Mathematics Commons^™

Simple Linear Interpolation Explains All Usual Choices In Fuzzy Techniques: Membership Functions, T-Norms, T-Conorms, And Defuzzification, Vladik Kreinovich, Jonathan Quijas, Esthela Gallardo, Caio De Sa Lopes, Olga Kosheleva, Shahnaz Shahbazova

Departmental Technical Reports (CS)

Most applications of fuzzy techniques use piece-wise linear (triangular or trapezoid) membership functions, min or product t-norms, max or algebraic sum t-conorms, and centroid defuzzification. Similarly, most applications of interval-valued fuzzy techniques use piecewise-linear lower and upper membership functions. In this paper, we show that all these choices can be explained as applications of simple linear interpolation.

Why Right-Brain Cultures Are More Flexible: A Possible Explanation Of Yu. Manin's Observation, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Yuri Manin, a renowned mathematician, observed that it is much easier for a person raised in a right-brain culture to adjust to the left-brain environment than vice versa. In this paper, we provide a possible explanation for this phenomenon.

When An Idea Comes, Write It Down Right Away: Mathematical Justification Of Vladimir Smirnov's Advice, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Among several advices to students, Vladimir Smirnov, a renowned Russian mathematician, suggested that when an idea comes, it is better to write it down right away. In this paper, we provide a quantitative justification for this advice.

Numerical Study Of The Supercritical Solution Of The Stationary Forced Korteweg-De Vries (Sfkdv) Equation, Sumi Dey

Open Access Theses & Dissertations

The time independent surface waves on a two dimensional incompressible and inviscid fluid flow passing over a bump on a flat bottom are considered. The stationary forced Korteweg-de Vries (fKdV) equation is defined in an unbounded interval. In this project, this unbounded interval is considered as three parts - on the bump, left hand side of the bump and right hand side of the bump. Exact boundary conditions are applied to the end points of the bump. All the possible combination of boundary conditions are used. A numerical approach is proposed to find the solitary wave solutions of fKdV equation …

Synchronization Of Bistatic Radar Using Chaotic Amplitude And Frequency Modulated Signals, Chandra Sekhar Pappu

Open Access Theses & Dissertations

The purpose of this work is to develop a synchronization scheme for bistatic radar that uses a 3-D chaotic system to generate and process wideband AM and FM signals, which allows for the extraction of high range-resolution information from targets. For AM bistatic radar, the setup includes a drive oscillator at the transmitter and a response oscillator at the receiver. The challenge is synchronizing the response oscillator to the drive oscillator with a scaled version of the transmitted signal sr(t, x) = αs t(t, x), where x is a chaotic state variable and α is a scaling factor. Here, α …

Contributions To The Solution Of Large Nonlinear Systems Via Model-Order Reduction And Interval Constraint Solving Techniques, Leobardo Valera

Open Access Theses & Dissertations

Many engineering problems boil down to solving partial differential equations (PDEs) that describe real-life phenomena. Nevertheless, efficiently and reliably solving such problems constitutes a major challenge in computational sciences and in engineering in general.

PDE-based systems can reach sizes so large after they are discretized. The large size in these problems generate several issues, among them we can mention: large space of storing, computing time, and the most important, lost of accuracy. A popular approach to solving such problems is assume that the PDE's solution is in a subspace, and the solution is sought there. This assumption and later searching …

Challenges In Assessing College Students' Conception Of Duality: The Case Of Infinity, Grace Olutayo Babarinsa-Ochiedike

Open Access Theses & Dissertations

Interpreting students' views of infinity posits a challenge for researchers due to the dynamic nature of the conception. There is diversity and variation among students' process-object perceptions. The fluctuations between studentsâ?? views however reveal an undeveloped duality conception. This study examined college students' conception of duality in understanding and representing infinity with the intent to design strategies that could guide researchers in categorizing students' views of infinity into different levels.

Data for the study were collected from N=238 college students enrolled in Calculus sequence courses (Pre-Calculus, Calculus I through Calculus III) at one of the southwestern universities in the U.S. …

Towards Analytical Techniques For Optimizing Knowledge Acquisition, Processing, Propagation, And Use In Cyberinfrastructure, Leonardo Octavio Lerma

Open Access Theses & Dissertations

For many decades, there has been a continuous progress in science and engineering applications.

A large part of this progress comes from the new knowledge that researchers acquire, propagate, and use. This new knowledge has revolutionized many aspects of our life, from driving to communications to shopping.

Somewhat surprisingly, there is one area of human activity which is the least impacted by the modern technological progress: the very processes of acquiring, processing, and propagating information. When we decide where to place sensors, which algorithm to use for processing the data – we rely mostly on our own intuition and on …

Associativity Forcing Commutativity In Left Nil Rings, Md Al Masum Bhuiyan

Open Access Theses & Dissertations

Our study in this thesis is concentrated on the AFC groups (Associativity Forces Commutativity) and observing the ring structure there. We are looking for abelian groups yielding ring structures (K) where associativity forces commutativity. We call them K-AFC groups.

We consider that the only conditions put on a ring multiplication are the distributive laws over the additive group addition. After [1], We say that an abelian group (G, +) is an AFC-group if (i) there exists a nonassociative and noncommutative ring (G, +, ˙) and (ii) all associative rings (G, +, ˙) are commutative.

We call a ring left-nil if …

When Can The Primitive Element Be Written As A Sum Of Two Algebraic Elements Adjoined To The Field Of The Rational Numbers, Mohamad Moussa

Open Access Theses & Dissertations

Given a field F and elements \alpha and \beta not in F, then F(\alpha, \beta) is the smallest field containing \alpha,\beta, and F. A simple extension is a field extension which is generated by the adjunction of a single element. The Primitive Element Theorem says that if F is a field of characteristic 0, and \alpha and \beta are algebraic over F, then there is an element \gamma in F(\alpha ,\beta ) such that

F(\alpha ; \beta ) = F(\gamma). When can we say that \gamma=\alpha+\beta? We will introduce some situations where \gamma=\alpha+\beta is true and some when this is …

Stochastic Differential Equation Applied To High Frequency Data Arising In Geophysics And Other Disciplines, Osei Kofi Tweneboah

Open Access Theses & Dissertations

Estimating future seismic hazards of a region constitutes an important study many scholars have shown a renewed interest in the past few decades. A good estimation of the seismic hazard in a region requires predicting the location and magnitude of future seismic events. As the knowledge of the geophysical mechanisms that drive seismic events have increased, so have the corresponding mathematical model representations.

This Thesis is devoted to the study of modeling geophysical data. We propose a stochastic differential equation arising on the superposition of independent Ornstein-Uhlenbeck processes driven by a Gamma process.

Superposition of independent Gamma Ornstein-Uhlenbeck processes offers …

Constructive Mathematics In St. Petersburg, Russia: A (Somewhat Subjective) View From Within, Vladik Kreinovich

Departmental Technical Reports (CS)

In the 1970 and 1980s, logic and constructive mathematics were an important part of my life; it's what I defended in my Master's thesis, it was an important part of my PhD dissertation. I was privileged to work with the giants. I visited them in their homes. They were who I went to for advice. And this is my story.

Fast Algorithm For Finding Lattice Subspaces In Rn And Its Implementation, Andrew Martin Pownuk

Open Access Theses & Dissertations

There are known necessary and sufficient conditions for a subspace of Rm to be lattice-ordered. Let Y = {y1,…,ym} and yi are rows of the matrix X. A subspace ⟨X⟩, of linear space generated by the set X of n linearly independent positive vectors is lattice-ordered if and only the set X admits a fundamental set of indices I, which means that the subset YI ⊆ Y of vectors indexed by I is linearly independent, and every vector from Y\YI is a nonnegative linear combination of vectors form YI.

In economics it is possible to prove that the minimum-cost insured …

Homeomorphisms Between Pairs Of Genus Two Handlebodies And Separating Circles In Their Boundaries, Daniel Orlando Vazquez

Open Access Theses & Dissertations

A known result in low-dimensional topology is that every incompressible surface properly embedded in a genus two handlebody is boundary compressible. This result produces a pair consisting of a genus two handlebody and a separating curve in its boundary.

We study particular examples of genus two handlebodies and essential simple closed curves which separate their boundaries. The aim of this paper is to give our own construction of separating curves on the boundary of genus two handlebodies and compare them with previously studied pairs. These constructions involve taking 2n arcs on either side of a waist disk of a genus …

Why 20? Why 40? A Possible Explanation Of A Special Role Of 20 And 40 In Traditional Number Systems, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Both historical and linguistic evidence shows that numbers 20 and 40 played a special role in many traditional numerical systems. The fact that, e.g., the same number 20 appears in unrelated cultures such as Romans and Mayans is an indication that this number must have a general explanation related to human experience. In this paper, we provide a possible explanation of 20 and 40 along these lines: namely, we show that these numbers can be identified as the smallest sample sizes for which we can extract statistically significant information.

Finding The Best Function: A Way To Explain Calculus Of Variations To Engineering And Science Students, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practical problems, we need to find the most appropriate function: e.g., we need to find a control strategy u(t) that leads to the best performance of a system, we need to find the shape of the car which leads to the smallest energy losses, etc. Optimization over an unknown function can be described by the known Euler-Lagrange equations. The traditional way of deriving Euler-Lagrange equations when explaining them to the engineering and science students is, however, somewhat over-complicated. We provide a new, simpler way to deriving these equations, a way in which we directly use the fact that …

Dialect Or A New Language: A Possible Explanation Of The 70% Mutual Intelligibility Threshold, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In most cases, linguists have a consensus on when people from different regions speak two different dialects of the same language (and can, thus, understand each other reasonably well) or two different languages (in this case, their mutual intelligibility is limited). In most cases, this intuitive consensus corresponds to a 70% mutual intelligibility threshold: if at least 70% of the words from one region are understandable to people from another region, then these are two dialects, otherwise these are two different languages. In this paper, we provide a possible explanation for this 70% threshold.

From Urysohn's Universal Metric Space To A Universal Space-Time, A. G. Aksoy, Z. Glassman, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

A known Urysohn's result shows that there exists a universal} metric space, i.e., a metric space into every other (separable) metric space can be isomorphically embedded. Moreover, this universal metric space can be selected to be ultra-homogeneous -- every isomorphism of its two finite subsets can be extended to the isomorphism of the whole space.

Starting with Einstein's theories of Special and General relativity, space-times are described by a different type of structure -- a set (of events) equipped with the proper time t(a,b) between points a and b; such spaces are known as space-times with kinematic metric, or k-space-times. …

Why Rozenzweig-Style Midrashic Approach Makes Rational Sense: A Logical (Spinoza-Like) Explanation Of A Seemingly Non-Logical Approach, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

A 20 century German Jewish philosopher Franz Rosenzweig promoted a new approach to knowledge, an approach in which in addition to logical reasoning, coming up with stories with imagined additional details is also important. This approach is known as midrashic since it is similar to the use of similar stories -- known as midrashes -- in Judaism. While stories can make the material interesting, traditionally, such stories are not viewed as a serious part of scientific discovery. In this paper, we show that this seemingly non-logical approach can actually be explained in logical terms and thus, makes perfect rational sense.

Why In Mayan Mathematics, Zero And Infinity Are The Same: A Possible Explanation, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In Mayan mathematics, zero is supposed to be, in some sense, equal to infinity. At first glance, while this statement may have a deep philosophical meaning, it does not seem to make much mathematical sense. In this paper, we show, that this statement may be made mathematically reasonable. Specifically, on a real line, it is often useful to consider both −∞ and +∞ as a single infinity. When we deal with very small and very large numbers, it makes sense to use floating point representation, i.e., in effect, consider logarithms of the original values. In terms of logarithms, the original …

Complete Description Of Idempotent Hedges In Fuzzy Logic, Jaime Nava

Departmental Technical Reports (CS)

In describing expert knowledge, it is often important to properly take into account hedges} like "very", "somewhat", etc. In particular, fuzzy logic provides a consistent way of describing hedges. For some of the hedges, a repetition changes the meaning: e.g., "very very small" is smaller than "very small". However, other hedges -- like "somewhat" -- are idempotent, in the sense that repeating this hedge twice does not change the meaning. In this paper, we provide a complete description of such idempotent hedges.

On Early Stages Of Idea Propagation, The Number Of Adopters Grows As N(T) ~ C * Ta: Theoretical Explanation Of The Empirical Observation, L. Octavio Lerma, Deana Pennington, Vladik Kreinovich

Departmental Technical Reports (CS)

New good ideas sometimes propagate too slowly. To speed up their propagation, we need to have a quantitative understanding of how ideas propagate. An intuitive understanding of ideas propagation has led to several reasonable first-approximation mathematical models. These models provide a good description of idea propagation on the later stages, when the ideas have already been adopted by a reasonably large number of people. However, at the critically important early stages, these models are not perfect: these models predict a linear growth with time, while empirical growth data is often better described by a power law. In this paper, we …

Vine Copulas As A Way To Describe And Analyze Multi-Variate Dependence In Econometrics: Computational Motivation And Comparison With Bayesian Networks And Fuzzy Approaches, Songsak Sriboonchitta, Jianxu Liu, Vladik Kreinovich, Hung T. Nguyen

Departmental Technical Reports (CS)

In the last decade, vine copulas emerged as a new efficient techniques for describing and analyzing multi-variate dependence in econometrics. Our experience has shown, however, that while these techniques have been successfully applied to many practical problems of econometrics, there is still a lot of confusion and misunderstanding related to vine copulas. In this paper, we provide a motivation for this new technique from the computational viewpoint. We show that other techniques used to described dependence -- Bayesian networks and fuzzy techniques -- can be viewed as a particular case of vine copulas.

Conservation Of Energy Implies Conservation Of Momentum: How We Can Explain Conservation Of Momentum To Before-Calculus Students, Eric Freudenthal, Eric Hagedorn, Olga Kosheleva

Departmental Technical Reports (CS)

In solving physics problems, it is often important to use the laws of conservation of energy and momentum. While most people have intuitive understanding of energy and of its conservation, there is usually no intuition behind momentum, and known textbook derivations of conservation of momentum use calculus -- which is usually taught after momentum. In this paper, we show how the law of conservation of momentum can be explained to before-calculus student: by using the fact that this law can be derived from the more intuitive conservation of energy if we consider energy in different coordinate systems.

How To Distinguish True Dependence From Varying Independence?, Marketa Krmelova, Martin Trnecka, Vladik Kreinovich, Berlin Wu

Departmental Technical Reports (CS)

A usual statistical criterion for the quantities X and Y to be independent is that the corresponding distribution function F(x,y) is equal to the product of the corresponding marginal distribution functions. If this equality is violated, this is usually taken to mean that X and Y are dependent. In practice, however, the inequality may be caused by the fact that we have a mixture of several populations, in each of which X and Y are independent. In this paper, we show how we can distinguish true dependence from such varying independence. This can also lead to new measures to degree …

Why Trapezoidal And Triangular Membership Functions Work So Well: Towards A Theoretical Explanation, Aditi Barua, Lalitha Snigdha Mudunuri, Olga Kosheleva

Departmental Technical Reports (CS)

In fuzzy logic, an imprecise ("fuzzy") property is described by its membership function μ(x), i.e., by a function which describes, for each real number x, to what degree this real number satisfies the desired property. In principle, membership functions can be of different shape, but in practice, trapezoidal and triangular membership functions are most frequently used. In this paper, we provide an interval-based theoretical explanation for this empirical fact.

Minimization Of Average Sensitivity As A Method Of Selecting Fuzzy Functions And Operations: Successes And Limitations, Riya George, Suresh Subramanian, Alejandro Vega, Olga Kosheleva

Departmental Technical Reports (CS)

Fuzzy logic is an extension of the standard 2-valued logic -- with two possible truth values 0 ("false") and ("true") -- to values (degrees of certainty) represented by arbitrary numbers from the interval [0,1]. One of the main challenges in fuzzy logic is that we need to extend the usual logical operations from the set {0,1} to the entire interval, and there are many possible extensions. One promising technique for selecting a reasonable extension is to take into account that the fuzzy degrees of certainty are themselves only known with uncertainty; so, it makes sense to select an operation which …

√(X2 + Μ) Is The Most Computationally Efficient Smooth Approximation To |X|: A Proof, Carlos Ramirez, Reinaldo Sanchez, Vladik Kreinovich, Miguel Argaez

Departmental Technical Reports (CS)

In many practical situations, we need to minimize an expression of the type |c₁| + ... + |c_n|. The problem is that most efficient optimization techniques use the derivative of the objective function, but the function |x| is not differentiable at 0. To make optimization efficient, it is therefore reasonable to approximate |x| by a smooth function. We show that in some reasonable sense, the most computationally efficient smooth approximation to |x| is the function √(x² + μ), a function which has indeed been successfully used in such optimization.

Towards A Better Understanding Of Space-Time Causality: Kolmogorov Complexity And Causality As A Matter Of Degree, Vladik Kreinovich, Andres Ortiz

Departmental Technical Reports (CS)

Space-time causality is one of the fundamental notions of modern physics; however, it is difficult to define in observational physical terms. Intuitively, the fact that a space-time event e=(t,x) can causally influence an event e'=(t',x') means that what we do in the vicinity of e changes what we observe at e'. If we had two copies of the Universe, we could perform some action at e in one copy but not in another copy; if we then observe the difference at e', this would be an indication of causality. However, we only observe one Universe, in which we either perform …

Imprecise Probabilities In Engineering Analyses, Michael Beer, Scott Ferson, Vladik Kreinovich

Departmental Technical Reports (CS)

Probabilistic uncertainty and imprecision in structural parameters and in environmental conditions and loads are challenging phenomena in engineering analyses. They require appropriate mathematical modeling and quantification to obtain realistic results when predicting the behavior and reliability of engineering structures and systems. But the modeling and quantification is complicated by the characteristics of the available information, which involves, for example, sparse data, poor measurements and subjective information. This raises the question whether the available information is sufficient for probabilistic modeling or rather suggests a set-theoretical approach. The framework of imprecise probabilities provides a mathematical basis to deal with these problems which …