Physical Sciences and Mathematics Commons^™

A Recent Result About Random Metrics Explains Why All Of Us Have Similar Learning Potential, Christian Servin, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In the same class, after the same lesson, the amount of learned material often differs drastically, by a factor of ten. Does this mean that people have that different learning abilities? Not really: experiments show that among different students, learning abilities differ by no more than a factor of two. This fact have been successfully used in designing innovative teaching techniques, techniques that help students realize their full learning potential. In this paper, we deal with a different question: how to explain the above experimental result. It turns out that this result about learning abilities -- which are, due to …

Why Geometric Progression In Selecting The Lasso Parameter: A Theoretical Explanation, William Kubin, Yi Xie, Laxman Bokati, Vladik Kreinovich, Kittawit Autchariyapanitkul

Departmental Technical Reports (CS)

In situations when we know which inputs are relevant, the least squares method is often the best way to solve linear regression problems. However, in many practical situations, we do not know beforehand which inputs are relevant and which are not. In such situations, a 1-parameter modification of the least squares method known as LASSO leads to more adequate results. To use LASSO, we need to determine the value of the LASSO parameter that best fits the given data. In practice, this parameter is determined by trying all the values from some discrete set. It has been empirically shown that …

Why Class-D Audio Amplifiers Work Well: A Theoretical Explanation, Kevin Alvarez, Julio Urenda, Vladik Kreinovich

Departmental Technical Reports (CS)

Most current high-quality electronic audio systems use class-D audio amplifiers (D-amps, for short), in which a signal is represented by a sequence of pulses of fixed height, pulses whose duration at any given moment of time linearly depends on the amplitude of the input signal at this moment of time. In this paper, we explain the efficiency of this signal representation by showing that this representation is the least vulnerable to additive noise (that affect measuring the signal itself) and to measurement errors corresponding to measuring time.

Why Black-Scholes Equations Are Effective Beyond Their Usual Assumptions: Symmetry-Based Explanation, Warattaya Chinnakum, Sean R. Aguilar

Departmental Technical Reports (CS)

Nobel-Prize-winning Black-Scholes equations are actively used to estimate the price of options and other financial instruments. In practice, they provide a good estimate for the price, but the problem is that their original derivation is based on many simplifying statistical assumptions which are, in general, not valid for financial time series. The fact that these equations are effective way beyond their usual assumptions leads to a natural conclusion that there must be an alternative derivation for these equations, a derivation that does not use the usual too-strong assumptions. In this paper, we provide such a derivation in which the only …

A "Fuzzy" Like Button Can Decrease Echo Chamber Effect, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

One of the big problems of the US political life is the echo chamber effect -- in spite of the abundance of materials on the web, many people only read materials confirming their own opinions. The resulting polarization often deadlocks the political situation and prevents politicians from reaching compromises needed to make needed changes. In this paper, we show, on a simplified model, that the echo chamber effect can be decreased if we simply replace the currently prevalent binary (yes-no) Like button on webpages with a more gradual ("fuzzy") one -- a button that will capture the relative degree of …

How To Explain The Anchoring Formula In Behavioral Economics, Laxman Bokati, Vladik Kreinovich, Chon Van Le

Departmental Technical Reports (CS)

According to the traditional economics, the price that a person is willing to pay for an item should be uniquely determined by the value that this person will get from this item, it should not depend, e.g., on the asking price proposed by the seller. In reality, the price that a person is willing to pay does depend on the asking price; this is known as the anchoring effect. In this paper, we provide a natural justification for the empirical formula that describes this effect.

Towards A Theoretical Explanation Of How Pavement Condition Index Deteriorates Over Time, Edgar Daniel Rodriguez Velasquez, Vladik Kreinovich

Departmental Technical Reports (CS)

To predict how the Pavement Condition Index will change over time, practitioners use a complex empirical formula derived in the 1980s. In this paper, we provide a possible theoretical explanation for this formula, an explanation based on general ideas of invariance. In general, the existence of a theoretical explanation makes a formula more reliable; thus, we hope that our explanation will make predictions of road quality more reliable.

How To Describe Conditions Like 2-Out-Of-5 In Fuzzy Logic: A Neural Approach, Olga Kosheleva, Vladik Kreinovich, Nguyen Hoang Phuong

Departmental Technical Reports (CS)

In many medical applications, we diagnose a disease and/or apply a certain remedy if, e.g., two out of five conditions are satisfied. In the fuzzy case, i.e., when we only have certain degrees of confidence that each of n statement is satisfied, how do we estimate the degree of confidence that k out of n conditions are satisfied? In principle, we can get this estimate if we use the usual methodology of applying fuzzy techniques: we represent the desired statement in terms of "and" and "or", and use fuzzy analogues of these logical operations. The problem with this approach is …

Quantum (And More General) Models Of Research Collaboration, Oscar Galindo, Miroslav Svitek, Vladik Kreinovich

Departmental Technical Reports (CS)

In the last decades, several papers have shown that quantum techniques can be successful in describing not only events in the micro-scale physical world -- for which they were originally invented -- but also in describing social phenomena, e.g., different economic processes. In our previous paper, we provide an explanation for this somewhat surprising successes. In this paper, we extend this explanation and show that quantum (and more general) techniques can also be used to model research collaboration.

How To Combine (Dis)Utilities Of Different Aspects Into A Single (Dis)Utility Value, And How This Is Related To Geometric Images Of Happiness, Laxman Bokati, Nguyen Hoang Phuong, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practical situations, a user needs our help in selecting the best out of a large number of alternatives. To be able to help, we need to understand the user's preferences. In decision theory, preferences are described by numerical values known as utilities. It is often not feasible to ask to user to provide utilities of all possible alternatives, so we must be able to estimate these utilities based on utilities of different aspects of these alternatives. In this paper, we provide a general formula for combining utilities of aspects into a single utility value. The resulting formula …

Theoretical Explanation Of Recent Empirically Successful Code Quality Metrics, Vladik Kreinovich, Omar A. Masmali, Nguyen Hoang Phuong, Omar Badreddin

Departmental Technical Reports (CS)

Millions of lines of code are written every day, and it is not practically possible to perfectly thoroughly test all this code on all possible situations. In practice, we need to be able to separate codes which are more probable to contain bugs -- and which thus need to be tested more thoroughly -- from codes which are less probable to contain flaws. Several numerical characteristics -- known as code quality metrics -- have been proposed for this separation. Recently, a new efficient class of code quality metrics have been proposed, based on the idea to assign consequent integers to …

Which Are The Correct Membership Functions? Correct "And"- And "Or"- Operations? Correct Defuzzification Procedure?, Olga Kosheleva, Vladik Kreinovich, Shahnaz Shahbazova

Departmental Technical Reports (CS)

Even in the 1990s, when many successful examples of fuzzy control appeared all the time, many users were somewhat reluctant to use fuzzy control. One of the main reasons for this reluctance was the perceived subjective character of fuzzy techniques -- for the same natural-language rules, different experts may select somewhat different membership functions and thus get somewhat different control/recommendation strategies. In this paper, we promote the idea that this selection does not have to be subjective. We can always select the "correct" membership functions, i.e., functions for which, on previously tested case, we got the best possible control. Similarly, …

Decision Making Under Interval Uncertainty: Towards (Somewhat) More Convincing Justifications For Hurwicz Optimism-Pessimism Approach, Warattaya Chinnakum, Laura Berrout Ramos, Olugbenga Iyiola, Vladik Kreinovich

Departmental Technical Reports (CS)

In the ideal world, we know the exact consequences of each action. In this case, it is relatively straightforward to compare different possible actions and, as a result of this comparison, to select the best action. In real life, we only know the consequences with some uncertainty. A typical example is interval uncertainty, when we only know the lower and upper bounds on the expected gain. How can we compare such interval-valued alternatives? A usual way to compare such alternatives is to use the optimism-pessimism criterion developed by Nobelist Leo Hurwicz. In this approach, we maximize a weighted combination of …

Scale-Invariance And Fuzzy Techniques Explain The Empirical Success Of Inverse Distance Weighting And Of Dual Inverse Distance Weighting In Geosciences, Laxman Bokati, Aaron A. Velasco, Vladik Kreinovich

Departmental Technical Reports (CS)

Once we measure the values of a physical quantity at certain spatial locations, we need to interpolate these values to estimate the value of this quantity at other locations x. In geosciences, one of the most widely used interpolation techniques is inverse distance weighting, when we combine the available measurement results with the weights inverse proportional to some power of the distance from x to the measurement location. This empirical formula works well when measurement locations are uniformly distributed, but it leads to biased estimates otherwise. To decrease this bias, researchers recently proposed a more complex dual inverse distance weighting …

Why Mean, Variance, Moments, Correlation, Skewness Etc. – Invariance-Based Explanations, Olga Kosheleva, Laxman Bokati, Vladik Kreinovich

Departmental Technical Reports (CS)

In principle, we can use many different characteristics of a probability distribution. However, in practice, a few of such characteristics are mostly used: mean, variance, moments, correlation, etc. Why these characteristics and not others? The fact that these characteristics have been successfully used indicates that there must be some reason for their selection. In this paper, we show that the selection of these characteristics can be explained by the fact that these characteristics are invariant with respect to natural transformations -- while other possible characteristics are not invariant.

New (Simplified) Derivation Of Nash's Bargaining Solution, Nguyen Hoang Phuong, Laxman Bokati, Vladik Kreinovich

Departmental Technical Reports (CS)

According to the Nobelist John Nash, if a group of people wants to selects one of the alternatives in which all of them get a better deal than in a status quo situations, then they should select the alternative that maximizes the product of their utilities. In this paper, we provide a new (simplified) derivation of this result, a derivation which is not only simpler -- it also does not require that the preference relation between different alternatives be linear.

Towards Making Fuzzy Techniques More Adequate For Combining Knowledge Of Several Experts, Nguyen Hoang Phuong, Vladik Kreinovich

Departmental Technical Reports (CS)

In medical and other applications, expert often use rules with several conditions, each of which involve a quantity within the domain of expertise of a different expert. In such situations, to estimate the degree of confidence that all these conditions are satisfied, we need to combine opinions of several experts -- i.e., in fuzzy techniques, combine membership functions corresponding to different experts. In each area of expertise, different experts may have somewhat different membership functions describing the same natural-language ("fuzzy") term like small. It is desirable to present the user with all possible conclusions corresponding to all these membership functions. …

Several Years Of Practice May Not Be As Good As Comprehensive Training: Zipf's Law Explains Why, Francisco Zapata, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Many professions practice certifications as a way to establish that a person practicing this profession has reached a certain skills level. At first glance, it may sound like several years of practice should help a person pass the corresponding certification test, but in reality, even after several years of practice, most people are not able to pass the test, while after a few weeks of intensive training, most people pass it successfully. This sounds counterintuitive, since the overall number of problems that a person solves during several years of practice is much larger than the number of problems solved during …

Why Ellipsoids In Mechanical Analysis Of Wood Structures, F. Niklas Schietzold, Julio Urenda, Vladik Kreinovich, Wolfgang Graf, Michael Kaliske

Departmental Technical Reports (CS)

Wood is a very mechanically anisotropic material. At each point on the wooden beam, both average values and fluctuations of the local mechanical properties corresponding to a certain direction depend, e.g., on whether this direction is longitudinal, radial or tangential with respect to the grain orientation of the original tree. This anisotropy can be described in geometric terms, if we select a point x and form iso-correlation surfaces -- i.e., surfaces formed by points y with the same level of correlation ρ(x,y) between local changes in the vicinities of the points x and y. Empirical analysis shows that for each …

Strength Of Lime Stabilized Pavement Materials: Possible Theoretical Explanation Of Empirical Dependencies, Edgar Daniel Rodriguez Velasquez, Vladik Kreinovich

Departmental Technical Reports (CS)

When building a road, it is often necessary to strengthen the underlying soil layer. This strengthening is usually done by adding lime. There are empirical formulas that describe how the resulting strength depends on the amount of added lime. In this paper, we provide a theoretical explanation for these empirical formulas.

Fusion Of Probabilistic Knowledge As Foundation For Sliced-Normal Approach, Michael Beer, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practical applications, it turns out to be efficient to use Sliced-Normal multi-D distributions, i.e., distributions for which the logarithm of the probability density function (pdf) is a polynomial -- -- to be more precise, it is a sum of squares of several polynomials. This class is a natural extension of normal distributions, i.e., distributions for which the logarithm of the pdf is a quadratic polynomial.

In this paper, we provide a possible theoretical explanation for this empirical success.

How To Gauge The Quality Of A Testing Method When Ground Truth Is Known With Uncertainty, Nicholas Gray, Scott Ferson, Vladik Kreinovich

Departmental Technical Reports (CS)

The quality of a testing method is usually measured by using sensitivity, specificity, and/or precision. To compute each of these three characteristics, we need to know the ground truth, i.e., we need to know which objects actually have the tested property. In many applications (e.g., in medical diagnostics), the information about the objects comes from experts, and this information comes with uncertainty. In this paper, we show how to take this uncertainty into account when gauging the quality of testing methods.

Why Squashing Functions In Multi-Layer Neural Networks, Julio Urenda, Orsoly Csiszár, Gábor Csiszár, József Dombi, Olga Kosheleva, Vladik Kreinovich, György Eigner

Departmental Technical Reports (CS)

Most multi-layer neural networks used in deep learning utilize rectified linear neurons. In our previous papers, we showed that if we want to use the exact same activation function for all the neurons, then the rectified linear function is indeed a reasonable choice. However, preliminary analysis shows that for some applications, it is more advantageous to use different activation functions for different neurons -- i.e., select a family of activation functions instead, and select the parameters of activation functions of different neurons during training. Specifically, this was shown for a special family of squashing functions that contain rectified linear neurons …

A Mystery Of Human Biological Development -- Can It Be Used To Speed Up Computations?, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

For many practical problems, the only known algorithms for solving them require non-feasible exponential time. To make computations feasible, we need an exponential speedup. A reasonable way to look for such possible speedup is to search for real-life phenomena where such a speedup can be observed. A natural place to look for such a speedup is to analyze the biological activities of human beings -- since we, after all, solve many complex problems that even modern super-fast computers have trouble solving. Up to now, this search was not successful -- e.g., there are people who compute much faster than others, …

How Quantum Cryptography And Quantum Computing Can Make Cyber-Physical Systems More Secure, Deepak Tosh, Oscar Galindo, Vladik Kreinovich, Olga Kosheleva

Departmental Technical Reports (CS)

For cyber-physical systems, cyber-security is vitally important. There are many cyber-security tools that make communications secure -- e.g., communications between sensors and the computers processing the sensor's data. Most of these tools, however, are based on RSA encryption, and it is known that with quantum computing, this encryption can be broken. It is therefore desirable to use an unbreakable alternative -- quantum cryptography -- for such communications. In this paper, we discuss possible consequences of this option. We also explain how quantum computers can help even more: namely, they can be used to optimize the system's design -- in particular, …

Can We Preserve Physically Meaningful "Macro" Analyticity Without Requiring Physically Meaningless "Micro" Analyticity?, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Physicists working on quantum field theory actively used "macro" analyticity -- e.g., that an integral of an analytical function over a large closed loop is 0 -- but they agree that "micro" analyticity -- the possibility to expand into Taylor series -- is not physically meaningful on the micro level. Many physicists prefer physical theories with physically meaningful mathematical foundations. So, a natural question is: can we preserve physically meaningful "macro" analyticity without requiring physically meaningless "micro" analyticity? In the 1970s, an attempt to do it was made by using constructive mathematics, in which only objects generated by algorithms are …

Need For Simplicity And Everything Is A Matter Of Degree: How Zadeh's Philosophy Is Related To Kolmogorov Complexity, Quantum Physics, And Deep Learning, Vladik Kreinovich, Olga Kosheleva, Andres Ortiz-Muñoz

Departmental Technical Reports (CS)

Many people remember Lofti Zadeh's mantra -- that everything is a matter of degree. This was one of the main principles behind fuzzy logic. What is somewhat less remembered is that Zadeh also used another important principle -- that there is a need for simplicity. In this paper, we show that together, these two principles can generate the main ideas behind such various subjects as Kolmogorov complexity, quantum physics, and deep learning. We also show that these principles can help provide a better understanding of an important notion of space-time causality.

Classification Of The Subalgebras Of The Algebra Of All 2 By 2 Matrices, Justin Luis Bernal

Open Access Theses & Dissertations

Classification of the subalgebras of the familiar algebra of all $n\times n$ real matrices over the real numbers can get quite unwieldy as all subalgebras are of dimension ranging from $1$ to $n^2$. Classification of the subalgebras of the algebra of all $2\times 2$ real matrices over the real numbers is an interesting first start.

Since $\2$ is of dimension $4$ then its possible subalgebras are of dimension $1, 2, 3,$ or $4$. The one-dimensional subalgebra and four-dimensional subalgebra need little to no attention. The two-dimensional and three-dimensional subalgebras however turn out to be of significance.

It turns out there …

Structural Analysis Of Deformed Caprock Associated With A Salt Shoulder At Gypsum Valley Salt Wall, Paradox Basin, Colorado, Hanah Draper

Open Access Theses & Dissertations

This study documents the style and distribution of deformation present in Triassic-age gypsic and carbonate caprock associated with the Gypsum Valley Salt Wall Salt Shoulder in the Paradox Basin, Colorado in order to determine the mechanism and timing of deformation. Mapping of gypsic and carbonate capstone facies shows that the caprock layering that was generated during formation has been deformed by large-and-small-scale folding and faulting, brecciation, and boudinage. Deformation within the caprock resulted in widely variable orientations along the length of the salt shoulder, that also varies locally in intensity of deformation and with proximity to the passively rising diapir …

Photovoltage Enhancement For Stable Perovskite Solar Cells With A Temperature-Controlled Grain Growth Technique, Luis Eduardo Valerio

Open Access Theses & Dissertations

By performing strong characterizations methods, one can begin to fully understand the chemistry and composition behind a great performing perovskite solar cell. Understanding how the interaction between layers inside a solar cell is driven by the temperature and overall environment is a key element to improve the fabrication process and overall efficiency of such cells. This Thesis will present a study of the hybrid organic-inorganic, mixed-cation, mixed-halide, temperature and thickness-controlled perovskite solar cell. A constant power conversion efficiency (PCE) ranging between 15-17% and an open circuit voltage V¬oc above 1.05 V for a wide-band gap perovskite cell is presented.