Physical Sciences and Mathematics Commons^™

How To Combine Probabilistic And Fuzzy Uncertainty: Theoretical Explanation Of Clustering-Related Empirical Result, Lázló Szilágyi, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In contrast to crisp clustering techniques that assign each object to a class, fuzzy clustering algorithms assign, to each object and to each class, a degree to which this object belongs to this class. In the most widely used fuzzy clustering algorithm -- fuzzy c-means -- for each object, degrees corresponding to different classes add up to 1. From this viewpoint, these degrees act as probabilities. There exist alternative fuzzy-based clustering techniques in which, in line with the general idea of the fuzzy set, the largest of the degrees is equal to 1. In some practical situations, the probability-type fuzzy …

Which Fuzzy Implications Operations Are Polynomial? A Theorem Proves That This Can Be Determined By A Finite Set Of Inequalities, Sebastia Massanet, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

To adequately represent human reasoning in a computer-based systems, it is desirable to select fuzzy operations that are as close to human reasoning as possible. In general, every real-valued function can be approximated, with any desired accuracy, by polynomials; it is therefore reasonable to use polynomial fuzzy operations as the appropriate approximations. We thus need to select, among all polynomial operations that satisfy corresponding properties -- like associativity -- the ones that best fit the empirical data. The challenge here is that properties like associativity mean satisfying infinitely many constraints (corresponding to infinitely many possible triples of values), while most …

Methodological Lesson Of Pythagorean Triples, Julio C. Urenda, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

There are many right triangles in which all three sides a, b, and c have integer lengths. The triples (a,b,c) formed by such lengths are known as Pythagorean triples. Since ancient times, it is known how to generate all Pythagorean triples: we can enumerate primitive Pythagorean triples -- in which the three numbers have no common divisors -- by considering all pairs of natural numbers m>n in which m and n have no common divisors, and taking a =m2 − n2, b = 2mn, and c = m2 + n2. Multiplying all elements of a triple by the same …

Why 6-Labels Uncertainty Scale In Geosciences: Probability-Based Explanation, Aaron Velasco, Julio C. Urenda, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

To describe uncertainty in geosciences, several researchers have recently proposed a 6-labels uncertainty scale, in which one the labels corresponds to full certainty, one label to the absence of any knowledge, and the remaining four labels correspond to the degrees of confidence from the intervals [0,0.25], [0.25,0.5], [0.5,0.75], and [0.75,1]. Tests of this 6-labels scale indicate that it indeed conveys uncertainty information to geoscientists much more effectively than previously proposed uncertainty schemes. In this paper, we use probability-related techniques to explain this effectiveness.

Fuzzy Mathematics Under Non-Minimal "And"-Operations (T-Norms): Equivalence Leads To Metric, Order Leads To Kinematic Metric, Topology Leads To Area Or Volume, Purbita Jana, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Most formulas analyzed in fuzzy mathematics assume -- explicitly or implicitly -- that the corresponding "and"-operation (t-norm) is the simplest minimum operation. In this paper, we analyze what happens if instead, we use other "and"-operations. It turns out that for such operations, a fuzzification of a mathematical theory naturally leads to a more complex mathematical setting: fuzzification of equivalence relation leads to metric, fuzzification of order leads to kinematic metric, and fuzzification of topology leads to area or volume.

Complex Numbers Explain Why In Chinese Tradition, 4 Is Bad But 8 Is Good, Luc Longpre, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In the traditional Chinese culture, 4 is considered to be an unlucky number, while the number 8 is considered to be very lucky. In this paper, we show that both "badness" and "goodness" can be explained if we take into account the role of complex numbers in the analysis of general dynamical systems.

Why Resilient Modulus Is Proportional To The Square Root Of Unconfined Compressive Strength (Ucs): A Qualitative Explanation, Edgar Daniel Rodriguez Velasquez, Vladik Kreinovich

Departmental Technical Reports (CS)

The strength of the pavement is determine by its resilient modulus, i.e., by its ability to withstand (practically) instantaneous stresses caused by the passing traffic. However, the resilient modulus is not easy to measure: its measurement requires a special expensive equipment that many labs do not have. So, instead of measuring it, practitioners often measure easier-to-measure Unconfined Compressive Strength (UCS) -- that describes the effect of a continuously applied force -- and estimate the resilient modulus based on the result of this measurement. An empirical formula shows that the resilient modulus is proportional to the square root of the Unconfined …

How To Estimate Unknown Unknowns: From Cosmic Light To Election Polls, Talha Azfar, Vignesh Ponraj, Vladik Kreinovich, Nguyen Hoang Phuong

Departmental Technical Reports (CS)

In two different areas of research -- in the study of space light and in the study of voting -- the observed value of the quantity of interest is twice larger than what we would expect. That the observed value is larger makes perfect sense: there are phenomena that we do not take into account in our estimations. However, the fact that the observed value is exactly twice larger deserves explanation. In this paper, we show that Laplace Indeterminacy Principle leads to such an explanation.

We Can Always Reduce A Non-Linear Dynamical System To Linear -- At Least Locally -- But Does It Help?, Orsolya Csiszar, Gábor Csiszar, Olga Kosheleva, Vladik Kreinovich, Nguyen Hoang Phuong

Departmental Technical Reports (CS)

Many real-life phenomena are described by dynamical systems. Sometimes, these dynamical systems are linear. For such systems, solutions are well known. In some cases, it is possible to transform a nonlinear system into a linear one by appropriately transforming its variables, and this helps to solve the original nonlinear system. For other nonlinear systems -- even for the simplest ones -- such transformation is not known. A natural question is: which nonlinear systems allow such transformations? In this paper, we show that we can always reduce a nonlinear system to a linear one -- but, in general, it does not …

What Was More Frequently Used -- "And" Or "Or": Based On Analysis Of European Languages, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Traditional logic has two main connectives: "and" and "or". A natural question is: which of the two is more frequently used? This question is easy to answer for the current usage of these connectives -- we can simply analyze all the texts, but what can we say about the past usage? To answer this question, we use the known linguistics fact that, in general, notions that are more frequently used are described by shorter words. It turns out that in most European languages, the word for "and" is shorter -- or of the same length -- as the word for …

Why Bump Reward Function Works Well In Training Insulin Delivery Systems, Lehel Dénes-Fazakas, Lásló Szilágyi, Gyorgy Eigner, Olga Kosheleva, Vladik Kreinovich, Nguyen Hoang Phuong

Departmental Technical Reports (CS)

Diabetes is a disease when the body can no longer properly regulate blood glucose level, which can lead to life-threatening situations. To avoid such situations and regulate blood glucose level, patients with severe form of diabetes need insulin injections. Ideally, the system should automatically decide when best to inject insulin and how much to inject. To find the optimal control, researchers applied machine learning with different reward functions. It turns out that the most effective learning occurred when they used the so-called bump function. In this paper, we provide a possible explanation for this empirical result.

Fuzzy Techniques Explain The Effectiveness Of Relu Activation Function In Deep Learning, Julio C. Urenda, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In the last decades, deep learning has led to spectacular successes. One of the reasons for these successes was the fact that deep neural networks use a special Rectified Linear Unit (ReLU) activation function s(x) = max(0,x). Why this activation function is so successful is largely a mystery. In this paper, we show that common sense ideas -- as formalized by fuzzy logic -- can explain this mysterious effectiveness.

Interpolation Problems And The Characterization Of The Hilbert Function, Bryant Xie

Mathematical Sciences Undergraduate Honors Theses

In mathematics, it is often useful to approximate the values of functions that are either too awkward and difficult to evaluate or not readily differentiable or integrable. To approximate its values, we attempt to replace such functions with more well-behaving examples such as polynomials or trigonometric functions. Over the algebraically closed field C, a polynomial passing through r distinct points with multiplicities m1, ..., mr on the affine complex line in one variable is determined by its zeros and the vanishing conditions up to its mi − 1 derivative for each point. A natural question would then be to consider …

Numerical Design And Optimization Of Near-Infrared Band- Pass Filter, Hafiza Syeeda Faiza, Ghazi Aman Nowsherwan, Basem A. Abu Izneid, Muhammad Azhar, Saira Riaz, Syed Sajjad Hussain, Saira Ikram, Mohsin Khan, Shahzad Naseem, Mohammad Kanan, Ibrahim M. Mansour

Applied Mathematics & Information Sciences

Band-pass filters functioning in the near-infrared (IR) range are desired for laser technology, multi-photon fluorescence, and IR imaging applications. In this study, we have designed four band-pass filters in the near Infrared spectrum (900-1200 nm) by vertically stacking different high and low-index materials. The band-pass filters are modelled by Essential Macleod software with different thicknesses. The layer’s thicknesses were optimized in such a way to provide the negligible reflectance and maximum transmission on the front side. All the simulated band-pass filters exhibit high transmittance, but TiO2/Al2O3 and Ta2O5/Al2O3 outperforms other modelled structure in terms of performance due to the better …

Modified Geometries, Clifford Algebras And Graphs: Their Impact On Discreteness, Locality And Symmetr, Roman Sverdlov

Mathematics & Statistics ETDs

In this dissertation I will explore the question whether various entities commonly used in quantum field theory can be “constructed". In particular, can spacetime be “constructed" out of building blocks, and can Berezin integral be “constructed" in terms of Riemann integrals.

As far as “constructing" spacetime out of building blocks, it has been attempted by multiple scientific communities and various models were proposed. But the common downfall is they break the principles of relativity. I will explore the ways of doing so in such a way that principles of relativity are respected. One of my approaches is to replace points …

Kinetic Particle Simulations Of Plasma Charging At Lunar Craters Under Severe Conditions, David Lund, Xiaoming He, Daoru Frank Han

Mathematics and Statistics Faculty Research & Creative Works

This paper presents fully kinetic particle simulations of plasma charging at lunar craters with the presence of lunar lander modules using the recently developed Parallel Immersed-Finite-Element Particle-in-Cell (PIFE-PIC) code. The computation model explicitly includes the lunar regolith layer on top of the lunar bedrock, taking into account the regolith layer thickness and permittivity as well as the lunar lander module in the simulation domain, resolving a nontrivial surface terrain or lunar lander configuration. Simulations were carried out to study the lunar surface and lunar lander module charging near craters at the lunar terminator region under mean and severe plasma environments. …

Why Deep Learning Is Under-Determined? Why Usual Numerical Methods For Solving Partial Differential Equations Do Not Preserve Energy? The Answers May Be Related To Chevalley-Warning Theorem (And Thus To Fermat Last Theorem), Julio C. Urenda, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In this paper, we provide a possible explanation to two seemingly unrelated phenomena: (1) that in deep learning, under-determined systems of equations perform much better than the over-determined one -- which are typical in data processing, and that (2) usual numerical methods for solving partial differential equations do not preserve energy. Our explanation is related to the intuition of Fermat behind his Last Theorem and of Euler about more general statements, intuition that led to the proof of Chevalley-Warning Theorem in number theory.

How To Best Retrain A Neural Network If We Added One More Input Variable, Saeid Tizpaz-Niari, Vladik Kreinovich

Departmental Technical Reports (CS)

Often, once we have trained a neural network to estimate the value of a quantity y based on the available values of inputs x1, ..., xn, we learn to measure the values of an additional quantity that have some influence on y. In such situations, it is desirable to re-train the neural network, so that it will be able to take this extra value into account. A straightforward idea is to add a new input to the first layer and to update all the weights based on the patterns that include the values of the new input. The problem with …

Topological Explanation Of Why Complex Numbers Are Needed In Quantum Physics, Julio C. Urenda, Vladik Kreinovich

Departmental Technical Reports (CS)

In quantum computing, we only use states in which all amplitudes are real numbers. So why do we need complex numbers with non-zero imaginary part in quantum physics in general? In this paper, we provide a simple topological explanation for this need, explanation based on the Second Law of Thermodynamics.

How To Make Decision Under Interval Uncertainty: Description Of All Reasonable Partial Orders On The Set Of All Intervals, Tiago M. Costa, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practical situations, we need to make a decision while for each alternative, we only know the corresponding value of the objective function with interval uncertainty. To help a decision maker in this situation, we need to know the (in general, partial) order on the set of all intervals that corresponds to the preferences of the decision maker. For this purpose, in this paper, we provide a description of all such partial orders -- under some reasonable conditions. It turns out that each such order is characterized by two linear inequalities relating the endpoints of the corresponding intervals, and …

Integrable Discretizations For A Generalized Sine-Gordon Equation And The Reductions To The Sine-Gordon Equation And The Short Pulse Equation, Han-Han Sheng, Bao-Feng Feng, Guo-Fu Yu

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

In this paper, we propose fully discrete analogues of a generalized sine-Gordon (gsG) equation utx=(1+ν∂2x)sinu. The bilinear equations of the discrete KP hierarchy and the proper definition of discrete hodograph transformations are the keys to the construction. Then we derive semi-discrete analogues of the gsG equation from the fully discrete gsG equation by taking the temporal parameter b→0. Especially, one full-discrete gsG equation is reduced to a semi-discrete gsG equation in the case of ν=−1 (Feng {\it et al. Numer. Algorithms} 2023). Furthermore, N-soliton solutions to the semi- and fully discrete analogues of the gsG equation in the determinant form …

Covering By Planks And Avoiding Zeros Of Polynomials, Alexey Glazyrin, Roman Karasev, Alexandr Polyanskii

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

We note that the recent polynomial proofs of the spherical and complex plank covering problems by Zhao and Ortega-Moreno give some general information on zeros of real and complex polynomials restricted to the unit sphere. As a corollary of these results, we establish several generalizations of the celebrated Bang plank covering theorem. We prove a tight polynomial analog of the Bang theorem for the Euclidean ball and an even stronger polynomial version for the complex projective space. Specifically, for the ball, we show that for every real nonzero d-variate polynomial P of degree n⁠, there exists a point in the …

Generation Of Cardiac Chamber Models Using Interpretable Generative Neural Networks For Electrophysiology Studies, Sunil Mathew

Dissertations (1934 -)

An Electrophysiology study is conducted to diagnose and treat heart rhythm disorders, such as arrhythmias (abnormal heartbeat) like atrial fibrillation. A catheter is inserted into the chamber of interest to acquire 3D location and electrical information to create an electroanatomical map. This dissertation explores the design of a mapping system based on interpretable generative neural networks for generating patient specific cardiac models. Chapter 1 provides an introduction to electroanatomical mapping, the need for interpretability in neural networks and other relevant topics that are discussed in detail in the chapters that follow. Neural networks are often very large models with millions …

May Graduation, Samuel Coskey

Mathematics Faculty Publications and Presentations

Here I narrate the story of the last few days of my graduate program in mathematics. After the completion of the thesis and the delivery of the defense, several twists and turns await in the hours and even minutes before the last deadline.

Widely Digitally Delicate Brier Primes And Irreducibility Results For Some Classes Of Polynomials, Thomas David Luckner

Theses and Dissertations

This dissertation considers three different sections of results. In the first part of the dissertation, a result on consecutive primes which are widely digitally delicate and Brier numbers is discussed. Making use of covering systems and a theorem of D. Shiu, M. Filaseta and J. Juillerat showed that for every positive integer k, there exist k consecutive widely digitally delicate primes. They also noted that for every positive integer k, there exist k consecutive primes which are Brier numbers. We show that for every positive integer k, there exist k consecutive primes that are both widely digitally …

Deep Learning Methods For Some Problems In Scientific Computing, Yuankai Teng

Theses and Dissertations

Deep learning has emerged as a powerful approach for solving complex problems in scientific computing due to the increasing availability of large-scale data and computational resources. This thesis explores the potential of deep learning methods for three specific problems in scientific computing: (i) reducing the dimensions of variables in function approximation, (ii) solving linear reaction-diffusion equations, and (iii) finding the parametric representations of parameters in the numerical schemes for solving time-dependent partial differential equations.

For the first problem, a novel deep learning architecture is developed for reducing the dimensions of variables in function approximation. The proposed method achieves state-of-the-art performance …

Fucik Spectrum With Weights And Existence Of Solutions For Nonlinear Elliptic Equations With Nonlinear Boundary Conditions, Nsoki Mavinga, Q. A. Morris, S. B. Robinson

Mathematics & Statistics Faculty Works

We consider the boundary value problem −Δu + c(x)u = αm(x)u⁺ − βm(x)u⁻ + f(x,u), x∈Ω, (∂u)/(∂η) + σ(x)u = αρ(x)u⁺ − βρ(x)u⁻ + g(x,u), x∈∂Ω, where (α,β) ∈R2, c, m ∈ L^∞(Ω), σ, ρ ∈ L^∞(∂Ω), and the nonlinearities f and g are bounded …

Special Case Of Partial Fraction Expansion With Laplace Transform Application, Laurie A. Florio, Ryan D. Hanc

CODEE Journal

Partial fraction expansion is often used with the Laplace Transforms to formulate algebraic expressions for which the inverse Laplace Transform can be easily found. This paper demonstrates a special case for which a linear, constant coefficient, second order ordinary differential equation with no damping term and a harmonic function non-homogeneous term leads to a simplified partial fraction expansion due to the decoupling of the partial fraction expansion coefficients of s and the constant coefficients. Recognizing this special form can allow for quicker calculations and automation of the solution to the differential equation form which is commonly used to model physical …

Applied Analysis For Learning Architectures, Himanshu Singh

USF Tampa Graduate Theses and Dissertations

Modern data science problems revolves around the Koopman operator Cφ (or Composition operator) approach, which provides the best-fit linear approximator to the dynamical system by which the dynamics can be advanced under the discretization. The solution provided by Koopman in the data driven methods is in the sense of strong operator topology, which is nothing better then the point-wise convergence of data (snapshots) in the underlying Hilbert space. Chapter 2 provides the details about the aforementioned issues with essential counter-examples. Thereafter, provable convergence guarantee phenomena is demonstrated by the Liouville weighted composition operators Af,φ over the Fock space by providing …

Applications Of Financial Mathematics: An Analysis Of Consumer Financial Decision Making, Alyssa Betterton

Honors Theses

Students always ask, “How can this be applied to the real world?” Mortgages, car loans, and credit card bills are things that almost everyone will have to make decisions about at some point in their lives. This research discusses the many different financial choices that consumers have to make. Consumers can use this information to understand how interest rates, the length of the loan, and the initial amount being borrowed affects the amount that is paid back to the companies. The intent of this thesis is to present the mathematical theory of interest. A web-based application has been built based …