsened model that is computationally feasible to perform simulations. These procedures require additional attention when coefficients in the system exhibit full-tensor anisotropy due to heterogeneity or not aligned with the computational grid. In this thesis, we generalize a multi-point finite volume scheme in several ways and benchmark it against the industry-standard routines. …

How The Amount Of Cracks And Potholes Grows With Time: Symmetry-Based Explanation Of Empirical Dependencies, Edgar Daniel Rodriguez Velasquez, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Empirical double-exponential formulas are known that describe how the amount of cracks and potholes in a pavement grows with time. In this paper, we show that these formulas can be explained based on natural symmetries (invariances) -- such as invariance with respect to changing the measuring unit or invariance with respect to changing a starting point for measuring time.

Why 3d Fragmentation Usually Leads To Cuboids: A Simple Geometric Explanation, Laxman Bokati, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

It has been empirically observed that the average shape of natural fragmentation results -- such as natural rock fragments -- is a distorted cube (known as cuboid). Recently, a complex explanation was provides for this empirical fact. In this paper, we propose a simple geometry-based physical explanation for the ubiquity of cuboid fragments.

Two Runners In The Time Of Social Distancing, Speedboats In The Gulf Of Finland: How To Best Pass?, Julio Urenda, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

If two runners follow the same running path, what is the best trajectory for the faster runner to pass the slower one, taking into account that they should always maintain a prescribed social distance? If a speedboat wants to pass a slower ship following a special canal in the Gulf of Finland, what is the best trajectory? In this paper, we provide answers to both questions.

The Similarity Between Earth's And Mars's Core-Mantle Boundary Seems To Be Statistically Significant, Laxman Bokati, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Latest, most accurate measurements of the depth of the Mars's core-mantle boundary shows that the ratio between this depth and Mars's radius is the same as for the Earth -- and with new measurements, this coincidence has become statistically significance. This coincidence seems to confirm a simple scale-invariant model in which for planets of Earth-Mars type, this depth is proportional to the planet's radius. Of course, we need more observations to confirm this model, but the fact that, for the first time, we got a statistically significant confirmation, is encouraging: it makes us believe that this coincidence is not accidental.

Under Limited Resources, Lottery-Based Tutoring Is The Most Efficient, Olga Kosheleva, Christian Servin, Vladik Kreinovich

Departmental Technical Reports (CS)

In the ideal world, every student who needs tutoring should receive intensive one-on-one tutoring. In practical, schools' resources are limited, so the students get only a portion of needed tutoring. It would have been not so bad if, e.g., half-time tutoring would be half as efficient as the intensive one. However, research shows that half-time tutoring is, on average, 15 times less efficient -- and, e.g., for math tutoring 20 times less efficient. To increase the efficiency, we propose to randomly divide the students who need tutoring into equal-size groups, and each year (or each semester) provide intensive tutoring to …

Exploring Food Deserts And Environmental Impacts On Health In Chicago And Oregon, Sivasomasundari Arunarasu, Paulina Grzybowicz

altREU Projects

Food deserts are defined as, “an impoverished area where residents lack access to healthy foods”. This lack of access can be due to a combination of socioeconomic, geographic, and food-related variables, and has been proven to impact the health of residents in the area. In this project, several statistical and machine learning techniques are used to model the impact of food desserts and various other factors on health outcomes, including diabetes and obesity rates, in both the different neighborhoods in the City of Chicago and the various counties in the state of Oregon. The models are then used to determine …

A Novel Correction For The Adjusted Box-Pierce Test — New Risk Factors For Emergency Department Return Visits Within 72 Hours For Children With Respiratory Conditions — General Pediatric Model For Understanding And Predicting Prolonged Length Of Stay, Sidy Danioko

Computational and Data Sciences (PhD) Dissertations

This thesis represents the results of three research projects that underline the breadth and depth of my interests.

Firstly, I devoted some efforts to the well-known Box-Pierce goodness-of-fit tests for time series models which has been an important research topic over the last few decades. All previously proposed tests are focused on changes of the test statistics. Instead, I adopted a different approach that takes the best performing test and modifying the rejection region. Thus, I developed a semiparametric correction of the Adjusted Box-Pierce test that attains the best I error rates for all sample sizes and lags and outperforms …

Combating Covid On College Campuses: The Impact Of Structural Changes On Viral Transmissions, Jared Knofczynski, Aria Killebrew Bruehl, Ben Warner, Ryne Shelton

altREU Projects

One of the most significant issues in the COVID-19 pandemic is the reopening of schools while minimizing the transmission of coronavirus. Opportunities for evaluating the effectiveness of policies that might be utilized at such institutions are limited, as the necessary empirical data has not been gathered yet. Agent-based modeling, where various entities within an environment are simulated as agents, offers an opportunity to examine the effectiveness of various policies in a way that drastically minimizes the health and economic risks involved. Agent-based modeling is common within biology, ecology and other fields; and has seen some use within the coronavirus literature. …

Analyzing Network Topology For Ddos Mitigation Using The Abelian Sandpile Model, Bhavana Panchumarthi, Monroe Ame Stephenson

altREU Projects

A Distributed Denial of Service (DDoS) is a cyber attack, which is capable of triggering a cascading failure in the victim network. While DDoS attacks come in different forms, their general goal is to make a network's service unavailable to its users. A common, but risky, countermeasure is to blackhole or null route the source, or the attacked destination. When a server becomes a blackhole, or referred to as the sink in the paper, the data that is assigned to it "disappears" or gets deleted. Our research shows how mathematical modeling can propose an alternative blackholing strategy that could improve …