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Articles 1 - 30 of 79
Full-Text Articles in Mathematics
Coarse-Gridded Simulation Of The Nonlinear Schrödinger Equation With Machine Learning, Benjamin F. Akers, Kristina O. F. Williams
Coarse-Gridded Simulation Of The Nonlinear Schrödinger Equation With Machine Learning, Benjamin F. Akers, Kristina O. F. Williams
Faculty Publications
A numerical method for evolving the nonlinear Schrödinger equation on a coarse spatial grid is developed. This trains a neural network to generate the optimal stencil weights to discretize the second derivative of solutions to the nonlinear Schrödinger equation. The neural network is embedded in a symmetric matrix to control the scheme’s eigenvalues, ensuring stability. The machine-learned method can outperform both its parent finite difference method and a Fourier spectral method. The trained scheme has the same asymptotic operation cost as its parent finite difference method after training. Unlike traditional methods, the performance depends on how close the initial data …
Some Studies On Mathematical Morphology In Remotely Sensed Data Analysis, Geetika Barman
Some Studies On Mathematical Morphology In Remotely Sensed Data Analysis, Geetika Barman
Doctoral Theses
The application of Mathematical Morphology (MM) techniques has proven to be beneficial in the extraction of shapebased and texture-based features during remote sensing image analysis. The characteristics of these techniques, such as nonlinear adaptability and comprehensive lattice structure, make them useful for contextual spatial feature analysis. Despite the advancements, there are still persistent challenges, including the curse of dimensionality, maintaining spatial correlation, and the adaptability of morphological operators in higher dimensions. The focus of this thesis is to explore the potential of MM-based methods to analyse spatial features in addressing these challenges, specifically in the context of spatialcontextual feature analysis …
Effective Wordle Heuristics, Ronald I. Greenberg
Effective Wordle Heuristics, Ronald I. Greenberg
Computer Science: Faculty Publications and Other Works
While previous researchers have performed an exhaustive search to determine an optimal Wordle strategy, that computation is very time consuming and produced a strategy using words that are unfamiliar to most people. With Wordle solutions being gradually eliminated (with a new puzzle each day and no reuse), an improved strategy could be generated each day, but the computation time makes a daily exhaustive search impractical. This paper shows that simple heuristics allow for fast generation of effective strategies and that little is lost by guessing only words that are possible solution words rather than more obscure words.
Sociomathematical Norms And Automated Proof Checking In Mathematical Education: Reflections And Experiences, Merlin Carl
Sociomathematical Norms And Automated Proof Checking In Mathematical Education: Reflections And Experiences, Merlin Carl
Journal of Humanistic Mathematics
According to a widely held view, mathematical proofs are essentially (indications of) formal derivations, and thus in principle mechanically checkable (this view is defended, for example, by Azzouni [3]). This should in particular hold for the kind of simple proof exercises typically given to students of mathematics learning to write proofs. If that is so, then automated proof checking should be an attractive option for math education at the undergraduate level. An opposing view would be that mathematical proofs are social objects and that what constitutes a mathematical proof can thus not be separated from the social context in which …
Stochastic Dominance: Cases Of Interval And P-Box Uncertainty, Kittawit Autchariyapanikul, Olga Kosheleva, Vladik Kreinovich
Stochastic Dominance: Cases Of Interval And P-Box Uncertainty, Kittawit Autchariyapanikul, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
Traditional decision theory recommendation about making a decision assume that we know both the probabilities of different outcomes of each possible decision, and we know the utility function -- that describes the decision maker's preferences. Sometimes, we can make a recommendation even when we only have partial information about utility. Such cases are known as cases of stochastic dominance. In other cases, in addition to not knowing the utility function, we also only have partial information about the probabilities of different outcomes. For example, we may only known bounds on the outcomes (case of interval uncertainty) or bounds on the …
If Subsequent Results Are Too Easy To Obtain, The Proof Most Probably Has Errors: Explanation Of The Empirical Observation, Olga Kosheleva, Vladik Kreinovich
If Subsequent Results Are Too Easy To Obtain, The Proof Most Probably Has Errors: Explanation Of The Empirical Observation, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
Many modern mathematical proofs are very complex, checking them is difficult; as a result, errors sneak into published proofs, even into proofs published in highly reputable journals. Sometimes, the errors are repairable, but sometimes, it turns out that the supposedly proven result is actually wrong. When the error is not noticed for some time, the published result is used to prove many other results -- and when the error is eventually found, all these new results are invalidated. This happened several times. Since it is not realistic to more thoroughly check all the proofs, and we want to minimize the …
For 2 X N Cases, Proportional Fitting Problem Reduces To A Single Equation, Olga Kosheleva, Vladik Kreinovich
For 2 X N Cases, Proportional Fitting Problem Reduces To A Single Equation, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
In many practical situations, for each of two classifications, we know the probabilities that a randomly selected object belong to different categories. For example, we know what proportion of people are below 20 years old, what proportion is between 20 and 30, etc., and we also know what proportion of people earns less than 10K, between 10K and 20K, etc. In such situations, we are often interested in proportion of people who are classified by two classifications into two given categories. For example, we are interested in the proportion of people whose age is between 20 and 30 and whose …
Is Alaska Negative-Tax Arrangement Fair? Almost: Mathematical Analysis, Chon Van Le, Vladik Kreinovich
Is Alaska Negative-Tax Arrangement Fair? Almost: Mathematical Analysis, Chon Van Le, Vladik Kreinovich
Departmental Technical Reports (CS)
In the State of Alaska there is no state income tax. Instead, there is a negative tex: every year every resident gets some money from the state. At present, every resident -- from the poorest to the richest -- gets the exact same amount of money: in 2024, it is expected to be around $1500. A natural question is: Is this fair? Maybe poor people should get more since their needs are greater? Maybe the rich people should get proportionally more, since fairness means equal added happiness for all, and for rich people, extra $1500 is barely noticeable? There have …
Why Angles Between Galactic Center Filaments And Galactic Plane Follow A Bimodal Distribution: A Symmetry-Based Explanation, Julio C. Urenda, Vladik Kreinovich
Why Angles Between Galactic Center Filaments And Galactic Plane Follow A Bimodal Distribution: A Symmetry-Based Explanation, Julio C. Urenda, Vladik Kreinovich
Departmental Technical Reports (CS)
Recent observations have shown that the angles between the Galaxy Center filaments and the Galactic plane follow a bimodal distribution: a large number of filaments are approximately orthogonal to the Galactic plane, a large number of filaments are approximately parallel to the Galactic plane, and much fewer filaments have other orientations. In this paper, we show this bimodal distribution can be explained by natural geometric symmetries.
Why Seismicity In Ireland Is Low: A Possible Geometric Explanation, Julio C. Urenda, Aaron Velasco, Vladik Kreinovich
Why Seismicity In Ireland Is Low: A Possible Geometric Explanation, Julio C. Urenda, Aaron Velasco, Vladik Kreinovich
Departmental Technical Reports (CS)
For each geographic location, its seismicity level is usually determined by how close this location is to the boundaries of tectonic plates. However, there is one notable exception: while Ireland and Britain are at approximately the same distance from such boundaries, the seismicity level in Ireland is much lower than in Britain. A recent paper provided a partial explanation for this phenomenon: namely, it turns out that the lithosphere under Ireland is unusually thick, and this can potentially lead to lower seismicity. However, the current explanation of the relation between the lithosphere thickness and seismicity level strongly depends on the …
Enhancing Tumor Classification Through Machine Learning Algorithms For Breast Cancer Diagnosis, Lawrence Agbota, Edmund Agyemang, Priscilla Kissi-Appiah, Lateef Moshood, Akua Osei- Nkwantabisa, Vincent Agbenyeavu, Abraham Nsiah, Augustina Adjei
Enhancing Tumor Classification Through Machine Learning Algorithms For Breast Cancer Diagnosis, Lawrence Agbota, Edmund Agyemang, Priscilla Kissi-Appiah, Lateef Moshood, Akua Osei- Nkwantabisa, Vincent Agbenyeavu, Abraham Nsiah, Augustina Adjei
School of Mathematical and Statistical Sciences Faculty Publications and Presentations
In cancer diagnosis, machine learning helps improve cancer detection by providing doctors with a second perspective and allowing for faster and more accurate determination and decisions. Numerous studies have used both classic machine learning approaches and deep learning to address cancer classification. In this study, we examine the efficacy of five commonly used machine learning algorithms; both traditional and deep learning models namely, Logistic Regression, Support Vector Machines (SVM), Random Forest (RF), Decision Tree and Deep Neural Networks (DNN). We analyze their ability to properly classify tumors as Benign or Malignant using the Wisconsin breast cancer dataset (WBCD). Random Forest …
For Discrete-Time Linear Dynamical Systems Under Interval Uncertainty, Predicting Two Moments Ahead Is Np-Hard, Luc Jaulin, Olga Kosheleva, Vladik Kreinovich
For Discrete-Time Linear Dynamical Systems Under Interval Uncertainty, Predicting Two Moments Ahead Is Np-Hard, Luc Jaulin, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
In the first approximation, when changes are small, most real-world systems are described by linear dynamical equations. If we know the initial state of the system, and we know its dynamics, then we can, in principle, predict the system's state many moments ahead. In practice, however, we usually know both the initial state and the coefficients of the system's dynamics with some uncertainty. Frequently, we encounter interval uncertainty, when for each parameter, we only know its range, but we have no information about the probability of different values from this range. In such situations, we want to know the range …
What To Do If An Inflexible Tolerance Problem Has No Solutions: Probabilistic Justification Of Piegat's Semi-Heuristic Idea, Olga Kosheleva, Vladik Kreinovich
What To Do If An Inflexible Tolerance Problem Has No Solutions: Probabilistic Justification Of Piegat's Semi-Heuristic Idea, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
In many practical situations, it is desirable to select the control parameters x1, ..., xn in such a way that the resulting quantities y1, ..., ym of the system lie within desired ranges. In such situations, we usually know the general formulas describing the dependence of yi on xj, but the coefficients of these formulas are usually only known with interval uncertainty. In such a situation, we want to find the tuples for which all yi's are in the desired intervals for all possible tuples of coefficients. But what if no such parameters are possible? Since we cannot guarantee the …
How To Make Ai More Reliable, Olga Kosheleva, Vladik Kreinovich
How To Make Ai More Reliable, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
One of the reasons why the results of the current AI methods (especially deep-learning-based methods) are not absolutely reliable is that, in contrast to more traditional data processing techniques which are based on solid mathematical and statistical foundations, modern AI techniques use a lot of semi-heuristic methods. These methods have been, in many cases, empirically successful, but the absence of solid justification makes us less certain that these methods will work in other cases as well. To make AI more reliable, it is therefore necessary to provide mathematical foundations for the current semi-heuristic techniques. In this paper, we show that …
Why Magenta Is Not A Real Color, And How It Is Related To Fuzzy Control And Quantum Computing, Victor L. Timchenko, Yuriy P. Kondratenko, Olga Kosheleva, Vladik Kreinovich
Why Magenta Is Not A Real Color, And How It Is Related To Fuzzy Control And Quantum Computing, Victor L. Timchenko, Yuriy P. Kondratenko, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
It is well known that every color can be represented as a combination of three basic colors: red, green, and blue. In particular, we can get several colors by combining two of the basic colors. Interestingly, while a combination of two neighboring colors leads to a color that corresponds to a certain frequency, the combination of two non-neighboring colors -- red and blue -- leads to magenta, a color that does not correspond to any frequency. In this paper, we provide a simple explanation for this phenomenon, and we also show that a similar phenomenon happens in two other areas …
How To Propagate Uncertainty Via Ai Algorithms, Olga Kosheleva, Vladik Kreinovich
How To Propagate Uncertainty Via Ai Algorithms, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
Any data processing starts with measurement results. Measurement results are never absolutely accurate. Because of this measurement uncertainty, the results of processing measurement results are, in general, somewhat different from what we would have obtained if we knew the exact values of the measured quantities. To make a decision based on the result of data processing, we need to know how accurate is this result, i.e., we need to propagate the measurement uncertainty through the data processing algorithm. There are many techniques for uncertainty propagation. Usually, they involve applying the same data processing algorithm several times to appropriately modified data. …
Why Empirical Membership Functions Are Well-Approximated By Piecewise Quadratic Functions: Theoretical Explanation For Empirical Formulas Of Novak's Fuzzy Natural Logic, Olga Kosheleva, Vladik Kreinovich
Why Empirical Membership Functions Are Well-Approximated By Piecewise Quadratic Functions: Theoretical Explanation For Empirical Formulas Of Novak's Fuzzy Natural Logic, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
Empirical analysis shows that membership functions describing expert opinions have a shape that is well described by a smooth combination of two quadratic segments. In this paper, we provide a theoretical explanation for this empirical phenomenon.
Why Fully Consistent Quantum Field Theories Require That The Space-Time Be At Least 10-Dimensional: A Commonsense Field-Based Explanation, Olga Kosheleva, Vladik Kreinovich
Why Fully Consistent Quantum Field Theories Require That The Space-Time Be At Least 10-Dimensional: A Commonsense Field-Based Explanation, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
It is known that quantum field theories that describe fields in our usual 4-dimensional space-times are not fully consistent: they predict meaningless infinite values for some physical quantities. There are some known tricks to avoid such infinities, but it is definitely desirable to have a fully consistent theory, a theory that would produce correct results without having to use additional tricks. It turns out that the only way to have such a theory is to consider space-times of higher dimensions, the smallest of which is 10. There are complex mathematical reasons for why 10 is the smallest such dimension. However, …
Why Is Grade Distribution Often Bimodal? Why Individualized Teaching Adds Two Sigmas To The Average Grade? And How Are These Facts Related?, Christian Servin, Olga Kosheleva, Vladik Kreinovich
Why Is Grade Distribution Often Bimodal? Why Individualized Teaching Adds Two Sigmas To The Average Grade? And How Are These Facts Related?, Christian Servin, Olga Kosheleva, Vladik Kreinovich
Departmental Technical Reports (CS)
To make education more effective, to better use emerging technologies in education, we need to better understand the education process, to gain insights on this process. How can we check whether a new idea is indeed a useful insight? A natural criterion is that the new idea should explain some previously-difficult-to-explain empirical phenomenon. Since one of the main advantages of emerging educational technologies -- such as AI -- is the possibility of individualized education, a natural phenomenon to explain is the fact -- discovered by Benjamin Bloom -- that individualization adds two sigmas to the average grade. In this paper, …
Towards A More Subtle (And Hopefully More Adequate) Fuzzy "And"-Operation: Normalization-Invariant Multi-Input Aggregation Operators, Yusuf Güven, Vladik Kreinovich
Towards A More Subtle (And Hopefully More Adequate) Fuzzy "And"-Operation: Normalization-Invariant Multi-Input Aggregation Operators, Yusuf Güven, Vladik Kreinovich
Departmental Technical Reports (CS)
Many reasonable conditions have been formulated for a fuzzy "and"-operation: idempotency, commutativity, associativity, etc. It is known that the only "and"-operation that satisfies all these conditions is minimum, but minimum is not the most adequate description of expert's "and", and it often does not lead to the best control or the best decision. Many other more adequate "and"-operations (t-norms) have been proposed and effectively used, but they do not satisfy the natural idempotency condition. In this paper, we show that a small relaxation of the usual description of "and"-operations leads to the possibility of non-minimum idempotent operations. We also show …
How To Make A Neural Network Learn From A Small Number Of Examples -- And Learn Fast: An Idea, Chitta Baral, Vladik Kreinovich
How To Make A Neural Network Learn From A Small Number Of Examples -- And Learn Fast: An Idea, Chitta Baral, Vladik Kreinovich
Departmental Technical Reports (CS)
Current deep learning techniques have led to spectacular results, but they still have limitations. One of them is that, in contrast to humans who can learn from a few examples and learn fast, modern deep learning techniques require a large amount of data to learn, and they take a long time to train. In this paper, we show that neural networks do have a potential to learn from a small number of examples -- and learn fast. We speculate that the corresponding idea may already be implicitly implemented in Large Language Models -- which may partially explain their (somewhat mysterious) …
How Can We Explain Empirical Formulas For Shrinkage Cracking Of Cement-Stabilized Pavement Layers, Edgar Daniel Rodriguez Velasquez, Vladik Kreinovich
How Can We Explain Empirical Formulas For Shrinkage Cracking Of Cement-Stabilized Pavement Layers, Edgar Daniel Rodriguez Velasquez, Vladik Kreinovich
Departmental Technical Reports (CS)
In pavement construction, one of the frequent defects is shrinkage cracking of the cement-stabilized pavement layer. To minimize this defect, it is important to be able to predict how this cracking depends on the quantities describing the pavement layer and the corresponding environment. Cracking is usually described by two parameters: the average width of the crack and the crack spacing. Empirical analysis shows that the dependence of the width on all related quantities is described by a power law. Power laws are ubiquitous in physics, they describe a frequent case when the dependence is scale-invariant -- i.e., does not change …
Topics In The Study Of The Pragmatic Functions Of Phonetic Reduction In Dialog, Nigel G. Ward, Carlos A. Ortega
Topics In The Study Of The Pragmatic Functions Of Phonetic Reduction In Dialog, Nigel G. Ward, Carlos A. Ortega
Departmental Technical Reports (CS)
Reduced articulatory precision is common in speech, but for dialog its acoustic properties and pragmatic functions have been little studied. We here try to remedy this gap. This technical report contains content that was omitted from the journal article (Ward et. al, submitted). Specifically, we here report 1) lessons learned about annotating for perceived reduction, 2) the finding that, unlike in read speech, the correlates of reduction in dialog include high pitch, wide pitch range, and intensity, and 3) a baseline model for predicting reduction in dialog, using simple acoustic/prosodic features, that achieves correlations with human perceptions of 0.24 for …
The Future Of Brain Tumor Diagnosis: Cnn And Transfer Learning Innovations, Shengyuan Wang
The Future Of Brain Tumor Diagnosis: Cnn And Transfer Learning Innovations, Shengyuan Wang
Mathematics, Statistics, and Computer Science Honors Projects
For the purpose of improving patient survival rates and facilitating efficient treatment planning, brain tumors need to be identified early and accurately classified. This research investigates the application of transfer learning and Convolutional Neural Networks (CNN) to create an automated, high-precision brain tumor segmentation and classification framework. Utilizing large-scale datasets, which comprise MRI images from open-accessible archives, the model exhibits the effectiveness of the method in various kinds of tumors and imaging scenarios. Our approach utilizes transfer learning techniques along with CNN architectures strengths to tackle the intrinsic difficulties of brain tumor diagnosis, namely significant tumor appearance variability and difficult …
The Forget Time For Random Walks On Trees Of A Fixed Diameter, Lola R. Vescovo
The Forget Time For Random Walks On Trees Of A Fixed Diameter, Lola R. Vescovo
Mathematics, Statistics, and Computer Science Honors Projects
A mixing measure is the expected length of a random walk on a graph given a set of starting and stopping conditions. We study a mixing measure called the forget time. Given a graph G, the pessimal access time for a target distribution is the expected length of an optimal stopping rule to that target distribution, starting from the worst initial vertex. The forget time of G is the smallest pessimal access time among all possible target distributions. We prove that the balanced double broom maximizes the forget time on the set of trees on n vertices with diameter …
Modeling The Neutral Densities Of Sparc Using A Python Version Of Kn1d, Gwendolyn R. Galleher
Modeling The Neutral Densities Of Sparc Using A Python Version Of Kn1d, Gwendolyn R. Galleher
Undergraduate Honors Theses
Currently, neutral recycling is a crucial contributor to fueling the plasma within tokamaks. However, Commonwealth Fusion System’s SPARC Tokamak is expected to be more opaque to neutrals. Thus, we anticipate that the role of neutral recycling in fueling will decrease. Since SPARC is predicted to have a groundbreaking fusion power gain ratio of Q ≈ 10, we must have a concrete understanding of the opacity
and whether or not alternative fueling practices must be included. To develop said understanding, we produced neutral density profiles via KN1DPy, a 1D kinetic neutral transport code for atomic and molecular hydrogen in an ionizing …
Formalization Of A Security Framework Design For A Health Prescription Assistant In An Internet Of Things System, Thomas Rolando Mellema
Formalization Of A Security Framework Design For A Health Prescription Assistant In An Internet Of Things System, Thomas Rolando Mellema
Electronic Theses and Dissertations
Security system design flaws will create greater risks and repercussions as the systems being secured further integrate into our daily life. One such application example is incorporating the powerful potential of the concept of the Internet of Things (IoT) into software services engineered for improving the practices of monitoring and prescribing effective healthcare to patients. A study was performed in this application area in order to specify a security system design for a Health Prescription Assistant (HPA) that operated with medical IoT (mIoT) devices in a healthcare environment. Although the efficiency of this system was measured, little was presented to …
Murmurations And Root Numbers, Alexey Pozdnyakov
Murmurations And Root Numbers, Alexey Pozdnyakov
University Scholar Projects
We report on a machine learning investigation of large datasets of elliptic curves and L-functions. This leads to the discovery of murmurations, an unexpected correlation between the root numbers and Dirichlet coefficients of L-functions. We provide a formal definition of murmurations, describe the connection with 1-level density, and provide three examples for which the murmuration phenomenon has been rigorously proven. Using our understanding of murmurations, we then build new machine learning models in search of a polynomial time algorithm for predicting root numbers. Based on our models and several heuristic arguments, we conclude that it is unlikely for …
Asteroidal Sets And Dominating Targets In Graphs, Oleksiy Al-Saadi
Asteroidal Sets And Dominating Targets In Graphs, Oleksiy Al-Saadi
Department of Computer Science and Engineering: Dissertations, Theses, and Student Research
The focus of this PhD thesis is on various distance and domination properties in graphs. In particular, we prove strong results about the interactions between asteroidal sets and dominating targets. Our results add to or extend a plethora of results on these properties within the literature. We define the class of strict dominating pair graphs and show structural and algorithmic properties of this class. Notably, we prove that such graphs have diameter 3, 4, or contain an asteroidal quadruple. Then, we design an algorithm to to efficiently recognize chordal hereditary dominating pair graphs. We provide new results that describe the …
Flipped Classroom For Linear Algebra At Undergraduate Level, M. Thulasidas
Flipped Classroom For Linear Algebra At Undergraduate Level, M. Thulasidas
Research Collection School Of Computing and Information Systems
In this article, we describe our experience in developing an undergraduate Linear Algebra course tailored to highlight its relevance and applicability in Computer Science. Over the course of three years, the course transitioned from a traditional direct-instruction format to a flipped-classroom design, resulting in positive student learning outcomes. This article covers the course design philosophy, its syllabus, learning objectives, and the incorporation of both quantitative and qualitative student feedback in shaping the course. Furthermore, the article shares the insights gleaned from our experience, which can serve as best practices for instructors aiming to deliver a successful Linear Algebra course for …