Mathematics Commons^™

How To Estimate The Present Serviceability Rating Of A Road Segment: Explanation Of An Empirical Formula, Edgar Daniel Rodriguez Velasquez, Vladik Kreinovich

Departmental Technical Reports (CS)

An accurate estimation of the road quality requires a lot of expertise, and there is not enough experts to provide such estimates for all the road segments. It is therefore desirable to estimate this quality based on easy-to-estimate and easy-to-measure characteristics. Recently, an empirical formula was proposed for such an estimate. In this paper, we provide a theoretical explanation for this empirical formula.

Education In The Era Of Google, Wikipedia, And Deep Learning: Are We Humans Still Needed And If Yes For What?, Miroslav Svitek, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

One of the main purposes of education is to teach skills needed in future life and future jobs. What is important and what is useful changes with time. Before the industrial revolution, routine mechanical work was an important part of human activity – now machines can do it (and do it better). Before printing, copying was an important activity – now copy machines do it. Before computers, humans were needed for computing – now computer do it better. With Wikipedia and Google, there is not much need for scholars being erudite. Even extracting dependencies from data – one of the …

Economy-Related Emotional Attitudes Towards Other People: How Can We Explain Them?, Christopher Reyes, Vladik Kreinovich, Chon Van Le

Departmental Technical Reports (CS)

Research has shown that to properly understand people's economic behavior, it is important to take into account their emotional attitudes towards each other. Behavioral economics shows that different attitudes results in different economy-related behavior. A natural question is: where do these emotional attitudes come from? We show that, in principle, such emotions can be explained by people's objective functions. Specifically, we show it on the example of a person whose main objective is to increase his/her country's GDP: in this case, the corresponding optimization problem leads exactly to natural emotions towards people who contribute a lot or a little towards …

The Primitive Root Problem: A Problem In Bqp, Shixin Wu

Mathematical Sciences Technical Reports (MSTR)

Shor’s algorithm proves that the discrete logarithm problem is in BQP. Based on his algorithm, we prove that the primitive root problem, a problem that verifies if some integer g is a primitive root modulo p where p is the largest prime number smaller than 2n for a given n, which is assumed to be harder than the discrete logarithm problem, is in BQP by using an oracle quantum Turing machine.

Data And Algorithmic Modeling Approaches To Count Data, Andraya Hack

Honors College Theses

Various techniques are used to create predictions based on count data. This type of data takes the form of a non-negative integers such as the number of claims an insurance policy holder may make. These predictions can allow people to prepare for likely outcomes. Thus, it is important to know how accurate the predictions are. Traditional statistical approaches for predicting count data include Poisson regression as well as negative binomial regression. Both methods also have a zero-inflated version that can be used when the data has an overabundance of zeros. Another procedure is to use computer algorithms, also known as …

What About Generic "And"- And "Or"-Operations: From Levels Of Certainty (Philosophical-Physical-Mathematical) To A Natural Interpretation Of Quantum-Like Negative Degrees Of Certainty, Miroslav Svitek, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Fuzzy techniques -- techniques designed to convert imprecise human knowledge into precise computer-understandable terms -- have many successful applications. Traditional applications of fuzzy techniques use only important general features of human reasoning and, to make an implement more efficient, ignore subtle details, details which are not important for the corresponding application. But from the more fundamental viewpoint, it is desirable to understand, in all the detail, how people actually reason. In this paper, we use general ideas of fuzzy approach to answer this question. Interestingly -- and somewhat unexpectedly -- the resulting analysis leads to a natural explanation of existence …

How Can The Opposite To A True Theory Be Also True? A Similar Talmudic Discussion Helps Make This Famous Bohr's Statement Logically Consistent, Miroslav Svitek, Vladik Kreinovich

Departmental Technical Reports (CS)

In his famous saying, the Nobelist physicist Niels Bohr claimed that the sign of a deep theory is that while this theory is true, its opposite is also true. While this statement makes heuristic sense, it does not seem to make sense from a logical viewpoint, since, in logic, the opposite to true is false. In this paper, we show how a similar Talmudic discussion can help come up with an interpretation in which Bohr's statement becomes logically consistent.

In The Absence Of Information, The Only Reasonable Negotiation Scheme Is Offering A Certain Percentage Of The Original Request: A Proof, Miroslav Svitek, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In the case of complete information, a reasonable solution to a negotiation process is Nash's bargaining solution, in which we maximize the product of all agents' utility gains. This is the only solution that does not depend on the order in which we list the agents, and does not change if we use a different scale for describing each agent's utility. In this paper, we apply similar invariance criteria to a situation when practically all information is absent, and all we know is the smallest and largest possible gains. We show that in this situation, the only invariant negotiation strategy …

Why Some Theoretically Possible Representations Of Natural Numbers Were Historically Used And Some Were Not: An Algorithm-Based Explanation, Christian Servin, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Historically, people have used many ways to represent natural numbers: from the original "unary" arithmetic, where each number is represented as a sequence of, e.g., cuts (4 is IIII) to modern decimal and binary systems. However, with all this variety, some seemingly reasonable ways of representing natural numbers were never used. For example, it may seem reasonable to represent numbers as products -- e.g., as products of prime numbers -- such a representation was never used in history. So why some theoretically possible representations of natural numbers were historically used and some were not? In this paper, we propose an …

Sinusoidal Projection For 360° Image Compression And Triangular Discrete Cosine Transform Impact In The Jpeg Pipeline, Iker Vazquez Lopez

Boise State University Theses and Dissertations

The equirectangular projection is commonly used to store and transmit 360' images. However, using the equirectangular projection to store and transmit 360° images is not efficient due to its natural topographic redundancy. To generate the 360° image, captured pixels that form a spherical point cloud in the 3D space are projected onto a 2D plane using the equirectangular projection; generating redundant pixels in the process. These extra pixels in the image add extra memory requirements that have low impact in the final image quality. This dissertation presents results of research into the compression of 360° spherical imagery. It examines and …

Security Analysis Of Lightweight Cryptographic Primitives, William Unger

Boise State University Theses and Dissertations

Symmetric key cryptographic primitives are essential to encrypt data and protect communication between parties. Due to resource constraints, some modern devices are not capable of executing traditional cryptographic algorithms. This fact necessitates new lightweight cryptographic algorithms. Current research into lightweight cryptology is vast, in part due to the National Institute of Standards and Technology's (NIST) lightweight cryptographic standardization process.

There is not much research into the vulnerability to a power analysis attack created by the choice of parameters of lightweight symmetric ciphers. This dissertation develops and demonstrates white box and black box cryptanalysis models for power analysis attacks on lightweight …

Machine Learning Classification Of Digitally Modulated Signals, James A. Latshaw

Electrical & Computer Engineering Theses & Dissertations

Automatic classification of digitally modulated signals is a challenging problem that has traditionally been approached using signal processing tools such as log-likelihood algorithms for signal classification or cyclostationary signal analysis. These approaches are computationally intensive and cumbersome in general, and in recent years alternative approaches that use machine learning have been presented in the literature for automatic classification of digitally modulated signals. This thesis studies deep learning approaches for classifying digitally modulated signals that use deep artificial neural networks in conjunction with the canonical representation of digitally modulated signals in terms of in-phase and quadrature components. Specifically, capsule networks are …

Volume 13, Payton Davenport, Audrey Lemons, Jacob Shope, Haley Smith, Cassandra Poole, Rachel Cannon, Rachel Boch, Suzanne Stetson

Incite: The Journal of Undergraduate Scholarship

Introduction Dr. Roger A. Byrne, Dean

From the Editor Dr. Larissa “Kat” Tracy

From the Designers Rachel English, Rachel Hanson

The Effect of Compliment Type on the Estimated Value of the Compliment by Payton Davenport, Audrey Lemons, and Jacob Shope

The Imperial Japanese Military: A New Identity in the Twentieth Century, 1853–1922 by Haley Smith

Longwood University’s campus: Human-cultivated Soil has Higher Microbial Diversity than Soil Collected from Wild Sites by Cassandra Poole

Reminiscent Modernism: Poetry Magazine’s Modernist Nostalgia for the Past by Rachel Cannon

Challenges Faced by Healthcare Workers During the COVID-19 Pandemic: A Preliminary Study of Age and …

Intra-Hour Solar Forecasting Using Cloud Dynamics Features Extracted From Ground-Based Infrared Sky Images, Guillermo Terrén-Serrano

Electrical and Computer Engineering ETDs

Due to the increasing use of photovoltaic systems, power grids are vulnerable to the projection of shadows from moving clouds. An intra-hour solar forecast provides power grids with the capability of automatically controlling the dispatch of energy, reducing the additional cost for a guaranteed, reliable supply of energy (i.e., energy storage). This dissertation introduces a novel sky imager consisting of a long-wave radiometric infrared camera and a visible light camera with a fisheye lens. The imager is mounted on a solar tracker to maintain the Sun in the center of the images throughout the day, reducing the scattering effect produced …

Why Rectified Linear Unit Is Efficient In Machine Learning: One More Explanation, Barnabas Bede, Vladik Kreinovich, Uyen Pham

Departmental Technical Reports (CS)

In many applications, in particular, in econometric application, deep learning techniques are very effective. In this paper, we provide a new explanation for why rectified linear units -- the main units of deep learning -- are so effective. This explanation is similar to the usual explanation of why Gaussian (normal) distributions are ubiquitous -- namely, it is based on an appropriate limit theorem.

Game-Theoretic Approach Explains -- On The Qualitative Level -- The Antigenic Map Of Covid-19 Variants, Olga Kosheleva, Vladik Kreinovich, Nguyen Hoang Phuong

Departmental Technical Reports (CS)

To effectively defend the population against future variants of Covid-19, it is important to be able to predict how it will evolve. For this purpose, it is necessary to understand the logic behind its evolution so far. At first glance, this evolution looks random and thus, difficult to predict. However, we show that already a simple game-theoretic model can actually explain -- on the qualitative level -- how this virus mutated so far.

A New Application Of The Central Limit Theorem, Kenneth Winters

Selected Honors Theses

This paper discusses the Central Limit Theorem (CLT) and its applications. The paper gives an introduction to what the CLT is and how it can be applied to real life. Additionally, the paper gives a conceptual understanding of the theorem through various examples and visuals. The paper discusses the applications of the CLT in fields such as computer science, psychology, and political science. The author then suggests a new mathematical theorem as an application of the CLT and provides a proof of the theorem. The new theorem relates to expected value and probabilities of random variables and provides a link …

Why Constraint Interval Arithmetic Works Well: A Theorem Explains Empirical Success, Barnabas Bede, Marina Tuyako Mizukoshi, Weldon Lodwick, Martine Ceberio, Vladik Kreinovich

Departmental Technical Reports (CS)

Often, we are interested in a quantity that is difficult or impossible to measure directly, e.g., tomorrow's temperature. To estimate this quantity, we measure auxiliary easier-to-measure quantities that are related to the desired ones by a known dependence, and use the known relation to estimate the desired quantity. Measurements are never absolutely accurate, there is always a measurement error, i.e., a non-zero difference between the measurement result and the actual (unknown) value of the corresponding quantity. In many practical situations, the only information that we have about each measurement error is the bound on its absolute value. In such situations, …

When Is Deep Learning Better And When Is Shallow Learning Better: Qualitative Analysis, Salvador Robles Herrera, Martine Ceberio, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practical situations, deep neural networks work better than the traditional "shallow" ones, however, in some cases, the shallow neural networks lead to better results. At present, deciding which type of neural networks will work better is mostly done by trial and error. It is therefore desirable to come up with some criterion of when deep learning is better and when shallow is better. In this paper, we argue that this depends on whether the corresponding situation has natural symmetries: if it does, we expect deep learning to work better, otherwise we expect shallow learning to be more effective. …

A New Metaphor: How Artificial Intelligence Links Legal Reasoning And Mathematical Thinking, Melissa E. Love Koenig, Colleen Mandell

Marquette Law Review

Artificial intelligence’s (AI’s) impact on the legal community expands exponentially each year. As AI advances, lawyers have more powerful tools to enhance their ability to research and analyze the law, as well as to draft contracts and other legal documents. Lawyers are already using tools powered by AI and are learning to shift their methodologies to take advantage of these enhancements. To continue to grow into their shifting role, lawyers should understand the relationship between AI, mathematics, and legal reasoning.

A Super Fast Algorithm For Estimating Sample Entropy, Weifeng Liu, Ying Jiang, Yuesheng Xu

Mathematics & Statistics Faculty Publications

: Sample entropy, an approximation of the Kolmogorov entropy, was proposed to characterize complexity of a time series, which is essentially defined as − log(B/A), where B denotes the number of matched template pairs with length m and A denotes the number of matched template pairs with m + 1, for a predetermined positive integer m. It has been widely used to analyze physiological signals. As computing sample entropy is time consuming, the box-assisted, bucket-assisted, x-sort, assisted sliding box, and kd-tree-based algorithms were proposed to accelerate its computation. These algorithms require O(N²) or …

Why Deep Neural Networks: Yet Another Explanation, Ricardo Lozano, Ivan Montoya Sanchez, Vladik Kreinovich

Departmental Technical Reports (CS)

One of the main motivations for using artificial neural networks was to speed up computations. From this viewpoint, the ideal configuration is when we have a single nonlinear layer: this configuration is computationally the fastest, and it already has the desired universal approximation property. However, the last decades have shown that for many problems, deep neural networks, with several nonlinear layers, are much more effective. How can we explain this puzzling fact? In this paper, we provide a possible explanation for this phenomena: that the universal approximation property is only true in the idealized setting, when we assume that all …

Why Menzerath's Law?, Julio Urenda, Vladik Kreinovich

Departmental Technical Reports (CS)

In linguistics, there is a dependence between the length of the sentence and the average length of the word: the longer the sentence, the shorter the words. The corresponding empirical formula is known as the Menzerath's Law. A similar dependence can be observed in many other application areas, e.g., in the analysis of genomes. The fact that the same dependence is observed in many different application domains seems to indicate there should be a general domain-independent explanation for this law. In this paper, we show that indeed, this law can be derived from natural invariance requirements.

How To Select A Representative Sample For A Family Of Functions?, Leobardo Valera, Martine Ceberio, Vladik Kreinovich

Departmental Technical Reports (CS)

Predictions are rarely absolutely accurate. Often, the future values of quantities of interest depend on some parameters that we only know with some uncertainty. To make sure that all possible solutions satisfy desired constraints, it is necessary to generate a representative finite sample, so that if the constraints are satisfied for all the functions from this sample, then we can be sure that these constraints will be satisfied for the actual future behavior as well. At present, such a sample is selected based by Monte-Carlo simulations, but, as we show, such selection may underestimate the danger of violating the constraints. …

Why Hate: Analysis Based On Decision Theory, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

At first glance, from the general decision-theory viewpoint, hate (and other negative feelings towards each other) makes no sense, since they decrease the utility (i.e., crudely speaking, level of happiness) of the person who experiences these feelings. Our detailed analysis shows that there are situations when such negative feelings make perfect sense: namely, when you have a large group of people almost all of whom are objectively unhappy. In such situations -- e.g., on the battlefield -- negative feelings help keep their spirits high in spite of the harsh situation. This explanation leads to recommendations on how to decrease the …

How To Solve The Apportionment Paradox, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

It is known that often, after it is proven that a new statement is equivalent to the original definition, this new statement becomes the accepted new definition of the same notion. In this paper, we provide a natural explanation for this empirical phenomenon.

How To Solve The Apportionment Paradox, Christopher Reyes, Vladik Kreinovich

Departmental Technical Reports (CS)

In the ideal world, the number of seats that each region or each community gets in a representative body should be exactly proportional to the population of this region or community. However, since the number of seats allocated to each region or community is whole, we cannot maintain the exact proportionality. Not only this leads to a somewhat unfair situation, when residents of one region get more votes per person than residents of another one, it also leads to paradoxes -- e.g., sometimes a region that gained the largest number of people loses a number of seats. To avoid this …

A Possible Common Mechanism Behind Skew Normal Distributions In Economics And Hydraulic Fracturing-Induced Seismicity, Laxman Bokati, Aaron Velasco, Vladik Kreinovich, Kittawit Autchariyapanitkul

Departmental Technical Reports (CS)

Many economic situations -- and many situations in other application areas -- are well-described by a special asymmetric generalization of normal distributions -- known as skew-normal. However, there is no convincing theoretical explanation for this empirical phenomenon. To be more precise, there are convincing explanations for the ubiquity of normal distributions, but not for the transformation that turns normal into skew-normal. In this paper, we use the analysis of hydraulic fracturing-induced seismicity to show explain the ubiquity of such a transformation.

Shall We Use Logical Approach Or More Traditional Mamdani Approach In Fuzzy Control: Pragmatic Analysis, R. Noah Padilla, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Fuzzy control methodology transforms the experts' if-then rules into a precise control strategy. From the logical viewpoint, an if-then rule means implication, so it seems reasonable to use fuzzy implication in this transformation. However, this logical approach is not what the first fuzzy controllers used. The traditional fuzzy control approach -- first proposed by Mamdani -- transforms the if-then rules into a statement that only contains and's and or's, and does not use fuzzy implication at all. So, a natural question arises: shall we use logical approach or the traditional approach? In this paper, we analyze this question on the …

Why Optimization Is Faster Than Solving Systems Of Equations: A Qualitative Explanation, Siyu Deng, Bimal K. C, Vladik Kreinovich

Departmental Technical Reports (CS)

Most practical problems lead either to solving a system of equation or to optimization. From the computational viewpoint, both classes of problems can be reduced to each other: optimization can be reduced to finding points at which all partial derivatives are zeros, and solving systems of equations can be reduced to minimizing sums of squares. It is therefore natural to expect that, on average, both classes of problems have the same computational complexity -- i.e., require about the same computation time. However, empirically, optimization problems are much faster to solve. In this paper, we provide a possible explanation for this …