Mathematics Commons^™

Decision Theory Explains "Telescoping Effect" -- That Our Time Perception Is Biased, Laxman Bokati, Vladik Kreinovich

Departmental Technical Reports (CS)

People usually underestimate time passed since distant events, and overestimate time passed since recent events. There are several explanations for this "telescoping effect", but most current explanations utilize specific features of human memory and/or human perception. We show that the telescoping effect can be explained on a much basic level of decision theory, without the need to invoke any specific ways we perceive and process time.

How To Generate "Nice" Cubic Polynomials -- With Rational Coefficients, Rational Zeros And Rational Extrema: A Fast Algorithm, Laxman Bokati, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Students feel more comfortable with rational numbers than with irrational ones. Thus, when teaching the beginning of calculus, it is desirable to have examples of simple problems for which both zeros and extrema point are rational. Recently, an algorithm was proposed for generating cubic polynomials with this property. However, from the computational viewpoint, the existing algorithm is not the most efficient one: in addition to applying explicit formulas, it also uses trial-and-error exhaustive search. In this paper, we propose a computationally efficient algorithm for generating all such polynomials: namely, an algorithm that uses only explicit formulas.

When Revolutions Happen: Algebraic Explanation, Julio Urenda, Vladik Kreinovich

Departmental Technical Reports (CS)

At first glance, it may seem that revolutions happen when life becomes really intolerable. However, historical analysis shows a different story: that revolutions happen not when life becomes intolerable, but when a reasonably prosperous level of living suddenly worsens. This empirical observation seems to contradict traditional decision theory ideas, according to which, in general, people's happiness monotonically depends on their level of living. A more detailed model of human behavior, however, takes into account not only the current level of living, but also future expectations. In this paper, we show that if we properly take these future expectations into account, …

Use Of Symmetries In Economics: An Overview, Vladik Kreinovich, Olga Kosheleva, Nguyen N. Thach, Nguyen Duc Trung

Departmental Technical Reports (CS)

In this paper, we show that many semi-heuristic econometric formulas can be derived from the natural symmetry requirements. The list of such formulas includes many famous formulas provided by Nobel-prize winners, such as Hurwicz optimism-pessimism criterion for decision making under uncertainty, McFadden's formula for probabilistic decision making, Nash's formula for bargaining solution -- as well as Cobb-Douglas formula for production, gravity model for trade, etc.

Why Bilingualism Helps Autistic Children Function: A Symmetry-Based Explanation, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

One of the main problems of autistic children is that it is very difficult for them to switch to a different state, to a different activity -- and such switches are often needed. Researchers have recently shown that bilingualism helps autistic children function: namely, it is somewhat easier for bilingual children to switch to a new activity. In this paper, we provide a possible explanation for this empirical phenomenon. Namely, we show that, in general terms, autism means difficulty with breaking symmetries of a state, and we describe how this general reformulation indeed explains the above recently discovered phenomenon.

Forecasting Crashes, Credit Card Default, And Imputation Analysis On Missing Values By The Use Of Neural Networks, Jazmin Quezada

Open Access Theses & Dissertations

A neural network is a system of hardware and/or software patterned after the operation of neurons in the human brain. Neural networks,- also called Artificial Neural Networks - are a variety of deep learning technology, which also falls under the umbrella of artificial intelligence, or AI. Recent studies shows that Artificial Neural Network has the highest coefficient of determination (i.e. measure to assess how well a model explains and predicts future outcomes.) in comparison to the K-nearest neighbor classifiers, logistic regression, discriminant analysis, naive Bayesian classifier, and classification trees. In this work, the theoretical description of the neural network methodology …

Robust Statistical Inference For The Gaussian Distribution, Andrews Tawiah Anum

Open Access Theses & Dissertations

The aim of robust statistics is to develop statistical procedures which are not unduly influenced by outliers or observations that are not representative of the underlying "true" data generating process. This thesis focuses on an estimator with this characteristic. The divergence function is introduced in Chapter 2 with the sole aim of taking the function f to be the univariate normal distribution and α - [0, 1]. The estimator fails when we rely on the classic Newton's method to converge to the minimum of the density power divergence (MDPD) function. There is a tendency of such estimator never to approach …

Inverse Gaussian Ornstein-Uhlenbeck Applied To Modeling High Frequency Data, Emmanuel Kofi Kusi

Open Access Theses & Dissertations

With about 226050 estimated deaths worldwide in 2010, an earthquake is considered as one of the disasters that records a great number of deaths. This thesis develops a model for the estimation of magnitude of future seismic events.

We propose a stochastic differential equation arising on the Ornstein-Uhlenbeck processes driven by IG(a,b) process. IG(a,b) Ornstein-Uhlenbeck processes offers analytic flexibility and provides a class of continuous time processes capable of exhibiting long memory

behavior. The stochastic differential equation is applied to geophysics and financial stock markets by fitting the superposed IG(a,b) Ornstein-Uhlenbeck model to earthquake and financial time series.

Cantor Sets, Cantorvals, And Their Topological Structure, Ángel Adrián Agüero

Open Access Theses & Dissertations

With interesting topological properties, the Cantor set is worth studying for itself. In other areas, topological structures arise that are in fact homeomorphic to the Cantor set. In particular, we see sets that are homeomorphic to the Cantor set which result from the subsums of particular series, as well as linear combinations of algebraic sums of Cantor sets. These also result in what has been termed a Cantorval, which we also investigate.

Formulation And Implementation Of Iterative Method For Generating Spatially-Variant Lattices, Manuel Fernando Martinez

Open Access Theses & Dissertations

The use of a matrix-free, memory-efficient approach to generate large-scale spatially variant lattices (SVL) was explored. A matrix-free iterative SVL generation algorithm was formulated and then implemented with a tremendous memory reduction observed. The algorithm consists of solving first-order central finite-differences along the entirety of the problem space point-by-point to obtain the grating phase function Φ(𝑠⃗) to which all desired spatially variant lattice properties are applied to. The algorithm was studied to identify key areas of data and task parallelism to exploit in heterogeneous computing systems consisting of clusters of central processing units (CPU) and graphics processing units (GPU) combinations. …

Towards Analytical Techniques For Systems Engineering Applications, Griselda Valdepeñas Acosta

Open Access Theses & Dissertations

One of the main objectives of systems engineering is to design, maintain, and analyze systems that help the users. To design an appropriate system for an application domain, we need to know: what are the users' desires and preferences (so that we know in what direction we should aim to change this domain), what is the current state and what is the dynamics of this application domain, and how to use all this information to select the best alternatives for the system design and maintenance. Designing a system includes selecting numerical values for many of the parameters describing the corresponding …

Earthquake Magnitude Prediction Using Support Vector Machine And Convolutional Neural Network, Esther Amfo

Open Access Theses & Dissertations

A deep learning-based method Convolutional Neural Network (CNN) and Support Vector Machine (SVM) for earthquake prediction is proposed. Large-magnitude earthquakes triggered by earthquakes can kill thousands of people and cause millions of dollars worth of economic losses. The accurate prediction of large-magnitude earthquakes is a worldwide problem.

In recent years, deep learning technology that can automatically extract features from mass data has been applied in image recognition, natural language processing, object recognition, etc., with great success. We explore to apply deep learning technology to earthquake prediction, we propose a deep learning method for continuous earthquake prediction using historical seismic events. …

Why Early Galaxies Were Pickle-Shaped: A Geometric Explanation, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

The vast majority of currently observed geometric shapes of celestial bodies can be explained by a simple symmetry idea: the initial distribution of matter is invariant with respect to shifts, rotations, and scaling, but this distribution is unstable, so we have spontaneous symmetry breaking. According to statistical physics, among all possible transitions, the most probable are the ones that retain the largest number of symmetries. This explains the currently observed shapes and -- on the qualitative level -- their relative frequency. According to this idea, the most probable first transition is into a planar (pancake) shape, then into …

Bhutan Landscape Anomaly: Possible Effect On Himalayan Economy (In View Of Optimal Description Of Elevation Profiles), Thach N. Nguyen, Laxman Bokati, Aaron A. Velasco, Vladik Kreinovich

Departmental Technical Reports (CS)

Economies of countries located in seismic zones are strongly effected by this seismicity. If we underestimate the seismic activity, then a reasonably routine earthquake can severely damage the existing structures and thus, lead to huge economic losses. On the other hand, if we overestimate the seismic activity, we waste a lot of resources on unnecessarily fortifying all the buildings -- and this too harms the economies. From this viewpoint, it is desirable to have estimations of regional seismic activities which are as accurate as possible. Current predictions are mostly based on the standard geophysical understanding of earthquakes as being largely …

Towards Optimal Implementation Of Decentralized Currencies: How To Best Select Probabilities In An Ethereum-Type Proof-Of-Stake Protocol, Thach N. Nguyen, Christian Servin, Vladik Kreinovich

Departmental Technical Reports (CS)

Nowadays, most financial transactions are based on a centralized system, when all the transaction records are stored in a central location. This centralization makes the financial system vulnerable to cyber-attacks. A natural way to make the financial system more robust and less vulnerable is to switch to decentralized currencies. Such a transition will also make financial system more transparent. Historically first currency of this type -- bitcoin -- use a large amount of electric energy to mine new coins and is, thus, not scalable to the level of financial system as a whole. A more realistic and less energy-consuming scheme …

Relativistic Effects Can Be Used To Achieve A Universal Square-Root (Or Even Faster) Computation Speedup, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In this paper, we show that special relativity phenomenon can be used to reduce computation time of any algorithm from T to square root of T. For this purpose, we keep computers where they are, but the whole civilization starts moving around the computer -- at an increasing speed, reaching speeds close to the speed of light. A similar square-root speedup can be achieved if we place ourselves near a growing black hole. Combining the two schemes can lead to an even faster speedup: from time T to the 4-th order root of T.

Translating Discrete Estimates Into A Less Detailed Scale: An Optimal Approach, Thongchai Dumrongpokaphan, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practical situations, we use estimates that experts make on a 0-to-n scale. For example, to estimate the quality of a lecturer, we ask each student to evaluate this quality by selecting an integer from 0 to n. Each such estimate may be subjective; so, to increase the estimates' reliability, it is desirable to combine several estimates of the corresponding quality. Sometimes, different estimators use slightly different scales: e.g., one estimator uses a scale from 0 to n+1, and another estimator uses a scale from 0 to n. In such situations, it is desirable to translate these estimates to …

A Symmetry-Based Explanation Of The Main Idea Behind Chubanov's Linear Programming Algorithm, Olga Kosheleva, Vladik Kreinovich, Thongchai Dumrongpokaphan

Departmental Technical Reports (CS)

Many important real-life optimization problems can be described as optimizing a linear objective function under linear constraints -- i.e., as a linear programming problem. This problem is known to be not easy to solve. Reasonably natural algorithms -- such as iterative constraint satisfaction or simplex method -- often require exponential time. There exist efficient polynomial-time algorithms, but these algorithms are complicated and not very intuitive. Also, in contrast to many practical problems which can be computed faster by using parallel computers, linear programming has been proven to be the most difficult to parallelize. Recently, Sergei Chubanov proposed a modification of …

Why Bohmian Approach To Quantum Econometrics: An Algebraic Explanation, Vladik Kreinovich, Olga Kosheleva, Songsak Sriboonchitta

Departmental Technical Reports (CS)

Many equations in economics and finance are very complex. As a result, existing methods of solving these equations are very complicated and time-consuming. In many practical situations, more efficient algorithms for solving new complex equations appear when it turns out that these equations can be reduced to equations from other application areas -- equations for which more efficient algorithms are already known. It turns out that some equations in economics and finance can be reduced to equations from physics -- namely, from quantum physics. The resulting approach for solving economic equations is known as quantum econometrics. In quantum physics, …

Soft Computing Ideas Can Help Earthquake Geophysics, Solymar Ayala Cortez, Aaron A. Velasco, Vladik Kreinovich

Departmental Technical Reports (CS)

Earthquakes can be devastating, thus it is important to gain a good understanding of the corresponding geophysical processing. One of the challenges in geophysics is that we cannot directly measure the corresponding deep-earth quantities, we have to rely on expert knowledge, knowledge which often comes in terms of imprecise ("fuzzy") words from natural language. To formalize this knowledge, it is reasonable to use techniques that were specifically designed for such a formalization -- namely, fuzzy techniques, In this paper, we formulate the problem of optimally representing such knowledge. By solving the corresponding optimization problem, we conclude that the optimal representation …

Economics Of Commitment: Why Giving Away Some Freedom Makes Sense, Vladik Kreinovich, Olga Kosheleva, Mahdokhat Afravi, Genesis Bejarano, Marisol Chacon

Departmental Technical Reports (CS)

In general, the more freedom we have, the better choices we can make, and thus, the better possible economic outcomes. However, in practice, people often artificially restrict their future options by making a commitment. At first glance, commitments make no economic sense, and so their ubiquity seems puzzling. Our more detailed analysis shows that commitment often makes perfect economic sense: namely, it is related to the way we take future gains and losses into account. With the traditionally assumed exponential discounting, commitment indeed makes no economic sense, but with the practically observed hyperbolic discounting, commitment is indeed often economically beneficial.

Why Bellman-Zadeh Approach To Fuzzy Optimization, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In many cases, we need to select the best of the possible alternatives, but we do not know for sure which alternatives are possible and which are not possible. Instead, for each alternative x, we have a subjective probability p(x) that this alternative is possible. In 1970, Richard Bellman and Lotfi Zadeh proposed a heuristic method for selecting an alternative under such uncertainty. Interestingly, this method works very well in many practical applications, while similarly motivated alternative formulas do not work so well. In this paper, we explain the empirical success of the Bellman-Zadeh approach by showing that its formulas …

How Interval Measurement Uncertainty Affects The Results Of Data Processing: A Calculus-Based Approach To Computing The Range Of A Box, Andrew Pownuk, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practical applications, we are interested in the values of the quantities y₁, ..., y_m which are difficult (or even impossible) to measure directly. A natural idea to estimate these values is to find easier-to-measure related quantities x₁, ..., x_n and to use the known relation to estimate the desired values y_i. Measurements come with uncertainty, and often, the only thing we know about the actual value of each auxiliary quantity x_i is that it belongs to the interval [X_i − Δ_i, X_i + Δ_{i …}

Why Under Stress Positive Reinforcement Is More Effective? Why Optimists Study Better? Why People Become Restless? Simple Utility-Based Explanations, Francisco Zapata, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In this paper, we use the utility-based approach to decision making to provide simple answers to the following three questions: Why under stress positive reinforcement is more effective? Why optimists study better? Why people become restless?

Why Zipf's Law: A Symmetry-Based Explanation, Daniel Cervantes, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practical situations, we have probability distributions for which, for large values of the corresponding quantity x, the probability density has the form ρ(x) ~ x^−αfor some α > 0. While, in principle, we have laws corresponding to different α, most frequently, we encounter situations -- first described by Zipf for linguistics -- when α is close to 1. The fact that Zipf's has appeared frequently in many different situations seems to indicate that there must be some fundamental reason behind this law. In this paper, we provide a possible explanation.

How To Explain The Results Of The Richard Thaler's 1997 Financial Times Contest, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In 1997, by using a letter published in Financial Times, Richard H. Thaler, the 2017 Nobel Prize winner in Economics, performed the following experiment: he asked readers to submit numbers from 0 to 100, so that the person whose number is the closest to 2/3 of the average will be the winner. An intuitive answer is to submit 2/3 of the average (50), i.e., 33 1/3. A logical answer, as can be explained, is to submit 0. The actual winning submission was -- depending on how we count -- 12 or 13. In this paper, we propose a possible explanation …

Italian Folk Multiplication Algorithm Is Indeed Better: It Is More Parallelizable, Martine Ceberio, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Traditionally, many ethnic groups had their own versions of arithmetic algorithms. Nowadays, most of these algorithms are studied mostly as pedagogical curiosities, as an interesting way to make arithmetic more exciting to the kids: by applying to their patriotic feelings -- if they are studying the algorithms traditionally used by their ethic group -- or simply to their sense of curiosity. Somewhat surprisingly, we show that one of these algorithms -- a traditional Italian multiplication algorithm -- is actually in some reasonable sense better than the algorithm that we all normally use -- namely, it is easier to parallelize.

Fuzzy Analogues Of Sets And Functions Can Be Uniquely Determined From The Corresponding Ordered Category: A Theorem, Christian Servin, Gerardo Muela, Vladik Kreinovich

Departmental Technical Reports (CS)

In modern mathematics, many concepts and ideas are described in terms of category theory. From this viewpoint, it is desirable to analyze what can be determined if, instead of the basic category of sets, we consider a similar category of fuzzy sets. In this paper, we describe a natural fuzzy analog of the category of sets and functions, and we show that, in this category, fuzzy relations (a natural fuzzy analogue of functions) can be determined in category terms -- of course, modulo 1-1 mapping of the corresponding universe of discourse and 1-1 re-scaling of fuzzy degrees.

The P-Laplacian Problem Via An Euler Equation And The Basis Properties Of Its Eigenfunctions, Luis Suarez Salas

Open Access Theses & Dissertations

The Laplacian operator is used in many fields of science, such as fluidodynamics, mechanics and elasticity. Mathematically, much research has been devoted to develop a theory with which it and other variations can be understood. In this work, we present the p-Laplacian problem via an Euler equation. We then study the properties of its eigenfunctions which generalize the trigonometric functions sine and cosine. In connection with a Fourier series, we then show the generalized trigonometric functions possess basis properties for L^r((0,1)^d), d=1,2,3. Finally, we introduce the spaces of variable exponent and the analogue p(x)-Laplacian problem which has immense applications such …

An Efficient Method For Online Identification Of Steady State For Multivariate System, Honglun None Xu

Open Access Theses & Dissertations

Most of the existing steady state detection approaches are designed for univariate signals. For multivariate signals, the univariate approach is often applied to each process variable and the system is claimed to be steady once all signals are steady, which is computationally inefficient and also not accurate. The article proposes an efficient online method for multivariate steady state detection. It estimates the covariance matrices using two different approaches, namely, the mean-squared-deviation and mean-squared-successive-difference. To avoid the usage of a moving window, the process means and the two covariance matrices are calculated recursively through exponentially weighted moving average. A likelihood ratio …