Mathematics Commons^™

Paradox Of Causality And Paradoxes Of Set Theory, Alondra Baquier, Bradley Beltran, Gabriel Miki-Silva, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Logical paradoxes show that human reasoning is not always fully captured by the traditional 2-valued logic, that this logic's extensions -- such as multi-valued logics -- are needed. Because of this, the study of paradoxes is important for research on multi-valued logics. In this paper, we focus on paradoxes of set theory. Specifically, we show their analogy with the known paradox of causality, and we use this analogy to come up with similar set-theoretic paradoxes.

Number Representation With Varying Number Of Bits, Anuradha Choudhury, Md Ahsanul Haque, Saeefa Rubaiyet Nowmi, Ahmed Ann Noor Ryen, Sabrina Saika, Vladik Kreinovich

Departmental Technical Reports (CS)

In a computer, usually, all real numbers are stored by using the same number of bits: usually, 8 bytes, i.e., 64 bits. This amount of bits enables us to represent numbers with high accuracy -- up to 19 decimal digits. However, in most cases -- whether we process measurement results or whether we process expert-generated membership degrees -- we do not need that accuracy, so most bits are wasted. To save space, it is therefore reasonable to consider representations with varying number of bits. This would save space used for representing numbers themselves, but we would also need to store …

Data Fusion Is More Complex Than Data Processing: A Proof, Robert Alvarez, Salvador Ruiz, Martine Ceberio, Vladik Kreinovich

Departmental Technical Reports (CS)

Empirical data shows that, in general, data fusion takes more computation time than data processing. In this paper, we provide a proof that data fusion is indeed more complex than data processing.

How To Fairly Allocate Safety Benefits Of Self-Driving Cars, Fernando Munoz, Christian Servin, Vladik Kreinovich

Departmental Technical Reports (CS)

In this paper, we describe how to fairly allocated safety benefits of self-driving cars between drivers and pedestrians -- so as to minimize the overall harm.

Using Known Relation Between Quantities To Make Measurements More Accurate And More Reliable, Niklas Winnewisser, Felix Mett, Michael Beer, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Most of our knowledge comes, ultimately, from measurements and from processing measurement results. In this, metrology is very valuable: it teaches us how to gauge the accuracy of the measurement results and of the results of data processing, and how to calibrate the measuring instruments so as to reach the maximum accuracy. However, traditional metrology mostly concentrates on individual measurements. In practice, often, there are also relations between the current values of different quantities. For example, there is usually an known upper bound on the difference between the values of the same quantity at close moments of time or at …

Somewhat Surprisingly, (Subjective) Fuzzy Technique Can Help To Better Combine Measurement Results And Expert Estimates Into A Model With Guaranteed Accuracy: Digital Twins And Beyond, Niklas Winnewisser, Michael Beer, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

To understand how different factors and different control strategies will affect a system -- be it a plant, an airplane, etc. -- it is desirable to form an accurate digital model of this system. Such models are known as digital twins. To make a digital twin as accurate as possible, it is desirable to incorporate all available knowledge of the system into this model. In many cases, a significant part of this knowledge comes in terms of expert statements, statements that are often formulated by using imprecise ("fuzzy") words from natural language such as "small", "very possible", etc. To translate …

How Difficult Is It To Comprehend A Program That Has Significant Repetitions: Fuzzy-Related Explanations Of Empirical Results, Christian Servin, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In teaching computing and in gauging the programmers' productivity, it is important to property estimate how much time it will take to comprehend a program. There are techniques for estimating this time, but these techniques do not take into account that some program segments are similar, and this similarity decreases the time needed to comprehend the second segment. Recently, experiments were performed to describe this decrease. These experiments found an empirical formula for the corresponding decrease. In this paper, we use fuzzy-related ideas to provide commonsense-based theoretical explanation for this empirical formula.

Why Bernstein Polynomials: Yet Another Explanation, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In many computational situations -- in particular, in computations under interval or fuzzy uncertainty -- it is convenient to approximate a function by a polynomial. Usually, a polynomial is represented by coefficients at its monomials. However, in many cases, it turns out more efficient to represent a general polynomial by using a different basis -- of so-called Bernstein polynomials. In this paper, we provide a new explanation for the computational efficiency of this basis.

How To Make A Decision Under Interval Uncertainty If We Do Not Know The Utility Function, Jeffrey Escamilla, Vladik Kreinovich

Departmental Technical Reports (CS)

Decision theory describes how to make decisions, in particular, how to make decisions under interval uncertainty. However, this theory's recommendations assume that we know the utility function -- a function that describes the decision maker's preferences. Sometimes, we can make a recommendation even when we do not know the utility function. In this paper, we provide a complete description of all such cases.

Shall We Place More Advanced Students In A Separate Class?, Shahnaz Shahbazova, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In every class, we have students who are more advanced and students who are more behind. From this viewpoint, it seems reasonable to place more advanced students in a separate class. This should help advanced students progress faster, and it should help other students as well, since the teachers in the remaining class can better attend to their needs. However, empirically, this does not work: when we form a separate class, the overall amount of gained knowledge decreases. In this paper, we provide a possible explanation for this seemingly counterintuitive phenomenon.

Towards An Optimal Design: What Can We Recommend To Elon Musk?, Martine Ceberio, Olga Kosheleva, Vladik Kreinovich, Hung T. Nguyen

Departmental Technical Reports (CS)

Elon Musk's successful "move fast and break things" strategy is based on the fact that in many cases, we do not need to satisfy all usual constraints to be successful. By sequentially trying smaller number of constraints, he finds the smallest number of constraints that are still needed to succeed -- and using this smaller number of constrains leads to a much cheaper (and thus, more practical) design. In this strategy, Musk relies on his intuition -- which, as all intuitions, sometimes works and sometimes doesn't. To replace this intuition, we propose an algorithm that minimizes the worst-case cost of …

Exploring Quaternion Neural Network Loss Surfaces, Jeremiah Bill, Bruce A. Cox

Faculty Publications

This paper explores the superior performance of quaternion multi-layer perceptron (QMLP) neural networks over real-valued multi-layer perceptron (MLP) neural networks, a phenomenon that has been empirically observed but not thoroughly investigated. The study utilizes loss surface visualization and projection techniques to examine quaternion-based optimization loss surfaces for the first time. The primary contribution of this research is the statistical evidence that QMLP models yield smoother loss surfaces than real-valued neural networks, which are measured and compared using a robust quantitative measure of loss surface “goodness” based on estimates of surface curvature. Extensive computational testing validates the effectiveness of these surface …

Wang Tilings In Arbitrary Dimensions, Ian Tassin

Rose-Hulman Undergraduate Mathematics Journal

This paper makes a new observation about arbitrary dimensional Wang Tilings,
demonstrating that any d -dimensional tile set that can tile periodically along d − 1 axes must be able to tile periodically along all axes.
This work also summarizes work on Wang Tiles up to the present day, including
definitions for various aspects of Wang Tilings such as periodicity and the validity of a tiling. Additionally, we extend the familiar 2D definitions for Wang Tiles and associated properties into arbitrary dimensional spaces. While there has been previous discussion of arbitrary dimensional Wang Tiles in other works, it has been …

Why Pavement Cracks Are Mostly Longitudinal, Sometimes Transversal, And Rarely Of Other Directions: A Geometric Explanation, Edgar Daniel Rodriguez Velasquez, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In time, pavements deteriorate, and need maintenance. One of the most typical pavement faults are cracks. Empirically, the most frequent cracks are longitudinal, i.e., following the direction of the road; less frequent are transversal cracks, which are orthogonal to the direction of the road. Sometimes, there are cracks in different directions, but such cracks are much rarer. In this paper, we show that simple geometric analysis and fundamental physical ideas can explain these observed relative frequencies.

Why Linear And Sigmoid Last Layers Work Better In Classification, Lehel Dénes-Fazakas, Lásló Szilágyi, Vladik Kreinovich

Departmental Technical Reports (CS)

Usually, when a deep neural network is used to classify objects, its last layer computes the softmax. Our empirical results show we can improve the classification results if instead, we have linear or sigmoid last layer. In this paper, we provide an explanation for this empirical phenomenon.

Why Two Fish Follow Each Other But Three Fish Form A School: A Symmetry-Based Explanation, Shahnaz Shahbazova, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Recent experiments with fish has shown an unexpected strange behavior: when two fish of the same species are placed in an aquarium, they start following each other, while when three fish are placed there, they form (approximately) an equilateral triangle, and move in the direction (approximately) orthogonal to this triangle. In this paper, we use natural symmetries -- such as rotations, shifts, and permutation of fish -- to show that this observed behavior is actually optimal. This behavior is not just optimal with respect to one specific optimality criterion, it is optimal with respect to any optimality criterion -- as …

Some Generalizations Of Corona Product Of Two Graphs, Aparajita Borah, Gajendra Pratap Singh

Applications and Applied Mathematics: An International Journal (AAM)

In this paper we are seeking to conceptualize the notion of corona product of two graphs to contrive some special types of graphs. That is, here our attempt is to regenerate a familiar graph as a product graph. We are considering seven familiar graphs here to reconstruct them with the help of corona product of two graphs. Such types of families of the graphs and operations can be used to study biological pathways as well as to find the optimal order and size for the special types of graphs.

Fuzzy Software Reliability And Optimal Release Policy With Log-Logistic Testing Effort: An Analysis, Seema Rani, Jitendra Kumar, N. Ahmad

Applications and Applied Mathematics: An International Journal (AAM)

We will discuss a Software Reliability Growth Model (SRGM) using fuzzy and imperfect debugging environments; we integrate Log-Logistic (LL) Testing Effort Function (TEF) into fuzzy SRGMs. Estimation methods, such as Least Square and Maximum Likelihood, are used to obtain the value of Testing-Effort and SRGMs parameters. It is not always possible and is constantly required to quantify the exact value of parameters. Due to human conduct, the value of Testing-Effort and SRGM parameters cannot be exactly quantified. In this scenario, parameters are supposed to be vague or fuzzy. To make the software consistent, the developer needs to propose some quantity …

Session 8: Machine Learning Based Behavior Of Non-Opec Global Supply In Crude Oil Price Determinism, Mofe Jeje

SDSU Data Science Symposium

Abstract

While studies on global oil price variability, occasioned by OPEC crude oil supply, is well documented in energy literature; the impact assessment of non-OPEC global oil supply on price variability, on the other hand, has not received commensurate attention. Given this gap, the primary objective of this study, therefore, is to estimate the magnitude of oil price determinism that is explained by the share of non-OPEC’s global crude oil supply. Using secondary sources of data collection method, data for target variable will be collected from the US Federal Reserve, as it relates to annual crude oil price variability, while …

Fuzzy Ideas Explain Fechner Law And Help Detect Relation Between Objects In Video, Olga Kosheleva, Vladik Kreinovich, Ahnaf Farhan

Departmental Technical Reports (CS)

How to find relation between objects in a video? If two objects are closely related -- e.g., a computer and it mouse -- then they almost always appear together, and thus, their numbers of occurrences are close. However, simply computing the differences between numbers of occurrences is not a good idea: objects with 100 and 110 occurrences are most probably related, but objects with 1 and 5 occurrences probably not, although 5 − 1 is smaller than 110 − 100. A natural idea is, instead, to compute the difference between re-scaled numbers of occurrences, for an appropriate nonlinear re-scaling. In …

There Is Still Plenty Of Room At The Bottom: Feynman's Vision Of Quantum Computing 65 Years Later, Alexis Lupo, Vladik Kreinovich, Victor L. Timchenko, Yuriy P. Kondratenko

Departmental Technical Reports (CS)

In 1959, Nobelist Richard Feynman gave a talk titled "There's plenty of room at the bottom", in which he emphasized that, to drastically speed up computations, we need to make computer components much smaller -- all the way to the size of molecules, atoms, and even elementary particles. At this level, physics is no longer described by deterministic Newton's mechanics, it is described by probabilistic quantum laws. Because of this, computer designers started thinking how to design a reliable computer based on non-deterministic elements -- and this thinking eventually led to the modern ideas and algorithms of quantum computing. So, …

From Quantifying And Propagating Uncertainty To Quantifying And Propagating Both Uncertainty And Reliability: Practice-Motivated Approach To Measurement Planning And Data Processing, Niklas R. Winnewisser, Vladik Kreinovich, Olga Kosheleva

Departmental Technical Reports (CS)

When we process data, it is important to take into account that data comes with uncertainty. There exist techniques for quantifying uncertainty and propagating this uncertainty through the data processing algorithms. However, most of these techniques do not take into account that in real world, measuring instruments are not 100% reliable -- they sometimes malfunction and produce values which are far off from the measured values of the corresponding quantities. How can we take into account both uncertainty and reliability? In this paper, we consider several possible scenarios, and we show, for each scenario, what is the natural way to …

Modifed Playfair For Text File Encryption And Meticulous Decryption With Arbitrary Fillers By Septenary Quadrate Pattern, N. Sugirtham, R. Sherine Jenny, B. Thiyaneswaran, S. Kumarganesh, C. Venkatesan, K. Martin Sagayam, Lam Dang, Linh Dinh, Helen Dang

Mathematics and Computer Studies: Faculty Publications

Cryptography secures data and serves to ensure the confidentiality of records. Playfair is a cryptographic symmetrical algorithm that encrypts statistics based on key costs. This secret is shared with an authorized person to retrieve data. In the conventional pattern, there is an area complexity and deficiency in letters, numbers, and special characters. This hassle has been overcome in previous studies by editing pattern dimensions. The fillers used throughout the enciphering were not eliminated during the retrieval process, which resulted in the indiscrimination of the retrieved statistics. The proposed method uses a separate quadrate pattern that strengthens the Playfair cipher and …

Does Chatgpt Know Calculus?, Kris H. Green

Journal of Humanistic Mathematics

Academics and educators across the world are grappling with how OpenAI’s new software, ChatGPT, will impact teaching and learning. This essay explores ChatGPT’s response to a typical calculus problem as a way of illustrating its functionality and limitations.

Machine Learning For Wireless Network Throughput Prediction, Gustavo A. Fernandez

School of Mathematical & Statistical Sciences Faculty Publications

This paper analyzes a dataset containing radio frequency (RF) measurements and Key Performance Indicators (KPIs) captured at 1876.6MHz with a bandwidth of 10MHz from an operational 4G LTE network in Nigeria. The dataset includes metrics such as RSRP (Reference Signal Received Power), which measures the power level of reference signals; RSRQ (Reference Signal Received Quality), an indicator of signal quality that provides insight into the number of users sharing the same resources; RSSI (Received Signal Strength Indicator), which gauges the total received power in a bandwidth; SINR (Signal to Interference plus Noise Ratio), a measure of signal quality considering both …

Every Feasibly Computable Reals-To-Reals Function Is Feasibly Uniformly Continuous, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

It is known that every computable function is continuous; moreover, it is computably continuous in the sense that for every ε > 0, we can compute δ > 0 such that δ-close inputs lead to ε-close outputs. It is also known that not all functions which are, in principle, computable, can actually be computed: indeed, the computation sometimes requires more time than the lifetime of the Universe. A natural question is thus: can the above known result about computable continuity of computable functions be extended to the case when we limit ourselves to feasible computations? In this paper, we prove that this …

From Normal Distribution To What? How To Best Describe Distributions With Known Skewness, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practical situations, we only have partial information about the probability distribution -- e.g., all we know is its few moments. In such situations, it is desirable to select one of the possible probability distributions. A natural way to select a distribution from a given class of distributions is the maximum entropy approach. For the case when we know the first two moments, this approach selects the normal distribution. However, when we also know the third central moment -- corresponding to skewness -- a direct application of this approach does not work. Instead, practitioners use several heuristic techniques, techniques …

Orthogonal Single-View And Multi-View Feature Selection Models Via Spectral Theory Based Methods, Zachary Chairez

Mathematics Dissertations - Archive

As the digital world continues to grow in the age of big data, there beckons a need for efficient and robust methods for data exploration. Under the umbrella of machine learning, feature selection positions itself as a fruitful approach that uncovers buried truths and illuminates important features of data, all while minimizing long-term storage requirements. In this work, we introduce a set of novel single-view and multi-view supervised feature selection models which are embedded with orthogonality constraints to maintain data's structural integrity while confining the optimal solution's search space.

Taking advantage of the underlying framework of these types of models, …

Inexact Fixed-Point Proximity Algorithm For The ℓ₀ Sparse Regularization Problem, Ronglong Fang, Yuesheng Xu, Mingsong Yan

Mathematics & Statistics Faculty Publications

We study inexact fixed-point proximity algorithms for solving a class of sparse regularization problems involving the ℓ₀ norm. Specifically, the ℓ₀ model has an objective function that is the sum of a convex fidelity term and a Moreau envelope of the ℓ₀ norm regularization term. Such an ℓ₀ model is non-convex. Existing exact algorithms for solving the problems require the availability of closed-form formulas for the proximity operator of convex functions involved in the objective function. When such formulas are not available, numerical computation of the proximity operator becomes inevitable. This leads to inexact iteration algorithms. We investigate in this …

Volume 15, Connor Thompson, Emily Steffenhagen, Emily Robertson, Luis Fernando Dos Reis, Emily Farmer, Samuel Villa, Robert Allison, Zachary Chessor, Megan Borden, Austin Burnett, Larry W. Grant Jr., Tristan Marowski, Emma Moore, Pearl Siff

Incite: The Journal of Undergraduate Scholarship

Introduction Dr. Amorette Barber, Director, Office of Student Research

From the Editor Dr. Hannah Dudley-Shotwell

Artist’s Statement Connor Thompson

On Mentorship Dr. John Miller

The Meat of the Matter: Alien, Human, and Animal in Terry Bisson’s “They’re Made Out of Meat” by Emily Steffenhagen

“Please REBLOG!”: An Ethical Analysis of Doxxing, Internet Vigilantism and Racists Getting Fired by Emily Robertson

Journaling: Paper Has More Patience Than People by Luis Fernando Dos Reis

The Effects of Climate Change on the Archaeological World by Emily Farmer

Lowered Seat Height Does Not Impair Wingate Performance in Untrained Cyclists by Samuel Villa, Robert Allison, …