Physical Sciences and Mathematics Commons^™

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.

How Accurate Are Fuzzy Control Recommendations: Interval-Valued Case, Juan Carlos Figueroa-Garcia, Vladik Kreinovich

Departmental Technical Reports (CS)

As a result of applying fuzzy rules, we get a fuzzy set describing possible control values. In automatic control systems, we need to defuzzify this fuzzy set, i.e., to transform it to a single control value. One of the most frequently used defuzzification techniques is centroid defuzzification. From the practical viewpoint, an important question is: how accurate is the resulting control recommendation? The more accurately we need to implement the control, the more expensive the resulting controller.

The possibility to gauge the accuracy of the fuzzy control recommendation follows from the fact that, from the mathematical viewpoint, centroid defuzzification is …

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.

Fuzzy Logic Beyond Traditional "And"-Operations, Vladik Kreinovich, Olga Kosheleva

Departmental Technical Reports (CS)

In the traditional fuzzy logic, we can use "and"-operations (also known as t-norms) to estimate the expert's degree of confidence in a composite statement A&B based on his/her degrees of confidence d(A) and d(B) in the corresponding basic statements A and B. But what if we want to estimate the degree of confidence in A&B&C in situations when, in addition to the degrees of estimate d(A), d(B), and d(C) of the basic statements, we also know the expert's degrees of confidence in the pairs d(A&B), d(A&C), and d(B&C)? Traditional ``and''-operations can provide such an estimate -- but only by ignoring …

Randomized Tax Deadlines Can Help Economy, Julio C. Urenda, Olga Kosheleva

Departmental Technical Reports (CS)

Purpose: While the main purpose of reporting -- e.g., reporting for taxes -- is to gauge the economic state of a company, the fact that reporting is done at pre-determined dates distorts the reporting results. For example, to create a larger impression of their productivity, companies fire temporary workers before the reporting date and re-hire then right away. The purpose of this study is to decide how to avoid such distortion.

Design/methodology/approach: We want to make our solution applicable for all possible reasonable optimality criteria. Thus, we use a general formalism for describing and analyzing all such criteria.

Findings: We …

Why Chomsky Normal Form: A Pedagogical Note, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

To simplify the design of compilers, Noam Chomsky proposed to first transform a description of a programming language -- which is usually given in the form of a context-free grammar -- into a simplified "normal" form. A natural question is: why this specific normal form? In this paper, we provide an answer to this question.

Godel's Proof Of Existence Of God Revisited, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In his unpublished paper, the famous logician Kurt Godel provided arguments in favor of the existence of God. These arguments are presented in a very formal way, which makes them difficult to understand to many interested readers. In this paper, we describe a simplifying modification of Godel's proof which will hopefully make it easier to understand. We also describe, in clear terms, why Godel's arguments are just that -- arguments -- and not a convincing proof.

Five Revolutionary Ideas In The 1950s-70s Science: 90th Birthday Of Revolt Pimenov, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

This year, Revolt Pimenov, a philosophical thinker whose main ideas were in geometry of space-time, would have turned 90. In this essay, we explain how in the 1950s-70s, when he was most productive, his were two of the five natural and important revolutionary scientific ideas -- along with fuzzy logic, constructive mathematics, and scalar-tensor theory of gravitation, ideas that, in our opinions, still have potential to change the world.

How To Teach Advanced Highly Motivated Students: Teaching Strategy Of Iosif Yakovlevich Verebeichik, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

The paper describes and explains the teaching strategy of Iosif Yakovlevich Verebeichik, a successful mathematics teacher at special mathematical high schools -- schools for students interested in and skilled in mathematics. The resulting strategy seems counterintuitive and contrary to all the pedagogical advice. Our explanation is not complete: it worked well for this teacher, but others who tried to follow seemingly the same strategy did not succeed. How he made it work, how can others make it work -- this is still not clear. In the words of Verebeichik himself, while mathematics itself is a science, teaching mathematics is an …

How To Extend Interval Arithmetic So That Inverse And Division Are Always Defined, Tahea Hossain, Jonathan Rivera, Yash Sharma, Vladik Kreinovich

Departmental Technical Reports (CS)

In many real-life data processing situations, we only know the values of the inputs with interval uncertainty. In such situations, it is necessary to take this interval uncertainty into account when processing data. Most existing methods for dealing with interval uncertainty are based on interval arithmetic, i.e., on the formulas that describe the range of possible values of the result of an arithmetic operation when the inputs are known with interval uncertainty. For most arithmetic operations, this range is also an interval, but for division, the range is sometimes a disjoint union of two semi-infinite intervals. It is therefore desirable …

Impact Of Various Hydrophobic Cargoes On The Structural Properties Of Pluronic-F127 Based Micelles Near The Standard Critical Transition Zone, Tahmida Raheen Iqbal

Open Access Theses & Dissertations

Pluronic F-127 is a triblock copolymer (PEO-PPO-PEO) that can form a micelle at critical micelle concentration (CMC). CMC for Pluronic F-127 is 0.7% at 25 0C. Micelles are lipid molecules that can arrange themselves in a spherical form in an aqueous solution with a hydrophobic core and hydrophilic shell. Pluronic F-127 can encapsulate poor water-soluble drugs and administrate the medicine to the targeted area with a controlled drug release rate and low toxicity. The purpose of this study is to understand the structural variation in Pluronic F-127 micelles after loading with different hydrophobic cargoes (alkanes, fatty acids, anticancer therapeutic agents). …

Robust Variable Selection In Multiple Linear Regression Via Penalized Least Trimmed Squares., Reagan Kesseku

Open Access Theses & Dissertations

Variable selection has been studied using different approaches. Its growing importance lies in numerous applications to high-dimensional data from experiments and natural phenomena. Often, models are to be constructed from such data based on significant variables for estimation or prediction purposes. This demands not just any variable selectionmethod, but one that is robust, computationally efficient and with other desirable statistical properties. Besides the high-dimensionality of such data, the presence of outliers is common due to heterogeneous sources. Though outliers often contain useful information, they can unduly influence non-robust estimators to produce misleading results. This is the case for ordinary least …

Synthesis And Characterization Of Functiona Materials For 3d Printed Composites, Elizabeth Irene Reza

Open Access Theses & Dissertations

The following body of work encompasses the synThesis, upscaling, additive manufacturing, and characterization of microspheres as functional fillers incorporated into polymer syntactic foams. Synthesizing microspheres permits control over morphology, dispersion of particles, and size while the use of direct write manufacturing allowed for the controllability of filler orientation. Two different syntactic foams were developed from similar methodologies: (1) a silicone syntactic foam (2) epoxy syntactic foam. The silicone epoxy foam showed an increasing in damping and elastic modulus. The epoxy syntactic foams showed an enhanced thermal and electrical properties.