Physical Sciences and Mathematics Commons^™

Baudelaire's Ideas Of Vagueness And Uniqueness In Art: Algorithm-Based Explanations, Luc Longpre, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

According to the analysis by the French philosopher Jean-Paul Sartre, the famous French poet and essayist Charles Baudelaire described (and followed) two main -- somewhat unusual -- ideas about art: that art should be vague, and that to create an object of art, one needs to aim for uniqueness. In this paper, we provide an algorithm-based explanation for these seemingly counter-intuitive ideas, explanation related to Kolmogorov complexity-based formalization of Garrett Birkhoff's theory of beauty.

How Much For A Set: General Case Of Decision Making Under Set-Valued Uncertainty, Laxman Bokati, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In real life, we often need to make a decision, i.e., we need to select one of the possible alternatives. In many practical situations, our objective is financial: we need to select an alternative that will bring us the largest financial gain. The problem is that usually, we only know the gain corresponding to each alternative with some uncertainty: instead of the exact numerical value of this gain, there is a whole set of possible values of this gain. How can we make decisions under such interval-valued uncertainty? An answer to this question is known for the case when these …

Why Fuzzy Techniques In Explainable Ai? Which Fuzzy Techniques In Explainable Ai?, Kelly Cohen, Laxman Bokati, Martine Ceberio, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

One of big challenges of many state-of-the-art AI techniques such as deep learning is that their results do not come with any explanations -- and, taking into account that some of the resulting conclusions and recommendations are far from optimal, it is difficult to distinguish good advice from bad one. It is therefore desirable to come up with explainable AI. In this paper, we argue that fuzzy techniques are a proper way to this explainability, and we also analyze which fuzzy techniques are most appropriate for this purpose. Interestingly, it turns out that the answer depends on what problem we …

How Void Ratio Depends On Grain Size In Soil Mechanics: Theoretical Explanation, Edgar Daniel Rodriguez Velasquez, Vladik Kreinovich

Departmental Technical Reports (CS)

When designing a road, it is important to know how many voids are in the underlying soil -- since these voids will affect the road stiffness. It is difficult to measure the voids ratio directly, so instead, we need to estimate it based on easier-to-measure characteristics such as grain size. There are empirical formulas for such estimation. In this paper, we provide a possible theoretical explanation for these empirical formulas.

Why Base-20, Base-40, And Base-60 Number Systems?, Sean R. Aguilar, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Historically, to describe numbers, some cultures used bases much larger than our usual base 10, namely, bases 20, 40, and 60. There are explanations for base 60, there is some explanation for base 20, but base 40 -- used in medieval Russia -- remains largely a mystery. In this paper, we provide a possible explanation for all these three bases, an explanation based on the natural need to manage large groups of people. We also speculate why different cultures used different bases.

Dimension Compactification Naturally Follows From First Principles, Julio C. Urenda, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

According to modern physics, space-time originally was of dimension 11 or higher, but then additional dimensions became compactified, i.e., size in these directions remains small and thus, not observable. As a result, at present, we only observed 4 dimensions of space-time. There are mechanisms that explain how compactification may have occurred, but the remaining question is why it occurred. In this paper, we provide two first-principles-based explanations for space-time compactification: based on Second Law of Thermodynamics and based on geometry and symmetries.

Additional Spatial Dimensions Can Help Speed Up Computations, Luc Longpre, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

While we currently only observe 3 spatial dimensions, according to modern physics, our space is actually at least 10-dimensional. In this paper, on different versions of the multi-D spatial models, we analyze how the existence of the additional spatial dimensions can help computations. It turns out that in all the versions, there is some speed up -- moderate when the extra dimensions are actually compactified, and drastic if extra dimensions are separated by a potential barrier.

Each Realistic Continuous Functional Dependence Implies A Relation Between Some Variables: A Theoretical Explanation Of A Fuzzy-Related Empirical Phenomenon, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In principle, one can have a continuous functional dependence y=f(x1,...,x_n) for which, for each proper subset of n+1 variable x1,...,x_n,y, there is no relation: i.e., for each selection of n variables out of these n+1, all combinations of these n values are possible. However, for fuzzy operations, there is always some non-trivial relation between y and one of the inputs xi; for example, for "and"-operations (t-norms) y=f(x1,x2), we have y ≤ x1; for "or"-operations (t-conorms) y=f(x1,x2) we have x1 ≤ y, etc. In this paper, we prove a general mathematical explanation for this empirical fact.

Fuzzy Logic Leads To A More Adequate Way Of Processing Likert-Scale Values: Case Study Of Burnout, Francisco Zapata, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Many phenomena like burnout are gauged by computing a linear combination of user-provided Likert-scale values. The problem with this traditional approach is that, while it makes sense to have linear combination of weights or other physical characteristics, a linear combination of Likert-scale values like "good" and "satisfactory" does not make sense. The only reason why linear combinations are used in practice is that the corresponding data processing tools are readily available. A more adequate approach would be to use fuzzy logic -- a technique specifically designed to deal with Likert-scale values. We show that fuzzy logic actually leads to a …

What Teachers Can Learn From Machine Learning, Christian Servin, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Decades ago, machine learning was not as good as human learning, so many machine learning techniques were borrowed from how we humans learn -- be it on the level of concepts or on the level of biological neurons, cells responsible for mental activities such as learning. Lately, however, machine learning techniques such as deep learning have started outperforming humans. It is therefore time to start borrowing the other way around, i.e., using machine learning experience to improve our human teaching and learning. In this paper, we describe several relevant ideas -- and explain how some of these ideas are related …

What Is The True Formula For Soil Permeability? Not Clear, Edgar Daniel Rodriguez Velasquez, Vladik Kreinovich

Departmental Technical Reports (CS)

To design and maintain pavements, it is important to know how fast water will penetrate the underlying soil. The speed of this penetration is determined by a quantity called permeability. There are several seemingly very different empirical and semi-empirical formulas that predict permeability. A recent attempt to select the formula that best fits the experimental data ended up in an unexpected conclusion that all three formula provide a good fit for the data. But these formulas are very different, how come that all three of them fit the same data? In this paper, we explain this somewhat paradoxical result.

How To Best Write Research Papers: Basic English? Sophisticated English?, Martine Ceberio, Christian Servin, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Instructors from English department praise our students when they use the most sophisticated grammatical constructions and the most appropriate (often rarely used) words -- as long as this helps better convey all the subtleties of the meaning. On the other hand, we usually teach the students to use the most primitive Basic English when writing our papers -- this way, the resulting paper will be most accessible to the international audience. Who is right? In this paper, we analyze this question by using a natural model -- inspired by Zipf's law -- and we conclude that to achieve the largest …

Low-Complexity Zonotopes Can Enhance Uncertainty Quantification (Uq), Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practical situations, the only information that we know about the measurement error is the upper bound D on its absolute value. In this case, once we know the measurement result X, the only information that we have about the actual value x of the corresponding quantity is that this value belongs to the interval [X − D, X + D]. How can we estimate the accuracy of the result of data processing under this interval uncertainty? In general, computing this accuracy is NP-hard, but in the usual case when measurement errors are relatively small, we can linearize the …

Why Romans Sometimes Wrote 8 As Viii, And Sometimes As Iix: A Possible Explanation, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Most of us are familiar with Roman numerals and with the standard way of describing numbers in the form of these numerals. What many people do not realize is that the actual ancient Romans often deviated from these rules. For example, instead of always writing the number 8 as VIII, i.e., 5 + 3, they sometimes wrote it as IIX, i.e., as 10 − 2. Some of such differences can be explained: e.g., the unusual way of writing 98 as IIC, i.e., as 100 − 2, can be explained by the fact that the Latin word for 98 literally means …

Why Do We Need Two Doses Of Covid-19 Vaccine: A Qualitative Explanation, Laxman Bokati, Julio Urenda, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

It is known that the most effective protection from Covid-19 comes if the vaccination is done in two doses separated by several weeks. In this paper, we provide a qualitative explanation for this empirical fact.

Why T-Duality: A Simple Explanation, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In many physical theories, there is a -- somewhat surprising -- similarity between events corresponding to large distances R and events corresponding to very small distances 1/R. Such similarity is known as T-duality. At present, the only available explanation for T-duality comes from a complex mathematical analysis of the corresponding formulas. In this paper, we provide an alternative explanation based on the fundamental notion of causality.

What Is The Logic Behind Cistercian Numbers?, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In the 13-15 centuries, many European monasteries used an unusual number system developed originally by the Cistercian monks; later on, this system was used by winemakers. In this paper, we provide a possible explanation of why these particular symbols were used.

Zadeh's Vision, Modern Physics, And The Future Of Computing, Vladik Kreinovich, Olga Kosheleva

Departmental Technical Reports (CS)

At first glance, Zadeh's ideas that everything is a matter of degree seem to be more appropriate for situations when we do not know the exact equations, when we only have expert rules for control and/or decision making. From this viewpoint, it may seem that in physics, where equations are ubiquitous and all the terms seem precise, there is not much place for fuzziness. But, as we show, in reality, fuzzy ideas can help -- and help dramatically -- in physics as well: in spite of the first impression, as physicists know well, many arguments in physics rely heavily on …

The Utep Corpus Of Dissatisfaction In Spoken Dialog, Jonathan E. Avila, Nigel Ward, Aaron Alarcon

Departmental Technical Reports (CS)

We present a corpus of spoken dialogs collected to support research in the automatic detection of times of dissatisfaction. We collected 191 mock customer-merchant dialogs in two conditions: one where the scripts guided the participants to a satisfactory, mutually agreeable outcome, and one where agreement was precluded. Most dialogs were 1 to 5 minutes in length. The corpus and metadata are freely available for research purposes.

How To Guarantee Fairness Of Grading Without Sacrificing Privacy?, Vladik Kreinovich, Olga Kosheleva, Christian Servin

Departmental Technical Reports (CS)

Everyone -– instructors and students –- want to make sure that grading of each test is fair, that the only thing that determines the students’ grade is their level of knowledge, that different students get the same penalty for the same mistake, irrespective of their gender, of their past grades, of their behavior in the class, of how many classes they missed, etc. How to help instructors achieve this goal? How to make sure that students are convinced that grading was indeed fair? In this paper, we describe possible measures: anonymous submissions, forming (and posting for all the student to …

How To Estimate Time Needed For Software Migration, Francisco Zapata, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practical situations, we need to migrate the existing software package to a new programming language and/or a new operating system. In such a migration, it is important to be able to accurately estimate time needed for this migration: if we underestimate this time, we will lose money and may go bankrupt; if we overestimate this time, other companies who estimate more accuracy will outbid us, and we will lose the contract. The formulas currently used for estimating migration time often lead to underestimation. In this paper, we start with the main ideas behind the existing formulas, and show …

Distributions On An Interval As A Scale-Invariant Combination Of Scale-Invariant Functions: Theoretical Explanation Of Empirical Marchenko-Pastur-Type Distributions, Vladik Kreinovich, Kevin Alvarez, Chon Van Le

Departmental Technical Reports (CS)

In many practical situations, we know the lower and upper bounds L and U on possible values of a quantity x. In such situations, the probability distribution of this quantity is also located on the corresponding interval [L, U]. In many such cases, the empirical probability distribution has the form d(x) = const * (x − L)α− * (U − x)α+ * xα. In the particular case α− = α+ = 0.5 and α = −1, we get the Marchenko-Pastur distribution that describes the distribution of the eigenvalues of a random matrix. However, in some cases, the empirical distribution corresponds …

When To Stop Computing And Start Investing, Sean R. Aguilar, Olga Kosheleva

Departmental Technical Reports (CS)

Purpose: The purpose of the study is to analyze when -- while predicting the future price of a financial instrument -- we should stop computations and start using this information for the actual investment.

Design/methodology/approach: We derive the explicit formulas explaining how the resulting gain depends on the duration of computations.

Findings: We provide an algorithm that enables us to decide the computation time that leads to the largest possible gain.

Originality/value: To the best of our knowledge, this is the first solution to the problem. Following our recommendations will allow investors to select the computation time for which the …

Tents Of Israel Revisited: Audio Privacy, Julio C. Urenda, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In one of the Biblical stories, prophet Balaam blesses the tents of Israel for being good. But what can be so good about the tents? The traditional Rabbinical interpretation is that the placement of the tents provided full privacy. In our previous paper, we considered the consequences of visual privacy: from each entrance, one cannot see what is happening at any other entrance. In this paper, we analyze the possible consequences of audio privacy: from each tent, you cannot hear what is going on in other tents.

How To Gauge Reliability Of A Binary Classification Result: A Simple Case, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practical situations, we need to make a binary decision based on the available data: whether an incoming email is a spam or not, whether to give a bank loan to a company, etc. In many such situations, we can (and do) use machine learning to come up with such a decision. The problem is that while the results of a machine learning model are not 100% reliable, the existing machine learning algorithms do not allow us to decide how reliable is each result. In this paper, for simple examples, we provide a technique for gauging this reliability.

Why Homogeneous Membranes Lead To Optimal Water Desalination: A Possible Explanation, Julio Urenda, Martine Ceberio, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

A recent experiment has shown that out of all possible biological membranes, homogeneous ones proved the most efficient water desalination. In this paper, we show that natural symmetry ideas lead to a theoretical explanation for this empirical fact.

Why Gradient Descent -- Not The Best Optimization Technique -- Works Best In Neural Networks: Qualitative Explanation, Jonatan Contreras, Martine Ceberio, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In a usual Numerical Methods class, students learn that gradient descent is not an efficient optimization algorithm, and that more efficient algorithms exist, algorithms which are actually used in state-of-the-art numerical optimization packages. On the other hand, in solving optimization problems related to machine learning -- and, in particular, in currently most efficient deep learning -- gradient descent (in the form of backpropagation) is much more efficient than any of the alternatives that have been tried. How can we reconcile these two statements? In this paper, we explain that, in reality, there is no contradiction here. Namely, in usual applications …

Why Question-Based Reasoning Leads To Constructive Approach To Knowledge, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Once we have partial knowledge, what next question do we usually pursue? Empirical study shows, e.g., that if we know that A \/ B is true, but we do not know whether A is true or B is true, then the usual next step is to ask whether A is true or B is true. This selection of the next step is in line with the constructive approach to knowledge, in which when A \/ B is true, this means that we either know that A is true, or we know that B is true. In this paper, we provide …

Can Ideas Behind Ancient Egyptian Fractions Speed Up Modern Computers?, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

To divide two numbers a and b, modern computers use an algorithm which is more efficient that what we humans normally do: they compute a*(1/b), where for all sufficiently small integers b, the inverse 1/b is pre-computed. For fractions, when both a and b are integers, this algorithm requires only one multiplication. Can we make the procedure even faster by not using multiplication at all? To do this, we need to represent each fraction as the sum of inverses -- which, interestingly, is how ancient Egyptians represented fractions.

So How Were The Tents Of Israel Placed? A Bible-Inspired Geometric Problem, Julio Urenda, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In one of the Biblical stories, prophet Balaam blesses the tents of Israel for being good. But what can be so good about the tents? A traditional Rabbinical interpretation is that the placement of the tents provided full privacy: from each entrance, one could not see what is happening at any other entrance. This motivates a natural geometric question: how exactly were these tents placed? In this paper, we provide an answer to this question.