Mathematics Commons^™

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 …

A Convex Optimization Algorithm For Sparse Representation And Applications In Classification Problems, Reinaldo Sanchez Arias

Open Access Theses & Dissertations

In pattern recognition and machine learning, a classification problem refers to finding an algorithm for assigning a given input data into one of several categories. Many natural signals are sparse or compressible in the sense that they have short representations when expressed in a suitable basis. Motivated by the recent successful development of algorithms for sparse signal recovery, we apply the selective nature of sparse representation to perform classification. Any test sample is represented in an overcomplete dictionary with the training sample as base elements. A given test sample can be expressed as a linear combination of only those training …

Generalizations Of Dirichlet Convolution, Juan Carlos Villarreal

Open Access Theses & Dissertations

The thesis reviews Dirichlet convolution of arithmetic functions and some generalizations. First, we present the main properties of the Dirichlet convolution on the set of arithmetic functions and some characterizations of prime numbers via arithmetic functions are given. Then we explore the K-convolution and the regular convolutions and finally we extend the study of Dirichlet convolution in categories. We give also an example for the evaluation of the Mobius function of a Mobius category.

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 …

Zadeh's Vision Of Going From Fuzzy To Computing With Words: From The Idea's Origin To Current Successes To Remaining Challenges, Vladik Kreinovich

Departmental Technical Reports (CS)

No abstract provided.

If Energy Is Not Preserved, Then Planck's Constant Is No Longer A Constant: A Theorem, Vladik Kreinovich, Andres Ortiz

Departmental Technical Reports (CS)

For any physical theory, to experimentally check its validity, we need to formulate an alternative theory and check whether the experimental results are consistent with the original theory or with an alternative theory. In particular, to check whether energy is preserved, it is necessary to formulate an alternative theory in which energy is not preserved. Formulating such a theory is not an easy task in quantum physics, where the usual Schroedinger equation implicitly assumes the existence of an energy (Hamiltonian) operator whose value is preserved. In this paper, we show that the only way to get a consistent quantum theory …

Towards Unique Physically Meaningful Definitions Of Random And Typical Objects, Luc Longpre, Olga Kosheleva

Departmental Technical Reports (CS)

To distinguish between random and non-random sequence, Kolmogorov and Martin-Lof proposed a new definition of randomness, according to which an object (e.g., a sequence of 0s and 1s) if random if it satisfies all probability laws, i.e., in more precise terms, if it does not belong to any definable set of probability measure 0. This definition reflect the usual physicists' idea that events with probability 0 cannot happen. Physicists -- especially in statistical physics -- often claim a stronger statement: that events with a very small probability cannot happen either. A modification of Kolmogorov-Martin-Lof's (KLM) definition has been proposed to …

Why Clayton And Gumbel Copulas: A Symmetry-Based Explanation, Vladik Kreinovich, Hung T. Nguyen, Songsak Sriboonchitta

Departmental Technical Reports (CS)

In econometrics, many distributions are non-Gaussian. To describe dependence between non-Gaussian variables, it is usually not sufficient to provide their correlation: it is desirable to also know the corresponding copula. There are many different families of copulas; which family shall we use? In many econometric applications, two families of copulas have been most efficient: the Clayton and the Gumbel copulas. In this paper, we provide a theoretical explanation for this empirical efficiency, by showing that these copulas naturally follow from reasonable symmetry assumptions. This symmetry justification also allows us to provide recommendations about which families of copulas we should use …

Orders On Intervals Over Partially Ordered Sets: Extending Allen's Algebra And Interval Graph Results, Francisco Zapata, Vladik Kreinovich, Cliff Joslyn, Emilie Hogan

Departmental Technical Reports (CS)

To make a decision, we need to compare the values of quantities. In many practical situations, we know the values with interval uncertainty. In such situations, we need to compare intervals. Allen's algebra describes all possible relations between intervals on the real line which are generated by the ordering of endpoints; ordering relations between such intervals have also been well studied. In this paper, we extend this description to intervals in an arbitrary partially ordered set (poset). In particular, we explicitly describe ordering relations between intervals that generalize relation between points. As auxiliary results, we provide a logical interpretation of …

Possible And Necessary Orders, Equivalences, Etc.: From Modal Logic To Modal Mathematics, Francisco Zapata, Olga Kosheleva

Departmental Technical Reports (CS)

In practice, we are often interested in order relations (e.g., when we describe preferences) or equivalence relations (e.g., when we describe clustering). Often, we do not have a complete information about the corresponding relation; as a result, we have several relations consistent with our knowledge. In such situations, it is desirable to know which elements a and b are possibly connected by the relation and which are necessarily connected by this relation. In this paper, we provide a full description of all such possible and necessary orders and equivalence relations. For example, possible orders are exactly reflexive relations, while necessary …

In Applications, A Rigorous Proof Is Not Enough: It Is Also Important To Have An Intuitive Understanding, Vladik Kreinovich

Departmental Technical Reports (CS)

From a purely mathematical viewpoint, once a statement is rigorously proven, it should be accepted as true. Surprisingly, in applications, users are often reluctant to accept a rigorously proven statement until the proof is supplemented by its intuitive explanation. In this paper, we show that this seemingly unreasonable reluctance makes perfect sense: the proven statement is about the mathematical model which is an approximation to the actual system; an intuitive explanation provides some confidence that the statement holds not only for the model, but also for systems approximately equal to this model -- in particular, for the actual system of …

From Unbiased Numerical Estimates To Unbiased Interval Estimates, Baokun Li, Gang Xiang, Vladik Kreinovich, Panagios Moscopoulos

Departmental Technical Reports (CS)

One of the main objectives of statistics is to estimate the parameters of a probability distribution based on a sample taken from this distribution. Of course, since the sample is finite, the estimate X is, in general, different from the actual value x of the corresponding parameter. What we can require is that the corresponding estimate is unbiased, i.e., that the mean value of the difference X - x is equal to 0: E[X] = x. In some problems, unbiased estimates are not possible. We show that in some such problems, it is possible to have interval unbiased estimates, i.e., …

Kansei Engineering: Towards Optimal Set Of Designs, Van-Nam Huynh, Octavio Lerma, Vladik Kreinovich

Departmental Technical Reports (CS)

In many engineering situations, we need to take into account subjective user preferences; taking such preference into account is known as {\em Kansei Engineering}. In this paper, we formulate the problem of selecting optimal set of designs in Kansei engineering as a mathematical optimization problem, and we provide an explicit solution to this optimization problem.

How To Define Mean, Variance, Etc., For Heavy-Tailed Distributions: A Fractal-Motivated Approach, Vladik Kreinovich, Olga Kosheleva

Departmental Technical Reports (CS)

In many practical situations, we encounter heavy-tailed distributions for which the variance -- and even sometimes the mean -- are infinite. We propose a fractal-motivated approach that enables us to gauge the mean and variance of such distributions.

Assessing Readability Of College-Level Statistics Textbooks: A Quantitative Lexical And Grammatical Analysis, Ariel I. Gonzalez, Amy E. Wagler, Lawrence M. Lesser

COURI Symposium Abstracts, Summer 2012

This research was conceived because of the need to create and identify college level statistics textbooks (or other instructional materials)) that are written in a manner that is less intimidating and easier to read and understand for students, including English language learners. We selected a representative set of current editions of 15-20 well-known statistics textbooks (that varied in their incorporation of reform or literacy approaches) to assess and compare aspects and levels of readability. We then chose two topics for a manageable focus: measures of center and line of fit. The parts of the textbooks corresponding to these topics were …

Generating Minimal T-Wise Test Suites, Luis C. Gutierrez, Carlos Nieto, Francisco Zapata, Martine Ceberio

COURI Symposium Abstracts, Summer 2012

As the use of computing devices increases every day, users rely on the adequate functioning of software. When software is not tested properly, it can yield erroneous information or a complete failure of the system. The NIST estimates that defective software cost the United States economy close to $60 billion a year. Therefore, there is a need to develop software testing techniques that are time and cost effective. Fully testing software under all possible combinations of parameters values cannot be reduced. However, testing can focus on covering all combinations of subsets of parameters and empirical data shows that doing so …

An Intelligent System To Manage Soft Constraints, Gabriel Higareda, Eduardo Breijo, Martine Ceberio

COURI Symposium Abstracts, Summer 2012

Most problems can be expressed in terms of requirements that must be met by their expected solutions. Constraints are requirements and constraint programming is a declarative programming paradigm that simply consists in modeling problems as lists of their requirements. Many constraint solvers exist that provide solutions to such list of requirements.

However, constraint solvers are stiff: if a problem was ill modeled, no solution will be returned and the original problem called over-constrained. Nevertheless there is hope even in these cases: such overconstrained problems can be handled as soft constraints, meaning that their processing will not be stiff anymore and …

Decision Making Under Interval And Fuzzy Uncertainty: Towards An Operational Approach, Rafik Aliev, Oleg H. Huseynov, Vladik Kreinovich

Departmental Technical Reports (CS)

Traditional decision theory is based on a simplifying assumption that for each two alternatives, a user can always meaningfully decide which of them is preferable. In reality, often, when the alternatives are close, the user is either completely unable to select one of these alternatives, or selects one of the alternatives only "to some extent". How can we extend the traditional decision theory to such realistic interval and fuzzy cases? In their previous papers, the first two authors proposed a natural generalization of the usual decision theory axioms to interval and fuzzy cases, and described decision coming from this generalization. …

Membership Functions Or Alpha-Cuts? Algorithmic (Constructivist) Analysis Justifies An Interval Approach, Vladik Kreinovich

Departmental Technical Reports (CS)

In his pioneering papers, Igor Zaslavsky started an algorithmic (constructivist) analysis of fuzzy logic. In this paper, we extend this analysis to fuzzy mathematics and fuzzy data processing. Specifically, we show that the two mathematically equivalent representations of a fuzzy number -- by a membership function and by alpha-cuts -- are not algorithmically equivalent, and only the alpha-cut representation enables us to efficiently process fuzzy data.

Estimating Correlation Under Interval And Fuzzy Uncertainty: Case Of Hierarchical Estimation, Ali Jalal-Kamali

Departmental Technical Reports (CS)

In many situations, we are interested in finding the correlation ρ between different quantities x and y based on the values x_i and y_i of these quantities measured in different situations i. The correlation is easy to compute when we know the exact sample values x_i and y_i. In practice, the sample values come from measurements or from expert estimates; in both cases, the values are not exact. Sometimes, we know the probabilities of different values of measurement errors, but in many cases, we only know the upper bounds Δ_xi and Δ_yi on …

Semi-Heuristic Target-Based Fuzzy Decision Procedures: Towards A New Interval Justification, Christian Servin, Van-Nam Huynh, Yoshiteru Nakamori

Departmental Technical Reports (CS)

To more adequately describe human decision making, V.-N. Nuynh, Y. Nakamori, and others proposed a special semi-heuristic target-based fuzzy decision procedure. A usual justification for this procedure is based on the selection of the simplest possible membership functions and "and"- and "or"-operations; if we use more complex membership functions and "and"- and "or"-operations, we get different results. Interestingly, in practical applications, the procedure based on the simplest choices most adequately describes human preferences. It is therefore desirable to come up with a justification that explains this empirical fact. Such a justification is proposed in this paper

How To Define Average Class Size (And Deviations From The Average Class Size) In A Way Which Is Most Adequate For Teaching Effectiveness, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

When students select a university, one of the important parameters is the average class size. This average is usually estimated as an arithmetic average of all the class sizes. However, it has been recently shown that to more adequately describe students' perception of a class size, it makes more sense to average not over classes, but over all students -- which leads to a different characteristics of the average class size. In this paper, we analyze which characteristic is most adequate from the viewpoint of efficient learning. Somewhat surprisingly, it turns out that the arithmetic average is the most adequate …

Matrix Product Application To Fibonacci Sequence, Abigail Damaris Muniz^, Jesus Loya^, Hamide Dogan*

COURI Symposium Abstracts, Spring 2012

No abstract provided.

Using Bioinformatics To Efficiently Organize And Analyze Significant Immunogenic Epitope Sequences In Various Stages Of Trypanosoma Cruzi, Karla Singh^, Andrea Wurm, Clemente Aguilar, Alexandre Marques, M-Y Leung*, Igor Almeida*

COURI Symposium Abstracts, Spring 2012

No abstract provided.