Mathematics Commons^™

How To Explain The Empirical Success Of Generalized Trigonometric Functions In Processing Discontinuous Signals, Pedro Barragan Olague, Vladik Kreinovich

Departmental Technical Reports (CS)

Trigonometric functions form the basis of Fourier analysis - one of the main signal processing tools. However, while they are very efficient in describing smooth signals, they do not work well for signals that contain discontinuities - such as signals describing phase transitions, earthquakes, etc. It turns out that empirically, one of the most efficient ways of describing and processing such signals is to use a certain generalization of trigonometric functions. In this paper, we provide a theoretical explanation of why this particular generalization is the most empirically efficient one.

How To Make Sure That Everyone Works Towards A Common Goal: Towards Optimal Incentives, Christian Servin, Vladik Kreinovich

Departmental Technical Reports (CS)

No abstract provided.

A Possible Utility-Based Explanation Of Deaton's Paradox (And Habits Of Mind), Hung T. Nguyen, Songsak Sriboonchitta, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

No abstract provided.

Oscillating Exam Averages And Their Control-Theory Explanation, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

When a student misses one of the exams, his overall grade for the class is often interpolated based on his available grades. This would have been a fair procedure if the grades for different tests were equally distributed. In practice, often, the average grades for different tests are oscillating. As a result, the usual interpolation techniques may inadvertently bias the student grade for the class. In this paper, we explain this oscillation, and analyze how to avoid the corresponding bias.

How To Compute Von Neumann-Morgenstern Solutions, Martha Osegueda Escobar, Vladik Kreinovich

Departmental Technical Reports (CS)

No abstract provided.

How To Modify Data Processing Algorithms So That They Detect Only Dependencies Which Make Sense To Domain Experts, Geovany Ramirez, Craig Tweedie, Jason Carlsson, Vladik Kreinovich

Departmental Technical Reports (CS)

No abstract provided.

How To Take Into Account Student's Degree Of Confidence When Grading Exams, Olga Kosheleva, Joe Lorkowski, Viannette Felix, Vladik Kreinovich

Departmental Technical Reports (CS)

When grading exams, it is important to take into account how confident the student is in the answer. If the answer is correct, then it is better �- and thus, deserves a better grade -- if the student is absolutely confident in this correct answer. On the other hand, if the answer is wrong, then, the more confident the student is in this wrong answer, the worse. The grading scheme should be such that provides an incentive for the students to report their true degree of confidence. In this paper, we explain how to design such a grading scheme.

How To Divide A Territory: An Argument In Favor Of Private Property, Mahdokhat Afravi, Vladik Kreinovich

Departmental Technical Reports (CS)

No abstract provided.

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 …

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.

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.

Combining Interval And Probabilistic Uncertainty: What Is Computable?, Vladik Kreinovich, Andrzej Pownuk, Olga Kosheleva

Departmental Technical Reports (CS)

In many practical problems, we need to process measurement results. For example, we need such data processing to predict future values of physical quantities. In these computations, it is important to take into account that measurement results are never absolutely exact, that there is always measurement uncertainty, because of which the measurement results are, in general, somewhat different from the actual (unknown) values of the corresponding quantities. In some cases, all we know about measurement uncertainty is an upper bound; in this case, we have an interval uncertainty, meaning that all we know about the actual value is that is …

Paradox Of Choice: A Possible Explanation, Vladik Kreinovich, Olga Kosheleva

Departmental Technical Reports (CS)

At first glance, we would expect that the more choices we have, the happier we will be. Experiments show, however, then when the number of choices increases, customers become less happy. In this paper, we provide a possible explanation for this paradox.

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.

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.

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.

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.

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 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.

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.

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.

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.

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 …

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 …

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 …

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 …

When Can We Simplify Data Processing: An Algorithmic Answer, Julio Urenda, Olga Kosheleva, Vladik Kreinovich, Berlin Wu

Departmental Technical Reports (CS)

In many real-life situations, we are interested in the values of physical quantities x₁, ..., x_n which are difficult (or even impossible) to measure directly. To estimate these values, we measure easier-to-measure quantities y₁, ..., y_m which are related to the desired quantities by a known relation, and use these measurement results to estimate x_i. The corresponding data processing algorithms are sometimes very complex and time-consuming, so a natural question is: are simpler (and, thus, faster) algorithms possible for solving this data processing problem? In this paper, we show that by using …

Why It Is Important To Precisiate Goals, Olga Kosheleva, Vladik Kreinovich, Hung T. Nguyen

Departmental Technical Reports (CS)

After Zadeh and Bellman explained how to optimize a function under fuzzy constraints, there have been many successful applications of this optimization. However, in many practical situations, it turns out to be more efficient to precisiate the objective function before performing optimization. In this paper, we provide a possible explanation for this empirical fact.