Mathematics Commons^™

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 …

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 …

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 …

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 …

Why ℓ1 Is A Good Approximation To ℓ0: A Geometric Explanation, Carlos Ramirez, Vladik Kreinovich, Miguel Argaez

Departmental Technical Reports (CS)

In practice, we usually have partial information; as a result, we have several different possibilities consistent with the given measurements and the given knowledge. For example, in geosciences, several possible density distributions are consistent with the measurement results. It is reasonable to select the simplest among such distributions. A general solution can be described, e.g., as a linear combination of basic functions. A natural way to define the simplest solution is to select a one for which the number of the non-zero coefficients c_i is the smallest. The corresponding "l₀-optimization" problem is non-convex and therefore, difficult to …

Estimating Third Central Moment C3 For Privacy Case Under Interval And Fuzzy Uncertainty, Ali Jalal-Kamali, Vladik Kreinovich

Departmental Technical Reports (CS)

Some probability distributions (e.g., Gaussian) are symmetric, some (e.g., lognormal) are non-symmetric ({\em skewed}). How can we gauge the skeweness? For symmetric distributions, the third central moment C₃ = E[(x - E(x))³] is equal to 0; thus, this moment is used to characterize skewness. This moment is usually estimated, based on the observed (sample) values x₁, ..., x_n, as C₃ = (1/n) * ((x₁ - E)³ + ... + (x_n - E)³), where E = (1/n) * (x₁ + ... + x_n). In many …

Use Of Grothendieck Inequality In Interval Computations: Quadratic Terms Are Estimated Accurately Modulo A Constant Factor, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

One of the main problems of interval computations is to compute the range of a given function f over given intervals. For a linear function, we can feasibly estimate its range, but for quadratic (and for more complex) functions, the problem of computing the exact range is NP-hard. So, if we limit ourselves to feasible algorithms, we have to compute enclosures instead of the actual ranges. It is known that asymptotically the smallest possible excess width of these enclosures is O(Δ²), where Δ is the largest half-width of the input intervals. This asymptotics is attained for the Mean …

Checking Monotonicity Is Np-Hard Even For Cubic Polynomials, Andrzej Pownuk, Luc Longpre, Vladik Kreinovich

Departmental Technical Reports (CS)

One of the main problems of interval computations is to compute the range of a given function over given intervals. In general, this problem is computationally intractable (NP-hard) -- that is why we usually compute an enclosure and not the exact range. However, there are cases when it is possible to feasibly compute the exact range; one of these cases is when the function is monotonic with respect to each of its variables. The monotonicity assumption holds when the derivatives at a midpoint are different from 0 and the intervals are sufficiently narrow; because of this, monotonicity-based estimates are often …

Why Complex-Valued Fuzzy? Why Complex Values In General? A Computational Explanation, Olga Kosheleva, Vladik Kreinovich, Thavatchai Ngamsantivong

Departmental Technical Reports (CS)

In the traditional fuzzy logic, as truth values, we take all real numbers from the interval [0,1]. In some situations, this set is not fully adequate for describing expert uncertainty, so a more general set is needed. From the mathematical viewpoint, a natural extension of real numbers is the set of complex numbers. Complex-valued fuzzy sets have indeed been successfully used in applications of fuzzy techniques. This practical success leaves us with a puzzling question: why complex-valued degree of belief, degrees which do not seem to have a direct intuitive meaning, have been so successful? In this paper, we use …

Bayesian Approach For Inconsistent Information, M. Stein, Michael Beer, Vladik Kreinovich

Departmental Technical Reports (CS)

In engineering situations, we usually have a large amount of prior knowledge that needs to be taken into account when processing data. Traditionally, the Bayesian approach is used to process data in the presence of prior knowledge. Sometimes, when we apply the traditional Bayesian techniques to engineering data, we get inconsistencies between the data and prior knowledge. These inconsistencies are usually caused by the fact that in the traditional approach, we assume that we know the {\it exact} sample values, that the prior distribution is {\it exactly} known, etc. In reality, the data is imprecise due to measurement errors, the …

Software Development For A 3d Gravity Inversion And Application To Study Of The Border Ranges Fault System, South-Central Alaska, Rolando Cardenas

Open Access Theses & Dissertations

The Border Ranges Fault System (BRFS) bounds the Cook Inlet and Susitna Basins, and is an important petroleum province within south-central Alaska. A primary goal of our research is to test several plausible models of structure along the BRFS using a novel three-dimensional inversion technique utilizing gravity data, constrained with other geophysical, borehole and surface geological information. This research involves the development of 3D inversion modeling software using C++ Builder from Embarcadero's XE2 Suite. The novel inversion approach directly models known geology with a priori uncertainties assigned to the geologic model to allow researchers to compare alternative interpretations. This technique …

Investigating The Validity Of A Survey Intended To Gauge Mathematics Problem-Solving Disposition Along The Impulsive-Analytic Dimension, Miguel Mendoza

Open Access Theses & Dissertations

This study investigates the validity of the Likelihood-to-Act (LtA) survey developed to assess respondents' problem-solving disposition along the impulsive-analytic dimension. Sixteen pre-service teachers were strategically selected from a pool of 495 LtA survey respondents to participate in a semi-structured task-based interview approximately one hour in length. The interview consisted of three main parts: a problem solving section, a consistency section, and a classification section.

* Part 1 of the interview was developed to get interviewees to solve five open-response problems meant to elicit an impulsive response. Interviewees were also asked to verbalize their thought process in order for the researchers …