Physical Sciences and Mathematics Commons^™

Occam's Razor Explains Matthew Effect, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Sociologists of science noticed that the results of many collaborative projects and discoveries are often attributed only to their most famous collaborators, even when the contributions of these famous collaborators were minimal. This phenomenon is known as the Matthew effect, after a famous citation from the Gospel of Matthew. In this article, we show that Occam's razor provides a possible explanation for the Matthew effect.

Constructive Mathematics Is Seemingly Simple But There Are Still Open Problems: Kreisel's Observation Explained, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In his correspondence with Grigory Mints, the famous logician Georg Kreisel noticed that many results of constructive mathematics seem easier-to-prove than the corresponding classical (non-constructive) results -- although he noted that these results are still far from being simple and the corresponding open problems are challenging. In this paper, we provide a possible explanation for this empirical observation.

Conditional Dimension In Metric Spaces: A Natural Metric-Space Counterpart Of Kolmogorov-Complexity-Based Mutual Dimension, Vladik Kreinovich, Luc Longpre, Olga Kosheleva

Departmental Technical Reports (CS)

It is known that dimension of a set in a metric space can be characterized in information-related terms -- in particular, in terms of Kolmogorov complexity of different points from this set. The notion of Kolmogorov complexity K(x) -- the shortest length of a program that generates a sequence x -- can be naturally generalized to conditionalKolmogorov complexity K(x:y) -- the shortest length of a program that generates x by using y as an input. It is therefore reasonable to use conditional Kolmogorov complexity to formulate a conditional analogue of dimension. Such a generalization has indeed been proposed, under …

Invariance Explains Multiplicative And Exponential Skedactic Functions, Vladik Kreinovich, Olga Kosheleva, Hung T. Nguyen, Songsak Sriboonchitta

Departmental Technical Reports (CS)

In many situation, we have an (approximately) linear dependence between several quantities y = f(x₁, ..., x_n). The variance v of the corresponding approximation error often depends on the values of the quantities x₁, ..., x_n: v = v(x₁, ..., x_n); the function describing this dependence is known as the skedactic function. Empirically, two classes of skedactic functions are most successful: multiplicative functions v = c * |x₁|^γ₁ * ... * |x_n|^γ_n and exponential functions v = exp(α + …

Why Linear (And Piecewise Linear) Models Often Successfully Describe Complex Non-Linear Economic And Financial Phenomena: A Fuzzy-Based Explanation, Hung T. Nguyen, Vladik Kreinovich, Olga Kosheleva, Songsak Sriboonchitta

Departmental Technical Reports (CS)

Economic and financial phenomena are highly complex and non-linear. However, surprisingly, in many cases, these phenomena are accurately described by linear models -- or, sometimes, by piecewise linear ones. In this paper, we show that fuzzy techniques can explain the unexpected efficiency of linear and piecewise linear models: namely, we show that a natural fuzzy-based precisiation of imprecise ("fuzzy") expert knowledge often leads to linear and piecewise linear models.

We also discuss which expert-motivated nonlinear models should be used to get a more accurate description of economic and financial phenomena.

Gazelle Companies: What Is So Special About The 20% Threshold?, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In business analysis, a special emphasis is placed on "gazelles", companies that grow by at least 20% per year for several years (usually four). While this 20% threshold is somewhat supported by empirical research, from the theoretical viewpoint, it is not clear what is so special about this value. In this paper, we provide a possible explanation for this empirical fact.

Al-Sijistani's And Maimonides's Double Negation Theology Explained By Constructive Logic, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Famous medieval philosophers Al-Sijistani and Maimonides argued that the use of double negation helps us to better understand issues related to theology. To a modern reader, however, their arguments are somewhat obscure and unclear. We show that these arguments can be drastically clarified if we take into account the 20 century use of double negation in constructive logic.

Why Political Scientists Are Wrong 15% Of The Time, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

An experimental study has shown that among situations when political scientists claimed that a political outcome was impossible, this outcome actually occurred in 15% of the cases. In this paper, we provide a possible explanation for this empirical fact.

How To Take Into Account A Student's Degree Of Certainty When Evaluating The Test Results, Joe Lorkowski, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

To more adequately gauge the student's knowledge, it is desirable to take into account not only whether the student's answers on the test are correct or nor, but also how confident the students are in their answers. For example, a situation when a student gives a wrong answer, but understands his/her lack of knowledge on this topic, is not as harmful as the situation when the student is absolutely confident in his/her wrong answer. In this paper, we use the general decision making theory to describe the best way to take into account the student's degree of certainty when evaluating …

In Engineering Classes, How To Assign Partial Credit: From Current Subjective Practice To Exact Formulas (Based On Computational Intelligence Ideas), Joe Lorkowski, Vladik Kreinovich, Olga Kosheleva

Departmental Technical Reports (CS)

When a student performed only some of the steps needed to solve a problem, this student gets partial credit. This partial credit is usually proportional to the number of stages that the student performed. This may sound reasonable, but in engineering education, this leads to undesired consequences: for example, a student who did not solve any of the 10 problems on the test, but who successfully performed 9 out of 10 stages needed to solve each problem will still get the grade of A ("excellent"). This may be a good evaluation of the student's intellectual ability, but for a engineering …

Why We Need Extra Physical Dimensions: A Simple Geometric Explanation, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

It is known that a consistent description of point-wise particles requires that we add extra physical dimensions to the usual four dimensions of space-time. The need for such dimensions is based on not-very-intuitive complex mathematics. It is therefore desirable to try to come up with a simpler geometric explanation for this phenomenon. In this paper, we provide a simple geometric explanation of why extra physical dimensions are needed.

Why Fuzzy Cognitive Maps Are Efficient, Vladik Kreinovich, Chrysostomos Stylios

Departmental Technical Reports (CS)

In many practical situations, the relation between the experts' degrees of confidence in different related statements is well described by Fuzzy Cognitive Maps (FCM). This empirical success is somewhat puzzling, since from the mathematical viewpoint, each FCM relation corresponds to a simplified one-neuron neural network, and it is well known that to adequately describe relations, we need multiple neurons. In this paper, we show that the empirical success of FCM can be explained if we take into account that human's subjective opinions follow Miller's seven plus minus two law.

What Is Computable? What Is Feasibly Computable? A Physicist's Viewpoint, Vladik Kreinovich, Olga Kosheleva

Departmental Technical Reports (CS)

In this paper, we show how the questions of what is computable and what is feasibly computable can be viewed from the viewpoint of physics: what is computable within the current physics? what is computable if we assume -- as many physicists do -- that no final physical theory is possible? what is computable if we consider data processing, i.e., computations based on physical inputs? Our physics-based analysis of these questions leads to some unexpected answers, both positive and negative. For example, we show that under the no-physical-theory-is-perfect assumption, almost all problems are feasibly solvable -- but not all of …

Towards A Physics-Motivated Small-Velocities Approximation To General Relativity, Vladik Kreinovich, Olga Kosheleva

Departmental Technical Reports (CS)

In the general case, complex non-linear partial differential equations of General Relativity are very hard to solve. Thus, to solve the corresponding physical problems, usually appropriate approximations are used. The first approximation to General Relativity is, of course, Newton's theory of gravitation. Newton's theory is applicable when the gravitational field is weak and when all velocities are much smaller than the speed of light. Most existing approximations allow higher velocities, but still limit us to weak gravitational fields. In this paper, he consider the possibility of a different approximation, in which strong fields are allowed but velocities are required to …

When Should We Switch From Interval-Valued Fuzzy To Full Type-2 Fuzzy (E.G., Gaussian)?, Vladik Kreinovich, Chrysostomos D. Stylios

Departmental Technical Reports (CS)

Full type-2 fuzzy techniques provide a more adequate representation of expert knowledge. However, such techniques also require additional computational efforts, so we should only use them if we expect a reasonable improvement in the result of the corresponding data processing. It is therefore important to come up with a practically useful criterion for deciding when we should stay with interval-valued fuzzy and when we should use full type-2 fuzzy techniques. Such a criterion is proposed in this paper. We also analyze how many experts we need to ask to come up with a reasonable description of expert uncertainty.

How To Gauge Disruptions Caused By Garbage Collection: Towards An Efficient Algorithm, Gabriel Arellano, Edward Hudgins, David Pruitt, Adrian Veliz, Eric Freudenthal, Vladik Kreinovich

Departmental Technical Reports (CS)

Comprehensive garbage collection is employed on a variety of computing devices, including intelligent cell phones. Garbage collection can cause prolonged user-interface pauses. In order to evaluate and compare the disruptiveness of various garbage collection strategies, it is necessary to gauge disruptions caused by garbage collection. In this paper, we describe efficient algorithms for computing metrics useful for this purpose.

Dow Theory's Peak-And-Trough Analysis Justified, Chrysostomos Stylios, Vladik Kreinovich

Departmental Technical Reports (CS)

In the analysis of dynamic financial quantities such as stock prices, equity prices, etc., reasonable results are often obtained if we only consider local maxima ("peaks") and local minima ("troughs") and ignore all the other values. The empirical success of this strategy remains a mystery. In this paper, we provide a possible explanation for this success.

Why Deep Neural Networks: A Possible Theoretical Explanation, Chitta Baral, Olac Fuentes, Vladik Kreinovich

Departmental Technical Reports (CS)

In the past, the most widely used neural networks were 3-layer ones. These networks were preferred, since one of the main advantages of the biological neural networks -- which motivated the use of neural networks in computing -- is their parallelism, and 3-layer networks provide the largest degree of parallelism. Recently, however, it was empirically shown that, in spite of this argument, multi-layer ("deep") neural networks leads to a much more efficient machine learning. In this paper, we provide a possible theoretical explanation for the somewhat surprising empirical success of deep networks.

Student Autonomy Improves Learning: A Theoretical Justification Of The Empirical Results, Octavio Lerma, Vladik Kreinovich

Departmental Technical Reports (CS)

In many pedagogical situations, it is advantageous to give students some autonomy: for example, instead of assigning the same homework problem to all the students, to give students a choice between several similar problems, so that each student can choose a problem whose context best fits his or her experiences. A recent experimental study shows that there is a 45% correlation between degree of autonomy and student success. In this paper, we provide a theoretical explanation for this correlation value.

We Live In The Best Of Possible Worlds: Leibniz's Insight Helps To Derive Equations Of Modern Physics, Vladik Kreinovich, Guoqing Liu

Departmental Technical Reports (CS)

To reconcile the notion of a benevolent and powerful God with the actual human suffering, Leibniz proposed the idea idea that while our world is not perfect, it is the best of possible worlds. This idea inspired important developments in physics: namely, it turned out that equations of motions and equations which describe the dynamics of physical fields can be deduced from the condition that the (appropriately defined) action functional is optimal. In practice, this idea is not always very helpful in physics applications: to fully utilize this fact, we need to how the action, and there are many possible …

Standing On The Shoulders Of The Giants: Why Constructive Mathematics, Probability Theory, Interval Mathematics, And Fuzzy Mathematics Are Important, Vladik Kreinovich

Departmental Technical Reports (CS)

Recent death of Ray Moore, one of the fathers of interval mathematics, inspired these thoughts on why interval computations -- and several other related areas of study -- are important, and what we can learn from the successes of these areas' founders and promoters.

Why Copulas?, Vladik Kreinovich, Hung T. Nguyen, Songsak Sriboonchitta, Olga Kosheleva

Departmental Technical Reports (CS)

A natural way to represent a 1-D probability distribution is to store its cumulative distribution function (cdf) F(x) = Prob(X ≤ x). When several random variables X₁, ..., X_n are independent, the corresponding cdfs F₁(x₁), ..., F_n(x_n) provide a complete description of their joint distribution. In practice, there is usually some dependence between the variables, so, in addition to the marginals F_i(x_i), we also need to provide an additional information about the joint distribution of the given variables. It is possible to represent this joint …

Analysis Of Random Metric Spaces Explains Emergence Phenomenon And Suggests Discreteness Of Physical Space, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practical situations, systems follow the pattern set by the second law of thermodynamics: they evolve from an organized inhomogeneous state into a homogeneous structure-free state. In many other practical situations, however, we observe the opposite emergence phenomenon: in an originally homogeneous structure-free state, an inhomogeneous structure spontaneously appears. In this paper, we show that the analysis of random metric spaces provides a possible explanation for this phenomenon. We also show that a similar analysis supports space-time models in which proper space is discrete.

Symbolic Aggregate Approximation (Sax) Under Interval Uncertainty, Chrysostomos D. Stylios, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practical situations, we monitor a system by continuously measuring the corresponding quantities, to make sure that an abnormal deviation is detected as early as possible. Often, we do not have ready algorithms to detect abnormality, so we need to use machine learning techniques. For these techniques to be efficient, we first need to compress the data. One of the most successful methods of data compression is the technique of Symbolic Aggregate approXimation (SAX). While this technique is motivated by measurement uncertainty, it does not explicitly take this uncertainty into account. In this paper, we show that we can …

Why Big-O And Little-O In Algorithm Complexity: A Pedagogical Remark, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In the comparative analysis of different algorithm, O- and o-notions are frequently used. While their use is productive, most textbooks do not provide a convincing student-oriented explanation of why these particular notations are useful in algorithm analysis. In this note, we provide such an explanation.

Why Some Families Of Probability Distributions Are Practically Efficient: A Symmetry-Based Explanation, Vladik Kreinovich, Olga Kosheleva, Hung T. Nguyen, Songsak Sriboonchitta

Departmental Technical Reports (CS)

Out of many possible families of probability distributions, some families turned out to be most efficient in practical situations. Why these particular families and not others? To explain this empirical success, we formulate the general problem of selecting a distribution with the largest possible utility under appropriate constraints. We then show that if we select the utility functional and the constraints which are invariant under natural symmetries -- shift and scaling corresponding to changing the starting point and the measuring unit for describing the corresponding quantity $x$. then the resulting optimal families of probability distributions indeed include most of the …

Once We Know That A Polynomial Mapping Is Rectifiable, We Can Algorithmically Find A Rectification, Julio Urenda, David Finston, Vladik Kreinovich

Departmental Technical Reports (CS)

It is known that some polynomial mappings φ: C^k --> Cⁿ are rectifiable in the sense that there exists a polynomial mapping α: Cⁿ --> Cⁿ whose inverse is also polynomial and for which α(φ(z₁, ...,z_k)) = (z₁, ...,z_k, 0, ..., 0) for all z₁, ...,z_k. In many cases, the existence of such a rectification is proven indirectly, without an explicit construction of the mapping α.

In this paper, we use Tarski-Seidenberg algorithm (for deciding the first order theory of real numbers) to design …

Sometimes, It Is Beneficial To Process Different Types Of Uncertainty Separately, Chrysostomos D. Stylios, Andrzej Pownuk, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practical situations, we make predictions based on the measured and/or estimated values of different physical quantities. The accuracy of these predictions depends on the accuracy of the corresponding measurements and expert estimates. Often, for each quantity, there are several different sources of inaccuracy. Usually, to estimate the prediction accuracy, we first combine, for each input, inaccuracies from different sources into a single expression, and then use these expressions to estimate the prediction accuracy. In this paper, we show that it is often more computationally efficient to process different types of uncertainty separately, i.e., to estimate inaccuracies in the …

A Corpus For Investigating English-Language Learners' Dialog Behaviors, Nigel Ward, Paola Gallardo

Departmental Technical Reports (CS)

We are interested in developing methods for the semi-automatic discovery of prosodic patterns in dialog and how they differ between languages and among populations. We are starting by examining how the prosody of Spanish-native learners of English differs from that of native speakers. To support this work, we have collected a new corpus of conversations among college students. This includes dialogs between a nonnative speaker of English and a native, dialogs between native speakers of English, and Spanish conversations.

A Simplified Explanation Of What It Means To Assign A Finite Value To An Infinite Sum, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Recently, a video made rounds that explained that it often makes sense to assign finite values to infinite sums. For example, it makes sense to claim that the sum of all natural numbers is equal to -1/12. This has picked up interested in media. However, judged by the viewers' and readers' comments, for many viewers and readers, neither the video, not the corresponding articles seem to explain the meaning of the above inequality clearly enough. One of the main stumbling blocks is the fact that the infinite sum is clearly divergent, so a natural value of the infinite sum is …