Computer Sciences Commons^™

Egyptian Triangle And Geometry Of Airplane Wings: A Simplified Explanation, Julio C. Urenda, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In historically first planes, wings were orthogonal to the fuselage. However, later it turned out that from the aerodynamic viewpoint, it is most efficient to place the wings at about 37 degrees from this orthogonal direction -- and this is where wings are placed in most modern planes. There exist theoretical explanations for this optimality -- explanations based on solving the equations of aerodynamics. In such situations when only a complex not-very-intuitive explanation exists, it is desirable to come up with a simpler more intuitive explanation. For the wing angles, such an explanation is provided in this paper. Namely, we …

"At Least K Out Of N" Under Fuzzy Uncertainty: Efficient Algorithm For General "And"-Operations, Olga Kosheleva, Vladik Kreinovich, Klaus-Peter Adlassnig

Departmental Technical Reports (CS)

In medicine, many diagnoses are made when, for some value k, at least k of n possible symptoms are present. Many of such symptoms -- such as fever -- are, in reality, fuzzy. For example, it makes no sense that say that 38.0 is fever while 37.9 is not a fever, both are fever to some degree. Once such degrees are given, we need to use them to estimate the degree to which the patient has the corresponding disease. For this problem, the usual fuzzy techniques require exponentially many computational steps -- so it is desirable to have a more …

Why Interval-Valued (And Type-2) Fuzzy Methods Are Often More Effective, Christian Servin, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Interval-valued and type-2 fuzzy techniques were designed to provide a more adequate representation of expert knowledge than the traditional (type-1) fuzzy techniques. Somewhat unexpectedly, they also often turn out to be more effective even when there is no expert knowledge at all -- when we are simply using fuzzy rules to fit experimental data. In precise terms, for the same number of parameters, interval-valued and type-2 systems often provide a better fit for the data and/or better quality control than traditional (type-1) fuzzy techniques. In this paper, we provide a theoretical explanation for this surprising phenomenon.

Why Convex Combinations Of Interval Endpoints: Related Explanations For Cases Of Data Processing And Decision Making, Christian Servin, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

There are two cases in which it has been empirically shown that a convex combination of the interval's endpoints works better than any other combination: processing interval data and dealing with situations in which we know both approximate probability and possibility and we need to make a decision. In this paper, we provide an explanation of both phenomena.

Stability Analysis In The Twist-Bend Nematic Liquid Crystal Model, Zhenqiang Li

Mathematics & Statistics Theses & Dissertations

The recently discovered twist-bend nematic liquid crystal (LC) phase is characterized by a nanoscale helical modulation of the nematic director n, forming a conical helix along the z-axis at an oblique angle θ. While many models assume a constant cone angle and equal elastic constants K11 = K22 = K33, this dissertation removes both assumptions by considering a fully anisotropic elastic energy with K11 ≠ K22 ≠ K33, and allowing θ to vary spatially. We analyze the stability of this system under frustrated and free boundary conditions using variational methods. …

Algorithms For Order Statistics In Farey Sequences: A Computational Study, Connor Weyers

School of Computing: Dissertations, Theses, and Student Research

Farey sequences are the sets of irreducible fractions in increasing order with denominator less or equal to some integer n. They are a well-known concept in number theory problems and are related to many other concepts in number theory including integer factoring, Fibonacci sequences, and Riemann’s Zeta function. In this paper, we investigate some known algorithms to solve certain problems in Farey sequences from a computational perspective. In particular, we implement established algorithms that have not been previously implemented with the goal of creating a package that can be used more broadly. We also develop a new algorithm for rational …

Robust Spacecraft Autonomy For Deep Space Exploration In Special Euclidean Group Se(3), Matthew Wittal

Doctoral Dissertations and Master's Theses

Over the past half-century, humanity has gained extensive experience conducting manned spaceflight near Earth. Arguably, "near Earth" could even include the Moon — the most distant destination humans have reached. However, "near" in this work primarily refers low Earth orbit (LEO). One could argue that we have not truly left Earth since the Apollo, as spacecraft in some LEOs remain subject to atmospheric drag thus emphasizing their continued connection to Earth's immediate environment. Reflecting on this, it becomes clear that humanity has largely remained bound to Earth’s immediate vicinity since the Apollo missions reached the Moon. However, that is set …

From Machine Learning To Human Learning: What Can Pedagogy Learn From Ai Successes, Victor L. Timchenko, Yury P. Kondratenko, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Many machine learning techniques -- including many techniques behind the current AI-based boom in machine learning -- come from the analysis of successful human learning strategies (and researchers expect that other human learning experiences can lead to even more effective AI-based systems). At this moment, so much experience have been accumulated in AI-based machine learning that it is time to start the analysis in the opposite direction -- to see what can human-based pedagogy learn from AI successes. In this chapter, we provide the first results of such an analysis -- some of which go somewhat against the current pedagogical …

Gurevich's Quizani Dialogs As An Example Of Explainable Mathematics, And How This Is Related To Quantum Space-Time Ideas That Can Speed Up Computations, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Everyone talks about the need for Explainable AI -- when, to supplement a long difficult-to-understand sequence of computational steps leading to AI's decision, we are looking for a shorter and understandable more-informal explanation for this decision. In this paper, we argue that this need is a particular case of what we call Explainable Mathematics -- when we want to supplement a long sequence of arguments and/or computations with a shorter and understandable more-informal explanation. Important instances of Explainable Mathematics are Yuri Gurevich's Quizani dialogs that help explain complex results from theoretical computer science and physicists' more-informal explanations of complex physical …

Unfortunately, The Universal Predictor Cannot Be Made Constructive, Julio C. Urenda, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

A recent article in the Notices of the American Mathematical Society reminded the mathematics community that, under the Axiom of Choice, it is possible to have a universal predictor: if we input, into this predictor, the values of a function for all moments t < to for some to, then, for almost all to, this predictor correctly predicts the next values of this function on some interval [to, to + ε). This predictor cannot be used for actual predictions: it is based on the Axiom of Choice and is, therefore, not constructive. A natural question is: maybe it is possible to have another universal predictor, which is constructive? In this paper we show that, unfortunately, it is not possible to have a constructive universal predictor. In other words, the above universal predictor result cannot be used for actual predictions.

Best Strategies For Bilingual Education: How Can We Explain Their Success?, Claudia Cabrera, Olga Kosheleva, Christian Servin, Vladik Kreinovich

Departmental Technical Reports (CS)

When designing AI-based tools for education, it is important to take into account the experience of human teachers. In this, it is necessary to distinguish between the education features that are justified by the general features of the corresponding education task -- these features should be taken into account in AI-based learning as well -- and features which are specific for traditional non-AI teaching. In this paper, on the important example of bilingual education, we show that several empirically successful teaching strategies can be explained in the general context -- and thus, should be implemented in AI-based teaching as well.

Ultrathin-Layer Strain-Based Electronic Devices: From-First-Principles Derivation Of The Corresponding Equation, Julio C. Urenda, Vladik Kreinovich

Departmental Technical Reports (CS)

Most information about the world comes from sensors -- and from the results of processing sensor data. In many practical situations -- e.g., in biomedical applications -- it is desirable to make sure that the sensors are as "invisible" as possible, in particular, that they are as small as possible. One way to achieve such small size is to use ultrathin-layer materials such as graphene. It is known that for such materials, strain causes electromagnetic effects -- which can be used to detect small strains. Interestingly, it turned out that the same equation describes the relation between strain and electric …

A Natural Extension Of F-Transform To Triangular And Triangulated Domains Necessitates The Use Of Triangular Membership Functions, Hana Zámečiková, Irina Perfilieva, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practical situations when we process 1-D data, the method of F-transform turned out to be very useful. In this method, we can use either triangular membership functions or more complex ones. Because this method has been so successful in 1-D applications, a natural idea is to extend it to functions defined on 2-D and higher-dimensional domains -- e.g., to images. This method allows natural generalization to rectangular domains, where it indeed turned out to be very effective. A recent paper showed that it can extended to more general domains -- e.g., to triangular domains and to more general …

Translation Of: Dupin'sche Hyperflächen In E^4, Manuscripta Math By Ulrich Pinkall, Thomas E. Cecil

Mathematics and Computer Science Department Faculty Scholarship

This is an English translation of the article "Dupin'sche Hyperflächen in E⁴" by Ulrich Pinkall, which was originally published in manuscripta math. 51 (1985), 89-119.

A note from Thomas E. Cecil, translator: This is an unofficial translation of the original paper which was written in German. All references should be made to the original paper.

Mathematics In Everyday Life: Exploring Practical Applications And Real-World Impact, Priyant Banerjee, Arshad Bhat

Himalayan Research Papers Archive

Mathematics is an essential part of daily life and influences decisions and problem-solving in various aspects of life. This study explores how mathematical concepts are embedded in daily activities such as financial management, cooking, travel planning, and technological interactions. We will show how arithmetic, algebra, geometry, and statistics are applied in real life to improve decision-making, efficiency, and productivity. Findings indicate that people with higher mathematical literacy solve problems more efficiently, especially in budgeting, as accurate calculations minimize financial mistakes and facilitate long-term financial planning. In cooking, proportional reasoning ensures the accuracy of recipes, thus providing consistent culinary results. Travel …

An Algorithm And Computation To Verify Legendre's Conjecture Up 7 · 1013, Jonathan Sorenson, Jonathan Webster

Computer Science and Software Engineering

We state a general purpose algorithm for quickly finding primes in evenly divided sub-intervals. Legendre’s conjecture claims that for every positive integer n, there exists a prime between n² and (n + 1)². Oppermann’s conjecture subsumes Legendre’s conjecture by claiming there are primes between n2 and n(n + 1) and also between n(n + 1) and (n + 1)². Using Cramér’s conjecture as the basis for a heuristic run-time analysis, we show that our algorithm can verify Oppermann’s conjecture, and hence also Legendre’s conjecture, for all n ≤ N in time O(N log N log …

On The Hölder Continuity Of The Brascamp-Lieb Constant, Ori Friesen

Mathematics, Statistics, and Computer Science Honors Projects

The Brascamp-Lieb inequality is a generalization of many well-known multilinear functional inequalities. The Brascamp-Lieb constant is the best constant that works for the Brascamp-Lieb inequality for a given tuple of input linear maps and powers. If we keep the powers constant while varying the input linear maps, the Brascamp-Lieb constant becomes a function of the linear maps. In this thesis, we explore the Hölder continuity of the Brascamp-Lieb constant. Specifically,we prove that the general 4-linear case of the Brascamp-Lieb inequality is locally Lipschitz continuous. Additionally, we provide an improvement of a previous result on the local Hölder continuity of the …

All We (And Llms) Need Is Fuzzy: An Argument, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Large Language Models (LLMs) like ChatGPT have spectacular successes -- but they also have surprising failures that an average person with common sense could easily avoid. It is therefore desirable to incorporate the imprecise ("fuzzy") common sense into LLMs. A natural question is: to what extent will this help? This way, we may avoid a few simple mistakes, but will it significantly improve the LLMs' performance? What portion of the gap between current LLMs and ideal perfect AI-based agents can be, in principle, covered by using fuzzy techniques? Judging by the fact that few researchers working on LLMs (and on …

How To Share A Success, How To Share A Crisis, And How All This Is Related To Fuzzy, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practical situations, a group of people needs to share a success. What is the fair way to share this success? Nobelist John Nash showed that under reasonable conditions, the group should select the alternative for which the product of utility gains is the largest possible. This solution makes perfect sense from the fuzzy-formalized commonsense viewpoint: it maximizes the degree of confidence that all participants are happy. A natural question is: can we extend this result to a different class of situations, when a group of people needs to share sacrifices caused by a crisis? In this paper, we …

How To Deal With High-Impact Low-Probability Events: Theoretical Explanation Of The Empirically Successful Fuzzy-Like Technique, Juan Ulloa, Aaron Velasco, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

When making decisions, it is important to take into account high-impact low-probability events. For such events, traditional probability-based approach -- which considers the product of the probability p that this event happens and the probability P that a randomly selected building will be destroyed -- often underestimates risks. Available data has lead to an empirical table that provides a more adequate risk estimate. Most of the entries in this table correspond to the fuzzy-like formula min(p,P). This paper explains this empirical result. Specifically, it explains both the effectiveness of the min formula -- and also explains deviations from this formula.

Gaps In Knowledge: Topological Insights Into The Structure Of Science, Gavin Engelstad

Mathematics, Statistics, and Computer Science Honors Projects

Understanding scientific development is essential to ascertaining the mechanisms leading us into the future. Building this understanding requires both methodological developments and empirical research. This thesis contributes in both aspects using a topological approach to examine scientific knowledge. The first section presents a new algorithm to find optimal cycle representatives for homological features in complex networks, a context for which we demonstrate existing algorithms can be inadequate. The second section applies a number of topological methods, including our cycle optimization algorithm, to data on individual scientific fields, demonstrating the value of topological approaches and highlighting new insights about how science …

Ms-Yolo: Infrared Object Detection For Edge Deployment Via Mobilenetv4 And Slideloss, Jiali Zhang, Thomas S. White, Haoliang Zhang, Wenqing Hu, Donald C. Wunsch, Jian Liu

Mathematics and Statistics Faculty Research & Creative Works

Infrared imaging has emerged as a robust solution for urban object detection under low-light and adverse weather conditions, offering significant advantages over traditional visible-light cameras. However, challenges such as class imbalance, thermal noise, and computational constraints can significantly hinder model performance in practical settings. To address these issues, we evaluate multiple YOLO variants on the FLIR ADAS V2 dataset, ultimately selecting YOLOv8 as our baseline due to its balanced accuracy and efficiency. Building on this foundation, we present MS-YOLO (MobileNetv4 and SlideLoss based on YOLO), which replaces YOLOv8's CSPDarknet backbone with the more efficient MobileNetV4, reducing computational overhead by 1.5% …

Optimizing Decision-Making In A Cerebral Palsy Model Using Reinforcement Learning, Richard Ampah

Pitzer Senior Theses

This study presents an original interdisciplinary investigation into how reinforcement learning (RL) can model motor and cognitive defects and potentially improve motor and cognitive functions in individuals with cerebral palsy (CP), a non-progressive neurological disorder that impairs movement and adaptability. Integrating computational neuroscience and machine learning, the research applies policy gradient methods and Markov Decision Processes (MDPs) to simulate adaptive learning in agents with and without CP-related constraints.

The central aim is to compare the cumulative rewards of optimal policies, derived from value iteration, and human-like learning policies using the REINFORCE algorithm, both with and without the Bellman baseline. The …

Empirical Analysis Of Political Districting Splitability Via Uniform Spanning Trees In Polynomial Time, Brooke C. Feinberg

Scripps Senior Theses

This work expands a recently proven conjecture that a polynomial fraction of all uniform spanning trees (USTs) are splittable into k balanced partitions on grid graphs to real-world political districting plans. We investigate whether similar structural properties hold for the planar dual graphs of U.S. counties (cnty) and tracts (t), using Wilson’s algorithm to generate uniform random spanning trees and Breadth- First Search (BFS) to check for splitability into balanced partitions. Our empirical findings suggest that real-world districting plans can be split into 2-balanced, connected partitions in a fraction of polynomial time. This result highlights the potential for scalable redistricting …

The Impact Of Loss Function Topology On Gradient Descent, Robert B. Skudnig Jr.

Theses and Dissertations

Gradient descent is a popular optimization method that utilizes a model’s prediction error to iteratively improve its parameters for a given task. The functions that measure this error can be defined to align with the user’s goals and sometimes satisfy metric or norm properties. It is common for these functions to measure over Rn, but any differentiable space allows for gradient descent to occur. There has been some research investigating the influence of topological spaces on optimization methods, but it is a limited field of study. This thesis further explores this phenomenon by applying a transformation prediction model to multiple …

Novel Generative And Language Model Architectures With Applications, Edison Mucllari

Theses and Dissertations--Mathematics

This dissertation investigates novel architectures to address fundamental challenges in machine learning, particularly focusing on transformer models, recurrent neural networks, GAN and continual learning and their applications in natural language processing and computer vision. We propose the Neumann-Cayley Gated Recurrent Unit (NC-GRU), which leverages a Neumann series-based Scaled Cayley transformation to maintain orthogonal weight matrices, effectively mitigating exploding gradients problems while improving long-term memory retention across prediction tasks. We demonstrate the practical applications of NC-GRU by implementing our proposed architecture into an autoencoder to derive neural molecular fingerprints. Building upon these advancements, we turn our attention to the transformer architecture, …

Action This Day: The Mathematics And Machinations That Bested The German Enigma, Jonah Weinbaum

Dartmouth College Master’s Theses

This thesis presents a comprehensive and chronological overview of cryptographic techniques designed to break Enigma, beginning in 1932 and culminating in the creation of the Turing-Welchman Bombe. We discuss the mathematical theory and electromechanical implements used to decode one of history's greatest ciphers.

Reexamining the Bombe through the lens of modern group theory, we critique Alan Turing's estimation of the number of "stops" that the Bombe produces for various plaintext-ciphertext pairing structures. To address its limitations, we introduce a new framework for estimating the number of stops by extending John Dixon's theorem concerning the probability that uniformly distributed elements of …

Machine Learning Methods For Intrusion Detection And Response In Network Security, Ayomide Oyemaja

College of Graduate Studies: Theses & Dissertations

Intrusion Detection Systems (IDS) play a crucial role in computer network security by identifying malicious activities and potential cyberattacks. This thesis combines machine learning and cybersecurity by applying Reinforcement Learning (RL) in intrusion detection and response using the NSL-KDD dataset.

We designed and implemented a Q-learning framework where an agent learns to classify network traffic over time by interacting with the environment and receiving rewards based on detection accuracy. We also look at the importance of feature selection and classification techniques and how effective they are in improving model performance, reducing the complexity of computation, and producing more desirable results. …

Isochronous And Period-Doubling Diagrams For Symplectic Maps Of The Plane, T. Zolkin, S. Nagaitsev, I. Morozov, S. Kladov, Y. -K. Kim

Physics Faculty Publications

Symplectic mappings of the plane serve as key models for exploring the fundamental nature of complex behavior in nonlinear systems. Central to this exploration is the effective visualization of stability regimes, which enables the interpretation of how systems evolve under varying conditions. While the area-preserving quadratic Hénon map has received significant theoretical attention, a comprehensive description of its mixed parameter-space dynamics remain lacking. This limitation arises from early attempts to reduce the full two-dimensional phase space to a one-dimensional projection, a simplification that resulted in the loss of important dynamical features. Consequently, there is a clear need for a more …

Extremal Trees For Random Walks, Ben Bridenbaugh

Mathematics, Statistics, and Computer Science Honors Projects

A random walk is a sequence of adjacent vertices that are chosen uniformly at random from the neighbors of the previous vertex. An access time is the average length of time that a random walk takes to reach a target probability distribution from a starting probability distribution, given an optimal stopping rule. This paper deals with characterizing the trees of diameter d and on n vertices that extremize three different types of access times.