Mathematics Commons^™

Towards Optimal Implementation Of Decentralized Currencies: How To Best Select Probabilities In An Ethereum-Type Proof-Of-Stake Protocol, Thach N. Nguyen, Christian Servin, Vladik Kreinovich

Departmental Technical Reports (CS)

Nowadays, most financial transactions are based on a centralized system, when all the transaction records are stored in a central location. This centralization makes the financial system vulnerable to cyber-attacks. A natural way to make the financial system more robust and less vulnerable is to switch to decentralized currencies. Such a transition will also make financial system more transparent. Historically first currency of this type -- bitcoin -- use a large amount of electric energy to mine new coins and is, thus, not scalable to the level of financial system as a whole. A more realistic and less energy-consuming scheme …

Relativistic Effects Can Be Used To Achieve A Universal Square-Root (Or Even Faster) Computation Speedup, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In this paper, we show that special relativity phenomenon can be used to reduce computation time of any algorithm from T to square root of T. For this purpose, we keep computers where they are, but the whole civilization starts moving around the computer -- at an increasing speed, reaching speeds close to the speed of light. A similar square-root speedup can be achieved if we place ourselves near a growing black hole. Combining the two schemes can lead to an even faster speedup: from time T to the 4-th order root of T.

Translating Discrete Estimates Into A Less Detailed Scale: An Optimal Approach, Thongchai Dumrongpokaphan, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practical situations, we use estimates that experts make on a 0-to-n scale. For example, to estimate the quality of a lecturer, we ask each student to evaluate this quality by selecting an integer from 0 to n. Each such estimate may be subjective; so, to increase the estimates' reliability, it is desirable to combine several estimates of the corresponding quality. Sometimes, different estimators use slightly different scales: e.g., one estimator uses a scale from 0 to n+1, and another estimator uses a scale from 0 to n. In such situations, it is desirable to translate these estimates to …

A Symmetry-Based Explanation Of The Main Idea Behind Chubanov's Linear Programming Algorithm, Olga Kosheleva, Vladik Kreinovich, Thongchai Dumrongpokaphan

Departmental Technical Reports (CS)

Many important real-life optimization problems can be described as optimizing a linear objective function under linear constraints -- i.e., as a linear programming problem. This problem is known to be not easy to solve. Reasonably natural algorithms -- such as iterative constraint satisfaction or simplex method -- often require exponential time. There exist efficient polynomial-time algorithms, but these algorithms are complicated and not very intuitive. Also, in contrast to many practical problems which can be computed faster by using parallel computers, linear programming has been proven to be the most difficult to parallelize. Recently, Sergei Chubanov proposed a modification of …

Why Bohmian Approach To Quantum Econometrics: An Algebraic Explanation, Vladik Kreinovich, Olga Kosheleva, Songsak Sriboonchitta

Departmental Technical Reports (CS)

Many equations in economics and finance are very complex. As a result, existing methods of solving these equations are very complicated and time-consuming. In many practical situations, more efficient algorithms for solving new complex equations appear when it turns out that these equations can be reduced to equations from other application areas -- equations for which more efficient algorithms are already known. It turns out that some equations in economics and finance can be reduced to equations from physics -- namely, from quantum physics. The resulting approach for solving economic equations is known as quantum econometrics. In quantum physics, …

Soft Computing Ideas Can Help Earthquake Geophysics, Solymar Ayala Cortez, Aaron A. Velasco, Vladik Kreinovich

Departmental Technical Reports (CS)

Earthquakes can be devastating, thus it is important to gain a good understanding of the corresponding geophysical processing. One of the challenges in geophysics is that we cannot directly measure the corresponding deep-earth quantities, we have to rely on expert knowledge, knowledge which often comes in terms of imprecise ("fuzzy") words from natural language. To formalize this knowledge, it is reasonable to use techniques that were specifically designed for such a formalization -- namely, fuzzy techniques, In this paper, we formulate the problem of optimally representing such knowledge. By solving the corresponding optimization problem, we conclude that the optimal representation …

Economics Of Commitment: Why Giving Away Some Freedom Makes Sense, Vladik Kreinovich, Olga Kosheleva, Mahdokhat Afravi, Genesis Bejarano, Marisol Chacon

Departmental Technical Reports (CS)

In general, the more freedom we have, the better choices we can make, and thus, the better possible economic outcomes. However, in practice, people often artificially restrict their future options by making a commitment. At first glance, commitments make no economic sense, and so their ubiquity seems puzzling. Our more detailed analysis shows that commitment often makes perfect economic sense: namely, it is related to the way we take future gains and losses into account. With the traditionally assumed exponential discounting, commitment indeed makes no economic sense, but with the practically observed hyperbolic discounting, commitment is indeed often economically beneficial.

Why Bellman-Zadeh Approach To Fuzzy Optimization, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In many cases, we need to select the best of the possible alternatives, but we do not know for sure which alternatives are possible and which are not possible. Instead, for each alternative x, we have a subjective probability p(x) that this alternative is possible. In 1970, Richard Bellman and Lotfi Zadeh proposed a heuristic method for selecting an alternative under such uncertainty. Interestingly, this method works very well in many practical applications, while similarly motivated alternative formulas do not work so well. In this paper, we explain the empirical success of the Bellman-Zadeh approach by showing that its formulas …

How Interval Measurement Uncertainty Affects The Results Of Data Processing: A Calculus-Based Approach To Computing The Range Of A Box, Andrew Pownuk, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practical applications, we are interested in the values of the quantities y₁, ..., y_m which are difficult (or even impossible) to measure directly. A natural idea to estimate these values is to find easier-to-measure related quantities x₁, ..., x_n and to use the known relation to estimate the desired values y_i. Measurements come with uncertainty, and often, the only thing we know about the actual value of each auxiliary quantity x_i is that it belongs to the interval [X_i − Δ_i, X_i + Δ_{i …}

Why Under Stress Positive Reinforcement Is More Effective? Why Optimists Study Better? Why People Become Restless? Simple Utility-Based Explanations, Francisco Zapata, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In this paper, we use the utility-based approach to decision making to provide simple answers to the following three questions: Why under stress positive reinforcement is more effective? Why optimists study better? Why people become restless?

Why Zipf's Law: A Symmetry-Based Explanation, Daniel Cervantes, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practical situations, we have probability distributions for which, for large values of the corresponding quantity x, the probability density has the form ρ(x) ~ x^−αfor some α > 0. While, in principle, we have laws corresponding to different α, most frequently, we encounter situations -- first described by Zipf for linguistics -- when α is close to 1. The fact that Zipf's has appeared frequently in many different situations seems to indicate that there must be some fundamental reason behind this law. In this paper, we provide a possible explanation.

How To Explain The Results Of The Richard Thaler's 1997 Financial Times Contest, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In 1997, by using a letter published in Financial Times, Richard H. Thaler, the 2017 Nobel Prize winner in Economics, performed the following experiment: he asked readers to submit numbers from 0 to 100, so that the person whose number is the closest to 2/3 of the average will be the winner. An intuitive answer is to submit 2/3 of the average (50), i.e., 33 1/3. A logical answer, as can be explained, is to submit 0. The actual winning submission was -- depending on how we count -- 12 or 13. In this paper, we propose a possible explanation …

Italian Folk Multiplication Algorithm Is Indeed Better: It Is More Parallelizable, Martine Ceberio, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Traditionally, many ethnic groups had their own versions of arithmetic algorithms. Nowadays, most of these algorithms are studied mostly as pedagogical curiosities, as an interesting way to make arithmetic more exciting to the kids: by applying to their patriotic feelings -- if they are studying the algorithms traditionally used by their ethic group -- or simply to their sense of curiosity. Somewhat surprisingly, we show that one of these algorithms -- a traditional Italian multiplication algorithm -- is actually in some reasonable sense better than the algorithm that we all normally use -- namely, it is easier to parallelize.

Fuzzy Analogues Of Sets And Functions Can Be Uniquely Determined From The Corresponding Ordered Category: A Theorem, Christian Servin, Gerardo Muela, Vladik Kreinovich

Departmental Technical Reports (CS)

In modern mathematics, many concepts and ideas are described in terms of category theory. From this viewpoint, it is desirable to analyze what can be determined if, instead of the basic category of sets, we consider a similar category of fuzzy sets. In this paper, we describe a natural fuzzy analog of the category of sets and functions, and we show that, in this category, fuzzy relations (a natural fuzzy analogue of functions) can be determined in category terms -- of course, modulo 1-1 mapping of the corresponding universe of discourse and 1-1 re-scaling of fuzzy degrees.

Why Rectified Linear Neurons Are Efficient: Symmetry-Based, Complexity-Based, And Fuzzy-Based Explanations, Olac Fuentes, Justin Parra, Elizabeth Y. Anthony, Vladik Kreinovich

Departmental Technical Reports (CS)

Traditionally, neural networks used a sigmoid activation function. Recently, it turned out that piecewise linear activation functions are much more efficient -- especially in deep learning applications. However, so far, there have been no convincing theoretical explanation for this empirical efficiency. In this paper, we show that, by using different uncertainty techniques, we can come up with several explanations for the efficiency of piecewise linear neural networks. The existence of several different explanations makes us even more confident in our results -- and thus, in the efficiency of piecewise linear activation functions.

How To Make A Proof Of Halting Problem More Convincing: A Pedagogical Remark, Benjamin W. Robertson, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

As an example of an algorithmically undecidable problem, most textbooks list the impossibility to check whether a given program halts on given data. A usual proof of this result is based on the assumption that the hypothetical halt-checker works for all programs. To show that a halt-checker is impossible, we design an auxiliary program for which the existence of such a halt-checker leads to a contradiction. However, this auxiliary program is usually very artificial. So, a natural question arises: what if we only require that the halt-checker work for reasonable programs? In this paper, we show that even with such …

Quantum Econometrics: How To Explain Its Quantitative Successes And How The Resulting Formulas Are Related To Scale Invariance, Entropy, Fuzzy, And Copulas, Hung T. Nguyen, Kittawit Autchariyapanitkul, Olga Kosheleva, Vladik Kreinovich, Songsak Sriboonchitta

Departmental Technical Reports (CS)

Many aspects of human behavior seem to be well-described by formulas of quantum physics. In this paper, we explain this phenomenon by showing that the corresponding quantum-looking formulas can be derived from the general ideas of scale invariance, fuzziness, and copulas. We also use these ideas to derive a general family of formulas that include non-quantum and quantum probabilities as particular cases -- formulas that may be more adequate for describing human behavior than purely non-quantum or purely quantum ones.

Do It Today Or Do It Tomorrow: Empirical Non-Exponential Discounting Explained By Symmetry And Fuzzy Ideas, Francisco Zapata, Olga Kosheleva, Vladik Kreinovich, Thongchai Dumrongpokaphan

Departmental Technical Reports (CS)

At first glance, it seems to make sense to conclude that when a 1 dollar reward tomorrow is equivalent to a D < 1 dollar reward today, the day-after-tomorrow's 1 dollar reward would be equivalent to D * D = D² dollars today, and, in general, a reward after time t is equivalent to D(t) = D^t dollars today. This exponential discounting function D(t) was indeed proposed by the economists, but it does not reflect the actual human behavior. Indeed, according to this formula, the effect of distant future events is negligible, and thus, it would be reasonable for a person to take on huge loans or get engaged in unhealthy behavior even when the long-term consequences …

An Ancient Bankruptcy Solution Makes Economic Sense, Anh H. Ly, Michael Zakharevich, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

While econometrics is a reasonable recent discipline, quantitative solutions to economic problem have been proposed since the ancient times. In particular, solutions have been proposed for the bankruptcy problem: how to divide the assets between the claimants? One of the challenges of analyzing ancient solutions to economics problems is that these solutions are often presented not as a general algorithm, but as a sequence of examples. When there are only a few such example, it is often difficult to convincingly extract a general algorithm from them. This was the case, for example, for the supposedly fairness-motivated Talmudic solution to the …

Efficient Parameter-Estimating Algorithms For Symmetry-Motivated Models: Econometrics And Beyond, Vladik Kreinovich, Anh H. Ly, Olga Kosheleva, Songsak Sriboonchitta

Departmental Technical Reports (CS)

It is known that symmetry ideas can explain the empirical success of many non-linear models. This explanation makes these models theoretically justified and thus, more reliable. However, the models remain non-linear and thus, identification or the model's parameters based on the observations remains a computationally expensive nonlinear optimization problem. In this paper, we show that symmetry ideas can not only help to select and justify a nonlinear model, they can also help us design computationally efficient almost-linear algorithms for identifying the model's parameters.

Maximum Entropy Beyond Selecting Probability Distributions, Thach N. Nguyen, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Traditionally, the Maximum Entropy technique is used to select a probability distribution in situations when several different probability distributions are consistent with our knowledge. In this paper, we show that this technique can be extended beyond selecting probability distributions, to explain facts, numerical values, and even types of functional dependence.

Almost All Diophantine Sets Are Undecidable, Vladik Kreinovich

Departmental Technical Reports (CS)

The known 1970 solution to the 10th Hilbert problem says that no algorithm is possible that would decide whether a given Diophantine equation has a solution. In set terms, this means that not all Diophantine sets are decidable. In a posting to the Foundations of Mathematica mailing list, Timothy Y. Chow asked for possible formal justification for his impression that most Diophantine equations are not decidable. One such possible justification is presented in this paper.

How To Use Absolute-Error-Minimizing Software To Minimize Relative Error: Practitioner's Guide, Afshin Gholamy, Vladik Kreinovich

Departmental Technical Reports (CS)

In many engineering and scientific problems, there is a need to find the parameters of a dependence from the experimental data. There exist several software packages that find the values for these parameters -- values for which the mean square value of the absolute approximation error is the smallest. In practice, however, we are often interested in minimizing the mean square value of the relative approximation error. In this paper, we show how we can use the absolute-error-minimizing software to minimize the relative error.

What Is The Optimal Bin Size Of A Histogram: An Informal Description, Afshin Gholamy, Vladik Kreinovich

Departmental Technical Reports (CS)

A natural way to estimate the probability density function of an unknown distribution from the sample of data points is to use histograms. The accuracy of the estimate depends on the size of the histogram's bins. There exist heuristic rules for selecting the bin size. In this paper, we show that these rules indeed provide the optimal value of the bin size.

Practical Need For Algebraic (Equality-Type) Solutions Of Interval Equations And For Extended-Zero Solutions, Ludmila Dymova, Pavel Sevastjanov, Andrzej Pownuk, Vladik Kreinovich

Departmental Technical Reports (CS)

One of the main problems in interval computations is solving systems of equations under interval uncertainty. Usually, interval computation packages consider united, tolerance, and control solutions. In this paper, we explain the practical need for algebraic (equality-type) solutions, when we look for solutions for which both sides are equal. In situations when such a solution is not possible, we provide a justification for extended-zero solutions, in which we ignore intervals of the type [−a, a].

Granular Approach To Data Processing Under Probabilistic Uncertainty, Andrzej Pownuk, Vladik Kreinovich

Departmental Technical Reports (CS)

In many real-life situations, uncertainty can be naturally described as a combination of several components, components which are described by probabilistic, fuzzy, interval, etc. granules. In such situations, to process this uncertainty, it is often beneficial to take this granularity into account by processing these granules separately and then combining the results.

In this paper, we show that granular computing can help even in situations when there is no such natural decomposition into granules: namely, we can often speed up processing of uncertainty if we first (artificially) decompose the original uncertainty into appropriate granules.

A Bad Plan Is Better Than No Plan: A Theoretical Justification Of An Empirical Observation, Songsak Sriboonchitta, Vladik Kreinovich

Departmental Technical Reports (CS)

In his 2014 book "Zero to One", a software mogul Peter Thiel lists the lessons he learned from his business practice. Most of these lessons make intuitive sense, with one exception -- his observation that "a bad plan is better than no plan" seems to be counterintuitive. In this paper, we provide a possible theoretical explanation for this somewhat counterintuitive empirical observation.

Possible Explanation Of Empirical Values Of The Matern Smoothness Parameter For The Temporal Covariance Of Gps Measurements, Gaël Kermarrec, Steffen Schön, Vladik Kreinovich

Departmental Technical Reports (CS)

The measurement errors of GPS measurements are largely due to the atmosphere, and the unpredictable part of these errors are due to the unpredictable (random) atmospheric phenomena, i.e., to turbulence. Turbulence-generated measurement errors should correspond to the smoothness parameter ν = 5/6 in the Matern covariance model. Because of this, we expected the empirical values of this smoothness parameter to be close to 5/6. When we estimated ν based on measurement results, we indeed got values close to 5/6, but interestingly, all our estimates were actually close to 1 (and slightly larger than 1). In this paper, we provide a …

How Accurate Are Expert Estimations Of Correlation?, Michael Beer, Zitong Gong, Francisco Alejandro Diaz De La O, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practical situations, it is important to know the correlation between different quantities -- finding correlations helps find the causes of different phenomena, and helps to find way to improve the situation. Often, there is not enough empirical data to experimentally determine all possible correlation. In such cases, a natural idea is to supplement this situation with expert estimates. Expert estimates are rather crude. So, to decide whether to act based on these estimates, it is desirable to know how accurate are expert estimates. In this paper, we propose several techniques for gauging this accuracy.

In Education, Delayed Feedback Is Often More Efficient Than Immediate Feedback: A Geometric Explanation, Francisco Zapata, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Feedback is important in education. It is commonly believed that immediate feedback is very important. That is why instructors stay often late at night grading students' assignments -- to make sure that the students get their feedback as early as possible. However, surprisingly, experiments show that in many cases, delayed feedback is more efficient that the immediate one. In this paper, we provide a simple geometric explanation of this seemingly counter-intuitive empirical phenomenon.