Physical Sciences and Mathematics Commons^™

Knowledge Geometry Is Similar To General Relativity: Both Mass And Knowledge Curve The Corresponding Spaces, Francisco Zapata, Vladik Kreinovich

Departmental Technical Reports (CS)

In this paper, we explain and explore the idea that knowledge is similar to mass in physics: similarly to how mass curves space-time, knowledge curves the corresponding knowledge space.

Approximate Nature Of Traditional Fuzzy Methodology Naturally Leads To Complex-Valued Fuzzy Degrees, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In the traditional fuzzy logic, the experts' degrees of confidence in their statements is described by numbers from the interval [0,1]. These degree have a clear intuitive meaning. Somewhat surprisingly, in some applications, it turns out to be useful to also consider different numerical degrees -- e.g., complex-valued degrees. While these complex-valued degrees are helpful in solving practical problems, their intuitive meaning is not clear. In this paper, we provide a possible explanation for the success of complex-valued degrees which makes their use more intuitively understandable -- namely, we show that these degrees naturally appear due to the approximate nature …

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 …

Why Triangular Membership Functions Work Well In F-Transform: A Theoretical Explanation, Jaime Nava, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practical applications, it is useful to represent a signal or an image by its average values on several fuzzy sets. The corresponding {\em F-transform} technique has many useful applications in signal and image processing. In principle, we can use different membership functions. Somewhat surprisingly, in many applications, the best results occur when we use triangular membership functions. In this paper, we provide a possible theoretical explanation for this empirical phenomenon.

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.

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.

Feasible Algorithms For Lattice And Directed Subspaces, Jennifer (Hamlyn) Del Valle, Vladik Kreinovich, Piotr Wojciechowski

Departmental Technical Reports (CS)

In some practical situations (e.g., in econometrics), it is important to check whether a given linear subspace of a space R^m with component-wise order is a lattice -- and if it is not, whether it is at least a directed ordered space. Because of the practical importance, it is desirable to have feasible algorithms for solving these problems -- which in Computer Science is usually interpreted as algorithms whose computation time does not exceed a polynomial of the length of the input. No such algorithms were previously known. In this paper, we present feasible algorithms for solving both problems.

Towards Designing Optimal Individualized Placement Tests, Octavio Lerma, Olga Kosheleva, Shahnaz Shahbazova, Vladik Kreinovich

Departmental Technical Reports (CS)

To find the current level of a student's knowledge, we use a sequence of problems of increasing complexity; if a student can solve a problem, the system generates a more complex one; if a student cannot solve a problem, the system generates an easier one. To find a proper testing scheme, we must take into account that every time a student cannot solve a problem, he/she gets discouraged. To take this into account, we define an overall effect on a student by combining "positive" and "negative" problems with different weights, and we design a testing scheme which minimizes this effect.

"And"- And "Or"-Operations For "Double", "Triple", Etc. Fuzzy Sets, Hung T. Nguyen, Vladik Kreinovich, Olga Kosheleva

Departmental Technical Reports (CS)

In the traditional fuzzy logic, the expert's degree of confidence d(A & B) in a complex statement A & B (or A\/B) is uniquely determined by his/her degrees of confidence d(A) and d(B) in the statements A and B, as f_&(d(A),d(B)) for an appropriate "and"-operation (t-norm). In practice, for the same degrees d(A) and d(B), we may have different degrees d(A & B) depending on the relation between A and B. The best way to take this relation into account is to explicitly elicit the corresponding degrees d(A & B) and d(A\/B), i.e., to come up with "double" …

Fuzziness And Bayesian Analysis In Engineering, Matthias Stein, Michael Beer, Vladik Kreinovich

Departmental Technical Reports (CS)

An engineering analysis requires a realistic quantification of all input information. The amount and quality of the available information dictates the uncertainty model and its associated quantification concept. For inconsistent information, a distinction between probabilistic and non-probabilistic characteristics is beneficial. In this distinction, uncertainty refers to probabilistic characteristics and non-probabilistic characteristics are summarized as imprecision. When uncertainty and imprecision occur simultaneously, the uncertainty model fuzzy randomness appears useful. In a Bayesian approach the fuzzy probabilistic model provides the opportunity to take account of imprecision in data and in prior expert knowledge. The Bayesian approach ex-tended to inconsistent information is demonstrated …

Finding Specifications Of While Statements Using Patterns, Aditi Barua, Yoonsik Cheon

Departmental Technical Reports (CS)

A formal correctness proof of code containing loops such as while statements typically uses the technique of proof-by-induction, and often the most difficult part of carrying out an inductive proof is formulating a correct induction hypothesis, a specification for a loop statement. An incorrect induction hypothesis will surely lead to a proof failure. In this paper we propose a systematic way for identifying specifications of while statements. The key idea of our approach is to categorize and document common patterns of while statements along with their specifications. This is based on our observation that similarly-structured while statements frequently have similarly-structured …

Algebraic Product Is The Only T-Norm For Which Optimization Under Fuzzy Constraints Is Scale-Invariant, Juan Carlos Figueroa Garcia, Martine Ceberio, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practical situations, we need to optimize under fuzzy constraints. There is a known Bellman-Zadeh approach for solving such problems, but the resulting solution, in general, depends on the choice of a not well-defined constant M. We show that this dependence disappears if we use an algebraic t-norm (and-operation) a * b, and we also prove that the algebraic product is the only t-norm for which the corresponding solution is independent on M.

A Vine Copula Approach For Analyzing Financial Risk And Co-Movement Of The Indonesian, Philippine And Thailand Stock Markets, Songsak Sriboonchitta, Jianxu Liu, Vladik Kreinovich, Hung T. Nguyen

Departmental Technical Reports (CS)

This paper aims at analyzing the financial risk and co-movement of stock markets in three countries: Indonesia, Philippine and Thailand. It consists of analyzing the conditional volatility and test the leverage effect in the stock markets of the three countries. To capture the pairwise and conditional dependence between the variables, we use the method of vine copulas. In addition, we illustrate the computations of the value at risk and the expected shortfall using Monte Carlo simulation with copula based GJR-GARCH model. The empirical evidence shows that all the leverage effects add much to the capacity for explanation of the three …

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. …

Studying Volatility And Dependency Of Chinese Outbound Tourism Demand In Singapore, Malaysia, And Thailand: A Vine Copula Approach, Jianxu Liu, Songsak Sriboonchitta, Hung T. Nguyen, Vladik Kreinovich

Departmental Technical Reports (CS)

This paper investigates the volatility and dependence of Chinese tourism demand for Singapore, Malaysia, and Thailand (SMT) destinations, using the vine copula based auto regression moving average-generalized autoregressive conditional heteroskedasticity (ARMA-GARCH) model. It is found that a jolt to the tourist flow can have long-standing ramifications for the SMT countries. The estimation of the vine copulas among SMT show that the Survival Gumbel, Frank, and Gaussian copulas are the best copulas for Canonical vine (C-vine) or Drawable vine (D-vine) among the possible pair-copulas. In addition, this paper illustrates the making of time-varying Frank copulas for vine copulas. Finally, there is …

Picture Fuzzy Sets - A New Concept For Computational Intelligence Problems, Bui Cong Cuong, Vladik Kreinovich

Departmental Technical Reports (CS)

Since Zadeh introduced fuzzy sets in 1965, a lot of new theories treating imprecision and uncertainty have been introduced. Some of these theories are extensions of fuzzy set theory, other try to handle imprecision and uncertainty in different way. In this paper, we introduce a new notion of picture fuzzy sets (PFS), which are directly extensions of fuzzy sets and of intuitonistic fuzzy sets (Atanassov). Then some operations on picture fuzzy sets are defined and some properties of these operations are considered. Here the basic preliminaries of PFS theory are presented.

Data Collection For The Similar Segments In Social Speech Task, Nigel G. Ward, Steven D. Werner

Departmental Technical Reports (CS)

Information retrieval systems rely heavily on models of similarity, but for spoken dialog such models currently use mostly standard textual-content similarity. As part of the MediaEval Benchmarking Initiative, we have created a new corpus to support development of similarity models for spoken dialog. This corpus includes 26 casual dialogs among members of two semi-cohesive groups, totaling about 5 hours, with 1889 labeled regions associated into 227 sets which annotators judged to be similar enough to share a tag. This technical report brings together information about this corpus and its intended uses.

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 …

Similarity Approach To Defining Basic Level Of Concepts Explained From The Utility Viewpoint, Joe Lorkowski, Martin Trnecka

Departmental Technical Reports (CS)

In many practical situations, it is necessary to describe an image in words. From the purely logical viewpoint, to describe the same object, we can use concepts of different levels of abstraction: e.g., when the image includes a dog, we can say that it is a dog, or that it is a mammal, or that it is a German Shepherd. In such situations, humans usually select a concept which, to them, in the most natural; this concept is called the basic level concept. However, the notion of a basic level concept is difficult to describe in precise terms; as a …

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.

Fuzzy Sets Can Be Interpreted As Limits Of Crisp Sets, And This Can Help To Fuzzify Crisp Notions, Olga Kosheleva, Vladik Kreinovich, Thavatchai Ngamsantivong

Departmental Technical Reports (CS)

Fuzzy sets have been originally introduced as generalizations of crisp sets, and this is how they are usually considered. From the mathematical viewpoint, the problem with this approach is that most notions allow many different generalizations, so every time we try to generalize some notions to fuzzy sets, we have numerous alternatives. In this paper, we show that fuzzy sets can be alternatively viewed as limits of crisp sets. As a result, for some notions, we can come up with a unique generalization -- as the limit of the results of applying this notion to the corresponding crisp sets.

Computing Covariance And Correlation In Optimally Privacy-Protected Statistical Databases: Feasible Algorithms, Joshua Day, Ali Jalal-Kamali, Vladik Kreinovich

Departmental Technical Reports (CS)

In many real-life situations, e.g., in medicine, it is necessary to process data while preserving the patients' confidentiality. One of the most efficient methods of preserving privacy is to replace the exact values with intervals that contain these values. For example, instead of an exact age, a privacy-protected database only contains the information that the age is, e.g., between 10 and 20, or between 20 and 30, etc. Based on this data, it is important to compute correlation and covariance between different quantities. For privacy-protected data, different values from the intervals lead, in general, to different estimates for the desired …

Note On Fair Price Under Interval Uncertainty, Joshua Mckee, Joe Lorkowski, Thavatchai Ngamsantivong

Departmental Technical Reports (CS)

Often, in decision making situations, we do not know the exact value of a gain resulting from making each decision, we only know the bounds on this gain. To make a reasonable decision under such interval uncertainty, it makes sense to estimate the fair price of each alternative, and then to select the alternative with the highest price. In this paper, we show that the value of the fair price can be uniquely determined from some reasonable requirements: e.g., the additivity requirement, that the fair price of two objects together should be equal to the sum of the fair prices …

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.

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.

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.

How To Faster Test A Device For Different Combinations Of Parameters, Francisco Zapata, Luis C. Gutierrez, Vladik Kreinovich

Departmental Technical Reports (CS)

A device has to function properly under all possible conditions: e.g., for all temperatures within a given range, for all possible humidity values within a given range, etc. Ideally, it would be nice to be able to test a device for all possible combinations of these parameters, but the number of such combinations is often so huge that such an exhaustive testing is not possible. Instead, it is reasonable to check the device for all possible values of each parameter, for each possible pairs of values of two parameters, and, in general, for all possible combinations of values of k …

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.

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 …