Published in: M. Papadrakakis, V. Papadopoulos, G. Stefanou (eds.), Proceedings of the 3rd ECCOMAS Thematic Conference on Uncertainty Quantification in Computational Sciences and Engineering UNCECOMP'2019, Crete, Greece, June 24-26, 2019, pp. 713-723.

In engineering simulation a black-box code is often a complex, legacy or proprietary (secret) black-box software used to describe the physics of the system under study. Strategies to propagate epistemic uncertainty through such codes are desperately needed, for code verification, sensitivity, and validation on experimental data. Very often in practice, the uncertainty in the inputs is characterised by imprecise probability distributions or distributions with interval parameters, also known as probability boxes. In this paper we propose a strategy based on line sampling to propagate both aleatory and epistemic uncertainty through black-box codes to obtain interval probabilities of failure. The efficiency of the proposed strategy is demonstrated on the NASA LaRC UQ problem.

Published in Reliable Computing, 2020, Vol. 27, pp. 12-20.

In many practical applications ranging from self-driving cars to industrial application of mobile robots, it is important to take interval uncertainty into account when performing odometry, i.e., when estimating how our position and orientation ("pose") changes over time. In particular, one of the important aspects of this problem is detecting mismatches (outliers). In this paper, we describe an algorithm to compute the rigid body transformation, including a provably optimal sub-algorithm for detecting mismatches.

Published in Foundations of Science, 2021, Vol. 26, pp. 703-725, doi 10.1007/s10699-020-09659-z

Joule's Energy Conservation Law was the first "meta-law": a general principle that all physical equations must satisfy. It has led to many important and useful physical discoveries. However, a recent analysis seems to indicate that this meta-law is inconsistent with other principles -- such as the existence of free will. We show that this conclusion about inconsistency is based on a seemingly reasonable -- but simplified -- analysis of the situation. We also show that a more detailed mathematical and physical analysis of the situation reveals that not only Joule's principle remains true -- it is actually strengthened: it is no longer a principle that all physical theories should satisfy -- it is a principle that all physical theories do satisfy.

Published in: Marie-Jeanne Lesot, Susana Vieira, Marek Z. Reformat, Joao Paulo Carvalho, Anna Wilbik, Bernadette Bouchon-Meunier, and Ronald R. Yager (eds.), Proceedings of the 18th International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems IPMU'2020, Lisbon, Portugal, June 15-19, 2020, pp. 70-79.

In many practical situations, we only know the interval containing the quantity of interest, we have no information about the probabilities of different values within this interval. In contrast to the cases when we know the distributions and can thus use Monte-Carlo simulations, processing such interval uncertainty is difficult -- crudely speaking, because we need to try all possible distributions on this interval. Sometimes, the problem can be simplified: namely, for estimating the range of values of some characteristics of the distribution, it is possible to select a single distribution (or a small family of distributions) whose analysis provides a good understanding of the situation. The most known case is when we are estimating the largest possible of Shannon's entropy: in this case, it is sufficient to consider the uniform distribution on the interval. Interesting, estimating other characteristics leads to the selection of the same uniform distribution: e.g., estimating the largest possible values of generalized entropy or of some sensitivity-related characteristics. In this paper, we provide a general explanation of why uniform distribution appears in different situations -- namely, it appears every time we have a permutation-invariant optimization problem with the unique optimum. We also discuss what happens if we have an optimization problem that attains its optimum at several different distributions -- this happens, e.g., when we are estimating the smallest possible value of Shannon's entropy (or of its generalizations).

Published in: Marie-Jeanne Lesot, Susana Vieira, Marek Z. Reformat, Joao Paulo Carvalho, Anna Wilbik, Bernadette Bouchon-Meunier, and Ronald R. Yager (eds.), Proceedings of the 18th International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems IPMU'2020, Lisbon, Portugal, June 15-19, 2020, pp. 59-69.

/ Current artificial neural networks are very successful in many machine learning applications, but in some cases they still lag behind human abilities. To improve their performance, a natural idea is to simulate features of biological neurons which are not yet implemented in machine learning. One of such features is the fact that in biological neural networks, signals are represented by a train of spikes. Researchers have tried adding this spikiness to machine learning and indeed got very good results, especially when processing time series (and, more generally, spatio-temporal data). In this paper, we provide a possible theoretical explanation for this empirical success.

Published in: Martine Ceberio and Vladik Kreinovich (eds.), How Uncertainty-Related Ideas Can Provide Theoretical Explanation for Empirical Dependencies, Springer, Cham, Switzerland, 2021, pp. 141-151.

In the usual 2-valued logic, from the purely mathematical viewpoint, there are many possible binary operations. However, in commonsense reasoning, we only use a few of them: why? In this paper, we show that fuzzy logic can explain the usual choice of logical operations in 2-valued logic.

Published in: Martine Ceberio and Vladik Kreinovich (eds.), How Uncertainty-Related Ideas Can Provide Theoretical Explanation for Empirical Dependencies, Springer, Cham, Switzerland, 2021, pp. 133-140.

In the traditional fuzzy logic, experts' degrees of confidence are described by numbers from the interval [0,1]. Clearly, not all the numbers from this interval are needed: in the whole history of the Universe, there will be only countably many statements and thus, only countably many possible degree, while the interval [0,1] is uncountable. It is therefore interesting to analyze what is the set S of actually used values. The answer depends on the choice of "and"-operations (t-norms) and "or"-operations (t-conorms). For the simplest pair of min and max, any finite set will do -- as long as it is closed under negation 1 &minus a. For the next simplest pair -- of algebraic product and algebraic sum -- we prove that for a finitely generated set, if the "and"-operation is exact, then the "or"-operation is almost always approximate, and vice versa. For other "and"- and "or"-operations, the situation can be more complex.

Published in: Martine Ceberio and Vladik Kreinovich (eds.), How Uncertainty-Related Ideas Can Provide Theoretical Explanation for Empirical Dependencies, Springer, Cham, Switzerland, 2021, pp. 43-50.

It is known that the distribution of seismic inter-event times is well described by the Gamma distribution. Recently, this fact has been used to successfully predict major seismic events. In this paper, we explain that the Gamma distribution of seismic inter-event times can be naturally derived from the first principles.

Published in: Evgeny Katz (ed.), DNA- and RNA-Based Computing Systems, Wiley, Hoboken, New Jersey, 2021, pp. 213-230.

The traditional DNA computing schemes are based on using or simulating DNA-related activity. This is similar to how quantum computers use quantum activities to perform computations. Interestingly, in quantum computing, there is another phenomenon known as computing without computing, when, somewhat surprisingly, the result of the computation appears without invoking the actual quantum processes. In this chapter, we show that similar phenomenon is possible for DNA computing: in addition to the more traditional way of using or simulating DNA activity, we can also use DNA inactivity to solve complex problems. We also show that while DNA computing without computing is almost as powerful as traditional DNA computing, it is actually somewhat less powerful. As a side effect of this result, we also show that, in general, security is somewhat more difficult to maintain than privacy, and data storage is more difficult than data transmission.

Published in: Martine Ceberio and Vladik Kreinovich (eds.), How Uncertainty-Related Ideas Can Provide Theoretical Explanation for Empirical Dependencies, Springer, Cham, Switzerland, 2021, pp. 21-26.

At present, we mostly use decimal (base-10) number system, but in the past, many other systems were used: base-20, base-60 -- which is still reflected in how we divide an hour into minutes and a minute into seconds -- and many others. There is a known explanation for the base-60 system: 60 is the smallest number that can be divided by 2, by 3, by 4, by 5, and by 6. Because of this, e.g., half an hour, one-third of an hour, all the way to one-sixth of an hour all correspond to a whole number of minutes. In this paper, we show that a similar idea can explain all historical number systems, if, instead of requiring that the base divides {\it all} numbers from 2 to some value, we require that the base divides all but one (or all but two) of such numbers.

To appear in: Proceedings of the 2020 4th International Conference on Intelligent Systems, Metaheuristics & Swarm Intelligence ISMSI'2020, Thimpu, Bhutan, March 21-22, 2020.

Successes of deep learning are partly due to appropriate selection of activation function, pooling functions, etc. Most of these choices have been made based on empirical comparison and heuristic ideas. In this paper, we show that many of these choices -- and the surprising success of deep learning in the first place -- can be explained by reasonably simple and natural mathematics.

To appear in: Proceedings of the 2020 4th International Conference on Intelligent Systems, Metaheuristics & Swarm Intelligence ISMSI'2020, Thimpu, Bhutan, March 21-22, 2020.

Several decades ago, traditional neural networks were the most efficient machine learning technique. Then it turned out that, in general, a different technique called support vector machines is more efficient. Reasonably recently, a new technique called deep learning has been shown to be the most efficient one. These are empirical observations, but how we explain them -- thus making the corresponding conclusions more reliable? In this paper, we provide a possible theoretical explanation for the above-described empirical comparisons. This explanation enables us to explain yet another empirical fact -- that sparsity techniques turned out to be very efficient in signal processing.

Published in Proceedings of the Annual Conference of the North American Fuzzy Information Processing Society NAFIPS'2020, Redmond, Washington, August 20-22, 2020, pp. 433-437.

It can be proven that linear dynamical systems exhibit either stable behavior, or unstable behavior, or oscillatory behavior, or transitional behavior. Interesting, the same classification often applies to nonlinear dynamical systems as well. In this paper, we provide a possible explanation for this phenomenon, i.e., we explain why a classification based on linear approximation to dynamical systems often works well in nonlinear cases.

Published in Proceedings of the Annual Conference of the North American Fuzzy Information Processing Society NAFIPS'2020, Redmond, Washington, August 20-22, 2020, pp. 373-378.

In this paper, we show that physicists' intuition about randomness is not fully consistent with their belief that every theory is only approximate. We also prove that there is no formal way to reconcile these two intuitions, this reconciliation has to be informal. Thus, there are fundamental reasons why informal knowledge is needed for describing the real world.

Published in Russian Digital Libraries Journal, 2019, Vol. 22, No. 6, pp. 773-779.

In some classes, students want to get a desired passing grade (e.g., C or B) by spending the smallest amount of effort. In such situations, it is reasonable for the instructor to assign the grades for different tasks in such a way that the resulting overall student's effort is the largest possible. In this paper, we show that to achieve this goal, we need to assign, to each task, the number of points proportional to the efforts needed for this task.

Published in Proceedings of the Annual Conference of the North American Fuzzy Information Processing Society NAFIPS'2020, Redmond, Washington, August 20-22, 2020, pp. 427-432.

In many application areas, it is important to predict the user's reaction to new products. In general, this reaction changes with time. Empirical analysis of this dependence has shown that it can be reasonably accurately described by a power law. In this paper, we provide a theoretical explanation for this empirical formula.

Published in Russian Digital Libraries Journal, Vol. 22, No. 6, pp. 763-768.

Ancient Egyptians represented each fraction as a sum of unit fractions, i.e., fractions of the type 1/n. In our previous papers, we explained that this representation makes perfect sense: e.g., it leads to an efficient way of dividing loaves of bread between people. However, one thing remained unclear: why, when representing fractions of the type 2/(2k+1), Egyptians did not use a natural representation 1/(2k+1) + 1/(2k+1), but used a much more complicated representation instead. In this paper, we show that the need for such a complicated representation can be explained if we take into account that instead of cutting a rectangular-shaped loaf in one direction -- as we considered earlier -- we can simultaneously cut it in two orthogonal directions. For example, to cut a loaf into 6 pieces, we can cut in 2 pieces in one direction and in 3 pieces in another direction. Together, these cuts will divide the original loaf into 2*3=6 pieces.

It is known that Egyptian fractions are an exciting topics for kids, helping them better understand fractions. In view of this fact, we plan to use our new explanation to further enhance this understanding.

Published in: Martine Ceberio and Vladik Kreinovich (eds.), How Uncertainty-Related Ideas Can Provide Theoretical Explanation for Empirical Dependencies, Springer, Cham, Switzerland, 2021, pp. 7-14.

It is well known that, according to the second law of thermodynamics, the entropy of a closed system increases (or at least stays the same). In many situations, this increase is the smallest possible. The corresponding minimum entropy production principle was first formulated and explained by a future Nobelist Ilya Prigogine. Since then, many possible explanations of this principle appeared, but all of them are very technical, based on complex analysis of differential equations describing the system's dynamics. Since this phenomenon is ubiquitous for many systems, it is desirable to look for a general system-based explanation, explanation that would not depend on the specific technical details. Such an explanation is presented in this paper.

Published in Journal of Uncertain Systems, 2019, Vol. 13, No. 3, pp. 180-185.

The famous EPR paradox shows that if we describe quantum particles in the usual way -- by their wave functions -- then we get the following seeming contradiction. If we entangle the states of the two particles, then move them far away from each other, and measure the state of the first particle, then the state of the second particle immediately changes -- which contradicts to special relativity, according to which such immediate-action-at-a-distance is not possible. It is known that, from the physical viewpoint, this is not a real paradox: if we measure any property of the second particle, the results will not change whether we perform the measurement on the first particle or not. What the above argument shows is that the usual wave function description of a quantum state does not always adequately describe the corresponding physics. In this paper, we propose a new, more physically adequate description of a quantum state, a description in which there is no EPR paradox: measurements performed at the first particle does not change the state of the remote second one.

Published in Journal of Uncertain Systems, 2019, Vol. 13, No. 3, pp. 193-196.

In many application areas, the dynamics of a system is described by a harmonic oscillator with quadratic damping. In particular, this equation describes the flapping of the insect wings and is, thus, used in the design of very-small-size unmanned aerial vehicles. The fact that the same equation appears in many application areas, from stability of large structures to insect flight, seems to indicate that this equation follows from some fundamental principles. In this paper, we show that this is indeed the case: this equation can be derived from natural invariance requirements.

Published in Journal of Uncertain Systems, 2019, Vol. 13, No. 3, pp. 186-188.

A statistical analysis of hundreds of successful and unsuccessful revolution attempts led historian to a very unexpected conclusion: that most attempts involving at least 3.5% of the population succeeded, while most attempts that involved a smaller portion of the population failed. In this paper, we show that this unexpected threshold can be explained based on the other two known rules of human behavior: the 80/20 rule (20% of the people drink 80% of the beer) and 7 plus minus 2 law according to which we naturally divide everything into 7 plus minus 2 classes.

Published in Journal of Uncertain Systems, 2019, Vol. 13, No. 3, pp. 220-223.

There are a lot of commonsense advices in decision making: e.g., we should consider multiple scenarios, we should consult experts, we should play down emotions. Many of these advices come supported by a surprisingly consistent quantitative evidence. In this paper, on the example of the above advices, we provide a theoretical explanations for these quantitative facts.

Published in Journal of Uncertain Systems, 2019, Vol. 13, No. 3, pp. 168-171.

Numerical experiments show that for classifying neural networks, it is beneficial to select a smaller deviation for initial weights that what is usually recommended. In this paper, we provide a theoretical explanation for these unexpected simulation results.

Published in Journal of Uncertain Systems, 2019, Vol. 13, No. 3, pp. 172-175.

If we place a can of coke that weigh 0.35 kg into a car that weighs 1 ton = 1000 kg, what will be the resulting weight of the car? Mathematics says 1000.35 kg, but common sense says 1 ton. In this paper, we show that this common sense answer can be explained by the Hurwicz optimism-pessimism criterion of decision making under interval uncertainty.

Published in: Martine Ceberio and Vladik Kreinovich (eds.), How Uncertainty-Related Ideas Can Provide Theoretical Explanation for Empirical Dependencies, Springer, Cham, Switzerland, 2021, pp. 1-5.

One of the main motivations for designing computer models of complex systems is to come up with recommendations on how to best control these systems. Many complex real-life systems are so complicated that it is not computationally possible to use realistic nonlinear models to find the corresponding optimal control. Instead, researchers make recommendations based on simplified -- e.g., linearized -- models. The recommendations based on these simplified models are often not realistic but, interestingly, they can be made more realistic if we "tone them down" -- i.e., consider predictions and recommendations which are close to the current status quo state. In this paper, we analyze this situation from the viewpoint of general system analysis. This analysis explain the above empirical phenomenon -- namely, we show that this "status quo bias" indeed helps decision makers to take nonlinearity into account.

Published in Matematical Structures and Modeling, 2020, Vol. 53, pp. 104-107.

Changes in the elderlies depression level result from a large number of small independent factors. Such situations are ubiquitous in applications. In most such cases, due to the Central Limit Theorem, the corresponding distribution is close to Gaussian. For the changes in the elderlies depression level, however, the empirical distribution is far from Gaussian: it is uniform. In this paper, we provide a possible explanation for the emergence of the uniform distribution.

Published in Matematical Structures and Modeling, 2020, Vol. 53, pp. 144-148.

One of the biases potentially affecting systems engineers is the confirmation bias, when instead of selecting the best hypothesis based on the data, people stick to the previously-selected hypothesis until it is disproved. In this paper, on a simple example, we show how important is to take care of this bias: namely, that because of this bias, we need twice as many experiments to switch to a better hypothesis.

Published in International Mathematical Forum, 2019, Vol. 14, No. 5, pp. 209-213.

To compare two different hypothesis testing techniques, researchers use the following heuristic idea: for each technique, they form a curve describing how the probabilities of type I and type II errors are related for this technique, and then compare areas under the resulting curves. In this paper, we provide a justification for this heuristic idea.

Published in International Mathematical Forum, 2019, Vol. 14, No. 5, pp. 205-208.

The 80/20 rule and the 7 plus minus 2 law are examples of difficult to explain empirical facts. According to the 80/20 rule, in each activity, 20% of the people contribute to the 80% of the results. The 7 plus minus 2 law means that we divide objects into 7 plus minus 2 groups -- i.e., into 5 to 9 groups. In this paper, we show that there is a relation between these two facts: namely, we show that, because of the 80/20 rule, the number of classes cannot be smaller than 5. Thus, the 80/20 rule explains the lower bound (5) on the 7 plus minus 2 law.

Published in Journal of Uncertain Systems, 2020, Vol. 13, No. 3, pp. 211-215.

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.

Published in International Journal of Unconventional Computing, 2020, Vol. 15, No. 3, pp. 193-218.

Traditional physics assumes that space and time are continuous. However, a deeper analysis shows that this seemingly reasonable space-time model leads to some serious physical problems. One of the approaches that physicists have proposed to solve these problems is to assume that the space-time is actually discrete. In this paper, we analyze possible computational consequences of this discreteness. It turns out that in a discrete space-time, it is probably possible to solve NP-complete problems in polynomial time: namely, this is possible in almost all physically reasonable models of dynamics in discrete space-time (almost all in some reasonable sense).

Published in Geombinatorics, 2021, Vol. 30, No. 4, pp. 208-213.

In the first approximation, the shape of our Galaxy -- as well as the shape of many other celestial bodies -- can be naturally explained by geometric symmetries and the corresponding invariances. As a result, we get the familiar shape of a planar spiral. A recent more detailed analysis of our Galaxy's shape has shown that the Galaxy somewhat deviates from this ideal shape: namely, it is not perfectly planar, it is somewhat warped in the third dimension. In this paper, we show that the empirical formula for this warping can also be explained by geometric symmetries and invariance.

Published in Applied Mathematical Sciences, 2019, Vol. 13, No. 16, pp. 775-779.

A recent patent shows that filtering out higher harmonics helps people sing in-tune. In this paper, we use the general signal processing ideas to explain this empirical phenomenon. We also show that filtering out higher harmonics is the optimal way of increasing the signal-to-noise ratio -- and thus, of making it easier for people to recognize when they are signing out of tune.

Published in Applied Mathematical Sciences, 2019, Vol. 13, No. 16, pp. 769-773.

To adequately describe the planets' motion, ancient astronomers used epicycles, when a planet makes a circular motion not around the Earth, but around a special auxiliary point which, in turn, performs a circular motion around the Earth -- or around a second auxiliary point which, in turns, rotates around the Earth, etc. Standard textbooks malign this approach by calling it bad science, but in reality, this is, in effect, trigonometric (Fourier) series -- an extremely useful tool in science and engineering. It should be mentioned, however, that the epicycles are almost as good as trigonometric series -- in the sense that in some cases, they need twice as many parameters to achieve the same accuracy.

Published in Journal of Uncertain Systems, 2020, Vol. 13, No. 3, pp. 233-235.

Testing is a very important part of quality control in education. To decide how to best test, it makes sense to use experience of other areas where testing is important, where there is a large amount of experimental data comparing the efficiency of different testing strategies. One such area is software engineering. The experience of software engineering shows that the most efficient approach to testing is to test thoroughly on every single stage of the project. In regards to teaching, the resulting recommendation means making testing as frequent as possible, preferably giving weekly quizzes. At first glance, this may seem difficult, since grading quizzes -- especially for big classes -- requires a lot of time, and instructors usually do not have that much time. This problem can be solved by giving multiple-choice quizzes for which grading can be automatic. Automatic grading also helps make grading more objective -- and thus, eliminate perceived grading subjectivity as a potential problem affecting student learning.

Published in Mathematical Structures and Modeling, 2019, Vol. 52, pp. 134-140.

To many students, the notion of a derivative seems unrelated to any previous mathematics -- and is, thus, difficult to study and to understand. In this paper, we show that this notion can be naturally derived from a more intuitive notion of invariance.

To appear in: Evgeny Dantsin and Vladik Kreinovich (eds.), Uncertainty Quantification and Uncertainty Propagation under Traditional and AI-Based Data Processing (and Related Topics): Legacy of Grigory Tseytin, Springer, Cham, Switzerland.

In their 1983 paper, C. Alsina, E. Trillas, and L. Valverde proved that distributivity, monotonicity, and boundary conditions imply that the "and"-operation on the interval [0,1] is min and the "or"-operation is max. In this paper, we extend this result to general partially ordered sets with the greatest element (denoted by 1) and the least element (denoted by 0), and we also show that all the above conditions are necessary for this result to be true.

Published in V. Kreinovich (ed.), Statistical and Fuzzy Approaches to Data Processing, with Applications to Econometrics and Other Areas, Springer, Cham, Switzerland, 2021, pp. 133-143.

To make an adequate decision, we need to know the probabilities of different consequences of different actions. In practice, we only have partial information about these probabilities -- this situation is known as imprecise probabilities. A general description of all possible imprecise probabilities requires using infinitely many parameters. In practice, the two most widely used few-parametric approximate descriptions are p-boxes (bounds on the values of the cumulative distribution function) and interval-valued moments (i.e., bounds on moments). In some situations, these approximations are not sufficiently accurate. So, we need more accurate more-parametric approximations. In this paper, we explain what are the natural next approximations.

Published in V. Kreinovich (ed.), Statistical and Fuzzy Approaches to Data Processing, with Applications to Econometrics and Other Areas, Springer, Cham, Switzerland, 2021, pp. 145-156.

In many practical situations, we only have partial information about the probabilities; this means that there are several different probability distributions which are consistent with our knowledge. In such cases, if we want to select one of these distributions, it makes sense not to pretend that we have a small amount of uncertainty -- and thus, it makes sense to select a distribution with the largest possible value of uncertainty. A natural measure of uncertainty of a probability distribution is its entropy. So, this means that out of all probability distributions consistent with our knowledge, we select the one whose entropy is the largest. In many cases, this works well, but in some cases, this Maximum Entropy approach leads to counterintuitive results. For example, if all we know is that the variable is located on a given interval, then the Maximum Entropy approach selects the uniform distribution on this interval. In this distribution, the probability density ρ(x) abruptly changes at the interval's endpoints, while intuitively, we expect that it should change smoothly with x. To reconcile the Maximum Entropy approach with our intuition, we propose to limit distributions to those for which the probability density's rate of change is bounded by some a priori value -- and to limit the search for the distribution with the largest entropy only to such distributions. We show that this natural restriction indeed reconciles the Maximum Entropy approach with our intuition.

Published in Geombinatorics, 2021, Vol. 30, No. 1, pp. 109-112.

It is known that, in general, a person keeps in mind between 5 and 9 objects -- this is known as the 7 plus minus 2 law. In this paper, we provide a possible simple geometric explanation for this psychological feature.

Published in Mathematical Structures and Modeling, 2019, Vol. 52, pp. 93-96.

In many graph-related problems, an obvious necessary condition is often also sufficient. This phenomenon is so ubiquitous that it was even named TONCAS, after the first letters of the phrase describing this phenomenon. In this paper, we provide a possible explanation for this phenomenon.

Published in: Songsak Sriboonchitta, Vladik Kreinovich, and Woraphon Yamaka (eds.), Behaviorial Predictive Modeling in Economics, Springer, Cham, Switzerland, 2021, pp. 141-144.

In the Bayesian approach, to describe a prior distribution on the set [0,1] of all possible probability values, typically, a Beta distribution is used. The fact that there have been many successful applications of this idea seems to indicate that there must be a fundamental reason for selecting this particular family of distributions. In this paper, we show that the selection of this family can indeed be explained if we make reasonable invariance requirements.

Published in: Songsak Sriboonchitta, Vladik Kreinovich, and Woraphon Yamaka (eds.), Behaviorial Predictive Modeling in Economics, Springer, Cham, Switzerland, 2021, pp. 195-201.

In many practical situations, for some components of the uncertainty (e.g., of the measurement error) we know the corresponding probability distribution, while for other components, we know only upper bound on the corresponding values. To decide which of the algorithms or techniques leads to less uncertainty, we need to be able to gauge the combined uncertainty by a single numerical value -- so that we can select the algorithm for which this values is the best. There exist several techniques for gauging the combination of interval and probabilistic uncertainty. In this paper, we consider the problem of gauging the combination of different types of uncertainty from the general fundamental viewpoint. As a result, we develop a general formula for such gauging -- a formula whose particular cases include the currently used techniques.

Published in: Nguyen Ngoc Thach, Vladik Kreinovich, and Nguyen Duc Trung (eds.), Data Science for Financial Econometrics, Springer, Cham, Switzerland, 2021, pp. 37-50.

In many practical situations, observations and measurement results are consistent with many different models -- i.e., the corresponding problem is ill-posed. In such situations, a reasonable idea is to take into account that the values of the corresponding parameters should not be too large; this idea is known as regularization. Several different regularization techniques have been proposed; empirically the most successful are LASSO method, when we bound the sum of absolute values of the parameters, and EN and CLOT methods in which this sum is combined with the sum of the squares. In this paper, we explain the empirical success of these methods by showing that they are the only ones which are invariant with respect to natural transformations -- like scaling which corresponds to selecting a different measuring unit.

Published in: Nguyen Ngoc Thach, Vladik Kreinovich, and Nguyen Duc Trung (eds.), Data Science for Financial Econometrics, Springer, Cham, Switzerland, 2021, pp. 115-120.

In many practical situations, we need to select a model based on the data. It is, at present, practically a consensus that the traditional p-value-based techniques for such selection often do not lead to adequate results. One of the most widely used alternative model selection techniques is the Minimum Bayes Factor (MBF) approach, in which a model is preferred if the corresponding Bayes factor -- the ratio of likelihoods corresponding to this model and to the competing model -- is sufficiently large for all possible prior distributions. Based on the MBF values, we can decide how strong is the evidence in support of the selected model: weak, strong, very strong, or decisive. The corresponding strength levels are based on a heuristic scale proposed by Harold Jeffreys, one of the pioneers of the Bayes approach to statistics. In this paper, we propose a justification for this scale.

Published in: Songsak Sriboonchitta, Vladik Kreinovich, and Woraphon Yamaka (eds.), Behaviorial Predictive Modeling in Economics, Springer, Cham, Switzerland, 2021, pp. 145-152.

In many practical situations, we need to make a group decision that takes into account preferences of all the participants. Ideally, we should elicit, from each participant, a full information about his/her preferences, but such elicitation is usually too time-consuming to be practical. Instead, we only elicit, from each participant, his/her ranking of different alternatives. One of the semi-heuristic methods for decision making under such information is Borda count, when for each alternative and each participant, we count how many alternatives are worse, and then select the alternatives for which the sum of these numbers is the largest. In this paper, we explain the empirical success of the Borda count technique by showing that this method naturally follows from the maximum entropy approach -- a natural approach to decision making under uncertainty.

Published in Applied Mathematical Sciences, 2019, Vol. 13, No. 14, pp. 681-684.

To improve the efficiency of artificial insemination, farmers equip cows with sensors, based on which a computer system determines the cow's insemination window. Analysis of the resulting calves showed an unexpected dependence of the calf's gender on the insemination time: cows inseminated earlier in their window mostly gave birth to female calves, while cows inseminated later in their window mostly gave birth to males. In this paper, we provide a general system-based explanation for this phenomenon.

Published in Mathematical Structures and Modeling, 2019, Vol. 52, pp. 27-32.

In their 1999 paper, psychologists David Dunning and Justin Kruger showed that, in general, experts not only provide better estimates of different situations, but they also provide a better estimates of the accuracy of their estimates. Which this phenomenon has been confirmed by many follow-up experiments, it remains largely unexplained. In this paper, we provide a simple system-based qualitative explanation for the Dunning-Kruger effect.

Published in Journal of Uncertain Systems, 2019, Vol. 13, No. 3, pp. 164-167.

Many computer-based services use recommender systems that predict our preferences based on our degree of satisfaction with the past selections. One of the most efficient techniques making recommender systems successful is matrix factorization. While this technique works well, until now, there was no general explanation of why it works. In this paper, we provide such an explanation.

Published in Geombinatorics, 2019, Vol. 29, No. 2, pp. 78-88.

It is known that seismic waves from a remote earthquake can trigger a small local earthquake. Recent analysis has shown that this triggering occurs mostly when the direction of the incoming wave is orthogonal to the direction of the local fault, some triggerings occur when these directions are parallel, and very few triggerings occur when the angle between these two directions is different from 0 and 90 degrees. In this paper, we propose a symmetry-based geometric explanation for this unexpected observation.

Published in Journal of Uncertain Systems, 2020, Vol. 13, No. 3, pp. 207-210.

It is known that a free neutron decays into a proton, an electron, and an anti-neutrino. Interesting, recent attempts to measure the neutron's lifetime has led to two slightly different estimates: namely, the number of decaying neutrons is somewhat larger than the number of newly created protons. This difference is known as the neutron lifetime puzzle. A natural explanation for this difference is that in some cases, a neutron decays not into a proton, but into some other particle. If this explanation is true, this implies that nuclei with a sufficiently large number of neutrons will be unstable. Based on the observed difference between the two estimates of the neutron lifetime, we can estimate the largest number of neutrons in a stable nucleus to be between 80 and 128. The fact that the number of neutrons (125) in the actual largest stable nucleus (lead) lies within this interval can serve as an additional argument is favor of the current explanation of the neutron lifetime puzzle.

Published in Mathematical Structures and Modeling, 2019, Vol. 51, pp. 114-117.

Researchers have found out that normally, we remember about 30% of the information; however, if immediately after reading, we get a test, the rate increases to 45%. In this paper, we show that Zipf law can explain this empirical dependence.

Published in Applied Mathematical Sciences, 2019, Vol. 13, No. 14, pp. 677-680.

Several researchers found out that acoustic stimulation during sleep enhances sleep and enhances memory. An interesting -- and somewhat mysterious -- part of this phenomenon is that out of all possible types of noise, the pink noise leads to the most efficient stimulation. In this paper, we use general system-based ideas to explain why in this phenomenon, pink noise works best.

Published in Mathematical Structures and Modeling, 2019, Vol. 51, pp. 109-113.

We expect that the quality of experts' decisions increases with their experience. This is indeed true for reasonably routine situations. However, surprisingly, empirical data shows that in unusual situations, novice experts make much better decisions than more experience ones. This phenomenon is especially unexpected for medical emergency situations: it turns out that the mortality rate of patients treated by novice doctors is a third lower than for patients treated by experience doctors. In this paper, we provide a possible explanation for this seemingly counterintuitive phenomenon.

Published in Journal of Uncertain Systems, 2019, Vol. 13, No. 3, pp. 176-179.

Researchers who monitor the average intelligence of human population have reasonably recently made an unexpected observation: that after many decades in which this level was constantly growing (this is known as the Flynn effect), at present, this level has started decreasing again. In this paper, we show that this reversed Flynn effect can be, in principle, explained in general system-based terms: namely, it is similar to the fact that a control system usually overshoots before stabilizing at the desired level. A similar idea may explain another unexpected observation -- that the Universe's expansion rate, which was supposed to be decreasing, is actually increasing.

Published in Journal of Uncertain Systems, 2020, Vol. 13, No. 3, pp. 201-206.

This article summarizes main ideas about education which were presented at the 2019 International Forum on Teacher Education in Kazan, Russia, on May 28--31, 2019.

Published in Asian Journal of Economics and Banking, 2019, Vol. 3, No. 2, pp. 17-28.

The traditional Markowitz approach to portfolio optimization assumes that we know the means, variances, and covariances of the return rates of all the financial instruments. In some practical situations, however, we do not have enough information to determine the variances and covariances, we only know the means. To provide a reasonable portfolio allocation for such cases, researchers proposed a heuristic maximum entropy approach. In this paper, we provide an economic justification for this heuristic idea.

Published in Journal of Uncertain Systems, 2020, Vol. 13, No. 3, pp. 197-200.

Research conferences are often organized by volunteers, namely, by people who are specialists in research but not in conference organization. As a result, some conferences are very successful, while some are not very successful. It is therefore desirable to provide newbie conference organizers with helpful recommendations. In this article, based on the experience of a successful recent conference, we provide the corresponding basic recommendations -- as well as their general-system explanation.

Published in Geombinatorics, 2020, Vol. 29, No. 4, pp. 104-110.

It is empirically known that roads built on clay soils have different nonlinear mechanical properties than roads built on granular soils (such as gravel or sand). In this paper, we show that this difficult-to-explain empirical fact can be naturally explained if we analyze the corresponding geometric symmetries.

Published in Applied Mathematical Sciences

At first glance, it seems that people should be paid in proportion to their contribution, so if one person produces a little more than the other one, he/she should be paid a little more. In reality, however, top performers are paid dis-proportionally more than those whose performance is slightly worse. How can we explain this from an economic viewpoint? We show that actually there is no paradox here: a simple economic analysis shows that in many area, it makes perfect economic sense to pay much more to top performers.

Published in Proceedings of the 2019 IEEE Symposium Series on Computational Intelligence SSCI'2019, Xiamen, China, December 6-9, 2019, pp. 828-832.

One of the main reasons why computations -- in particular, engineering computations -- take long is that, to be on the safe side, models take into account all possible affecting features, most of which turn out to be not really relevant for the corresponding physical problem. From this viewpoint, it is desirable to find out which inputs are relevant. In general, the problem of checking the input's relevancy is itself NP-hard, which means, crudely speaking, that no feasible algorithm can always solve it. Thus, it is desirable to speed up this checking as much as possible. One possible way to speed up such a checking is to use quantum computing, namely, to use the Deutsch-Josza algorithm. However, this algorithm is just a way to solve this problem, it is not clear whether a more efficient (or even different) quantum algorithm is possible for solving this problem. In this paper, we show that the Deutsch-Josza algorithm is, in effect, the only possible way to use quantum computing for checking which inputs are relevant.

Published in: Martine Ceberio and Vladik Kreinovich (eds.), How Uncertainty-Related Ideas Can Provide Theoretical Explanation for Empirical Dependencies, Springer, Cham, Switzerland, 2021, pp. 63-67.

Formal implication does not capture the intuitive idea of "if A then B", since in formal implication, every two true statements -- even completely unrelated ones -- imply each other. A more adequate description of intuitive implication happens if we consider how much the use of A can shorten a derivation of B. At first glance, it may seem that the number of bits by which we shorten this derivation is a reasonable degree of implication, but we show that this number is not in good accordance with our intuition, and that a natural formalization of this intuition leads to the need to use, as the desired degree, the ratio between the shortened derivation length and the original length.

Published in Proceedings of the 2019 IEEE Symposium Series on Computational Intelligence SSCI'2019, Xiamen, China, December 6-9, 2019, pp. 814-817.

Deep learning and deep reinforcement learning are, at present, the best available machine learning tools for use in engineering problems. However, at present, the use of these tools is limited by the fact that they are very time-consuming, usually requiring the use of a high performance computer. It is therefore desirable to look for possible ways to speed up the corresponding computations. One of the time-consuming parts of these algorithms is softmax selection, when instead of selecting the alternative with the largest possible value of the corresponding objective function, we select all possible values, with probabilities increasing with the value of the objective function. In this paper, we propose a significantly faster quantum-computing alternative to softmax selection.

Published in Applied Mathematical Sciences, 2019, Vol. 13, No. 12, pp. 585-589.

When people have several possible investment instruments, people often invest equally into these instruments: in the case of n instruments, they invest 1/n of their money into each of these instruments. Of course, if additional information about each instrument is available, this 1/n investment strategy is not optimal. We show, however, that in the absence of reliable information, 1/n investment is indeed the best strategy.

Java and Android applications can be written in the same programming language. Thus, it is natural to ask how much code can be shared between them. In this paper, we perform a case study to measure quantitatively the amount of code that can be shared and reused for a multiplatform application running on the Java platform and the Android platform. We first configure a development environment consisting of platform-specific tools and supporting continuous integration. We then propose a general architecture for a multiplatform application under a guiding design principle of having clearly defined interfaces and employing loose coupling to accommodate platform differences and variations. Specifically, we separate our application into two parts, a platform-independent part (PIP) and a platform-dependent part (PDP), and share the PIP between platform-specific versions. Our key finding is that 37%--40% of code can be shared and reused between the Java and the Android versions of our application. Interestingly, the Android version requires 8% more code than Java due to platform-specific constraints and concerns. We also learned that the quality of an application can be improved dramatically through multiplatform development.

Published in Panos Pardalos, Varvara Rasskazova, and Michael N. Vrahatis (eds.), Black Box Optimization, Machine Learning and No-Free Lunch Theorems, Springer, Cham, Switzerland, 2021, p. 195-220. One of the main objectives of science and engineering is to predict the future state of the world -- and to come up with devices and strategies that would make this future state better. In some practical situations, we know how the state changes with time -- e.g., in meteorology, we know the partial differential equations that describes the atmospheric processes. In such situations, prediction becomes a purely computational problem. In many other situations, however, we do not know the equation describing the system's dynamics. In such situations, we need to learn this dynamics from data. At present, the most efficient way of such learning is to use deep learning -- training a neural network with a large number of layers. To make this idea truly efficient, several trial-and-error-based heuristics were discovered, such as the use of rectified linear neurons, softmax, etc. In this chapter, we show that the empirical success of many of these heuristics can be explained by optimization-under-uncertainty techniques.

Published in: Vladik Kreinovich and Nguyen Hoang Phuong (eds.), Soft Computing for Biomedical Applications and Related Topics, Springer Verlag, Cham, Switzerland, 2021, pp. 67-79.

The more information we have about a quantity, the more accurately we can estimate this quantity. In particular, if we have several estimates of the same quantity, we can fuse them into a single more accurate estimate. What is the accuracy of this estimate? The corresponding formulas are known for the case of probabilistic uncertainty. In this paper, we provide similar formulas for the cases of interval and fuzzy uncertainty.

Published in: Vladik Kreinovich and Nguyen Hoang Phuong (eds.), Soft Computing for Biomedical Applications and Related Topics, Springer Verlag, Cham, Switzerland, 2021, pp. 61-65.

At present, one of the main ways to gauge the quality of a researcher is to use his or her h-index, which is defined as the largest integer n such that the researcher has at least n publications each of which has at least n citations. The fact that this quantity is widely used indicates that h-index indeed reasonably adequately describes the researcher's quality. So, this notion must capture some intuitive idea. However, the above definition is not intuitive at all, it sound like a somewhat convoluted mathematical exercise. So why is h-index so efficient? In this paper, we use known mathematical facts about h-index -- in particular, the results of its fuzzy-related analysis -- to come up with an intuitive explanation for the h-index's efficiency.

Published in Journal of Uncertain Systems, 2019, Vol. 13, No. 3, pp. 216-219.

Machine learning techniques have been very efficient in many applications, in particular, when learning to classify a given object to one of the given classes. Such classification problems are ubiquitous: e.g., in medicine, such a classification corresponds to diagnosing a disease, and the resulting tools help medical doctors come up with the correct diagnosis. There are many possible ways to set up the corresponding neural network (or another machine learning technique). A direct way is to design a single neural network with as many outputs as there are classes -- so that for each class i, the system would generate a degree of confidence that the given object belongs to this class. Instead of designing a single neural network, we can follow a hierarchical approach corresponding to a natural hierarchy of classes: classes themselves can usually be grouped into a few natural groups, each group can be subdivided into subgroups, etc. So, we set up several networks: the first classifies the object into one of the groups, then another one classifies it into one of the subgroups, etc., until we finally get the desired class. From the computational viewpoint, this hierarchical scheme seems to be too complicated: why do it if we can use a direct approach? However, surprisingly, in many practical cases, the hierarchical approach works much better. In this paper, we provide a possible explanation for this unexpected phenomenon.