Mathematics Commons^™

Shall We Be Foxes Or Hedgehogs: What Is The Best Balance For Research?, Miroslav Svitek, Olga Kosheleva, Shahnaz Shahbazova, Vladik Kreinovich

Departmental Technical Reports (CS)

Some researchers have few main ideas that they apply to many different problems -- they are called hedgehogs. Other researchers have many ideas but apply them to fewer problems -- they are called foxes. Both approaches have their advantages and disadvantages. What is the best balance between these two approaches? In this paper, we provide general recommendations about this balance. Specifically, we conclude that the optimal productivity is when the time spent on generating new ideas is equal to the time spent on understanding new applications. So, if for a researcher, understanding a new problem is much easier than generating …

As Complexity Rises, Meaningful Statements Lose Precision -- But Why?, Miroslav Svitek, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

One of the motivations for Zadeh's development of fuzzy logic -- and one of the explanations for the success of fuzzy techniques -- is the empirical observation that as complexity rises, meaningful statements lose precision. In this paper, we provide a possible explanation for this empirical phenomenon.

Predicting (Economic) Trends: Why Signature Method In Machine Learning, Vladik Kreinovich, Chon Van Le

Departmental Technical Reports (CS)

In many practical situations, we can predict the trend -- i.e., how the system will change -- but we cannot predict the exact timing of this change: this timing may depend on many unpredictable factors. For example, we may be sure that the economy will recover, but how fast it will recover may depend on the status of the pandemic, on the weather-affected agriculture input, etc. In such trend predictions, one of the most efficient methods is signature method, which is based on applying machine learning techniques to several special characteristics of the corresponding time series. In this paper, we …

How To Work? How To Study? Shall We Cram For The Exams? And How Is This Related To Life On Earth?, Olga Kosheleva, Vladik Kreinovich, Nguyen Hoang Phuong

Departmental Technical Reports (CS)

If we follow the same activity for a long time, our productivity decreases. To increase productivity, a natural idea is therefore to switch to a different activity, and then to switch back and resume the current task. On the other hand, after each switch, we need some time to get back to the original productivity. As a result, too frequent switches are also counterproductive. Natural questions are: shall we switch? if yes, when? In this paper, we use a simple model to provide approximate answers to these questions.

Correcting Interval-Valued Expert Estimates: Empirical Formulas Explained, Laura A. Berrout Ramos, Vladik Kreinovich, Kittawit Autchariyapanitkul

Departmental Technical Reports (CS)

Experts' estimates are approximate. To make decisions based on these estimates, we need to know how accurate these estimate are. Sometimes, experts themselves estimate the accuracy of their estimates -- by providing the interval of possible values instead of a single number. In other cases, we can gauge the accuracy of the experts' estimates by asking several experts to estimates the same quantity and using the interval range of these values. In both situations, sometimes the interval is too narrow -- e.g., if an expert is overconfident. Sometimes, the interval is too wide -- if the expert is too cautious. …

Why Moving Fast And Breaking Things Makes Sense?, Francisco Zapata, Eric Smith, Vladik Kreinovich

Departmental Technical Reports (CS)

In the traditional approach to engineering system design, engineers usually come up with several possible designs, each improving on the previous ones. In coming up with these designs, they try their best to make sure that their designs stay within the safety and other constraints, to avoid potential catastrophic crashes. The need for these safety constraints makes this design process reasonably slow. Software engineering at first followed the same pattern, but then realized that since in most cases, failure of a software test does not lead to a catastrophe, it is much faster to first ignore constraints and then adjust …

Why 70/100 Is Satisfactory? Why Five Letter Grades? Why Other Academic Conventions?, Christian Servin, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Why 70/100 is usually a threshold for a student's satisfactory performance? Why there are usually only five letter grades? Why the usual arrangement of research, teaching, and service is 40-40-20? We show that all these arrangements -- and other similar academic arrangements -- can be explained by two ideas: the Laplace Indeterminacy Principle and the seven plus minus two law.

Blessings, God, Sacrifices: Possible Rational Explanations Of Biblical Ideas, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In this paper, we show that many seemingly irrational Biblical ideas can actually be rationally interpreted: that God is everywhere, that we can only say what God is not, that God's name is holy, why cannot you bless as many people as you want, etc. We do not insist on our interpretations, there probably are many others, our sole objective was to show that many Biblical ideas can be rationally explained.

How To Gauge Students' Ability To Collaborate?, Christian Servin, Olga Kosheleva, Shahnaz Shahbazova, Vladik Kreinovich

Departmental Technical Reports (CS)

Usually, we mostly gauge individual students' skills. However, in the modern world, problems are rarely solved by individuals, it is usually a group effort. So, to make sure that students are successful, we also need to gauge their ability to collaborate. In this paper, we describe when it is possible to gauge the students' ability to collaborate; in situations when such a determination is possible, we explain how exactly we can estimate these abilities.

What Is The Uncertainty Of The Result Of Data Processing: Fuzzy Analogue Of The Central Limit Theorem, Julio C. Urenda, Olga Kosheleva, Shahnaz Shahbazova, Vladik Kreinovich

Departmental Technical Reports (CS)

It is known that, due to the Central Limit Theorem, the probability distribution of the uncertainty of the result of data processing is, in general, close to Gaussian -- or to a distribution from a somewhat more general class known as infinitely divisible. We show that a similar result holds in the fuzzy case: namely, the membership function describing the uncertainty of the result of data processing is, in general, close to Gaussian -- or to a membership function from an explicitly described more general class.

So How To Make Group Decisions? Arrow's Impossibility Theorem 70 Years After, Hung T. Nguyen, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In 1951, Kenneth Arrow proved that it is not possible to have a group decision making procedure that satisfies reasonable requirements like fairness. From the theoretical viewpoint, this is a great result -- well-deserving the Nobel Prize that was awarded to Professor Arrow. However, from the practical viewpoint, the question remains -- so how should we make group decisions? A usual way to solve this problem is to provide some reasonable heuristic ideas, but the problem is that different seemingly reasonable idea often lead to different group decision -- this is known, e.g., for different voting schemes. In this paper, …

What Fuzzy And Quantum Computing Can Learn From The Success Of Deep Learning, Shahnaz Shahbazova, Vladik Kreinovich

Departmental Technical Reports (CS)

How can we apply the ideas that made deep neural networks successful to other aspects of computing? For this purpose, we reformulate these ideas in a more general form -- and we show that this generalization also covers fuzzy and quantum computing. This enables us to suggest that similar ideas can be helpful for fuzzy and quantum computing as well. In this suggestion, we are encouraged by the fact that as we show, to some extent, these ideas are already helpful.

Why Quantum Techniques Are A Good First Approximation To Economic Phenomena, And What Next, Vladik Kreinovich, Olga Kosheleva

Departmental Technical Reports (CS)

Somewhat surprisingly, several formulas of quantum physics -- the physics of micro-world -- provide a good first approximation to many social phenomena, in particular, to many economic phenomena, phenomena which are very far from micro-physics. In this paper, we provide three possible explanations for this surprising fact. First, we show that several formulas from quantum physics actually provide a good first-approximation description for many phenomena in general, not only to the phenomena of micro-physics. Second, we show that some quantum formulas represent the fastest way to compute nonlinear dependencies and thus, naturally appear when we look for easily computable models; …

Why Cauchy Membership Functions: Efficiency, Javier Viana, Stephan Ralescu, Kelly Cohen, Anca Ralescu, Vladik Kreinovich

Departmental Technical Reports (CS)

Fuzzy techniques depend heavily on eliciting meaningful membership functions for the fuzzy sets used. Often such functions are obtained from data. Just as often they are obtained from experts knowledgable of the domain and the problem being addressed. However, there are cases when neither is possible, for example because of insufficient data, or unavailable experts. What functions should one choose and what should guide such choice? This paper argues in favor of using Cauchy membership functions, thus named because their expression is similar to that of the Cauchy distributions. The paper provides a theoretical explanation for this choice.

Many Known Quantum Algorithms Are Optimal: Symmetry-Based Proofs, Vladik Kreinovich, Oscar Galindo, Olga Kosheleva

Departmental Technical Reports (CS)

Many quantum algorithms have been proposed which are drastically more efficient that the best of the non-quantum algorithms for solving the same problems. A natural question is: are these quantum algorithms already optimal -- in some reasonable sense -- or they can be further improved? In this paper, we review recent results showing that many known quantum algorithms are actually optimal. Several of these results are based on appropriate invariances (symmetries).

Why Rectified Linear Neurons: Two Convexity-Related Explanations, Jonatan Contreras, Martine Ceberio, Olga Kosheleva, Vladik Kreinovich, Nguyen Hoang Phuong

Departmental Technical Reports (CS)

At present, the most efficient machine learning technique is deep learning, in which non-linearity is attained by using rectified linear functions s(x)=max(0,x). Empirically, these functions work better than any other nonlinear functions that have been tried. In this paper, we provide a possible theoretical explanation for this empirical fact. This explanation is based on the fact that one of the main applications of neural networks is decision making, when we want to find an optimal solution. We show that the need to adequately deal with situations when the corresponding optimization problem is feasible -- i.e., for which the objective function …

Invariance-Based Approach: General Methods And Pavement Engineering Case Study, Edgar Daniel Rodriguez Velasquez, Vladik Kreinovich, Olga Kosheleva

Departmental Technical Reports (CS)

In many application areas such as pavement engineering, the phenomena are complex, and as a result, we do not have first-principle models describing the corresponding dependencies. Luckily, in many such areas, there is a lot of empirical data and, based on this data, many useful empirical dependencies have been found. The problem is that since many of these dependencies do not have a theoretical explanation, practitioners are often hesitant to use them: there have been many cases when an empirical formula stops being valid when circumstances change. To make the corresponding empirical formulas more reliable, it is therefore desirable to …

Why Dilated Convolutional Neural Networks: A Proof Of Their Optimality, Jonatan Contreras, Martine Ceberio, Vladik Kreinovich

Departmental Technical Reports (CS)

One of the most effective image processing techniques is the use of convolutional neural networks that use convolutional layers. In each such layer, the value of the output at each point is a combination of input data corresponding to several neighboring points. To improve the accuracy, researchers have developed a version of this technique, in which only data from some of the neighboring points is processed. It turns out that the most efficient case -- called dilated convolution -- is when we select the neighboring points whose differences in both coordinates are divisible by some constant l. In this paper, …

How General Is Fuzzy Decision Making?, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practical situations, users describe their preferences in imprecise (fuzzy) terms. In such situations, fuzzy techniques are a natural way to describe these preferences in precise terms.

Of course, this description is only an approximation to the ideal decision making that a person would perform if we took time to elicit his/her exact preferences. How accurate is this approximation? When can fuzzy decision making -- potentially -- describe the exact decision making, and when there is a limit to the accuracy of fuzzy approximations?

In this paper, we show that decision making can be precisely described in fuzzy terms …

Green Computing: Three Examples Of How Non-Trivial Mathematical Analysis Can Help, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Environment-related problems are extremely important for mankind, the fate of humanity itself depends on our ability to solve these problems. These problems are complex, we cannot solve them without using powerful computers. Thus, in the environmental research, environment-related computing is one of the main computing-related research directions. Another direction is related to the fact that computing itself can be (and currently is) harmful for the environment. How to make computing more environment-friendly, how to move towards green computing -- this is the second important direction. A third direction is motivated by the very complexity of environmental systems: it is difficult …

Is It Fair That Advanced Workers Get Paid Disproportionally More: Economic Analysis, Olga Kosheleva, Sean R. Aguilar

Departmental Technical Reports (CS)

On the one hand, everyone agrees that economics should be fair, that workers should get equal pay for equal work. Any instance of unfairness causes a strong disagreement. On the other hand, in many companies, advanced workers -- who produce more than others -- get paid dispropotionally more for their work, and this does not seem to cause any negative feelings. In this paper, we analyze this situation from the economic viewpoint. We show that from this viewpoint, additional payments for advanced workers indeed make economic sense, benefit everyone, and thus -- in contrast to the naive literal interpretation of …

Revolt Pimenov Taught Us How To Be Scientists, Vladik Kreinovich

Departmental Technical Reports (CS)

In 2021, we are celebrating the 90th birthday of Revolt Pimenov, a specialist in space-time geometry. He was my teacher. In this article, I am trying to summarize what he taught to his students.

Why Kappa Regression?, Julio C. Urenda, Orsolya Csiszár, József Dombi, György Eigner, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

A recent book provide examples that a new class of probability distributions and membership functions -- called kappa-regression distributions and membership functions -- leads to better data processing results than using previously known classes. In this paper, we provide a theoretical explanation for this empirical success -- namely, we show that these distributions are the only ones that satisfy reasonable invariance requirements.

Fuzzy Techniques, Laplace Indeterminacy Principle, And Maximum Entropy Approach Explain Lindy Effect And Help Avoid Meaningless Infinities In Physics, Julio C. Urenda, Sean R. Aguilar, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In many real-life situations, the only information that we have about some quantity S is a lower bound T ≤ S. In such a situation, what is a reasonable estimate for S? For example, we know that a company has survived for T years, and based on this information, we want to predict for how long it will continue surviving. At first glance, this is a type of a problem to which we can apply the usual fuzzy methodology -- but unfortunately, a straightforward use of this methodology leads to a counter-intuitive infinite estimate for S. There is an empirical …

What Is Wrong With Micromanagement: Economic View, Sean R. Aguilar, Olga Kosheleva

Departmental Technical Reports (CS)

Purpose: It is well known that micromanagement -- excessive control of employees -- is detrimental to the employees' morale and thus, decreases their productivity. But what if the managers keep people happy -- will there still be negative consequences of micromanagement? This is the problem analyzed in this paper.

Design/methodology/approach: To analyze our problem, we use general -- but simplified -- mathematical models of how productivity depends on the working rate.

Findings: We show that even in the absence of psychological discomfort, micromanagement is still detrimental to productivity. Interestingly, the negative effect of micromanagement increases as the population becomes more …

Is Our World Becoming Less Quantum?, Lidice Castro, Vladik Kreinovich

Departmental Technical Reports (CS)

According to the general idea of quantization, all physical dependencies are only approximately deterministic, and all physical "constants" are actually varying. A natural conclusion -- that some physicists made -- is that Planck's constant (that determines the magnitude of quantum effects) can also vary. In this paper, we use another general physics idea -- the second law of thermodynamics -- to conclude that with time, this constant can only decrease. Thus, with time (we are talking cosmological scales, of course), our world is becoming less quantum.

Order Relations Are Ubiquitously Fundamental: Alexandrov(-Zeeman) Theorem Extended From Space-Time Physics To Logical Reasoning And Decision Making, Vladik Kreinovich, Olga Kosheleva, Laxman Bokati, Laura Berrout

Departmental Technical Reports (CS)

In all areas of human activity, there are natural ordering relations: causality in space-time physics, preference in decision making, and logical inference in reasoning. In space-time physics, a 1950 theorem by A. D. Alexandrov proved that causality relation is fundamental: many other features, including numerical characteristics of time and space, can be reconstructed from this relation. In this paper, we provide simple proofs that, similarly, the corresponding ordering relations are fundamental in decision making and in logical reasoning.

Shall We Ignore All Intermediate Grades?, Christian Servin, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In most European universities, the overall student's grade for a course is determined exclusively by this student's performance on the final exam. All intermediate grades -- on homework, quizzes, and previous texts -- are, in effect, ignored. This arrangement helps gauge the student's performance by the knowledge that the student shows at the end of the course. The main drawback of this approach is that some students do not start studying until later, thinking that they can catch up and even get an excellent grade -- and this hurts their performance. To motivate students to study hard throughout the semester, …

Extension To Multidimensional Problems Of A Fuzzy-Based Explainable & Noise-Resilient Algorithm, Javier Viana, Stephan Ralescu, Kelly Cohen, Anca Ralescu, Vladik Kreinovich

Departmental Technical Reports (CS)

While Deep Neural Networks (DNNs) have shown incredible performance in a variety of data, they are brittle and opaque: easily fooled by the presence of noise, and difficult to understand the underlying reasoning for their predictions or choices. This focus on accuracy at the expense of interpretability and robustness caused little concern since, until recently, DNNs were employed primarily for scientific and limited commercial work. An increasing, widespread use of artificial intelligence and growing emphasis on user data protections, however, motivates the need for robust solutions with explainable methods and results. In this work, we extend a novel fuzzy based …

Why Too Much Interaction Between Different Parts Of The Brain Leads To Unhappiness, Ricardo Alvarez, Yamel Hernandez, Vladik Kreinovich

Departmental Technical Reports (CS)

Reasonably recent experiments show that unhappiness is strongly correlated with the excessive interaction between two parts of the brain -- amygdala and hippocampus. At first glance, in situations when outside signals are positive, additional interaction between two parts of the brain that get signals from different sensors should only reinforce the positive feeling. In this paper, we provide a simple explanation of why, instead of the expected reinforcement, we observe unhappiness.